For β = 0, we have from our previous equations: d ln ν d ln r + d ln v2 r

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1 Isotropic models Now simplify the equation, and assume β = 0, or isotropic velocity dispersions - i.e., they are the same in all directions. Below we show how the observed light distribution and observed velocity dispersion can be used to derive the internal mass distribution. In order to do that, we have to calculate the projection effects. For β = 0, we have from our previous equations: M(< r) = r v2 r G d ln ν d ln r + d ln v2 r d ln r BT87 : 4 56 Now use the geometry of the projection along the line-of-sight to calculate the projected surface brightness I(R), and the projected velocity dispersion σ p : 1

2 If I is observed surface brightness, ν is luminosity density, then I(R) = 2 ν( z 2 + R 2 ν(r)rdr )dz = 2 0 R r 2 R 2 Similarly, we derive for the observed dispersion: I(R)σ 2 p (R) = 2 R ν(r) v 2 r (r)rdr r 2 R 2 These are Abel integral equations and can be easily inverted: ν(r) = 1 π r di dr dr R 2 r 2 2

3 ν(r) v 2 r = 1 π r d(iσp 2) dr dr R 2 r 2 Hence, the observables σ p and I can be used directly to infer the intrinsic σ and ν. With the Jeans equation above we can immediately calculate the mass distribution from these two. 3

4 An example is M87 (see also BT). Three main questions: a), b), and c). 4

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8 Sargent et al. 1978: long-slit spectroscopy of M87 yields a galaxy spectrum for positions along the slit. The strong absorption is due to the sum of the Ca II H and K absorption lines of individual stars in the galaxy. Also is show measurement for a reference star. Comparing the galaxy and stellar spectrum yields the projected velocity dispersion at the observed location of the galaxy. 8

9 From this analysis, Sargent et al conclude that the observed velocity dispersion is rising towards the center: 9

10 Based on this velocity dispersion profile, the mass distribution and mass-to-light ratios were calculated. Υ(r) is ratio of mass with radius r to luminosity with radius r = M(< r)/[4π r 0 νr 2 dr] 10

11 The mass-to-light ratio increases strongly towards the center. The authors argued that this proved that a black hole had to be present (with a mass of order M sun ). 11

12 Anisotropic models for M87 (BT-87 p ) However, it is far from clear that these isotropic models are correct. There is no law of physics saying that models have to be isotropic - the velocity dispersions can also be anisotropic. This can give large changes in the observed velocity dispersions for the same mass distributions. Correspondingly, the true mass distribution can be quite different from the mass distribution derived under the assumption of isotropy. Previous underlying assumption: M/L is free parameter β(r) = 0 These assumptions are not necessarily correct. 12

13 Alternative assumption: M/L ratio is fixed! β(r) is free parameter How to proceed? The detailed derivation is shown in the Appendix (this is not explained during the course). Here we take a simpler road: we show how different assumptions for the anositropy cause different dispersion profiles to be observed. Take as an example M87. Van der Marel (1994, MNRAS 270, p271) observed and analyzed the galaxy. He used the surface brightness distribution from literature. 13

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15 He inverted this to obtain the following luminosity density distribution:

16 Next, he calculated the intrinsic and observed velocity dispersion profiles assuming a constant anisotropy β.

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18 Notice the strong variations in velocity dispersion profiles for the same density distribution. Now, we can compare the observed velocity dispersion profile to the models Hence a model with constant anisotropy β = 0.5 fits the data well - without needing a black hole!

19 HOW TO SOLVE THIS PROBLEM? The problem shown above seems to indicate that it is nearly impossible to derive mass distributions if the velocity dispersion has been measured. Fortunately, this is not the case. Several aspects can help us to get good measurements of the mass distributions: 1. We can measure more than just the dispersion. M easurements of the higher order moments (v 4 ) can help to get better mass measurements (van der Marel & Franx 1993). 2. With regards to black holes, observations with the Hubble Space Telescope really help. They can go so far in, that suddenly the freedom in the models is seriously curtailed. In modern applications, Schwarzschild models are constructed to fit the observations. See for an example below. 14

20 3. Measurement of gas velocities can sometimes be used with great success. This is obviously a completely different method - and raises its own problems. For example: is the gas really nicely in equilibrium and rotating in a circular disk?

21 Example of HST observations of a nearby galaxy with a Black Hole: M32 van der Marel et al. performed observations, and constructed Schwarzshild models (1998, ApJ 493, 613). The observations are shown below: 15

22 The authors constructed detailed models, for a range of mass-to-light ratios and black hole masses. The residuals from the observations are shown in next figure. The parameter χ 2 is defined as: (residuals/measurement error) 2. As can be seen, a super massive black hole must be present

23 HOME WORK 1) Derive the black hole mass in M32 from the figure above. 2) Calculate the circular velocity around the black hole mass at 0.1 arcsec away from the center of M32, and at 1 arcsec away from the center. Assume that the distance to M32 is the same as the distance to M31. 3) How do these circular velocities compare to the stellar velocity dispersion at 1 arcsec What do you conclude from that? 16

24 APPENDIX. How to derive the anisotropy under assumption of a constant mass-to-light ratio First, calculate mass density: ρ(r) = Υν(r), with ρ(r) is mass density, ν(r) is luminosity density, Υ is M/L ratio. Υ follows from observed ν(r) and σ p (r) (Virial Theorem) The problem is now reduced to finding a β(r) and v 2 r (r) that fit the observed σ p (r) Remember β = 1 v 2 θ / v2 r v 2 θ and v2 r project to produce the following observed velocity dispersion σ p 17

25 v p = v r cos α v θ sin α. No rotation, hence = 2 = 2 Σ(R)σ 2 p (R) (v r cos α v θ sin α) 2 νrdr R r 2 R 2 ( R v2 r cos2 α + vθ 2 sin2 α) 2 νrdr r 2 R 2 = 2 R (1 βr2 r 2 ) ν v2 r rdr r 2 R 2

26 We can eliminate β from this equation with the Jeans equation = R 2 + GMR2 r 3 v 2 r Σσ 2 p (R) + R2 d ln(ν vr 2 ) r 2 d ln r ν v2 r rdr r 2 R 2 This equation can be solved for v 2 r. After solution, one has to verify that v 2 r > 0, and β can be calculated

27 The figure below shows an old application to M87. The solution reproduces the observed velocity dispersion for M87. We still need to verify whether a physical distribution function is likely related to this β(r). 18

28 The resulting model fits the observations well, as shown below: 19