Studying the magnetic fields of cool stars

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 2014 Studying the magnetic fields of cool stars Christene Rene Lynch University of Iowa Copyright 2014 Christene Rene Lynch This dissertation is available at Iowa Research Online: Recommended Citation Lynch, Christene Rene. "Studying the magnetic fields of cool stars." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 STUDYING THE MAGNETIC FIELDS OF COOL STARS by Christene Rene Lynch A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa August 2014 Thesis Supervisor: Professor Robert L. Mutel

3 Copyright by CHRISTENE RENE LYNCH 2014 All Rights Reserved

4 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Christene Rene Lynch has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the August 2014 graduation. Thesis Committee: Robert Mutel, Thesis Supervisor Kenneth Gayley Steven Spangler Cornelia Lang Tom Ray

5 To Q, who waited for me at mile 24 with a sign reading THE END IS NEAR! ii

6 Not just beautiful, though the stars are like the trees in the forest, alive and breathing. Haruki Maurakami, Kaf ka on the Shore iii

7 ACKNOWLEDGEMENTS First I would like to thank my thesis advisor, Robert Mutel, for his invaluable guidance throughout my graduate career. He has not only shown me how to be a better scientist and ask interesting questions but also how to be a better communicator and share my ideas with others. I would also like to thank the rest of my thesis committee for their support and comments throughout the progression of my thesis work and specifically Cornelia Lang for all her career advice. Thanks must also be given to my group of Iowa friends, who accepted me in all my awkwardness and always found a way to cheer me up when I most needed it. Finally I want to thank the Lynch Family, for all their love and for teaching me that anything can be achieved as long as you are willing to work hard. iv

8 ABSTRACT Magnetic fields are prevalent in a wide variety of low mass stellar systems and play an important role in their evolution. Yet the process through which these fields are generated is not well understood. To understand how such systems can generate strong field structures characterization of these fields is required. Radio emission traces the fields directly and the properties of this emission can be modeled leading to constraints on the field geometry and magnetic parameters. The new Karl Jansky Very Large Array (VLA) provides highly sensitive radio observations. My thesis involves combining VLA observations with the development of magnetospheric emission models in order to characterize the magnetic fields in two fully convective cool star systems: (1) Young Stellar Objects (YSOs); (2) Ultracool dwarf stars. I conducted multi epoch observations of DG Tau, a YSO with a highly active, collimated outflow. The radio emission observed from this source was found to be optically thick thermal emission with no indication of the magnetic activity observed in X-rays. I determined that the outflow is highly collimated very close to the central source, in agreement with jet launching models. Additionally, I constrained the mass loss of the ionized component of the jet and found that close to the central source the majority of mass is lost through this component. Using lower angular resolution observations, I detected shock formations in the extended jet of DG Tau and modeled their evolution with time. Taking full advantage of the upgraded bandwidth on the VLA, I made widev

9 band observations of two UCDs, TVLM and 2M Combining these observations with previously published and archival VLA observations I was able to fully characterize the spectral and temporal properties of the radio emission. I found that the emission is dominated by a mildly polarized, non-thermal quiescent component with periodic strongly polarized flare emission. The spectral energy distribution and polarization of the quiescent emission is well modeled using gyrosynchrotron emission with a mean field B 100 G, mildly relativistic power-law electrons with a density n e cm 3, and source size of R 2R. We were able to model the pulsed emission by coherent electron cyclotron radiation from a small number of isolated loops of high magnetic field (2-3 kg) with scale heights R. The loops are well-separated in magnetic longitude, and are not part of a single dipolar magnetosphere. The overall magnetic configuration of both stars appears to confirm recent suggestions that radio over-luminous UCD s have weak field non-axisymmetric topologies (Cook et al. 2013; Williams et al. 2013b), but with isolated regions of high magnetic field. vi

10 TABLE OF CONTENTS LIST OF TABLES x LIST OF FIGURES xi CHAPTER 1 STELLAR MAGNETIC FIELDS Generation of Magnetic Fields in Low Mass Stars Characterizing Magnetic Activity in Low Mass Star Systems Magnetic Activity in Main Sequence Stars Zeeman Techniques Tracers of Chromospheric and Coronal Activity Magnetic Activity in Young Stellar Objects Accretion Outflows YSO Large Scale Fields Summary and Central Science Question RADIATIVE PROCESSES Radiation at Radio Frequencies Radiative Transfer Equation Bremsstrahlung Gyromagnetic Radiation Gyrosynchrotron Emission from Thermal Plasma Gyrosynchrotron Emission from Power-Law Electrons Gyrosynchrotron Emission Model TECHNICAL INFORMATION The Response of an Interferometer Aperture Synthesis Radio Data Reduction vii

11 3.2.1 Data Editing Calibration Formalism Coordinate Systems for Imaging Imaging and Deconvolution RADIO OBSERVATIONS OF CLASSICAL T TAURI THERMAL JETS Preface Abstract Introduction VLA Observations Results Imaging the DG Tau Jet Spectral energy distribution Discussion Comparison between radio and optically-derived electron densities Mass-loss rate of the ionized component Radio Knots in the Extended Jet Knot proper motion Knot evolution: Expanding post-shock gas? Summary WIDEBAND DYNAMIC RADIO SPECTRA OF TWO ULTRA-COOL DWARFS Preface Abstract Introduction Target Star Properties Observations and Data Analysis RESULTS Dynamic spectra Pulse Periods M J pulse period TVLM pulse period Quiescent emission Spectral indices Circular polarization Brightness temperature Coronal Models Pulsed Emission: ECM sources on isolated loops ECM Source locations viii

12 Rotation axis inclination angles Beaming angles Quiescent Emission: Gyrosynchrotron Model Conclusions CONCLUSIONS AND FUTURE WORK YSO Further Investigation of Radio Jet Structure UCD Expanding the Frequency Range of UCD Dynamic Spectra Gyrosynchrotron Models Adding Thermal Gyrosynchrotron Mechanism Development of GS-Model Looking for Thermal GS APPENDIX A PYTHON CODE FOR CREATING LIGHT CURVES IN CASA B PYTHON CODE FOR MODELING THE RADIO EMISSION FROM STELLAR CORONA B.1 EMISSION-LIBRARY.py B.2 PwrLawGS-WAngle-Approx.py REFERENCES ix

13 LIST OF TABLES Table 4.1 VLA Results UCD Observing Epochs Period Analysis Results UCD Spectral Indices ECM pulsed emission model parameters UCD Quiescent Emission Model Parameters x

14 LIST OF FIGURES Figure 1.1 The properties of the large-scale magnetic field topologies of cool stars as a function of rotation period and mass Magnetic activity traced in Hα, X-ray, and radio emission for M and L dwarfs Radio, X-ray, and Hα activity as a function of projected rotation velocity The Güdel-Benz relationship between L x ( kev) and L ν,r The distance from a linear fit to the Güdel-Benz relationship of rotation rate and spectral type Three examples of radio light curves from UCDs Illustration of the stages of low-mass young stellar evolution Sketch of the different infall and outflow patterns in CTTs D profiles of log(n e ), x e, and log(n H ) along the flow of DG Tau in discrete velocity intervals Radial velocity profiles across the jet of DG Tau in near-uv and optical emission lines Measurements of the strength and topology of T Tauri fields using Zeeman techniques overlaid on an H-R diagram for objects with masses M Geometry for radiative transfer equation Schematic spectra of brightness temperature and flux density The spectral energy distribution for thermal and power-law gyrosynchrotron emission calculated from the numerical and the simplified power-law expressions Simplified schematic diagram of a two-element interferometer xi

15 3.2 Position vectors used to derive the interferometer response due to a source Example display of uncalibrated and unedited data using the plotms GUI application in CASA Position vectors used to derive the interferometer response due to a source The dirty beam is the Fourier transform of the sampling function of the interferometer An example of an interactive CLEAN of 2012 X-band observation of DG Tau The two-absorber X-ray spectrum of DG Tau Constraining mass loss and opening angle using Reynolds (1986) VLA 8 GHz naturally-weighted contour map of DG Tau s inner jet at epoch VLA contour maps of DG Tau during the two spring 2012 observations Flux density of DG Tau as a function of frequency for observations at epochs Analytic functions used to model the electron density and velocity profiles across DG Tau s jet at 0.35 from the central source DG Tau contour maps at 8 GHz and 5 GHz at epochs and respectively Projected knot location along jet vs. epoch with observed centroid locations at epochs and Evolution of the observed radio knots in the extended jet of DG Tau Plasma parameters for the magnetosphere of a typical low mass star, including Faraday Rotation, the magneto-ionic parameters, the Razine ratio, and propagation of polarization vector Composite dynamic spectra of 2M J Composite dynamic Spectra of TVLM xii

16 5.4 Normalized chi-square of the summed difference between the published main pulse period and trial period for 2M J Reduced chi-square of weighted phase differences as a function of period relative to the published period of Doyle et al. (2010) for TVLM Comparison between model and measured values for the total flux and polarization of the observed quiescent emission Comparison between observed and modeled dynamic spectra for 2M J Comparison between observed and modeled dynamic spectra for TVLM Comparison between observed and modeled dynamic spectra for sloped feature of 2M J Perspective views of coronal loop models Light curves for quiescent radio component of TVLM513-46and 2M J Spectral Energy Distribution for a Stellar Corona; includes both Powerlaw and Thermal Gyrosynchrotron Emission xiii

17 1 CHAPTER 1 STELLAR MAGNETIC FIELDS Magnetic fields are present in a wide variety of stellar objects, from the very low mass dwarf stars to most massive O stars. They influence all phases of stellar evolution: from star-forming molecular clouds and protostellar accretion disks to evolved giant and supergiants as well as magnetic white dwarfs and neutron stars. Additionally, they play a crucial part in many aspects of stellar physics, including the accretion and outflow of material during the protostellar stage (Pudritz et al. 2012; Cai et al. 2008; Ferreira 2008) as well as the spin down of main sequence stars (e.g. Collier Cameron & Robinson 1989; Donati et al. 2000) and influence fundamental stellar quantities such as mass, rotation rate, and chemical composition. Even though magnetic fields play such an important role, the processes through which these fields are generated and impact the host star system are still not well understood. In this thesis, we focus on the magnetic fields of low mass star systems and their role in the formation and early evolution of such stellar objects. Most cool stars exhibit solar-like activity phenomena. This includes dark spots on the surface of the star as well as prominences that are detected as emission or absorption transients (e.g. Collier Cameron & Robinson 1989; Donati et al. 2000). Similar to the Sun, cool stars are also surrounded by low density coronal plasma at MK temperatures and are associated with frequent flaring and recurrent coronal mass ejections and winds (e.g Hartmann & Noyes 1987; Hall 2008).

18 2 Enhanced activity phenomena from cool stars is understood to be associated with magnetic fields generated through a dynamo process (e.g. Moffatt 1978). Alternatively, the fields could be fossil remnants from some prior evolutionary stage. Models predict that dissipation of these fossil fields due to convection would occur in 1000 years. However, cool star fields are often observed to be variable on timescales of months to decades. Such variability cannot result from an evolutionary process that ended several million years beforehand. Thus it is unlikely that these fields are fossil fields (Donati & Landstreet 2009). The observed trend of increased magnetic activity with increasing rotation rate is predicted by conventional dynamo theories and further supports dynamo generated magnetic fields in cool stars (Reiners 2012). 1.1 Generation of Magnetic Fields in Low Mass Stars The dynamo process for solar-like stars is thought to be concentrated in the thin layer (called the tachocline) located between the outer convective layer of the stellar atmosphere and the inner radiative layer (e.g. Ossendrijver 2003). Helioseismology reveals that this thin layer is the site of strong shearing. The shearing results from the angular velocity profile transitioning from differential rotation to solid body rotation (e.g. Thompson et al. 1996). The shearing stretches and amplifies poloidal fields generated in the convective layer and gives rise to organized toroidal fields (Ωeffect). The generated toroidal fields may become unstable to magnetic buoyancy and rise through the convective layer. These rising fields could either appear on the surface of the star as spots or become shredded by the convection to create the global

19 3 poloidal field (α-effect), thereby completing the dynamo cycle (Browning 2008). Most models of the dynamo process have focused on the Sun, where the model parameters have been finely tuned to reproduce solar observations. However, main sequence stars with masses below 0.35M and young solar-type stars (T Tauri stars) are fully convective and therefore lack a tachocline (Donati & Landstreet 2009). Yet both dwarf and pre-main sequence stars are observed to have strong fields and magnetic activity (e.g. Donati et al. 2008; Reiners & Basri 2007; Berger 2002; Audard et al. 2007; Gizis et al. 2000). If the tachocline is key in the generation of magnetic fields for solar-like stars then fully convective stars must generate their magnetic fields by non-solar processes. Early investigators proposed that because fully convective stars lack a tachocline, their fields should be dominated by small scale dynamo action on the spatial scale of order the size of convective cells (Durney et al. 1993). Newer models determined that fully convective stars could potentially generate purely non-axisymmetric large-scale fields if the convective layer is assumed to rotate as a rigid body (Küker & Rüdiger 1999; Chabrier & Küker 2006). The most recent simulations find that axisymmetric poloidal fields can be produced with significant differential rotation (Dobler et al. 2006) but that toroidal fields are usually dominant and differential rotation is rather weak in the limit of small Rossby number (ratio of rotation to the convective turnover time, R o ) (Browning 2008). These models are hard to reconcile with observations of fully convective stellar magnetic fields, which find the fields to be axisymmetric with little differential rotation and purely poloidal instead of toroidal topology(donati

20 4 et al. 2006; Morin et al. 2008). If dynamo theories are to explain the generation and sustainment of low mass stellar magnetic fields, they must account for the properties and behavior of fields associated with a spectrum of stellar masses, rotation rates, and environments. In order to develop such theories, observations of a variety of low mass stellar systems are important because they provide a means of studying how the dynamo process depends on fundamental stellar properties. 1.2 Characterizing Magnetic Activity in Low Mass Star Systems Stellar magnetic fields are not directly visible, but we can measure the effect the existence of these fields has on observable properties. From observations of the Sun we know that the presence of magnetic regions leads to enhanced emission in both the solar chromosphere and corona. This magnetically induced emission can be traced in radio, X-ray, and Hα emission from cool stars. Additionally, direct evidence for the presence of magnetic fields is provided through Zeeman splitting of magnetically sensitive molecular lines. Magnetic field measurements using the Zeeman effect utilize the impact magnetic fields have on the profiles and polarization of stellar spectral lines (Donati & Landstreet 2009; Reiners 2012). Each of these tracers is not only sensitive to specific properties of the magnetic field but are also able to probe different physical size scales of the field.

21 Magnetic Activity in Main Sequence Stars Zeeman Techniques Zeeman techniques are able to directly measure the global properties of the surface field for a cool star. Measurements of the Zeeman splitting in Stokes I are sensitive to the entire stellar magnetic field independent of geometry and provides a measure of the integrated surface magnetic flux, Bf, where B is the magnetic field strength and f is the field covering fraction (Reiners & Basri 2006; Reiners 2012). Using Zeeman broadening of the FeH band, Reiners & Basri (2007) made direct measurements of the fields of several M dwarfs with spectral types ranging from M2 down to M9. They find that magnetic fields are prevalent throughout the M spectral class ranging in strength from 0.1 kg to 3 kg, similar to more massive solar-type stars. Additionally, comparisons of the magnetic flux across spectral type do not show a strong break in the field strength at the fully convective limit. However, measurements of the magnetic flux using circularly polarized lines indicates much stronger fields for fully convective stars. Additionally, the ratio of magnetic fluxes measured from circularly polarized and unpolarized lines change over the fully convection limit. This may be evidence that fully convective fields are organized on larger spatial scales than the fields of partially convective stars and possibly indicates a change in the dynamo mechanism at this point (Reiners & Basri 2009). Zeeman Doppler Imaging (ZDI) techniques combine polarization measurements of Zeeman split lines with Doppler Imaging. This technique is sensitive to the field geometry, giving information about the field orientation and how it splits

22 6 Figure 1.1. The properties of the large-scale magnetic field topologies of cool stars as a function of rotation period and mass (updated from Donati (2011)). Larger symbols indicate larger fields (with a range of 1.5 kg to 3.0 kg), the shape depicts the degree of axisymmetry (decagons for purely axisymmetric and pointed stars for non-axisymmetric), and the color represents the field configuration (blue is purely toroidal and red is purely poloidal). The solid, dashed, and dash-dot lines indicate contours of constant Rossby number R o =1.0, R o =0.1, the saturation threshold, and R o =0.01. Additionally the theoretical full-convection limit is indicated by the dashed line as well as the three stellar groups discussed in the text. into its poloidal and toroidal components (Donati & Landstreet 2009). ZDI of M dwarfs reveal a transition from mainly toroidal and non-axisymmetric fields in M0- M3 objects to predominately poloidal axisymmetric fields for M4 dwarfs. This shift in the field geometries coincides with the transition to full convection and may be additional evidence for a change in the dynamo mechanism (Morin et al. 2008; Donati et al. 2008). However the situation for late-m dwarfs (M5-M8) is much more complex. As

23 7 seen in figure 1.1, two distinct magnetic field geometries, including strong axisymmetric dipolar fields and weak fields featuring a non-axisymmetric component or toroidal field, are found for objects with similar masses and rotation periods (Morin et al. 2010). Since Zeeman techniques only trace the large-scale or integrated field, they are unlikely to uncover variability on short timescales or small physical scales. Furthermore, neither technique provides information on how the field is coupled to the stellar atmosphere and dissipates energy Tracers of Chromospheric and Coronal Activity An independent test for the existence of the magnetic fields in cool stars is to observe various radiative processes that trace plasma heating and particle accelerations due to magnetic fields. Observations of Hα, X-ray and radio emission reveal field energy dissipation on short timescales and small physical scales (Berger et al. 2010). The source of Hα and X-ray emission is plasma heated by the dissipation of magnetic energy. The standard scenario involves input energy from the magnetic field driving an outflow of hot plasma into the corona through evaporation of the underlying chromosphere (Neupert 1968; Hawley et al. 1995; Güdel et al. 1996). As shown in figure 1.2, surveys of chromospheric Hα and coronal X-rays show a sharp decline in quiescent emission after spectral type M7 (e.g. Neuhäuser et al. 1999; Gizis et al. 2000; West et al. 2004); this includes the lowest mass stars and brown dwarfs (collectively called ultracool dwarfs: UCDs). The decline in magnetic activity traced in either X-rays or Hα is thought to be the result of decreasing ionization of the stellar atmospheres with spectral type. The magnetic fields become decoupled from the

24 8 atmosphere, reducing the dissipation of magnetic field energy and thus the heating of plasma (Mohanty et al. 2002). Since rotation plays a key role in many stellar dynamo models, studies of the relationship between rotation and magnetic activity can shed light on the dynamo process. For solar-type stars, rapid rotation leads to increasing X-ray and Hα emission relative to the stellar bolometric luminosity. These activity indicators are found to saturate at v sin(i) 5 kms 1 (Pizzolato et al. 2003; Wright et al. 2011). This trend is also observed in early and mid M dwarfs (Delfosse et al. 1998; Mohanty & Basri 2003; McLean et al. 2012). The rotation-activity relation breaks down for objects beyond M6, where for many fast rotators (v sin(i) >15 km s 1 ) the activity drops by 1-2 orders of magnitude and the scatter increases by a factor of 3 (Berger et al. 2010; Reiners et al. 2010; Williams et al. 2013a) (see figure 1.3). In a recent survey of 38 UCDs, Cook et al. (2013) measure a supersaturation anti correlation between rotation and X-ray activity, as opposed to the no correlation for rapidly rotating early type stars. Cook et al. (2013) argue that this trend in the X-ray emission is unlikely due to centrifugal stripping. Such a mechanism would require UCDs to have extremely large coronae in order for the centrifugal effects to be significant. They note that the overall scatter in the quiescent X-ray activity between stars with similar spectral types and rotational velocities could be explained by the results of Morin et al. (2010) who find that UCDs may have one of two different magnetic topologies: strong axisymmetric dipolar fields or weak non-axisymmetric toroidal fields.

