Dust Dynamics in the Galactic Disk-Halo Vicinity

Transcription

1 Dust Dynamics in the Galactic Disk-Halo Vicinity by Guillaume Rivest A thesis submitted to the Department of Physics, Engineering Physics and Astronomy in conformity with the requirements for the degree of Master of Science Queen s University Kingston, Ontario, Canada September 2007 Copyright c Guillaume Rivest, 2007

2 Abstract We report on the dynamics of interstellar dust grains above the plane of the Milky Way. Our Galactic model closely matches its large-scale observed properties, namely the luminosity output, the gas content and distribution, as well as the extinction of starlight by dust. The gravitational model is composed of a central bulge, a disk and a dark matter halo. In spite of the fact that the emphasis in the results is for our Galaxy, we also discuss the effects of varying key galactic parameters, such as the total luminosity output and gas distribution. The parameter space of the main dust grain properties is also explored; these include the grain type (graphite and silicate) and size (0.001 µm µm). The grains were launched at various positions in the Milky Way, but always in the disk-halo connection region; namely at three Galactocentric radii (5, 8 and 11 kpc) and three initial heights (150, 300 and 1000 pc). The grains were subject to radiation pressure, a gravitational force, as well as Coulomb and collisional drag with the gas. Due to the large size of the parameter space, there is a wide variety of possible grain dynamics. Indeed, depending on the physical conditions in which the grains were launched, the grains could either: (1) fall down towards the midplane, (2) stay at the same height, (3) rise at a more or less constant speed, (4) rise and then fall down, (5) rise and then stabilize at some height, and (6) be quickly expelled into the i

3 intergalactic medium. In general, graphite grains reach greater heights than silicate grains. The smaller grains (of radius a = 0.01 and µm) tend to stay at the same height they started at. The classical grain (a = 0.1 µm) is the most sensitive to radiation pressure and usually reaches the highest heights, even if they are modest. The largest grain we have studied (a = 0.3 µm) also responds well to radiative forces, but its large mass prevents it from going as high as the classical grain, and it even falls down towards the midplane under some circumstances. ii

4 Acknowledgments I am grateful to my supervisors Richard N. Henriksen and Judith Irwin for their continuous support and guidance, as well as to my family and friends for everything else. iii

5 Table of Contents Abstract i Acknowledgments iii Table of Contents iv List of Tables vii List of Figures viii Chapter 1: Introduction Chapter 2: Observations at High Altitudes Dust Absorption Dust Emission Dark Filaments Polarization of Starlight by Dust Magnetic Field iv

6 Chapter 3: Galactic Environment Gas Dust and Stellar Radiation Interaction Interstellar Dust Chapter 4: Forces Acting on Dust Gravitational Force Drag Force Radiative Force Lorentz Force Chapter 5: Procedure and Initial Conditions Chapter 6: Results Parameter Dependence Discussion Chapter 7: Conclusion Appendix A: Additional Graphs v

7 Appendix B: Alternate Galactic Gas Model vi

8 List of Tables 3.1 Grain Properties Gravitational potential constants. The numbers between parentheses correspond to the subscripts in the text Summary of the four Cases. The corresponding object to each Case is loosely defined Summary of the initial conditions for each Case described in Table B.1 Gas parameters according to the Wolfire et al. (2003) model vii

9 List of Figures 2.1 Optical view of the edge-on galaxy NGC 891, from APOD Milky Way, from APOD The JCMT 850 µm contour map overlayed on a DSS optical image. The contour levels are 6, 10, 14, 20, 30, 40, 50, 60,70 and 80 mjy beam 1. (Figure copied from Brar & Irwin 2003) Left: Sketches of the various forms of dark filaments for the galaxy NGC 253. Right: Sketches of the various dark filaments shapes seen in the left Figure. From top to bottom, Sofue et al. (1994) coined them as: arcs, loops and/or bubbles, vertical streamers, and short dust filaments. (Figure copied from Sofue et al. 1994) Polarization of optical starlight by dust grains. The magnetic field is in the same direction as Ω, the axis of rotation. (Figure from the Planck website -but the actual webpage does not exist anymore) Magnetic field direction from starlight polarization by dust. (Figure from Fosalba et al. 2002) viii

10 2.7 Strength of the total magnetic field in the Galaxy, averaged from the deconvolved surface brightness of the synchrotron emission at 408 MHz (Beuermann et al., 1985), assuming energy equipartition between magnetic field and cosmic ray energy densities (Berkhuijsen, personal communication). The accuracy is about 30%. The Sun is assumed to be located at R=8.5 kpc. (Caption and figure copied from Beck 2001) VLA radio continuum at 4.86 GHz maps with total power contours (left) as well as contours of polarized intensity (right) and with observed B-vectors (B = magnetic) of polarized intensity overlaid onto the Hα image of NGC Contours are 3, 8, 21, 55, 144, 377 and µjy for total power and 3, 5, 8, 13, 21 and 34 7 µjy for polarized intensity. The B-vectors of length of 1 correspond to 10 µjy. The beam size is 16. (Caption and figure copied from Dettmar et al. 2001) GHz total intensity map with 40 resolution, contours: -1, 1, 2, 4,..., 100 µjy/beam, overlaid on an optical R-band image. Vectors indicate the magnetic field orientation. (Caption and figure copied from Golla et al. 1994) Radial dependence of the gas density in the Galactic plane. He is not included Height dependence of the gas density at the solar radius. He is not included The effective cross-section Q ext as a function of the size parameter X. Also shown are the effective absorption and scattering cross-sections Q abs and Q sca, respectively. (Figure copied from Whittet 2003) ix

