DIPARTIMENTO DI FISICA ED ASTRONOMIA Sezione di Astronomia e Scienza dello Spazio. Dottorato di ricerca in Astronomia Ciclo XIII. Dr.

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1 UNIVERSITÀ DEGLI STUDI DI FIRENZE Facoltà di Scienze Matematiche, Fisiche e Naturali DIPARTIMENTO DI FISICA ED ASTRONOMIA Sezione di Astronomia e Scienza dello Spazio Dottorato di ricerca in Astronomia Ciclo XIII Spectroastrometry of rotating gas disks: from local supermassive black holes to high redshift galaxies Candidato: Dr. Alessio Gnerucci Tutore e Responsabile Scientifico: Prof. Alessandro Marconi Coordinatore del Dottorato: Prof. Alberto Righini Firenze, Dicembre 2010

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3 Contents 1 Introduction: gas dynamical studies in galaxies and the technique of spectroastrometry Observational techniques for the gas dynamics and the spectroastrometry Supermassive Black Holes in local galaxy nuclei High redshift galaxies dynamics Outline of the Thesis Spectroastrometry of rotating gas disks for the detection of supermassive black holes in galactic nuclei: method and simulations Introduction The standard gas kinematical method The spectroastrometric gas kinematical method The spectroastrometric curve Simulations Practical application of the method The use of multiple spectra of the source: the spectroastrometric map The effect of noise on spectroastrometry Estimate of the BH mass from the spectroastrometric map Integral Field Spectroscopy Combining Spectroastrometric and Rotation Curves Summary and Conclusions A Appendix. The measure of the light profile centroid B Appendix. Building the spectroastrometric map from multiple slits observations Spectroastrometry of rotating gas disks for the detection of supermassive black holes in galactic nuclei: application to the galaxy Centaurus A Introduction Previous measurements of the black hole mass in Centaurus A Spectroastrometric measurements of BH masses Longslit spectra: observations and data analysis The data Analysis of the spectra The spectroastrometric map of the source

4 ii Estimate of the BH mass from the spectroastrometric map Integral field spectra: observations and data analysis Data and spectra analysis The spectroastrometric map of the source Estimate of the BH mass from the spectroastrometric map Discussion and conclusions Dynamical properties of AMAZE and LSD galaxies from gas kinematics and the Tully-Fisher relation at z Introduction Observations and data reduction Extraction of the gas kinematics Results Kinematical modeling The model Fit strategy Fit Results Discussion Rotating galaxies, turbulence and dynamical masses The Tully-Fisher relation at z Conclusions A dynamical mass estimator for high z galaxies based on spectroastrometry Introduction Data and previous dynamical modeling Measuring masses with spectroastrometry Virial mass estimator Spectroastrometric mass estimator Simulations Calibration of the spectroastrometric estimator Application of the spectroastrometric estimator to the AMAZE and LSD complete samples Discussion and conclusions A Appendix. Simulations Conclusions 120 Bibliography 124

5 Chapter 1 Introduction: gas dynamical studies in galaxies and the technique of spectroastrometry One of the fundamental open questions of modern astrophysics is understanding the physical processes that transformed the nearly homogeneous primordial medium into the present-day universe, characterized by a wealth of complex structures such as galaxies and clusters of galaxies. This question is tightly linked to many fundamental and yet obscure astrophysical and cosmological topics such as hierarchical models of structure formation, the nature and the role of dark matter and the star formation history in galaxies. Understanding how galaxies formed and how they become the complex systems we observe today is therefore a major theoretical and observational effort. The dynamical properties of galaxies play a fundamental role in the context of galaxy formation and evolution. The observed dynamics of galaxies represent a fundamental test to theoretical models and are the most direct way to probe the content of dark matter. In particular, the dynamical mass of a galaxy, inferred from its rotation curve, is the most direct way to constrain mass and angular momentum of dark matter haloes, which can be compared with the predictions of cosmological cold dark matter models of hierarchical structure formation. Being our principal aim to understand the mechanisms and the processes underlying galaxies formation and evolution, we need to study those properties that provide direct or indirect informations on the galaxies evolution history. Therefore we focus on the study of the dynamical properties of galaxies in their past, when these processes were active, and therefore on the study of high redshift galaxies evolving in a young universe. Present day galaxies show some physical properties that have a memory of the past galaxy evolution and that can provide indirect informations on the galaxy past history. One of these is related to the existence of tight links between supermassive black holes (BH) present in the center of most nearby galaxies, the phenomenon of Active Galactic Nuclei (AGN) and galaxy evolution. These links reveal what is now called as co-evolution of black holes and their host galaxies which is manifested in the discovery of relic BHs in the nuclei of most nearby galaxies and of tight correlations between that BH masses and the structural parameters of the host spheroid.

6 2 In this Thesis we will conduct a parallel study of two different physical systems in the local and past universe, in the framework of galaxy formation and evolution, but sharing the same observational features: the study of supermassive black holes in the nuclei of local galaxies and the study of high redshift galaxy dynamics. These systems will be investigated with gas dynamical studies. Gas is a fundamental component of galaxies and its motions are often good tracers of the gravitational potential of the entire galaxy and therefore of the dynamical state of the galaxy itself. In some cases, however, gas motions can also have a non gravitational origin and be very difficult to interpret. Stellar dynamics is also a good tool to trace the gravitational potential of the galaxy, since stellar motions are always gravitational, but this kind of studies are more challenging from the observational point of view. In fact stellar absorption lines are intrinsically more difficult to observe than gas emission lines. Despite the advantage of stellar dynamics of not being contaminated by non gravitational motions, the more difficult observations, especially for sources with relatively low fluxes, make gas dynamics the preferable choice for the topics and the systems we want to study. In general, gas kinematics can be very complex and can influenced by many factors difficult to interpret and model. Therefore we focus on the study of the physical context in which gas kinematics are more easily modeled with a more direct link between gas motions and the galaxy gravitational potential: this is the case of circularly rotating disks. In this cases gas kinematics and dynamics, through the study of rotation curves, allow us to get information on the galaxy dynamical state and in particular to model the galaxy gravitational potential and to measure its dynamical mass. In both the physical systems we will study we use the observational tool of gas dynamics and in both systems we adopt, for the gas motions, the modeling approach based on rotating disks. Also, from an observational point of view, in both systems we study object with the same apparent angular dimension on the plane of the sky, with a similar ratio between apparent dimensions and instrumental spatial resolution: the apparent dimensions of the central 10pc of a local galaxy (distance 4Mpc) are the same of the entire gas disk ( 2 4kpc) of a z 3 galaxy. An important observational difference of the two systems is on the signal to noise ratio (S/N) of the emission line which is obviously lower for high redshift galaxies. 1.1 Observational techniques for the gas dynamics and the spectroastrometry The basic observational technique usually adopted for the study of gas dynamics is longslit spectroscopy, but the development in recent years of Integral Field Unit (hereafter IFU) spectrographs has allowed many improvement. The use of IFUs has proven to be a powerful tool to study galaxy dynamics as it provides a two dimensional coverage of the source without the restrictions of longslit studies, like unavoidable light losses or systematic effects due to the positioning of the spectroscopic slit relatively to the source. The standard approach to a spectroscopical study of the kinematics and dynamics of rotating gas disks passes trough the study of the gas rotation curve. This method is based

7 Introduction: gas dynamical studies and the spectroastrometry 3 on spatially resolved spectra of the galaxy and in its basic application to longslit spectra consists in measuring the mean velocity of the gas (from a gas emission line) as a function of the position along the spectroscopic slit. The modeling assumes that the gas is rotating in a thin disc configuration (neglecting hydrodynamical effects) under the effect of the gravitational potential of the stellar mass (and of a pointlike dark mass M BH if we study the presence of a BH in the galaxy nucleus). The principal properties of the stellar mass distribution (and the value of M BH ) are obtained by fitting the rotation curves. The basic and classical application of this method is to longslit spectra. Usually, to better constrain the many unknown parameters of the model, many spectra of the galactic nucleus are obtained each with a different orientation of the spectroscopic slit and a simultaneous fit of all of them is performed. On the other hand with integral field spectra it is possible to derive, in place of a one dimensional rotation curve of the gas, a two dimensional map of the line of sight velocity field of the gas. The fit of the rotation curves becomes in this case a fit of the observed velocity field with a model that reproduces the line of sight velocity field of a gas disk rotating under the galaxy gravitational potential. Clearly, the ability to recover the mass distribution parameters (and to detect the presence of a BH and measuring its mass) strongly depends on the signal-to-noise ratio of the data and only poor constraints on these parameters can be obtained from low S/N data. However, even with high S/N data, the fundamental limit of the standard gas kinematical method ( rotation curves method hereafter) resides in the ability to spatially resolve the characteristic spatial scale in which the velocity field (or the rotation curve) is expected to vary due to the underlying gravitational potential (the ability to spatially resolve the region where the gravitational potential of the BH dominates with respect to the contribution of the stars for the case of the nuclear BH). If the instrumental spatial resolution is low compared to this scale distance the important kinematic features are mixed and diluted by the instrumental beam smearing and the knowledge of the underlying gravitational potential is no longer possible. Moreover, the model of the rotation curve, apart from the gravitational model must also take into account some secondary but important effects: the positioning of the spectroscopic slit relatively to the gas disk, the instrumental spatial resolution, and the intrinsic flux distribution on the plane of sky of the gas emission lines we are studying. These effects make the modeling more complex and problematic, in fact some of them are often difficult to model (e.g. the positioning of the slit on the sky plane in some cases can be recovered with low accuracies and the intrinsic flux distribution of the gas emission lines in many cases shows very complex shapes). Motivated by the need to overcome the shortcomings of the standard gas kinematic method observed above, in particular the limit imposed by the spatial resolution of the observations, in this Thesis we study the application of the a new method to kinematical studies of the rotating gas disks in galaxies: the spectroastrometry. The use of spectroastrometry was originally introduced by Beckers (1982), Christy et al. (1983) and Aime et al. (1988) to detect unresolved binaries. However these earlier studies required specialist instrumentation, and it was not until the work of Bailey (1998) that this method was exploited using standard common user instrumentation: a longslit CCD spectrograph. Subsequently spectroastrometry has been used by several authors

8 4 to study pre main sequence binaries (Baines et al. 2004; Porter et al. 2004, 2005) and the presence of inflow or outflow or the disk structure on the gas surrounding pre main sequence stars (Takami et al. 2003; Whelan et al. 2005). More recently, Brannigan et al. (2006) discussed the presence and the detection of artifact in the output of the method and Pontoppidan et al. (2008) used spectroastrometry on 4.7µm CO lines to study the kinematical properties of proto-planetary disks. The spectroastrometry consists in measuring the mean position of the source as a function of wavelength (in its basic application to longslit spectra, this is the measurement of the mean position along the spectroscopic slit in function of wavelength). While this method has been applied to unresolved pointlike sources as binary stars and protostellar systems, it has never been applied to the study of kinematics and dynamics of extended sources like the gas emission in galaxies. Motivated by the need to overcome the shortcomings of the standard gas kinematical method described above, in this Thesis we study the application of the spectroastrometric method to kinematical studies of the rotating gas disks in galaxies. In Figure 2.1 we report an example of the rotation curve and spectroastrometry measuring approach to a gas emission line spectrum. We can observe from figure how the spectroastrometrical approach is orthogonal and complementary to the standard rotation curve approach and therefore it can provide informations on the observed spectrum that the rotation curve alone can not provide. This is the first point that convinced us to exploit the application of this method. We will also see in the following that the fundamental advantage of the spectroastrometric method is that it can provide position measurements on scales smaller than the spatial resolution of the observations, thus overcoming the principal limit of rotation curve method. In addition to the basic application to longslit spectra, the spectroastrometric approach can be also applied to integral field spectra. We will show in the following that in this case its application is even more simpler and direct and that the application to different data type gives consistent results, demonstrating the versatility of this method. We will apply the spectroastrometric technique to the study of gas dynamics in both the physical systems of BHs in local galaxy nuclei and high redshift galaxies. We will perform a near infrared spectroscopic study of gas emission lines in galaxies. The choice of this band is primarily due to the fact that the presence of dust in galaxy nuclei is responsible for a heavy absorption of the light emitted in the central regions. Moreover many lines from ionized gas in high redshift objects fall in this wavelength range. 1.2 Supermassive Black Holes in local galaxy nuclei In the last few years, strong evidence has emerged for the existence of tight links between supermassive black holes (BH), nuclear activity and galaxy evolution. These links reveal what is now called as co-evolution of black holes and their host galaxies. Strong evidence is provided by the discovery of relic BHs in the center of most nearby galaxies, and that BH masses (M BH M ) are tightly related to the structural parameters of the host spheroid like mass, luminosity and stellar velocity dispersion (e.g. Kormendy & Richstone 1995, Gebhardt et al. 2000, Ferrarese & Merritt 2000, Marconi & Hunt 2003, Ferrarese & Ford 2005, Graham 2008, Sani et al. 2010). Moreover, while it is widely

9 Introduction: gas dynamical studies and the spectroastrometry 5 Longslit spectrum Position profile at v= 153km/s Position along the slit (pixel) Section at v= 153km/s Section at pixel 178 Position along the slit (pixel) Position centroid Velocity (km/s) Counts (a.u.) Velocity profile at pixel 178 Longslit spectrum Counts (a.u.) Mean velocity Position along the slit (pixel) Velocity (km/s) Velocity (km/s) Figure 1.1: Example of the measurements on which the construction of rotation and spectroastrometric curves is based. Upper left panel: the longslit spectrum of a given continuum-subtracted emission line (isophotes) with superimposed two particular sections at fixed slit position and velocity (dotted lines). Bottom left panel: the profile relative to the Sect. at fixed slit position with superimposed the position of its centroid (dashed line). Upper right panel: the profile relative to the Sect. at fixed velocity with superimposed the position of its centroid (dashed line). Bottom right panel: the position-velocity diagram (PVD) of the upper left panel with superimposed rotation (blue dots with error bars) and spectroastrometric curves (red points with error bars). In all panels velocities are measured relative to the systemic velocity of the galaxy V S YS which is set to the zero point of the velocity scale. accepted that Active Galactic Nuclei (AGN) are powered by accretion of matter on a supermassive BH, it has been possible to show that BH growth is mostly due to accretion of matter during AGN activity, and therefore that most galaxies went through a phase of strong nuclear activity (Soltan 1982, Yu & Tremaine 2002, Marconi et al. 2004). It is believed that the physical mechanism responsible for this co-evolution of BHs an their galaxies is probably the feedback by the AGN, i.e. the accreting BH, on the host galaxy (Silk & Rees 1998, Fabian & Iwasawa 1999, Granato et al. 2004, Di Matteo et al. 2005, Menci 2006, Bower et al. 2006). In order to proceed further it is important to secure the most evident sign of coevolution, the correlations between BH mass and galaxy properties which can be achieved by increasing the number, accuracy and mass range of existing measurements. Supermassive BHs are detected and their masses measured by studying the kinematics of gas or

10 6 stars in galaxy nuclei and, currently, about 50 BH mass measurements most of which in the M range and only very few measurements below and above those limits. In Fig. 1.2 we show the distribution of the measured BH masses and a recent observation of the correlation between the BH mass and the host spheroid stellar velocity dispersion from the work of Sani et al. (2010). We can observe from left panel of figure that the most measurements ( 80%) fall in the M range. Figure 1.2: Left panel. distribution of the more recent BH masses measurements (Sani et al. 2010): the continuous vertical line represents the median value, the dot-dashed lines represent the indicated percentiles. Right panel. Scaling relations: the M BH is plotted as a function of velocity dispersion of the host spheroid (Sani et al. 2010). As mentioned above, the fundamental limit of the standard gas kinematical method for measuring BH masses resides in the ability to spatially resolve the region where the gravitational potential of the BH dominates with respect to the contribution of the stars. This region corresponds to the so-called sphere of influence of the BH and its radius (r BH ) can be estimated as (Binney & Tremaine 1987): r BH = GM BH σ 2 ( ) ( = 0.42 MBH σ 10 8 M 200 km/s ) 2 ( ) 1 D (1.1) 5 Mpc We can observe that the smaller is the BH mass and larger the distance of the galaxy, the smaller is the apparent dimension of the BH sphere of influence and so the detection of the BH is more difficult and less accurate (a typical good spatial resolution available from ground based observations is of the order of 0.5 ). Therefore our ability to detect BHs and measure their masses is limited to moderately high masses and nearby galaxies; this is a strong limitation from the point of view of a demographic study of BHs in galactic nuclei. In Fig we show again the M BH -σ relation but with superimposed the regions for which, for a given distance of the galaxy, the apparent dimension of the BH sphere of influence is lower than the reference spatial resolution for ground based observations

11 Introduction: gas dynamical studies and the spectroastrometry 7 of 0.5 and thus for which the BH detection becomes very difficult. From equation 2.1, in fact, we see that the apparent dimensions on the plane of sky of r BH is a function of the BH mass M BH, of the stellar velocity dispersion σ and of the distance of galaxy D. Therefore, for a given distance of the galaxy, the condition that r BH is lower or equal than 0.25 (i.e. the radius of the sphere of influence has to be compared with half of the spatial resolution) allow us to individuate a corresponding region on the M BH -σ plane (i.e. the gray regions corresponding to each indicated distance value plotted in Figure 2.18). Figure 1.3: Scaling relation M BH -σ (Sani et al. 2010). The gray regions under each line represents the half plane for which, given the distance D of the galaxy, the apparent dimension of the BH sphere of influence is lower than 0.5 (typical spatial resolution of seeing limited ground based observations). Each region refers to the indicate distance of the galaxy. From the figure we can observe this important limitation and demographic bias of the BH mass measurements with the standard gas dynamics method. The low mass region in the M BH -σ plane can only be studied for very nearby galaxies and, on the contrary, for more distant galaxies we can only detect BHs with larger masses. The majority of the BH mass measurement measurements are made with longslit spectroscopy, but the development in recent years of Integral Field Unit (hereafter IFU) spectrographs has allowed some improvements in this field. Indeed the use of IFUs have proven to be powerful tools to study galaxy dynamics as they provide two dimensional coverage of the source without the restrictions of longslit spectrographs, also plagued by unavoidable light losses. Recent works has been presented on the measurement of BH masses in galactic nuclei by using integral field spectroscopy of gas or stellar spectral features (e.g. Davies et al. 2006, Nowak et al. 2007, Nowak et al. 2010,Krajnović et al. 2007, Krajnović et al. 2009, Cappellari et al. 2009, Neumayer et al. 2010, Rusli et al. 2010).

12 8 1.3 High redshift galaxies dynamics The dynamical properties of galaxies represent fundamental informations in the context of galaxy formation and evolution. The observed dynamics of galaxies represent a test to theoretical models and are the most direct way to probe the content of dark matter. In particular, the dynamical mass of a galaxy inferred from its rotation curve is the most direct way to constrain mass and angular momentum of dark matter haloes, which can be compared with the predictions of cosmological cold dark matter models of hierarchical structure formation. In such models (Blumenthal et al. 1984; Davis et al. 1985; Springel et al. 2006, Mo et al. 1998) mergers are believed to play an important role for galaxy formation and evolution, in particular them are believed to be an important trigger of the intense events of star formation observed in galaxies at redshift z 2 3. However, the observational evidence for the existence of rotating disks with high star formation rates at z 2 suggests that smooth accretion of pristine gas is also an important mechanism that drives star formation and mass assembly at high redshift (Epinat et al. 2009; Wright et al. 2009; Förster Schreiber et al. 2009; Cresci et al. 2009). In recent years, our observational knowledge in this field has increased enormously. Many dynamical studies have been performed on extended samples of z objects. A very important contribution has been given by the SINS project (Genzel et al. 2006; Genzel et al. 2008; Förster Schreiber et al. 2006b; Förster Schreiber et al. 2009; Cresci et al. 2009) that studies an extended sample of 63 galaxies at z 2 by means of integral field spectroscopy and perform, for a selected subsample of rotating disks, a detailed dynamical modeling. Law et al. (2007, 2009) takes advantage of the higher spatial resolution of Adaptive Optics observations for integral field spectroscopy of a sample of 12 objects at z 2.5. Other works concentrates on slightly lower redshift samples: Epinat et al perform integral field spectroscopy of 12 galaxies in the redshift range ; Wright et al. 2007, 2009 obtain AO assisted integral field spectroscopy of 6 objects at a z 1.6. Other works make use of longslit spectra: Erb et al makes near-infrared longslit spectroscopy of 114 star-forming galaxies at z 2. However, little is known on the dynamics of galaxies at z 2.5, where only a few selected objects have been investigated (Nesvadba et al. 2006; Nesvadba et al. 2007; Nesvadba et al. 2008; Law et al. 2009; Lemoine-Busserolle et al. 2009), in some cases taking advantage of gravitational lensing that can provide magnified images of even more distant galaxies (z 4 5) (Jones et al. 2010; Swinbank et al. 2007; Swinbank et al. 2009). The redshift range z 3 4 is particularly important to study since it is before the peak of the cosmic star formation rate (see, for example, Dickinson et al. 2003; Rudnick et al. 2006; Hopkins & Beacom 2006; Mannucci et al. 2007), when only a small fraction ( 15%, Pozzetti et al. 2007) of the stellar mass in present-day galaxies has been assembled. It is also the redshift range when the most massive early-type galaxies are expected to form (see, for example, Saracco et al. 2003). Moreover, the number of galaxy mergers is much larger than at later times (Conselice et al. 2007; Stewart et al. 2008). As a consequence the predictions of different models tend to diverge significantly at z 2.5. Our work is based on two projects focused on studying metallicity and dynamics of high-redshift galaxies: AMAZE (Assessing the Mass-Abundance redshift Evolution)

13 Introduction: gas dynamical studies and the spectroastrometry 9 (Maiolino et al. 2008a, Maiolino et al. 2008b) and LSD (Lyman- break galaxies Stellar populations and Dynamics) (Mannucci & Maiolino 2008, Mannucci et al. 2009). Both projects use integral field spectroscopy of samples of z 3 4 galaxies in order to derive their chemical and dynamical properties. In both projects we make use of data obtained with the Spectrograph for Integral Field Observations in the Near Infrared (SINFONI) at the Very Large Telescope (VLT) of the European Southern Observatory (ESO). The AMAZE sample consists of 30 Lyman Break Galaxies in the redshift range 3 < z < 4.8 (most of which at z 3.3), with deep Spitzer/IRAC photometry (3.6 8µm), an important piece of information to derive reliable stellar masses. These galaxies were observed with SINFONI in seeing-limited mode. The LSD sample is a representative, albeit small, sample of 10 LBGs at z 3 with available Spitzer and HST imaging from the Steidel et al. (2003) catalogue. For LSD SINFONI observations were performed with the aid of adaptive optics in order to improve spatial resolution since this project was aimed to obtain spatially-resolved spectra for measuring kinematics and gradients in emission lines. Lyman Break Galaxies are objects selected based on the Ly-break technique and based on their UV rest-frame blue color. As a consequence these samples are biased both against dust reddened systems and against aged stellar populations, which are characterized by redder colors. Therefore, the conclusions inferred from our results apply only to a sub-population (about half) of galaxies at z 3 (Reddy et al. 2005, van Dokkum et al. 2006, Grazian et al. 2007). It is important to remark that despite the many cosmological model of hierarchical structure formation there are in literature few observations of galaxies at z 3 to be compared with models, and an extended sample of 30 galaxies at this redshift constitutes an important step forward for the knowledge of this fundamental but yet obscure topic. 1.4 Outline of the Thesis In this Thesis we will perform a study on the applicability of spectroastrometry to the characterization of rotating gas disks in order to estimate dynamical masses. We will investigate the method with simulations and we will calibrate it with real data, assessing its reliability and limitations. In Chapters 2 and 3 we deal with BH mass measurements from gas kinematics and presents the application to this context of our new method, based on spectroastrometry, which can provide a simple but accurate way to estimate BH mass and which partly overcomes the limitations due to spatial resolution which plague the classical gas (or stellar) kinematical methods. In particular in Chapter 2 we illustrate how the technique of spectroastrometry can be used to measure the black hole masses focusing on explaining the basis of the spectroastrometric approach and showing with an extended and detailed set of simulations its capabilities and its limits. We deal principally with the application of spectroastrometry to longslit spectra, but also show that the extension of the technique to integral field spectra is straightforward. In Chapter 3 we consider real data to which apply our spectroastrometric method and estimate the BH mass. As a benchmark for the our spectroastrometric approach to the study of local BHs, we selected the galaxy Centaurus A because it has

14 10 Figure 1.4: Kinematical maps for the objects SSA22A-C16 from the AMAZE sample. Respectively, from the left: flux, velocity and velocity dispersion map. the X Y coordinates are in arcseconds referred to an arbitrary object centre position. The North direction is the positive Y axis. The vertical color bars are in arbitrary units for the flux map and in km s 1 for the velocity and sigma maps. Overplotted on the flux map the continuum flux distribution (brown isophotes) for the object. been extensively studied with the gas kinematical method, the gas is circularly rotating and BH mass and other free parameters are well constrained from the observed kinematics (Marconi et al. 2006, Neumayer et al. 2007). Chapter 4 is dedicated to the study of kinematical and dynamical properties of the AMAZE and LSD galaxies at z 3 while other studies (e.g. integrated properties, metallicity, etc.) have been or will be presented in dedicated papers (Maiolino et al. 2008a, Mannucci et al. 2009, Mannucci et al. 2010, Cresci et al and Troncoso et al. 2010, in prep.). In particular we first study the gas kinematics of these galaxies. In Fig. 1.4 we report an example of the two dimensional kinematical maps we obtain from integral field spectra for a galaxy of the AMAZE sample. Then, for a selected subsample of objects consistent with a rotating disk, we perform a complete (rotation curve based) two dimensional dynamical modeling. One of the lesson learned from our own work and from the literature is that the data for objects at such high redshifts often suffer of a poor signal to noise ratio (hereafter S/N) which does not allow spatially resolved kinematical studies and complete dynamical modeling. For this reason one can only estimate the dynamical mass of a galaxy by applying the virial theorem to its integrated spectrum. The principal problems of virial mass estimates are due to the presence of large systematic errors. Apart those due to the not-verified assumption that the system is virialized, one of the more important problems is the estimate of the size of the galaxy, that often suffers of the low intrinsic spatial resolution (i.e. measuring the size of a spatially unresolved source taking into account the effect of beam smearing leads to high systematic errors). In Chapter 5 we reconsider the technique of spectroastrometry and apply it also to integral field spectra of high redshift galaxies. We present an alternative to the classical virial mass estimate based on spectroastrometry with the aim to obtain a more accurate dynamical mass estimator for the cases in which a spatially resolved two dimensional dynamical modeling is not possible. We follow up from the results explained in Chapters 2 and 3 on the application of the spectroastrometry technique to the study of the dynamics of rotating gas disks. Finally, in Chapter 6 we draw our conclusions and summarize the refereed papers

15 Introduction: gas dynamical studies and the spectroastrometry 11 which has been published or submitted from the work presented in this Thesis. In the following, we adopt a ΛCDM cosmology with H 0 = 70kms 1 Mpc 1, Ω m = 0.3 and Ω Λ = 0.7.

