In a spherical, time-independent potential, the energy `E`
and angular momentum `J` of each star are conserved. The Jeans
theorem implies that any function `f(E,J)` is a solution of the
time-independent collisionless Boltzmann Equation.

Let `I(x,v)` be a function which depends on the phase-space
coordinates `x` and `v`, and let `x(t)` and
`v(t)=dx/dt` be a parametric representation of an orbit. If

d (1) -- I(x(t),v(t)) = 0 dtfor

Any integral of the motion is a solution of the time-independent CBE. The proof is straightforward:

dI dI dx dI dv dI dPhi dI (2) 0 = -- = -- . -- + -- . -- = v . -- - ---- . -- , dt dx dt dv dt dx dx dvwhere the first equality follows because

(3) F(x,v) = F(I_1(x,v),I_2(x,v), ...) ,is

The simplest use of the Jeans Theorem is the construction of
*isotropic* models of spherical galaxies; in this case, the
distribution function `f(x,v)` is a function of `E`
*only*. In a self-consistent system, the gravitational field is
related to the mass density by Poissons equation; adopting spherical
coordinates, we have

-2 d 2 dPhi (4) r -- (r ----) = 4 pi G rho(r) . dr drThe mass density

/ 2 (5) rho(r) = 4 pi | dv v f(E(r,v)) . /Assuming that the system has a finite total mass, it follows that the stars must have velocities less than the local

/ 0 | 1/2 (6) rho = 4 pi | dE (2E - 2Phi) f(E) . | / Phi

Given any functional form for `f(E)` which is non-negative
for all `E < 0`, use Eq. 6 to calculate the function
`rho(Phi)`, and insert the result in Eq. 4. This now gives an
ODE for `Phi` as a function of `r`:

-2 d 2 dPhi (7) r -- (r ----) = 4 pi G rho(Phi) . dr drSolving this equation yields

**Plummer's model** is perhaps the simplest example of this
procedure (BT87, Chapter 4.4.3(a)). The distribution function has the
form

7/2 f(E) = F (-E) , E < 0 , (8) = 0 , E > 0 ,where

-2 d 2 dPhi 5 (9) r -- (r ----) = K (-Phi) , dr drwhere

3 -3 2 2 -5/2 (10) rho(r) = --- M a (1 + r /a ) , 4piwhere

Other models derived by starting with a distribution function
include the polytropes (where `f` is a power-law in
`E`), the `isothermal sphere' (velocities have a Maxwellian
distribution), and the King (1966) model or `lowered isothermal'.

Suppose instead that the density profile `rho(r)`is known;
the goal is now to solve for the distribution function. Eq. 4 yields
the potential `Phi(r)`, and inverting this function allows us
to calculate `rho(Phi)`. Eq. 6 relates `rho(Phi)` to
the distribution function; differentiating this equation by
`Phi` yields

/ 0 d rho 1/2 | -1/2 (11) ----- = 8 pi | dE (E - Phi) f(E) , d Phi | / Phiwhich is an Abel integral equation; it can be solved to obtain

**Jaffe's model** (Jaffe 1983) is one of a number which have
been proposed as approximations to the density profiles of elliptical
galaxies. This model has the form

1 -2 -2 (12) rho(r) = --- M a r (r + a) , 4piwhere again

M r (13) Phi(r) = G - ln(-----) . a r + aThe distribution function which generates this model may be expressed in terms of Dawson's integral.

If the velocity distribution is not isotropic then `f(x,v)`
is a function of `E` and the angular momentum `J`. Most
interesting are systems where `f` depends on the magnitude of
`J` but not on its direction. Eq. 5 is then replaced by a
double integral:

/ / pi | 2 | (14) rho(r) = 2 pi | dv v | deta sin(eta) f(E,J) , | | / / 0where

Proceeding by analogy with the isotropic case, one can pick a form
for `f(E,J)` and calculate the corresponding density profile;
examples include the `generalized polytropes' (power laws in
`E` and `J`) as well as many more physically motivated
models.

Likewise, one may pick a `rho(r)` and find an anisotropic
distribution function. Because `f(E,J)` is a function of two
variables, there are an enormous number of possible anisotropic
distribution functions which produce a given density profile. One
choice are the Osipkov-Merritt models (Osipkov 1979, Merritt 1985),
which assume that `f = f(Q)`, where

1 2 2 2 (15) Q = Phi(r) + - v (1 + (r/r_a) sin (eta)) , 2and

Due date: 2/16/95

13. Starting with Eq. 10, find the functional form for the gravitational potential of the Plummer model (up to a multiplicative constant), and verify that it obeys Eq. 9.

- Jaffe, W. 1983,
*M.N.R.A.S***202**, 995. - King, I.R. 1966,
*A.J.***71**, 64. - Merritt, D. 1985,
*M.N.R.A.S***214, 25p.** **Osipkov, L.P. 1979,***Pis'ma Astr. Zh.***5**, 77.

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

Last modified: February 7, 1995