11 3.4 Coordinate system used for optical depth calculations Optical depth as a function of z and x (= R, since y and y s = 0). The sample star is located at the origin The perpendicular force, K z, as a function of z (thick solid line). The points shown are the data of Oort (1960) for two values of the local density, ρ = 0.15 M pc 3 (dots) and ρ = 0.18 M pc 3 (triangles). Also shown are the values obtained by Bahcall (1984) for his proportionate model with a massive halo (crosses). (Caption and figure copied from Allen & Santillán 1991) Circular velocity Coulomb and collisional drag as a function of the relative velocity V between the dust and gas, for a = 0.1 µm, z = 300 pc and R = R Total drag as a function of the relative velocity and Galactic height. The color scheme represents log F drag (dynes) Location of the bins in the xy-plane. All the bins are separated by 250 pc, radially and azimuthally. Since the circumferences at various radii are not exact factors of 250 pc, there exist larger gaps along y s = 0, for x s < 0. This effect is clearly negligible, because the grains were launched at x s > Schematic of the extended magnetic lobes inflated outward from both faces of the gaseous disk of the galaxy. (Caption and figure copied from Parker 1992) Graphite grains, R i = 8 kpc, z i = 300 pc Silicate grains, R i = 8 kpc, z i = 300 pc x

12 6.3 Graphite grains, R i = 5 kpc, z i = 300 pc Graphite grains, R i = 8 kpc, z i = 300 pc Graphite grains, R i = 11 kpc, z i = 300 pc Graphite grains, R i = 8 kpc, z i = 150 pc Graphite grains, R i = 8 kpc, z i = 1 kpc Graphite grains, R i = 8 kpc, z i = 1 kpc, u z = 2 km/s Graphite grains, R i = 8 kpc, z i = 1 kpc, u z = 30 km/s Graphite grains, R i = 8 kpc, z i = 1 kpc, L 0 = 5 L 0,MW Graphite grains, R i = 8 kpc, z i = 1 kpc, L 0 = 10 L 0,MW A.1 Silicate grains, R i = 8 kpc, z i = 150 pc A.2 Silicate grains, R i = 8 kpc, z i = 1 kpc A.3 Silicate grains, R i = 5 kpc, z i = 300 pc A.4 Silicate grains, R i = 11 kpc, z i = 300 pc A.5 Graphite grains, R i = 8 kpc, z i = 150 pc, u z = 2 km/s A.6 Graphite grains, R i = 8 kpc, z i = 150 pc, u z = 30 km/s A.7 Graphite grains, R i = 8 kpc, z i = 300 pc, u z = 2 km/s A.8 Graphite grains, R i = 8 kpc, z i = 300 pc, u z = 30 km/s A.9 Graphite grains, R i = 8 kpc, z i = 150 pc, L 0 = 5 L 0,MW A.10 Graphite grains, R i = 8 kpc, z i = 150 pc, L 0 = 10 L 0,MW A.11 Graphite grains, R i = 8 kpc, z i = 300 pc, L 0 = 5 L 0,MW A.12 Graphite grains, R i = 8 kpc, z i = 300 pc, L 0 = 10 L 0,MW xi

13 B.1 Radial dependence of the gas density in the Galactic plane. He is included. The sudden drop of atomic hydrogen at R = 13 kpc is caused by the different parameters Wolfire et al. (2003) used for the HI beyond this radius B.2 Height dependence of the gas density at the solar radius. He is included.103 B.3 Graphite grains, R i = 8 kpc, z i = 300 pc, Wolfire et al. (2003) gas model xii

14 Chapter 1 Introduction It is now well known that there is an inflow/outflow of matter between galactic disks and halos, as gas, dust and heavy elements have been observed in the halos of several spiral galaxies. More specifically, various forms of filaments and loops of dust above the disk have been reported (Sofue et al. 1994), thus suggesting that energetic events such as supervona(e), stellar winds and/or instabilities in a given toroidal (azimuthal) galactic magnetic field might play a significant role in the expulsion mechanism. Indeed, since one does not expect to observe dust and gas in the halo, some form of ejection mechanism has to originate from the disk. In this report, we investigate the effect of radiation pressure on the dust dynamics in order to see if starlight alone can expel the grains into the halo of the Galaxy. However, being aware of the aforementioned possible expulsion mechanisms, we also try and simulate these mechanisms by using different initial conditions. Studies of the expulsion of dust grains under radiation pressure have been made in the past (Davies et al. 1998, Shustov & Vibe 1995 and Ferrara et al. 1991), but since the parameters considered by these authors were different, they sometimes obtained different results; thus showing 1

15 CHAPTER 1. INTRODUCTION 2 the sensitivity of the dust dynamics upon the galactic and dust model used. Shustov & Vibe (1995) launched graphite and silicate grains from the Milky Way at a Galactic radius of 2 kpc and a height of 1 kpc, and found that both types of grains are expelled from the Galaxy at speeds ranging from 10 2 to 10 3 km/s. In their model, they took the dust sputtering into account, but they neglected the influence of the starlight extinction caused by dust. Also, their luminosity output from O,B and A stars seems to be overestimated for the Milky Way, with a total luminosity of L within 14 kpc, about a factor of ten too large. Ferrara et al. (1991) did a similar analysis to that of Shustov & Vibe (1995), and consequently obtained similar results. However, the model developed by Shustov & Vibe (1995) is more detailed than the one of Ferrara et al. (1991); thus, henceforth we will refer to the Shustov & Vibe (1995) model and not to the one of Ferrara et al. (1991). Ferrara et al. (1991) did not include in their study the size distribution of the grains, and, according to Shustov & Vibe (1995), they underestimated the interstellar gas density (0.1 cm 3 at the Galactic center). The analysis of Davies et al. (1998) was done for a typical spiral galaxy, and their study clearly showed that the absorption of starlight by dust along the line of sight is an important factor in the dust dynamics even at a few hundred pc from the midplane of the disk. Depending on the extinction normalization used, the dust grains could either fall into the plane or rise to 1 kpc above the disk, a significant difference compared to the kpc obtained from the aforementioned models. Unlike the Shustov & Vibe (1995) model, Davies et al. (1998) neglected the Coulomb drag, which dominates the collisional drag at velocities around and below the thermal gas velocity. Thus, with the addition of the Coulomb drag, it is harder for a grain to accelerate when it moves at sub-thermal gas speeds, hence justifying the need to