16 Chapter 2 Spectroastrometry of rotating gas disks for the detection of supermassive black holes in galactic nuclei: method and simulations 2.1 Introduction In this chapter we deal with the detection and mass measurement of the supermassive black holes present in the nuclei of local galaxies. The importance of this topic in the context of galaxy formation and evolution lies in the existence of tight correlations between BH masses and many structural parameters of the host spheroid like mass, luminosity and stellar velocity dispersion (e.g. Kormendy & Richstone 1995, Gebhardt et al. 2000, Ferrarese & Merritt 2000, Marconi & Hunt 2003, Ferrarese & Ford 2005, Graham 2008, Sani et al. 2010). These correlations reveal the existence of tight links between supermassive black holes (BH), nuclear activity and galaxy evolution that are interpreted as a co-evolution of black holes and their host galaxies. Increasing the number, accuracy and mass range of existing BH mass measurements is fundamental in order to better constrain this correlations that represents the most evident sign of this co-evolution between the BHs and their host galaxies. We introduce our new method for the detection and mass measurement of the BH based on spectroastrometry and present an extended and detailed set of simulations aimed at assessing its limits and advantages such as the capability to overcome the limit imposed by the spatial resolution of the observations. Our work on this physical context will continue in Chapter 3 where we will apply our spectroastrometrical method to real data and compare our results with those obtained with classical gas dynamical studies. In Sect. 2.2 we introduce the standard method for gas kinematical studies, that is based on the gas rotation curves, and briefly discuss its characteristics and limitations. In Sect. 2.3 we introduce the new gas kinematical method based on spectroastrometry. We explain the basis of this approach that consist on measuring spectroastrometric curves (Sect ) and show simulations based on a model of the gas dynamics from which

17 Spectroastrometry of rotating gas disks: method and simulations 13 we want to learn how the spectroastrometric curve changes in function of some parameters of the model (Sect ). In Sect. 2.4 we explain the practical application of the method. We start by presenting a method for using simultaneously several spectroastrometric curves of the same source (Sect ) and we consider its application on noisy data (Sect ). In Sect. 2.5 we present a trivial fitting method for recovering the values of the various model parameters using long slit spectroscopy with noisy data. Finally, in Sect. 2.6 we describe the practical application of the method with Integral Field Units (IFU s) and in Sect. 2.8 we draw our conclusions. In Appendix 2.A we discuss in details of the spectroastrometric measurements with special regard to the determination of the light centroids and in Appendix 2.B we describe in detail the method to recover 2D spectroastrometric maps from multiple slit spectra. 2.2 The standard gas kinematical method The standard method of gas kinematics is based on spatially resolved spectra of the nuclear region of a galaxy. This method consists in recovering the rotation curve of a given gas emission line from a longslit spectrum. The modeling assumes that the gas is rotating in a thin disc configuration (neglecting hydrodynamical effects) under the effect of the gravitational potential of the stellar mass and of a pointlike dark mass M BH that is the BH. The value of M BH and other unknown parameters of the model are obtained by a fitting the rotation curves (see Marconi et al and references therein for details). Because of the many unknown parameters of the model, to better constrain the fit, usually many spectra of the galactic nucleus are obtained each with a different orientation of the spectroscopic slit and a simultaneous fit of all of them is performed. As observed in Chapter 1, the ability to detect the presence of a BH and measuring its mass depends on the signal-to-noise ratio of the data However the fundamental limit of the standard gas kinematical method ( rotation curves method hereafter) resides in the ability to spatially resolve the sphere of influence of the BH (i.e. the region where the gravitational potential of the BH dominates with respect to the contribution of the stars).the radius of the sphere of influence (r BH ) can be estimated as (Binney & Tremaine 1987): r BH = GM BH σ 2 ( ) ( = 0.7 MBH σ 10 8 M 200 km/s ) 2 ( ) 1 D (2.1) 3 Mpc where G is the gravitational constant, M BH the BH mass and σ is the velocity dispersion of the stars in the galaxy. For a galaxy with BH mass M BH 10 8 M and stellar velocity dispersion σ 200km/s we obtain r BH 11.2pc which, for a very nearby galaxy at distance D 3 Mpc, provides an apparent size of 0.7. If a typical very good spatial resolution available from ground based observations is of the order of 0.5 (this Full Width Half Maximum (hereafter FWHM) value should then be compared with twice r BH ), we can notice that the sphere of influence is only marginally resolved even for very nearby galaxies with moderately large BHs. For a galaxy distance of 30 Mpc the sphere of influence becomes marginally resolved even with the Hubble Space Telescope (HST) which provides the best spatial resolution currently available from space. Therefore, the

18 14 rotation curves method can detect only BHs with moderately high masses and located in nearby galaxies; this is a strong limitation from the point of view of a demographic study of BHs in galactic nuclei, because the adopted investigation tool cannot reveal the entire BH population. 2.3 The spectroastrometric gas kinematical method Longslit spectrum Position profile at v= 153km/s Position along the slit (pixel) Section at v= 153km/s Section at pixel 178 Position along the slit (pixel) Position centroid Velocity (km/s) Counts (a.u.) Velocity profile at pixel 178 Longslit spectrum Counts (a.u.) Mean velocity Position along the slit (pixel) Velocity (km/s) Velocity (km/s) Figure 2.1: Example of the measurements on which the construction of rotation and spectroastrometric curves is based. Upper left panel: the longslit spectrum of a given continuum-subtracted emission line (isophotes) with superimposed two particular sections at fixed slit position and velocity (dotted lines). Bottom left panel: the profile relative to the Sect. at fixed slit position with superimposed the position of its centroid (dashed line). Upper right panel: the profile relative to the Sect. at fixed velocity with superimposed the position of its centroid (dashed line). Bottom right panel: the position-velocity diagram (PVD) of the upper left panel with superimposed rotation (blue dots with error bars) and spectroastrometric curves (red points with error bars). In all panels velocities are measured relative to the systemic velocity of the galaxy V S YS which is set to the zero point of the velocity scale. As observed in Section 1.1, the technique of spectroastrometry has so far been used to study motions in stellar sources, i.e. to detect unresolved binaries or to study the motions of the gas surrounding pre-main sequence stars.

19 Spectroastrometry of rotating gas disks: method and simulations 15 The fundamental advantage of the spectroastrometric method is that, in principle, it can provide position measurements on scales smaller than the spatial resolution of the observations. We will now explain the general principle of the spectroastrometric method with a simple example: consider two point-like sources located at a distance smaller than the spatial resolution of the telescope; these sources will be seen as spatially unresolved with their relative distance not measurable from a conventional image. However, if in the two sources are present spectral features, such as absorption or emission lines at different wavelengths, the light profiles extracted from a longslit spectrum at these wavelengths will show the two sources separately. From the difference in the centroid of the light profiles at these two wavelengths one can estimate the separation between the two sources even if this is much smaller that the spatial resolution. This overcoming of the spatial resolution limit is made possible by the spectral separation of the two sources. While this method has been applied to unresolved pointlike sources as binary stars and protostellar systems, it has never been applied to the problem of measuring BH masses from the nuclear gas emission in galaxies. Motivated by the need to overcome the shortcomings of the standard gas kinematic method described above in Sect. 2.2, in this chapter we study the application of the spectroastrometric method to gas kinematical studies of the mass of BH s in galactic nuclei. We will start presenting the spectroastrometric method based on the same longslit spectra used for the rotation curves method but instead of the rotation curves we recover the spectroastrometric curve. However, as shown in Sect. 2.6, the application of spectroastrometry to integral field data is even more simple and straightforward than to longslit spectra. Here we focus on the theory of the method and the development of a practical framework for its application, exploring its capabilities and limitations using simulated data in order to understand how the spectroastrometric curve is affected by the object itself or by the instrumental setup. In Chapter 3 we will apply the method to the kinematical data of real galaxies to yield new improved BH mass measurements. In this chapter we concentrate on the application of the spectroastrometric method to continuum subtracted spectra in order to focus exclusively on the gas kinematics. Indeed, among other things, the underlying continuum only dilutes or modifies the spectroastrometric signal expected for the spatial distribution and kinematics of emission line gas The spectroastrometric curve Here we explain how the spectroastrometric curve is obtained and its main differences with classical rotation curves. A longslit spectrum of a continuum subtracted emission line provides a pixel array whose axes map the dispersion and slit directions, the so-called position-velocity diagram (hereafter PVD). The slit axis maps the observed position along the slit. The dispersion axis maps the wavelength of emission from which the line of sight velocity can be derived. The upper left panel in Fig. 2.1 shows the isophotes of an emission line from a PVD. Ideally, for infinite S/N and perfectly circularly rotating gas, the rotation curve denotes the mean gas velocity as a function of the position along the slit. In practice, that curve is obtained by fitting the observed line profiles along the slit with gaussian functions which

20 16 provide an estimate of the average velocity after discarding components which are clearly not circularly rotating. The lower left panel of Fig. 2.1 displays the line profile extracted at the slit position marked by the horizontal dotted line in the PVD. The spectroastrometric curve provides the mean position of the emitting gas as a function of velocity. In ideal data that curve is derived by taking the light profile of the line at given velocities along the dispersion direction and measuring the corresponding mean emission centroids. In practice, the finite S/N of real data requires more complex measurements, which are described below. The upper right panel of Fig. 2.1 displays the light profile extracted at the velocity marked by the vertical dotted line in the PVD. The two curves can be compared in lower right panel superimposed on the isophotes in the PVD diagram. Clearly, the two methods analyze the same spectrum from complementary points of view Simulations In the following we show the results of the tests based on simulations of rotating gas disks, outlining the effect of varying the free parameters of the simulation and of the spectroscopic observations. The model we use in our simulations is based on a thin gas disk rotating in a plane under the effect of the gravitational potential produced by a mass distribution of stars in a galaxy nucleus and by a pointlike dark mass, the central BH M BH. Therefore, we assume that the gas is circularly rotating in the disk plane with the rotational velocity uniquely determined by the combination of M BH and the stellar mass distribution. This model depends on several parameters: the dynamical parameters (the BH mass, the shape of the mass density function of the stars and the systemic velocity of the galaxy) and the geometrical parameters that establish the position and orientation of the gas disk (the distance of the galaxy or the angular distance scale, the inclination of the disk plane with respect to the line of sight and the orientation of the line of modes of the disk). The model takes into account the effect of the shape of the intrinsic light distribution of the emission line on the sky plane, that is modeled analytically with a combination of gaussian or exponential functions. The model takes also into account the effect of the instrumental Point Spread Function that is modeled with a gaussian function with a given FWHM, and the effect of the others instrumental setup parameters: the spectral resolution, the slit width and position angle, the detector s pixel size. With this model we can simulate a longslit spectrum of a gas emission line in a particular galaxy nucleus (see Marconi et al. 2006, and references therein for a detailed description). For simplicity, in order to illustrate the technique, we make use of a basic reference model which is meant to closely match the physical parameters of a BH of mass 10 8 M in an elliptical galaxy in the local universe observed with a typical longslit spectrograph, like ISAAC at the VLT (Moorwood et al. 1999). The key model parameters are: The distance of the galaxy is set to 3.5 Mpc that corresponds to an angular distance scale of 17 pc/ (e.g. the same of the galaxy Centaurus-A). The disk inclination is set to 35.

21 Spectroastrometry of rotating gas disks: method and simulations 17 The disk line of nodes position angle (with respect to the North direction) is set to 0 The amplitude of rotational velocity in the disk due to the stellar mass component at ±1 is set to ± 200 km/s. The FWHM of the spatial PSF is set to 0.5 as for typical high quality ground based observations. The spectral resolution is set to 10 km/s. The detector s angular pixel size is set to , resulting in a spatial oversampling of 3.5. The spectrograph slit is usually set parallel to the disk line of nodes unless otherwise specified. We have chosen a basic set of parameters which results in a resolved BH sphere of influence with the aim of clearly showing the spectroastrometric features of disk rotation. In the rest of the chapter we will of course consider more extreme sets of parameter values which will result in non-resolved BH spheres of influence in order to show the full power of spectroastrometry. BH mass We first consider the effect of changing the value of the central BH mass on the spectroastrometric curve. We simulated the spectrum of a particular rotating gas disk model with different values of the mass of the central BH, and then we derived the spectroastrometric curves from these simulated spectra (see Fig. 2.2). The spectroastrometric curve in Fig. 2.2 is centered at the systemic velocity of the galaxy. In these noiseless simulations, used for illustration purposes, the spectroastrometric curves are drawn only when the mean flux of the light profile along the slit is larger than 10 3 of the maximum flux of the spectrum. This is meant to give a representation of the low signal-to-noise regions that should be used in real data for the measurement of the centroid of the light profile, without considering unrealistically low flux levels. We defer a discussion of the practical cut off fluxes imposed by the signal-to-noise in real data to Sect In the M BH = 0 case (Fig. 2.2 upper right panel) the gas kinematics is due only to the gravitational potential of the stellar mass distribution and the spectroastrometric curve is monotonically decreasing at increasing velocity. In all other cases (Fig. 2.2 other panels), the points at high velocities (v 300km/s and v 300km/s) tend to approach the 0 value for increasing velocities (we consider increasing referring to v V sys ; V sys = 0 for this simulation), giving the curve the characteristic S -shape. The reason for this behavior is that when the gas kinematics is dominated by the BH s gravitational potential, the gas at high velocity is located close to the BH and the light centroid of the high velocity gas has to be near to the BH position.

22 Position along the slit (arcsec) pixel Position along the slit (arcsec) pixel Velocity (km/s) Velocity (km/s) Position along the slit (arcsec) pixel Position along the slit (arcsec) pixel Velocity (km/s) Velocity (km/s) Figure 2.2: Spectroastrometric curve (black solid line) obtained from a simulated spectrum for a rotating gas disk with various values of the central BH and stellar mass. In gray we plot the isophotes of the simulated line spectrum. Upper left panel: case of M BH = 10 8 M BH without mass contribution from stars. Upper right panel: stars only, with M BH = 0. Lower panels: M BH = 10 8 M (left) and M BH = M (right) with included the same stellar mass distribution as in model in the upper right panel. Note that the differences in the kinematics observed in the left panels (both BH dominated, but with and without stellar contributions) are marginal, and only visible i.e. beyond 0.2 from the nucleus. Operatively the high velocity points are those where the line emission along the slit is spatially unresolved. In Appendix 2.A we describe how to find these points as a by product of centroid determination. We can also recall that the presence of a turnover in the spectroastrometric curve is the signature of the presence of a BH. Therefore, the high velocity range is also characterized by a 75% drop of the spectroastrometric signal with respect to the maximum shift reached at the turnover point. The velocity in a circular orbit of radius r around a pointlike mass (the BH) is well approximated by a keplerian law with v = (GM BH /r) 1/2 in the innermost region of the gas disk where we can neglect the contribution of the gravitational potential of the stellar mass distribution. Combined with the above considerations, this also suggests that the spectroastrometric curve asymptotic value at high velocities provides an estimate of the BH position along the slit. For increasing M BH values the spectroastrometric curve extends to higher velocities

23 Spectroastrometry of rotating gas disks: method and simulations 19 (higher values of v V sys ) since this has the effect of increasing the amount of emission in the high velocity bins. However the limited extension in the velocity axis of the spectroastrometric curve with real data is due to the presence of noise. At velocities where the flux of the line is too low with respect to noise we will not be able to calculate a reliable value of the centroid of the light profile along the slit. In conclusion the spectroastrometric curve reveals the presence of a pointlike mass contribution to the gravitational potential when it shows an S-shaped structure, with a turn-over of the high velocities components that get closer to the 0. It can then be concluded that the information about the BH resides predominantly in the high velocity part of the curve. Spatial resolution 0.3 Spectroastrometric curve 300 Rotation curve Position along the slit (arcsec) pixel Velocity (km/s) pixel Velocity (km/s) Position along the slit (arcsec) Figure 2.3: Left panel: spectroastrometric curves from spectra differing only in spatial resolution. Solid line: spatial resolution of 0.1. Dotted line: spatial resolution of 0.5. Dashed line: spatial resolution of 1.0. The vertical long-dashed lines denote the high velocities. Right panel: corresponding rotation curves. Here we consider the effects of the spatial resolution on the spectroastrometric curve. The spatial resolution is the width (FWHM) of the point spread function which, in the model, is approximated by a gaussian function. The results of this test are shown in Fig. 2.3 where we compare spectroastrometric and normal rotation curves. In the left panel we present the spectroastrometric curves for models differing only in spatial resolution while in the right panel we display the corresponding rotation curves. The spectroastrometric curves differ in the low velocity range (-300 km/s v 300 km/s), as they show a steeper gradient and a large amplitude. However, in the high velocities range differences are negligible, 0.1 pix at most. In Section we showed how the information on the BH mass is encoded in the high velocity range and, in particular, the presence of the BH is revealed by the fact that the spectroastrometric curve approaches 0 at high velocities. The spectroastrometric curve

24 20 at the high velocities is almost unchanged by worsening the spectral resolution, leaving the BH signature unaltered. For the standard rotation curves the information about the BH is also encoded in the presence of points at high velocities at low distances from the center. However, a lower spatial resolution the BH signature is effectively canceled in the rotation curves when the sphere of influence of the BH is not resolved. In this simulation the apparent radius of the sphere of influence is 0.6, which is resolved in the case of 0.1 spatial resolution, partially resolved for 0.5 case, and unresolved for 1.0. Clearly, even in the case of the spectroastrometric curves a better spatial resolution is desirable. In fact, a poorer spatial resolution results in a broadening of the light profile along the slit and in a decreased accuracy with which the centroid position (and consequently the BH mass) can be measured. Note, however, that the spatial resolution is not a free parameter, since it can be measured directly from the data. Its effects can be modeled and taken into proper account. We have just shown that the the rotation curve changes drastically when varying the spatial resolution while the spectroastrometric curve does not. Now we conclude this section with a last set of simulations to showing in a qualitative but more accurate way how this method can really allow us to overcome the spatial resolution limit. 0.3 Spectroastrometric curve 150 Rotation curve Position along the slit (arcsec) pixel Velocity (km/s) pixel Velocity (km/s) Position along the slit (arcsec) Figure 2.4: Left panel: spectroastrometric curves for spectra differing in the value BH mass. Solid line: M BH = 0. Dotted line: M BH = M. Right panel: rotation curves for the same models. The results of these simulations are shown in Fig In the left panel we display the spectroastrometric curves for two models with the same spatial resolution (0.5 ) but different BH mass; in the right panel we display the corresponding rotation curves. By considering only the rotation curves, the case M BH = 0 is effectively indistinguishable from that with M BH = M and indeed the sphere of influence is unresolved in this case (the apparent radius of the sphere of influence is r BH 0.02 ). Instead, for this particular set of simulations, a M BH = M mass appears to be still distinguishable from the M BH = 0 case; in the high velocity range the differences between the two models reach a value of 0.2, i.e. 1.3 pixels. Such difference can be measured in real data, assuming that we can achieve a reasonable accuracy of 1 pixel in the measure of the photocenter at these velocities. In conclusion, according to these

25 Spectroastrometry of rotating gas disks: method and simulations 21 simulations we are able to detect a BH whose apparent size of the sphere of influence is less than 1/10 of the spatial resolution. Slit position angle Position along the slit (arcsec) (θ slit θ LON )=0 (θ slit θ LON )=30 (θ slit θ LON )=45 (θ slit θ LON )=60 (θ slit θ LON )=90 (θ slit θ LON )=110 (θ slit θ LON )=130 1 pixel Amplitude of the curve (arcsec) Velocity (km/s) (θ slit θ LON ) (deg) Figure 2.5: Left panel: simulated spectroastrometric curves from spectra differing only in slit position angle (referred to the disk line of nodes θ slit θ LON ). Right panel: measured amplitude of the spectroastrometric curve (distance between the maximum and the minimum) as a function of the slit position angle referred to the disk line of nodes; the dashed line denotes a cosine function opportunely rescaled to match the value of the amplitude of θ slit θ LON = 0. Here we consider how the spectroastrometric curve is affected by changing the position angle of the slit. We simulated the spectrum of a particular rotating gas disk model with different values of the position angle of the slit and then we get the spectroastrometric curves from this simulated spectra. In the left panel of Fig. 2.5 we can see the comparison of the spectroastrometric curves relative to different values of the slit position angle referred to the disk line of nodes position angle (θ slit θ LON where θ slit and θ LON are the position angles of the slit and disk line of nodes respectively). We can see that for θ slit θ LON = 0 (slit aligned with the disk line of nodes) the spectroastrometric curve has the maximum amplitude. For increasing values of θ slit θ LON the amplitude decreases reaching a null value for θ slit θ LON = 90. For values larger than 90 the curve invert itself. This is clearly a geometrical projection effect just like the one affecting the amplitude of normal rotation curves as a function of θ slit θ LON. In the right panel of Fig. 2.5 we show the curve amplitude as a function of θ slit θ LON which is well approximated by a cosine function (dashed line). Slit width Another parameter which influences the spectroastrometric curve is the width of the slit with which the spectra are obtained. Each point of the spectroastrometric curve represents the centroid of the light profile along the slit at a given velocity but one can only sample

26 22 the fraction of light emitted by the gas at that specific velocity that is intercepted by the slit. M BH =10 8 M sun 0.6 Position (arcsec) Position (arcsec) Position along the slit (arcsec) Velocity (km/s) 1 pixel Figure 2.6: Left panel: iso-velocity contour map of the line of sight velocity field on the plane of sky for a rotating gas disk with the of line of nodes along the x axis, with superposed a 0.5 wide slit. Right panel: comparison of spectroastrometric curves for simulated spectra differing only in slit width (we remind that we assumed a spatial resolution of 0.5 ). Solid line: 0.2 wide slit. Dotted line: 0.5 wide slit. Dashed line: 1.0 wide slit. In Fig. 2.6, left panel, we show the map of line of sight velocity field of a rotating gas disk. The gas in a given velocity bin lies in the locus delimited by two subsequent isovelocity contours. The impact of superimposing the slit on the iso-velocity contour map is to artificially truncate the spatial regions contributing to line emission at a given velocity that are not confined within the slit extension. This results in distorted and displaced photo-centers in velocity space. In Fig. 2.6, right panel, we show the spectroastrometric curves obtained from models differing only in slit width. As long as the slit width is smaller or equal to the spatial resolution of the observations (which is, in this case, 0.5 FWHM) then differences among spectroastrometric curves in the high velocity range are negligible (e.g., slits with 0.2, 0.5 width). On the contrary the spectroastrometric curve is significantly affected for larger slit widths, even in the high velocity range, because of the inclusion of more extended emission. This comparison indicates that one should select for the observations a slit width smaller or at most equal to the spatial resolution of the observations. As already noted discussing the effects of spatial resolution, the small residual differences in the spectroscopic curves corresponding to various slit widths can be effectively modeled out. Spectral resolution We now focus on the effects of the finite spectral resolution of the observations on the spectroastrometric curves. In Fig. 2.7 we show models differing only in spectral resolution. By decreasing the spectral resolution, the amplitude of the spectroastrometric curves is decreased and they show, at a given spatial offset, higher velocities. Effectively, due to the spectral convolution, the curves are stretched along the velocity scale. To understand

27 Spectroastrometry of rotating gas disks: method and simulations Position along the slit (arcsec) pixel Velocity (km/s) Figure 2.7: Comparison of spectroastrometric curves for spectra differing only in spectral resolution. Solid line: spectral resolution of 10 km/s. Dotted line: spectral resolution of 50 km/s. Dashed line: spectral resolution of 100 km/s. Dot-dashed line: spectral resolution of 150 km/s. this effect we can approximate the true velocity profile and the instrumental line profile with gaussian functions. The resulting line profile is the convolution of these two functions and is therefore a gaussian function with standard deviation σ wide = σ 2 + σ 2 0 (2.2) where σ wide is the resulting standard deviation, σ is the standard deviation of the velocity profile and σ 0 is the standard deviation of the instrumental response function. We here consider as a source of the line broadening the unresolved rotation of the gas that originates from the fact that one is observing with finite spatial resolution and is then not able to spatially resolve the high velocity regions close to the BH. In our simulations, the line width due to unresolved rotation reaches σ 250 km/s at the galaxy s center. As expected, the spectroastrometric curves are almost unchanged at high spectral resolution (σ 0 = 10, 50 km/s) but they are significantly altered for the lower spectral resolutions considered (σ 0 = 100, 150 km/s) when σ 0 approaches the value of σ. As a consequence, the BH mass estimates derived from data of insufficient spectral resolution are systematically overestimated, since at a given position, one measures an artificially increased velocity. This result underscores the importance of using data of as high as possible spectral resolution. The optimal value must be however derived trading-off with the level of signalto-noise necessary to build well defined spectroastrometric curves. Finally, we note that the value of σ can be estimated by modeling the classical rotation curves. It is then possible to establish, a posteriori, whether the BH measurement is affected by such an effect and eventually to validate its value.

