1. SIGRAV Graduate School in Contemporary Relativity and Gravitational Physics Laura Ferrarese Rutgers University Lecture 2: Stellar Dynamics

2. Lecture 2: Outline 1. As spectacular as unique: the Milky Way 2. Observational requirements (beyond the Milky Way) and Instrumentation - ground and space based 3. Mathematical formalism 4. Application: M87 5. Biases and Systematics 6. Results and Concluding Remarks

3. The Milky Way  The galactic center is only 8kpc away. Because of heavy dust absorption, a clear view of the stars in the vicinity of the galactic center can only be obtained in the near- infrared.  In the K-band (2.1  m), the diffraction limited resolution is 0.06 arcsec:  A star which travels at a speed of 27 km/s, would move relative to the background stars by one resolution element in one month. Proper motion studies are therefore feasible.

4. The Milky Way  Schödel et al. 2002 (Nature 419, 694): CONICA/NAOS adaptive optics images at the ESO/VLT.  Movie showing the proper motion of the ‘S2’ star over a period of 10 years. The star is on a bound, highly elliptical orbit around SgrA*, with an orbital period of 15.2 years, a pericenter distance of 17 light hours, and an orbital semi-major axis of 5.5 light days.  This single stellar orbit allowed to measure the enclosed mass: 3.7  10 6 M  ! There are virtually no assumptions made, except for those regarding the stellar mass distribution (which needs to be known to determine how much of the enclosed mass is in a central point source.)  The measurements imply a central mass density in excess of 10 17 M  pc -3, excluding with high confidence that the central dark mass consists of a cluster of unusual stars or elementary particles. This leaves little doubt of the presence of a supermassive black hole at the centre of the galaxy in which we live.

5. The Milky Way  Further support to the existence of a SBH in the MW come from the detection of an X-ray flare, using the Chandra satellite (Baganoff et al. 2001, Nature, 413, 45). The flare is interpreted as due to instability in the mass accretion rate. The timescale of the event is consistent with the light crossing time for the inner accretion disk (~10 Schwartzchild radii) September 1999 October 2000

6. Beyond the Milky Way  The most nearby large galaxy, M31, is 100 times farther away than the galactic center.  proper motion timescales are a factor 100 longer;  stellar mass density is a factor 100 3 larger, therefore individual stars are not resolved.  In (almost) every galaxy beyond the Milky Way, proper motion studies of individual stars are not feasible, and we have to resort to extract the velocity field from the integrated (along the line of sight) stellar light. This presents several problems:  Stellar absorption lines are faint and long exposure times are required to collect the necessary signal-to-noise. For instance, it would take the Hubble Space Telescope more than 100 orbits (more than any TAC would allocate to the project) to obtain spectra suitable for stellar dynamical studies in M87, the cD galaxy in the Virgo cluster.  The modeling is difficult: the orbital structure is unknown and difficult to derive from the data because of line of sight projections  There might be a fundamental mathematical limitation to the accuracy of the mass measurements that can be derived from dynamical modeling.

7. RequirementsRequirements  Stellar kinematics (bottom figure): need to cover the Ca or Mg absorption lines at 8500 and 5200 Å respectively, with spectral resolution > 60 km/s (depending on the mass of the BH).  Gas kinematics (top figure): H  +NII region at ~6500 Å, with spectral resolution ~100 km/s

8. Instrumentation - HST  Challenge:  to detect the dynamical signature of a supermassive black hole, we need to observe the region of space - the sphere of influence - within which the black hole gravitational potential dominates over that of the surrounding stars. The farther in we are within this region, the more markedly keplerian the rotation and velocity dispersion curves will be. The ‘black hole radius of influence’ is: r infl = GM BH /  2 ~ 50 (M BH /10 9 M  )/(  2 /300 km s -1 ) pc ~ 0.7  at Virgo where  is the stellar velocity dispersion and M BH is the black hole mass.  Even at the distance of Virgo (15 Mpc), the BH sphere of influence is very small.  This explains why most secure SBH detections have been made using the Hubble Space Telescope: Primary mirror is 2.4m in diameter (small for ground-based standards) but Resolution (in the optical): 0.1 arcsec (5-10 times better than ground based telescopes) Point Spread Function is very stable