16 CHAPTER 1. INTRODUCTION 3 include this form of drag. It is also interesting to note that the grains also move in the radial direction, up to a few kpc inwards in the Davies et al.(1998) model and up to several tens of kpc outwards in the Shustov & Vibe (1995) model. Our results will be compared with those of these two papers in Section 6.2. Thus, in order to study such a dust ejection mechanism based on radiation pressure, one has to model very carefully the various fundamental properties of the dust grains -such as their size distribution, charge and chemical composition-, not to mention the importance of knowing the details of the galactic environment in which they propagate, the interstellar medium (ISM). Hence we have modeled the various components of the Milky Way according to its observed properties on large scales, thus creating a realistic environment in which the dust grains should move. Following the Introduction, Chapter 2 summarizes the observations of dust and magnetic fields in the halos of a few spiral galaxies, as the discussion of dust expulsion by starlight is certainly not limited to the Milky Way. In Chapter 3 we define the details of our model in terms of the galactic environment and dust grain properties, whilst in Chapter 4 the forces included in the dust equation of motion are described. Chapter 5 solely displays the initial conditions used and the results are shown in Chapter 6. A short discussion and conclusion in Chapter 7 will bring this thesis to a close.

17 Chapter 2 Observations at High Altitudes There exist various ways to detect the presence of dust in interstellar environments, both for the Milky Way and distant galaxies. In this Chapter, although the results of our theoretical model are mostly worked out for our own Galaxy, dust and magnetic field observations in other spirals are also considered. Such observations allow us to see that the dust distribution perpendicular to various types of galactic disks varies from one another, as some are dust rich and others dust depleted. 2.1 Dust Absorption The presence of dust can be inferred from its obscuration of optical and ultraviolet (UV) light. This observational technique typically provides us with higher resolution images of the various dust features than the emission technique, as can be seen by comparing Figure 2.1 and Figure 2.3. Indeed, the details of the extraplanar dust features in Figure 2.1, an optical image of NGC 891, are finer than in the emission case. However, since this method requires the starlight from the galaxy in the background to be present, it allows us to see dust at limited heights from the galactic midplane; 4

18 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 5 Figure 2.1: Optical view of the edge-on galaxy NGC 891, from APOD. one must then resort to the infrared dust emission to detect their presence at greater heights. Moreover, it should be mentioned that one must account for the inclination of the observed galaxy in order to obtain the correct dust height. A view of the Milky Way is provided in Figure 2.2, as seen from Stagecoach, Colorado, USA. In Figures , although most of the dust clearly lies close to the midplane, extraplanar dust is also evident in both cases. In fact, Howk & Savage (1999) observed several spiral galaxies, and although some were excluded due to observational limitations, 5 out of the 7 spiral galaxies of their final sample exhibited high-z dust features. It is also interesting to note that they not only concluded that

19 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 6 Figure 2.2: Milky Way, from APOD high-z dust is a common property of large spiral galaxies; they also observed high-z Diffuse Ionized Gas (DIG) in these same 5 galaxies (whilst the DIG was not observed in the 2 where dust was not observed). This suggests that when halo material is present, it tends to be present in various phases. 2.2 Dust Emission As is well known, due to their internal vibrational motions, dust grains re-emit in the infrared the optical and UV light they have absorbed. The exact wavelength at which a given grain emits this radiation varies according to a variety of factors, such as the grain composition, shape, size and temperature. Figure 2.3 is a contour map of this emission in the far-infrared (850 µm) for the nearly edge-on galaxy NGC For this galaxy, dust grains have been observed at heights as high as 5 kpc above the

20 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 7 Figure 2.3: The JCMT 850 µm contour map overlayed on a DSS optical image. The contour levels are 6, 10, 14, 20, 30, 40, 50, 60,70 and 80 mjy beam 1. (Figure copied from Brar & Irwin 2003)

21 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 8 Figure 2.4: Left: Sketches of the various forms of dark filaments for the galaxy NGC 253. Right: Sketches of the various dark filaments shapes seen in the left Figure. From top to bottom, Sofue et al. (1994) coined them as: arcs, loops and/or bubbles, vertical streamers, and short dust filaments. (Figure copied from Sofue et al. 1994) galactic plane. It should also be noted that dust has been detected at large galactic radii, even further than the star forming disk; thus suggesting that supernovae are not the only expulsion mechanism. 2.3 Dark Filaments As can be seen in the sketches of Figure 2.4, dust above galactic planes is observed in various forms. Indeed, the dust features comprise dark arcs, loops and/or bubbles, vertical streamers, and short dust filaments. Sofue et al. (1994) describe the arcs as being arch-like structures, usually extending between different dust clouds in the disk. They are mostly observed in the dense spiral arm regions, and their size ranges from a few hundred pc to 1 kpc in length and about 100 to 300 pc in height. They also argue that their shape suggests a relation with Parker-type magnetic instabilities, which we will discuss in Section