28 Position along the slit (arcsec) pixel Velocity (km/s) Figure 2.8: Comparison of spectroastrometric curves for spectra differing only in intrinsic velocity dispersion. Solid line: intrinsic velocity dispersion of 10 km/s. Dotted line: intrinsic velocity dispersion of 50 km/s. Dashed line: s intrinsic velocity dispersion of 100 km/s. Dot-dashed line: intrinsic velocity dispersion of 150 km/s. Intrinsic velocity dispersion Another source of line broadening might be due to non circular or chaotic motions in the gas which can be modeled by adding an intrinsic velocity dispersion to the gas motions (e.g. see the discussion in Marconi et al. 2006). In Fig. 2.8 we show the spectroastrometric curves for models differing only in intrinsic velocity dispersion. The comparison with Fig. 2.7 clearly shows that, when adopting the same values of σ 0, no significant difference is found regardless of whether the source of line broadening is poor spectral resolution or an intrinsic velocity dispersion in the source. In general, when spurious line broadening is present, the high velocity points in the spectroastrometric curve are not entirely due to the gravitational potential of the BH, and the curve is artificially stretched in the velocity direction leading to a possible overestimate of the BH mass with the method described in Sect From the simulations presented in Figs. 2.7 and 2.8 we can verify that the effect of line broadening on spectroastrometric curves is indeed approximable as a x-axis stretching. Following equation 2.2 we verified that can then recover the de-stretched velocities as: F ( ) v Vsys = v 1/2 obs Vsys 1 + σ2 0 (2.3) σ 2 thus correcting the spectroastrometric curve. A detailed analysis of the effects of spurious line broadening is beyond the scope of this thesis and will be presented elsewhere but, briefly, we can in principle estimate the spurious line broadening σ 0 and that due to unresolved rotation σ by modeling the classical rotation curves (e.g. Marconi et al and references therein) and then correct the spectroastrometric curves by de-stretching the velocity axis as indicated in eq Clearly, the most accurate approach will be that

29 Spectroastrometry of rotating gas disks: method and simulations 25 of a combined fit of the classical rotation and spectroastrometric curve, much beyond the simple analysis presented here in Sect Flux distribution Figure 2.9: Comparison of spectroastrometric curves for spectra differing only in intrinsic flux distribution. Left panel. Solid line: base model flux distribution (explained in the text). Dotted line: model with r 0 = 0.1. Dashed line: model with r 0 = Dot-dashed line: base model with a central hole of radius r h = Right panel. Solid line: base model flux distribution (explained in the text). Dotted line: base model centered on (0.05, 0.05 ) position. Dashed line: base model centered on (0.1, 0.1 ) position. Dot-dashed line: base model centered on (0.15, 0.15 ) position. In this section we present simulations based on simple intrinsic flux distribution models, with the aim of showing how sub-resolution variations of the flux distribution influence the spectroastrometric curve. The basic model adopted for the flux distribution in these simulations is an exponential function I(r) = Ae r/r 0 where r is the radial distance from the symmetry center located at the position (x 0, y 0 ) = (0, 0 ) and r 0 is a characteristic radius which we assume equal to At the spatial resolution of our simulations (FWHM 0.5 ), this emission is spatially unresolved. In the first set of simulations, shown in left panel of Fig. 2.9, we vary the characteristic radius r 0 in size and introduce a central hole in the flux distribution with radius r h. In the second set of simulations, shown in right panel Fig. 2.9, we vary the position of the center of the flux distribution. In the high velocity range the variation in the centroid position with respect to the base model is lower than 0.06 and 0.09 for the first and second set of simulations, respectively; these values should be compared with the pixel size, 0.125, and the spatial resolution, FWHM = 0.5. As we will show in Sect. 2.5, these variations will produce only small changes (up to ±0.2 dex) on the BH mass which can be inferred from spectroastrometric data. On the contrary, much larger variations are present in the low velocity range. This behavior, observed also in previous simulations, is explained by the fact that that the gas with lower velocities is located in an extended (spatially resolved) region, and the

30 26 light profiles along the slit are influenced by the shape of the flux distribution. Conversely the gas at higher velocities is confined in a small (spatially unresolved) region close to the BH and the light profiles along the slit are not influenced by the shape of the flux distribution. Only in the case of a central hole in the flux distribution does the largest variation in the spectroastrometric curve occur in the high velocity range. This behavior is explained by the fact that that the gas with higher velocities (located in a small region close to the BH) is not illuminated because of the central hole of the flux distribution and the centroid position is influenced by the flux distribution at larger distance from the BH. Summary of simulation results We can now summarize the results from the above simulations and the considerations that can be derived. The presence of a black hole is revealed by a turn-over in the spectroastrometric curve, with the high velocity components approaching a null spatial offset from the center of the galaxy. Conversely, in the case of no black hole, the curve shows a monotic behavior. The information about the BH is predominantly encoded in the high velocity range of the spectroastrometric curve which is generated by spatially unresolved emission. The high velocity range of the curve is not influenced strongly by the spatial resolution of the observations, leaving the BH signature unaltered. According to our simulations we are able to detect a BH whose apparent size of the sphere of influence is as small as 1/10 of the spatial resolution. Instead, for the standard rotation curves method the information about the BH is effectively canceled when the sphere of influence of the BH is not spatially resolved. The amplitude of the spectroastrometric curve decreases by increasing the angle between the slit and the line of nodes of the gas disk, according to a cosine law. The slit width affects the spectroastrometric curve, due to the truncation of the isovelocity contours. This effect can be reduced to a negligible level selecting for the observations a slit width smaller than (or equal to) the spatial resolution. The effect of the finite spectral resolution is to stretch (in velocity space) the spectroastrometric curve. A robust BH mass estimates requires a velocity resolution smaller than the line width due to (spatially) unresolved rotation. Variations of the intrinsic line flux distribution at sub-resolution scales, do not greatly affect the spectroastrometric curve in the high velocity range.

31 Spectroastrometry of rotating gas disks: method and simulations Practical application of the method The use of multiple spectra of the source: the spectroastrometric map In order to improve the constraints on model parameters with the standard method, it is common practice to obtain several long-slit spectra at different slit orientations. Also spectroastrometry benefits from this approach as it is possible to recover for the same source several spectroastrometric curves, one for each position angle (PA) of the slit. With respect to the standard rotation curves method, where spectra taken along parallel slits have been often used, here the fundamental requirement is that the various slits must be centered on the expected BH position. We simulated the case of three longslit spectra, oriented parallel (PA1), perpendicular (PA2) and forming an angle of 45 with the line of nodes (PA3). In Fig we can see the spectroastrometric curves obtained from the three spectra. 0.2 Position along the slit (arcsec) Velocity (km/s) Figure 2.10: Spectroastrometric curves for three simulated spectra of the same model with different PA of the slit. Solid line: slit with PA (referred to the disk line of nodes position angle) θ slit θ LON = 0. Dotted line: slit with θ slit θ LON = 45. Dashed line: slit with θ slit θ LON = 90. Each spectroastrometric curve provides the photocenter position along one slit, i.e. the position of the photocenter projected along the axis identified by the slit. Combing the spectroastrometric curves we can thus obtain the map of photocenter positions on the plane of the sky for each velocity bin. In principle, the spectroastrometric curves from two non-parallel slits should suffice but we can use the redundant information from the three slits to recover the 2D sky map as described in more detail in Appendix 2.B. In Fig (left panel) we present the 2D spectroastrometric map on the plane of the sky. All the points of the map lies on a straight line which identifies the line of nodes. Indeed, whenever the intrinsic distribution of line emission is circularly symmetric and centered on the BH position (as assumed in our simulations), the light centroids at a given velocity are located on the disk line of nodes. In the right panel of Fig we show

32 y East (arcsec) Velocity (km/s) x North (arcsec) Figure 2.11: Spectroastrometric 2D map derived from the three spectroastrometric curves of Fig Left panel: derived photocenter positions on the sky plane, the black open diamonds are the points actually used for the χ 2 minimization, the dashed lines indicate the three slits each with its derived centre location (filled squares). Right panel: the 3D plot of the map, where the z axis is velocity. the three-dimensional representation of the spectroastrometric curve combining the 2D spatial map of the left panel with the velocity axis. In the 3D representation all the points lie on a plane parallel to the velocity axis and aligned with the disk line of nodes. On that plane, the spectroastrometric curve present the usual S shape symmetric with respect to the zero point of velocities. We remark that in this work we only apply the spectroastrometric method to continuum subtracted spectra. When dealing with real data an accurate continuum subtraction will be an important requirement to obtain an accurate spectroastrometric curve, especially for the high velocity points where the line-continuum contrast is lower. Any residual continuum emission can alter the spatial profile of the emission line, and the turn over signature of a point mass in the spectroastrometric curve. In the latter case, the spectroastrometric displacement of the emission line will be shifted towards the continuum photocenter in the line wings no matter the actual spatial distribution of the high velocity gas. The accuracy of continuum subtraction, as well as the signal to noise ratio at high velocities, is therefore a fundamentally limit to the ability of detecting the central BH. A more detailed analysis of the effects of continuum emission will be presented in forthcoming works The effect of noise on spectroastrometry We now consider the effect of noise on spectroastrometric curves and maps, as in real data. We simulate noise by adding normally distributed random values to the simulated position-velocity spectral maps. These random numbers are characterized by zero mean and standard deviation σ noise which is chosen in the following way. We first extract a nuclear spectrum by co-adding line emission over an aperture equal to the PSF FWHM, centered on the nucleus position. The S/N ratio of the spectrum is then defined on the

33 Spectroastrometry of rotating gas disks: method and simulations S/N= Position along the slit (arcsec) y East (arcsec) Velocity (km/s) S/N= x North (arcsec) Position along the slit (arcsec) y East (arcsec) Velocity (km/s) S/N= x North (arcsec) Position along the slit (arcsec) y East (arcsec) Velocity (km/s) x North (arcsec) Figure 2.12: Left row: spectroastrometric curves for the model of Fig with added noise for a S/N of 20, 50, and 100 (from top to bottom). Solid line: slit at PA = 0. Dotted line: slit at PA 45. Dashed line: slit at PA = 90. For simplicity we plot the error bars of the points only for the PA= 0 curve. Right row: spectroastrometric 2D map derived from the three spectroastrometric curves of the left row. The red filled circle marks the derived position of the BH, while the red solid line represents the position angle of the line of nodes (θ LON ) obtained by fitting a straight line to the X-Y position of the high velocity points. peak of the line profile, and the actual σ noise value per pixel is selected to provide a given S/N. In Fig we show the spectroastrometric curves and maps for the same model presented in Fig to which we added an artificial noise in order to obtain S/N of 20,

34 30 Table 2.1: Lines of Nodes and BH position estimates from noisy data Model x BH ( ) y BH ( ) θ LON ( ) Noiseless S/N= ± ±0.03-5±4 S/N= ± ±0.01-1±1 S/N= ± ± ± and 100, respectively. A key point to emphasize is that the curves (and maps) with higher S/N tend to have more measured points. This is because at the high velocities we have a low flux with respect to the line core and the presence of noise limits our ability to obtain a reliable light centroid position. In detail we stop estimating the light centroids when the mean flux of the light profile for that particular velocity bin is lower than the noise level. Thus, a spectrum with higher S/N enables us to obtain more accurate measurements on individual data-points, but also extends the coverage at higher velocities, crucial to detect a BH and to measure its mass. The 2D spectroastrometric maps can be used to estimate the geometrical parameters of the gas disk. In the following, we will always make use of simulations with noise. As noted in Sect , if the gas kinematics is dominated by rotation around a pointlike mass (the BH), the position of the light centroid at the highest velocities approaches the position of the BH. This consideration, which is also valid for the 2D spectroastrometric map, allows us to estimate the position on the plane of the sky of the BH as the average position of the high velocity points. In practice, we calculate this position by first taking the averages of the coordinates of the points in the blue and red high velocity ranges and then taking the average coordinates of the blue and red positions. The inferred BH position is marked by a red filled circle in the right row diagrams of Fig Also, the high velocity points in Fig (and also Fig. 2.11) lie on a straight line which identifies the direction of the disk line of nodes (see Fig. 2.6 and Sect ). The red solid line shown in the right row of Fig. 2.4 represents the position angle of the line of nodes (θ LON ) obtained by fitting a straight line to the high velocity points. In the Table 2.1 we report the BH position and disk line of nodes position angle values obtained from the 2D map of the model of Fig We can observe that the accuracy of these measurements scales approximately with the square root of the S/N of the spectrum. Moreover in the maps with lower S/N the scatter of the points around the line of nodes increases (the points are almost perfectly aligned on the line of nodes direction in the noise free model of Fig. 2.10). This fact decreases the accuracy of the BH position and line of nodes PA estimates for lower S/N. 2.5 Estimate of the BH mass from the spectroastrometric map In this section we present a simple and straightforward method to recover the BH mass from the spectroastrometric measurements. The method is based on the following as-

35 Spectroastrometry of rotating gas disks: method and simulations 31 sumptions: (i) the gas is circularly rotating in a thin disk and (ii) each point of the spectroastrometric map represents a test particle on the disk line of nodes and is characterized by a line of sight ( channel ) velocity which is given by the center of the velocity bin where the light centroid was estimated. Under these assumptions it is trivial to relate the spectroastrometric map to the BH mass. The circular velocity of a gas particle with distance r from the BH is given by: G[MBH + M star (r)] V rot = (2.4) r where M star (r) is the stellar mass enclosed in a sphere of radius r (assuming spherical symmetry) and can be written as: M star (r) = M/L L(r) (2.5) with L(r) representing the radial luminosity density distribution in the galactic nucleus and M/L is the mass to light ratio of the stars. The line of sight velocity (hereafter V ch for channel velocity ) of a test particle located on the disk line of nodes at distance r from the BH is then given by V ch = V rot sin(i) + V sys (2.6) where i is the inclination of the disk plane and V sys is the systemic velocity of the galaxy. The centroid positions on the spectroastrometric map are denoted by (x ch, y ch ) and, in general, they are not perfectly aligned along the line of nodes as described in Sects and 2.B. We select the high velocity points and we then identify the line of nodes by a linear fit of the (x ch, y ch ) spectroastrometric map. To reduce the correlation between the values of intercept and slope of the fitted line we consider y = (x x mean ) tan(θ LON ) + b (2.7) where x mean is the mean of the x ch positions. θ LON and b denote respectively the position angle and the intercept of the line of nodes to be determined by the fit. Then θ LON and b are recovered by minimizing the following χ 2 where we take into account both errors in x and y: [ χ 2 ych (x ch x mean )tan(θ LON ) b ] 2 = y 2 ch + tan(θ (2.8) LON) 2 x 2 ch ch where x chan and y chan are the uncertainties on the position of the points of the spectroastrometric map and, as observed in Sect , the sum is extended only over the high velocities range. The second step requires to associate each velocity bin to its position within the rotating disk. As described in Sect , the 2D map points lie with good approximation on the line of nodes, particularly in the high velocities range. We then project the (x ch, y ch ) position of the 2D-map points on the line of nodes and calculate their coordinate S with respect to this axis of reference, with S defined as: S ch = x ch cos(θ LON ) + [y ch b + x mean tan(θ LON )]sin(θ LON ) (2.9)

36 32 Each S ch has an uncertainty S ch estimated by taking into account the uncertainties on the 2D map points position ( x ch and y ch as well as the uncertainties on the line of nodes parameters ( θ LON, b) resulting from the fit. The distance r of a test particle from the BH used in eq. 2.4 differ from S ch only by a constant S 0, the unknown coordinate of the BH along the line of nodes, i.e. r ch = S ch S 0 (2.10) Combining eqs. 2.4, 2.5, 2.6, and 2.10 we finally obtain the expected velocity in channel ch: G[M BH + M/L L( S ch S 0 )] V ch = sin(i) + V sys (2.11) S ch S 0 The unknown model parameters are: M BH M/L S 0 V sys i mass of the BH mass to light ratio of the nuclear stars line of nodes coordinate of the BH systemic velocity of the galaxy inclination of the gas disk and can be determined by minimizing the following χ 2 [ ] 2 χ 2 Vch V sys V ch (par) V sys = (2.12) (S ch ; par) ch where (S ch ; par) is the uncertainty of the numerator computed as a function of the unknown parameters values (par) and the uncertainties S ch on the positions along the line of nodes. We remark that the channel velocity V ch has no associated uncertainty since it is not a measured quantity but is the central value of a velocity bin where the spectroastrometric curve is measured. As discussed in the previous sections, we restrict the fit (i.e. the sum over the velocity channels) to the high velocities range. The (S ch ; par) factor in equation 2.12 is much smaller for the points at lower velocities (i.e. smaller V ch V sys ) which are closer to the peak of the line profile and have therefore much larger S/N than the points at high velocity. However, from the discussion in the previous sections, we know that the spectroastrometry is less reliable for the determination of BH properties like position and mass. Therefore, to avoid being biased by potentially faulty points we add in quadrature a constant error sys to (S ch ; par). The value of sys is found by imposing that the reduced χ 2 is equal to 1. Therefore, if the points at lower velocity with higher S/N are problematic they will not bias the final fit results, because their high weight will be greatly reduced by the addition of a much larger sys. After determining the best fit values of the free parameters, the position of the BH on the plane of the sky is simply given by x BH = S 0 cos(θ LON ) (2.13) y BH = S 0 sin(θ LON ) + b x mean tan(θ LON ) (2.14)

37 Spectroastrometry of rotating gas disks: method and simulations 33 We have performed the fit of the spectroastrometric data for the models of Figs. 2.12, which are computed with the same parameters but different spectrum S/N. The results of the fitting procedure are presented in Fig where we plot the high velocity points of the spectroastrometric map in the V ch V sys vs r ( S ch S 0 ) diagram. The solid red lines represent the curves expected from the rotating disk model. In Fig we present the direct comparison between the observed spectroastrometric curve along the line of nodes (e.g. S chan vs V chan ) with the best fit model (for S/N = 100) V (km/s) V (km/s) r (arcsec) r (arcsec) 600 V (km/s) r (arcsec) Figure 2.13: Fit results for models of Fig with S/N = 20 (top left panel), S/N = 50 (top right) and S/N = 100 (bottom). Points with error bars are the result of observations. The red line is the best fit model and the dotted line is the expected contribution from the stellar mass. In all cases, for each test particle, we are able to measure photocenter positions on the plane of the sky with an accuracy of better than 0.025, corresponding to 1/20 of the spatial resolution (FWHM = 0.5 ). The comparison between best fit and simulation parameters is shown in Table 2.2. In particular, we can recover the BH mass value with an accuracy of 0.1 dex. The M/L value is not well constrained by the models because with the adopted parameters the contribution to the gravitational potential of the stellar mass is negligible. In the test cases considered here (M BH = M, stellar velocity dispersion of σ star 200 km/s and D 3.5Mpc), the radius of the BH sphere of influence is r BH 10 pc corresponding to 0.6, barely resolved with the adopted 0.5 resolution. With the

38 S (arcsec) V (km/s) Figure 2.14: Observed and model spectroastrometric curves at S/N = 100 (e.g. S chan vs V chan ). The horizontal dotted line is the BH S coordinate S 0. The vertical dotted line is the systemic velocity V sys. spectroastrometric technique, we have instead probed down to radii of the order of 0.05 ( 1 pc), a factor 10 smaller than r BH. This clearly demonstrates that with spectroastrometry we can recover information on spatial scales which are much smaller than the limit imposed by the spatial resolution, thus opening the possibility of probing the gravitational potential inside smaller BH spheres of influence. Table 2.2: Fit results from the baseline model with different S/N. Parameter Fit result Model value S/N=100 S/N=50 S/N=20 θ LON [ ] (LON f it) 0.9 ± ± 1 5 ± b [ ] (LON f it) ± ± ± log 10 (M BH /M ) 8.05 ± ± ± log 10 (M/L) [M /L ] 9.02 a 7.90 a 0.2 ± S 0 [ ] 0.02 ± ± ± V sys [km/s] 528 ± ± ± i [ ] 35.0 b 35.0 b 35.0 b 35.0 b sys [km/s] χ 2 red (χ 2 /D.O.F.) (5.12/5) (2.976/3) (0.031/3) x BH [ ] 0.02 ± ± ± y BH [ ] ± ± ± Notes. (a) Parameter not constrained from the fit. (b) Parameter hold fixed. We now present a set of simulations where we vary the intrinsic flux distribution of the gas at sub-resolution scales and observe how this affects the recovery of the input model parameter values from the fit. As in Section we adopt a basic model for the flux distribution that is an exponential function (I(r) = Ae r/r 0 ) where the symmetry is center

39 Spectroastrometry of rotating gas disks: method and simulations 35 located at the position (x 0, y 0 ) = (0, 0 ) and the characteristic radius is set to r 0 = In Figs we show the cases of: the basic flux distribution, flux distribution with characteristic radius r 0 = 0.15, flux distribution with central hole of radius r h = 0.04, flux distribution with characteristic radius r 0 = 0.05 but centered at the position (x 0, y 0 ) = (0.1, 0.1 ), all with a simulated noise for a S/N of 100. In Table 2.3 we report the best fit values of the free parameters. These simulations are chosen to represent extreme cases of sub resolution variation of the intrinsic flux distribution. Figure 2.15: Effect of varying the intrinsic flux distribution at sub-resolution scales: results for various intrinsic flux distributions. Top left panel: exponential flux distribution centered on (0, 0 ) with r 0 = Top rigth panel: exponential flux distribution centered on (0, 0 ) with r 0 = Bottom left panel: exponential flux distribution centered on (0, 0 ) with r 0 = 0.05 and a central hole of radius r h = Bottom rigth panel: exponential flux distribution centered on (0.1, 0.1 ) with r 0 = 0.2. We can observe that with the fit we can recover the correct BH mass value with an accuracy of better than 0.16dex, showing that even extreme variations of the flux

40 36 Table 2.3: Effect of varying the intrinsic flux distribution at sub-resolution scales: fit results. Parameter Fit result Model value flux distribution A f flux distribution B f flux distribution C f flux distribution D f θlon [ ] (LON f it) 1.0 ± ± ± ± b [ ] (LON f it) ± ± ± ± log10(mbh/m ) 7.91 ± ± ± ± log10(m/l) [M /L ] 8.23 a 7.41 a 8.15 a 6.68 a 0.00 S 0 [ ] ± ± ± ± Vsys [km/s] 534 ± ± ± b 500 i [ ] 35.0 b 35.0 b 35.0 b 35.0 b 35.0 sys [km/s] reduced χ 2 (χ 2 /D.O.F.) 1.05 (5.25/5) 1.05 (4.19/4) 1.05 (6.28/6) (1.006/1) xbh [ ] ± ± ± ± ybh [ ] ± ± ± ± Notes. ( f ) Flux distribution. A: exponential centered on (0, 0 ) with r0 = B: exponential centered on (0, 0 ) with r0 = C: exponential centered on (0, 0 ) with r0 = 0.05 and a central hole of radius rh = D: exponential centered on (0.1, 0.1 ) with r0 = (a) Parameter not constrained from the fit. (b) Parameter hold fixed.