9. Instrumentation - HST/STIS  The Space Telescope Imaging Spectrograph uses two-dimensional detectors operating from the ultraviolet to the near infrared. First-order gratings cover the full spectral range and are designed for spatially resolved spectroscopy using a long slit. The echelle grating, available only in the ultraviolet, are designed to maximize the spectral coverage in single observations of point sources. The STIS optical design includes corrective optics to compensate for HST's spherical aberration. 1024  1024 CCD Detector, 0.05 arcsec pixels 52  52 arcsec FOV 2000 - 10,300 Å MAMA Detectors, 0.024 arcsec pixels 25  25 arcsec FOV 1150 - 3100 Å (combined) Apertures (for regular gratings): 52 arcsec long, 0.05,0.1,0.2,0.5,2 arcsec wide

10. Instrumentation - HST/STIS

11. Instrumentation - Ground Based  Compared to HST, ground based telescopes have the great advantage of a larger collective area, but they generally lack the spatial resolution for systematic SBH searches.  this will change when adaptive optics will be implemented in the optical.  Integral Field Units (IFU) are much more suited than long-slit spectroscopy for kinematical (both stellar and gas) studies on large scales. Available IFUs:  WHT: Sauron http://www.strw.leidenuniv.nl/sauron/  CFHT: GrIF (Near IR) http://www.cfht.hawaii.edu/Instruments/Spectroscopy/GriF/  Gemini: GMOS http://www.gemini.edu/sciops/instruments/gmos/gmosIFU.htmlhttp://www.gemini.edu/sciops/instruments/gmos/gmosIFU.html  VLT: VIRMOS http://www.eso.org/instruments/vimos/http://www.eso.org/instruments/vimos/  AAT: SPIRAL http://www.aao.gov.au/astro/spiral.htmlhttp://www.aao.gov.au/astro/spiral.html  Sauron data for NGC 4365, a galaxy known to have a kinematically decoupled core (from the Sauron’s webpage)

12. Mathematical Formulation  Galaxies are composed of several hundred billion stars. In the most general case, each of these stars:  moves through the gravitational potential of the entire galaxy (produced by the effect of every other star, and any “dark” mass distribution).  interacts with nearby stars through stellar encounters.  Observationally, except for a handful of cases, we do not observe individual stars. All we can observe is the integrated light of stars along the line of sight:  stellar surface brightness (projected on the plane of the sky).  projected stellar rotation and velocity dispersion (integrated along the line of sight).  higher moments of the absorption line profiles.  From this, we want to arrive to a self-consistent description of the stellar system: derive the mass density (luminous + dark matter) which reproduces the observables.  This seems hopelessly complicated! Thankfully, there are a number of simplifications we can make.

13. Collisionless Stellar-Dynamical Systems  Real galaxies can be approximated as collisionless systems, i.e. the time scale over which interactions between stars lead to a significant change in each star’s velocity is long compared to the galaxy’s age.  The Crossing Time is defined as the typical time scale for orbital motion; i.e. the time it takes a typical star to cross the system: where  h is the average stellar density inside the half mass radius.  The Relaxation Time is defined as the characteristic time scale over which, due to the cumulative effect of stellar encounters, a typical star acquires a transverse velocity comparable to its initial velocity: where N is the total number of stars in the galaxy.

14. Collisionless Stellar-Dynamical Systems  A typical galaxy has N = 10 11 stars but is only t = 100 crossing times old, so the cumulative effect of stellar encounters is insignificant:  In this limit, each star moves in the smooth gravitational field  (x,t) of the galaxy. This greatly simplifies the problem, since instead of thinking about the motion of one point in a phase space of 6N dimensions (3 spatial coordinates + 3 velocity coordinates, times N stars), we can think about the motion of N points in a phase space of just 6 dimensions.  Therefore, a complete description of the system is given by the one-particle distribution function: f is equal to the number of stars in the phase space volume d 3 xd 3 v centered on (x,v) at time t.  The distribution function obeys a continuity equation, i.e. the rate of change of the number of stars within a phase space volume d 3 xd 3 v is equal to the amount of inflow minus the amount of outflow:  This is known as the Collisionless Boltzmann Equation (CBE)

