11. Jeans Theorem & Applications

Astronomy 626: Spring 1997

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.


11.1 Jeans Theorem

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
            dt
along all orbits (r(t), v(t) = dr/dt). In a time-independent system the specific energy E = 0.5|v|^2 + Phi(r) is always an integral; in spherical systems the specific angular momentum J is another.

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              dv
where the first equality follows because I is an integral, the second by the chain rule, and the third by substitution of the equations of motion. Moreover, any function F(r,v) which depends on (r,v) only through one or more integrals of motion,
(3)         F(r,v) = F(I_1(r,v),I_2(r,v), ...) ,
is also a solution of the time-independent CBE (BT87, Chapter 4.4). Jeans theorem is useful in constructing equilibrium models of stellar systems.

11.2 Isotropic Models

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       dr
It's convenient to use the boundary condition Phi -> 0 as r -> infinity; then the escape energy is zero and all stars at radius r have energies between Phi(r) and 0.

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)) ,
                       |         
                       / 0
where v_e = sqrt(-2 Phi(r)) is the escape velocity at radius r. Using the definition of the binding energy to change the integration variable gives
                       / 0
                       |               1/2
(6)         rho = 4 pi | dE (2E - 2Phi)    f(E) .
                       |
                       / Phi

From f to rho

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       dr
Once the solution Phi(r) is determined, the density profile is just rho(r) = rho(Phi(r)).

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 F is a constant. Using Eq. 5 to get rho(Phi), Eq. 7 becomes
             -2 d    2 d(Phi)            5
(9)         r   -- (r  ------) = K (-Phi)  ,
                dr       dr
where K is another constant. The solution is a model with the density profile
                      3     -3       2  2 -5/2
(10)        rho(r) = --- M a   (1 + r /a )     ,
                     4pi
where M is the total mass and a is a scale radius. This model was originally devised to describe observations of star clusters. It is actually not a very good model for elliptical galaxies; most of the mass lies within a nearly-constant-density core, and at large r the density falls off as r^-5, which is steeper than the density profiles of E galaxies.

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_0
where rho_1 is a parameter with units of density, sigma is a parameter with units of velocity, and E_0 is a parameter with units of energy. One more parameter is needed to construct a solution; this is the value of the potential at the center, Phi_0 = Phi(0). However, there is a degeneracy among these four parameters, so these models actually form a three-parameter sequence. Two of the parameters effectively fix the total mass and radius of the model. The third parameter, which may be expressed as W_0 = (E_0 - Phi_0)/sigma^2, determines the form of the density profile. Models with W_0 = 3, for example, are almost `all core', while for larger values of W_0 an extended envelope is added surrounding the core.

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).

From rho to f

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)           |
                             / Phi
The 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)
                                   / E 
where rho'(Phi) is the derivative of rho(Phi).

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)  ,
                      4pi
where again M is the total mass and a is a scale radius. The distribution function which generates this model may be expressed in terms of Dawson's integral.

11.3 Anisotropic Models

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     / 0
where eta is the angle between the velocity and radius vectors and J = |r v sin(eta)| is the magnitude of the angular momentum.

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)) ,
                         2
and r_a is the anisotropy radius. For r < r_a the velocity distribution is approximately isotropic, while for r > r_a it is increasingly anisotropic. At any given radius the distribution function is stratified on spheroidal shells in velocity space.


Homework

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.


References


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

Last modified: February 27, 1997