25 Figure 1.2. Magnetic activity traced in Hα, X-ray, and radio emission for M and L dwarfs. For UCDs the quiescent emission is marked as squares, upper limits as inverted triangles, and flares as asterisks; objects with both flare and quiescent emission are connected with dotted lines. (combined figures 3, 5, and 7 from Berger et al. (2010)). (T op) Magnetic activity traced by radio emission shows clear increase with later spectral type. (M iddle) Activity traced in X-rays show saturation down to spectral type M6 and a significant decline in activity for objects with spectral type beyond M7. (Bottom) The activity traced by Hα also shows saturation down to spectral type M6 and then a clear decline beyond spectral type M7. 9

26 10 The reduction in magnetic activity traced in Hα or X-rays in UCDs does not imply a drop in magnetic field strength or filling factor. In fact the detection of both quiescent and flaring non-thermal radio emission from some late-m dwarfs, L dwarfs and recently T dwarfs (Berger et al. 2001; Berger 2002, 2006; Berger et al. 2009; Burgasser & Putman 2005; Phan-Bao et al. 2007; Osten & Jayawardhana 2006; McLean et al. 2012; Burgasser et al. 2013; Route & Wolszczan 2012; Williams et al. 2013a) confirms that at least some UCDs are still capable of generating strong magnetic fields. Stellar radio emission provides an important probe for studying the plasma processes and physical environment of stellar coronae. The most intense radio emission is associated with non-thermal processes driven by electrons accelerated in coronal magnetic fields, such as gyro-synchrotron acceleration and the cyclotron maser instability. For the very late-type dwarfs, where rapid rotation blends Zeeman signatures, and where the overall optical, X-ray, and Hα luminosity decrease greatly, radio emission is key in understanding the magnetic fields of these objects. Combining their results with those from the literature, Antonova et al. (2013) find a detection rate of a large sample of radio-loud dwarfs between M7 and L3.5 of 9%. All active dwarfs are found to have high v sin(i) values ( 20 km s 1 ) suggesting two possibilities: A detection rate dependence on rotation velocity or on inclination angle. Currently there is no conclusive evidence which is more important for detecting radio-loud UCDs (Antonova et al. 2013). Similiar to X-rays and Hα emission, the radio activity-rotation relation for M0-M6 dwarfs saturates at v sin(i) 5 km s 1. For spectral types M7, contrasting to the trends in Hα and X-rays, the radio activity

27 11 Figure 1.3. Radio (top), X-ray (middle), and Hα (bottom) activity as a function of projected rotation velocity (figure 6 from McLean et al. (2012)). In this figure both flares (squares) and quiescent emission (circles) are shown. Additionally, upper limits are placed on both the activity and projected velocity (left arrows). The red symbols represent objects with spectral type later than M7 (UCDs) and the black symbols are for spectral types M0-M6.5. All three activity indicators saturate at 5 km s 1 for objects earlier than M6. However, for spectral types M7-M9 the activity-rotation relation breaks down and the radio activity and the Hα/X-ray trends diverge; the radio emission is found to increase with larger projected rotation velocity while the other two emission types drop steeply beyond v sin(i) 20 km s 1. increases greatly regardless of rotation velocity. Additionally, the radio luminosity for most rapid rotators remains the same as a function of both rotation and spectral type to at least spectral type L4 (see top panel of figures 1.3 and 1.2). However, not all rapidly rotating UCDs are detected in the radio indicating a range in the level of

28 12 Figure 1.4. The Güdel-Benx relationship between L x ( kev) and L ν,r (figure 6 from Williams et al. (2013a)). The gray circles reproduce the original data of Benz & Guedel (1994): objects with [L ν,r ]<12 are solar flares; 12<[L ν,r ]<14.5 are dme and dke stars; L ν,r ]<14.5 are active binaries. The color symbols represent three different stellar groups: (green) spectral types M6 or earlier; (red) M6.5-M9; (blue) spectral types later than L0. From this figure we see that UCDs display a wide range of behavior with respect to the GBR. magnetic activity for such objects (McLean et al. 2012). There is an observed tight correlation between radio and X-ray luminosities of magnetically active stellar systems. For most active F-M stars the ratio of the X-ray to radio luminosity is approximately Lx L ν,r Hz (Gudel et al. 1993). This relation can be extended to include solar flares and active binaries; in this case L x L α ν,r, where α 0.73 over 10 orders of magnitude of radio spectral luminosity (Benz & Guedel 1994)

29 13 (the original data from Benz & Guedel (1994) are represented by the gray circles in figure 1.4). This relation is called the Güdel-Benz relation (GBR). The continuity of this relation over such a large range of stellar systems suggest that the radio and X- ray emission have a common driver and there is a common physical process over the range of emitters (Williams et al. 2013a). The standard interpretation of the GBR is that magnetic reconnection, occurring in the atmosphere of a star, accelerates a population of energetic non-thermal particles. In the presence of a magnetic field this non-thermal population will produce gyro-radiation at radio frequencies (see section 2.3). These energetic electrons deposit some of their energy in the chromosphere leading to evaporation. The evaporated material concentrates in coronal loops and emits thermally as X-rays (Güdel 2002). This interpretation is called the Neupert model (Neupert 1968) and predicts that the change in the X-ray luminosity over time is proportional to the radio luminosity. This suggests that the X-ray emission tracks the total energy deposited by the particle acceleration process. While the Neupert model is well cited it has only been observed in few solar and stellar flares (e.g. Dennis & Zarro 1993; Güdel 2002; Osten et al. 2004). Early radio observations of UCDs found that the GBR breaks down at spectral type M7, evolving from L ν,rad /L x to (Berger et al. 2010). However a more recent study, carried out by Williams et al. (2013a), of a comprehensive sample of UCDs with both radio and X-ray observations shows that UCDs display a range of behavior with respect to the GBR (see color symbols in figure 1.4). Some UCDs violate the GBR by several orders of magnitude. Williams et al. (2013a) show that

30 14 Figure 1.5. The distance from a linear fit to the Güdel-Benz relationship as a function of rotation rate (upper) and spectral type (lower) (adapted from Williams et al. (2013a)). even if you assume the X-ray luminosity to be the saturated value L x = 10 3 L bol, several UCDs still violate GBR. This is indicates that under-luminous X-ray emission is not enough to account for the violation of GBR, but also requires over-luminous radio emission. Other UCDs are consistent with GBR within 2 dex of the relation. Williams et al. (2013a) investigated the variation from GBR as a function of

31 15 spectral type and rotation rate, finding variation from the GBR becomes possible for later spectral types and rotational velocities >20 km s 1 (see figure 1.5). They find that UCDs with similar spectral types and rotation rates diverge from GBR by varying amounts. This contrasts with Stelzer et al. (2012), who used rotation velocities to divide UCDs into two groups: radio-loud, rapidly rotating UCDs that violate GBR, and radio-quiet slowly rotating UCDs that agree with GBR. Williams et al. (2013a) find that the dichotomy in UCD behavior is not strictly a function of rotation. They instead propose that the range in UCD behavior may be due to varying magnetic field topology. They suggest UCDs with strong, axisymmetric fields would have values of L x L ν,r consistent with the GBR and UCDs with weak, non-axisymmetric fields are radio over-luminous. A similar proposal was made to explain the varying levels of radio activity at high rotation rates, where some UCDs were observed to exhibit high levels of magnetic activity, while others were undetected in the radio band (McLean et al. 2012). This concept is supported by ZDI observations of late-m dwarfs that show varying magnetic topology for similar spectral type (Morin et al. 2010). UCDs with detectable levels of radio emission display a wide variety of behaviors (see figure 1.6). A few UCDs are observed to be in a quiescent state, where radio emission is broadband, does not display any variability, and has low levels of circular polarization (CP) (e.g. Ravi et al. 2011; Berger et al. 2008). Several UCDs have light curves that show variation with the same period as their rotation. These light curves either have smooth modulations (e.g. McLean et al. 2011) or have short highly CP peaks lasting for a few minutes (e.g. Hallinan et al. 2007). Other UCDs

32 16 (A) 2M J (Berger et al. 2009) (B) TVLM (Hallinan et al. 2008) (C) J (McLean et al. 2011) Figure 1.6. Three examples of radio light curves from UCDs (adapted from Hallinan et al. (2008); Berger et al. (2009); McLean et al. (2011)). (A) Radio emission observed from 2M J at 8.46 GHz; this light curve shows a single strong, right circularly polarized flare at about UT 4 hrs and a steady, weak quiescent component. (B) Radio emission observed at 8.46 GHz from TVLM showing strong periodic double pulses (left and then right circularly polarized) that occur ever 2 hours and a weak quiescent component. (C) Radio light curve from J at 4.9 GHz; this light curve only has a sinusoidally varying quiescent component and no evidence of periodic/episodic flares. show large, isolated flares that are generally 100% CP and last a few minutes (Burgasser & Putman 2005). The short duration of these pulses place strong constraints

33 17 on both the brightness temperature and directivity of the associated emission mechanism (Hallinan et al. 2006). In both types of pulsed emission, the UCDs generally exhibit interpulse quiescent emission as well. Furthermore, the type of radio emission exhibited by UCDs changes over timescales of years, varying between undetectable, quiescent, and periodic (Berger 2002; Berger et al. 2008; Hallinan et al. 2006, 2007; Osten & Wolk 2009). The changes in the radio emission properties indicate that the magnetic structures giving rise to the radio emission have finite lifetimes and vary over yearly timescales (Osten & Wolk 2009; Berger et al. 2008). The electron cyclotron maser instability (CMI) is generally accepted to be the source of the pulsed emission since it can account for the high brightness temperature, directivity, and CP of the emission (Hallinan et al. 2006). The CMI mechanism is a particle-wave plasma instability that is caused by the resonance interaction between gyrating electrons in an ambient magnetic field and the electric field of electromagnetic waves at frequencies close to the electron cyclotron frequency (Treumann 2006). CMIdriven radio emission has been detected from all of the magnetized planets in our solar system (Zarka 1998; Ergun et al. 2000) and thought to be the source for certain classes of solar and stellar bursts (Melrose & Dulk 1982; Güdel 2002). There is still some debate over the nature of the quiescent emission. Both depolarized CMI radiation (Hallinan et al. 2006, 2008) and incoherent gyrosynchrotron emission from a nonthermal population of mildly relativistic electrons (Berger 2002; Burgasser & Putman 2005; Berger et al. 2005; Osten & Jayawardhana 2006) have been proposed as sources for this emission.

34 18 Infrared Class: Class 0 Class I Class II Class III Sketch: Description: Protostar deeply embedded in circumstellar envelope where both a disk and outflow forms. Evolved protostar surrounded by an accretion disk, outflow and remnant infalling envelope. YSO evacuated most of remnant circumstellar envelope;; surrounded by an optically thick disk and bipolar jets. Classification includes Classical T Tauri stars. YSO surrounded by an optically thin disk and weak or absent jet. Classification contains Weak Line T Tauri stars. Age ~ 10 5 years Age ~ 10 4 years Age ~ years Age ~ years No known X-ray emission;; only thermal radio emission. Known X-ray emission;; both thermal and non-thermal radio emission. Strong X-ray emission;; only thermal radio emission Strong X-ray emission;; only non-thermal radio emission. Figure 1.7. Illustration of the stages of low-mass young stellar evolution; references for information are from Feigelson & Montmerle (1999) and Andre (1996) Magnetic Activity in Young Stellar Objects Stars form from the gravitational collapse of dense proto-stellar cores in giant molecular clouds. Mass infall leads to the formation of a young stellar object (YSO) that is embedded in an envelope of gas and dust. There are four observationally well-defined stages of early stellar evolution (see Figure 1.7 ). Class 0 objects are young YSOs embedded in massive, cold envelopes which are collapsing towards the central region (Feigelson & Montmerle 1999). Due to conservation of angular momentum, material from the envelope does not accrete directly on the YSO but forms an accretion disk and accompanying collimated outflow (Günther 2011). These objects are primarily seen in the far-infrared and sub-millimeter range (Andre 1996). The Class I stage corresponds to evolved, near-ir YSO surrounded by a disk and

35 19 a remnant in-falling envelope (Andre 1996). At this point most of the material in the circumstellar envelope has accreted onto the disk. The outflow is still present but has a larger opening angle and lower mass-loss rate than during the Class 0 stage (Feigelson & Montmerle 1999). The Class II stage comprises YSOs that have evacuated most of their remnant circumstellar envelope becoming optically visible (Dougados et al. 2000). These objects are surrounded by an optically thick disk and the youngest members still drive bipolar jets (Feigelson & Montmerle 1999). Classical T Tauri stars (CTTS) are included in this class of YSOs and are optically defined as young, low-mass stars with Hα emission lines that have equivalent widths >10Å. The wide emission lines are due to the magnetically-funneled accretion of material from the disk to the YSO (Günther 2011). The final stage, Class III, contains YSOs that have a simple blackbody spectral-energy distribution implying little to no disk accretion (Feigelson & Montmerle 1999). These objects are thought to be surrounded by an optically thin disk and contain a weak or absent jet (Andre 1996). Due to the decrease in disk accretion the equivalent width of Hα decreases and these stars are optically classified as weak line T Tauri stars (WTTS) (Günther 2011) Accretion Mass transfer onto YSOs is thought to occur through magnetically funneled accretion. The simplest models assume a dipole field aligned with the rotation axis of the star. The field extends several stellar radii and truncates the inner edge of the stellar disk at the co-rotation radius. Truncation occurs at this point because it is here that the ram pressure of the accreted material is less than the magnetic pressure

36 20 of the field. The material is then channeled from the disk along magnetic field lines to impact points at hight stellar latitudes. The material then passes through a strong accretion shock and is heated to 2-3 MK. This shocked material produces soft X- ray radiation, the bulk of which is absorbed in the shock and contributes to further heating. It is possible that some X-ray emission escapes and heats the pre shock gas. Observations of the soft X-ray component find that the densities of the postshock gas is higher than expected for non-flaring coronal plasma in main sequence stars (>10 11 cm 3 ) (Günther 2013). Additionally excess in soft X-rays is only seen in accreting objects. There seems to be, however, a correlation between the soft excess and the coronal heating which is only reconciled if the soft excess is due to an interaction of accretion streams with coronal magnetic field (Güdel & Nazé 2009). In other wavebands, the magnetically funneled accretion model can account for characteristics of the emission from YSOs. Photometric variability and modulation observed in Balmer lines are explained in terms of accretion funnels that pass along the line of sight and a magnetically warped disk that periodically occults the star (Bouvier et al. 2007). Surveys of CTTs show wide and asymmetric Hα lines with wings that reach out to km s 1 ; such profiles can be explained using radiative transfer models for the magnetic funnels (Günther 2013) Outflows In order for accretion to occur onto the YSO there must be some mechanism to remove angular momentum from the system. Outflows and jets from the YSO are widely accepted to be the mechanism that controls angular momentum transfer in

37 21 Figure 1.8. Sketch of the different fall and outflow patterns in CTTs. The arrow heads on the field lines (black) give the direction of mass flux. The total contribution to the outflow of material from CTTs through stellar wind, magnetic ejections, and disk wind is unclear. In this sketch it is suggested that the innermost jet component might be launched from the star and heated in a decollimation shock (from Günther (2013)). forming stars. These outflows are associated with Herbig-Haro (HH) objects, forming where the fast streams of material collide with the slower material along the outflow (McGroarty et al. 2007; McGroarty & Ray 2004). The inner most part of YSO outflows can be a very fast and highly collimated jet. Typical velocities for these inner jets are km s 1 which is close to the gravitational escape velocity from the surface of the young star. The temperature of this gas is on the order of 10 4 K with corresponding sound speeds of 10 km s 1. From this sound speed it follows that the jets are highly supersonic with Mach numbers around The opening angles of YSO jets vary but are typically only a few degrees; this high collimation exists for all stages of YSO evolution. They are dynamical, evolving on the timescales of a few years and contain shock features which have shock velocities considerably less than the corresponding jet velocity. These features are thought to arise from velocity

38 22 variations in the outflow close to the source (Ray 2009). The acceleration and collimation of these jets requires the presence of magnetic fields and the circumstellar disk. Magneto-centrifugal acceleration is generally accepted as the mechanism that accelerates the jet material out to the Alfvén radius after which the material is collimated via built up magnetic hoop stress. This process includes three general steps. First, because on the disk the rotational kinetic energy is greater than the magnetic energy, the disk carries the field lines around the central object. Next, above the disk the magnetic field is strong enough to entrap the surrounding plasma and forces corotation out to the Alfvén distance. Then, after the Alfvé distance the plasma has gained enough energy to move independently of the magnetic field. This results in the plasma winding up the magnetic field and creates a strong azimuthal field. The magnetic hoop stress from this azimuthal component of the field finally collimates the flow (Tsinganos 2007). There are three different launching regions proposed for YSO outflows (see figure 1.8): (1) There could be a stellar wind analogous to the solar wind (Kwan & Tademaru 1988; Matt & Pudritz 2005). YSOs could accelerate stronger stellar winds than main sequence stars because the accretion shock provides additional heating in the upper layers of the atmosphere (Cranmer 2009). There is, however, a fundamental limit to the mass loss through a stellar wind. For a hot, optically thin plasma with a density n the cooling increases as n 2. For higher mass loss rates a higher density is required in the corona. However, there is a threshold density where the cooling rate

39 23 Figure D profiles of log(n e ), x e, and log(n H ) along the flow of DG Tau in discrete velocity intervals. Low velocity interval (LVI) defined as -120 to +25 km s 1, medium velocity interval (MV() defined as -270 to -120 km s 1, and high velocity interval (HVI) defined as -420 to -270 km s 1. These profiles show that there is significant variation in the plasma parameters along the jet (adapted from Maurri et al. (2014)). becomes high enough to drop the temperature below the point where it can drive a wind. An upper limit of M yr 1 was derived for a hot stellar wind based on this argument (Matt & Pudritz 2007). Typical mass loss rates for YSO jets are between M yr 1 so a stellar wind alone could not account for the total

40 24 mass-loss observed for these sources. (2) Outflows could be driven along the interface of the stellar magnetic field and the disk field at the edge of the inner disk; these winds are called X-winds (Shu et al. 1994). (3) Winds could be driven from the disk (Blandford & Payne 1982; Pudritz & Norman 1983; Anderson et al. 2005). In the case where the outer layers of the disk are warm enough to ionize the gas, it can then be loaded onto field lines of the disk field which are stretching outwards and material traveling along the field will be accelerated magneto-centrifugally (Günther 2013). Since CTTS have evacuated most of their remnant circumstellar envelope, this allows an unobscured line of sight to the central 100 AUs of their jet where acceleration is thought to occur. For these sources measurements of forbidden line emission are possible (Dougados et al. 2000) Using the spectroscopic diagnostic technique described by Bacciotti et al. (1999) (BE technique), the forbidden-line emission can provide constraints for the electron density, total hydrogen density, ionization fraction, and the average excitation temperature of the gas in these jets. Typical parameter values are shown in figure 1.9 which gives the electron density, ionization fraction, and neutral density along the jet associated with DG Tau. From these measurements we can determine fundamental jet parameters such as the mass loss rate of the jet. Imaging of YSO outflows can provide information about the typical lengthscales of these features. The sizes of YSO jets can be large and comparable to the size of the parent cloud. From the velocity of extended outflows and their statistically

41 25 Figure Radial velocity profiles across the jet of DG Tau in near-uv and optical emission lines. The optical data sets are overlaid on the near-uv data to illustrate how well the results from the two wavebands agree. For DG Tau s approaching jet we see that the central region of the jet has the highest velocity gas. At the edges of the jet slower moving gas is found. This velocity profile is not symmetrical: velocities on one side of the jet are lower than the other. This is a natural outcome if the jet is rotating. (from Coffey et al. (2007)). estimated lifetimes it is likely that many outflows, especially those for more evolved young stars, extend beyond their associated cloud (McGroarty & Ray 2004). If jets are launched and collimated through the magneto-centrifugal scenario, the base of the flow would maintain a record of rotation during propagation. For favorable inclination angles with respect to the line of sight, a trace of this rotation should be seen in high angular resolution spectra taken close to the central source with the slit parallel to the outflow axis (Bacciotti et al. 2002a). Jet rotation was initially discovered using the Space Telescope Imaging Spectrograph (STIS) on board the Hubble Space Telescope (HST) (Bacciotti et al. 2002a). The average radial velocities

42 26 laterally across the jets were found to differ from one side to the other by km s 1. Similar results were found in the UV (Coffey et al. 2007) and in the nearinfrared (Chrysostomou et al. 2008). Using these radial velocity measurements across the jet and assuming that they are resulting from centrifugal acceleration, the angular momentum of the jet material is consistent with the material being launched within a few AU of the the central source (Coffey et al. 2007). The jet structure for CTTS resembles an onion where the inner layers are consecutively faster and can reach velocities up to a few hundred km s 1 (Bacciotti et al. 2000). Additionally, the temperature and density of the gas varies across the jet, where the hotter more dense gas is concentrated in the inner layers along the central axis of the jet (see figure 1.9). X-ray emission has been detected from the hotter gas found in the inner jet for two CTTS (Güdel et al. 2005, 2008; Güdel et al. 2011; Skinner et al. 2011), with a higher detection rate for younger objects with higher mass loss rates. In addition to the jet X-ray emission, CTTS have an X-ray spectrum with two spectral components: (1) a weakly absorbed component from a cool plasma (2-3 MK), (2) a strongly absorbed component from a very hot plasma (20-30 MK). The hot component is associated with a coronal source and the softer component is thought to be associated with shocks formed in the acceleration region of the jet (Güdel et al. 2005; Güdel et al. 2007a). Near-infrared spectroscopic observations suggest that dust is launched with jets. This is plausible since close to the source the dust is not entirely destroyed by jet shocks (Podio et al. 2006). Chrysostomou et al. (2007) observed HH 135/136 and

43 27 measured circular polarization for the dust component of the jet. Using a Monte Carlo simulation, they also determined that the most likely cause of the polarization is aligned non-spherical dust grains in a helical magnetic field. A large number of YSOs have been detected at radio wavelengths. The dominant radio emission mechanism is thought to be thermal bremsstrahlung from the shock-heated gas in the jet. However, evidence for non-thermal emission from several YSOs has also been discovered. Circular polarization (CP) has been associated with several YSOs, including objects in the ρ Ophiuchi molecular cloud (Andre et al. 1988; White et al. 1992; Andre et al. 1992), the Taurus-Auriga molecular cloud (Phillips et al. 1993; Skinner 1993; Feigelson et al. 1994; Ray et al. 1997), the R Coronae Australis region (Choi et al. 2008), the Orion region (Zapata et al. 2004), and HH 7-11 (Rodríguez et al. 1999). Furthermore, a few YSOs have been associated with linearly polarized (LP) radio emission, these include HH (Carrasco-Gonzalez et al. 2010), HD (Phillips et al. 1996), and the Orion Streamers (Yusef-Zadeh et al. 1990). Additional characteristics of non-thermal emission, including strong variation in flux density on timescales of hours to days, a negative spectral index, and VLBI measurements of high brightness temperatures (T B 10 7 K), have also been found (Curiel et al. 1993; Hughes 1997; Andre et al. 1992; Wilner et al. 1999; Rodríguez et al. 2005). A suggested source of this non-thermal emission is the gyrosynchrotron (GS) mechanism (Andre 1996). Non-thermal radio emission is almost always associated with the more evolved WTTS. This is consistent with the idea that the non-thermal

44 28 coronal emission is revealed only after the optically thick mass outflows have evolved away (Eislöffel et al. 2000). However, this idea may be too simple: Non-thermal radio emission has recently been reported from several less-evolved YSO s that have infrared evidence for a disk, further supporting the possibility that non-thermal diagnostics of magnetic activity could be produced in Class I and II objects (Osten & Wolk 2009) YSO Large Scale Fields As mentioned previously, X-ray emission associated with a hot corona is observed in CTTS. X-ray flares are common and the most energetic examples reach T 10 8 K. The characteristics of this X-ray emission are comparable to more evolved main sequence stars. Coronal X-ray emission are also observed in Class I protostars suggesting significant magnetic activity even in these younger stars. However our ability of detecting coronal X-ray emission is limited due to attenuation by circumstellar material (Güdel & Nazé 2009). Zeeman techniques are also used to study YSO coronal magnetic fields. ZDI maps for a sample of YSOs with different masses and ages show that these stars possess magnetic fields with a range of strengths and topologies. The topologies range from simple, almost axisymmetric fields to much more complex non-axisymmetric fields, similar to main sequence M dwarfs. The strength and topology of the YSO fields appear to be related to the internal structure of the host star.ysos that have developed a radiative core have weak, complex non-axisymmetric fields while fully convective YSOs have strong simple axisymmetric fields (Gregory et al. 2012). Since the truncation radius of the accretion disk depends on the strength of this component

45 29 Figure Measurements of the strength and topology of T Tauri fields using Zeeman techniques overlaid on an H-R diagram for objects with masses M. The top two figures show the results of Zeeman Doppler imaging of a sample of T Tauri stars. (top lef t) They symbol size scales with the magnetic intensity and the different symbols denote dominate field orientation: circles=dipole-dominant; squares=octupole-dominate; triangles=complex multipolar fields. (top right) The symbol size scales with the strength of the dipole field component. The bottom row shows Zeeman broadening measurements of T Tauri mean magnetic field strengths (bottom lef t) and mean magnetic fluxes (bottom right). In the left-hand plot the symbol sizes reflect the mean field strengths ranging from kg. Similarly, in the right-hand plot the symbol size represents the mean flux for each star (adapted from Hussain & Alecian (2014)).