22 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 9 The loops are like the arcs, but they are more developed in the sense that although their length is similar, the loops reach higher heights (for NGC 253, up to 700 pc). Due to observational limitations, the three-dimensional structure of these loops is hard to discern; hence they may well be bubbles. Now, the vertical streamers are elongated dust lanes perpendicular to the plane and, assuming that the diffusion of charged grains perpendicular to a given horizontal magnetic field is negligible, these features imply the presence of vertical magnetic field lines. They are typically a few tens of pc wide and can reach between 200 pc and 1 kpc in height. They are numerous in the central galactic region, but are observed outside the nuclear region as well. Finally, the short vertical filaments are like the vertical streamers, but with smaller dimensions: ten pc or less wide. Their smaller size thus puts serious constraints on the resolution needed to detect them, but their existence is nonetheless indisputable. The number and size of these dark dust lanes, as well as the variety in their morphologies, vary from one spiral galaxy to another. Nevertheless, NGC 253 is certainly not a special case, in the sense that these features indeed grossly represent the variety of morphologies in which dust is observed. It should be mentioned, however, that in addition to these aforementioned dust structures, there also exists a more extensive distribution. Indeed, Figure 2.3 hints at such a smoother distribution; although one could argue that it appears thus simply due to poor resolution. Still, the variety of morphologies in which dust lanes are seen tells us that dust is expelled in the diskhalo region by more than one physical phenomenon and that all types of dust grains do not react the same way to these physical phenomena. Surely, dust grains expelled by supernovae will reach greater heights than those expelled by one supernova, or by stellar radiation alone for example. Thus, as the age of the Galaxy, or any disk galaxy

23 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 10 in our surroundings, is several times larger than the time scales involved in these ejection phenomena (typically yr), it is very likely that we often observe the result of a mixture of past events, which smoothes out the dust distribution. This interpretation then leads to the fact that dust is not necessarily in molecular clouds; we can also consider the dynamics of intercloud dust. Moreover, the gas to dust ratio is approximately constant in the Milky Way (Binney & Merrifield 1998); therefore dust also resides outside of the discrete features. 2.4 Polarization of Starlight by Dust Yet another way to infer the presence of dust is from the polarization of starlight. Clearly, the grains have to be non-spherical for them to polarize light, and a magnetic field has to be present so that the alignment between the field lines and the grains can take place. Indeed, elongated dust grains are aligned with their long axis perpendicular to a given magnetic field line. The physical mechanism behind this phenomenon has been a matter of debate for several decades. The main processes that were proposed originally (e.g. Davis et al. 1951, Gold 1952) fall in two main categories: (i) processes involving magnetic relaxation in spinning grains and (ii) dynamical alignment caused by the streaming of gas relative to dust (Hoyle et al and references therein). The main issue with the former hypothesis is that for a typical µg galactic magnetic field, the magnetic time scale is greater than the relaxation time of the grains by 2-3 orders of magnitude. The alignment is not possible in this scenario, because the situation is repeatedly randomised before dissipative effects can become important (Hoyle et al. 1991). Thus, the magnetic forces at play are too weak to produce the observed alignment. In terms of the latter proposition, namely

24 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 11 Figure 2.5: Polarization of optical starlight by dust grains. The magnetic field is in the same direction as Ω, the axis of rotation. (Figure from the Planck website -but the actual webpage does not exist anymore) the alignment of the grains due to the gas streaming, it turns out that the alignment it predicts is along the magnetic field lines, which contradicts the observations. More recently, Draine & Weingartner (1997) showed that the alignment can be caused by an anisotropic distribution of starlight, which naturally occurs in a galactic context. Thus, the anisotropic starlight will make the grain spin at very high speeds, and it is the torque induced from this rotation that will gradually align the grain. This process is not instantaneous; it takes around 1 Myr to align a dust grain. Although this may seem like a long time, it is still small compared with the lifetime of a grain and hence the grains spend most of their lifetime aligned. As is shown in Figure 2.5, when a magnetic field is perpendicular (or has a perpendicular component) to the optical starlight we receive, the dust grain will block the component along the longer side of the grain. Hence only the vertical component of the optical light will be transmitted, and the grain will emit in the infrared the optical light it had absorbed along its long axis (horizontal in the figure). It is also

25 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 12 Figure 2.6: Magnetic field direction from starlight polarization by dust. (Figure from Fosalba et al. 2002) important to note that if the magnetic field is parallel to the line of sight, no polarization will be produced since the dust grain would now be rotating in the plane perpendicular to the line of sight. Although this technique is a good one to find magnetic field directions, it cannot provide any information about its strength or about the dust density. Moreover, observations are limited to the solar neighbourhood, since dust blocks significant amounts of the light from stars further away. This can be seen in Figure 2.6, in which we see that most of the information is provided by sources within 1 kpc of the sun.