41 Spectroastrometry of rotating gas disks: method and simulations 37 distribution at sub-resolution scales, do not greatly affect the BH mass estimate. Finally we consider a set of simulations where we decrease the M BH value. In Figs we show the cases of BH masses M BH = 10 7 M and M BH = M respectively, both with a simulated noise for a S/N of 100. In Table 2.4 we report the best fit values of the free parameters Position along the slit (arcsec) Position along the slit (arcsec) Velocity (km/s) Velocity (km/s) y East (arcsec) 0.0 y East (arcsec) x North (arcsec) x North (arcsec) V (km/s) 150 V (km/s) r (arcsec) r (arcsec) Figure 2.16: Application of the method to low BH masses: results from simulations with S/N = 100, M BH = 10 7 M (left) and M (right). Top: spectroastrometric curves. Middle: 2D spectroastrometric maps. Bottom: fit results.

42 38 Table 2.4: Application of the method to low BH masses: fit results. Parameter MBH = 10 7 M model MBH = M model Fit result Model parameters Fit result A Fit result B Model parameters θlon [ ] (line o f nodes f it) 0 ± ± ± b [ ] (line o f nodes f it) ± ± ± log10(mbh/m ) 7.20 ± ± ± log10(m/l) [log10(m /L )] 0.00 b b 0.00 b 0.00 S 0 [ ] 0.03 ± b ± 0.00 c 0.00 Vsys [km/s] 493 ± b 503 ± i [ ] 35.0 b b 35.0 b 35.0 sys [km/s] χ 2 red (χ 2 /D.O.F.) 0.04 (0.04/1) 0.26 (0.26/1) 1.13 (1.13/1) xbh [ ] 0.03 ± ± 0.00 d ± 0.00 d 0.00 ybh [ ] ± ± ± Notes. (A) Simulation of the model with 50km/s velocity bins. (B) Simulation of the model with 10km/s velocity bins. (b) Parameter hold fixed. (c) Parameter at the edge of allowed range. (d) Parameter calculated from fixed parameters.

43 Spectroastrometry of rotating gas disks: method and simulations 39 The case of lower black hole mass is particularly significative. We choose a M BH = M value because, as observed in Section 2.3.2, the standard rotation curve method cannot distinguish between a case of M BH and no BH. By looking at Fig we note that the BH signature is still present in a realistic spectroastrometric curve Position along the slit (arcsec) y East (arcsec) Velocity (km/s) x North (arcsec) V (km/s) r (arcsec) Figure 2.17: Application of the method to low BH masses: results from the simulations with BH mass of M BH = M and velocity bins of 10km/s. Top left: Spectroastrometric curves (for simplicity we plot the error bars of the points only for the PA= 0 curve). Top right: spectroastrometric map. Bottom panel: The result of the fit. Nonetheless, the fit is not well constrained by the few data-points. Furthermore, we obtain a value of M BH of M (Table 2.4), substantially larger than the model mass of M. This is due to the fact that in our simulation the size of the velocity bins in the PVD diagram is 50 km/s, matched to the assumed spectral resolution (R 6000). This value is insufficient to adequately resolve the emission line profile. As observed in Sect , the spectroastrometric curves are stretched along the velocity axis with artificially high velocity values which result in an overestimated BH mass. This problem can be overcome by increasing the spectral resolution of the simulated data to a higher value, R 30000, still well within reach of existing spectrographs. We then repeated the simulation setting the size of the velocity bin of the longslit spectra to 10 km/s. In Fig we show the spectroastrometric curves and the result of the fit for this

44 40 model. The sampling of the spectroastrometric curves is largely improved and the fit is consequently better constrained. The derived BH mass value, log(m BH /M ) = 6.8 ± 0.1 is now in better agreement with the model value. The results of these simulations confirm that to assess the accuracy of BH masses from the spectroastrometric method one should first check that the line widths in the central emission region are well resolved spectrally. In the last fits reported in Table 2.4 we hold fixed the mass-to-light ratio M/L to the model value. The number of high velocity points in the spectroastrometric curves is small (only 2 for the model M BH = M with 50km/s velocity bin) and it is not possible to constraint the radial dependence of the mass distribution. In general, when dealing with real data with very few high velocity points, one can only measure the total mass enclosed in the smaller radius which can be estimated from spectroastrometry. If it is possible to analyze the classical rotation curve, one could determine the mass-tolight ratio M/L from that and use it in the spectroastrometric analysis. The fixed mass-tolight ratios used in the above analysis should be interpreted as resulting from a classical analysis of the extended rotation curves. 2.6 Integral Field Spectroscopy The extension of the spectroastrometric technique from multi-slit observations to Integral Field Units (IFU) is straightforward, and carries with it many advantages. With IFUs the complex issue of the limited spatial coverage of the slits (leading to the problem of velocity truncation) is partially removed and a full 2D map can be directly derived. Another substantial advantage is the reduced observing time requested to obtain a given S/N level, since there is no need to obtain observations of the same galaxy at different silt position angle. The analysis of IFU data now reduces to fitting a 2D gaussian to each channel map in turn yielding the X,Y photocenters as a function of velocity, i.e. the spectroastrometric map. Armed with these derived photocenters one proceeds directly to the application of Sect In practice it is important to determine the accuracy of each photocenter. This can easily be accomplished by using Monte Carlo realizations drawn from the original data in each channel giving a distribution of i values for X and Y from which one can determine a median and inter-quartile spread. In order to provide a quick analysis of advantages and disadvantages in the use of IFUs vs. longslit spectra, we first recall the mandatory requirements needed to obtain an accurate and useful spectroastrometric curve. The spectra must be characterized by good spectral resolution, signal-to-noise ratio and spatial sampling. The emission line profile must be well spectrally resolved otherwise, as discussed in Sect , one would overestimate the BH mass. High signal-to-noise and spatial sampling are required for a robust and accurate robust estimate of the spatial centroid of the emission line, as discussed in detail in Appendix 2.A. Compared to longslit spectrographs, IFUs have the advantage of a two-dimensional spatial covering of the source which provides a more direct and accurate determination of the spectroastrometric map on the plane of the sky. However, for a given number of

45 Spectroastrometry of rotating gas disks: method and simulations 41 detector pixels, this is usually done at the expense of spectral resolution thus limiting the application of the spectroastrometric analysis to larger BHs. As a compromise between field-of-view and spectral resolution, an IFU can have a lower spatial sampling compared to longslit spectra which limits the accuracy of the photocenter determination. However, with IFUs it is possible to integrate longer, since one does not need to obtain spectra at different position angles. As an example, let us compare the seeing-limited performances of ISAAC and SIN- FONI (both at the ESO VLT) for spectroastrometric use. ISAAC has a spatial sampling along the slit of 0.147, where for SINFONI in the seeing limited mode spatial pixels (spaxels) have angular dimensions of There exist modes with finer spatial sampling ( ) but at the expense of a smaller field of view which might be insufficient in seeing limited observations. Overall, the spatial sampling in ISAAC and SINFONI are not extremely different, at least for moderately good seeing ( 0.5 ). Regarding the spectral resolution, let consider the case of the K (J) band, at 2.2µm ( 1.2µm). ISAAC, in the Short Wavelength Medium Resolution configuration offer a spectral resolution of 8900 (10500) while SINFONI with the K (J) grating offers a spectral resolution of 4000 (2000). Clearly, the advantage of the 2D coverage provided by SINFONI is obtained at the expense of spectral resolution, and therefore SINFONI is not a good choice to detect small BHs (or large BHs in more distant galaxies). In conclusions, IFUs have the advantage of the two-dimensional spatial covering which allows longer integrations on source and more accurate spectroastrometric maps. On the other hand, longslit spectrographs can provide a better combination of high spectral resolution and spatial sampling for the detection of smaller BHs. Regardless of the use of IFUs or longslit spectrographs, the super-resolution provided by spectroastrometry allows BH mass determinations in galaxies at distances substantially higher than those that can be studied with the standard rotation curves method. 2.7 Combining Spectroastrometric and Rotation Curves In this chapter we have shown that by means of spectroastrometric curves we can recover information at scales smaller than the spatial resolution of the observations and we have provided a simple method to estimate BH masses. However, spectroastrometry is not a replacement of the standard rotation curve method. As shown in Sec. 3 (e.g. Fig. 1), the spectroastrometric curve is complementary to the rotation curve and it is clear that any kinematical model must account for both curves at the same time. For example, any contribution from extended mass distributions (e.g. stars) can be well constrained with rotation curves but not with spectroastrometric curves which sample very small scales. A detailed analysis of the constraints posed simultaneously by spectroastrometric and rotation curves will be discussed in forthcoming works. The reader might wonder why a combined modeling of spectroastrometric and rotation curves should be better than than modeling the full Position-Velocity diagram. The reason is very simple and is related to the fact that the full PVD depends on the unknown intrinsic flux distribution of the emission line. Marconi et al. (2006) showed that line profiles do depend on the assumed flux distribution while, on the contrary, such dependence

46 42 is much weaker on the first moment of the line profile (i.e. the mean velocity). In this chapter we have shown that the spectroastrometric curve depends on the assumed flux distribution in the low velocity range, while such dependence almost disappears in the high velocity range. Therefore, by selecting the rotation curve and the spectroastrometric curve we can greatly diminish the effects of the unknown intrinsic flux distribution which plague the full PVD. 2.8 Summary and Conclusions In this chapter we have discussed the application of the spectroastrometric method in the context of kinematical studies aimed at measuring the masses of supermassive black holes in galactic nuclei and we have presented its advantages compared to classical rotation curves method. We have conducted an extensive set of simulations which have shown that the presence of a supermassive black hole is revealed by a turn-over in the spectroastrometric curve, with the high velocity component approaching a null spatial offset from the location of the galaxy nucleus. All the relevant information about the BH is encoded in the high velocity range of the spectroastrometric curve, which is almost independent of the spatial resolution of the observations. According to our simulations, the use of spectroastrometry can allow the detection of BHs whose apparent size of the sphere of influence is as small as 1/10 of the spatial resolution. We have then provided a simple method to estimate BH masses from spectroastrometric curves. This method consist in the determination of the spectroastrometric map, that is the positions of emission line photocenters in the available velocity bins. This is trivially obtained from IFUs but can also be obtained by combining longslit spectra centered on a galaxy nucleus and obtained at different position angles. From the high velocity points in the spectroastrometric map one can then obtain a rotation curve to trivially estimate the BH mass. We have provided practical applications of this method to simulated data but considering noise at different levels. From this analysis, we confirm that with seeing limited observations ( 0.5 ) we are able to detect a BH with mass M (D/3.5 Mpc), where D is the galaxy distance. This is a factor 10 better than can be done with the classical method based on rotation curves. Finally, we have discussed advantages and disadvantages of IFU vs longslit spectrographs concluding that IFUs have the advantage of the two-dimensional spatial covering which allows longer integrations on source and more accurate spectroastrometric maps. On the other hand, longslit spectrographs can provide a better combination of high spectral resolution and spatial sampling for the detection of smaller BHs. Regardless of the adopted type of spectrograph, the super-resolution provided by spectroastrometry allows BH mass determinations in galaxies at distances substantially higher than those that can be studied with the standard rotation curves method.

47 Spectroastrometry of rotating gas disks: method and simulations 43 2.A Appendix. The measure of the light profile centroid The first fundamental step in the application of the spectroastrometric method is to find the best possible way to estimate the centroid position of the light profile for a given wavelength (or velocity). In principle one could simply estimate the centroid position as the weighted mean of the position with the corresponding emission line flux. If y i denotes the position and I i the flux of the i-th pixel along the slit (for a given velocity), the centroid position y cent is simply: i y i I i y cent = (2.15) i I i The problem in using equation 2.15 resides in the presence of noise and in the fact that the value of y cent depends on the choice of the position range used to compute the weighted mean. In Fig we exemplify the effect of choosing different ranges to calculate the centroid of real data. This effect will not be negligible especially because one is looking for precisions which are much lower than the spatial resolution (e.g., 2.5 pixels for the data shown in the figure). Position profile at v=410km/s 60 Counts (a.u.) Position along the slit (pixel) Figure 2.18: Example of a light profile along the slit for a given velocity bin for real data. Three different position ranges are indicated as horizontal lines and the corresponding centroid positions are indicated with vertical lines with the same style. Another possibility to estimate the centroid position is provided by a parametric fitting of the light profile with an appropriate function. For instance, we have tried a Gauss- Hermite expansion: F(x) = Ae y2 2 [1 + h3 H 3 (y) + h 4 H 4 (y)] (2.16) where y = (x x 0 )/σ and H 3 and H 4 are the Gauss-Hermite polynomials of the third and fourth order. The advantage of the fitting method is that with the parametric fit the results are much less sensitive to noise and choice of the position range for the analysis but it is difficult to find an appropriate parametric function which can reproduce the profile shape without a large increase of the free parameters which, in turn, decreases the accuracy of the centroid

48 44 position. The light profile along the slit has often a complex shape, sometimes with multiple peaks and there is not any physical reason to suggest a particular parametric function. Moreover, the shape of the light profile for a given velocity bin is determined not only by velocity field of the rotating gas disk but also by many other factors, like the intrinsic light distribution of the emission line, the slit width and position and many others. Therefore one has to investigate what is the signature which is most related to the velocity field. The piece of information we seek is the average position of the gas rotating at a given observed velocity which can be converted into the gravitational potential in the nuclear region, with the usual assumption of a thin, circularly rotating disk. As explained above, the use of the entire light profile along the slit for a given velocity bin to calculate the photocenter position is extremely complex, moreover such position might be influenced by many features which do are not connected with the rotation of the disk. After testing the method with real and simulated data, we have reached the conclusion that the quantity we need to measure is not the centroid of the whole light profile but only of the central spatially unresolved component which is related to gas located very close to the galaxy nucleus (e.g. gas rotating around the BH). Thus, the problem of measuring the light centroid along the slit can be transformed into measuring the position of the peak of the central unresolved emission. 100 Position profile at v=577km/s 100 Position profile at v=452km/s Counts (a.u.) Counts (a.u.) Position along the slit (pixel) Position along the slit (pixel) Figure 2.19: Example of centroid determination on a real spectrum. The dashed line denotes the fitted gauss-hermite functions and the relative centroid position (vertical dashed line). The solid lines denotes the gaussian function which has been fitted in a window around the principal peak as described in the text. The vertical solid lines represent the edges of the window and the gaussian center respectively. The profile peak at the location of the nucleus is due to the spatially unresolved emission from the innermost parts of the rotating gas disk. Because it is spatially unresolved, the width of central peak is expected to be of the order of the spatial resolution, and its shape can be well approximated with a gaussian function. This component is easy to identify and fit. Moreover this unresolved component is emitted from the innermost parts of the gas disk, where the gravitational potential of the BH is strongest with a larger influence on the gas kinematics. Conversely, the spatially resolved features are very complex to fit and are influenced by many factors not directly connected with the gravitational

49 Spectroastrometry of rotating gas disks: method and simulations 45 potential. In practice, we first obtain an initial guess of the principal peak position and we select the light profile within a box centered on this initial guess and whose width is of the order of the spatial resolution. We then accurately locate the peak position by fitting a simple gaussian function to this selected portion of the light profile. In details, we usually set the width of the box as kσ 0 where σ 0 is the spatial resolution (standard deviation of the gaussian approximating the point spread function) and k a constant between 1 and 3. Using values of k too close to 1 results in too few fit points but k should also not be too large if we want to consider only the unresolved component. From our tests, the best choice has come out to be k 2. This method used to obtain the light centroid provides us with an important control parameter, that is the width of the fitted gaussian: if this is much larger than the spatial resolution, this is an indication that line emission is dominated by a spatially extended component and the obtained value of the centroid is less reliable. In Fig we show an example of centroid determination obtained by fitting a gausshermite function to the whole profile and a simple gaussian in a 2σ 0 wide window centered around the principal peak. The centroid values obtained with the two methods might or might not be consistent but it is clear that the centroid value with the first method is shifted to the right by the contribution of the profile wings, which, moreover, are not even well reproduced by the gauss-hermite function. In the bottom panel of Fig none of the two methods can well reproduce the profile peak, but the width of the fitted gaussian turns out to be 2.5 times larger than the spatial resolution indicating that the principal peak is partially resolved, and the value of the centroid is less reliable FWHM (arcsec) Velocity (km/s) Figure 2.20: FWHM of the gaussian fitted to the principal peak of the light profile principal peak for spectroastrometric curves of Fig Solid line: PA1 curve. Dotted line: PA2 curve. Dot-dashed lines: PA3 curve. The horizontal dashed line denotes 1.1 times the PSF FWHM. The gray dashed lines are the estimated limits of the high velocities. Following these considerations, we found that the most robust and objective way of defining the high velocity range is through the width of the gaussian fitted to the prin-

50 46 cipal peak of the light profile: in Fig we display the FWHMs (Full Width Half Maxima) of the gaussian fitted to the principal peak of the light profile for spectroastrometric curves of Fig We can observe that at low velocities the FWHM increases because the peak is no longer unresolved. We compare the FWHM to the spatial resolution ( 0.5 ) because the FWHM of the light profile of an unresolved source should be of the order of the spatial resolution. We selected the high velocity range by imposing that the FWHM is lower than 1.1 times the spatial resolution (FWHM of the PSF) resulting in the range v 250 km/s and v 250 km/s for the spectroastrometric curves. 2.B Appendix. Building the spectroastrometric map from multiple slits observations. In this section we describe how to build a 2D spectroastrometric map, i.e. as to estimate the photocenter position on the plane of the sky for each velocity bin, from multiple slits observations. We first consider a reference frame in the plane of the sky which is centered on the center of PA1 slit and which has the X axis along the North direction. For a given velocity bin v i the position of the light centroid on the sky plane is [x(v i ), y(v i )]; the expected photocenter position along a given slit (i.e. a point of the spectroastrometric curve) is simply the position [x(v i ), y(v i )] projected along the slit direction that is: P slit theor (v i) = [x(v i ) x slit 0 ]cos(θ slit) + [y(v i ) y slit 0 ]sen(θ slit) (2.17) where x slit 0 and y slit 0 are the coordinates of the slit centre and θ slit is the position angle of the slit referred to the X axis (i.e. the North direction as in the usual definition of PA). The spectroastrometric curve provides the measured centroid position along a given slit P slit obs (v i). Thus, in order to determine the free parameters x(v i ), y(v i ), x slit 0, yslit 0 (θ slit is known), one can minimize the following χ 2 : χ 2 i = slit [ P slit theor (v i) P slit P slit obs (v i) obs (v i) ]2 (2.18) It is worth noticing that the relevant quantities are not the absolute positions of the slit centers, but the relative ones. Indeed we have chosen the center of the reference frame coincident with the center of the PA1 slit. The number of free parameters is then 6 (2 slit centers and the photocenter position on the sky, i.e. 6 coordinates) compared to 3 data points (the photocenter positions from the 3 slits). However, the problem is not undefined since many velocity bins are available and the slit center positions must be the same for all velocity bins. Therefore, for each set of slit center positions x slit 0, yslit 0 (slit = 1, 2, 3) we minimize separately all χ 2 i. The best xslit 0, yslit 0 values are then those which the minimize χ 2 = i χ 2 i (x slit 0, yslit 0 ) (2.19) if N points are available from each spectroastrometric curve, the number of degrees of freedom is then d.o. f. = 3N 4 2N = N 4, where 4 is the number of unknown slit

51 Spectroastrometry of rotating gas disks: method and simulations 47 center coordinates and 2N is the number of unknown photocenter positions. Since N is larger than 4 the problem is well posed. The final spectroastrometric map on the plane of the sky is that given by the best fitting set of slit centers. The sum over v i in (2.19) is extended only over the high velocity range. Indeed, in Sect we concluded that the high velocity range of the spectroastrometric curve is more robust, and less affected by slit losses which artificially change the photocenter position in the low velocity range. As explained in Appendix 2.A, where we discuss in detail, centroid determination, we find the spectroastrometric curve by fitting a gaussian to the principal peak of line emission along the slit (that centered on the nucleus position). Then the high velocity range is selected by imposing that the FWHM of the fitted gaussian is lower than 1.1 times the spatial resolution (FWHM of the PSF). This ensures that we consider only velocities where the line emission along the slit is spatially unresolved.

52 Chapter 3 Spectroastrometry of rotating gas disks for the detection of supermassive black holes in galactic nuclei: application to the galaxy Centaurus A 3.1 Introduction In this third chapter we deal again with gas kinematical BH mass measurements based on spectroastrometry. In particular, we apply the method developed and presented in Chapter 2 to real data. In the previous chapter we focused on the basis of the spectroastrometric approach and investigated its capabilities and limits with an extended and detailed set of simulations. We also showed that our method, mainly developed for the application to longslit spectra, can also be straightforwardly applied to integral field spectra. As a benchmark for our spectroastrometric approach to the study of local BHs, we selected the galaxy Centaurus A because it has been extensively studied with classical methods showing that the gas is circularly rotating and that BH mass and other free parameters of the rotating disk are well constrained from the observed kinematics. In Sect. 3.2 we summarize the existing measurements of BH mass for Centaurus A. In Sect. 3.3 we briefly resume the results of Chapter 2. In Sect. 3.4 we apply the method to the longslit ISAAC spectra of the nucleus of Centaurus A. In Sect. 3.5 we apply the method to integral field SINFONI spectra of the nucleus of Centaurus A, both with and without the assistance of Adaptive Optics. In Sect. 3.6 we compare and discuss the results from the different datasets and draw some conclusions on the reliability of the method. 3.2 Previous measurements of the black hole mass in Centaurus A Existing measurements of the black hole mass in Centaurus A are summarized in the top panel of Fig. 3.1.

53 Spectroastrometry of rotating gas disks: application to Centaurus A 49 The supermassive black hole in Centaurus A was first detected and its mass measured with a near infrared gas kinematical study using seeing limited spectra obtained with ISAAC at the ESO VLT (Marconi et al. 2001). Subsequent higher spatial resolution gas kinematical studies based on longslit spectroscopy were performed using STIS on the HST (Marconi et al. 2006) and AO assisted observations with NAOS-CONICA at the ESO VLT Häring-Neumayer et al. (2006). More recent studies based on integral field spectroscopy were performed by Krajnović et al. (2007) using seeing limited observations with CIRPASS at the Gemini South telescope and by Neumayer et al. (2007) using AOassisted observations obtained with SINFONI at VLT. On the other hand, Silge et al. (2005) and Cappellari et al. (2009) performed near infrared stellar kinematical studies based, respectively, on seeing limited longslit spectra (GNIRS at Gemini South) and AOassisted integral field spectra (SINFONI at the ESO VLT). The top panel of Fig. 3.1 shows the different M BH values obtained by the previous authors, spread over almost a order of magnitude. To understand the origin of these differences, in the bottom panel of Fig. 3.1 we plot the M BH sin 2 i values, i.e. the values constrained by the observed velocity fields and not dependent on the inclination of the rotating gas disks. In the case of the stellar kinematical studies, the authors assumed edge on axisymmetric potentials, therefore no correction is made to obtain M BH sin 2 i. After removing the inclination effect, it is clear that all gas kinematical measurements are all in agreement within the errors; as noted several times, the inclination of the rotating disk is the major source of uncertainty in gas kinematical measurements. Figure 3.1: BH mass measurements for Centaurus A from the works mentioned in the text (top panel) and the corresponding M BH sin 2 i values (bottom panel). Note that the uncertainties on the measurement by Marconi et al. (2001) are reduced because they were including uncertainties on i.