15. Collisionless Stellar Systems  The distribution function tells us all we ever want to know about the dynamics of the system: once we know the distribution function, the CBE can be solved to give the total gravitational potential, which is related to the total mass density (including dark matter and a putative central SBH) by Poisson’s equation:  But how can we derive the distribution function from the observables?  Apart for a multiplicative factor (which we will assume of order unity for the sake of argument), the stellar mass density is just the integral of the DF over all velocities:  The mean stellar velocity is:  So, in principle, if we know the stellar mass density, the three components of the streaming motion, and the three diagonal components of the velocity dispersion, then we can derive the distribution function, the total gravitational potential (from the CBE), and the total mass density (from Poisson’s equation).  Unfortunately, even assuming that none of the observables depends on time (i.e. the system is in a steady state), we simply cannot measure all seven quantities. Velocity dispersion, arising because not all the stars near a given point x have the same velocity Streaming motion

16. Collisionless Stellar-Dynamical Systems  To gain further insight onto the system, it is often convenient to take moments of the CBE, which directly relate the gravitational potential to the mean streaming velocity, velocity dispersion and stellar mass density. The zeroth and 1st order moments give the Jeans equations:  Integrating the CBE over all velocities:  Multiplying the CBE by a velocity component v j and then integrating over all velocities:  If further assumptions are made, the Jeans equations can sometimes be simplified to such an extent that they can be solved analytically or numerically.

17. Spherical Systems  Let’s consider the very simple case of a spherically symmetric, steady-state stellar system for which v r = v  = 0. In this case, all derivatives with respect to t,  and  vanish, and the first moment (wrt v r ) of the CBE becomes (in spherical coordinates): Since d  /dr=GM(r)/r 2, this equation can be rearranged as:  Therefore, M(r) is a function of five variables, namely the stellar mass density, the streaming circular velocity, and the three components of the velocity dispersion.  Unfortunately, real observations only allow us to measure three quantities, namely the surface brightness profile at each projected radius, the projected circular velocity, and the line of sight velocity dispersion. It follows that we cannot build a unique model of an external galaxy (even in the oversimplified case of a spherical system) based on the observables only. More assumptions need to be made.

18. Spherical, Isotropic Systems  The easiest case is to assume that the stellar mass density follows the luminosity density, the galaxy does not rotate and that the velocity ellipsoids are spherical throughout the galaxy, i.e. the velocity field is isotropic. In this case:  The stellar mass density and radial velocity dispersion are simply related to the surface brightness profile I(r) and the line of sight velocity dispersion  p :  These are Abel’s integrals which can be easily inverted to give (r) and (r)  2 r (r), which substituted into equation (2) immediately give the mass enclosed within a radius R. r z R (2) (1)

19. Spherical, Anisotropic Systems  Unfortunately, there is no good theoretical reason to assume that a real galaxy is isotropic. Quite the opposite:  the rotation speeds of giant elliptical galaxies (which are more closely approximated by spherical systems) are smaller than predicted for an isotropic model. This indicates that the velocity dispersion tensor of giant ellipticals must be anisotropic.

20. Anisotropy in Giant Ellipticals From Illingworth 1981 Relation between v/  and ellipticity for an isotropic system

21. Spherical, Anisotropic Systems  In the previous example, we have given ourselves complete freedom in deriving the mass density (i.e. the mass to light ratio of the system), at the cost of an extreme assumption about the velocity dispersion tensor. Let’s consider the opposite scenario:  let’s assume that the velocity structure is invariant with respect to rotations about the galactic center (i.e. the mean square velocities in  and  are identical). Equation (1) can then be written as:  This aside, let’s give ourselves freedom in deriving the velocity dispersion tensor, at the cost of making an extreme assumption on the mass density. In particular, let’s assume that the mass density follows the luminosity density, i.e.  (r) =  (r), with  = constant. (3)

22. Spherical, Anisotropic Systems  Then, simple geometry gives: In conjunction with equation (3), this can be used to evaluate  (r) uniquely. r (r 2 -R 2 ) 1/2 R VrVr VV (3)

23. Two Simple Cases  The easiest case is to assume that the stellar mass density follows the luminosity density, the galaxy does not rotate and that the velocity ellipsoids are spherical throughout the galaxy, i.e. the velocity field is isotropic. In this case:  The second case is to assume that the mass density follows the luminosity density, i.e.  (r) =  (r), with  = constant. In this case, (1) becomes:  The problem is: both cases can reproduce exactly the same observational constraints! (2) (1) (3)

24. Application: M87  M87 was the first galaxy for which the presence of a supermassive black hole was claimed (Sargent et al. 1978, ApJ, 221, 731).  Central galaxy in the Virgo Cluster.  Very well known radio galaxy.