26 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 13 Consequently, even though this figure only provides us with the magnetic field direction, the fact that there is data at high Galactic latitudes reveals the presence of both magnetic fields and aligned dust grains at these high latitudes. 2.5 Magnetic Field In astrophysical contexts, magnetic fields are important from the scale of the earth up to, perhaps, the scale of clusters of galaxies. On galactic scales, magnetic fields play an important role in the vertical equilibrium of the gas, because the magnetic pressure has a similar contribution as the thermal and cosmic ray pressure. Its structure in the galactic halo, i.e. whether it is open-ended or closed, has implications for the origin of the cosmic rays that we detect. Indeed, a closed magnetic configuration would both confine the cosmic rays (or any other charged particle, dust for example) produced in our Galaxy to stay in our Galaxy, and block extragalactic cosmic rays. There are, of course, many other examples. The study of the Galactic magnetic field is one of those subjects in astrophysics in which the observations have better constraints on reality than the theories. There are several techniques used to detect the presence of magnetic fields in the Galaxy; some only detect its direction, others its magnitude only, while some can do both. The four main techniques are Zeeman splitting, synchrotron radiation, Faraday rotation and the polarization of starlight by dust. Describing the global structure of the magnetic field in the Galaxy is a very challenging task; the major difficulty comes from the fact that we are embedded in it and hence we do not have a face-on view of it. Still, looking at Figure 2.6, which displays the polarization of optical starlight by dust, we clearly see that the field in

27 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 14 Figure 2.7: Strength of the total magnetic field in the Galaxy, averaged from the deconvolved surface brightness of the synchrotron emission at 408 MHz (Beuermann et al., 1985), assuming energy equipartition between magnetic field and cosmic ray energy densities (Berkhuijsen, personal communication). The accuracy is about 30%. The Sun is assumed to be located at R=8.5 kpc. (Caption and figure copied from Beck 2001) the disk is mainly horizontal. There are also several hints of vertical field lines (i.e. perpendicular to the disk), which strongly suggest that some form of ejection process is taking place Strength In this Section, we will look at the magnitude of the magnetic field in the disk as a function of Galactic radius. The results from the synchrotron emission, and hence from the equipartition assumption between the magnetic field and the cosmic rays, are shown in Figure 2.7. It is important to note that in this graph, it is the total magnetic field that is plotted, namely the regular plus the random component. One also has to use the polarization measurements in order to disentangle the contribution from each component. It turns out that the regular component, B reg, is approximately equal

28 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 15 to 0.6B t, where B t is the total magnetic field magnitude (Beck 2001). However, this approximation is not necessarily true everywhere, i.e. in the spiral arms where there is more star formation and supernovae explosions, the turbulent component can be stronger than the regular one. Nonetheless, if we assume this approximation holds, we find: B reg (3kpc) 6µG, B reg (R ) 4µG and B reg (15kpc) 2µG. Also, although the results are not shown in Figure 2.7, Vallée (1997) confirmed that the field in the center of the Milky Way can be as high as a few mg (within 100 pc), while it is 130µG within 500 pc. The field in the Galactic center, however, is seen in vertical filaments, very likely due to outflows and/or jets. The magnitude of the magnetic field is an important factor in the dust dynamics, because as we will see shortly in Section 3.3, dust grains are charged particles. Therefore the grains will be more or less influenced by the field depending on its strength NGC 5775 and NGC 4631 Distant, edge-on galaxies, can also provide us with valuable information with respect to possible magnetic field configurations in the disk-halo connection. The two galaxies shown in this Section, NGC 5775 in Figure 2.8 and NGC 4631 in Figure 2.9, are wellknown for their vertical magnetic field configuration in the halo. Indeed, the vertical field lines reach 5 kpc and more from the midplane. It is also interesting to note that, according to the authors of these papers (Dettmar et al and Golla et al. 1994), there exists a correlation between the magnetic structure in the halo and the streaming of the gas. We will discuss this very important interaction in more detail in Section 4.4. This relation between the field structure and the gas can be seen in Figure 2.8 by comparing the two radio-continuum maps: the

29 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 16 Figure 2.8: VLA radio continuum at 4.86 GHz maps with total power contours (left) as well as contours of polarized intensity (right) and with observed B-vectors (B = magnetic) of polarized intensity overlaid onto the Hα image of NGC Contours are 3, 8, 21, 55, 144, 377 and µjy for total power and 3, 5, 8, 13, 21 and 34 7 µjy for polarized intensity. The B-vectors of length of 1 correspond to 10 µjy. The beam size is 16. (Caption and figure copied from Dettmar et al. 2001)

30 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 17 Figure 2.9: 4.86 GHz total intensity map with 40 resolution, contours: -1, 1, 2, 4,..., 100 µjy/beam, overlaid on an optical R-band image. Vectors indicate the magnetic field orientation. (Caption and figure copied from Golla et al. 1994)

31 CHAPTER 2. OBSERVATIONS AT HIGH ALTITUDES 18 left panel shows the total power whilst the right panel displays the polarized intensity only. The polarization features are clearly anchored in the disk, most likely to a star forming region or a supernova.

32 Chapter 3 Galactic Environment Studying the dynamics of interstellar dust grains from their formation sites, through their evolution into the ISM and possibly into the intergalactic medium (IGM) obviously requires a significant amount of knowledge and approximations on various physical aspects. Indeed, extensive information on the galactic environment and dust grain properties is necessary in order to obtain realistic grain dynamics, in whichever environment the grain is. Typical examples of these environments are molecular clouds, expanding shells/bubbles, supernovae shocks, or in our case in the disk-halo interface and halo. Dust grains are either formed, destroyed, or else they simply evolve with ever-changing properties (e.g. grain charge, mass, composition) depending on the environment they are in. In this Section, we describe how we modeled the various components of this galactic environment, which comprises gas, stars and dust. 19

