Jeans theorem states that any function `F = F(I_1,I_2,...)`,
where the quantities `I_i` are integrals of motion, is a
solution of the time-independent collisionless Boltzmann Equation.
Here this theorem is used to construct models of spherical
galaxies.

Recall from Lecture 8 that an *integral of
motion* is any function `I(r,v)` of the phase-space
coordinates `r` and `v` such that

d (1) -- I(r(t),v(t)) = 0 dtalong

Jeans theorem states that any integral of the motion is a solution of the time-independent CBE. The proof is straightforward:

dI dr dI dv dI dI dI (2) 0 = -- = -- . -- + -- . -- = v . -- - grad Phi . -- , dt dt dr dt dv dr dvwhere the first equality follows because

(3) F(r,v) = F(I_1(r,v),I_2(r,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(r,v) = f(E)` is a function of
*only* the specific energy `E`. 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 d(Phi) (4) r -- (r ------) = 4 pi G rho(r) . dr drIt's convenient to use the boundary condition

The mass density `rho(r)` is the integral of `f(r,v)`
over all velocities; since the velocity distribution is isotropic,
this integral is

/ v_e | 2 (5) rho = 4 pi | dv v f(E(r,v)) , | / 0where

/ 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 either Eq. 5 or 6 to
calculate the function `rho(Phi)`, and insert the result in Eq.
4. This yields an ODE for `Phi` as a function of `r`:

-2 d 2 d(Phi) (7) r -- (r ------) = 4 pi G rho(Phi) . dr drOnce the solution

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

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

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

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

The **King models** (King 1966) are based on a `lowered
isothermal' distribution function,

-3/2 -(E - E_0)/sigma^2 { rho_1 (2 pi sigma^2) (e - 1) , E < E_0 (11) f(E) = { { 0 , E > E_0where

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

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 the Abel integral equation

/ 0 d(rho) 1/2 | -1/2 (12) ------ = 8 pi | dE (E - Phi) f(E) . d(Phi) | / PhiThe solution (BT87, Appendix 1.B.4) is

/ 0 1 d | rho'(Phi) (13) f(E) = ------------ -- | d(Phi) ------------- , sqrt(8) pi^2 dE | sqrt(E - Phi) / Ewhere

Apart from its application to specific problems, Eq. 13 illustrates
a general and important point: *not all spherical density profiles
can be realized with physically meaningful isotropic distribution
functions*. If the integral in Eq. 13 is a decreasing function of
`E`, the distribution function is negative, which is physically
impossible. For example, `hollow' models in which the density has a
local *minimum* at the center can't be realized using isotropic
distribution functions.

**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 (14) rho(r) = --- M a r (r + a) , 4piwhere again

If the velocity distribution is not isotropic then `f(r,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:

/ v_e / pi | 2 | (15) rho(r) = 2 pi | dv v | d(eta) sin(eta) f(E,J) , | | / 0 / 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 (16) Q = Phi(r) + - v (1 + (r/r_a) sin (eta)) , 2and

Due date: 2/27/97

11. Starting with Eq. 8, find the density `rho(Phi)` for
the Plummer model (*hint: start with Eq. 5 and make the
substitution* `v^2 = -2 Phi cos^2(theta)`).

12. Insert your result for Prob. 11 into Eq. 7 to get a
differential equation for `Phi[r]`. If you can, solve this
equation with the boundary conditions (a) `Phi = 0` at `r =
infinity`, and (b) `d(Phi)/dr = 0` at `r = 0`. If
you don't know how to solve this equation, show that the Plummer model
potential from Lecture 7 is a solution and
express `F` and `Phi(0)` in terms of the scale radius
`a` and the total mass `M`; then use dimensional
analysis to check that `F (-E)^(7/2)` has units of phase-space
density.

- 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@galileo.ifa.hawaii.edu)

Last modified: February 27, 1997