25. Application: M87  Projected surface brightness profile (left) and velocity dispersion profile (right) for M87 (from Young et al. 1978, ApJ, 221, 721; Sargent et al. 1978, ApJ, 221, 731) Steady-state, spherical, isotropic dynamical models fits to the data: a) with black hole b) with black hole & seeing convolved c) no black hole

26. Application: M87  Steady-state, spherical, isotropic dynamical models applied to M87 by Sargent et al. (1979, ApJ, 221, 731). The data is interpreted as evidence of a central 5  10 9 M  SBH.

27. Application: M87  However, lets now consider a case in which we impose the mass to light ratio throughout the galaxy to be constant: M(R)/L(R)=constant.  Notice that M(R) here is not the stellar mass, but includes a central supermassive black hole (and anything else).  In this case, the system is not isotropic and equation (2) can be used to determine the degree of anisotropy.  Right: The computed profile for the radial velocity dispersion and anisotropy parameter  in M87. This model assumes a constant M(R)/L(R) (i.e. no central SBH!) and gives just as good a fit to the observables as the isotropic model by Sargent et al. (1979). From Binney & Mamon (1982, MNRAS, 200, 361)

28. Application: M87  Spherical, isotropic and anisotropic models for M87, using more recent data (from van der Marel 1994, MNRAS, 270, 271).

29. M/L vs Anisotropy  In other words, anisotropy in the velocity dispersion, and a varying mass to light ratio, can mimic the same observables. Here is a more intuitive way to see how this ambiguity comes about. Consider the Jeans equation once more:  In a real galaxy, - dln /dlnr is of order or less than +1, while - dln  r 2 /dlnr must always be smaller than +1.  If the galaxy is isotropic, then the last two terms drop out, and we are left with a positive solution for M(r).  Let’s take the extreme case of a highly anisotropic galaxy for which v  =v  =0. In this case, the last two terms offset the first two, with the effect of reducing M(r)! (1)

30. Breaking the Degeneracy  Can we constrain the degree of anisotropy from the observations?  yes: the line of sight velocity distribution (LOSVD), which is reflected in the shape of the absorption lines, is different depending on the level of anisotropy in the system (van der Marel & Franx 1993, ApJ, 407, 525; Gerhard 1993 MNRAS, 265, 213) where  = projected surface brightness at position (x,y)  Therefore, we can: 1) estimate  from the LOSVD and 2) use the dispersion profile to constrain the potential.  Analytically, the LOSVD is analyzed in term of Gauss-Hermite moments, which arise from an expansion in terms of orthogonal functions. The zeroth order moment represents a Gaussian with dispersion . Higher order terms represent deviations from a Gaussian; odd and even higher order moments describe asymmetric and symmetric deviations from a Gaussian profile shape. The second moment in particular is sensitive to .

31. Breaking the Degeneracy  Tangentially anisotropic DFs are generally more flat-topped and those from radially anisotropic models are more peaked than an isotropic profile (figure from van der Marel & Franx 1993, ApJ, 407, 525).

32. Breaking the Degeneracy  The presence of a central SBH also influences the line profiles, producing wings that are more extended that those of a Gaussian, due to high velocity stars orbiting close to the SBH (figure from van der Marel 1994, MNRAS, 270, 271).

33. More General Galaxy Models  In terms of the distribution function, how do isotropic and anisotropic models differ?  Let’s define an integral of motion as any function of velocity and coordinates whose time derivative vanishes for all orbits: It can be shown easily that any integral of motion, or any function which depends on (x,v,t) only through an integral of motion, is a solution of the time dependent CBE. Again, let’s limit ourselves to time-independent systems.  A spherical, isotropic system, admits only one integral of motion: the total energy. In other words, the distribution function depends only on E : f = f(E) In this case, there is a one-to-one correspondence between mass density and distribution function: given  (r) we can analytically find f(E) which self-consistently generates  (r) (and vice versa).