33 CHAPTER 3. GALACTIC ENVIRONMENT Gas The gas content and distribution in the ISM are very important factors for dust dynamics studies. Indeed, since the gas and dust are coupled through their collisional and Coulomb encounters, their interaction changes the grain charge, mass and chemical composition. Thus, it is crucial to have a reasonable estimate of the global gas content and distribution in the Galaxy. However, even though the distribution of the molecular component, for example, is known to be clumpy, one has to approximate its distribution in the disk in a smoother fashion. Hence all the gas components and dust in this study will be represented by either an exponential function or a gaussian in both the radial and vertical directions (with respect to the disk plane), as these are known to simulate relatively well the gas distribution of the Galaxy on large scales. The gas components included are the atomic hydrogen HI, molecular hydrogen H 2 (with temperature T = 15 K), as well as ionized hydrogen H +. It is also important to keep in mind that helium (He) is not included in the equations and graphs of this section; so effectively, the gas densities are multiplied by 1.36 in order to include He. The 1.36 factor corresponds to 73% H and 27% He by mass, and all the He is assumed to be neutral (Ferrière 1998). The molecular gas distribution is based on the Wolfire et al. (2003) model, and is described by equation 3.1, in which ρ 0,H2 (0.33 M pc 3 ) is the density, R 0 (2.89 kpc) the radial scale length and z 0 (63.4 pc) the vertical scale height. ρ = ρ 0 e R R 0 + z z 0 + Rm R (3.1) The parameter R m (3.3 kpc) accounts for the central molecular gas depletion in the inner regions of our Galaxy (a few kpc). It is important to note as well that the peak concentration of gas in the center is not included in these equations; in fact, this

34 CHAPTER 3. GALACTIC ENVIRONMENT 21 contribution is not required because we did not consider dust dynamics in this area. The dust density distribution that is used in the optical depth calculations is taken from the Misioritis et al. model (2006). However, since it is simply defined by an exponential law, it can also be defined by equation (3.1), in which the vertical and radial scale length are, respectively, 100 pc and 5 kpc (and the suppression factor R m = 0). These values are consistent with the radial and vertical scale lengths of the stars (see Section 4.3), with a smaller scale height ( 1/2) and a larger scale length ( 1.4) (Xilouris et al. 1999). The density of dust at the origin, ρ 0,d = g cm 3, was adjusted so that the dust to gas mass ratio approximately corresponds to the value quoted in Allen (2000), namely 1/160. The atomic hydrogen component, on the other hand, is represented by a set of gaussian equations developed by Ferrière (1998). While the choice of a gaussian or an exponential distribution is not really a source of concern, it is important to have a realistic distribution in the disk-halo vicinity. A gaussian distribution shows a smoother, or less abrupt, distribution near the Galactic midplane, and thus appears to be more realistic. However, since we are only interested in the motion of dust grains at high latitudes, the distribution near z = 0 is not really an issue. The model of Ferrière (1998) was thus chosen over the one of Wolfire et al. (2003), not really because of its gaussian distribution, but rather because it uses three different vertical scale heights (versus one). Thus, the total HI distribution proposed by Ferrière (1998) is described by: [ ( ) ] 2 z n HI (R, z) = cm { exp [ ( ) ] 2 ]} z exp exp [ zh3, (3.2) H 2 where H 1 = 127 pc, H 2 = 318 pc and H 3 = 403 pc (Dickey & Lockman 1990). H 1

35 CHAPTER 3. GALACTIC ENVIRONMENT 22 Moreover, the HI thickness increases towards the outer Galaxy (Narayan & Jog 2002), hence the scale heights are in fact functions of R. Thus, the parameter α(r) shown below accounts for the flaring of the warm component in the outer disk, namely, it makes the scale height grow linearly starting from the solar radius and outwards. Since the total HI scale height is known to be almost independent of radius within the solar radius, the factor α(r) is equal to 1 in this region. Equation (3.3) summarizes the behaviour of α: { kpc R R, α(r) = R/R R R 20 kpc (3.3) The atomic gas distribution was left undefined for R 3.5 kpc in the Ferrière model, so we kept α(r) = 1 in this region for simplicity (it does not matter since we do not study any grain dynamics in this inner region). The scale heights H 1, H 2 and H 3 are then, of course, functions of α(r): H 1 (R) = (127 pc) α(r), (3.4) H 2 (R) = (318 pc) α(r), (3.5) H 3 (R) = (403 pc) α(r), (3.6) Also, the model of Ferrière accounted for a cold (T = 80 K) and a warm component (T = 10 4 K); the gas temperature also has an impact on the drag force caused by collisions between gas particles and dust grains. So the space-averaged HI distributions of the cold n c and warm n w components are described by the following equations: { [ ( ) ] 2 n c (R, z) = 0.340cm 3 z exp α 2 (R) H 1 (R) [ ( ) ] 2 [ z exp exp z ]}, (3.7) H 2 (R) H 3 (R)

36 CHAPTER 3. GALACTIC ENVIRONMENT 23 and n w (R, z) = 0.226cm 3 α(r) { [ ] [ ( ) ] 2 z exp α(r) H 1 (R) [ ] [ ( ) ] 2 z exp α(r) H 2 (R) [ ] [ exp z ]}, (3.8) α(r) H 3 (R) In fact, the cold and warm components cannot be easily disentangled, but it is clear that the cold component lies closer to the plane than the warm component (i.e. the warm component has a higher scale height). The ionized component of the interstellar medium is not to be neglected, because it has a large vertical scale height and it also creates Coulomb interactions with the charged grains. Thus, although one could use the radially flat distribution of electrons that was described in the Misioritis et al. (2006) model, we chose to use a more likely distribution that reproduces the peak gas density around R = 4 kpc, namely the one proposed by Ferrière (1998). The ionized hydrogen component included here is the diffuse component only; meaning that it does not include the contribution from discrete, classical HII regions near the midplane (Reynolds 1991 and references therein). The ionized hydrogen number density n + is based on the electron number density equation (equation (3.9)), which comes from pulsar rotation measures. [ ( ) ] 2 ( R n e = (0.025 cm 3 )exp exp z ) 37 kpc 1 kpc [ ( ) ] 2 ( R 4 kpc +(0.2 cm 3 )exp exp z ) 2 kpc 150 pc (3.9) We use a slightly modified equation for n +, but it is virtually the same as can be seen from Figures where these two components are plotted, together with the neutral gas.