54 Spectroastrometric measurements of BH masses In Chapter 2 we illustrated how the technique of spectroastrometry can be used to measure the black hole masses at the center of galaxies. We focused in explaining the basis of the spectroastrometric approach and in showing with an extended and detailed set of simulations how this method is able to probe the principal dynamical parameters of the nuclear gas disk. The spectroastrometrical method consists in measuring the photocenter of emission lines in different wavelength or velocity channels. We compared it with the standard method for gas kinematical studies based on the gas rotation curves and showed that the two methods have complementary approaches to the analysis of spectral data (i.e. the former measures mean positions for given spectrum velocity channels while the latter measures mean velocities for given slit position channels). The principal limit of the rotation curves method resides in the ability to spatially resolve the region where the gravitational potential of the BH dominates with respect to the contribution of the stars; in Chapter 2 we showed that the fundamental advantage of spectroastrometry is its ability to provide information on the galaxy gravitational potential at scales significantly smaller than the spatial resolution of the observations ( 1/10, and more as we will also show in the present work). To better understand the general principle of the spectroastrometric method and its capability in overcoming the spatial resolution limit, we report a simple example: consider two point-like sources located at a distance significantly smaller than the spatial resolution of the observations; these sources will be seen as spatially unresolved with their relative distance not measurable from a conventional image. However, if spectral features, such as absorption or emission lines at different wavelengths, are present in the spectra of two sources the light spatial profiles extracted at these wavelengths will show the two sources separately. From the difference in centroid positions at these two wavelengths one can estimate the separation between the two sources even if this is much smaller that the spatial resolution. This overcoming of the spatial resolution limit is made possible by the spectral separation of the two sources. In Chapter 2 we concentrated mainly on the application of spectroastrometry to longslit spectra, but also showed that the extension of this technique from to integral field observations is straightforward and discussed the advantages it carries. Finally we explained the practical application of the method and presented a trivial fitting method for estimating the values of the principal dynamical parameters, in particular the BH mass. In the present work we apply this same method to real longslit and integral field spectra. For any detail on the method refer to Chapter Longslit spectra: observations and data analysis The data We use available near infrared spectra of the nucleus of Centaurus A obtained with ISAAC at the ESO VLT telescope (see Marconi et al. 2006, for details). Briefly, the spectra were obtained with a 0.3 wide slit and cover a wavelength range of µm with a

55 Spectroastrometry of rotating gas disks: application to Centaurus A 51 resolving power of λ/ λ = 10500, corresponding to a spectral resolution of 1.2Å at the central wavelength (λ=1.274 µm). The spatial scale of the spectra is /pixel along the slit axis and the dispersion is 0.58Å/pixel. The spatial resolution of the spectra is estimated as 0.5 (FWHM of the PSF). There are three different spectra characterized by a different slit position angle: the one we will refer to as PA1 has a position angle of 32.5, PA2 has 44.5 and PA3 has 83.5 (see Marconi et al. 2006, for details) Analysis of the spectra In Fig. 3.2 we display an example of a typical spectrum of Centaurus A extracted from one pixel along one of the slits: in particular, this corresponds to the PA1 continuum subtracted spectrum, extracted at the position of the continuum peak. Several gas emission lines can be identified: Paβ at 1.284µm, [FeII] and HeI at 1.281µm, [FeII] at 1.259µm and [S IX] at 1.285µm. The lines used for the gas kinematics by Marconi et al. (2006) are the Paβ and the [FeII] and we will also concentrate on these lines. As clearly visible in the figure, both Paβ and [Fe II] lines are blended on the blue sides with [Fe II]+ HeI and [S IX], respectively. Potentially this constitute a problem in measuring the spectroastrometric curve because the blended lines will certainly affect the position of the light centroid along the slit for a given blue velocity and the position centroid will not be indicative anymore of the mean position of the gas at a given line of sight velocity Counts (a.u.) Wavelenght (Å) Figure 3.2: Continuum subtracted spectrum extracted at the at the position of the continuum peak along the slit of PA1. Solid red line: simultaneous fit of all four lines. Dashed red lines: the deblended Paβ and [Fe II] components. In order to solve this problem it is necessary to deblend the lines under examination and therefore we performed a simultaneous fit of the 4 lines, each with a Gauss-Hermite function (the red solid line), at all slit positions along the PA1, PA2 and PA3 slits. Following Marconi et al. (2006) we impose that all lines, hence Paβ and [Fe II], share the same

56 52 kinematics: in the simultaneous fit all lines are constrained to the same velocity, velocity dispersion and Hermite parameters (h 3 and h 4 ), while they can have different line fluxes. Fig. 3.2 shows an example of such a fit. Position along the slit (pixel) Original data longslit spectrum Position along the slit (pixel) Wavelenght (µm) Synthetic reconstructed longslit spectrum Wavelenght (µm) Figure 3.3: Position velocity diagram of the Paβ line at PA1. Upper panel: observed. Bottom panel: synthetic reconstruction. Isophotes denote the same values in both panels. To obtain the spectroastrometric curve we considered all fitted profiles of Paβ and [FeII] and we reconstructed synthetic longslit spectra of the emission lines, cleaned in terms of noise and blended lines. In Fig. 3.3 we show an example of one original Paβ longslit spectrum compared with its synthetic version where one can notice that the noise has been smoothed away and the [FeII] + HeI complex has been removed. It should be noticed that whereas we constrained Paβ and [FeII] to the same kinematics, their fluxes can be different therefore, in terms of the spectroastrometric analysis, the two lines are different can still provide different results. Also the synthetic reconstructed spectrum is noise-free because each row represents the fitted parametric profile but one has to take into account the errors on the free parameters in order to estimate the errors on each synthetic pixel counts. Therefore, for each spectral fit, we simulated 1000 synthetic spectra from 1000 realization of the set of the five parameters distributed following a pentavariate distribution with the fit correlation matrix. The flux and error of each pixel is then estimated from mean and standard deviation of the 1000 realizations. We note that for each profile along the slit, the errors on fluxes are uncorrelated because they originate from independent fits to the line profiles. From the synthetic Paβ and [FeII] spectra for the three slits (PA1, PA2 and PA3),

57 Spectroastrometry of rotating gas disks: application to Centaurus A 53 PA1 PA2 PA3 0.2 Position along the slit (arcsec) Velocity (km/s) Figure 3.4: Spectroastrometric curves of the Paβ (black points) and [FeII] (red points) lines at the three slit position angles. Left panel: PA1. Central panel: PA2. Right panel: PA3. following the method outlined above, one can derive the spectroastrometric curves which are shown in Fig Fig. 3.4 suggests several considerations. First, the extremely large error bars observed in the Paβ curve at PA1 (400km/s v 550km/s) are due to the fact that the light profile is spatially resolved and our method to measure the centroid position is not reliable. As observed in section 2.3 this is expected in the low velocity range, but the relevant information are concentrated in the high velocity range of the curve. Except for those points, errors on photocenter positions range from 0.01 to 0.05 that is from 1/50 to 1/10 of the spatial resolution of the data and this is the accuracy with which we can measure centroid positions. As expected, the spectroastrometric curves for the two lines are marginally different. Their differences are due to the intrinsic flux distribution of the lines on the sky plane but, as observed in section 2.3.4, these differences tend to disappear in the high velocity range The spectroastrometric map of the source For each emission line we have obtained three spectroastrometric curves, one for each PA of the slit. Each spectroastrometric curve provides the photocenter position along one slit, i.e. the position of the photocenter projected along the axis identified by the slit. Combining the spectroastrometric curves we can thus obtain the map of photocenter positions on the plane of the sky for each velocity bin. In principle, the spectroastrometric curves from two non-parallel slits should suffice but we can use the redundant information from the three slits to recover the 2D sky map as described in Section We have chosen a reference frame in the plane of the sky centered on the center of PA1 slit and with the X axis along the North direction. For a given velocity bin we then determined the position of the light centroid on the sky plane resulting in the 2D spectroastrometric map shown in Fig. 3.5.

58 54 Note that also the coordinates of the origin of each slit must be considered as free parameters. These unknowns are estimated simultaneously with the position of the photocenter following a χ 2 minimization procedure. The final spectroastrometric map on the plane of the sky is that given by the best fitting set of slit centers. The error bars on the point represent the uncertainties resulting from the fit. The black points correspond to the high velocity range (i.e. v 380 km/s and v 800 km/s) which were actually used to determine the location of slit centers. Indeed, in Chapter 2 we concluded that the high velocity range of the spectroastrometric curve is more robust, and less affected by slit losses which artificially change the photocenter position in the low velocity range y East (arcsec) Velocity (km/s) x North (arcsec) Figure 3.5: Spectroastrometric 2D map derived from the Paβ line. Left panel: derived photocenter positions on the sky plane, the black points are those actually used for the minimization.right panel: the 3D plot of the map, where the z axis is velocity. As observed in appendix 2.A, we used the width of the gaussian fitted to the principal peak of the light profile to select the high velocity range: in Fig. 3.6 we display the FWHMs (Full Width Half Maxima) of the gaussian fitted to the principal peak of the light profile for spectroastrometric curves of Centaurus A. We can observe that at low velocities the FWHM increases because the emission peak is no more spatially unresolved. We compare the FWHM to the spatial resolution ( 0.5 ) because the FWHM of the light profile of an unresolved source should be of the order of the spatial resolution. We selected the high velocity range by imposing that the FWHM is lower than 1.1 times the spatial resolution (FWHM of the PSF), resulting in v 380km/s and v 800km/s for the spectroastrometric curves of both lines. The two-dimensional spectroastrometric map just described (and shown in Fig. 3.5) can now be used to estimate some geometrical parameters of the nuclear gas disk. If the gas kinematics is dominated by rotation around a point-like mass (the BH), the position of the light centroid at the high velocities should lie on a straight line (which identifies the direction of the disk line of nodes) and should approach, at increasing velocities, the position of the BH. This consideration allows us to estimate the position on the plane of the sky of the BH as the average position of the high velocity points in the 2d spec-

59 Spectroastrometry of rotating gas disks: application to Centaurus A FWHM (arcsec) Velocity (km/s) 2.0 FWHM (arcsec) Velocity (km/s) Figure 3.6: FWHM of the gaussian fitted to the principal peak of the light profile principal peak in the case of Centaurus A data. Upper panel: Paβ line. Bottom panel: [FeII] line. Solid lines: PA1 curves. Dotted lines: PA2 curves. Dot-dashed lines: PA3 curves. The horizontal dashed lines denotes 1.1 times the PSF FWHM. troastrometric map 1 as well as the position angle of the line of nodes (θ LON ), obtained by fitting a straight line to the high velocity points (see Fig. 3.5). We report below the BH position and disk line of nodes obtained from the two lines separately and that are consistent with each other within the uncertainties. The uncertainty on BH position is 0.01, i.e. 1/20 of the spatial resolution of the data. Line (x BH, y BH ) [ ] θ LON [ ] Paβ (0.096 ± 0.012, ± 0.008) 18.3 ± 1.8 [Fe II] (0.100 ± 0.009, ± 0.006) 20.3 ± 1.2 It should be noticed that all low velocities lie outside of the line of nodes, as expected Estimate of the BH mass from the spectroastrometric map Here we recover the BH mass value from the spectroastrometric map, following the method outlined in Sect In practice, we calculate this position by first taking the averages of the coordinates of the points in the blue and red high velocity ranges and then taking the average coordinates of the blue and red positions.

60 56 Briefly, under the assumption that the gas lies in a thin disk configuration inclined by i with respect to the plane of the sky (i = 0 face-on) and the disk line of nodes has a position angle θ LON, the circular velocity of a gas particle with distance r from the BH is given by: V rot = G[MBH + M/L L(r)] r (3.1) where L(r) is the radial luminosity density distribution in the galactic nucleus and M/L is the mass to light ratio of the stars (see Chapter 2 for more details). The component of V rot along the line of sight (hereafter V ch for channel velocity ) is: V ch = V rot sin(i) + V sys (3.2) where i is the inclination of the disk and we also added the systemic velocity of the galaxy V sys (see Marconi et al. 2006, for details). As explained in Sections and and Sections 2.3 and 2.5, we only make use of the high velocities of the spectroastrometric 2d map to estimate the BH mass. The first step is to recover the disk line of nodes by fitting a straight line on the 2d map. We then project the position of the 2d map points (x ch, y ch ) on the line of nodes, calculating their coordinate with respect to this reference axis (S ch ) and then their distance r from the BH used in eq. 3.1 (i.e. r = k S ch S 0 where S 0 is the coordinate along the line of nodes of the BH and k is a scale factor to transform arcsec in the right distance unit 2 ; see Chapter 2 for details). Then from eqs. 3.1 and 3.2 we obtain the model channel velocity: V ch = G(M BH + M/L L(k S ch S 0 ) sin(i) + V sys (3.3) k S ch S 0 The unknown parameters of this model are found minimizing the quantity [ ] 2 χ 2 Vch V ch = (3.4) (S ch ; par) ch where (S ch ; par) is the uncertainty of the numerator. As previously discussed, we restrict the fit (i.e. the sum over the velocity channels) to the high velocities range. As explained in sect. 2.5, the channel velocity V chan has no associated uncertainty since it is not a measured quantity but the central value of the velocity bin. Finally we add a constant error ( sys ) in quadrature to the quantity (S ch ; par) in order to obtain a reduced χ 2 close to 1 (see sect. 2.5 for a detailed explanation of this choice). Finally, using the best fit values of the model parameters we can compute the (x, y) position of the BH in the sky plane. In the rotation curve model shown in eq. 3.3, disk inclination i and BH mass are coupled since coordinates along the line of nodes (S ch ) do not depend on i and this parameter 2 To be consistent with all previous measurements we assume a distance to Centaurus A of 3.5Mpc. At this distance 1 corresponds to 17pc.

61 Spectroastrometry of rotating gas disks: application to Centaurus A 57 appears only as a scaling factor on the velocity. The reason for this coupling is that we are effectively measuring velocities of rotating material located on the line of nodes, thus removing any dependence on i except for the projection of the velocity along the line of sight. Therefore our fitting method can only measure M BH sin 2 i, and we need to assume a value for the inclination to obtain a value of the mass. In conclusion the free parameters in our fit are: M BH sin 2 i M/L sin 2 i S 0 V sys mass of the BH; mass to light ratio of the nuclear stars; line of nodes coordinate of the BH; systemic velocity of the galaxy; in the following we will only report M sin 2 i values and we will discuss the inclination values we assume and the relative mass values we obtain. We have performed the fit of the spectroastrometric data from the [FeII] and Paβ lines, both separately and simultaneously. Fit results are tabulated in Table 3.1 and presented graphically in Fig. 3.7 where we plot the S ch S 0 vs V ch V sys rotation curve and the line of nodes projected rotation curve (e.g. S ch vs V ch ). The solid red lines represent the curves expected from the model (r vs V ch V sys and S ch vs V ch respectively). The first result is that assuming a disk inclination of 25 as in Marconi et al. (2006) we obtain M BH = ±0.02 M, perfectly consistent with the result presented in that work (Marconi et al report M BH = ±0.04 M ). Since we are actually using the same ISAAC data of Marconi et al. 2006, we can draw the important conclusion that the spectroastrometric method provides results perfectly consistent with the rotation curve method, but with a much simpler approach which does not require to take into account complex instrumental effects. Furthermore, we note that M/L is not constrained, a results which has been already found with the classical rotation curve fitting. Computing the radius of the BH sphere of influence for Centaurus A and using a stellar velocity dispersion of σ star 200km/s, one obtains r BH 14.9pc, a factor of 10 larger than the distances we are considering here (at the distance of 3.5Mpc 1 corresponds to 17pc and so the apparent dimension of r BH is 0.9 ). This implies that at these small radii the contribution of the stellar mass to the gravitational potential is negligible and consequently M/L cannot be constrained with the fit. Another important result derived from the application of the spectroastrometric method is that the minimum distance from the BH at which there is a velocity estimate is 0.05 corresponding to 0.8 pc, while with the standard rotation curve method the minimum distance from the BH which can be observed is of the order of half the spatial resolution (0.25 ). This clearly shows how spectroastrometry can overcome the spatial resolution limit. In Centaurus A the sphere of influence of the BH is already well resolved in ground based observations with good seeing and this enables us to obtain a BH measurement already from the standard methods. The potential of spectroastrometry clearly relies on the possibility to use the extra resolution to measure BH with smaller spheres of influence, like those with lower masses, or located in more distant galaxies.

62 58 S (arcsec) V (km/s) V (km/s) r (arcsec) S (arcsec) V (km/s) V (km/s) r (arcsec) S (arcsec) V (km/s) V (km/s) r (arcsec) Figure 3.7: Results of the for the [Fe II] data (upper panels), Paβ data (central panels) and simultaneous fit of the [FeII] and Paβ data (lower panels). The left panels show the line of nodes projected rotation curve S ch vs. V chan. The right panels show the S ch S 0 vs. V ch V sys rotation curve. The solid red lines represent the curves expected from the model 3.5 Integral field spectra: observations and data analysis Data and spectra analysis We use available near infrared spectra of the nucleus of Centaurus A obtained with SIN- FONI at the ESO VLT (see Neumayer et al for details). In particular we make use

63 Spectroastrometry of rotating gas disks: application to Centaurus A 59 of H band spectra observed in seeing limited mode and H and K band spectra obtained with the assistance of the adaptive optics (AO) system MACAO. The seeing of the observations as measured by the seeing monitor was FWHM V 0.5 (transformed to K band, FWHM K 0.38 ). Seeing limited spectra use a pixel scale of and cover a 8 8 field of view. The SINFONI AO module used as guide star an R 14 mag star 36 southwest of the nucleus providing a spatial resolution of 0.12 (FWHM). AO assisted spectra use a pixel scale of and cover a 3 3 field of view. Spectral resolution is R 4000 for the K band and a slightly lower for the H band, R The total on-source exposure for the K band data cube was 13500s whereas for H band was 3600s. For all details of observations and data reduction the reader should refer to Neumayer et al. (2007). All data cubes were continuum subtracted by fitting a power law function to the spectrum of each spatial pixel (with emission and absorption lines masked) which was then subtracted. As observed in sections 2.3 and 2.4.1, for this application of spectroastrometry it is mandatory to use continuum subtracted spectra. The focus is exclusively on gas kinematics and the presence of an underlying continuum can significantly alter the spectroastrometric measurement of the emission line gas by modifying the spatial light distribution in each velocity channel. In the following spectroastrometric analysis we will use [FeII](1.6435µm) observed in the H band and the H 2 (2.1213µm) line observed in K band The spectroastrometric map of the source As observed in section 2.6, the extension of the spectroastrometric technique to integral field spectra is straightforward, and the problem of deriving a 2d spectroastrometric map with longslit spectra becomes trivial with integral field spectra due to their 2d spatial coverage. The analysis of data now reduces to fitting a 2d Gaussian to each channel map in turn, yielding the X, Y positions of the photocenters as a function of velocity, i.e. the spectroastrometric map. Therefore, we can directly derive the 2d spectroastrometric map from the continuum subtracted SINFONI data cubes, overcoming the problems related to the uncertainties in slit positioning with respect to the galaxy nucleus. To select the high velocity range in the case of integral field data we also make use of the widths of the 2d Gaussians fitted to each velocity channel map. As explained in sect. 3 and appendix 2.B we search for unresolved spatial emission. In Fig. 3.8 we show as an example the FWHMs (Full Width Half Maxima) of the fitted 2d gaussian 3 for the seeing limited [FeII] data in the H band where the estimated seeing is 0.4. We can observe an evident peak in the FWHM at low velocities due to the presence of spatially resolved emission. As previously, we identify the high velocity range by considering FWHMs lower than 1.1 times the spatial resolution ( 0.4 ). The resulting range in this particular case is v 370km/s and v 370km/s. In Figs. 3.9 we show the derived spectroastrometric maps for the H band [FeII] data (both AO assisted and not) and for the K band H 2 AO assisted data. Typical uncertainties in the light centroid positions are of the order of 0.01 for the seeing limited data 3 Actually we fit a non circularly symmetric 2D gaussian function. In Fig. 3.8 we show the minimum of the two FWHM values along the proper axis.

64 60 Figure 3.8: Example of the FWHM of the fitted 2D gaussian to each velocity channel map for the H band seeing limited [FeII] data. The horizontal dashed lines denotes 1.1 times the PSF FWHM. (spatial resolution 0.4 and pixel scale ), and a factor 2.5 lower for the AO assisted data ( with spatial resolution of 0.12 and pixel scale of ). It should also be noticed that all low velocities lie outside the line of nodes, as expected. These 2d spectroastrometric maps can now be used to obtain, as previously explained, the direction of the disk line of nodes and a first estimate of the BH position on the plane of the sky (see the table below). The derived BH positions are not directly comparable among different datasets. In fact the origin of the sky plane coordinates in Fig. 3.9 is an arbitrary position near to the continuum peak position, chosen independently for each spectrum. Line (x BH, y BH )(arcsec) θ LON [Fe II] (no AO) (0.127 ± 0.007, ± 0.008) 47 ± 2 [Fe II] (AO) (0.024 ± 0.002, ± 0.002) 39 ± 1 H 2 ( ± 0.005, ± 0.006) 28 ± 1 We can observe that the derived θ LON values for the [Fe II] line observed with or without the AO are consistent within 8. Otherwise the H 2 line spectroastrometric map shows an orientation of line of the nodes offset by 15. While the differences between [FeII] (AO) and H 2 are due to the different kinematics of the two lines, the differences between [FeII] (AO) and [FeII] (no AO) might be accounted for by a warping in the gas disk which results in different orientations of the line of nodes at different spatial scales.

65 Spectroastrometry of rotating gas disks: application to Centaurus A 61 Figure 3.9: 2D spectroastrometric map for the ISAAC longslit J band data (upper panels): Paβ line (upper left panel) and [FeII] line (upper right panel). Spectroastrometric map for the SINFONI H band data (middle panels): [FeII] line for the seeing limited (middle left panel) and AO assisted observations (middle right). Spectroastrometric map from the H 2 molecular line (lower panel) from AO assisted observations. The red point marks the inferred BH position while the red solid line represents the line of nodes of the gas disk. All boxes (except the lower one) have the same angular dimension on the plane of sky ( ) and are centered on the BH positions. Note that the x and y coordinates of each box are different because the origin of the sky plane is an arbitrary position near to the continuum peak positions chosen independently in each spectrum analysis and it is the same only for the ISAAC data.

66 Estimate of the BH mass from the spectroastrometric map Here we estimate the BH mass using the spectroastrometric map as described in Sect The application is exactly the same as in the case of the longslit spectra presented in Sect , because in both cases we use as input the 2d spectroastrometric map (i.e. the positions of the light centroids in the plane of sky x ch, y ch as a function of the correspondent channel velocity V ch ). We have performed the fit of the data from the three spectroastrometric maps shown in Fig Fit results are tabulated in Table 3.1 and presented graphically in Fig where we plot either the S ch S 0 vs V ch V sys rotation curve and the line of nodes projected rotation curve (e.g. S ch vs V ch ). The solid red lines represent the model rotation curves (r vs. V ch V sys and S ch vs. V ch ). From the [FeII] data we estimate a value of the BH mass of log 10 (M BH sin 2 i/m ) = 7.5, perfectly consistent between AO-assisted and seeing limited (Tab. 3.1) whereas from the H 2 line we obtain a value ( 0.2 dex) larger. Note that the higher accuracy of the light centroid positions of the spectroastrometric map for the AO assisted [FeII] data with respect to the seeing limited ones results in a lower uncertainty on the M BH sin 2 i best fit value. M/L is not constrained from the fit even with SINFONI data. As observed in Section 3.4.4, the radius of the BH sphere of influence for Centaurus A is r BH 14.9pc corresponding to an apparent dimension of 0.9 ; here we are studying the rotation curve at 1/20 smaller scales where the contribution of the stellar mass to the gravitational potential is negligible. An impressive result of the spectroastrometric method is that the minimum radii at which we can probe the rotation curve are 25mas for seeing limited data and 20mas for AO assisted data. The latter are only slightly smaller but have a much better positional accuracy, as shown in Fig These values are 1/16 and 1/6, respectively, of the spatial resolution ( 0.4 for the seeing limited and 0.12 for the AO assisted observations) and correspond to distances from the BH of 0.42pc and 0.35pc, respectively, 1/40 of the radius of the BH sphere of influence. This is a clear demonstration of the great potentials of spectroastrometry in overcoming the spatial resolution limit. The fit of the H 2 spectroastrometric map provides a larger value of M BH sin 2 i, while all the best fit parameters are characterized by lower accuracy. This is mainly due to the fact that there is just a small number (only 4) of spectroastrometric points which can be used for the fit. As previously mentioned, these high velocity points where selected by requiring that the corresponding spatial light distribution be unresolved. For the H 2 line, the combined effect of lower S/N and of more extended spatial emission prevent us to have a larger number of high velocity points. With only 4 points (and 1 degree of freedom, after fixing the M/Lsin 2 i parameter) the quality and reliability of the fit is obviously worse and more sensible to biases and systematic errors. Therefore, we do not consider problematic the observed different estimate obtained from H 2.

67 Spectroastrometry of rotating gas disks: application to Centaurus A 63 Table 3.1: Fit Results. Parameter Best fit value±error ISAAC data SINFONI data [FeII] line fit Paβ line fit both lines fit NO AO [FeII] line fit AO [FeII] line fit AO H2 line fit θlon [ ] (1) 20.3 ± ± ± ± ± ± 0.8 b [ ] (1) ± ± ± ± ± ± log10(mbh sin 2 i/m ) 7.39 ± ± ± ± ± ± 0.09 log10(m/l sin 2 i/m ) 10.1 ± 0.0 (2) 9.6 ± 0.0 (2) 7.9 ± 0.0 (2) 11.8 ± 0.0 (2) 18.8 ± 0.0 (2) 1.1 ± 0.0 (3) S 0 [ ] ± ± ± ± ± ± 0.06 Vsys [km/s] ± ± ± ± ± ± 47 sys [km/s] χ 2 red (χ 2 /D.O.F.) 1.01 (9.11/9) 0.82 (8.20/10) 0.93 (21.49/23) 0.17 (1.57/9) 1.03 (11.3/11) 1.05 (1.05/1) xbh [ ] (4) ± ± ± ± ± ± 0.05 ybh [ ] (4) ± ± ± ± ± ± 0.03 Notes. (1) Best fit parameters estimated in the fit of the line of nodes. (2) Parameter not constrained from the fit. (3) Parameter hold fixed. (1) The origin of the plane of sky (x,y) reference system is the same for the ISAAC data fits, but is different for the three SINFONI data fits.

68 64 Figure 3.10: Results of the fit for the [Fe II] data seeing limited (upper panels), [Fe II] data AO assisted (middle panels) and H 2 data AO assisted (lower panels). The left panels show the line of nodes projected rotation curve S ch vs. V ch. The right panels show the S ch S 0 vs. V ch V sys rotation curve. The solid red lines represent the curves expected from the model.