34. More General Galaxy Models  If the distribution function depends on more than one integral of motion, the velocity dispersion tensor cannot be isotropic.  The simplest case is one where f=f(E,L z ), with L z the z -component of the angular momentum.  The Jeans equations in this case are greatly simplified: since the distribution function depends on the r and z -components of the velocity only through the combination v r 2 + v z 2 the velocity dispersion in the r and z -directions must be equal to each other. This defines an axisymmetric system.  Unlike the isotropic case, in an anisotropic axisymmetric system there are infinite f(E, L z ) which generate a given  (r). Physically, this is because the contribution of a star to the density does not depend on the sense of the star’s rotation around the symmetry axis. In other words, if f(E, L z ) yields the desired  (r), so does f(E, L z )+f 0 (E, L z ) where f 0 (E, L z )=  f 0 (E,  L z ) is an odd function of L z.  While the odd part of the distribution function determines the net streaming motion if the  direction and therefore can in principle be constrained from the kinematics, in practice this approach does not work: any small errors in the observables translate into large errors in the distribution function.  Dynamical models in which the distribution function depends on two integrals of motions are referred to as ‘2-integral models’. They can yield reasonable approximations to some stellar systems.

35. 2-Integral Models  Two integral models can be handled through the Jeans equation. Since the radial and vertical velocity dispersions must always be the same, the Jeans equations take the simplified form:  The procedure is then the following: 1. Observe the 2-D surface brightness (projected onto the plane of the sky). 2. Deproject to obtain the luminosity density. The deprojection is not unique, and an inclination angle must be assumed. 3. Compute the potential , given a reasonable assumption for the mass-to-light ratio. 4. Solve the Jeans equations to derive the mean square velocities. 5. Project the mean velocities onto the plane of the sky to get the line of sight velocity and velocity dispersion (this means that the azimuthal velocity has to be divided into a streaming and random part first). 6. Compare the predicted and the observed velocities. 7. Vary the inclination angle and the mass to light ratio. Go back to #2 and iterate until a good fit is reached. z r 

36. More General Galaxy Models  Some galaxies are indeed well fit by 2-integral models. Is this always the case? No!  In most galaxies, 2-integral models predict major-axis velocity dispersions in excess of those observed.  Furthermore, remember that if f = f(E,L z ), then the velocity dispersion in the r and z directions must be equal to each other. This is not true in the solar neighborhood!  This means that in most systems, two integrals are not sufficient to fully characterize the stellar orbits. In fact, numerical calculations show that most orbits are not completely described by just two integrals, i.e. they admit a third integral. There is no general expression for this integral of motion, although in nearly spherical systems it is approximated by the magnitude of the total angular momentum.  Dynamical models in which the distribution function depends on three integrals of motions are referred to as ‘3-integral models’

37. 3 Integral Models  The challenge here is to build distribution functions f=f(E,L z,I 3 ) that generate axisymmetric models with specified observational properties. This is difficult for a number of reasons:  There is no simple analytical form for the third integral  the addition of a third variable for f makes the computations more complicated.  In the spherical and isotropic or in the 2-integral axisymmetric models, there is only one distribution function which corresponds to a given mass density and azimuthal streaming velocity. In the less restricted 3-integral case, there are many distribution functions which correspond to a given  (r) and v  (r). Therefore, if we wish to derive the distribution function f from the observations, we need to know some further moment of f than just  (r) and v  (r). For example, we might know the distribution over the galaxy of the line of sight velocity dispersion, but this is very hard to do.

38. 3-Integral Models  Schwarzschild (1979) devised a powerful method for constructing equilibrium models of galaxies without explicit knowledge of the integrals of motions: 1. Observe the 2-D surface brightness (projected onto the plane of the sky) 2. Deproject to obtain the luminosity density. The deprojection is not unique, and an inclination angle must be assumed. 3. Compute the potential using Poisson’s equation, given a reasonable assumption for the mass-to-light ratio, which will be a function of the spatial coordinates (so far, so good). 4. Construct a grid of k cells in position space 5. Choose initial conditions for a set of N orbits, and for each one: a) integrate the equations of motion over many orbital periods b) keep track of how much time the orbit spends in each cell, which is a measure of how much mass the orbit contributes to that cell. 6. Determine non-negative weights for each orbit such that the summed mass and velocity structure in each cell, when integrated along the line of sight, reproduce the observed surface brightness and kinematical constraints.

