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 all orbits, then I(x,v) is an integral of motion (BT87, Chapter 3.1).
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 I is an integral, the second by the chain rule, and the third by substitution of the equations of motion. Moreover, any function F(x,v) which depends on (x,v) only through one or more integrals of motion,
(3) F(x,v) = F(I_1(x,v),I_2(x,v), ...) ,is also a solution of the time-independent CBE (BT87, Chapter 4.4). The Jeans theorem is useful in constructing equilibrium models of stellar systems.
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 rho(r) is the integral of f(x,v) over all velocities; since the velocity distribution is isotropic,
/ 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 escape velocity; thus f(E) must vanish for energies less than the corresponding escape energy. It is convenient to adopt the boundary condition Phi -> 0 at infinity; then the escape energy is zero and all bound stars at radius r have energies between Phi(r) and 0. Using the definition of the binding energy E = Phi(r) + 0.5 v^2 to change the integration variable gives
/ 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 rho and Phi as functions of r.
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 F is a constant. Eq. 6 gives rho ~ (-Phi)^5, and Eq. 7 becomes
-2 d 2 dPhi 5 (9) r -- (r ----) = K (-Phi) , dr drwhere K is a constant. The solution yields a model with the density profile
3 -3 2 2 -5/2 (10) rho(r) = --- M a (1 + r /a ) , 4piwhere 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, because at large r the density is rho ~ r^-5, considerably steeper than the density profiles of E galaxies.
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 f(E) (BT87, Chapter 4.4.3(d)).
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 is the total mass and a is a scale radius. The potential is given by
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 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 (15) Q = Phi(r) + - v (1 + (r/r_a) sin (eta)) , 2and 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.
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.
Last modified: February 7, 1995