69 Spectroastrometry of rotating gas disks: application to Centaurus A Discussion and conclusions We have applied the spectroastrometric method to longslit and IFU spectra of Centaurus A, both seeing limited and AO assisted. Compared to the standard one, the spectroastrometric method is much simpler from the modeling point of view; it only requires the determination of the spectroastrometric map and is very little affected by the problems that plague the standard approach based on rotation curves like, e.g., the effect of beam smearing, the intrinsic flux distribution of the line, and the biases due to the slit positioning in longslit observations. With our proposed spectroastrometric approach we can derive two-dimensional planeof-the-sky spectroastrometric maps of the source characterized by accuracies of position measurements much lower than the spatial resolution of the observations. The mean accuracies of our light centroid position estimates are 0.01 for the ISAAC data and seeing limited SINFONI data ( 1/40 of the spatial resolution) and for AO assisted SINFONI data ( 1/30 of the spatial resolution). The position angles of the disk line of nodes estimated from those maps are 19 for the two ISAAC maps (consistent within 2 ). The estimates from SINFONI maps are 20 offset from the previous, but we note that different lines have been observed, and in principle those differences can reflect intrinsic kinematical differences of the different emitting regions. The two SINFONI [FeII] observations (seeing limited and AO assisted), in fact, give us line of nodes P.A. estimates consistent within 8, whereas the H 2 line shows a line of the nodes orientation different for 15 from the others. The most important result in this work is the demonstration of the capability of spectroastrometry to overcome the spatial resolution limit and estimate BH masses in a simple and neat way. The minimum distance from the BH at which we can probe the gas rotation curve is 50mas for the ISAAC data ( 1/10 of the spatial resolution) and 20mas for the SINFONI data ( 1/15 of the spatial resolution for the seeing limited data and 1/6 for the AO assisted data). In the case of Centaurus A, these corresponds to 1/18 and 1/40, respectively, of the radius of the BH sphere of influence indicating that is is possible to probe deep in the BH potential well, where the contribution from the mass in stars is totally negligible. In Table 3.2 we compare M BH sin 2 i values resulting from recent applications of the classical rotation curve method to various data sets of Centaurus A (including those analyzed here) analyzed here with our results from the application of the spectroastrometric method. From Table 3.2 several considerations can be made. Our new simple method based on spectroastrometry is perfectly in agreement with the classical method based on the rotation curves, at least when comparing the results obtained from the same dataset (cf. the ISAAC Marconi et al data). This is a fundamental indication for the robustness of our new method: using the same dataset we can apply indifferently the classical and the spectroastrometrical method obtaining perfectly consistent results. When applying the two methods to different datasets but with the same target line we obtain consistent estimates (cf. the M BH estimate by Häring-Neumayer et al targeting the H band [FeII] line and the one presented in this chapter, and that

70 66 Table 3.2: Mass estimates for Centaurus A Classical method applications log 10 (M BH sin 2 i/m ) ISAAC J band [FeII] and Pa β lines (Marconi et al. 2006) (i = 25 ) 7.39 ± 0.04 NACO H band AO [FeII] line (Häring-Neumayer et al. 2006) (i = 45 ) CIRPASS J band Pa β line (Krajnović et al. 2007) (i = 25 ) SINFONI K band AO H 2 line (Neumayer et al. 2007) (i = 34 ) 7.1 ± 0.1 Spectroastrometric method applications ISAAC J band [FeII] line 7.39 ± 0.02 ISAAC J band Pa β line 7.39 ± 0.03 ISAAC J band [FeII] and Pa β simultaneous fit 7.39 ± 0.02 SINFONI H band no AO [FeII] line 7.53 ± 0.07 SINFONI H band AO [FeII] line 7.52 ± 0.01 SINFONI K band AO H 2 line 7.77 ± 0.09 by Krajnović et al based on the J band Pa β, with our own which is within 0.2 dex). The differences observed in Table 3.2 ( dex) are mainly due to the fact that we are observing different lines and therefore different mass estimates can arise from different lines kinematics and not from the application of a particular method. The largest discrepancy comes from the the spectroastrometry of AO-assisted K band H 2 observations, but in this case the application of spectroastrometry is made difficult by lower S/N and more extended spatial emission resulting in a small number of useful spectroastrometric measurements. The application of our spectroastrometric method to different data type (IFU and longslit) give consistent result (within only 0.1 dex) and this demonstrates the versatility of our method. The application to IFU data is obviously much simpler but this agreement also show that our method of reconstruction of the 2d map from multiple longslit spectroastrometry is correct. The application of the method to IFU data with and without AO also produces consistent results at similar spatial scales. This clearly demonstrates how spectroastrometry is much less sensitive to spatial resolution than the classical method. As expected, the accuracy of measurement positions without AO is worse. This is due to the fact that for each velocity channel we recover the centroid position by fitting a two dimensional gaussian function and the width of this function clearly increases with the spatial resolution of the data. Being this function center symmetric we can always recover the correct center position and increasing the width of the function only makes the center position uncertainty larger but does not alter its position. The weak point of spectroastrometry is its insensitivity to disk inclination. In our fit, in fact, mass and disk inclination are coupled. This is because coordinate along the line of nodes (S ch ) do not depend on i, which only appears as an unavoidable projecting factor on the velocity. Therefore our method can return only a M BH sin 2 i value with the need of

71 Spectroastrometry of rotating gas disks: application to Centaurus A 67 assuming an inclination value to derive the value of the mass. In this work we decided to present only the M BH sin 2 i values and to move the discussion on the M BH values to a discussion on the i values estimated or assumed by the various authors. For completeness we report in Table 3.2 the inclinations estimated or assumed in those previous works. In conclusion, we have applied the spectroastrometric method presented in Chapter 2 to several datasets of the nucleus of the Centaurus A galaxy. This method, which is much simpler and straightforward than the classical method based on rotation curves, provides consistent estimates of M BH, and allow to probe spatial scales which are much smaller that the spatial resolution of the observations. So far we have applied the spectroastrometric method independently from the classical method. However as discussed in Chapter 2, it is clear that the spectroastrometric and classical rotation curves are complementary and orthogonal descriptions of the position velocity diagram. Therefore, a future development of this method will be its application in combination to the classical method based on rotation curves. This will also allow us to mitigate the problem of the disk inclination which can be constrained using twodimensional spatially resolved kinematical maps.

72 Chapter 4 Dynamical properties of AMAZE and LSD galaxies from gas kinematics and the Tully-Fisher relation at z Introduction In this chapter we deal with the second physical system considered in this thesis. We move away in distance and time from the study of gas dynamics in the nuclei of nearby galaxies to the study of gas dynamics of high redshift galaxies. The dynamical properties of galaxies play a fundamental role in the context of galaxy formation and evolution and the study of high redshift galaxies is the more direct way to understand the physical mechanism that were acting in the past of galaxies. The observed dynamics of galaxies represent a fundamental test to the predictions of cosmological cold dark matter models of hierarchical structure formation (e.g. Blumenthal et al. 1984; Davis et al. 1985; Springel et al. 2006, Mo et al. 1998). In such models mergers are believed to play an important role for galaxy formation and evolution. However, the observational evidence for the existence of rotating disks with high star formation rates at z 2 suggests that smooth accretion of pristine gas is also an important mechanism that drives star formation and mass assembly at high redshift (Epinat et al. 2009; Wright et al. 2009; Förster Schreiber et al. 2009; Cresci et al. 2009). In recent years many dynamical studies have been performed on extended samples of z objects (Genzel et al. 2006; Genzel et al. 2008; Förster Schreiber et al. 2006b; Förster Schreiber et al. 2009; Cresci et al. 2009; Erb et al. 2006; Law et al. 2007; Law et al. 2009; Epinat et al. 2009; Wright et al. 2007; Wright et al. 2009). However, little is known on the dynamics of galaxies at z 2.5, where only a few particular objects have been investigated (Nesvadba et al. 2006; Nesvadba et al. 2007; Nesvadba et al. 2008; Jones et al. 2010; Law et al. 2009; Lemoine-Busserolle et al. 2009; Swinbank et al. 2007; Swinbank et al. 2009). Moreover, the redshift range z 3 4 is particularly important to study since it is before the peak of the cosmic star formation rate (Dickinson et al. 2003; Rudnick et al. 2006; Hopkins & Beacom 2006; Mannucci et al. 2007), it is the redshift range when the most massive early-type galaxies are expected to form (Saracco et al. 2003) and the number of galaxy mergers is much larger than at later times (Conselice

73 Dynamical properties of AMAZE and LSD z 3 galaxies 69 et al. 2007; Stewart et al. 2008). The work presented in this chapter is based on the projects AMAZE (Assessing the Mass-Abundance redshift Evolution) (Maiolino et al. 2008a, Maiolino et al. 2008b) and LSD (Lyman- break galaxies Stellar populations and Dynamics) (Mannucci & Maiolino 2008, Mannucci et al. 2009). Both projects use integral field spectroscopy (i.e. with SINFONI at the ESO VLT) of samples of z 3 4 galaxies in order to derive their chemical and dynamical properties. The AMAZE sample consists of 30 Lyman Break Galaxies in the redshift range 3 < z < 4.8 (most of which at z 3.3), with deep Spitzer/IRAC photometry (3.6 8µm), an important piece of information to derive reliable stellar masses. These galaxies were observed with SINFONI in seeing-limited mode. The LSD sample is a representative, albeit small, sample of 10 LBGs at z 3 with available Spitzer and HST imaging from the Steidel et al. (2003) catalogue. For LSD SINFONI observations were performed with the aid of adaptive optics in order to improve spatial resolution since this project was aimed to obtain spatially-resolved spectra for measuring kinematics and gradients in emission lines. The global sample is listed in Tab We thus focus on the dynamics of AMAZE and LSD galaxies at z 3 in order, among other things, to obtain a sizable sample of high z galaxies with reliable dynamical mass estimates which will be used in the next chapter to calibrate the spectroastrometrical method in this context. In Sect. 4.2 we present observations, data reduction and the method followed to extract kinematical maps from integral field spectra. In Sect. 4.3 we present the results and a simple method to individuate the objects with a velocity map consistent with a rotating disk. In Sect. 4.4 we describe the rotating disk model adopted in this paper, explain our fitting strategy to constrain model parameters and estimate their errors. In Sect we discuss the results of model fitting on the selected subsample of objects consistent with rotating disks. In Sect. 4.5 we discuss fit results and present the Tully-Fisher relation at z 3 (Sect ). 4.2 Observations and data reduction Complete descriptions of the AMAZE and LSD programs, of their observations and data reduction are presented in Maiolino et al. (2008a) and Mannucci et al. (2009). Here we report a brief summary on observations and data reduction. The near-ir spectroscopic observations were obtained by means of SINFONI, the integral field spectrometer at VLT (Eisenhauer et al. 2003). For AMAZE galaxies, SIN- FONI was used in its seeing-limited mode, with the pixel scale and the H+K grism, yielding a spectral resolution R 1500 over the spectral range µm. For the LSD galaxies, SINFONI was used with the Adaptive Optics module using a bright star close to the galaxy to guide the wavefront correction system. The pixel scale used is of (for all object except SSA22a-C6 and SSA22a-M4 for which was used the pixel scale). The (K-band) seeing during the observations was generally about In the AO assisted observations the spatial resolution obtained is 0.2. Data were reduced by using the ESO-SINFONI pipeline (version 3.6.1). The pipeline

74 70 Table 4.1: Galaxy properties and observation setups. Object sample R.A. (1) Dec. (1) z/scale(kpc/ ) (2) Texp(min) pixel scale( ) line (3) SSA22a-M38 AMAZE 22:17: :19: / [OIII] SSA22a-C16 AMAZE 22:17: :13: / [OIII] CDFS-2528 AMAZE 03:32: :53: / [OIII] SSA22a-D17 AMAZE 22:17: :18: / [OIII] CDFA-C9 AMAZE 00:53: :32: / [OIII] CDFS-9313 AMAZE 03:32: :47: / [OIII] CDFS-9340 (4) AMAZE 03:32: :47: / [OIII] CDFS AMAZE 03:32: :45: / [OIII] 3C324-C3 AMAZE 15:49: :27: / [OIII] DFS2237b-D29 AMAZE 22:39: :55: / [OIII] CDFS-5161 AMAZE 03:32: :51: [OIII] DFS2237b-C21 AMAZE 22:39: :50: / [OIII] SSA22a-aug96M16AMAZE 22:17: :13: / [OIII] Q1422-D88 AMAZE 14:24: :00: / [OIII] CDFS-6664 AMAZE 03:32: :50: / [OIII] SSA22a-C36 AMAZE 22:17: :16: / [OIII] CDFS-4414 AMAZE 03:32: :51: / [OIII] CDFS-4417 (4) AMAZE 03:32: :51: / [OIII] CDFS AMAZE 03:32: :45: / [OIII] CDFS AMAZE 03:32: :44: / [OIII] CDFS AMAZE 03:32: :43: / [OIII] CDFS AMAZE 03:32: :42: / [OIII] CDFS AMAZE 03:32: :41: / [OIII] Cosmic eye (5) AMAZE 21:35: :01: / (5) [OIII] Abell (5) AMAZE 13:11: :19: / (5) [OIII] Abell (5) AMAZE 13:11: :20: / (5) [OII] Abell (5) AMAZE 13:11: :19: / (5) [OIII] SSA22b-C5 LSD 22:17: :04: / (AO) [OIII] SSA22a-C6 LSD 22:17: :11: / (AO)[OIII] SSA22a-M4 (4) LSD 22:17: :11: / (AO)[OIII] SSA22a-C30 LSD 22:17: :15: / (AO) [OIII] Q0302-C131 LSD 03:04: :11: / (AO) [OIII] Q0302-C171 LSD 03:04: :08: / (AO) [OIII] DSF2237b-D28 LSD 22:39: :55: / (AO) [OIII] Q0302-M80 LSD 03:04: :13: / (AO) [OIII] DSF2237b-MD19 LSD 22:39: :48: / (AO) Hα Notes. (1) J2000 (2) Redshifts are measured from the observed [OIII] line wavelength. (3) [OIII] denotes the [OIII]λλ 5007,4959Å doublet; Hα the Hα λ 6563Å line. (4) The object is in the same field of view of the object on the previous line. (5) Lensed objects not analyzed in this paper. The spatial scale (pc/ ) depends on the lensing model.

75 Dynamical properties of AMAZE and LSD z 3 galaxies 71 subtracts the sky from the temporally contiguous frames, flat-fields the images, spectrally calibrates each individual slice and then reconstructs the cube. Individual cubes were aligned in the spatial direction using the offsets of the position of the [OIII] or Hα line emission peak. Within the pipeline the pixels are resampled to a symmetric angular size of or The atmospheric absorption and instrumental response were taken into account and corrected by dividing with a suitable standard star. In Table 4.1 we summarize all the relevant properties and observation setups of the object presented in this paper. In the AMAZE sample there is a subsample of 4 lensed objects. We decided not to analyze these objects in this paper because of the uncertainties introduced by the lensing model and the associated de-projection. They will be discussed in a forthcoming paper Extraction of the gas kinematics We extract the kinematics of the gas by fitting the emission line spectrum for each spatial pixel of the cube corresponding to a given position on the sky. In order to improve the signal-to-noise ratio (hereafter S/N) we first perform a smoothing of the cube spatial planes, by using a gaussian filter with FWHM of 3 pixels. This is smaller than the spatial resolution of the observations which usually corresponds to a Point Spread Function with a FWHM of at least 4 pixels. In all cases, but one, we fit the profile and shift of [OIII]λλ5007, 4959Å doublet. The two [OIII] lines are parametrized using single gaussian functions with the following, physically motivated constraints: the two lines are forced to have the same average velocity v and velocity dispersion σ, while their flux ratio is fixed, F 4959 /F 5007 = 0.33, since both the lines are emitted from the same upper energy level of the O 2+ ion. For one object at z 2.6 we fit the Hα line with a single gaussian function. Since we perform the fit of the spectrum within a spectral window of only 0.2µm around the line, the continuum level is simply set to the mean continuum flux in this band. To automatically exclude noise fluctuations or bad pixels and to take into account the effect of instrumental broadening we constrain the velocity dispersion of the lines to be larger than the instrumental resolution estimated from the sky emission lines. We define the signal-to-noise ratio (S/N) of each line as the peak of the line model divided by the r.m.s. of the spectrum estimated in regions with no emission lines. Finally, we reject all fits where the S/N is lower than 2. Uncertainties on measured fluxes, velocities and velocity dispersions are obtained from the formal errors on best fit parameters. Such errors are computed after two iterations of the line fitting procedure: in the second fit we adopt pixel by pixel errors equal to the r.m.s. of the residuals in the first fit. 4.3 Results In Fig. 4.1 to 4.6 we present the kinematical maps for all the objects described in Tab This paper is dedicated only to the study of galaxy dynamical properties while other studies (e.g. integrated properties, metallicity, etc.) have been or will be presented in

76 72 Figure 4.1: Kinematical maps for objects SSA22A-M38, SSA22A-C16, CDFS-2528 and SSA22A-D17 (from the top respectively). Respectively, from the left: flux, velocity and sigma map. the X Y coordinates are in arcseconds referred to an arbitrary object centre position. The North direction is the positive Y axis. The vertical color bars are in arbitrary units for the flux map and in km s 1 for the velocity and sigma maps. Overplotted on the flux map the continuum flux distribution (brown isophotes) for the object whenever we detect a continuum flux component with sufficient S/N. other dedicated papers (Maiolino et al. 2008a, Mannucci et al. 2009, Mannucci et al. 2010, Cresci et al and Troncoso et al. 2010, in prep.). In the majority of the AMAZE sample (17 out of 23 objects, i.e. 72%, excluding the four lensed objects) the morphology of line flux maps is simple, characterized by single peak distributions with typical dimensions of and relatively low ellipticities of the isophotes. These morphologies are likely the consequence of beam smearing since

77 Dynamical properties of AMAZE and LSD z 3 galaxies 73 Figure 4.2: Kinematic maps for the objects (respectively from the top) CDFa-C9, CDFS-9313 (in the same field of view the fainter source at the North-West is CDFS-9340), CDFS-11991, 3C324-C3 and DFS2237b-D29. Panels, axis and color bar indications same as in Fig. 4.1.

78 74 Figure 4.3: Kinematic maps for the objects (respectively from the top) CDFS-5161, DFS2237b-C21, SSA22a-aug96M16, Q1422-D88 and CDFS Panels, axis and color bar indications same as in Fig. 4.1.

79 Dynamical properties of AMAZE and LSD z 3 galaxies 75 Figure 4.4: Kinematic maps for the objects (respectively from the top) SSA22A-C36, CDFS- 4414/4417(CDFS-4417 is the brighter object at the north), CDFS-12631,CDFS and CDFS Panels, axis and color bar indications same as in Fig. 4.1.

80 76 Figure 4.5: Kinematic maps for the objects (respectively from the top) CDFS-16272, CDFS-16767, SSA22b-C5, SSA22a-C6/M4 (SSA22a-M4 is the fainter northern object) and SSA22a-C30. Panels, axis and color bar indications same as in Fig. 4.1.

81 Dynamical properties of AMAZE and LSD z 3 galaxies 77 Figure 4.6: Kinematic maps for the objects (respectively from the top) Q0302-C131, Q0302-C171, DSF2237b-D28, Q0302-M80 and DSF2237b-MD19. Panels, axis and color bar indications same as in Fig. 4.1.

82 78 the typical seeing in this data is FWHM (see below for a more accurate estimation). However, in some cases we find a secondary clump or asymmetric extended emission (e.g. in SSA22-M38). In the galaxies from the LSD sample we find more complex morphologies which are likely the consequence of the AO-assisted observations providing higher spatial resolution ( 0.2 FWHM) which allows us to spatially resolve more complex structures, but lower sensitivity to extended sources given the higher surface brightness detection threshold. There are three cases of close pairs of objects with similar brightnesses. CDFS-9313 and CDFS-9340 have a projected plane of the sky separation of 1.0 ( 7kpc at the average redshift of the sources); CDFS-9340 is redshifted by 280kms 1 with respect to CDFS-9313, the brightest of the pair. CDFS-4414 and CDFS-4417 have a projected separation of 1.2 ( 9kpc); CDFS-4414 is blueshifted by 120kms 1 with respect to the brighter companion. SSA22a-C6 and SSA22a-M4, from the LSD sample, have a projected separation of 1.6 ( 12kpc) and SSA22a-M4 is redshifted by 90kms 1 with respect to the brighter companion. Inspection of the velocity maps in Fig. 4.1 to 4.6 reveals a number of objects with a clear and regular velocity gradient (12 out of 22 objects in the AMAZE sample and only one out of 9 in the LSD sample). The presence of such velocity gradients is an indication of possible rotating-disk kinematics. The identification of rotating objects and their distinction from complex kinematics cases or mergers is important for the comparison with theoretical models. Some authors (Förster Schreiber et al. 2009, Cresci et al. 2009) used the technique developed by Shapiro et al. (2008) based on kinemetry (Krajnović et al. 2006) to quantify asymmetries in both the velocity and sigma maps in order to empirically differentiate between rotating and non rotating systems. This technique is not useful when applied to the data presented in this paper due to the low S/N that does not allow us to recover with sufficient accuracy the needed kinemetry parameters. We assess the presence of such velocity gradients by using a different approach. We fit the velocity map with a plane (i.e. v = ax + by + v 0 ) and classify the object as rotating or not rotating if the velocity map is well fitted (or not) by a plane. In detail we use the χ 2 statistic to accept or reject the hypothesis that the velocity map is well fitted by a plane: we accept the hypothesis if there is a probability lower than the 4.6% (adopting a χ 2 distribution with the correct number of degree of freedom) to obtain a χ 2 value greater than the value obtained. For objects with velocity maps well fitted by a plane we calculate its inclination relative to the X Y plane (i.e the magnitude of the velocity gradient). We confirm its rotating classification only if this inclination is inconsistent with zero at least a significance level of 4σ. On the other hand a plane inclination consistent with zero might correspond either to genuinely non rotating object or to a rotating one seen face-on. In this case we cannot identify it as a rotating disk. In conclusion we classify the observed galaxies with the following criteria. If the velocity map shows a non zero gradient from plane fitting, we classify the object as rotating. If the velocity map is well fitted by a plane but its inclination is consistent with zero within 4σ, we label the object as not classifiable.

83 Dynamical properties of AMAZE and LSD z 3 galaxies 79 If the velocity map cannot be fitted with a plane, we classify the object as not rotating. We choose to fit a plane to the velocity gradient because, due to the limited S/N, we are sampling preferentially the linear part of the rotation curve of our targets. By using a plane we can adopt a simple model with few free parameters and therefore we can rely on more simple statistics. In principle we could have used also velocity dispersion maps to classify galaxies as rotating or non rotating : in fact, a circularly symmetric dispersion map is the expected signature of spatially unresolved rotation at the center of a rotating disk. However, we decided not to use them principally because we would have missed galaxies with regular rotation patterns in velocity, but whose velocity dispersion maps are distorted by turbulent and/or non-gravitational motions, as verified a posteriori with our modeling. In Fig. 4.7 we show an example of our method. We plot the projection of the velocity map along the direction (s) of maximum inclination of the fitted plane (i.e. the fitted plane seen from its edge). We also show the 1,2,3, and 4σ slopes of the plane inclination. Figure 4.7: Example of the method (applied to object CDFA-C9) for assessing the presence of a smooth gradient in the velocity map. The solid line shows the fitted plane seen from the edge, while the filled points show the observed velocity map on the projection. The dashed lines represent the 1,2,3, and 4σ on the plane inclination. The classification for all the objects is reported in Table 4.2. Note that the object SSA22a-M38 is classified as rotating even if its velocity map is not fitted by a plane (see the χ 2 value in Tab. 4.2). In this case the smooth velocity gradient is very clear and the S/N is high; the fit with a simple plane fails because of the flattening of the velocity field at the red and blue edges. The next step is to model the gas kinematics of rotating galaxies to relate observed kinematics with the mass distribution.

84 80 Table 4.2: Classification of the objects observed. AMAZE Objects χ 2 /d.o. f. Classification SSA22a-M /176 Rotating SSA22a-C16 182/136 Rotating CDFS /52 Rotating SSA22a-D /42 Rotating CDFA-C9 157/142 Rotating CDFS /91 Rotating CDFS /29 Rotating CDFS /99 Not rot. 3C324-C3 45.3/76 Rotating DFS2237b-D29 229/107 Not rot. CDFS /12 Not clas. DFS2237b-C21 725/141 Not rot. SSA22a-aug96M16 680/119 Not rot. Q1422-D88 478/127 Not rot. CDFS /40 Not clas. SSA22a-C36 102/86 Not clas. CDFS /91 Not rot. CDFS /94 Not clas. CDFS /70 Not rot. CDFS /91 Not rot. CDFS /68 Rotating CDFS /82 Not clas. CDFS /46 Rotating LSD Objects SSA22b-C5 216/138 Not rot. SSA22a-C6 101/54 Not rot. SSA22a-M4 21.8/36 Not clas. SSA22a-C /243 Not rot. Q0302-C /95 Rotating Q0302-C /102 Not clas. DSF2237b-D /203 Not rot. Q0302-M80 206/87 Not rot. DSF2237b-MD19 268/95 Not rot. 4.4 Kinematical modeling The model The adopted dynamical model assumes that the ionized gas is circularly rotating in a thin disk. We neglect all hydrodynamical effects, therefore the disk motion is entirely determined by the gravitational potential. In principle turbulent motions, often observed in high z rotating disk galaxies as well as in our own data, could provide significant dynamical support. Indeed Epinat et al.