46 30 it is important to constrain the strength of this component of the field for YSOs. It is clear from the top of figure 1.11 that as the complexity of the field increases, the dipole field strength drops and this should have a corresponding effect on the accretion state of the star (Hussain & Alecian 2014) The mean magnetic field strengths measured through Zeeman broadening range between kg. There is no clear trend in the measured Bf values with stellar parameter. The highest and lowest values are both found in stars with similarly spectral types (bottom of figure 1.11). The field strengths measured for these objects are much larger than those required by pressure equipartition considerations: B eq =(8πP g ) 1/2 ; where P g is the gas pressure at the atmospheric height associated with the observed line formation (Johns-Krull 2007). This implies that no equilibrium can exist between magnetic and non-magnetic regions in T Tauri stars. Thus caution must be used when interpreting activity phenomena for T Tauri stars as a simple scaled up version of those observed on the sun and solar-type stars. The mean magnetic flux (F B =4πR 2 B) measured using Zeeman broadening decreases with age. This decrease is attributed to the shrinking radii of the evolving YSO (bottom right of figure 1.11). 1.3 Summary and Central Science Question Magnetic fields are important to stellar physics, playing a role the formation, evolution and composition of the star. They are prevalent in stars throughout their evolutionary history and over a wide range of stellar parameters; here we concentrate on low mass star systems. Generation of magnetic fields in solar-like stars involves

47 31 dynamo action in the tachocline, a region of great shear located between the radiative and convection zones in the star s atmosphere. Fully convective stellar systems do not have such a region but still exhibit the presence of strong magnetic fields. Dynamo models need to be able to explain the ability of such stellar systems to generate and maintain strong fields without the tachocline region. To further develop dynamo models to account for all low mass stellar systems, constraints on the geometry and physical parameters of the magnetospheres of a variety of low mass stellar systems are necessary. Such constraints are acquired through observations of magnetic activity; low mass star systems have been well studied through optical line emission, X-rays, radio, and by use of the Zeeman effect. However for the fully convective cool stars, where the X-ray and optical emission drop off significantly and rapidly rotating objects obscure Zeeman split lines, radio observations are key in understanding magnetic activity in these stars. Thus it is important to understand how to use the properties of the radio emission to constrain the field parameters and geometry. The following thesis project uses radio observations from the Karl Jansky Very Large Array (VLA) to characterize the magnetic features of a low-mass class II YSO as well as two UCDs on the main sequence. From the previous sections, it is obvious that the magnetic properties of both these systems are well documented. However with the upgrade to the VLA both the increase in sensitivity and observing bandwidth, future radio observations of YSOs and UCDs are more likely to place meaningful constraints on the magnetospheres in these stellar systems. The detection of X-ray emission in the extended jet and central source of DG

48 32 Tau motivated the radio campaign presented in this thesis. Strong shocks in the jets of YSOs can heat the gas to X-ray emitting temperatures, T v100, 2 where v 100 is the shock velocity relative to the target in unites of 100 km s 1 (Raga et al. 2002). The X-ray emission in DG Tau s jet reveals electron temperatures of 3-4 MK, incompatible with the modest shock velocities of km s 1 derived from optical line ratios. The additional heating may be related to the helical magnetic fields thought to collimate and launch the jets (Güdel et al. 2008). Furthermore, the central source X-ray emission is observed to have two spectral components, one of which is attributed to a hot corona. Thus the purpose of the radio campaign of DG Tau was to use high angular resolution radio images of the inner region of the jet and central source to look for signatures of magnetic fields, including negative spectral indices and polarization measurements. With the upgrade to the VLA, these observations are 10 times more sensitive than previous observations of DG Tau. Thus they allow for a better characterization of the inner radio jet and higher possibility of detecting non-thermal radio emission. Since the discovery of radio emission in UCDs, the two UCDs, TVLM and 2M , chosen for this thesis project have been well studied. However, the upgrade to the VLA allows for a more compressive study of not only these sources temporal variation but also the spectral variation of the radio emission. Previous radio observations of UCDs were bandwidth limited and could only characterize the temporal variation within a 50 MHz band. Yet previous subarray and multi-frequency observing campaigns indicated that there was complex frequency structure for the

49 33 radio emission The new wideband system of the VLA allows simultaneous 2 GHz bandwidth coverage, and the observations presented in this thesis reveal the complex structure previous observations only hinted at. The newly revealed frequency structure of both the quiescent and pulsed component of the radio emission for TVLM513-46and 2M J , are key in better constraining the magnetic properties of the source regions for the radio emission. By developing emission models to reproduce the complex features observed in the radio emission we can determine what parameters and orientations can possibly lead to the radio emission we observe.

50 34 CHAPTER 2 RADIATIVE PROCESSES In order to constrain the physical parameters of a region we must understand how the observed properties of the radio emission, including the spectral energy distribution, polarization, and turnover frequency, depend on the size of the source region R, number density n e, energy distribution index δ, and magnetic field strength B. In the following I give a brief derivation of the radiative transfer equation. This equation is key in understanding how radiation propagates from an emitting source to an observer and takes into consideration interactions the radiation has with intervening material. The derivation follows the work done by Dulk (1985), Rybicki & Lightman (1986), and Melrose & McPhedran (2005). Once the radiative transfer equation is established, the primary task is to find forms for the absorption and emission coefficients corresponding to particular physical processes. In sections 2.3 and 2.4 I give the coefficients for Bremsstrahlung (free-free) and gyrosynchrotron emission respectively. These two emission types are common in low mass star systems and solving the radiative transfer in these two cases will all me to compare the predicted radio emission properties to those observed for the two studied fully convective systems. Further details on the derivation of these coefficients are given in Dulk (1985) and Melrose & McPhedran (2005).

51 Radiation at Radio Frequencies Much of the continuum radio radiation from astrophysical objects is due to the acceleration of electrons through collisions with ions or by spiraling in a magnetic field; the resulting radiation in these cases is incoherent. Additionally, in some circumstances it is possible for an efficient conversion of electron energy into some natural wave mode of the plasma (e.g. electron-cyclotron waves or Langmuir waves). These waves are in the radio-frequency domain because the characteristic frequencies of the plasma are typically GHz. These frequencies include the electron plasma frequency and the electron-cyclotron frequency ν p = [n e e 2 /πm e ] 1/2 9000n 1/2 e (2.1) ν B = eb/2πm e c B (2.2) where n e is the electron density, e the electric charge, m e electron mass, c the speed of light, and B the magnetic field strength. Resonances between particles and these characteristic frequencies can then extract any free energy that exists in the electron distribution of the plasma. Plasma with such free energy can only exist when the electron-electron and electron-ion collision frequencies are not as high as the resonance frequencies nor so high as to restore the plasma quickly to equilibrium. Such restrictions are the major reasons that amplified radiation is generally confined to radio frequencies. For the case of resistive instabilities, amplification carries the intensity of particular wave modes to very high levels and leads to coherent emission as in the case of electron-cyclotron maser emission.

52 Radiative Transfer Equation In free space the specific intensity I ν of radiation is conserved along a ray, di ν ds = 0 (2.3) where s is the coordinate along the ray between the source and the observer. However, for a ray passing through a medium, energy may be added to or subtracted from it by emission or absorption. To solve for the intensity of the ray as it travels through emitting and absorbing media we use the radiative transfer equation. The spontaneous emission coefficient j ν is defined as the energy emitted per unit time, per unit solid angle, per unit volume, and per unit frequency. For a ray of cross-section da traveling a distance ds, the intensity added to the beam is di ν = j ν ds (2.4) The absorption coefficient α ν is the loss of intensity in a beam as it travels a distance ds di ν = α ν I ν ds (2.5) Incorporating the effects of emission and absorption into a single equation, the variation of the specific intensity along a ray is given by di ν ds = α νi ν + j ν (2.6) We can re-write the transfer equation in terms of the variable dτ ν =α ν ds, the optical depth of the medium the ray passes through. If τ ν >1 the medium is called optically thick; in this medium the average photon of frequency ν cannot traverse the

53 37 entire medium without being absorbed. Alternatively, if τ ν <1 the medium is called optically thin and a typical photon of frequency ν can traverse the entire medium without being absorbed. The radiative transfer equation is now written di ν dτ ν = I ν + S ν (2.7) where S ν is the source function defined as the ratio of the emission coefficient to the absorption coefficient S ν = j ν α ν (2.8) From equation (2.7) we see that if I ν > S ν then diν dτ ν < 0 and I ν tends to decrease along the ray. Similarly, if I ν < S ν then I ν increases along the ray. So the source function is the quantity the specific intensity tries to approach and does approach given sufficient optical depth. In this regard the transfer equation describes a relaxation process. One way to characterize the specific intensity at a frequency ν is to specify the brightness temperature T b, or the temperature a black body in thermal equilibrium with its surrounding would have to be to produce this specific intensity. A black body is an idealized physical body that absorbs all incident electromagnetic radiation. The radiation the black body emits when in thermal equilibrium is called black body radiation. There are two important properties associated with I ν if we consider the case of black body radiation: (1) the specific intensity is isotropic and (2) only depends on temperature T. In this case the radiation is emitted according to Planck s law, I ν = B ν (T ) (2.9)

54 38 If we place a thermally emitting material at temperature T, whose emission depends solely on its temperature and internal properties, inside the black body enclosure, the presence of the material cannot alter the radiation since the new configuration is a blackbody enclosed at temperature T. Thus we have S ν = B ν (T ) (2.10) From equation (2.8) j ν = α ν B ν (T ) (2.11) Equation (2.11) is called Kirchhoff s law and relates α ν, j ν, and the temperature T for a thermally emitting material. The radiative transfer equation for thermal radiation is then di ν dτ ν = I ν + B ν (T ) (2.12) Note the distinction between thermal radiation and black body radiation: for thermal radiation S ν =B ν (T ) and black body radiation I ν =B ν (T ); thermal radiation only becomes black body radiation for optically thick media. The Planck function gives the energy per solid angle, per volume, per frequency and is written B(T ) = 2hν3 c 2 1 e ( hν kt ) 1 (2.13) In the limit hν<<kt the exponential in equation (2.13) can be expanded to give the Rayleigh-Jeans law: I ν (T ) = 2kT ν2 c 2 (2.14)

55 39 Figure 2.1. Geometry for radiative transfer equation. Illustrated is a source of optical depth τ ν located in front of a possible background of brightness temperature T bo ; from Dulk (1985). Equation (2.14) applies at low frequencies and almost always applies in the radio regime. So for a black body in the radio regime: T b = c2 2ν 2 k I ν (2.15) Additionally, its convent to replace the source function S ν by T eff, the effective temperature of the radiating electrons, using S ν = kt effν 2 c 2 (2.16) In the case of a Maxwellian electron distribution of temperature T, T eff is equal to T independent of the emission mechanism, frequency, or polarization mode. Yet for a non-thermal electron distribution, T eff is generally a function of both frequency and polarization mode. The usual factor of 2 is omitted in equations (2.15) and (2.16) since I ν and S ν are defined for each of the orthogonal polarizations separately, I tot ν = I p1 ν + I p2 ν (2.17) Using these expressions for the source function and specific intensity we can

56 40 write the transfer equation as or from the geometry in figure (2.1) dt b dτ ν = T b + T eff (2.18) T b = τν 0 T eff e tν dt ν + T bo e τν (2.19) In the case of an isolated source with constant T eff, equation (2.19) reduces to T b = T eff [1 e τν ] (2.20) T b = T eff (if τ ν >> 1) (2.21) T B = T eff τ ν = c2 kν 2 j νr (if τ ν << 1) (2.22) where R is the size of the source along the line of sight. Equations (2.21) and (2.22) show that incoherent radiation cannot attain a value of T b higher than T eff, where T eff is related to the average energy of the emitting particles by < E >=kt eff. For stellar cases the field strengths tend to be large and the electron energies low so that T eff and T b of incoherent emission are usually limited to about K; observational values substantially higher imply a coherent mechanism. The flux density S for one polarization of a radio source is related to the brightness temperature by S = kν2 c 2 T b dω (2.23) where dω is a differential solid angle and the integral is over the projected area of the source.

57 41 A useful quantity for diagnostics is the turnover frequency ν peak. This is the frequency at which the radiation goes from being optically thick to optically thin, or where τ ν = α ν L 1. For gyrosynchrotron emission the turnover frequency depends strongly on the magnetic field strength and average electron energy but weakly on the electron number density. Determining the polarization of the radiation is fairly complex because of coupling among the various Stokes parameters. Under stellar conditions, however, the characteristic modes of the plasma (the o- and x-modes) are usually circular; this is not true when the propagation is nearly perpendicular to the magnetic field (see section 2.6). The sense of the polarization usually reflects the sense of the magnetic field in the source region. The degree of circular polarization r c is r c = T b,x T b,o T b,x + T b,o (2.24) In the optically thin case (τ ν << 1) we have r c = j ν,x j ν,o j ν,x + j ν,o (2.25) Alternatively for τ ν >>1 in both modes, the polarization of free-free and gyrosynchrotron radiation goes to zero for a thermal plasma; for a non-thermal plasma the polarization is low in this case (<20%). In some circumstances the polarization ellipse of a characteristic mode differs significantly from circular. In these cases the righthand side of equation (2.24) must be multiplied by a correction factor of 2T x /(T 2 x +1).

58 Bremsstrahlung Bremsstrahlung or free-free emission is generated from the acceleration of a charged particle in the coulomb field of another particle. Coulomb interactions in a plasma are considered to be collisions. This is clearly appropriate for close encounters but is it also appropriate for distant interactions? If we consider the affect of coulomb fields for surrounding particles on that of a test charge it is clear that a single test charge will interact simultaneously with all other particles in the plasma. This is due to the fact that the coulomb field falls off with distance r from the test charge as r 2, however the number of field particles between distances r and r+dr increases as r 2 dr so that the field at the test charge has equal contributions from particles at all distances. So then bremsstrahlung due to distant encounters involves many particles influencing a single charge at one time. To treat this problem we can regard the effect of distant encounters as perturbations on the orbit of the test charge, where the unperturbed orbit of the test charge is taken to be constant rectilinear motion. Radio frequency emission results from distant electron-ion encounters in which the motion of the electron is perturbed only slightly by the coulomb field. Electronelectron encounters usually produce negligible bremsstrahlung emission due to the fact that the power in electric dipole emission is proportional to the 2nd-order time derivative of the electric dipole moment. In the case of collisions between like particles, the electric dipole moment is a constant of the motion. The average emissivity of the electrons in a plasma is calculated by first considering the energy radiated by a single electron passing an ion of charge Z i at a

59 43 distance of d and velocity v, and then multiplying by the rate of such encounters and integrating over d and v. In the following we only consider a Maxwellian velocity distribution, where v=(kt/m) 1/2. However, note that the resulting bremsstrahlung emission is found to be different for a variety of limiting cases. The simplest way to describe bremsstrahlung emission in the general case is to consider the emission from a single electron and then use what is called the Gaunt factor to account for the details in each limiting case; this factor tends to be proportional to the logarithm of d max /d min. Collisions at a given distance d will lead to emission only at ω v/d and so d max v/ω. The minimum impact parameter will vary depending on the average energy of the electrons. For low-energy electrons the minimum distance is approximately the distance for which the electron suffers a 90 deflection, giving a Gaunt factor of: G(T, ν) = 3 ( 2(kT ) 3/2 ) π ln ΓωZ i m 1/2 e 2 (2.26) where Γ is Euler s constant. In the case of higher energy electrons the minimum distance is related to the de Broglie wavelength where the Gaunt factor is: G(T, ν) = 3 ( ) 2kT π ln hν (2.27) At frequencies where hν kt the above two forms for the Gaunt factor do not apply and instead a quantum mechanical form of the Gaunt Factor must be used.

60 44 Figure 2.2. Schematic spectra of brightness temperature and flux density. The upper three curves are for gyrosynchrotron emission. The top is for highly relativistic electrons and the third for mildly-relativistic electrons (from Dulk (1985)) For thermal electrons the absorption coefficient is given by: κ ν i 1 3c ( ) 1/2 2 νp 2 4πZi 2 n i e 4 π 3 G(T, ν) π ν 2 m 1/2 (kt ) 3/ n e Z 2 ν 2 T 3/2 i n i ln( T 3/2 ν ) (T < K) ln( T ν ) (T > K) i (2.28) The emissivity j ν is then related to κ ν by Kirchhoff s law (equation 2.11). Optically thick bremsstrahlung shows a ν 2 dependence, while optically thin emission nearly independent of ν (see figure 2.2). In the case of a homogeneous optically thin magnetized source, the emission is polarized in the sense of the magnetic x-mode. If the same source is instead optically thick the emission is unpolarized.