37 CHAPTER 3. GALACTIC ENVIRONMENT HI cold HI warm H2 HI + H 2 H + e 1 Gas Density (cm -3 ) R (kpc) Figure 3.1: Radial dependence of the gas density in the Galactic plane. He is not included. The gas density as a function of Galactic radius is shown in Figure 3.1. We can see that the H 2 distribution has a peak value at R 3 kpc, and that the depletion near the center is also taken into account by this model. The atomic hydrogen component shows, as is observed (Narayan & Jog 2002), a relatively constant density throughout the radial extent of the disk. The molecular gas component is dominant in the region R 6 kpc, beyond which the atomic hydrogen starts to dominate the total gas contribution. Figure 3.2 shows the vertical gas distribution at the Solar radius, which is assumed to be at R = 8.5 kpc from the Galactic center. This Figure was plotted using a log scale in order to show that the diffuse H + component starts to dominate the neutral HI and H 2 components at z 700 pc (at R = R ). It is also clear from Figure 3.2 that the cold component of the HI gas falls faster than the warm component, and

38 CHAPTER 3. GALACTIC ENVIRONMENT HI cold HI warm H2 HI + H 2 H + e -2 log (Gas Density) (cm -3 ) Figure 3.2: Height dependence of the gas density at the solar radius. He is not included. that the molecular gas mostly lies close to the plane since its density decreases rapidly with z. Neglecting any contribution from the very center of our Galaxy, the HI mass thus modeled adds up to M HI M within R, which is consistent with the maximum HI mass estimated by Wolfire et al. (2003) for that same region, namely M HI M. 3.2 Dust and Stellar Radiation Interaction Stars As we will see shortly in Section 3.2.2, the radiation pressure felt by dust comes from radiation in the UV and optical range. Clearly, most of this radiation comes from

39 CHAPTER 3. GALACTIC ENVIRONMENT 26 various types of stars, although it is mainly the O,B and A stars that make up most of the light output of the Galaxy (even though the smaller stars account for most of the stellar mass). For example, it is the B stars that dominate the radiation in the UV, as they account for 82% of the total radiation at λ = 156.5nm (O stars make up 4%, A stars 10% and the others 3%)(Gondhalekar 1990). As one goes to larger wavelengths the contribution from the smaller stars gradually increases. Fortunately, although each type of star radiates with its very own spectrum of light, for the purposes of our study we can approximate the effect of these different types of radiation on dust grains as if all the light was coming from only one type of star. This is a good approximation, because (i) as we have just said most of the light output comes from young stars and (ii) as we will see in the next Section (3.2.2), the absorption and scattering of starlight by dust grains is nearly constant for a wide range of wavelengths. Moreover, even if there existed only one type of star, this type of star would still emit over a wide range of wavelengths; therefore the problem of adjusting the dust extinction efficiency factor, Q ext, to each wavelength would still be present Extinction of Radiation Astronomers have learned in the course of the 20 th century that dust can block significant fractions (if not all) of the UV and optical light coming from a star, depending on the amount of dust there is between the star and the observer. The shape of the extinction curves which have been observed tell us about the size distribution of the grains; extinction in the UV part of the spectrum implies that there exist small grain sizes similar to that of the radiation wavelength, and similarly the extinction in the optical reveals the presence of larger grains. Thus, it is crucial to include the amount

40 CHAPTER 3. GALACTIC ENVIRONMENT 27 of starlight extinction there is between a given star and a given dust grain if one wants to know the amount of radiation pressure that is imparted to the grain from that star. The differential equation expressing this loss of intensity in the radiation beam, both due to absorption and scattered light, is di λ = I λ k λ dl, (3.10) where I λ is the intensity at a given wavelength λ, k λ (cm 1 ) is the extinction coefficient, and l is the distance between the source and a given grain. Now, letting k λ = nσ λ and integrating, one finds that I λ = I 0,λ e τ λ, (3.11) where I 0,λ is the intensity without extinction, and τ λ, the optical depth, is defined by the following integral τ λ τ = l 0 σ n dl. (3.12) In the above equations, n is the space number density of dust and σ λ σ is the absorption and scattering cross-section. The cross-section can thus be rewritten in terms of the dimensionless extinction efficiency factor Q ext as: σ = πa 2 Q ext, where Q ext = Q abs + Q sca. Q abs and Q sca refer to the absorption and scattering efficiency factors, respectively. These Q values are dimensionless because they are equal to the ratio of the usual cross-section σ (units of cm 2 ) to the geometrical cross-section of the grain, namely πa 2. Now, while the optical depth due to the dust grains is wavelength dependent, it can be approximated as being independent of it in the UV to optical range. The scattering regime of a particle is usually determined using the size parameter X = 2πa/λ. When X is large, the size of the grain is larger than the wavelength of the radiation, and vice versa. Now, looking at Figure 3.3, we see that when X is greater than 7, Q ext becomes relatively constant; meaning that the