39. Biases and Systematics  To recap, dynamical models applied to real data have evolved from 1)Models assuming spherical symmetry and an isotropic velocity tensor: f=f(E) 2)Two-integral, axisymmetric models: f = f(E, L z ) 3)Axisymmetric models admitting 3 integral of motion f = f(E, L 2, I 3 ) (Cretton et al. 1998, Gebhardt et al. 2002).  Has this increased complexity in the modeling algorithms lead to a increasingly tighter constrained on the gravitational potential and mass density? This is subject of heated debate at the moment!  It has been claimed (and not yet disproved) that there is an intrinsic mathematical indeterminacy which precludes us from pinning down the gravitational potential uniquely when using 3-integral models (Valluri, Merritt & Emsellem, 2002, astro-ph/0210379). If correct, this result undermines any stellar dynamical study to date.

40. Biases and Systematics  Here is an intuitive way to understand the origin of such indeterminacy.  Consider a spherical, isotropic case. Given a choice of potential  (r), there is only one distribution function f=f(E) which reproduces the observed stellar mass density (r). Such distribution function f uniquely defines the velocity dispersion profile  2 (r)   f(E)v 2 d 3 v Changing the potential (for instance by changing the mass-to-light ratio  or by adding a central mass concentration M  ) changes both f and  2 (r) in a very well defined manner. It follows that the goodness of fit of  2 (r) to the observed velocities will vary continuously with the parameters ( , M  ) which define the potential, i.e. there is a well defined minimum in the chi square space which identifies the ‘best’ ( , M  ).  Now, consider an axisymmetric model with f=f(E,I z,I 3 ). Given a choice of potential  (r,  ), there are many distribution functions f=f(E, I z,I 3 ) which reproduces the observed stellar mass density (r,  ), because many 3D functions f project to the same 2D function. Changing the potential can therefore always be compensated by a change in f so as to leave the fit to the observed kinematics unaltered: there will be a range of potentials all of which provide equally good fits to the data.  Because the choice of distribution functions allowed by 3I models critically depends on the number of orbits used to represent the distribution function, the suggestion is that an insufficient number of orbits might create a spurious minimum in the chi-square plots.

41. Anisotropy in Giant Ellipticals From Illingworth 1981 Relation between v/  and ellipticity for an isotropic system

42. More General Galaxy Models  Let’s look at the systems we just discussed in terms of the distribution function.  Let’s define an integral of motion as any function of velocity and coordinates whose time derivative vanishes for all orbits: It can be shown easily that any integral of motion, or any function which depends on (x,v,t) only through an integral of motion, is a solution of the time dependent CBE. Again, let’s limit ourselves to time-independent systems.

43. More General Galaxy Models  A spherical, isotropic system, admits only one integral of motion: the total energy. In other words, the distribution function depends only on E : f = f(E) In this case, there is a one-to-one correspondence between mass density and distribution function: given  (r) we can analytically find f(E) which self-consistently generates  (r) (and vice versa).  If the distribution function depends on more than one integral of motion, the velocity dispersion tensor cannot be isotropic. The simplest case is one where f=f(E,L z ), with L z the z -component of the angular momentum. These “2-Integral” models can yield reasonable approximations to some stellar systems. As in the spherical, isotropic model. there is only one distribution function that can generate a given  (r) and the net streaming motion if the  direction  However, in most systems, two integrals are not sufficient to fully characterize the stellar orbits. In fact, numerical calculations show that most orbits are not completely described by just two integrals, i.e. they admit a third integral. There is no general expression for this integral of motion, although in nearly spherical systems it is approximated by the magnitude of the total angular momentum. In full ‘3-integral models’, there is not a unique distribution function which generates a given  (r) and net streaming motion. In other words, a change in the potential can always be compensated with an appropriate change in the distribution functions to give exactly the same observables

44. Biases and Systematics  This indeterminacy might preclude us from pinning down the gravitational potential uniquely when using 3-integral models (Valluri, Merritt & Emsellem, 2002, astro- ph/0210379).  Simulated dataset designed to reproduce HST/STIS data for M32 - Results obtained for N c = number of constraints = 571 (masses, v, , h 3, h 4 ) and varying number of orbits N o N 0 /N c =15.6 N 0 /N c =10.1 N 0 /N c =5.0 N 0 /N c =2.5 Valluri, Merritt & Emsellem (2002) N 0 /N c =15.6N 0 /N c =10.1 N 0 /N c =5.0N 0 /N c =2.5