61 Gyromagnetic Radiation For a magnetized plasma, accelerations resulting from particle collisions are often negligible in comparison to those due to gyration around the field lines. For nonrelativistic plasmas the emission is termed gyroresonance. In this case the population of electrons is usually thermal because the average energy of the electrons is low and leads to frequent collisions and generally to Maxwellian distributions. The emission is concentrated at the fundamental frequency ω ω B and directed mainly along the magnetic field, with a power-pattern P (θ) cos 2 (θ). For the opposite limit of highly relativistic electrons, there are few collisions and a non-maxwellian tail is generally dominate. The electron distribution in this case can be described as a power-law and the radiation is termed synchrotron emission. The emission is distributed over a broad range of frequencies near ω ν B γ 2 sin(θ) and directed strongly in the direction of the instantaneous electron motion. In the intermediate case of mildly relativistic electrons, the radiation is termed gyrosynchrotron emission and can be generated by either a power-law (non-thermal) or thermal distribution of electrons. In the case of fully convective stellar systems the electrons of the magnetized plasmas are usually mildly relativistic and their non-thermal emission is generally attributed to the gyrosynchrotron mechanism. However there are some cases of highly relativistic plasmas occurring for such systems and evidence of synchrotron emission has been observed (Phillips et al. 1996; Carrasco-Gonzalez et al. 2010). Thus in the following two sections, I will only go over the absorption and emission coefficients for thermal and power-law gyrosynchrotron emission.the derivation of the volume emis-

62 46 sion (η ν ) and linear absorption (κ ν ) coefficients for GS radiation for either power-law or thermal electron distributions involves integrals over infinite sums of Bessel functions, making this calculation difficult. Several approximation methods have been developed, however, allowing for a derivation of analytical expressions for the coefficients of both thermal and power-law GS (Trubnikov 1958; Dulk et al. 1979; Petrosian 1981; Robinson & Melrose 1984); these are given in the following two sections Gyrosynchrotron Emission from Thermal Plasma Analytical expressions for the radiation coefficients for thermal gyrosynchrotron were first derived by Petrosian (1981) and were quite accurate over a large range of harmonics and angles θ (angle between magnetic field vector and line of sight). These were further expanded to include higher harmonics by Robinson & Melrose (1984) resulting in the analytical expressions given in 2.29; these expressions have an accuracy better than 20% for ν/ν B >5, θ 10, and T 10 7 K. κ ν B N 2.67 µ2 (1 15/18µ) γ 10 9 o 3/2 (γo 2 1) 1/2 sin(θ) Tσ 2 ζ 2 o (ζ 2 o 1) s 3/2 o x 1/2 [ { c 2 ( s c /s o ) 1/3 + ( ) 1 βo 2 1/2 ( 1 β 2 o cos 2 (θ) ) } ] 1/2 2 β 2 + otσ 2 ζ o sin 4 (θ) 2(s o + s c ) ( 1 β 2 o cos 2 (θ) ) ( s c s o ) 1/6 Z 2so exp [ µ (γ o 1)] (2.29)

63 47 where µ = [ , γ o = 1 + 2ν ( 1 + 9x T µν B 2 ) 1/3 ] 1/2 β o = ( 1 1 ) 1/2, x = ν sin 2 (θ) γo 2 ν B µ T o = T 1 x [ = a + ( 1 + a 2) ] 1/2, a = ν B sin 2 (θ) ν 2 cos(θ) (2.30) s o = γ o ν ν B ( 1 β 2 o cos 2 (θ) ) ζ o = ( 1 β 2) 1/2, β = β o sin(θ) (1 β 2 o cos 2 (θ)) 1/2 s c = 3 2 ζ3 o, c 2 = T σ cos(θ) ( 1 β 2 o ), Z = β exp 1/ζo 1 + 1/ζ o Dulk et al. (1979) was able to further simplify the radiation coefficients to have the form of a power-law in frequency and are given below. The range of validity for these expressions is 10 8 K T 10 9 K. Figure 2.3 shows comparison between the numerically calculated values (solid) and these power-law fits (dashed). In the case of thermal gyrosynchrotron emission these power-law expressions accurately represent the numerical values for a small range of frequencies. κ ν B N 50 T 7 sin 6 (θ) B 10 ν 10 (2.31)

64 48 j ν BN T ( ν ν B ) 2 κ ν B N (2.32) Additionally, expressions for the frequency of maximum flux density (ν peak ) and degree of circular polarization (r c ) have been derived: r c 13.1 T cos(θ) cos2 (θ) ( ν ν B ) cos(θ) (τ ν << 1) (2.33) ν peak 1.4 ( ) NL 0.1 B (sin(θ)) 0.6 T 0.7 B (10 8 < T < 10 9 K) 475 ( ) NL 0.05 B (sin(θ)) 0.6 T 0.5 B (10 7 < T < 10 8 K) (2.34) For the case of thermal electrons, ν peak is the frequency at which τ 2.5. This frequency depends sensitively on the magnetic field strength and temperature, making it is a good diagnostic for B if the temperature is independently known Gyrosynchrotron Emission from Power-Law Electrons Now consider an electron distribution that is isotropic in pitch angle and power-law in energy: n(e) = KE δ (2.35) Here K is related to N, the number of electrons per cubic centimeter with E > E o by the relation: K = (δ 1)E δ 1 O N (2.36)

65 49 νb/n Thermal Gyrosynchrotron ν B/N for θ=40 T=10 8 K T= K T=10 9 K ν/ν B Power-Law Gyrosynchrotron ν B/N for θ=40 δ=2.0 δ=2.5 δ= νb/n ν/ν B Figure 2.3. The spectral energy distribution for thermal (top) and power-law (bottom) gyrosynchrotron emission calculated from the numerical (solid line) and the simplified powerlaw expressions (dashed line). Its clear for thermal emission that there is a very small range of frequencies that the simplified expressions accurately represent the numerically calculated values. The simpler expressions for gyrosynchrotron emission generated from a power-law distribution of electrons are much better representations of the numerically calculated values. The emission and absorption coefficients for this distribution of electrons in

66 50 the presence of a magnetic field are given by: j ν /BN ( = 2πc e sin(θ) νb ) 2 (δ 1) 3 8 ν 1 + T κ ν B/N 2 (1 + 1γo ) { ( 1 + T cot(θ) ) } 2 2T 2 + 3(a + b + 2) γ o (2.37) ( ) 1 δ γo 1 Z 2s ɛ 1 δ+2.0 ɛ ( γo 1 ) 1 γ o ɛ 1+γ o where a + b = { δ for j(ν, θ) δ for κ(ν, θ) ( ) 1/2 4ν γ o = 3ν B (a + b) sin(θ) β o = ( 1 1 ) 1/2, x = ν sin 2 (θ) γo 2 ν B µ T o = T 1 x [ = a + ( 1 + a 2) ] 1/2, a = ν B sin 2 (θ) ν 2 cos(θ) (2.38) s o = γ o ν ν B ( 1 β 2 o cos 2 (θ) ) ζ o = ( γ 2 o sin 2 (θ) + cos 2 (θ) ) 1/2 γo sin(θ) Z = β exp 1/ζo 1 + 1/ζ o

67 51 In figure 2.3 the quantities κ ν B/N, and r c are calculated numerically for the X-mode (solid). The slopes of the curves approximately constant for ν/ν B 10, so like the thermal case, the spectra can be approximated by power-law expressions (dashed): κ ν B N ( ) δ ν δ (sin(θ)) δ (2.39) ν B ( ) δ j ν ν BN δ (sin(θ)) δ (2.40) ν B These power-law expressions fit the numerically calculated values much better than for thermal gyrosynchrotron, however the fit does deviate at high and low frequencies. Thus in my own research I use only the more complicated numerical expressions to model the observed radio emission. The fractional polarization and peak frequency are given by: ( ) cos(θ) ν r c δ cos(θ) (2.41) ν B ν peak δ (sin(θ)) δ (NL) δ B δ (2.42) Gyrosynchrotron Emission Model Using the absorption and emission coefficients derived by Robinson & Melrose (1984), we can calculate the expected gyrosynchrotron radio emission from a low mass stellar corona. Comparisons between the expected spectral energy distribution (SED) and polarization of the calculated radio emission and that of the observed

68 52 radio emission place constraints on the stellar corona field strength and non-thermal electron density. Using radio observations of the quiescent emission from two UCDs, I develop a power-law gyrosynchrotron emission model to constrain the magnetospheric properties of these two objects (see chapter 5). In the emission model developed during this thesis, I assumed the corona to be an isothermal plasma with temperature T c = 10 6 K and radially symmetric electron density of the form ( ) 2 r n e = n eo (2.43) R s where R s = 0.1R and we are constraining n eo. The energy distribution of these electrons is a power-law with index δ that is related to the SED index by α = δ. I also assume the magnetic field to be dipolar with an initial surface field of B 0 and angle between the magnetic axis and line of sight (LOS) of θ B. The orientation of the magnetic field for the two UCD sources is constrained through emission models of the pulsed radio emission observed from these sources (see section 5.6.1). Using this inclination angle, we can use the developed gyrosynchrotron emission model to constrain B 0. To calculate the specific intensity, I integrate the radiative transfer equation along LOS through the coronal plasma for each plasma mode (O and X) S(ν) = 2 π 0 0 j ν (b, φ) ( ) 1 e τ ν(b,φ) b db dφ (2.44) κ ν (b, φ) where b is the radial distance from the center of the star to each LOS and φ is the azimuthal angle measured in a plane containing the star center and normal to the direction to the observer. This integral is approximated by summing numerically over

69 53 a 2-dimensional grid of LOS spaced by db = 0.05R s and dφ=5 from the star center (0.1R s ) to 3 stellar radii. This requires first calculating the optical depth for each LOS in this grid. The absorption coefficient derived by Robinson & Melrose (1984) is dependent on the local angle between the LOS and the local magnetic field vector. Thus for each point along each LOS I calculate this angle and the corresponding absorption coefficient. The optical depth then is then calculated by approximating the integral τ ν (b, φ) = κ ν (b, φ, s)ds (2.45) S where s is the distance along the LOS, by summing over a set of dκ ν = κ ν (b, φ, s)ds points spaced by distance ds = 0.1R s from -3R s to 3R s. After determining the optical depth along each LOS in the 2-dimensional grid of (b, φ)-points, I sum over the contribution ds ν (b, φ) for each point. This sum is carried out for both X and O-mode and the total flux density (Stokes I) is then the sum of these fluxes. Since the Faraday depth on all LOS with significant flux contribution is large for a typical low mass stellar corona (see section 5.1), the emergent fractional linear polarization is negligible. Hence the total polarization measured from these sources is circular (Stokes V ), given by the difference between the X and O-mode fluxes. A simplified version of this model was used in the UCD section of this thesis because the model above does not account for the transfer of Stokes V flux to Stokes U and Q flux. Incorporating such propagation of the polarization vector was beyond the scope of that paper (see section 5.1). A simplified version of the emission model

70 54 above was used where instead of calculating the local angle between the LOS and the local magnetic field vector, we assumed a constant, weighted angle. To determine this angle for each LOS, we first noted that the observed SED indicates that we are in the optically thin regime. Thus the total flux from the source will be proportional to B 1.5 N e. We take this factor to be the weight for each point along each LOS and calculate a weighted angle to be assumed as the constant orientation of the magnetic field. Additionally, we assume that the magnetic field strength and electron density is constant along each LOS, using the values of B and N e that gave the greatest weight as the constant values. Since we expect that only a fraction of the total source extent along each LOS to contribute significantly to the total flux, I take the emitting region size to be the half-maximum width of the weighted angle distribution. Once the field strength, density, and orientation are assumed, we can then approximate each LOS by a rectangle with a length equal to the half-maximum width, height of db and width of dφ. The total flux is then the sum over the contribution from each rectangle. The results of this model for the two UCDs are given in section and while the SED of the total flux is well matched, the expected polarization is less than that measured. However, this discrepancy is most likely due to pulsed radio emission contamination rather than inaccuracies in this model.

71 55 CHAPTER 3 TECHNICAL INFORMATION My doctoral thesis research involves analyzing several radio observations from Karl G. Janksy Very Large Array (VLA). The VLA is a 27-element array arranged in a Y-shape and is located on the plains of San Agustin in southwestern New Mexico. It is managed from the Pete V. Domenici Science Operations Center (SOC) in Socorro, New Mexico by the National Radio Astronomy Observatory (NRAO). The angular resolution (θ) of an interferometer is related to both the observing frequency (λ) and the longest distance between any two antennas in the array (D), where θ = λ/d. Each of the radio antennas in the VLA are located on rails and can be moved. This functionality allows the VLA to vary its resolution over a range exceeding a factor 50. All of the VLA antennas are outfitted with eight receivers providing continuous frequency coverage from 1-50 GHz. In the future, the antennas are going to be outfitted with additional low frequency receivers to cover frequencies 479 MHz. In the following section I give an overview of the basics behind radio interferometry and producing calibrated images to use for analysis. The material presented in these sections is from Wilson et al. (2009) and Taylor et al. (1999). 3.1 The Response of an Interferometer The most basic interferometer is a pair of radio telescopes whose voltage outputs are correlated. Large arrays of radio telescopes with N 2 elements can be treated as N(N-1)/2 independent interferometer pairs. Thus the results of a two-

72 56 element interferometer can be adopted to understand larger arrays. A simple block diagram of a two-element interferometer is given in figure 3.1. The two antennas point towards a radio source whose direction is indicated by unit vector s. These antennas are separated by a distance b called the baseline. The wavefront from the source reaches one antenna at a time τ g later than the other; τ g is called the geometrical delay and is given by: τ g = b s/c (3.1) where c is the speed of light. The signal received by each antenna passes through amplifiers which incorporate filters to select the required frequency band of width ν centered on ν. The correlator combines the signal via a voltage multiplier followed by a time averaging circuit. If the voltages induced in the antennas are V 1 and V 2 the output from the correlator is proportional to V 1 (t)v 2 (t), where the angular brackets denote time averaging. The received signals can be represented by quasi-monochromatic Fourier components of frequency ν and have the form V 1 = v 1 cos(2πν(t τ g )) and V 2 = v 2 cos(2πνt). The correlated signal is then r(τ g ) = v 1 v 2 cos(2πντ g ) (3.2) From equation 3.2 it is clear that the output of the correlator varies periodically with τ g. If the orientation of the baseline and the wave propagation direction remain constant, the correlated signal will also be a constant. However, due to the rotation of the Earth, τ g varies slowly. The resulting oscillation of the cosine term in equation 3.2 represents the motion of the source through the interferometer fringe pattern.

73 57 Figure 3.1. et al. (1999). Simplified schematic diagram of a two-element interferometer; from Taylor Note that v 1 v 2, which represents the fringe pattern amplitude, is proportional to the power received by the antennas. The interferometer output can be written in terms of measurable quantities. The radio brightness in the direction of s at frequency ν is represented by I ν (s). The signal power received in bandwidth ν from the source element dω is A(s)I ν (s) νdω, where A(s) is the effective collecting area in the direction s. The correlator output is then dr = A(s)I ν (s) νdω cos(2πντ g ) (3.3)

74 58 Figure 3.2. Position vectors used to derive the interferometer response due to a source. The source is represented by radio contours of brightness I ν (s); from Taylor et al. (1999). In terms of the baseline and source position vector the correlated signal is R(b) = ν S ( ) 2πν b s A(s)I ν (s) cos dω (3.4) c The integral in equation 3.4 is taken over the entire surface S of the celestial sphere but in practice the integrand usually falls to very low values outside a small angular field as a result of the antenna beam width and the finite dimensions of the radio source. We assume that the bandwidth is sufficiently small so that variation of A and I can be ignored. Furthermore, in deriving equation 3.4 we must also make the following assumptions: (1) that the source is in the far field of the interferometer so that the incoming wavefronts can be considered to be in a plane and (2) that the source is spatially incoherent so that the responses from different points in the source can be added independently.

75 Aperture Synthesis Aperture synthesis is a method used to determine the intensity distribution I ν (s) of a part of the sky from the measured visibility function R(b). It requires inverting the integral equation 3.4 via a Fourier transform. To carry out this solution it is convenient to introduce a new coordinate system for s and b. In this coordinate system the image center is chosen to be the position of zero phase and is called the phase tracking center or the phase reference position. We can represent this position by the vector s 0 as shown in figure 3.2 and write s = s 0 + σ. With substitution into equation 3.4 we have ( ) 2πνb s0 R(b) = ν cos c ( ) 2πνb s0 ν sin c S S ( ) 2πν b σ A(σ)I ν (σ) cos dω c ( ) 2πν b σ A(σ)I ν (σ) sin dω c (3.5) The integrals in equation 3.5 makeup the the complex visibility of the source. This quantity is the un-normalized measure of the coherence of the source field, modified by the characteristics of the interferometer. The visibility is a complex quantity whose magnitude has the dimensions of spectral power flux density and is defined as, V = V e iφ V = A(σ)I ν (σ)e 2πib σ/c dω (3.6) S where A(σ)=A(σ)/A 0 is the normalized antenna reception pattern, A 0 being the response at the beam center. Splitting the real and imaginary parts of V, A 0 V cos(φ V ) = S ( ) 2πν b σ A(σ)I ν (σ) cos dω (3.7) c

76 A 0 V sin(φ V ) = A(σ)I ν (σ) sin S ( 2πν b σ and substituting back into equation 3.5, the output from the correlator is c 60 ) dω (3.8) ( ) 2πν b s0 R(b) = A 0 ν V cos φ V c (3.9) The general procedure for interpreting interferometer measurements is to first measure the amplitude and phase of the interferometer fringe pattern. Then, using appropriate calibration, the amplitude and phase of V is derived. The brightness distribution of the source is then obtained from the visibility data by inversion of the transformation in equation 3.6. This procedure requires V to be measured over sufficiently wide range of νb σ/c, which is the component of the baseline normal to the direction of the source. 3.2 Radio Data Reduction An array of antennas samples the true visibility function V ij at many discrete locations; here i and j indicate the pair of antennas involved. The data from each antenna pair are then recorded and the resulting set of numbers is called the observed viability data, Ṽij. The recorded data are in general different from the desired visibility and the purpose of calibration is to recover the true visibility. For the majority of my doctoral thesis research I used NRAO s Common Astronomy Software Application (CASA) package to reduce radio data Data Editing Editing the data involves identifying and discarding discrepant and severely corrupted data. This process is commonly referred to as flagging the data. Data

77 61 editing is considered part of the data calibration procedure in the sense that there may be periods of time when the complex gains cannot be determined accurately for some or all the antenna pairs. The data corresponding to these times should be removed. Examples of problems that may require flagging the affected data include interference from terrestrial sources, antenna tracking inaccuracies, inclement weather, malfunctioning receivers, incorrect observing parameters and data recording errors. I used the CASA task flagdata to remove corrupted data in my observations of DG Tau and the UCD sources. This task has several modes that allow a spectrum of automated flagging. In manual mode, this task allows the user to identify the discrepant data in several ways, including by source, frequency, time period, polarization, or antenna. Modes like quack, elevation, and clip allow the user to specify certain ranges of the data, such as times, elevations or weights, to remove. Modes like rflag and tfcrop remove outliers from the data based on user supplied thresholds. Initial editing of the VLA observations for my thesis involved using the observer log, which lists known problems with the VLA antennas during the observation, to identify possible bad data and then flag it using f lagdata. Further flagging was carried out using the quack mode of flag data to remove the first 10 seconds of each observation scan during which the antennas are settling down. For low-frequency data known to be severely affected by radio frequency interference (RFI) the rflag mode of flagdata was very useful. The flagdata task in the rflag mode becomes an auto-flag algorithm based on a sliding window statistical filter. In this mode, the data

78 62 is iterated-through in chunks of time, the local statistics are calculated, and flags are applied based on the user supplied thresholds. To further identify bad visibilities, I used the plotms GUI in CASA. Combinations of colorize and plot-axes allow for easy and thorough searches for corrupted data in CASA. Using the display-colorize feature, plotms is able to effectively display a third dimension of your dataset on a 2D plot. For example in figure 3.3 I plot amplitude versus frequency and use colorize to represent scan number, in other words time, for the gain calibrator during my 2012 X-band observation of DG Tau. This plot then gives both spectral and temporal information about the data. It is obvious from figure 3.3, that there is significant RFI between GHz for this source, all of which occurs in the same observation scan. RFI also occurs between GHz and GHz but in different observation scans. Using the information from this window we can now identify and flag this corrupted data using flagdata in manual mode Calibration Formalism The relationship between V ij and Ṽij can be arbitrary, however, most arrays are linear devices so that the output is a linear function of the input. Additionally, the response associated with one antenna pair does not depend on the response of any other antenna pair. The basic calibration formula is then written as Ṽ ij (t) = G ij (t)v ij (t) + ɛ ij (t) + η ij (t) (3.10) where t is the time of the observation, G ij (t) is the baseline-based complex gain, ɛ ij (t) is the baseline-based complex offset, and η ij (t) is the stochastic complex noise.

79 63 Figure 3.3. Example display of uncalibrated and unedited data using the plotms GUI application in CASA. Shown is the visibility amplitudes as a function of frequency for J during my 2012 X-band observations of DG Tau. The color represents the observation scan number (or time), and so this plot shows not only spectral information but also temporal information about the data. Most data corruption occurs before the signal pairs are correlated. Thus the baseline-based complex gain can be approximated by G ij (t) = g i (t)g j (t)g ij (t) (3.11) where g i (t) is the antenna-based complex gain for antenna i and g ij (t) is the residual baseline-based complex gain. If the baseline gain can be perfectly factored into a product of the antenna gains then g ij (t)=1. This residual for well-designed systems

80 64 is within one percent of unity. There are several advantages of factoring the based-line based gain factors into antenna-based gain factors. First, most of the variation in the instrument are related to particular antennas and the errors due to a well-designed correlator are always much smaller. Secondly, for an array of N elements there are N(N-1)/2 different baselines. So for N 4, the number of baselines is far larger than the number of elements. Thus the computer capacity needed to store the calibration parameters and apply them to the observed visibilities is reduced when using antenna-based relations. Lastly, the antenna-based calibrations can be determined without the full set of baseline data. This is especially important in the case of a partially resolved calibrator sources where full calibration solutions can only be determined by approximating the calibrator as a point source over a limited range of baseline lengths, reducing the number of antennas usable for calibration. We can separate G ij into an amplitude and phase A ij (t) = a i (t)a j (t)a ij (t) (3.12) Φ ij (t) = φ i (t) φ j (t) + φ ij (t) (3.13) and represent the true visibility as V ij = A ij e iϕ ij and the observed visibility as Ṽij = à ij e i ϕ ij. For a point-source calibrator of flux density S we have A ij =S and ϕ ij =0. The calibration equations become à ij = a i a j a ij S (3.14)

81 65 and ϕ ij = φ i φ j + φ ij (3.15) Provided that the baseline gain g ij is close to unity, these two equations can be solved for a i and φ i for all N antennas. In order to solve equations 3.14 and 3.15 observations of a point-like source with an accurately known position and location near the target source are required. Observations of this calibrator and the target source are usually made alternately where the frequency with which the switch occurs depends on the stability of the electronics and weather. For millimeter and sub millimeter wavelengths calibrator observations need to be made every few minutes since atmospheric effects are more prevalent at these wavelengths. Additionally, observations of a thermal calibrator are required to calibrate the response in units of flux density or brightness temperature. For the VLA there are several primary calibrators (3C286, 3C48 and 3C147) that are well observed and used to set the flux density for observations with this array. For the new wideband system of the VLA calibration of the receiver passband is also required. For continuum observations this calibration step is not that important and usually the flux density calibrator is strong enough to also use it as the passband calibrator. 3.3 Coordinate Systems for Imaging We can define the baseline vector to have components (u, v, w) measured in units of wavelength at the centered frequency of the signal band, where w points in the direction of s 0 towards the source, u points in the direction towards east, and

82 66 v points in the direction towards north. The positions on the sky are defined in l and m which are direction cosines with respect to the u and v axes. A image in the (l, m) plane represents a projection of the celestial sphere onto a tangent plane at the (l,m) origin (see figure 3.4). In these coordinates the parameters used to derive the interferometer response in terms of visibility become, νb s c = ul + vm + wn dω = νb s 0 = w c dl dm n = dl dm 1 l2 m 2 and the complex visibility in this set of coordinates is then V (u, v, w) = A(l, m)i(l, m)e 2πi[ul+vm+w 1 l 2 m 2 1] dl dm (3.16) 1 l2 m 2 where the integrand is zero outside the full width of the primary beam, in other words for l 2 +m 2 1. Equation 3.16 can be reduced to the form of a two-dimensional Fourier transform by assuming we are mapping a small region of the sky so that l and m are small. The dependence of the visibility upon w is then small and can be omitted. Additionally if l and m are small, 1 l 2 m 2 1. Such a reduction makes inverting the visibility to solve for I(l, m) simpler. The visibility in this case becomes V (u, v) = A(l, m)i(l, m)e 2πi[ul+vm+] dl dm (3.17) 3.4 Imaging and Deconvolution To invert equation 3.17 a fast Fourier transform algorithm is generally used. This requires an interpolation of the visibility function onto a regular rectangular grid.