41 CHAPTER 3. GALACTIC ENVIRONMENT 28 Figure 3.3: The effective cross-section Q ext as a function of the size parameter X. Also shown are the effective absorption and scattering cross-sections Q abs and Q sca, respectively. (Figure copied from Whittet 2003) extinction becomes independent of wavelength, thus justifying the aforementioned assumption (i.e. σ λ σ). Also, it is the bigger grains (a µm) that make up most of the extinction (in the optical that is), as the smaller grains (a µm) have X values below 1 and in this case the Q values are at least a factor of ten smaller than when X 7. Consequently, the grain radius that is typically chosen to represent the ensemble of the grains that make up the extinction is the classical one, namely the one with a = 0.1 µm. Strickly speaking, the cross-section in equation (3.12) is the sum of the contributions from dust and gas. There are indeed several absorption lines in the optical and UV, and one might ask whether the sum of all these absorption lines turns out to be a significant amount of extinction. In order to address this issue, we considered the strongest absorption line, namely the Lyman alpha (Ly α) line. The cross-section of

42 CHAPTER 3. GALACTIC ENVIRONMENT 29 this Ly α absorption line is 4 orders of magnitude smaller than that of the dust: cm 2 compared to πa 2 Q ext cm 2 for a grain with a = 0.1 µm and Q ext = 2. Moreover, it turns out that, assuming a gaussian line shape and including Doppler broadening, the full width at half maximum of the absorption line is 0.87 nm at T = 10 4 K. This value is certainly negligible with respect to the large spectral range covered by the dust absorptive and reflective properties (optical and UV). Thus, even if we were to take into account all the other gas absorption lines, their summed effect is negligible. The integration in equation 3.12 is done along the line of sight, and although it is a one-dimensional equation, any given line of sight between a given star and the position of the dust grain is in three dimensions. Thus, one has to use the spatial equation of line of sight in order to have an expression for the optical depth as a function of one coordinate only (z in our case). It is vital to start with a well-defined coordinate system, since the calculations, though straightforward, can quickly get rather messy. Hence we chose to use cartesian coordinates centered in the Galactic center, so that when y = 0, ˆx is radially outwards and ẑ points along the rotation axis (ŷ is then defined by the right hand rule). Let us now define a vector d r, which has the Galactic center as its origin and the dust grain position as its ending point. Also, as is shown in Figure 3.4 we need to define a vector that goes from a given star to the dust grain: r d r s, where r d goes from the origin to the dust grain and r s from the origin to the star. We are now ready to apply the condition that will make d r lie along the same line of sight as the line of sight between the star and the dust grain ( r d r s ), namely the spatial equation of line of sight: d r ( r d r s ) = 0 (3.13) Now, using the fact that r d r s = (x d - x s, y d - y s, z d - z s ), where x s, y s and z s refer

43 CHAPTER 3. GALACTIC ENVIRONMENT 30 Figure 3.4: Coordinate system used for optical depth calculations. to the star position and x d, y d and z d correspond to the dust grain position, we have: dx (y d y s ) dy (x d x s ) = 0, (3.14) dx (z d z s ) + dz (x d x s ) = 0 and (3.15) dy (z d z s ) dz (y d y s ) = 0. (3.16) Then, using the appropriate boundary conditions, one can solve equations ( ) for x = x(z) and y = y(z) and get: x(z) = x s + (x d x s )(z z s ) z d z s, (3.17) y(z) = y s + (y d y s )(z z s ) z d z s. (3.18) Next, since dl 2 = dx 2 + dy 2 + dz 2, we can let dl = dl dz, where: dz (dx dl dz = = ) 2 ( ) 2 dy + + 1, (3.19) dz dz (xd ) 2 ( ) 2 x s yd y s (3.20) z d z s z d z s

44 CHAPTER 3. GALACTIC ENVIRONMENT 31 Then, letting σ in equation 3.12 equal to the geometrical cross-section πa 2 times the extinction factor Q ext, the optical depth can now be integrated in terms of one variable only, here written as z in the equation below: where n 0,d = ρ 0,d m d (µm). τ = πa 2 Q ext zd previously defined in equation (3.1). z s n 0,d e x(z) 2 +y(z) 2 R 0,d e z z 0,d dl dz, (3.21) dz The values of ρ 0,d, z 0,d and R 0,d are the same as those In Figure 3.5, a star has been placed in the Galactic center, and hence the plot shows the optical depth at any given field point along one radial line of sight, as y d was set to 0 as well. As was expected from the optical and UV views of our and of external galaxies, the extinction is very significant in the plane, but practically negligible directly above the star (meaning in the z direction only). The normalisation of the Galactic dust content was made so that the optical depth matches the amount of extinction A λ observed in two different lines of sight. The extinction coefficient is related to the optical depth τ λ as follows: A λ = τ λ, but we have previously argued that the wavelength dependence can, in fact, be neglected for the range of wavelengths we are considering. At the North Galactic pole, our model yields an extinction of A B,V = 0.15, which corresponds to the average between the observed values of A V = 0.1 from Allen (2000) and A B = 0.2 from de Vaucouleurs & Buta (1983). In the line of sight towards the Galactic center, it has been determined that the optical depth is equal to 1 at a radial distance of only 0.6 kpc (Binney & Tremaine 1994), and in our model τ 1 at 0.7 kpc from R. In terms of the normalization, it is in fact the combination of the grain radius a, ρ 0,d and Q ext that has to be normalized. Since we had already chosen to use the aforementioned classical grain radius, and that the value of ρ 0,d was also already adjusted (so that the dust to gas mass ratio is 1/160), Q ext was the only parameter left to be adjusted. Thus,