Manipulation of the collisionless Boltzmann equation (CBE) yields a
set of relationships, the *Jeans equations*, which link
observables such as velocity dispersion profile to the gravitational
potential.

To reduce the collisionless Boltzmann equation to something more manageable, we can project out the velocity dimensions. There are an infinite number of ways to make such projections (BT87, Chapter 4.2). The first of these is simply to integrate over all velocities:

/ / / | df | df | dPhi df (1) | dv -- + | dv v_i ---- - | dv ---- ---- = 0 . | dt | dr_i | dr_i dv_i / / /Rearranging the first two terms, and using the divergence theorem (BT87, Eq. 1B-44) to get rid of the third term, the result is a continuity equation for the 3-D stellar density,

d d ___ (2) -- nu + ---- (nu v_i) = 0 , dt dr_iwhere the stellar mass and momentum density are

/ ___ / (3a,b) nu = | dv f , nu v_i = | dv f v_i , / /with the integrals taken over all velocities.

To obtain the next (three) equations in this series, multiply the CBE
by the velocity component `v_j`, and integrate over all
velocities. Proceeding much as above, the result is

d ___ d _______ dPhi (4) -- (nu v_j) + ---- (nu v_j v_i) + nu ---- = 0. dt dr_i dr_jwhere

_______ / (5) nu v_j v_i = | dv f v_j v_i . /This may be placed in a `more familiar' form by defining the dispersion tensor

2 _______ ___ ___ (6) sigma_ij = v_j v_i - v_j v_iwhch represents the distribution of stellar velocities with respect to the mean at each point. The result is called the equation of stellar hydrodynamics,

___ ___ dv_j ___ dv_j dPhi d 2 (7) nu ---- + nu v_i ---- = - nu ---- - ---- (nu sigma_ij) , dt dr_i dr_j dr_ibecause it resembles Euler's equation of fluid flow, with the last term on the right representing an anisotropic pressure. Note that because there is no equation of state, this pressure is not related in any simple way to the mass and momentum density defined in Eq. 3.

By multiplying the CBE by `v_i v_j ... v_l` and integrating
over all velocities, yet higher order equations can be formed; each,
alas, makes reference to quantities of one order higher than those it
describes. To close this hierarchy of equations, one must make some
physical assumptions.

**Caveat:** Any *physical* solution of the CBE must have a
non-negative phase-space distribution function. The velocity moments of
the CBE describe possible solutions, but do *not* guarantee that a
non-negative `f(r,v;t)` exists. You've been warned!

Perhaps the simplest application of the equations of stellar hydrodynamics is to equilibrium spherical systems. Assuming that all properties are invariant with respect to rotation about the center, the radial Jeans equation is

d _______ 2nu _______ _______ dPhi (8) -- (nu v_r v_r) + --- (v_r v_r - v_t v_t) = - nu ---- , dr r drwhere

2 _______ 2 _______ (9) sigma_r = v_r v_r , sigma_t = v_t v_t ,and the anisotropy profile,

2 2 (10) beta(r) = 1 - sigma_t / sigma_r ,which is

1 d 2 2 2 dPhi (11) -- -- (nu sigma_r) + - beta(r) sigma_r = - ---- . nu dr r dr

If `nu(r)`, `beta(r)`, and `Phi(r)` are known
functions, Eq. 11 may treated as a differential equation for the
radial dispersion profile. Thus one application of this equation is
making models of spherical systems. From an observational point of
view, however, `nu(r)` and `sigma_r(r)` may be known,
and the goal is to obtain the mass profile, `M(r)`. Using the
relation

dPhi M(r) (12) ---- = G ---- , dr 2 rthe mass profile is given by

2 2 r sigma_r d ln nu d ln sigma_r (13) M(r) = - --------- (------- + ------------ + 2 beta(r)) . G d ln r d ln rThe catch here is that we don't know

There is one galactic component which is *known* to possess an
isotropic velocity distribution: the hot gas responsible for extended
X-ray emission in ellipticals. In this case the observable quantities
are the gas density and temperature profiles, `n_e(r)` and
`T(r)`, respectively. In terms of these the mass profile is
given by

k r T(r) d ln n_e d ln T (14) M(r) = - -------- (-------- + ------) , G m_p mu d ln r d ln rwhere

This method has several advantages: (1) no assumptions about velocity anisotropy, (2) it probes the mass distribution at radii where the stellar surface brightness is too low to measure velocity dispersions, and (3) good statistics can be accumulated, whereas tracers such as globular clusters are limited in number. But until recently very few galaxies were well enough resolved to yield detailed temperature profiles.

This galaxy is important in searches for nonluminous matter at
radii large and small. Early studies (*e.g.* Sargent *et
al.* 1978) *assumed* an isotropic velocity distribution, and
used measurements of surface brightness and line-of-sight velocity
dispersion to try and measure the mass profile; the results indicated
the presence of a large amount of unseen mass at small radii.
However, Binney & Mamon (1982) showed that the observations could
just as well be fit by an anisotropic velocity distribution with no
unseen mass.

The extensive X-ray halo around M87 provided more conclusive
evidence for unseen matter at large radii; essentially, the galaxy
must have a very deep potential well to hang on to its hot gas. The
mass profile appears to rise almost linearly with radius, enclosing of
order `3 * 10^13` solar masses within a radius of `300
kpc`. Curiously, if the stars at these radii have isotropic
velocities, the dispersion profile should *rise* by approximately
a factor of two in the outer regions (*e.g.* Sarazin 1987). Such
a rising dispersion profile has so far been seen in only one system,
the cD galaxy IC 1101 (Dressler 1979, Fisher *et al.* 1995).

Optical and X-ray data for NGC 4472 have been combined to try and
constrain the mass distribution of this galaxy (Lowenstein 1992). The
gas is significantly `hotter' than the stars and samples the potential
at large radii, while the stars provide information at smaller scales.
It proves rather difficult to fit all the observations with a single
model; the main uncertainty appears to be the temperature of the hot
gas. However, the best-fitting models contain large amounts of dark
matter, and models containing *only* visible stars and hot gas
seem to be strongly excluded.

More recent data for NGC 4636 constrain the temperature profile of
this galaxy and provide rather clear indications of dark mass at large
radii (Mushotzky *et al.* 1994). The local value of the
mass-to-light ratio is approximately constant within the effective
radius `R_e`, and then rises smoothly, increasing by an order
of magnitude at `12 R_e`. The stellar dispersion profile
predicted for isotropic orbits is essentially flat throughout this
entire radial range.

- Binney, J.J. & Mamon, G.A. 1982
*M.N.R.A.S.***200**, 361. - Dressler, A. 1979,
*Ap.J.***231**, 659. - Fisher, D.
*et al.*1995,*Ap.J.***438**, 539. - Lowenstein, M. 1992,
*Ap.J.***384**, 474. - Mushotzky, R.F.
*et al.*1994,*Ap.J.***436**, L79. - Sarazin, C.L. 1978, in de Zeeuw 1978, p. 179.
- Sargent, W.L.
*et al.*1978,*Ap.J.***221**, 731.

Joshua E. Barnes (barnes@galileo.ifa.hawaii.edu)

Last modified: March 1, 1997