83 67 Figure 3.4. Position vectors used to derive the interferometer response due to a source. The source is represented by radio contours of brightness I ν (s); from Taylor et al. (1999). If the measured points are randomly distributed, this interpolation is best carried out using a convolution scheme. For V (u, v) sampled at M different points (u k, v k ) in the uv-plane, the result of the Fourier transform is the convolution of the true sky brightness I ν with the point spread function, B(l, m), I D ν = I ν B (3.18) where I D ν is known as the dirty image and B(l, m) = S(u, v)e 2πi(ul+vm) dudv (3.19)

84 68 B(l,m) S(u,v) Figure 3.5. The dirty beam is the Fourier transform of the sampling function of the interferometer. This is the dirty beam (lef t) and sampling due to the antenna tracks (right) for the 2012 X-band observation of DG Tau. is the Fourier transform of the sampling function, also known as the dirty beam. So then Iν D (l, m) = S(u, v)v ν (u, v)e 2πi(ul+vm) dudv (3.20) The sampling function can be approximated by a two-dimensional Dirac delta function M S(u, v) = δ(u u k, v v k ) (3.21) k=1 which is 1 wherever the uv-plane was sampled by the interferometer and 0 elsewhere. The sampling function in the case of the interferometer is the tracks traced out by the antennas as the Earth rotates. From the top right of figure 3.5 it is clear that there is incomplete sampling of the uv-plane by the interferometer.to offset the high concentration of (u, v) tracks near the center and to simulate more uniform coverage a weighted sampling function

85 69 can be used, M W (u, v) = g(u k, v k )δ(u u k, v v k ) (3.22) k=1 where g(u k, v k ) is the weighting function. The weighting function can be used to change the effective beam shape and noise level in the resulting image. There are two commonly used weighting functions: g(u k, v k ) N = 1/σ 2 k (3.23) is called natural weighting where σ k is the rms noise on visibility k. Natural weighting gives the best signal-to-noise ratio for detecting weak sources. However, since the inner regions of the uv-plane are more sampled than the outer regions, natural weighting emphasizes the data from the short spaces and produces a beam lower angular resolution. The other common weighting is uniform weighting: g(u k, v k ) U = 1/N s (k) (3.24) where N s (k) is the number of data points within a symmetric region of the (u, v) plane of characteristic width s centered on the k th data point. The default value of s is usually a function of the physical image dimensions and so by changing the image or pixel size during deconvolution you also change the uniform density weights. Uniform weighting decreases the beam width but increases the rms image noise. There is not a unique solution to the convolution equation If spatial frequencies present in the true brightness distribution, I ν, are not sampled, then changing the amplitudes of the corresponding visibilities will have no effect on the

86 70 reconstructed intensity distribution. Essential the dirty beam filters out these unsampled spatial frequencies. Mathematically, this amounts to the following. Let Z be an intensity distribution that contains only the unmeasured spatial frequencies. Then B Z=0. So then if I is a solution of the convolution equation, so too is I + αz where α is any number. Thus the existence of homogenous solutions implies the general non-uniqueness of any solution in the absence of boundary conditions. To carry out image reconstruction requires non-linear techniques since a linear solution I = A D, where A is some matrix, do not include the homogeneous solutions Z. The solution Z = 0 is called the principal solution and differs from the true intensity distribution by some unknown invisible distribution or ghosts. The aim of image reconstruction is then to obtain reasonable approximations to these ghosts by using additional knowledge or plausible extrapolations. However, there is no direct way to select the best or correct image from all possible images. There are several methods to carry out the deconvolution process however for my thesis I only used the CLEAN method. This method approximates the actual but unknown intensity distribution I(x, y) by the superposition of a finite number of point sources with positive intensity A i placed at positions (x i,y i ). The aim of CLEAN is to determine the A i (x i, y i ) such that I D (x, y) = i A i B(x x i, y y i ) + I ɛ (x, y) (3.25) where I ɛ (x, y) is the residual brightness distribution after decomposition and should be on the order of the noise in the measured intensities. This decomposition cannot be done analytically. Instead an iterative technique must be used which functions as

87 71 follows: First the peak intensity of the dirty image is identified. A fraction γ of this peak intensity with the shape of the dirty beam is subtracted from the image. This is repeated n times until the intensities of the remaining peaks are below some limit. The resulting point source model is convolved with a restoring beam that is usually a Gaussian fitted to the central peak of the point spread function of the dirty beam and then added back to the residual image. In CASA both the Fourier transform of the visibility data and deconvolution using the CLEAN method are done using the clean task. The size of the grid that the visibilities are interpolated onto is set by the parameter imsize and the number of pixels along each side of the grid is set by cell. The most effective CLEANing occurs with 3-5 pixels across the synthesize beam whose angular size is given by θ F W HP = λ/d (3.26) where λ is the observing wavelength and D is the maximum baseline distance. The weighting to be used in deconvolution can be set in the clean task using the weighting parameter. In addition to natural and uniform weighting, a hybrid form of the two is also available in the form of the Briggs robustness parameter, set in clean using the parameter robust. The values for this parameter range from -2.0 (close to uniform) to 2.0 (close to natural). For the majority of my thesis research I used natural weighting to maximize the signal-to-noise in the images, however, uniform weighting was used to study the radio knots in DG Tau s extended jet (see section 4.6.3). To carry out the iterative CLEAN process, you have the option of running

88 72 (A) Dirty Map (B) First Clean Cycle (D) Third Clean Cycle (C) Second Clean Cycle (E) Clean Map Figure 3.6. An example of an interactive CLEAN of 2012 X-band observation of DG Tau. This process starts with a dirty image, where the user is allowed to select clean regions using a GUI window (B). The CLEAN task then goes through n iterations of subtracting the bright emission before allowing the user to edit the clean regions again (C & D). This process continues until a user specified stop point and produces a CLEAN map (E).

89 73 the clean task interactively or not using the interactive parameter. To produce an initial CLEAN map of my observations, I would always first run clean interactively to see where the bright emission might be located. An example of an interactive CLEAN of my 2012 X-band DG Tau data is shown in figure 3.6. In both interactive and non-interactive you start with the dirty image (figure 3.6 A), which is the source intensity distribution convolved with the dirty beam. For an interactive clean, a GUI window will appear showing the residual map of the observation and allow you to mark clean regions of known bright emission (figure 3.6 B). Then the clean task runs through n iterations, set by the niter parameter, of subtracting a fraction of the bright emission in the clean regions from the residual map. After n iterations the GUI window appears again with the new residual map and allows the user to edit the clean regions if needed (figure 3.6 C & D). This process is carried out until either the user decides the bright regions visually appear to be approaching the noise level or until the peak flux in the image is less than the threshold you set using the threshold parameter. During each interactive iteration you have the option of carrying out n more iterations before seeing a new version of the residual map, or you can carry out the rest of the CLEAN process non-interactively. In the case of a non-interactive CLEAN, you can use the parameter mask to set the clean regions to be used throughout the CLEAN process. This process will stop when it reaches the point where the remaining peak emission is less than the value set by the parameter threshold. The fraction of the peak intensity subtracted from the image throughout both the interactive and non-interactive CLEAN process

90 74 is set by the gain parameter. During my thesis research I found using the default value of 0.1 for the gain to be sufficient. For the threshold value I usually set this to the expected thermal noise in the map, which I estimated using the online VLA exposure calculator.

91 75 CHAPTER 4 RADIO OBSERVATIONS OF CLASSICAL T TAURI THERMAL JETS 4.1 Preface The evolution of young low mass stars onto the main sequence involves both accretion and the generation of outflows. It is generally accepted that magnetic fields play in important role in both these process. Yet the generation of these fields and the exact details of how they interact with the surround plasma are not well known. In an effort to better understand the role fields play in the launching and collimation of young star outflows, I conducted an observational campaign of the highly active CTTS DG Tau. This project involved several collaborators, where I played a central role in setting up the radio observations, reducing and analyzing these observations, as well as writing the manuscript for publication. The results of this campaign were published in 2013 (Lynch et al. 2013) and are presented in this following chapter. The initial motivation for a radio study of DG Tau was the discovery of an extended X-ray jet and two X-ray components close to the optical position of the star. The X-ray emission close to DG Tau manifests itself as superposition of two distinct spectral components that are subject two different hydrogen absorption column densities (figure 4.1). The hard component is thought to be associated with coronal magnetosphere and is strongly absorbed by dust-depleted accretion streams. The weaker absorbed soft component is thought to be emission from the jet base (Güdel et al. 2005; Güdel et al. 2007a). Chandra observations of DG Tau between 2006 and

92 showed that this inner soft component had not changed its position. If it is related to shocked, we are observing a prominent standing shock possibly related to the collimation region at the base of the jet (Güdel et al. 2011). The purpose of the radio observations was to identify shock structures of the inner jet that would be associated with the two X-ray spectral components. This would involve seeking possible circular polarization and non-thermal spectra and thereby potentially uncover magnetic fields around the star and in the jet. Although DG Tau has several previous radio observations, the recent VLA upgrade provided more sensitive high angular resolution observations ( 10 increase in signal-to-noise) of the inner region of the jet. This significant increase in the sensitively is important since these sources are radio faint and any circularly polarized emission would be difficult to detect even with this upgrade. Our 2011 observation of DG Tau at 8.5 GHz showed extended radio emission in the direction of the larger X-ray and optical jet, which had a thermal spectrum and no polarization ( 3%). The elongated nature of the radio emission combined with the thermal spectrum and lack of polarization suggests that the radio emission traces a collimated thermal outflow. Adopting this interpretation, we initially used the expressions derived by Reynolds (1986) for an isothermal, fully ionized, constant velocity thermal jet to constrain both the half opening angle and mass-loss rate of the ionized material in the jet of DG Tau. The mass-loss rate is given by: Ṁ = M yr 1 [V ] 100 S 3/4 mjy ν 9/20 GHz d 1.5 kpc θ 3/4 s T sin(i) 1/4 (4.1)

93 77 Figure 4.1. The two-absorber X-ray spectrum of DG Tau. Comprised of two components with different absorption column densities and offset by a measured 0.2 in the direction of the jet(from Güdel & Nazé (2009)) where V 100 is the jet velocity in 100 km s 1, S mjy is the flux density in mjy, ν GHz is the observing frequency (GHz), d kpc is the distance to the source (kpc), θ s is the opening half-angle in radians, T 4 is the gas temperature in units of 10 4 K, and i is the jet inclination angle to the line of sight. The distance along the jet to the unity optical depth (τ=1) surface is given by: R τ = cm [Ṁ 2/3 6 V 100 2/3 ν GHz 0.7 T θ s 1 sin(i) 1/3 ] (4.2) where Ṁ6 is the mass loss rate in 10 6 M yr 1. If both the flux density and R τ (length of the jet to the τ = 1 surface) are determined from observations, the only remaining free parameters are the mass-loss rate and the jet opening angle. We use the epoch GHz observation to fix these parameters viz. total flux density S = 1.1 mjy, and the de-projected distance

94 78 Figure 4.2. Constraining mass loss and opening angle using Reynolds (1986). The massloss rate is given as a function of the opening angle for both a fixed flux density of 1.1 mjy (blue solid line) and a fixed deprojected jet distance to tau = 1 surface R τ = 47 AU (green solid line). In addition we use the size of the jet at its error bounds (±20 AU) to plot the mass-loss rate as a function of opening angle for fixed a fixed jet size plus and minus the error in the size estimate (dashed lines); shaded region denotes the range of values possible for the mass-loss rate given this error. to the τ=1 surface from the sourcer τ = 47 ± 20 AU (Figure 4.3). Figure 4.2 shows the allowed values of mass-loss as a function of opening angle for these values of fixed flux density (solid blue line) and fixed R τ (solid green line, with dashed green lines encompassing the uncertainty in R τ ). The intersection of the solid lines provide an estimate of opening angle and the mass-loss rate, while the intersection of the dashed lines with the blue line map the uncertainty in these values. We only consider the uncertainty in the distance to the τ=1 surface because it

95 79 is much greater than that of the flux density and thus dominates the total uncertainty in both the mass-loss rate and the half opening angle. The allowed mass-loss rate and opening angle ranges are Ṁ=(2+3 1) 10 9 M yr 1, and θ=(3 +6 2) respectively. Recent mass loss of the DG Tau jet from recent optical line emission is Ṁ = (1 3) 10 8 M yr 1 (Agra-Amboage et al. 2011; Maurri et al. 2014). This estimate is an order of magnitude higher than the one given using the expressions from Reynolds (1986). This discrepancy is most likely due to the assumption of a uniform density and velocity profile across the jet. We know from optical line measurements that the density and velocity vary across the jet, where fast, dense gas is found towards the center of the jet and both the velocity and density decrease towards the edges of the jet. Using the results of Maurri et al. (2014) we modeled the variation of the density and velocity across the jet and calculate the mass loss of the ionized component using these modeled profiles. This calculation gives a mass loss Ṁ M yr 1, which roughly agrees with the optical results. More details about the velocity and density profile models and the resulting calculation of the mass loss are given in sections and While initially we were only interested in characterizing radio emission of the inner jet, we were motivated by the results of Rodríguez et al. (2012) to look for features in the extended jet. In a 2009 VLA observation of DG Tau at 8.5 GHz, Rodríguez et al. (2012) detected shock features located 7 along the extended jet. In our 2011 observations we made no detection of these features and could not determine from this observation alone whether these features actually disappeared by

96 or were over-resolved in this higher resolution observation. Motivated by this discrepancy we conducted two observations in 2012 using a less extended array at 4.5 GHz and 8.5 GHz for lower angular resolution. We detected the 7 knot features in the 2012 observations. Additionally, from these two observations we found that the shock features had significantly decreased in radio flux between 2009 and From comparisons between the angular size and radio flux from the shock features in 2009 and 2012 we are able to model the evolution of these features and predict that these features will soon be undetectable by the VLA; more details about this analysis are given in section 4.6.3

97 81 VLA Observations of DG Tau s Radio Jet: A highly collimated thermal outflow C. Lynch, 1 R. L. Mutel, 1 M. Güdel, 2 T. Ray, 3 S. L. Skinner, 4 P. C. Schneider 5, K. G. Gayley 1 1 Department of Physics and Astronomy, University of Iowa, Iowa 52240, USA 2 Department of Astrophysics, University of Vienna, Vienna, AT 3 Dublin Institute for Advanced Studies, Astronomy and Astrophysics Section, Dublin 2, Ireland 4 Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, Colorado 80309, USA 5 Hamburger Sternwarte, Gojenbergsweg 112, Hamburg, Germany 4.2 Abstract The active young protostar DG Tau has an extended jet that has been well studied at radio, optical, and X-ray wavelengths. We report sensitive new VLA fullpolarization observations of the core and jet between 5 GHz and 8 GHz. Our high angular resolution observation at 8 GHz clearly shows an unpolarized inner jet with a size 42 AU (0.35 ) extending along a position angle similar to the optical-x ray outer jet. Using our nearly coeval 2012 VLA observations, we find a spectral index α = ± 0.05, which combined with the lack of polarization, is consistent with bremsstrahlung (free-free) emission, with no evidence for a non-thermal coronal component. By identifying the end of the radio jet as the optical depth unity surface, and calculating the resulting emission measure, we find our radio results are in agreement with previous optical line studies of electron density and consequent mass-loss rate. We also detect a weak radio knot at 5 GHz located 7 from the base of the jet, coin-

98 82 cident with the inner radio knot detected by Rodríguez et al. (2012) in 2009 but at lower surface brightness. We interpret this as due to expansion of post-shock ionized gas in the three years between observations. 4.3 Introduction The evolution of a young stellar object (YSO) involves not only mass accretion, through a circumstellar molecular disk, but also the loss of angular momentum and mass flux through a narrowly collimated jet, thought to be launched close to the YSO. Magnetic fields are suspected of playing an important role in both the launching and collimation of these jets (Pudritz et al. 2012; Cai et al. 2008). However, the processes through which these fields act are not well understood (e.g. Carrasco-Gonzalez et al. 2010). Classical T Tauri stars (CTTS) are YSOs which have evacuated most of their remnant circumstellar envelope. This allows an unobscured line of sight to the central 100 AUs of their jet and measurements of forbidden line emission are possible (Dougados et al. 2000). Using the spectroscopic diagnostic technique described by Bacciotti et al. (1999), the forbidden-line emission can provide constraints for the electron density, total hydrogen density, ionization fraction, and the average excitation temperature of the gas in these jets. Consequently, probing the inner wind structure of CTTS is crucial in understanding the mechanisms by which these YSO jets are launched and collimated. Some CTTS have associated parsec-scale jets, with Herbig-Haro (HH) objects forming where the fast streams of material collide with the slower material along

99 83 the jet (McGroarty et al. 2007; McGroarty & Ray 2004). These jets are dynamical, evolving on the timescales of a few years, and contain knots which have proper motions in the range of a few 100 km s 1. In addition, about 10 have been discovered to have X-ray emission. These X-rays most likely trace the fastest shocks in the jets of YSOs. If the material in the YSO jet is heated to X-ray emitting temperatures by shocks, shock velocities around 500 km s 1 required (Schneider et al. 2011). A large number of the compact jets associated with YSOs have been detected at radio wavelengths. The dominant radio emission mechanism is thought to be thermal bremsstrahlung from the shock-heated gas. However, evidence for non-thermal emission from several YSOs has also been discovered. Polarization has been associated with several YSOs, including objects in the ρ Ophiuchi molecular cloud (Andre et al. 1988; White et al. 1992; Andre et al. 1992), the Taurus-Auriga molecular cloud (Phillips et al. 1993; Skinner 1993; Feigelson et al. 1994; Ray et al. 1997), the R Coronae Australis region (Choi et al. 2008), the Orion region (Zapata et al. 2004), and HH 7-11 (Rodríguez et al. 1999). Furthermore, a few YSOs have been associated with linearly polarized radio emission, these include HH (Carrasco-Gonzalez et al. 2010), HD (Phillips et al. 1996), and the Orion Streamers (Yusef-Zadeh et al. 1990). Additional characteristics of non-thermal emission, including strong variation in flux density on timescales of hours to days, a negative spectral index, and VLBI measurements of high brightness temperatures (T B 10 7 K), have also been found (Curiel et al. 1993; Hughes 1997; Andre et al. 1992; Wilner et al. 1999; Rodríguez et al. 2005). A suggested source of this non-thermal emission is the gyrosynchrotron

100 84 mechanism (Andre 1996). Non-thermal emission is almost always associated with the more evolved weaklined T Tauri stars (WTTS). This is consistent with the idea that the non-thermal coronal emission is revealed only after the optically thick mass outflows have evolved away (Eislöffel et al. 2000). However, this idea may be too simple: Apparently nonthermal emission has recently been reported from several less-evolved YSO s that have infrared evidence for a disk (Osten & Wolk 2009). This paper focuses on DG Tau, a highly active CTTS driving a well studied energetic bipolar jet. Located in the Taurus Molecular Cloud (estimated distance 140 pc; Torres et al. (2009)), DG Tau is of spectral type K5-M0, with a mass of 0.67 M, a luminosity of 1.7 L and an estimated age of yr (Kitamura et al. 1996b; Güdel et al. 2007a). It was one of the first T Tauri stars to be associated with an optical jet (Mundt & Fried 1983) and has been studied extensively with adaptive optics, interferometry and the Hubble Space T elescope. The micro-jet, associated with the HH 158 knot, extends out to 12 (e.g. Eislöffel & Mundt 1998) at a position angle of 223 (Lavalley et al. 1997). Eislöffel & Mundt (1998) calculated the proper motion of 4 knots observed in the HH 158 jet located at distances between ( au), velocities around 150 km s 1. Furthermore, Dougados et al. (2000) calculated the proper motions of knots located within 4.0 from the source of the HH 158 jet; they determine velocities of 200 km s 1. The jet of DG Tau has an onion-like structure within 500 AU of the star, where faster, highly collimated gas is nested in wider slower material, with maximum

101 85 bulk gas speeds reaching 500 km s 1 (Bacciotti et al. 2000; Lavalley-Fouquet et al. 2000). This velocity structure is expected if the jet material is launched from a range of disk radii (Agra-Amboage et al. 2011). Moreover, for the slower jet material in the jet the velocity between the two sides of the jet is between 6-15 km s 1 ; this velocity shift may indicate that the jet is rotating (Bacciotti et al. 2002b). Using [Fe II] observations and averaging over the central 1.0, the mass-loss rate from the high velocity gas is determined to be (1.6 ± 0.8) 10 8 M yr 1 and from the medium velocity gas (1.7 ± 0.7) 10 8 M yr 1, giving a total mass-loss rate for the velocity range of 50 to 300 km s 1 of (3.3 ± 1.1) 10 8 M yr 1. However, this value is a lower limit to the mass-loss rate from the atomic component of the jet since the [Fe II] emission does not probe the whole range of velocities seen at optical wavelengths (Agra-Amboage et al. 2011). The mass-loss rate of DG Tau has also been estimated using [OI]λ6300, giving a rate of M yr 1 (Hartigan et al. 1995). The most recent estimate for the mass loss through the jet atomic component is Maurri et al. (2014), with Ṁ = (1.2 ± 0.4) 10 8 M yr 1. Near-infrared evidence for a counterjet has been reported by Pyo et al. (2003). The redshifted emission appears suddenly at -0.7 which suggests that the inner part of the counter-jet is hidden behind an optically thick circumstellar disk. DG Tau shows strong millimeter continuum emission thought to arise from a compact dust disk around the star. Assuming a dust opacity coefficient κ ν = cm 2 g 1 at 147 GHz, the spectrum is consistent with thermal emission from a disk having a radius of about 110 AU and a mass M (Kitamura et al. 1996b).