45. Biases and Systematics  This indeterminacy becomes the more severe the fewer the observational constraints, For instance, if the data does not resolve the sphere of influence of the central SBH (FWHM/2r h > 1), the problem is completely indeterminate. Simulated data for M32 with effective resolution FWHM/2r h = 0.5; spatial coverage and S/N significantly better than those of most HST/STIS nuclear data. Valluri, Merritt & Emsellem (2002) N 0 /N c =19.4 N 0 /N c =10.1 N 0 /N c =4.6 N 0 /N c =10.1 N 0 /N c =19.4

46. A Complete Census of SBH Masses from Stellar Dynamical Studies GalaxyType DistanceM BH +  -  Reference (Mpc) (10 8 solar masses) I1459 E3 30.326.01.11.1Cappellari et al. 2002, 578, 787 N221 cE2 0.80.0250.0050.005Verolme et al. 2002, MNRAS, 335, 517 N3115 S0 9.89.23.03.0Emsellem et al. 1999, MNRAS, 303, 495 N3379 E110.81.350.730.73Gebhardt et al. 2000, AJ, 119, 1157 N821 E624.70.370.240.08Gebhardt et al. 2003, ApJ, 583, 92 N1023 S010.70.440.060.06Gebhardt et al. 2003, ApJ, 583, 92 N2778 E23.30.140.080.09Gebhardt et al. 2003, ApJ, 583, 92 N3377 E511.61.00 0.90.1Gebhardt et al. 2003, ApJ, 583, 92 N3384 SB(s)0- 11.90.160.010.02 Gebhardt et al. 2003, ApJ, 583, 92 N3608 E223.61.91.00.6 Gebhardt et al. 2003, ApJ, 583, 92 N4291 E26.93.10.82.3 Gebhardt et al. 2003, ApJ, 583, 92 N4473 E516.11.10.50.8 Gebhardt et al. 2003, ApJ, 583, 92 N4564 E14.90.56 0.03 0.08 Gebhardt et al. 2003, ApJ, 583, 92 N4649 E217.320.04.06.0 Gebhardt et al. 2003, ApJ, 583, 92 N4697 E611.91.70.20.3 Gebhardt et al. 2003, ApJ, 583, 92 N5845 E*28.52.40.41.4 Gebhardt et al. 2003, ApJ, 583, 92 N7457 SA(rs)0- 13.5 0.0350.0110.014 Gebhardt et al. 2003, ApJ, 583, 92

47. Points to Bring Home 1) Dynamical measurements which do not resolve the SBH sphere of influence can severely bias the mass measurement. This has in fact happened in the vast majority of ground based studies (e.g. Kormendy et al. 1992, 1997,1998; Lauer et al. 1992; Fisher et al. 1995; Gebhardt et al. 1997; Magorrian et al. 1998)

48. R (arcsec)  ( km s -1 ) R (arcsec)  ( km s -1 ) R (arcsec) Data from Magorrian et al. (1998, AJ, 115, 2285) Points to Bring Home

49. 2.The more complex the dynamical models, the larger the number of observational constraints needed. The combination of HST data (in the innermost regions) and ground based/2D data (in the outer regions) for M32 constitutes the best suite of data for any galaxy so far (Verolme et al. 2002, MNRAS, 335, 517) V  h3h3 h4h4 STIS data V  h3h3 h4h4  SAURON data

50. SAURON + STIS Four slits + STIS 3  level Points to Bring Home  Model parameters and internal structure are much more strongly constrained with the use of the 2D data (Verolme et al. 2002, MNRAS, 335, 517)

51. Suggested Readings  The SBH in the Milky Way:Schodel, et al. 2002, Nature, 419, 694 Ghez et al. 2003, astro-ph/0303151  Stellar Dynamics: Binney & Tremaine, ‘Galactic Dynamics’, Princeton University Press, Chapter 4  M87, a practical case:Sargent et al. 1979, ApJ, 221, 731 Binney & Mamon 1982, MNRAS, 200, 361 van der Marel 1994, MNRAS, 270, 271