85 Dynamical properties of AMAZE and LSD z 3 galaxies 81 (2009) apply an asymmetric drift correction assuming that turbulent motions support mass. However, while stellar chaotic motions always support mass because they have gravitational origin, this is not always true for gas motions. The gas velocity dispersion might be strongly increased because of non gravitational motions (like, e.g., in winds driven by starburst activity), and in that case it would not be directly linked to the dynamical mass. Therefore we adopted a conservative approach and neglected the contribution of the gas velocity dispersion. In any case we will, at most, underestimate the dynamical mass by a factor lower than 2 at the disk scale length (Epinat et al. 2009). The galaxy gravitational potential is generated by an exponential disk mass distribution with surface density given by: Σ(r) = Σ 0 e r/r e (4.1) The rotation curve of a thin disk with due to such mass distribution is (Binney & Tremaine 1987): V c (r) 2 = 4πG Σ 0 r e y 2 [I 0 (y)k 0 (y) I 1 (y)k 1 (y)] (4.2) where y = r/2r e and I n, K n are the modified Bessel functions of the first and second kind. By defining the dynamical mass of the galaxy as the total mass enclosed in a 10kpc radius (e.g. Cresci et al. 2009), we can finally write where ( ) 0.5 ( ) V c (r) = 6.56 km s 1 Mdyn r A(y) (4.3) M kpc [ ] 0.5 I 0 (y)k 0 (y) I 1 (y)k 1 (y) A(y) = (4.4) I 0 (y 10 )K 0 (y 10 ) I 1 (y 10 )K 1 (y 10 ) and y 10 corresponds to y computed for r = 10 kpc. The velocity along the line of sight for a given position on the sky is derived from V c (r) by taking into account geometrical projection effects. Another important element of the model is the intrinsic flux distribution (hereafter IFD) of the emission line, since it acts as a weighting function in the calculation of the observed velocities and velocity dispersions when taking into account instrumental effects. Following Cresci et al. (2009) the IFD in the disk plane is modeled with an exponential function. I(r) = I 0 e r/r 0 (4.5) where r is the distance from the disk center and r 0 is the scale radius. Such IFD is then projected onto the plane of the sky. Several objects in our sample have surface brightness distributions clearly different from those of exponential disks. However, given the low S/N and poor spatial resolution of our data, we decided to keep the IFD modeling as simple as possible (see also Cresci et al. 2009). In computing the observed velocity shift and velocity dispersion we take into account the instrumental beam smearing. The spatial PSF is modeled with a two-dimensional gaussian function with full width half maximum FWHM PS F, sampled over the detector x, y pixels. The instrumental spectral response is also described by a Gaussian function with sigma σ inst. Finally, we calculate the model values for the three observed quantities:

86 82 F model (x i, y j ), v model (x i, y j ) and σ model (x i, y j ) where x i, y j represent a spatial pixel in the data cube. For a detailed description of the model refer to appendix B of Marconi et al. (2003). Summarizing, the model parameters are: x c, y c θ i M dyn r e V sys coordinates in the plane of the sky of the disk dynamical centre. position angle (PA) of the disk line of nodes. inclination of the disk. dynamical mass. characteristic radius of the exponential disk. systemic velocity of the galaxy. By using these parameters values we derive V max that is the maximum velocity of the rotation curve. We will use this parameter in Sect to build the Tully-Fisher relation for our data sample Fit strategy Preliminary steps The first preliminary step of the model fitting is to estimate the spectral and spatial resolution of the observations, characterized by the corresponding σ spec and FWHM PS F of the gaussian broadening functions. The spectral resolution (σ spec ) is simply estimated by using the profile of isolated lines in sky spectra. Estimating the spatial resolution (FWHM PS F ) is more complex since we miss datacubes of stars obtained with the same instrumental setting at the same time and at a similar position on the sky as the object datacube. We have therefore devised a method to estimate FWHM PS F directly from the datacubes of the objects showing rotating-disk kinematics. In the assumption that the kinematics is well approximated by a rotating disk we expect an intrinsic line of sight velocity field on the sky plane as shown in Fig. 4.8, which illustrates the iso-velocity contours projected on the sky of a simulated galaxy with mass M (the so called spider diagram ). The gas with line of sight velocity in a given velocity bin lies in the locus delimited by two subsequent iso-velocity contours. We can observe that the spatial extent of such region is always lower along the line of nodes direction (the X-axis) than in the perpendicular direction, and the central velocity bins show the minimal extent. For this reason we expect that an image of the object in the central velocity bins will be spatially unresolved along the line of nodes direction (see the example of Fig. 4.8: the central velocity bin of 30km s 1 has an extent along the line of nodes direction lower than 0.1, well below the typical value of our spatial resolution of 0.6 FWHM). In practice, we select the spatial planes of the object datacube with wavelength close the line centroid, and for each of these spatial planes we fit the line surface brightness with a two dimensional gaussian function. According to what is stated above, we take the minimum FWHM value from these two dimensional gaussians as an estimate of FWHM PS F.

87 Dynamical properties of AMAZE and LSD z 3 galaxies log(mdyn/msun)=10 Scale=7482pc/arcsec i=60deg Y (arcsec) X (arcsec) Figure 4.8: Velocity field on the sky plane ( spider diagram ) for a simulated rotating disk at redshift z 3.3 with mass of M, an exponential mass profile with characteristic radius of 2.2kpc, inclination of 60 and velocities binned in steps of 30km s 1. We could test our method with one object, CDFa-C9, for which there is a star in the SINFONI field of view. The FWHM of this star is consistent within 0.1 with our estimate of the spatial resolution. We also verified that our estimates are in good agreement with the FWHM of the telluric standard stars observed after the science exposures, although with different airmasses. Moreover, in the following, we will test how systematic errors on FWHM PS F can affect the best fit model parameters. The second preliminary step is the determination of the intrinsic flux distribution (IFD) to be used in the computation of the velocity maps. The exponential IFD in the disk plane is projected onto the plane of the sky, convolved by the PSF, averaged over the detector binsize and matched to the observed flux map by minimizing the following χ 2 χ 2 f lux = i, j F(x i, y j ) F model (x i, y j ) F(x i, y j ) 2 (4.6) the free parameters in this fit are the center position of the IDF in the plane of the sky (x c,y c ), the scale radius r 0 and the position angle and inclination of the elliptical isophotes (I 0, the flux scale, is only a normalization factor). From the fit of the IFD we also estimate some parameters needed in the model. As explained in the following section, we set the disk center position equal to the IFD center position (x c,y c ) and the mass distribution scale radius r e equal to the IFD scale radius r 0. The inclination and position angle of the disk are instead determined from the fit of the velocity maps, but we use the estimates from the IFD fit as first guesses.

88 84 Fitting of velocity maps To estimate the free model parameters we minimize the following χ 2 : 2 χ 2 vel = v(x i, y j ) v model (x i, y j ) v(x i, j i, y j ) (4.7) In principle, all model parameters (disk center position, orientation and inclination, systemic velocity and dynamical mass) should be left free to vary. However, the moderate S/N of the data combined with the limited spatial coverage suggest to adopt a different approach in which some of these parameters are constrained a priori. The mass distribution should be traced by the continuum flux distribution, therefore the disk center position should be identified by the continuum peak. Only in some objects we can detect continuum emission with high enough S/N to locate the disk center. In such cases we use that position in the fit allowing a variation of ±0.1 consistent with measurement errors. For all other objects we use the position inferred from the fitting of the emission line flux maps. For those objects where we detect a continuum component the average distance between line and continuum peaks is Therefore, when we do not detect any continuum, we allow for a variation of the disk center position of ±0.25. The scale radius of the mass distribution, r e, cannot be derived from the velocity maps. Following Cresci et al. (2009), we fix r e to the value of r 0, the scaling radius of the continuum flux map. When this is not detected, we adopt the scale radius of the emission line flux map. Estimating the scale radius r e by fitting the IFD with a particular model (i.e. exponential) can of course lead to systematic errors. Therefore we will analyze for every object how systematic errors on r e affect dynamical mass values. Another potential source of systematic errors is the use of the line flux map to estimate disk center position and scale radius when no continuum is detected. Therefore we also evaluated how this choice affects the dynamical mass: for the objects where the continuum is detected we performed the fit of the velocity maps using the disk center position and scale radii estimated either from continuum and line flux maps. We find an average variation of 0.2 dex for the best fit M dyn values. This can be considered an estimate of the systematic error associated to the use of the line flux map instead of the continuum one. Since the disk inclination is coupled with the total dynamical mass, we decide to keep it as a fixed parameter in the fitting procedure to avoid convergency problems with the χ 2 minimization. Its best value and confidence interval is then identified by using a grid of i values and performing the fit for each of them. In this way we construct the χ 2 curve as a function of i which allows us to identify both the best i value and the confidence intervals (for details see Avni 1976). In the cases where the inclination is not constrained by the fit, we set its value to 60, which represents the mean value for a population of uniformly randomly oriented disks. The intrinsic velocity dispersion Our model computes the observed line velocity dispersion by taking into account the unresolved rotation, the intrinsic instrumental broadening (σ spec ) and the broadening due

89 Dynamical properties of AMAZE and LSD z 3 galaxies 85 to the finite size of the spatial pixels and of the spatial PSF. However, the disk can be characterized by an intrinsic velocity dispersion due to turbulent and non-gravitational motions in general. We estimate such intrinsic velocity dispersion of the gas as σ int (x i, y j ) 2 = max[ (σ observed (x i, y j ) 2 σ model (x i, y j ) 2 ) ; 0] (4.8) where σ observed (x i, y j ) is the observed map of velocity dispersion and σ model (x i, y j ) is that computed with the model which best fits the velocity map. Error estimates To estimate the errors on the best fitting model parameters we use the bootstrap method (Efron & Tibshirani 1994). The dataset of each galaxy consists of n data points each characterized by spatial position, flux, velocity and velocity dispersion. We randomly extract from each dataset a subsample of n data points. Due to the random extraction, each subsample will have some data points that are replicated a few times and some data points that are entirely missing. We perform the fit on 100 subsamples and we then estimate errors on parameters by taking the standard deviation of the best fit values which are usually normally distributed Fit Results The fit procedure outlined in the previous section is applied to the subsample of rotating objects in Tab In the following we present and discuss the results of the fit for each of these objects. The observed, model and residual velocity maps are presented in the top panels of Figs The bottom panels of the same figures show the observed and model and intrinsic velocity dispersion maps. The best fit parameters are presented in Tab SSA22a-M38 (Fig. 4.9). In this object, as shown in Fig. 4.1, we detect continuum emission which we use to estimate the position of the disk center. The inclination is well constrained by the observations to i = at the 1σ confidence level indicating that the disk is seen edge-on. This is a clear consequence of the fact that the observed iso-velocity contours do not show any curvature as expected when the rotating disk is seen at intermediate inclinations. The best fitting dynamical mass is then little affected by systematic errors due to uncertainties on the disk inclination. The FWHM of the spatial PSF estimated as described in Sec is 0.5. To evaluate how an incorrect estimate of this parameter can affect the M dyn value we repeated the fit using FWHM in the range The value of M dyn differ in this case by only 0.07 dex. The characteristic radius r e of each object depends on the value of the adopted PSF FWHM (e.g. setting a narrower PSF corresponds to an intrinsically broader flux distribution, with a larger r e ). Therefore we have also evaluated how an incorrect estimate of the scale radius r e affects the M dyn best fit value. We repeated the fit using r e different by ±30% but keeping the PSF width fixed: the value of M dyn changes by only ±0.1 dex.

90 86 Figure 4.9: Fit results for the object SSA22A-M38. Respectively on top panels from the left: data, model and residuals map. On bottom panels from the left: data, model and intrinsic sigma map. The plus sign on the middle panels represents the position of the model dynamical center. The X Y coordinates are in arcsec referred to an arbitrary object center position. The North direction is the positive y axis. The vertical color bars are in km s 1. By adding in quadrature all the systematic errors we obtain the values for the systematic errors on the dynamical masses reported in Tab In the bottom right panel of Fig. 4.9 we present the intrinsic velocity dispersion that is not accounted for by the ordered rotation of the gas. Most of the map is characterized by a very low intrinsic dispersion with average value of σ int 50kms 1. This means that most of the rotating disk is dynamically cold. However, a North-East region is apparently characterized by high turbulence (σ int 250kms 1 ), although it is not clear whether the large velocity dispersion at this location is a consequence of low S/N of the emission lines. There is the possibility that the weak clump observable in the flux map at the south of the principal peak (Fig. 4.1) could be a distinct object. In our analysis we assume that all emission comes from an unique disk, but if the southern clump is really a distinct object we have to exclude its emission from the disk. However our assumption is strengthened by the continuum flux map that do not show any secondary clump, hence the secondary southern clump is likely a star forming region within the disk. SSA22a-C16 (Fig. 4.10). In this object we detect continuum emission that allows us to fix the center of rotation, but the S/N does not allow us to estimate the scaling radius, which is then inferred from the flux distribution of the emission line. The inclination is constrained to be i = at the 1σ confidence level resulting in a systematic error on the mass of dex. The PSF width estimated is FWHM PS F = 0.5. Considering a range of variation of FWHM PS F = , the value of M dyn has a systematic error of 0.15 dex. A variation of 30% of the disk

91 Dynamical properties of AMAZE and LSD z 3 galaxies 87 Figure 4.10: Fit results for the object SSA22A-C16. Respectively on top panels from the left: data, model and residuals map. On bottom panels from the left: data, model and intrinsic sigma map. Axis, symbols and color bars indication same as in Fig scale radius produces a variation of 0.1dex on the best fit M dyn value. By combining all the contributions to the systematic error we obtain a total error of dex. The intrinsic velocity dispersion maps in the bottom right panel of Fig indicates that this object is turbulent, with a quite constant value of σ int 100kms 1. Figure 4.11: Fit results for the object CDFS Panels, axis, symbols and color bars indication same as in Fig. 4.9.

92 88 CDFS-2528 (Fig. 4.11). The continuum emission detected in this object can only be used to constrain the position of the center of rotation. The inclination is only constrained to be i < 70, therefore we can only provide a lower limit on the dynamical mass of log (M dyn /M ) > In this case, when inclination is partially or totally unconstrained, we estimate the mass by adopting the fiducial value of i = 60. The PSF estimated width is FWHM PS F = 0.7. Allowing a variation in the range FWHM PS F = the value of M dyn varies by 0.07 dex. A variation of 30% of the disk scale radius produces a variation on M dyn of 0.1dex. The intrinsic dispersion is quite low on average ( σ int 20kms 1 ) with an excess of 100kms 1 in the East part. Figure 4.12: Fit results for the object SSA22a-D17. Panels, axis, symbols and color bars indication same as in Fig SSA22a-D17 (Fig ). For this object we do not detect any continuum emission. The inclination is constrained to be i > 70 with a best fit value of 85. The systematic error on the dynamical mass due to the inclination is then dex. The PSF width is FWHM PS F = Varying FWHM PS F by ±0.1 we find a negligible systematic error on the dynamical mass. Varying by 30% the disk scale radius produces a variation on M dyn of 0.08dex. As previously noted, since we do not detect any continuum in this object we have to consider an additional systematic error of 0.2 dex due to the use of the line flux distribution. The disk is quite turbulent with an average intrinsic dispersion of σ int 130kms 1 and a southern peak of 250kms 1. CDFa-C9 (Fig. 4.13). For this object we detect the continuum, but we can only constrain the disk center position. The inclination is constrained to be i > 20 with a best fit value of 60. The systematic error on the dynamical mass due to the inclination is then dex. The PSF width is FWHM PS F = 0.5. Varying FWHM PS F

93 Dynamical properties of AMAZE and LSD z 3 galaxies 89 Figure 4.13: Fit results for the object CDFa-C9. Panels, axis, symbols and color bars indication same as in Fig by ±0.1 we find a systematic error on M dyn of 0.22 dex. Varying by 30% the disk scale radius produces a variation on M dyn of only 0.02dex, therefore we can neglect this contribution to the systematic error. The disk is quite turbulent with an average intrinsic dispersion of σ int 120kms 1. CDFS-9313 (Fig. 4.14). (In the same field as CDFS-9340). The continuum is not detected. We set the inclination to i = 60 because it is totally unconstrained. The PSF width is FWHM PS F = 0.7. Varying FWHM PS F by ±0.1 we find a M dyn systematic error of 0.88 dex. Varying by 30% the disk scale radius produces a variation on M dyn of only 0.02dex, so we can neglect this contribution to the systematic error. The intrinsic velocity dispersion is quite constant over the entire map with an average value of σ int 100kms 1. CDFS-9340 (Fig. 4.15). (In the same field as CDFS-9313). The continuum is not detected. The inclination best fit value is i = 70, but it is unconstrained at the 1σ confidence level. The PSF width is the same estimated for CDFS Varying FWHM PS F by ±0.1 we find a M dyn systematic error contribution of 0.1 dex. Varying by 30% the disk scale radius produces a variation on M dyn of 0.1dex. The intrinsic dispersion has an average value of σ int 43kms 1. The small separation of this object from CDFS-9313 either in position ( 1, 0 7.2kpc) and in redshift ( 280kms 1 ) suggests to consider them as an interacting pair. The most massive object is in this case the least luminous, CDFS-9340 (see Tab. 4.3). We estimate a mass ratio for the pair of 10, indicative of a minor merging event with a relatively large separation of the two objects (i.e. 7.2kpc compared to the disk scale radii of 0.7kpc and 1.4kpc), consistently with the non perturbed rotating disk kinematics observed in the two objects.

94 90 Figure 4.14: Fit results for the object CDFS Panels, axis, symbols and color bars indication same as in Fig Figure 4.15: Fit results for the object CDFS Panels, axis, symbols and color bars indication same as in Fig C324-C3 (Fig. 4.16). In this object we do not detect any continuum emission. The inclination is constrained to be 30 < i < 70 with a best fit value of 60 providing a systematic error on the dynamical mass of dex. The PSF width is FWHM PS F = 0.7. Varying FWHM PS F by ±0.1 we find a negligible M dyn systematic error contribution (0.001 dex). Varying by 30% the disk scale radius results in a variation of M dyn by 0.1dex. We note that the observed velocity field although

95 Dynamical properties of AMAZE and LSD z 3 galaxies 91 Figure 4.16: Fit results for the object 3C324-C3. Panels, axis, symbols and color bars indication same as in Fig showing a regular gradient is quite different from what expected in the case of a rotating disk. The intrinsic velocity dispersion is quite constant over the entire map with an average value of σ int 125kms 1. Figure 4.17: Fit results for the object CDFS Panels, axis, symbols and color bars indication same as in Fig CDFS (Fig. 4.17). For this object we do detect a continuum emission and use it to constrain the disk center position and scale radius. The inclination is constrained to

96 92 be 80 < i < 88 with a best fit value of 85. The dynamic mass systematic error due to the inclination is then negligible (±0.04 dex). The PSF width is FWHM PS F = 0.6. Varying FWHM PS F by ±0.1 we find a M dyn systematic error contribution of 0.17 dex. Varying by 30% the disk scale radius varies M dyn by 0.12dex. The intrinsic velocity dispersion rises in the center (σ int 110kms 1 ). The mean value over the entire map is σ int 65kms 1. Figure 4.18: Fit results for the object CDFS Panels, axis, symbols and color bars indication same as in Fig CDFS (Fig. 4.18). In this object we detect the continuum emission and use it to constrain the disk center position and scale radius. The inclination best value is i = 15, but it is unconstrained at the 1σ confidence level. The PSF width is FWHM PS F = Varying FWHM PS F by ±0.1 we find a M dyn systematic error contribution of 0.21 dex. Varying of 30% the disk scale radius produces a negligible variation on M dyn (0.04 dex). The disk is characterized by high turbulence with an average intrinsic dispersion of σ int 100kms 1. Q0302-C131 (Fig. 4.19). This is the only rotating object in the LSD sample. In this object we do not detect any continuum emission. The inclination is constrained to be i < 60, with a best fit value of 30 therefore we can only provide a lower limit on the dynamical mass of log (M dyn /M ) > The PSF width is FWHM PS F = 0.3. Varying FWHM PS F by ±0.1 we find a M dyn systematic error contribution of 0.88 dex. Varying by 30% the disk scale radius results in a negligible variation on M dyn (0.01 dex). The disk has a quite low average intrinsic dispersion ( σ int 43kms 1 ).

97 Dynamical properties of AMAZE and LSD z 3 galaxies 93 Table 4.3: Fit Results. Object FWHMPS F[ ] Model parameter θ [ ] i [ ] (1) [log10(mdyn/m )] re[kpc] (2) Vmax[km s 1 ] SSA22a-M ± ± 0.02 (±0.12) (3) SSA22a-C ± ± 0.02 ( )(3) CDFS ± 6 60 (< 70) ± 0.05 (> 10.65) (4) > 197 (4) SSA22a-D ± 5 80 (> 70) ± 0.06 ( )(3) CDFA-C ± 2 60 (> 20) 9.93 ± 0.03 ( )(3) CDFS ± 4 60 (5) 9.29 ± 0.04 (5) (5) CDFS ± (5) 10.3 ± 0.1 (5) (5) 3C324-C ± ± 0.12 ( )(3) CDFS ± ± 0.04 (±0.21) (3) CDFS ± (5) ± 0.11 (5) (5) Q0302-C ± 5 30 (< 60) 9.99 ± 0.07 (> 9.59) (4) > 117 (4) Notes. (1) Parameter hold fixed, confidence interval estimated as explained in the text. (2) Parameter hold fixed. (3) Systematic error. Obtained combining the systematic error contributes due to the inclination estimate and to the PSF width estimate. (4) Upper-lower limit. (5) Parameter unconstrained: we can not give a systematic error contribute due to this parameter.