102 86 At larger scales, 13 CO observations have revealed a gas disk with radius of 2800 au, oriented with its major axis perpendicular to the jet of DG Tau (Kitamura et al. 1996a). The X-ray jet of DG Tau was first discovered by Güdel et al. (2005) using a Chandra ACIS-S observation, which showed very faint soft emission along the jet out to a distance of about 5.0 to the SW with position angle 225. In addition, Güdel et al. (2005) found that DG Tau reveals a new type of X-ray spectrum that includes two emission components with vastly different absorbing column densities: A weakly attenuated soft spectral component associated with a plasma with electron temperatures of no more than a few MK and a strongly absorbed hard spectral component associated with a hot plasma of several tens of MK.The soft X-ray component is thought to originate at the base of the jet where the first shocks form, while the hard component is attributed to emission arising from a magnetospheric corona. Moreover, Schneider & Schmitt (2008) demonstrated that the soft and hard X-ray components have a separation of 0.2 and that the soft X-ray emission is coincident with emission from optical lines indicating that this X-ray component is indeed from the jet. There are few previous radio observations of DG Tau. It was first detected in a VLA survey of the Taurus-Auriga region at 5 GHz (Cohen et al. 1982). Further observations at 15 GHz and 1.5 GHz (Cohen & Bieging 1986) found that the structure is elongated. Cohen & Bieging (1986) suggested that the radio structure and spectrum is consistent with free-free emission from ionized gas in an outflowing jet. Rodríguez

103 87 et al. (2012) reported radio knots in the extended jet at 0.42 and 6.98 respectively. They determine that these radio components were coincident with previously detected optical knots. In this paper we present the results of a Jansky Very Large Array 1 (VLA) multi-frequency campaign of DG Tau. The goals of these observations were to determine the morphology of the inner jet, search for possible non-thermal coronal radio emission, and investigate the nature of the radio knots at large distance from the star. 4.4 VLA Observations Table 4.1. VLA Results Epoch Array Freq BW Time Total Flux Peak Flux Sky RMS Θ F W HM (GHz) (GHz) (min) (mjy) (mjy/beam) (µjy) (arcsec) A A C C The radio campaign comprised three observing epochs between June 2011 and April The June 18, 2011 observation used two 128 MHz bands centered on 8.33 GHz and 8.46 GHz in A configuration. In 2012 we conducted two C-array observations using the newly-available 2 GHz bandwidth capability of the VLA. The March 22, 2012 observation spanned the frequency range of 4.5 GHz to 6.5 GHz, while the April 1 The VLA is operated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

104 88 15, 2012 observation spanned 7.9 GHz to 9.9 GHz. The details of these observations are listed in Table 4.1. For all three observations both the receiver bandpass correction and the absolute flux density scale was set using the amplitude calibrator 3C48. The phase calibration used the angularly nearby phase calibrator source J In addition, we made short observations of both 3C84 and 3C138 in our June 2011 observation. These sources were used to solve for the antenna polarization leakage terms and linear polarization position angle. Data reduction and imaging was carried out using Common Astronomy Software Application (CASA) package distributed by the National Radio Astronomy Observatory (NRAO). Additionally, we used the Astronomical Imaging Processing System (AIPS) to calibrate and map an archival 4-hour A-array VLA observation of DG Tau from November 1, 2000 at 43 GHz. 4.5 Results Imaging the DG Tau Jet After flagging bad visibilities and applying the normal amplitude and phase calibration, clean maps in total intensity (Stokes I, see Figure 4.3), circularly polarized intensity (Stokes V ) and linear polarized intensity were produced using the standard CASA data calibration procedures. The peak and integrated flux density of both the total and polarized intensity was obtained using the CASA task IMFIT (see Table 4.1). We did not detect either circular or linear polarization, with upper limits 0.03 mjy beam 1 (3%) for circular and 0.02 mjy beam 1 (2%) for linear polarization. Both the rising spectral index and lack of polarization suggest that any non-thermal

105 89 coronal component is absent or very weak. Figure 4.3. VLA 8 GHz naturally-weighted contour map of DG Tau s inner jet at epoch The contours are -3, 3, 6, 10, 15, 20, 25, 30, 35, and µjy beam 1, the RMS noise level of the image. The restoring beam is shown in the bottom left corner with dimensions The A-array 8 GHz total intensity contour map at epoch is shown in Figure 4.3. In this high-angular resolution image, the emission extends approximately 0.35 SW in approximately same direction as the optical jet. Both the asymmetry and elongation of the emission suggest that the source is the collimated jet mapped by optical observations (Bacciotti et al. 2000; Agra-Amboage et al. 2011) which have resolved the inner jet to within 0.1 of the central source and determined that collimation must occur within this region.

106 90 (a) (b) A A C C Figure 4.4. VLA contour maps of DG Tau during the two spring 2012 observations. The 8.5 GHz map at epoch is shown on the right, while the GHz emission (epoch ) map is on the left. Both maps were made used natural weighting with contour levels -3, 3, 4, 6, 8, 10, 12, 15, 20, 40, and and 6.2 µjy beam 1, the RMS values of the respective maps. The restoring beam for each map is given in the bottom left corner with the dimensions given in Table 4.1. The cross indicates the location of knot A detected by Rodríguez et al. (2012) in 2009 radio observations. Feature C, located 20.0 SW of the inner centriod, is visible in both 2012 maps and on the 2009 map of Rodríguez et al. (2012), but is displaced from the optical jet. The 2012 observations were taken in C-array, with much lower angular resolution but higher sensitivity to larger, low-surface brightness components. Contour maps of the 5 GHz (epoch ) and 8 GHz (epoch ) observations are shown in Figure 4.4. The location of the central source for the (5.5 GHz) observation is coincident with the (8.5 GHz) central source within the centroid uncertainty (± 0.01 ). Note that the integrated flux density at 8 GHz apparently increased 17% from epoch to , but it is possible that the increase results from faint extended structure over-resolved by the smaller beam at epoch (cf. Table 1). The 5.5 GHz map shows a component (labeled A) extending 7 SW whose centroid is coincident with a much weaker feature seen in the 8.5 GHz map.

107 91 This component is nearly coincident with the knot component reported by Rodríguez et al. (2012) but at a much lower flux density; we discuss this in more detail in section 4.3. There is also a more distant feature in the 5 GHz map (labeled C) with a flux density S 100 µjy about 14 from the stellar position, but well-displaced from the jet axis. This feature is also weakly seen in the Rodriguez et al. map, and is certainly real. We do not know if it is associated with the DG Tau jet, although it is unlikely that an unrelated background source this strong would be located this close to the jet. We speculate that the feature may be a part of an extended bow shock associated with the optical knot seen near 13 (Rodríguez et al. 2012), but do not discuss it further in this paper Spectral energy distribution Figure 4.5 shows a radio spectrum of DG Tau using our VLA data, previously published radio data (Cohen et al. 1982; Cohen & Bieging 1986; Rodríguez et al. 2012), and an archival VLA observation of DG Tau at 43 GHz (project code AW545, PI David Wilner) that we calibrated and mapped. The spectrum monotonically increases over the frequency range of 4.5 to 43 GHz. However, it is clear from the multiple observations at 8.5 GHz and 4.5 GHz that the flux density is significantly variable over a period of years. Therefore we use only our nearly coeval 2012 observations to fit a power-law to the spectrum, giving a spectral index α= 0.46 ± This spectral index is typical of collimated thermal jets, whose spectral indices are close to +0.6 for constant velocity isothermal outflows, but whose overall spectral index can vary from

108 S (µjy) Freq (GHz) Figure 4.5. Flux density of DG Tau as a function of frequency for observations at epochs (Cohen & Bieging 1986), , , and (Rodríguez et al. 2012), (archival VLA), and (Lynch et al. 2013). The solid line, a power-law fit to the nearly coeval and points only, has a spectral index α = 0.46 ± nearly flat to +2 depending on the physical conditions in the jet, such as velocity gradients and recombination in the flow (Reynolds 1986). To better constrain the spectral index of the radio emission we need nearly coeval observations over a much larger range of frequencies.

109 Discussion Comparison between radio and optically-derived electron densities Figure 4.6. Analytic functions used to model the electron density (left) and velocity (right) profiles across DG Tau s jet at 0.35 from the central source. The points are derived values of electron density and velocity from (Maurri et al. 2014). Free-free emission depends only on the plasma temperature, the observing frequency, and the linear emission measure i.e., the square of the electron density integrated along the line of sight to the observer. If the plasma temperature is known (e.g., from optical line observations), and the optical depth at a given location can be estimated (e.g., from source structure), the emission measure at that location can be estimated. The entire detected radio jet is the optically thick surface. We therefore identify the outermost detectable region of the radio jet with the unity optical depth surface (τ = 1). We then can calculate the emission measure and compare with estimates of electron density and jet width at this location based on optical line

110 94 ratios. Inspection of Figure 1 shows that the radio emission becomes undetectable 0.35 (50 AU projected) from the base of the jet. Assuming a thermal jet, this should correspond to the τ = 1 surface, meaning inside 0.35 the jet is fully optically thick while outside this distance the jet is optically thin. The temperature of the jet gas is not well-constrained, but if the jet is launched via a quasi-steady centrifugal MHD disk wind, as suggested by the transverse velocity gradients (Bacciotti et al. 2002a), the gas temperature should be not much higher than the photospheric temperature (4800 K). However, the innermost, high-velocity core is likely shock-heated and may have a temperature near 8000 K, typical to other stellar jets. The lower-speed, broader flow associated with warm molecular emission is cooler, with a temperature of 2000 K, determined from infrared line ratios (Takami et al. 2004). In the following we assume a mean temperature across the flow T = 5000 K. The free-free optical depth can be written τ(ν, T, EM) = 1.06 ( ν ) ( 2.1 T ) 1.35 ( EM ) GHz 10 4 K cm 5 (4.3) Assuming τ = 1 at ν= 8.5 GHz, the emission measure at 0.35 is EM n e 2 ds = cm 5 (4.4) High angular resolution observations of optical line ratios have be used to determine both the electron density and jet width as a function of velocity bin along the jet (e.g., Coffey et al. 2008; Maurri et al. 2014). By combining these measurements, we can estimate the axial electron density profile. Figure 4.6(a) shows a simple

111 95 Gaussian fit to the electron density as a function of axial distance, where the points represent the mean density in each velocity bin and the uncertainties are the FWHM width uncertainty in each bin, using data from Maurri et al. (2014). We have combined these measurements to fit a simple analytic function to the electron density as a function of jet axial distance ρ, n e (ρ) = n 0 e ( ρ ρe )2 (4.5) where n 0 = cm 3 and ρ e = 5.5 au. Integrating this density profile, we find a linear emission measure EM = cm 5, in excellent agreement with the radio data Mass-loss rate of the ionized component We next use the axial electron density derived in the previous subsection to estimate the mass-loss rate of the ionized component of the jet outflow at τ = 1 (50 AU projected distance). Figure 4.6(b) shows a Gaussian fit to the FWHM widths of the velocity bins given in Maurri et al. (2014), V (ρ) = V 0 e ( ρ ρv )2 (4.6) where V 0 = 400 km s 1, and ρ v = 12 au. Using this velocity profile, along with the density profile determined in the previous sub-section, we determine the mass-flux of the ionized gas at 50 AU projected distance, Ṁ = 2 π m i 0 V (ρ) n e (ρ) ρ dρ. (4.7)

112 96 where m i 1.2 m p is the average ion mass. Using the approximate analytic models shown in Figure 4.6 for n e (ρ) and V (ρ), we find mass flux Ṁ M yr 1, with more than half of the mass flux in the high-velocity component within 5 AU of the jet center. Since the high-velocity component is almost completely ionized (Maurri et al. 2014), this mass-loss estimate should be less than a factor of two smaller than the mass-loss of both the ionized and neutral gas. Recent mass loss estimates of the DG Tau jet include that of Agra-Amboage et al. (2011), Ṁ = (3.3 ± 1.1) 10 8 M yr 1 and Maurri et al. (2014), Ṁ = (1.2 ± 0.4) 10 8 M yr 1. While these are both somewhat lower than the present model prediction, the differences are probably not significant given the poorlyconstrained functional forms used for the axial density and velocity profiles Radio Knots in the Extended Jet Radio observations of DG Tau at epoch (Rodríguez et al. 2012) show a radio knot located approximately 7 along the jet with an integrated flux density 150 µjy, coincident with an optical [S II] knot. There is also a much weaker feature near 12, also coincident with an optical knot. If these radio structures are associated with shock-compressed gas in the jet flow, we would expect them to evolve both in flux and position on the dynamical timescale of the shock and the expanding post-shock gas. Our observations confirm the existence of the inner knot. Figure 4.7 shows 5.4 GHz and 8.5 GHz images of DG Tau at epochs and respectively. At each frequency we show maps made with two different uv-plane density weighting

113 97 (a) (c) A B C A B C (b) (d) A A B B C C Figure 4.7. DG Tau contour maps at 8 GHz (a, c) and 5 GHz (b, d) at epochs and respectively. The two maps in the first column of this figure, (a) & (b), are made using natural weighting. The contour levels are -3, 3, 4, 6, 10, 20, 40, 60, 80, 100, 120, and 140 times the RMS in each map. The RMS for (a) is 8.0 µjy with beam dimensions The RMS for (b) is 10 µjy with beam dimensions The second column maps, (c) & (d), are made using uniform weighting in order to maximize angular resolution of the maps. The contour levels are -3, 3, 6, 10, 20, 40, and 60 times the RMS in each map. The RMS for (c) is 15 µjy with beam dimensions ; (d) has a RMS of 13 µjy and beam dimensions of The crosses in each of these maps indicate the locations of the two knots detected by Rodríguez et al. (2012). Feature C, located 20 SW of the inner centriod, is visible in both 2012 maps and on the 2009 map of Rodríguez et al. (2012), but is displaced from the optical jet. functions: uniform weighting to maximize angular resolution (panels c, d), and natural (a.k.a. unity) weighting to maximize sensitivity to low-brightness features (panels

114 98 a, b). The locations of the 7 knot (labeled A) and 12 knot (labeled B) detected by Rodríguez et al. (2012) are indicated by crosses. Both the 8.5 and 5.4 GHz natural weighted maps show radio emission coincident with knot A but not with knot B. We did not detect either knot A or B in our epoch A-array observation, probably because the knots were over-resolved, as discussed below Knot proper motion The DG Tau jet optical knots move outward with sky-plane proper motions 0.3 per year (V 200 km/s), at least within 10 arcsec of the stellar position (Eislöffel & Mundt 1998; Dougados et al. 2000). Rodríguez et al. (2012) found radio knots that are approximately cospatial with the optical knots, and also move with this speed, suggesting that the radio and optical knots are two manifestations of the same traveling shocks. Here we examine whether our more recent knot observations are consistent with this hypothesis. We consider only the motion of knot A, since we did not detect knot B at any epoch. Note that all positions discussed below are angular separations from the stellar position in the sky plane. The optical position of knot A has recently been determined from HST STIS observations at epoch (Schneider et al. 2012, in preparation). Fitting a Gaussian to the spatial profile of the [S II] 6731Å line within v= -265 ± 65 km/s results in a distance 7.2 ± 0.1, compared with a predicted position 7.4 using the Rodríguez et al. (2012) proper motion. The measured position is only slightly offset from the predicted position, and could indicate that the peak radio and optical intensities occur along different lines of sight in the shocked emission region. This is plausible,

115 99 10 V = 200 km/s 9 Projected Distance (arcsec) Epoch Figure 4.8. Projected knot location along jet vs. epoch (solid line) with observed centroid locations at epochs and since the radio and optical line emergent intensities have different dependencies on the density and temperature of the gas Knot evolution: Expanding post-shock gas? In addition to moving with the predicted proper motion, the flux density of knot A decreased dramatically. In order to avoid differences associated with spectral index and angular resolution, we can compare the epoch measurement of Rodríguez et al. (2012) with our observation, both of which were at 8.5 GHz and had the same angular resolution (VLA at C-array). The epoch integrated flux was (150 ± 20) µjy while the integrated flux was (46 ± 26) µjy, both measured using naturally-weighted maps. This is a 70% decrease in 2.7 years, and is certainly real given the matched frequency and angular resolutions of the two

116 100 observations. This behavior is not unprecedented: Radio knots in several other YSO jets have been observed to decrease with time and become undetectable within years of their initial ejection (e.g. Martí et al. 1998). The total flux density at 5.5 GHz (epoch ) was (73 ± 25) µjy. This implies a spectral index 0.4 ± 0.4, consistent with optically thin thermal emission, but highly uncertain because of the large uncertainty in each measurement, and the differing angular resolutions. Rodríguez et al. (2012) proposed a model consisting of periodic generation of shocks moving outward at at projected speed 200 km s 1 in a conical outflow. The model predicts a periodic variation of knot flux density caused by corresponding periodic velocity variations in a bipolar outflow. Fig. 8 of Rodríguez et al. (2012) predicts a flux for knot A near that observed at epoch , but occurring 1.5 years after the flux level observed at epoch rather than 2.7 years. However, since their model is parameterized by several geometrical parameters which are not wellconstrained, a detailed comparison is not appropriate with only a single additional flux measurement. Instead, guided by Occam s razor, we interpret the flux decrease using a much simpler conceptual model of an expanding volume of post-shock gas characterized by a single variable, the electron density. We model the knot as an optically-thin, free-free emitting isotropic sphere whose density scales exponentially with radial distance ρ, [ ( ) ] 2 N ρ n e (ρ) = ρ 3 F 3π exp ln(2), (4.8) 3 ρ F where 2ρ F is the full width at half maximum scale, and the normalization ensures that the integrated density is a constant N, i.e., the total number of electrons does not

117 (a) Flux density 3000 (b) Electron density Flux density (µjy) Surface brightness (µjy/arcssec 2 ) Epoch Surface Brightness (c) Epoch Optical depth Axial distance (AU) Optical depth (d) 5 GHz 8 GHz Epoch Figure 4.9. Evolution of the observed radio knots in the extended jet of DG Tau. (a) DG Tau knot A flux density vs. epoch calculated using optically thin expanding sphere model (see text), with observed values at epochs (Rodríguez et al. 2012) and (this paper). (b) Model electron density vs. projected axial distance (arc sec) at epochs (solid line) and (dashed line). (c) Model surface brightness vs. epoch (solid line) with observed values ( ) and upper limits ( ). (d) Optical depth at 5 GHz (solid line) and 8 GHz (dashed line) vs. epoch, confirming the optically-thin model assumption. change as the sphere expands. We also assume the knot is isothermal, a simplistic but reasonable assumption since for optically-thin emission, the flux density dependence on temperature is much smaller than for density (S T 0.35 n 2 e ). We assume T = 10 4 K, based on optical line ratio observations (Agra-Amboage et al. 2011). The sphere is assumed to expand linearly with time, ρ(t) = ρ 0 + ρ(t t 0 ). The flux and size of the sphere were calculated as a function of time and

118 102 compared with observations. We solved for best-fit model parameters N = , ρ 0 = 336 AU(2.4 at d = 140 pc), and ρ = 21 AU yr 1 (0.15 yr 1 ). Figure 4.9 shows the flux density, electron density profile, surface brightness, and optical depth as a function of epoch for the fitted model, along with observed values and upper limits. The agreement is well within the measurement uncertainty of the radio knot at all epochs, and has τ 1, as expected. Maurri et al. (2014) measure electron densities n e 10 3 cm 3 along the jet at a projected distance near 4-5. For a conical flow with constant ionization fraction, the density scales as r 2, so we expect n e 500 cm 3 outside the radio knot at a projected distance of 7. According to the model, the maximum knot density varied from 2000 to 1000 cm 3 from epochs , implying a density enhancement factor of 4 in , decreasing to 2 in Assuming that the present expansion continues, the model predicts that the density contrast will vanish and that the knot will disappear within a few years. 4.7 Summary We report multi-epoch VLA observations of the pre-main sequence star DG Tau s radio jet. The radio spectrum (α = 0.46 ± 0.05) and lack of polarization indicate that the emission is bremsstrahlung, with no evidence for a non-thermal coronal component. Assuming the end of the radio jet at 0.35 (50 AU projected distance) is the τ = 1 surface, we calculate the column emission measure. Assuming an onion skin ionization model and azimuthal symmetry, we find that the mean electron density in the centre of the jet is n = cm 3. This agrees well

119 103 with optical estimates at this location (Maurri et al. 2014). We model the electron density and velocity axial profiles to calculate the mass loss of the ionized component, Ṁ ion M yr 1, with more than half of the mass flux in the high-velocity component within 5 AU of the jet center. This mass loss is comparable to the total mass loss calculated using optical line observations (e.g., Agra-Amboage et al. 2011; Maurri et al. 2014), indicating that most of the mass loss in the jet at this location is in the ionized component. We confirm the existence of a radio knot near 7 recently reported by Rodríguez et al. (2012). The knot proper motion is consistent with a projected speed V = 200 km s 1, as suggested by Rodríguez et al. (2012) and previous optical estimates. The flux density of the knot dramatically decreased between and We present a simple model for radio emission from the knot consisting of an optically-thin ionized sphere which expands linearly with time. The model predicts that the FWHM size of the radio knot increases from 340 AU (2.4 ) to 390 AU (2.75 ) from to , while the central electron density decreases from 2000 to 1000 cm 3. The resulting radio flux decreases from 150 µjy to 50 µjy, in agreement with observations. By scaling from previously published density measurements at closer distances along the jet, we find that the density enhancement factor of the knot decreased from 4 in to 2 in , and that the knot will disappear completely within a few years.

120 104 CHAPTER 5 WIDEBAND DYNAMIC RADIO SPECTRA OF TWO ULTRA-COOL DWARFS 5.1 Preface Surveys of chromospheric Hα and coronal X-rays, indicators for the presence and dissipation of magnetic fields, have shown a sharp decline in emission beyond spectral type M7 (e.g. Neuhäuser et al. 1999; Gizis et al. 2000; West et al. 2004). This reduction in activity is associated with a decrease in plasma heating through the dissipation of magnetic fields (Mohanty et al. 2002). However, this reduction in heating does not imply a drop in magnetic field strength or filling factor. In fact the detection of both quiescent and flaring non-thermal radio emission from some UCDs (Berger et al. 2001; Berger 2002, 2006; Berger et al. 2009; Burgasser & Putman 2005; Phan-Bao et al. 2007; Osten & Jayawardhana 2006; McLean et al. 2012) confirms that at least some of these objects are still capable of generating strong magnetic fields. All radio loud UCDs have a quiescent component that in some cases is shown to vary with the rotation of the star. Additionally, some radio loud UCDs are observed to have strong radio flares that can be periodic. The electron cyclotron maser (ECM) mechanism is generally accepted to be the source of the pulsed emission since it can account for the high brightness temperature, directivity, and circular polarization of this emission (Hallinan et al. 2006). There is still some debate over the nature of the quiescent component where both depolarized ECM (Hallinan et al. 2007) and

121 105 gyrosynchrotron emission from a non-thermal population of electrons (Berger 2002; Burgasser & Putman 2005; Osten et al. 2006b) are proposed sources for this emission. In order to understand how UCDs are able to produce magnetic fields, the magnetospheric parameters and geometry need to be determined. This requires detailed information about how the radio pulse and quiescent emission vary over time and with frequency. The new wideband capabilities of the VLA allow for both increased sensitivity as well as greater frequency coverage during a single observation. Taking full advantage of these capabilities, I observed two UCDs, TVLM and 2M , in These observations were complimented by archival wideband VLA observations of TVLM in 2010 and 2M in In all of the observations the two sources had light curves that exhibited weak non-variable quiescent emission with strong periodic radio flares. From these observations I was able to create radio dynamic spectra of both the total flux and percent circular polarization of the pulsed component and better sample the spectral energy distribution for the quiescent component of the two UCDs. A written report of the results from these observations is submitted for publication in the Astrophysical Journal as well as presented in the following chapter. I used the analysis package CASA to reduce and analyze the wideband observations of TVLM and 2M J Since CASA does not have a task to generate light curves from a radio dataset, I developed a python script to perform such a task (see Appendix A). Given information about how the user wants to split the dataset in time and frequency, the script will use the clean task to create CLEAN

122 106 images in both Stokes I and V for each time-frequency bin. It then fits the UCD in each CLEAN image using a user specified pixel coordinate and the imfit task. It additionally fits a dark region specified by the user to measure the RMS noise in each CLEAN image. The stokes I and V values from the source fits as well as the RMS values are written into a file and the CLEAN images are then deleted. The written file is then used in a separate python script to produce either dynamic spectra or light curves. The dynamic spectra uncover complex frequency structure for the radio pulses, including frequency drifting as well as double or triple pulse structure. Additionally, the polarization of these pulse can be complex: the helicity of the polarization is found to reverse for some pulses and depolarization of short duration periodic pulses, similar to the pulses observed for J by Hallinan et al. (2007), is found to occur. Additional details about these spectra are given in section To develop a geometric magnetospheric model for UCDs the features of the pulsed radio emission are used to probe the conditions of the magnetospheric plasma. Provided that the source-dependent parameters (e.g. angular beaming, refraction) can be successfully modeled, the frequency of the ECM emission directly maps the magnetic field strength at the emission source and the rotation of the star provides time-lapsed spatial slices of the source magnetic structure. This model is based on a self-consistent description of the coronal plasma and magnetic field, taking into account ECM growth rates and ray propagation. The free parameters of the model (e. g., rotation and magnetic field axes orientation, magnetic field strength, fre-

123 107 quency dependence of emission beam) are adjusted for to best fit the radio features in the observations. My advisor, R. Mutel, is responsible for developing the geometric magnetospheric emission model for the pulses observed from TVLM and 2M J and so I will refer the reader to section for a more in depth discussion of this model. The periodicity of the pulsed radio emission is believed to be due to the rotation of the star and so using the observed times of the pulsed emission we can determine the rotational period. Using a combination of the observed times in and previously published times for the radio pulses of TVLM and 2M J I use a minimization scheme to constrain the rotational period for both these stars (see section 5.5.2). For 2M J the new period is in agreement with previous estimates and improves upon their precision. TVLM on the other hand is not in agreement with a previous estimate by Doyle et al. (2010) but our period covers a larger range of time periods and can account for the pulses they observe for TVLM Additionally, an optical estimate was published recently and is 30 seconds shorter than our radio estimate. Comparing the optical period with the radio observations of the pulse times, I determine that this 30 second difference is real. In addition to constraining the UCD rotational period, I was also responsible for modeling the quiescent component of the radio emission. From the spectral energy distribution (section ), measured circular polarization (section ), and estimated brightness temperatures (section ), I concluded that the source

124 108 ν/ν R Plasma X, Y ν (GHz) 10 2 Plasma X & Y Razine Ratio R = 1.1 Rs R = 1.5 Rs R = 3.0 Rs X (R=1.1 Rs) X (R=3.0 Rs) Y (R=1.1 Rs) Y (R=3.0 Rs) ν (GHz) A C Rotation (rads) Fractional Polarization R = 1.1 Rs R = 1.5 Rs R = 3.0 Rs Faraday Rotation ν (GHz) Propagation of Fractional Polarization Circular Linear r (Rs) Figure 5.1. Plasma parameters for the magnetosphere of a typical low mass star, including Faraday Rotation (A), the magneto-ionic parameters (B), the Razine ratio (C), and propagation of polarization vector (D). The star is assumed to have a surface field of B 0 =2000 Gauss, surface density of N e0 =10 11 cm 3, and the angle between the magnetic axis and line of sight is θ=89. B D of this component is most likely power-law gyrosynchrotron. When developing this emission model, I had to consider several effects the magnetospheric plasma has on the radiation. Firstly, for gyrosynchrotron radiation propagating through a magnetized plasma, the emission is suppressed whenever the index of refraction deviates significantly from unity. For mildly relativistic electrons this suppression will occur when the frequency of the radiation is less than the Razine frequency (Dulk 1985),

125 109 given by ν R = ν2 pe ν ce (5.1) Here ν pe = 9 KHz n e is the plasma frequency and ν ce = 2.8 MHz B is the cyclotron frequency. In panel (A) of figure 5.1, the ratio of the radiation frequency ν to the Razine frequency ν R, as a function of frequency for a typical low mass stellar magnetosphere is shown. I assume the field is dipolar (B r 3 ) and the density has a quadratic dependence with radial distance (n e r 2 ). This ratio is calculated at three different distance from the surface of the star with an assumed surface field strength of B 0 =2000 Gauss, surface density of N e0 =10 11 cm 3, radius of the star R s =0.1R, and angle between the magnetic field axis and line of sight of θ B =89. From this figure is is clear that close to the star, suppression of gyrosynchrotron radiation occurs at frequencies 1 GHz, while further from the star this suppression occurs at higher frequencies ( 3-4 GHz). In our analysis we are using observations at 4-7 GHz and the radiation we observe should not experience significant Razine suppression at these frequencies. Interpreting the polarization of the quiescent radio emission is not simple since the apparent polarization can change as the emission propagates from the source. This is the result of mode coupling in the corona of the source. There are two electromagnetic modes that can propagate in a cold plasma, the ordinary (O) mode and the extraordinary (X) mode. These wave modes are generally elliptically polarized and propagate independently. If there is no coupling between the two modes during

126 110 propagation they will remain in their original modes upon arriving at the telescope but the circular polarization can change sign, this is called weak coupling. In the case of strong mode coupling the radiation would retain the circular polarization instead of the mode (White et al. 1992). The strength of mode coupling is described by the coupling ratio C: C = 4 dθ XY 3 dz (5.2) where X, Y are the magneto-ionic parameters: X = ν2 pe ν Y = ν ce ν (5.3) (5.4) For strong coupling C 1 and for weak C 1 (Melrose & Robinson 1994). One well-known example of mode coupling is Faraday depolarization. To show that we do not expect to detect linear polarization from the two UCDs due to this effect, I calculated the Faraday rotation that would result from a typical UCD magnetosphere. For astrophysical systems, Faraday rotation occurs when light passes through magnetized plasma. In the case of a plasma which is spatially inhomogeneous, the Faraday effect can lead to a loss of linearly polarized intensity. Faraday rotation of the plane of polarization of linearly polarized light is given by χ = RM λ 2 (5.5) where λ is the wavelength of the radiation and RM is the rotation measure RM = l 0 B n e dz (5.6)

127 111 having units of rad m 2. The strength of the magnetic field parallel to the line of sight is B, the electron density is n e, and z is the coordinate (in parsec) along the line of sight (Trippe 2014). For modulations RM of the rotation measure on spatial scales smaller than the source, the radiation experiences different Faraday rotation depending on the position. If the observations of the source do not resolve the RM structure, they will spatially superimpose waves with different orientations of their planes of linear polarization. This effect will partially average out the polarization signal and reduce the degree of linear polarization observed. In the case of RM λ 2 1, the plasma is considered to be Faraday thick and complete depolarization occurs (Trippe 2014). Panel (B) in figure 5.1 shows the Faraday rotation as a function of frequency for the same typical low mass stellar magnetosphere used to calculate the Razine suppression. For these values we see that the magnetosphere of a typical low mass star is Faraday thick and do not expect to detect linear polarization from these sources. The transfer of polarized radiation from circular to linear may be described using the Stokes parameters, I, Q, U, and V, written as the Stokes vector S a with S 1 =I, S 2 =Q, S 3 =U, S 4 =V. In non-dissipative media the total intensity I is constant along the ray and the polarized part of the radiation may be described by the transfer equation ds a dz = ρ abs B (5.7) where z is the distance along the ray path and the sum over the repeated index

128 112 B = Q, U, V is implied. The matrix ρ ab has the form 0 ρ V ρ U ρ ab = ρ V 0 ρ Q (5.8) ρ U ρ Q 0 The expressions for the parameters in equation 5.8 are determined by properties of the natural frequency modes in the plasma (Melrose & Robinson 1994). If we assume that these properties are given by magneto-ionic theory and neglecting collisions ρ V = X cos(θ) (5.9) 1 Y 2 ρ U = XY 2 1 Y 2 sin(θ)2 sin(2φ) (5.10) ρ Q = XY 2 1 Y 2 sin(θ)2 cos(2φ) (5.11) where θ, φ are the polar angels of the ray with respect to the magnetic field. Figure 5.1 panel (C) shows X and Y as a function of frequency for the same magnetosphere used for the Razine suppression and Faraday rotation calculations. For frequencies 4-7 GHz, its clear that mode coupling will have more of an effect close to the star than further from the star since the strength of the coupling varies inversely with X and Y 3 (see equation 5.2). To determine how much circular polarization is lost during propagation close to the star, I use the values of X and Y from panel (C) for R = 1.1 R s, where X=0.25 and Y =0.75. I calculate the propagation of the stokes vectors for radiation which is 100% right circularly polarized at the surface of the star as it propagates

129 113 through a plasma that has a constant density and magnetic field strength but a slowly changing magnetic orientation varying from 0-π with dθ = π10 5. The result of this calculation is shown in panel (D) of figure 5.1. For these plasma parameters the circular polarization does transfer to linear polarization as it propagates through the plasma and so a complete model would take into account the propagation of the polarization. The initial model I developed was for a full magnetosphere, where I assumed a dipolar magnetic field, whose magnetic axis has an orientation θ B with the line of sight, and a radially symmetric electron density N e = N e0 ( r R s ) 2 (5.12) with a power-law energy distribution with index δ. Using the orientation constrained by the ECM models of the pulsed emission, the surface magnetic field, N e0 and δ were varied in order to fit both the spectral energy distribution and polarization measurements. To calculate the specific intensity I integrated the equation of radiative transfer along lines of sight using the Robinson & Melrose (1984) expression for the absorption and emission coefficients. Since these coefficients depend on the local angle between the propagation direction and the local magnetic field vector, I calculated this angle at each step along each line of sight. After determining the optical depth for each line of sight, I integrate over radius and azimuth angle to obtain the net flux density in each mode. This model does not take into account mode coupling and how the Stokes vector propagates along the line of sight. As seen in figure 5.1 the propagation of the polarization needs to be taken into account, however we believed

130 114 that incorporation of this effect was beyond the scope of the project and would be handled in a future paper. Instead of using a full magnetosphere, I then approximated the emission region with the weighted mean scheme described in section In this new model I still assume a dipolar field and isothermal plasma with a radially symmetric electron density and energy distribution with index δ. Instead of using the local angle at each point along the line of sight, I calculate a weighted angle to use for the whole line of sight. To determine this angle, note that the emission from each point is S B 1.5 local N local. I calculated this factor for each point along the line of sight and then weighted each local angle by this factor. The angle with the highest weight is then assumed to be the constant angle between the magnetic field axis and the line of sight for that line of sight. Additionally, I take the magnetic field and density along each line of sight to be constant and equal to the values that give the highest weight. This gyrosynchrotron model fits the total flux measurements of the quiescent emission well but is not able to fit the polarization measurements. This is most likely due to some low-level pulse emission that is not easily separated from the quiescent emission. The magnetospheric environment described by both the gyrosynchrotron and ECM emission models suggest that radio loud UCDs, like TVLM and 2M J , have weak non-axisymmetric magnetic topologies leading to the quiescent component and small regions with strong fields giving rise to the pulsed emission.

131 115 Wideband Dynamic Radio Spectra of Two Ultra-cool dwarfs C. Lynch 1, R. L. Mutel 1, M. Güdel 2 1 Department of Physics and Astronomy, University of Iowa, Iowa 52240, USA 2 Department of Astrophysics, University of Vienna, Vienna, AT 5.2 Abstract A number of radio-loud ultra cool dwarf (UCD) stars exhibit both continuous broadband and highly polarized pulsed radio emission. In order to determine the nature of the emission and the physical characteristics in the source region, we have made multi-epoch, wideband spectral observations of two UCD stars, TVLM and 2M J We combine these observations with previously published and archival radio data to fully characterize both the temporal and spectral properties of the radio emission. The continuum spectral energy distribution and fractional polarization can be well modeled using gyrosynchrotron emission with mean magnetic field B G, mildly relativistic power-law electrons of number density n e cm 3, and source size R 2R s. The pulsed emission exhibits a variety of time-variable characteristics, including large frequency drifts, high and low frequency cutoffs, and multiple pulses per period. For 2M J we determine a main pulse period consistent with previously determined values, but for TVLM the radio pulse period is 30 sec longer than the optically-determined period. We modeled the locations of pulsed radio emission using an oblique rotating magnetospheric model with beamed electron cyclotron maser (ECM) sources on local magnetic loops. We adjusted the ECM source locations and beaming characteristics to fit the observed spectra. The best-fit models have ECM beaming angles that are very narrow and

132 116 aligned with the magnetic field tangent direction at the source, except for one isolated burst from 2M J For TVLM513-46, the best-fit rotation axis inclination is nearly orthogonal to the line of sight. For 2M J we found a good fit using a fixed inclination i = 36, as recently determined from optical observations. For both stars the ECM sources are located near the feet of intense, isolated magnetic loops with scale heights 1.2R s 2.7R s and surface fields kg. These results support recent suggestions that radio over-luminous UCD stars have a weak field non-axisymmetric magnetic topologies. 5.3 Introduction The discovery of intense, non-thermal radio emission from stars at the lowmass end of the main sequence (e.g., Berger et al. 2001; Berger 2002; Hallinan et al. 2006; Osten & Jayawardhana 2006; Phan-Bao et al. 2007) implies the presence of strong magnetic fields. These fields are unexpected given the fully convective stellar interior and observed sharp decline in chromospheric Hα, and coronal X-ray emission for dwarf stars (e.g. Neuhäuser et al. 1999; Gizis et al. 2000; West et al. 2004). The α-ω dynamo, driven by shearing motions at the radiative-convective boundary, is the canonical model for magnetic field generation in solar-type stars, but this mechanism clearly cannot apply to fully convective dwarfs. Instead, magnetic field in these stars may be generated by α 2 or turbulent dynamos, which are driven by turbulent motions associated with internal convection or on both stratification and rotation (Raedler et al. 1990; Durney et al. 1993; Browning 2008). Radio surveys of ultra-cool dwarfs (UCDs, spectral class M8 and cooler) have

133 117 found that about 10% of these system are radio luminous (Berger 2006; Antonova et al. 2008, 2013). The radio luminosity of the detected systems is far in excess of the well-known Güdel-Benz relation (Güdel & Benz 1993, GB), an empiricallyderived ratio between radio and X-ray luminosity (log Lr/Lx -15.5) that applies to magnetically active stars over a wide range of spectral types. A theoretical model that has been suggested to explain the GB correlation is the chromospheric evaporation model (Machado et al. 1980; Allred et al. 2006), in which heating and evaporation of chromospheric plasma (X-ray emission) is caused by non-thermal beamed electrons (radio emission) via the Neupert effect (Neupert 1968). However, the dozen UCDs detected in the radio to date (Berger 2002, 2006; Burgasser & Putman 2005; Phan- Bao et al. 2007; Antonova et al. 2008; McLean et al. 2011; McLean et al. 2012; Route & Wolszczan 2012) all violate the Güdel - Benz relation by orders of magnitude, suggesting that the chromospheric evaporation model does not apply to these stars. In 2006, an even more unexpected discovery was made: A few UCD stars had periodic, pulsar-like radio emission (e.g., Hallinan et al. 2007; Berger et al. 2009). The pulsed emission was 100% circularly polarized and occurred either once or twice per rotational period, depending on the observed frequency. The emission mechanism was attributed to the electron-cyclotron maser instability (ECM, Hallinan et al. 2008), which would account for both the high circular polarization and the apparent beaming of the radiation. The ECM mechanism has been well-studied in space and planetary environments, both theoretically (e.g., Treumann 2006, and references) and observationally

134 118 (e.g., Zarka 2004). The ECM instability couples the kinetic energy of electrons spiraling in converging magnetic fields to the ambient radiation field at the electrons gyro frequency, resulting in exponential growth of the radiation field at the local gyro-frequency. The radiation pattern is strongly beamed, since the growth rate is strongly peaked for wave vectors oriented normal the the magnetic field (Mutel et al. 2007). However, ECM growth is quenched unless the ratio of the electron plasma frequency is much less than the cyclotron frequency this implies that the source region has a high magnetic field and/or low density plasma. In a planetary magnetosphere such as the Earth, the ECM growth rate is highest in density-depleted auroral cavities at high magnetic latitudes (Ergun et al. 1998; Ergun et al. 2000), resulting in radiation at frequencies between 50 KHz and 500 KHz, i.e. magnetic fields between 0.02 and 0.2 Gauss. The radiation is dominantly in the extraordinary mode and is initially linearly polarized (Ergun et al. 1998), but becomes circular as the radiation is refracted upward and parallel to the magnetic field. The beaming pattern of terrestrial ECM emission, known as auroral kilometric radiation, is strongly modified by the auroral cavity, so that the resulting far-field pattern resembles a cigar-shape, with the long axis parallel to the auroral cavity (Mutel et al. 2008). Whether this also occurs in stellar magnetospheres is not known. For UCD stars, the pulses have been observed between 1.4 GHz and 10 GHz, which implies magnetic fields of order several kilogauss. These strong magnetic fields, although surprising in fully convective stars, may not be completely unexpected. Reiners & Basri (2007) studied the magnetically sensitive Wing-Ford FeH band in a

135 119 wide spectral range of M dwarfs, and found kilogauss surface fields, thus extending the direct measurement of magnetic field strengths to spectral type M9. Morin et al. (2010) proposed that late-m dwarfs have magnetic field topologies that can be classified into one of two types: (1) strong axisymmetric dipolar fields or (2) weak non-axisymmetric fields, with perhaps strong localized regions. These two topologies can be associated with distinctly different emissions. The UCDs that are radio-dim, X-ray bright, and closely follow the GB relation are believed to have strong axisymmetric fields, while the radio-bright, X-ray dim UCDs that break the GB relation are thought to have weak non-axisymmetric fields (McLean et al. 2012; Stelzer et al. 2012; Williams et al. 2013b). This paper reports multi-epoch radio observations of two well studied UCDs, 2MASS J (hereafter 2M J ) and TVLM (hereafter TVLM513-46). We used the new wideband capabilities of the Very Large Array 1 (VLA) to characterize both the temporal and spectral properties of the continuum and pulsed radio emission. For both stars, we constructed time-frequency (a.k.a. dynamic) spectra of Stokes I and V emission over several pulse periods at multiple epochs. We compared the pulse morphologies in these spectra with synthetic dynamic spectra generated using an oblique rotating magnetospheric model. By fitting the geometrical and beaming free parameters of the model, we determined possible source locations for the pulsed emission regions. We also fit spectral energy distributions of 1 The VLA is operated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

136 120 the non-pulsed continuum emissions to power-law gyrosynchrotron models to estimate the number density and mean magnetic field strength in the magnetosphere Target Star Properties 2M J is a dwarf binary system (L0 + L1.5) at a distance of 12.2 pc (Dahn et al. 2002). The orbit is elliptical (e = 0.49, Konopacky et al. 2010) with a semi-major axis 2.7 AU (Reid et al. 2001). The orbital inclination of the binary system is 41.8 ± 0.5 (Konopacky et al. 2012), and the equatorial and orbital planes are coplanar within 10 (Harding et al. 2013b). In addition to continuous radio emission (Antonova et al. 2008), 2M J has pulsed radio emission with a 2.07 hour period (Berger et al. 2009). Harding et al. (2013b) recently analyzed v sin i and periodic light curve variations and determined that the primary has a 3.32 hour period, implying that the less-massive secondary must be the origin of the 2.07 hr periodic radio emission. TVLM is an M9 dwarf located at a distance of 10.6 pc (Dahn et al. 2002). It was first detected at radio wavelengths by (Berger 2002) and has been extensively studied since (e.g., Osten et al. 2006b; Hallinan et al. 2006; Berger et al. 2008). Hallinan et al. (2007) found periodic, highly polarized radio pulses and both 5 GHz and 8 GHz, but with opposite circular polarizations. Berger et al. (2008) confirmed the presence of polarized flares at 8 GHz. and showed the pulse period was the same as periodic changes in H α equivalent width, although intensity variations of individual pulses were not correlated with H α variations. They suggested that both the pulsed radio and optical emission originates in a co-rotating chromospheric hot

137 121 Table 5.1. UCD Observing Epochs Epoch Array UT Range ν(ghz) 2M Nov 12 C 08:07 12: ; Dec 16 C 04:20 11: ; Dec 08 A 06:24 08: ; Dec 11 A 06:26 08: ; Dec 14 A 05:30 08: ; TVLM May 07 B 06:25 12: ; May 08 B 06:21 12: ; July 02 A 02:45 08: ; Dec 22 A 12:52 15: ; spot or an extended magnetic structure with a covering fraction 50%. 5.4 Observations and Data Analysis We used the wideband WIDAR correlator system at the VLA to observe 2M J at three epochs (8, 11, 14 Dec 2012) and TVLM at one epoch (22 Dec 2012), all during A array configuration. We observed simultaneously in two 1 GHz wide bands centered on 5.0 and 7.0 GHz. The receiver bandpass correction and absolute flux density was set using the amplitude calibrator 3C48. The angularly nearby source J was used for phase calibration. In addition, we calibrated and analyzed five archival wideband VLA observations of these stars (10B-209, Table5.1). These observations also used two 1 GHz bands, centered on 4.8 and 7.5 GHz. The data editing, calibration, and mapping was done using NRAO s Common Astronomy Software Application (CASA) using standard data reduction techniques. Observing details are summarized in Table 5.1.