98 94 Figure 4.19: Fit results for the object Q0302-C131. Panels, axis, symbols and color bars indication same as in Fig Discussion Rotating galaxies, turbulence and dynamical masses. In the AMAZE sample 40% of the galaxies show a smooth velocity gradient (10 out of 24) consistent with rotating disk kinematics. In the LSD sample only 13% of the galaxies show a smooth velocity gradient (1 out of 9). The lower fraction of rotating objects in the LSD sample is likely due to the lower sensitivity of these observations to the outer, low surface brightness region (which dominate the rotation signal ) because of the finer pixel scale, which makes the detector read-out noise more important relative to the AMAZE observations performed with larger pixel scale. Moreover, the AMAZE sample also tend to have a higher fraction of bright and redder targets (generally translating into higher masses), which may imply a larger fraction of settled massive systems. We note that the fraction of rotating galaxies is fully consistent, within statistical fluctuations, with the fraction of rotating galaxies ( 33%) found in the SINS sample at z (Förster Schreiber et al. 2009). This is particularly interesting, given that the latter sample includes not only rest-frame UV selected galaxies, but also near- and mid-ir selected, and that the fraction of rotating galaxies does not appear to depend on the selection criteria. Taken at face value, the comparison between the SINS analysis at z 2 and the AMAZE analysis at z 3.3 suggests that the fraction of rotating objects does not evolve within this redshift interval. Epinat et al. (2009) derive a fraction of rotating galaxies of 22% in a z sample, but a direct comparison with this value is more difficult considering that their definition of rotating objects is quite different from the one adopted here. In any case, these results should be taken with great caution since several observational effects may easily affect the capability of identifying rotating objects. More specifically,

99 Dynamical properties of AMAZE and LSD z 3 galaxies 95 the lack of angular resolution may prevent the detection of a rotation pattern or, vice versa, may blend two merging systems mimicking a rotation curve. Moreover, the limited sensitivity to low surface brightness disks (which is a strong function of redshift) may prevent us to identify large rotating disks in distant galaxies. In the following we compare the ordered rotation (the velocity gradient v over the map) with the random motions (the mean velocity dispersion). Figure 4.20: Maximum rotational velocity V max versus intrinsic velocity dispersion σ int. Objects from the AMAZE and LSD samples are marked with black and blue symbols, respectively. The σ int value is obtained fitting the observed velocity dispersion map and error bars represent 1 σ uncertainties. Overplotted the V max = σ int locus (dashed line), the V max = k σ int loci (dotted lines), the mean value of V max /σ for the local galaxies (continuous line) and the V max /σ interval for the SINS sample at z 2 (Förster Schreiber et al. 2009) (gray region). By using the results of our dynamical modeling we obtain the maximum rotational velocity (V max ) and the intrinsic gas velocity dispersion σ int (i.e. the turbulent gas motions). In Fig we plot V max versus σ int for our rotating objects. σ int is a constant value added in quadrature to the model velocity dispersion and determined, as in Cresci et al. (2009), by fitting the observed velocity dispersion map. The line V max = σ int discriminate the loci of rotation dominated (V max > σ int ) and dispersion dominated objects (V max < σ int ) and, for comparison, in local disk galaxies V max /σ 10. From Fig it is clear that, with V max /σ int < 2, the majority of the rotating galaxies at z 3 (6 out of 11) is composed of dynamically hot disks, in contrast with local galaxies. Two galaxies have a larger V max /σ int ratio but still smaller then local galaxies, while three galaxies are dynamically cold. The finding that high-z disk galaxies are much more turbulent than local galaxies was already obtained by other studies at z 1-2 (e.g. Genzel et al. 2006, Förster Schreiber et al. 2006b) and suggestive of high gas fractions making disks gravitationally unstable (Tacconi et al. 2010, Daddi et al. 2010a). Förster Schreiber et al. (2009) obtain with the SINS sample V max /σ 1 7 with a mean value of 4.5. Our result at z 3.3

100 96 Table 4.4: Stellar masses and Star formation rates. Object [log 10 (M /M )] SFR [M /y] SSA22a-M SSA22a-C CDFS SSA22a-D CDFA-C CDFS CDFS C324-C CDFS CDFS Q0302-C ( V max /σ int z=3.3 2) goes in the same direction and is even more extreme considering that the majority of our galaxies have V max /σ int comparable with the smallest values observed in the SINS survey. We do find however a few galaxies which are dynamically cold even when compared to local galaxies.the missing link between high redshift and local galaxies seems to be provided by Epinat et al. (2009) who obtain V max /σ int 3.5 (at z ) when considering the objects classified as rotating disks and perturbed rotators. This number rises to 7.2 when considering only the rotating disks. In Fig we compare dynamical and stellar masses for the rotating objects. We estimate dynamical masses in the range M M with a mean value of M. The stellar masses are reported in Table 4.4 and are estimated from standard broad-band SED fitting (see Sect. 6 of Maiolino et al. (2008a) and Sect. 4.3 of Mannucci et al. (2009) for more details). All of the objects are consistent with having stellar masses lower than dynamical masses within the 1σ uncertainties. By ascribing the difference between dynamical and stellar mass to the gas mass, the loci of constant gas fraction are given by the dashed lines. Here we are assuming negligible contribution of Dark Matter, justified by the fact that the spatial region sampled in our dynamical study is of the order of only 2-3 characteristic radii of the exponential disk, where the contribution of dark matter is still small. In any case, if this assumption is not appropriate, we can call this quantity a gas and dark matter fraction. Three objects require at high confidence a gas fraction of 90% or higher. One object (SSA22a-C16) has little room for any gas mass. Most of the other galaxies have the gas fraction poorly constrained or totally unconstrained, except for SSA22a-M38 (with a gas fraction constrained within 30% and 80%) and CDFS (with a gas fraction constrained to <40%). However, due to the large uncertainties, it is not possible to draw firm conclusions on the observed wide range of gas fractions, and more data are needed to set tighter constraints on this property. It is interesting to compare the amount of gas inferred from the comparison between dynamical and stellar mass with the amount of gas that can be inferred from the Schmidt-

101 Dynamical properties of AMAZE and LSD z 3 galaxies 97 Figure 4.21: Comparison of the dynamical and stellar mass for the objects analyzed in this paper. The masked gray region represent the non physical condition of M > M dyn. The brown dashed lines represents the loci of constant gas fraction (90%, 60% and 30% gas fraction, from left to right, respectively). Kennicut law. More specifically, if we assume the validity of the Schmidt-Kennicut law (hereafter SK) for our z 3 galaxies, we can calculate the gas surface density from the star formation rate surface density, and therefore derive the gas mass. In particular we use the relation presented in Mannucci et al. (2009) (eqs. 1 and 2) that adopt for the slope of the SK relation the value 1.4 given by Kennicutt (1998). To infer the SFR surface density we use the characteristic radius of the mass distribution r e estimated in our dynamical modeling as the radius of the galaxy. Then, under the assumption of a negligible contribution in mass of the dark matter, and by adding the stellar mass to the gas mass we obtain the baryonic mass of the galaxy (M bar ). In this way we obtain for our objects an estimate of the mass by using only the results of the SED fitting. In Fig we show the comparison of the M dyn estimated from the gas dynamics modeling presented in this paper and the M bar estimated from the SK law, combined with the stellar mass. The two estimates of the mass, considering the error bars, are generally quite consistent. The mean difference (in module) (i.e. the mean scatter of the points around the M dyn = M bar condition) is of 0.5 dex and there is no particular bias of one of the two values on respect of the other. The mean residual of the points on respect of the M dyn = M bar condition (i.e. the difference in module divided by the errors) is 1.2. We can conclude that the results of the SED fitting give a reliable estimate of the dynamical mass. This also implies that the two assumptions of the validity of the SK law and of a negligible contribution in mass of the dark matter appear to be valid. Daddi et al. (2010b) using CO observations of galaxy samples at low and high redshifts (z 0.5 and z 1.5) find a 1.42 slope for the SK relation, very similar to that by Kennicutt (1998). Genzel et al. (2010) using systematic data sets of CO molecular line emission in z 1 3 normal star-forming galaxies find instead an index of In

102 98 the present work we have used the Kennicutt (1998) index for comparison with other data in the literature, but using the smaller Genzel et al. (2010) index we would have obtained, on average, gas masses larger by only 0.3 dex. Figure 4.22: Comparison of the dynamical mass M dyn obtained from the gas dynamical modeling and the baryonic mass M bar obtained by combining the stellar mass and the gas mass inferred from the SK law. The dashed line represents the condition of equality of the two masses. The dot-dashed lines represents the mean scatter of the points around the M dyn = M bar condition ( 0.5 dex) The Tully-Fisher relation at z 3 The Tully-Fisher relation (hereafter TFR) is the relation between luminosity (or stellar mass) and maximum rotational velocity of disk galaxies (Tully & Fisher 1977). This relation, originally used as a distance indicator for disk galaxies, represents an important feature for understanding galaxy formation and evolution because it links directly the angular momentum of the dark matter halo with the stellar mass content of disk galaxies. In fact, according to theoretical models, galaxy disks form out of gas cooling down from a surrounding dark matter halo, maintaining its specific angular momentum and settling in a rotationally supported disk. Therefore, the structure and dynamics of disk galaxies at different cosmic epochs are expected to be closely correlated with the properties of the dark matter halo in which they are embedded. In recent years, an increasing number of dynamical studies of disk galaxies in the local and intermediate-redshift Universe allowed the investigation of the evolution of the TFR with redshift. The evolution of the TFR is expected to be related both to the conversion of gas into stars and to the inside-out growth of the dark matter halo by accretion. In fact, while the extent of the halo grows considerably with time, the circular velocity of the halo grows

103 Dynamical properties of AMAZE and LSD z 3 galaxies 99 less, keeping the rotation curve approximately flat to larger and larger distances. The accretion of the dark matter is followed by accretion of baryonic gas, which is subsequently converted into stars by ordinary star formation in the disk. However, the details of the process and the amount of evolution expected depends strongly on the model assumptions, on the accretion mechanism and the timescale needed to convert the gas into stars. Any successful model of disk formation should then be able to reproduce the slope, zero-point, scatter, and redshift evolution of the TFR. There are still discrepant results on a possible evolution at intermediate redshifts of the tight relation observed in local galaxies (e.g., Haynes et al. 1999; Pizagno et al. 2007). For example Vogt et al. (1996) reported very little evolution of the B-band TFR up to z 1, but other groups (e.g. Barden et al and Nakamura et al. 2006) found a strong brightening of 1 2 mag in B-band luminosity over the same redshift range. Furthermore, the interpretation of a possible evolution of the luminosity TFR is difficult since the luminosity and the angular momentum might be evolving together. As a consequence, the stellar mass TFR, which correlates the stellar mass and the maximum rotational velocities of disks, offers a physically more robust comparison as it involves more fundamental quantities. In this context, Flores et al. (2006), Kassin et al. (2007), and Conselice et al. (2005) have found no evolution in both the slope and zero point of the stellar mass TFR up to z = 1.2. In contrast Puech et al. (2008) detect an evolution of the TFR zero point of 0.36 dex between z 0.6 and z = 0, by using a sample of 18 disk like galaxies observed with the integral field spectrograph GIRAFFE at the VLT. At z 2 Cresci et al. (2009) build a TFR at z 2 based on the SINS sample and find an evolution of the TFR zero point of 0.4 dex relative to the local TFR. By using the dynamics of z 3 galaxies inferred from our data, we are now able to derive the TFR at z 3. In Fig we present the z 3 TFR obtained by using the data presented in this paper (obviously only for the rotating objects). The maximum rotational velocities have been computed from the best fit models as reported in Tab The stellar masses are those reported in Table 4.4. All our data points are consistent with having a stellar-to-dynamical mass ratio smaller than the local value. We can actually observe an evolution of the TFR with the redshift and this evolution is in the direction predicted by the models: the dynamical mass is already in place at this cosmic epoch, but the stellar mass has yet to be formed. The large scatter ( 1.5 dex) of the TFR at z 3 suggests that at this redshift the relation is not yet in place, probably due to the young age of the galaxies at this epoch (between 50Myr and 500Myr). However we can not exclude that our V max measurements are affected by an incorrect assumption of a rotating disk kinematics for some objects of our samples, which introduces a large scatter in the relation. Assuming that the TFR is in place, despite the large data scatter, and that the slope is the same of the local universe, we can fit the relation obtaining a zero point for our data of 0.97 ± This implies an evolution of the zero point of 1.29 dex relative to the local TFR and 0.88 dex relative to the z 2 TFR. Given the high intrinsic velocity dispersion observed in our z 3 sample, we also explored the S 0.5 = 0.5V 2 + σ 2 estimator introduced by Kassin et al. (2007) to study

104 100 Figure 4.23: The Tully-Fisher relation at z 3 reconstructed by using the dynamical and stellar masses of the rotating galaxies found in our sample. The solid black line represents the local relation, the magenta line represents the z 2 relation (Cresci et al. 2009) and blue solid line represents the fit to our data (z 3) by keeping the same slope as for the local and z 2 cases. The dotted blue lines represents the formal 1σ uncertainty on the zero point of our fitted relation. Figure 4.24: The S 0.5 Tully-Fisher relation at z 3. The solid black line represents the z 1 relation (Kassin et al. 2007), the dashed line represents the fit to our data (z 3) by keeping the same slope as for the z 1 relation.

105 Dynamical properties of AMAZE and LSD z 3 galaxies 101 the impact of turbulent motions. In Fig we present the S 0.5 TFR obtained by using our V max and σ int values. Kassin et al. (2007) found a tight relation ( 0.3 dex scatter in M ) in a sample of galaxies with redshift up to z = 1.2, with no significant evolution. By only allowing for a change in zero point of the z 1 relation we estimate a mean scatter of 0.5 dex in our galaxies at z 3.3. The reduction of the scatter in the S 0.5 TFR compared to the classical one ( 1.5 dex) suggests that, at least for some objects, part of the mass might be supported by turbulent or not ordered motions. We observe a significant evolution of the zero point of the relation between z 1 and z 3 ( 0.8 ± 0.1 dex in M ), providing further support for the observed evolution on the classical TFR. The use of SINFONI integral field spectra represents a major step forward in the modeling of the dynamics of high-z galaxies, as the two-dimensional mapping is more complete and not biased by a priori assumptions about the kinematic major axis and inclination of the system as in long-slit spectroscopy. Also SINFONI data can allow a better selection of the rotating systems and ensure a much lower contamination from complex dynamics and irregular motions than in longslit data sets. This might be a reason for the apparent inconsistency of the results of kinematical studies at redshift greater than Conclusions We conducted an extensive investigation of the dynamics of z 3 galaxies using near infrared integral field spectra obtained with SINFONI at the ESO VLT. This is the first time that such dynamical analysis is conducted for a relatively large sample (33 galaxies) at z 3. We estimated galaxy masses by fitting rotating disk models to selected galaxies. Due to the relatively low S/N of the data, we implemented a simple but quantitative method to identify smooth velocity gradients, which are the signature of possible disk rotation. After this pre-selection, we performed complete dynamical modeling using a rotating disk model with an exponential mass profile. We found that the kinematics of 30% of the objects (11 out of 33) is consistent with that of a rotating disk. This fraction is fully consistent with that found with z 2 samples, like SINS, suggesting that the fraction of rotating objects does not evolve between z 2 and 3. When considering only the galaxies observed with AO and finer pixel scale (LSD sample), only 1 out of 9 shows a smooth velocity gradient. This is likely the consequence of such observational setup providing higher spatial resolution but lower sensitivity to the outer, low surface brightness regions which dominate the rotation signal. The 11 rotating objects have dynamical masses in the range M M with a mean value of M. By comparing the amplitude of the inclination-corrected velocity gradient with the intrinsic gas velocity dispersion derived from models, we found that the majority of the rotating galaxies at z 3 are dynamically hot disks with an average

106 102 v depro j /σ int 0.9. This confirms the results obtained by other studies at z 1 2 ( v/σ 2 4), indicating that high-z disk galaxies are more turbulent than local galaxies and even more hot than galaxies at z 2. We compared stellar and dynamical masses for the rotating objects, and derived an estimate of the gas and dark matter fraction. Using independent estimates for stellar mass and star formation rate, and applying the Schmidt-Kennicut law, we obtained a SED-based estimate of the total baryonic mass of our objects. This estimate is consistent with our estimate of the mass from gas dynamics (rms scatter lower than 0.5dex). This shows that SED fitting gives a reliable estimate of the dynamical mass and also implies that our assumptions on the validity of the SK law and of a negligible contribution of dark matter are well justified. Finally, we obtained the Tully-Fisher relation at z 3. All our data points are consistent with a stellar-to-dynamical mass ratio smaller than the value in the local universe confirming the redshift evolution of the relation already found at z 2. This is consistent with models predictions, according to which stellar mass is still building up compared to the total dynamical mass. However, the large scatter of the points may also suggest that, at this redshift, the relation is not yet in place, probably due to the young age of the galaxies. The Tully-Fisher relation based on the S 0.5 indicator has a significantly smaller scatter that the classical one and also shows an evolution in zero point compared to z 1. The reduction in scatter suggests that, at least for some objects, part of the mass might be supported by turbulent motions.

107 Chapter 5 A dynamical mass estimator for high z galaxies based on spectroastrometry 5.1 Introduction In this chapter we use the results obtained in the previous chapters and apply the spectroastrometric technique to the study of gas dynamics in high redshift galaxies. In Chapters 2 and 3 we developed and discussed the application of spectroastrometry to the study of the gas dynamics in the nuclear region of nearby galaxies with the aim of measuring the nuclear mass concentration (i.e. the BH mass). We developed a new method based on spectroastrometry to estimate masses from the kinematics of rotating gas disks. Our aim is to apply this method, or a variation of it, to the kinematics of rotating gas disks in high z galaxies. In Chapter 4, we studied a sample of high redshift galaxies and measured their dynamical masses using standard techniques for gas dynamical studies. In this chapter we obtain dynamical mass estimates for high redshift galaxies based on a spectroastrometric approach. In Chapter 4 we have measured of dynamical masses in a sizable sample of galaxies at z 3 (see also Gnerucci et al. 2010, Cresci et al. 2010) from the AMAZE and LSD samples. One of the lessons learned from our own work and from the literature is that data for objects at such high redshifts often suffer of a poor signal to noise ratio (hereafter S/N) which does not allow spatially resolved kinematical studies and complete dynamical modeling. For this reason one can only estimate the dynamical mass of a galaxy by applying the virial theorem to its integrated spectrum. Virial mass estimates are affected by large systematic errors. Apart those due to the unverified assumption that the system is virialized, one of the principal problems is the estimate of the size of the galaxy, that often suffers from the low intrinsic spatial resolution. Therefore, in this chapter, we present an alternative to the classical virial mass estimate based on the technique of spectroastrometry with the aim to obtain a more accurate dynamical mass estimator. In Sect. 5.2 we briefly introduce the data we use in this chapter and their previous analysis presented in Chapter 4. In Sect. 5.3 we briefly summarize the main features of the spectroastrometry technique presented in Chapters 2 and 3. In Sect. 5.4 we introduce the classical virial mass estimator and calibrate it by comparing with the more

108 104 accurate dynamical mass estimates. In Sect. 5.5 present our spectroastrometric dynamic mass estimator, we use simulations to understand how our estimator is affected by various dynamical or instrumental parameters (Sect ) and in Sect we calibrate it by comparing with the more accurate dynamic mass measurements. In Sect. 5.6 we apply our method to galaxies of the AMAZE and LSD samples and discuss the results. Finally, in Sect. 5.7 we draw our conclusion. 5.2 Data and previous dynamical modeling We consider the sample of z 3 galaxies from the AMAZE and LSD samples, described in the previous chapter. A general description of the AMAZE and LSD programs, of their sample selection, observations and data reduction has already been presented but for a more detailed and complete description refer to Maiolino et al. (2008a) and Mannucci et al. (2009). In Chapter 4 we performed a complete dynamical modeling of the two dimensional kinematical maps derived from the integral field spectra (SINFONI at the ESO VLT) of the AMAZE/LSD galaxies. We selected a subsample of galaxies showing a velocity field on the sky plane consistent with a rotating disk. Then, for these rotating objects, we modeled the observed velocity field with a thin gas disk rotating under the gravitational potential of the galaxy. In such way we estimated the dynamical mass of the galaxy. In this chapter we will use the best fit dynamical masses obtained for the various objects analyzed in Chapter 4 as reference values to be compared with our estimator prediction. In Table 5.1 (last column) we report that dynamical masses. 5.3 Measuring masses with spectroastrometry In Chapters 2 and 3 we presented the application of the spectroastrometric technique to the study of the dynamics of rotating gas disks in the nuclear region of local galaxies. We observed that the spectroastrometrical approach is orthogonal and complementary to the standard rotation curve approach and therefore it can provide informations on the observed spectrum that the rotation curve alone can not provide. We also predicted with our tests and simulations in Chapter 2 and verified in the application to real data in Chapter 3 that the fundamental advantage of the spectroastrometric method is that it can provide position measurements on scales smaller than the spatial resolution of the observations (i.e. we are able to probe the gas kinematic down to disk radii of 1/10 of the spatial resolution), thus overcoming the principal limit of rotation curve method. We also provided a simple method to estimate BH masses that requires only the spectroastrometric map of the source, that is the positions on the sky plane of emission line photocenters in the available velocity bins. This method gives results in perfect agreement with the classical method based on the rotation curves. In the following we make use of these results to apply the spectroastrometric technique to the study of the gas dynamics on high redshift galaxies. In particular we use the spectroastrometry to improve the classical virial mass estimates working on IFU data (SINFONI integral filed spectra).

109 A dynamical mass estimator for high z galaxies based on spectroastrometry 105 The principal difference with the application to local BHs is that we deal with rotating gas disk dynamics driven by extended mass distributions instead of point-like ones. But in both cases spatial resolution is an issue, since these rotating disks are often barely resolved. 5.4 Virial mass estimator The main and more reliable technique to estimate the dynamical mass of a high redshift galaxy is the modeling of spatially resolved gas kinematics. This technique requires that an object is well spatially resolved and has a high S/N to constrain all the model parameters with sufficient accuracy. Unfortunately these two requirements are not often met for high redshift galaxies, that are in many cases only marginally spatially resolved and with poor S/N. In this cases a method to estimate the dynamical mass of the object is to use the integrated spectrum of the object and apply the virial theorem. Applying the virial theorem, the dynamical mass enclosed in a sphere of radius r c is given by the simple following equation. M(r c ) = f V2 r c (5.1) G where V is the velocity, provided, e.g., by FWHM or velocity dispersions of the line profile and f is a factor that take into account the geometry of the mass distribution. The principal problem of the virial mass estimate lies in the estimate of the characteristic dimension r c. First one needs a continuum image of the source with large enough S/N to estimate, e.g., the half light radius. This size has then to be corrected to take into account the finite spatial resolution of the observations; indeed one is often dealing with objects which are marginally spatially resolved and the estimate of r c can lead to high uncertainties and systematic errors. Second, one needs to estimate the velocity dispersion of the gas from the integrated spectrum. By combining gas velocity dispersion with continuum sizes one has to assume that gas emission is co-spatial with continuum emission and, in general, this assumption cannot be straightforwardly verified. Using the data of the AMAZE and LSD projects we can calculate the classical virial mass for several galaxies (using eq. 5.1 ) and compare it with the more accurate estimates from dynamical modeling of spatially resolved gas kinematics presented in Chapter 4 (see Sect. 5.2). In particular we estimate the characteristic radius r c from the gaussian HWHM of the line surface brightness (not having detected any continuum emission for most objects) and correct for the PSF by using the HWHM of the seeing disk (Bouché et al. 2007). In the left panel of Fig. 5.4 we compare virial products (Vir = FWHM 2 r c /G) with masses resulting from full dynamical modeling (M dyn ) in Chapter 4. The numbers plotted in the figure are also reported in Table 5.1. We also extend this comparison to a subsample of 8 rotating disks from the SINS survey at z 2 (Förster Schreiber et al. 2006a, Förster Schreiber et al. 2009, Genzel et al. 2008, Cresci et al. 2009) selected for their high S/N, rotating disk gas kinematic and robust dynamical modeling. For the comparison we do not use directly the dynamical mass value M dyn, but the M dyn sin 2 i value, where i is the inclination of the galaxy disk. Due to the coupling of mass

110 106 and inclination, M dyn sin 2 i is the quantity which is less affected by systematic errors due to the uncertain disk inclination which is difficult to constrain especially with low S/N and barely resolved sources (see Chapter 4). Moreover, the uncertain disk inclination also affects the virial mass estimate because the FWHM is determined by the line of sight velocity dispersion. We note that the virial product Vir and M dyn sin 2 i have a non-linear relation with large scatter: this is not what one would have liked for a reliable prediction of M dyn sin 2 i from Vir. By imposing the expected linear relation log 10 (Vir) = log 10 (M dyn sin 2 i) log 10 f, the mean scatter is 0.3 dex. By letting the slope free to vary, one obtains log 10 (Vir) = (0.66 ± 0.05)log 10 (M dyn sin 2 i) log 10 f and the mean scatter reduces to 0.21 dex. The classical virial mass estimate can be biased by systematic errors principally originating from dimension estimates and therefore it is not a robust proxy for the dynamical mass. 5.5 Spectroastrometric mass estimator The spectroastrometric mass estimator presented in this chapter is based on measuring the characteristic dimension of the object by means of the spectroastrometric technique. As for the classical virial mass estimates we assume that the object is a rotating disk. Following this assumption, we expect that the redshifted gas is located principally in a side of the disk, whereas the blueshifted gas is located in the other side. Therefore, if we ideally obtain images of the object in the red and blue sides, these two images have to be spatially shifted due to the rotation. We estimate the characteristic radius of the object by measuring this shift between the red and blue sides with the spectroastrometry. The first step is to obtain a spatially integrated spectrum of the source.then we fit a simple gaussian function to the emission line in the integrated spectrum and we estimate from this the line velocity dispersion (σ). Using the fitted line model we select the red and the blue side of the emission line. The red side is identified by the wavelength range bordered by the central wavelength of the line and the most extreme red wavelength for which all this conditions are satisfied: (i) the line model is greater or equal to the 5% of the maximum amplitude, (ii) the data spectrum is greater or equal than the r.m.s. of the fit residuals and (iii) such wavelength is less than 3σ from the central wavelength. We similarly define the wavelength range identifying the blue side of the line (see Fig. 5.1). We then collapse all the velocity planes in the red and blue side of the line to obtain the red and blue images of the object, respectively. The spatial plane related to the central wavelength bin is added to the red and blue images but with a weight given by the fraction of the bin laying on the red and blue sides (see Fig. 5.1). The next step is to fit such images with a two-dimensional Gaussian function and determine the position of the blue and red light centroids (see Fig. 5.3). We can now define the spectroastrometric characteristic radius (r spec ) as the half of the distance between the two centroids. Note that the orientation of the direction connecting the two centroids, in the assumption of a rotating disk kinematics, identifies the disk line of nodes. Figures 5.1 to 5.3 shows as an example the application of the method to the object SSA22a- C16 of the AMAZE sample. We can understand the physical meaning of the spectroastrometric characteristic ra-

111 A dynamical mass estimator for high z galaxies based on spectroastrometry 107 Figure 5.1: Example of the integrated spectrum for an object (SSA22a-C16). Overplotted is the fitted model of the line (red continuous line), the central wavelength (red dashed vertical line), the r.m.s. of residuals (two dotted horizontal lines) and the selected bins for the blue and red side (respectively blue and red filled bins) and the central bin (brown filled bin). Figure 5.2: Example of the blue and red image for the same object of Fig White isophotes and black dashed lines represent the best model and axis orientation for the two dimensional gaussian function fitted to the images. The blue and red filled circles represent the position of the centroid of respectively the other image. dius (r spec ) in the assumption of rotating disk kinematics. We obtain the blue and red images by binning the spectral dimension of the data cube, and then measuring the light centroid position for each of the two resulting bins. Therefore we can consider the blue