138 122 After editing and calibrating, we made Stokes I and V snapshot maps for each source and epoch using time-frequency sampling intervals of 1 minute and 128 MHz. We then used CASA task IMSTAT to determine the peak flux density in a box centered on the source position and with dimensions equal to the restoring beam. These binned data samples were used to construct time-frequency plots for both sources. As discussed above, the radio emission has both a continuum and a pulsed component. In order to visually accentuate the (more intense) pulsed component in dynamic spectra plots, we filtered the binned data, replacing low signal-to-noise (SNR) bins (I < 6σ, V/I < 4σ) with zero flux density. On the other hand, In order to determine the spectral characteristics of the continuum emission, we analyzed only those time intervals with no pulsed emission. 5.5 RESULTS Dynamic spectra For both TVLM and 2M J , the dynamic spectra reveal pulsed emission with a complex, epoch-dependent frequency dependence. The temporal variations are presumably caused by the emergence and decay of active regions in each star s magnetosphere that are the origin of the pulses. The characteristic lifetime of these regions, inferred from the persistence of the pulse profiles, is of order months to several years. In section we model the the location and physical environment of these active regions. Dynamic spectra of the star 2M J in Stokes I and V/I are shown in

139 123 Fig. 5.2 for five epochs from 12 Nov 2010 to 14 Dec Note that for several epochs more than one pulse is shown. These correspond to observing times spanning several pulses periods. Also, only the lower observed frequency band is shown, since there was no detectable radio emission in the upper band except at epoch 2010 Dec 16, which we discuss separately in section The spectra have been phase-shifted so that the brightest pulse in each spectrum is located at phase 0 on average. These pulses which we refer to as the main pulse, show a clear evolution with epoch. The pulse phase was computed using the period hr (see section 5.5.2). At early epochs, the main pulse is weak and right-circularly polarized, but becomes left-circularly polarized with a frequency drift df/dt 2 MHz/sec starting at epoch 2010 Dec 16. This frequency drift becomes a stable feature of the main pulse for several years, up through at least the last observing epoch on 14 Dec There is also a secondary pulse at phase 0.75 at later epochs (2012 Dec 8,11,14). Curiously, this pulse is not significantly polarized. This is similar to the UCD star 2M , which Hallinan et al. (2008) found has a highly circularly polarized main pulse but a largely unpolarized interpulse. TVLM dynamic spectra for four epochs is shown if Fig. 5.3, with several epochs with longer observing intervals showing multiple pulses. As with 2M J , we phase-shifted the strongest pulse using a calculated period hr, as described in section Both the low and high bands are shown for TVLM since there is detectable emission in both bands at all epochs. The main pulse, which is nearly always right-circularly polarized, is brightest in the upper band, but is also

140 124 detectable in the lower band. It is also almost always a double pulse, with a phase separation 0.1 phase. There is also a secondary pulse near phase 0.35 seen most intensely in the lower band at three epochs. These pulse characteristics are significantly different from previously published pulse profiles for TVLM For example, Hallinan et al. (2007) detected two repeating LCP pulses at 4.9 GHz, but separated by 0.5 in phase, and two oppositely polarized pulses at 8.4 GHz, separated by 0.1 phase. The pulse characteristics are also systematically different from 2M J in that we detect the strongest pulses at much higher frequencies and do not detect any frequency drift. We discuss implications of these differences in terms on different rotation axis inclinations in section Pulse Periods In this section, we combine our observed main pulse (MP) times with previously published MP timings to calculate an improved estimate of the pulse period. To do this, we compute the sum of the absolute value of the phase difference between the observed and model MP weighted by the reciprocal timing uncertainty (σ) for each observed MP, f(ɛ) = N 1 min[φ i (ɛ), 1 φ i (ɛ)] σ i, (5.13) where min[] is the minimum function and φ i (ɛ) is the observed MP phase at epoch i computed using a trial period P (ɛ) = P 0 + ɛ, where P 0 is the previously published period for each star. We varied ɛ over the range ±10 sec in increments of 1 msec to find a minimum in the difference function f(ɛ).

141 Figure 5.2. Composite dynamic spectra of 2M J for each of the observations; the date of the observation and UT of the MP are given in the caption. The left panel displays Stokes I spectra while the right panel shows Stokes V/I. The phases are calculated using the period derived in section

142 Figure 5.3. Composite dynamic spectra of TVLM513-46for each of the observations; the date of the observation and the UT of the MP is given in the inset. The left panel show the Stokes I spectra while the right panel shows Stokes V/I. The phases are calculated using the period derived in section

143 127 It is important to note that for both stars, the observed MP timing datasets are unevenly sampled with several large gaps of several thousand periods between sampled epochs. For such datasets, it is difficult to determine whether phase jumps occurred in the gaps. This is a result of the ambiguity concerning the number of pulse counts between observations. Consider a series of consecutive pulses separated by a nominal period P and uncertainty σ. Now consider another series of pulses with the same period and uncertainty, but observed n periods later, where n 1. The latter pulses can be rectified with the previous series by fractionally adjusting the period by 1/n. If σ > 1/n, all pulses will be compatible with the adjusted period whether or not there was a phase jump in the gap. Hence, the period estimates given below may not provide a reliable ephemeris for prediction of the MP arrival time at future epochs M J pulse period In addition to the MP times determined in this study (shown in Fig.5.2 insets), we used MP times from Berger et al. (2009). We note that although Antonova et al. (2008) also observed a radio pulse from 2M J , we did not use it because it was unclear whether the pulse was a MP or an inter pulse. We used the published period of Berger et al. (2009) 2.072±0.001 hr as a starting point for the minimization search. The summed differences as a function of ɛ are shown in Figure 5.4. The minimum value corresponds to a correction ɛ = seconds, resulting in a revised period ± hr where the error is determined from a χ 2 analysis of the residuals. This period is consistent with the period uncertainty determined by

144 128 Figure 5.4. Normalized chi-square of the summed difference between the published main pulse period and trial period for 2M J The lowest minimum corresponds to a correction ɛ = seconds giving a revised period hr ± hr. This value lies within the period uncertainty range of Berger et al. (2009, blue bar). Berger et al. (2009) but improves its precision TVLM pulse period For TVLM there are several previously published observations with detected radio pulses (Berger 2002; Hallinan et al. 2007; Berger et al. 2008; Doyle et al. 2010). However, we only used pulse timings from Hallinan et al. (2007) in May 2006 and Doyle et al. (2010) in June The light curves in these papers show clear, sharp pulses with times that can be measured precisely. The observations also span more than one period, allowing us to determine whether the pulses are periodic or

145 129 Table 5.2. Period Analysis Results Star Period (hr) σ (hr) χ 2 Reference 2M J Berger et al. (2009) 2M J This paper TVLM Doyle et al. (2010) TVLM Harding et al. (2013a) TVLM This paper episodic flares. In addition to the published pulses, we use timings for main pulses presented in this paper. These times are listed in the insets of Figure 5.3. The reduced chi-square of weighted phase differences for all pulses as a function of period is shown in Figure 5.5(a). The lowest minimum has a correction ɛ = seconds relative to the period reported by Doyle et al. (2010), resulting in a revised period ± hr. This period is not within with the period uncertainty published by Doyle et al. (2010), shown by the blue uncertainty bars on the left panel of Figure 5.5. However, the Doyle et al. (2010) period is based solely on their 2007 observations, while our period is calculated using the much larger set of observed MP spanning more than four years. In addition to radio variability, TVLM is known to also show periodic optical variations. Harding et al. (2013a) measured an optical period ± hr using light curves at several epochs from The optical period is sec shorter than the best-fit radio period. To investigate whether this period could also account for the radio pulses, we plot in Figure 5.5(b) the reduced chi-square of weighted phase differences versus a time range that includes the optical period. We

146 130 Figure 5.5. (a) Reduced chi-square of weighted phase differences as a function of period relative to the published period of Doyle et al. (2010) for TVLM The smallest minimum differs by sec from the Doyle et al. (2010) period, resulting in a revised period ± hr. (b) Same as (a), but over a wider period range that includes the optical variability period observed by Harding et al. (2013b). The optical period, which does not fit the radio data, differs from the best-fit radio period by nearly 31 sec. indicate the optical period with a green arrow, the radio period from Doyle et al. (2010) with a blue arrow, and the period estimated in this paper with a red arrow. While all three periods correspond to minima, the hypothesis that the optical period can account for the radio data can be rejected at high confidence (χ 2 r = 5.5, df = 10, p < 10 5 ). Hence, we conclude that the 31 second difference between the radio and optical periods is real. This result can be compared with the optical-radio period difference found for the UCD 2MASS J McLean et al. (2011) found that the optical period is 6 ± 3 min shorter than the radio period. They suggest that the difference may be due to differential rotation between the equatorial and polar regions of this

147 131 object. This same phenomena has been observed in the magnetic chemically peculiar star CU Virginis as well, but the period difference is much smaller ( 1 sec, Ravi et al. 2010; Pyper et al. 2013) and is attributed to an unseen companion or instability in the emission region. We note that some of the light curves used by Harding et al. (2013a) correspond to dates where we have radio observations of TVLM (2011 May 7, 8). For these dates we find that the pulse peak in the optical is shifted from the peak in the radio by 0.5 phase. This phase shift between the radio and optical peaks has also been observed in other UCD s (Berger et al. 2009, 2008). All period determinations, both from this paper and by previous studies, are summarized in Table Quiescent emission In addition to pulsed emission, broad-band quiescent radio emission is detected from each source. To study this component, we first identified time intervals that do not include discernible pulsed emission. Cleaned maps were made by averaging over these time intervals in 512 MHz frequency windows. In each frequency window we measured the peak Stokes I and V flux density by fitting Gaussian profiles at the source location using the CASA task IMSTAT. Figure 5.6 shows the continuum Stokes I and V/I spectral energy distributions for each source at all observed epochs (cf. Table 5.1), along with gyro-synchrotron model fits as described in section below.

148 132 Table 5.3. UCD Spectral Indices Epoch α GHz 2M J Nov ± Dec ± Dec ± Dec ± Dec ±0.20 TVLM May ± May ± July ± Dec ± Spectral indices We used the peak I-flux measurements in the four 512 MHz bins to determine the spectral index for each observation (Table 5.3). The spectral indices are consistent with previously published measurements of -0.4 ± 0.1 (Osten et al. 2006b) for TVLM and -0.7 ± 0.3 (Berger et al. 2009) for 2M J Circular polarization Circular polarization (Stokes V/I, hereafter π c ) was not detected from TVLM for any of the quiescent emission observations, to a limiting fraction between 0.1 and 0.2, depending on epoch, This is in accord with Osten et al. (2006a) who found π c < 0.15 at both 5 GHz and 8.4 GHz. Likewise, we did not detect significant polarization from 2M J , except at epochs 2010 Nov 12 and 2010 Dec 16, where we detected π c 0.20 in the lower frequency band (Figure 5.6). Assuming gyrosynchrotron emission, this provides an approximate estimate of the angle-averaged

149 133 energetic electron energy, E 1 π c m e c MeV. (5.14) Brightness temperature The brightness temperature of an incoherent radio source as a function of distance d, flux density S ν, frequency ν, and effective diameter D can be conveniently expressed as, ( ) T b = Sν ( ( ) ν 2 2 ( ) 2 d D K, (5.15) mjy GHz) pc R J where R J is the radius of Jupiter. Using the measured quiescent emission flux densities and assuming the emission region sizes are of order a stellar diameter, the brightness temperatures for both UCDs are in the range (1 5) 10 9 K. These high brightness temperatures, combined with the fractional circular polarization detected in 2M J and the power-law spectra all rule out thermal emission, but support a model of optically-thin gyro-synchrotron radiation from a population of mildly relativistic power-law electrons (see section 5.6.2). We note that the spectral energy distributions for both stars are relatively stable over years. This long-term stability has also been observed for quiescent emission form other UCDs (e.g., Osten & Wolk 2009), and is in marked contrast to other radio-loud stellar sources, such as RS CVn binaries. A power-law gyrosynchrotron process is also usually invoked for these sources, but unlike the UCD s, they produce large flares, with dramatic changes in flux density, spectral index, and fractional polarization (e.g., Mutel et al. 1998; Richards et al. 2003). This may reflect a difference

150 134 in energization, wherein flares in active binaries may be energized by large-scale magnetic interactions between components (Richards et al. 2012), while UCD s may be energized by quasi-continuous, low-level magnetic reconnection events Williams et al. (2013a). 5.6 Coronal Models Pulsed Emission: ECM sources on isolated loops The temporal behavior of frequency and polarization of ECM-driven pulses provide robust constraints on the topology of the stellar magnetosphere, provided the source-dependent parameters (e.g. angular beaming, refraction) can be tenably modeled. The key idea is that each ECM emission frequency directly maps the local magnetic field strength, and the rotation of the star provides time-lapsed spatial slices of the region of the magnetosphere favorably aligned to the beamed radiation. The properties of ECM-driven radiation depend on the plasma conditions at the emission site. For low density plasma (ratio of electron plasma to cyclotron frequency ω pe /Ω ce << 1), and a loss-cone electron phase distribution, the dominant mode is R-X at the fundamental harmonic. This implies the radiation is rightcircularly polarized at a frequency very close to the electron cyclotron frequency. This is the most commonly observed ECM mode in planetary magnetospheres (e.g. Zarka 1998). However, for stellar magnetospheres, this may not apply. For example, stellar ECM regions could have larger plasma to cyclotron frequency ratios that result in higher harmonics and/or L-O mode being the dominant emission (Lee et al. 2013). Since we have no independent information on the plasma conditions at the

151 TVLM Spectral Energy Distribution 190 2M J Spectral Energy Distribution Flux density (µjy) Flux density (µjy) (S X - S O )/(S X + S O ) A Frequency [GHz] C TVLM Fractional Circular Polarization Frequency [GHz] (S X - S O )/(S X + S O ) B Frequency [GHz] D 2M J Fractional Circular Polarization Frequency [GHz] Figure 5.6. Comparison between model and measured values for the total flux and polarization of the observed quiescent emission. (A) TVLM peak I-flux for: May (blue), May (red), July (green), and December (magenta). The model values are given by the solid line where the color corresponds to the fitted measured values; since May 7 and have similar measured peak fluxes we model both sets of data with the black line. The corresponding model and upper limits to the CP for TVLM are shown in (C). (B) 2M J peak I-flux for: November (green), December (magenta), December (cyan), December (red) and December (blue). The modeled and constrained CP for the 2010 observations of 2M J is given in (D). While our model fits the total flux measurements for each of the objects, the modeled CP does not accurately represent that measured for 2M J ECM sites, we have applied Occam s razor and assumed the planetary case for the model viz. R-X mode at the fundamental harmonic.

152 136 We modeled the observed dynamic spectra with a set of ECM emission sources fixed in an oblique rotating coordinate system (i.e., the star s rotation axis inclination). Since the pulses have a short duty cycle, we assume there exists a small number of localized magnetic loops on which the ECM instability is active. Each source location is chosen so that the emission frequency is equal to the local electron gyro-frequency. Since unstable electron beams will populate an entire L-shell field line, we also assume that there exists a pair of ECM sources at conjugate points on each active L-shell field line. As viewed by an observer whose location intercepts the angular pattern of both sources, the pulses from conjugate locations will have opposite circular polarizations. For globally dipolar fields, helicity is used to map opposite magnetic hemispheres e.g. at Saturn (Lamy et al. 2008). The angular beaming pattern of stellar ECM emission is unknown, but multispacecraft studies of terrestrial ECM emission (known as auroral kilometric radiation, Gurnett 1974) have shown that the emission is elliptically beamed with the major axis zonally aligned, i..e, along the density-depleted auroral cavities (Mutel et al. 2008). The beaming angular width is frequency-dependent, with higher frequencies subtending large opening angles, consistent with refraction in a dispersive medium (e.g., Menietti et al. 2011). The physical conditions in planetary magnetospheres are likely much different from stellar coronae, but the narrowness of the pulses suggests that stellar ECM radiation is also beamed. The beaming opening angle could also be frequency dependent, since the coronal plasma will be dispersive. Hence we include a frequency-dependent

153 137 elliptical beaming weighting function, Table 5.4. ECM pulsed emission model parameters Geometrical parameters Beaming parameters Star Loop θ los θ B B eq L N,S φ m ν 0 θ 0 σ θ φ σ φ 2M J A kg 1.7 N GHz M J B kg 2.5 N GHz M J C kg 1.25 S 60 5 GHz TVLM A kg 2.7 N GHz TVLM B kg 1.7 N GHz TVLM C kg 2.0 N GHz { ( ) } { 2 ( ) } 2 θ θc φ W (θ, ϕ) = exp exp, (5.16) σ θ where angles (θ, σ θ ) are the emission cone opening angle and width measured with respect to the magnetic field tangent direction at the source, (φ, σ φ ) are the tangent plane angle and width, measured in a plane normal to the magnetic field, with φ = 0 along the direction given by the cross product of the tangent vector and a vector pointing toward the magnetic pole (Mutel et al. 2008). The frequency scaling of the cone opening angle is given by σ φ θ c (ν) = θ 0 ( ν ν 0 ) β. (5.17) The parameters θ 0, ν 0, β, σ θ, θ c, and σ φ are adjusted for best-fit to the observed dynamic spectra. In addition, we varied the rotation axis inclination angle (θ los ), the inclination of the magnetic field loop with respect to the rotation axis (θ B ), the loop

154 A A Loop B : : : : : :45 UT :27 9: : : : Loop C B B Loop A : : : : : :45 UT : : : : : Figure 5.7. Comparison between observed and modeled dynamic spectra for 2M J (left) Observed dynamic spectrum of 2M on 16 Dec 2010 from 04:20 to 11:11 UT in the frequency bands GHz and GHz. All pulses are left-circularly polarized, except the pulse labeled 1, which is rightcircularly polarized. This is shown in more detail in Figure 5.9. (right) Model dynamic spectrum from 4 8 GHz from rotating oblique coronal model using parameters in Table 5.4.

155 A 8.0 A Loop B : : : : : : : : : Loop A :45 B : : : :36 UT : : : : B Loop C Figure 5.8. Comparison between observed and modeled dynamic spectra for TVLM (left) Observed dynamic spectrum of TVLM on 2 July 2011 from 02:45 to 08:26 UT in the frequency bands GHz and GHz. (right) Model dynamical spectrum from 4 8 GHz from rotating oblique coronal model with parameters in Table 5.4

156 140 I1 Loop B I1 Loop C I2 Loop A I2 V1 Loop B V1-1.0 V2 Stokes V/I Loop C Loop A V2 1.0 Figure 5.9. Comparison between observed and modeled dynamic spectra for sloped feature of 2M J Dynamic spectrum of 2M J on 16 Dec 2010 from 04:40 to 06:25 UT in the frequency bands GHz (I1: Stokes I, V1: Stokes V) and GHz (I2: Stokes I, V2: Stokes V). The corresponding model dynamical spectra from 4 8 GHz from a rotating oblique coronal model with parameters given in Table 5.4 are shown in the right-hand panels.

157 141 Loop A Loop A Loop B Loop C Loop B Loop A Loop C Loop B Loop C TVLM Orientation 1 TVLM Orientation 2 TVLM Orientation 3 Loop A Loop B Loop B Loop B Loop A Loop A Loop C Loop C 2M J Orientation 1 2M J Orientation 2 2M J Orientation 3 Figure Perspective views of coronal loop models for TVLM (top) and 2M J (bottom) described by parameters in Table 5.4. Location of ECM sources at 4 GHz (purple dots), 6 GHz (green dots), and 8 GHz (red dots) are shown at the locations on the coronal loop corresponding to the local electron gyro-frequencies. The locations of sources beamed toward the observer are shown with yellow dots.