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 | dx_i | dx_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 dx_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 dx_i dx_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 dx_i dx_j dx_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(x,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 v_r and v_t are speeds in the radial and tangential directions, respectively. Define the radial and tangential dispersions,
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 -infinity for a purely tangential velocity distribution, 0 for an isotropic distribution, and +1 for a purely radial velocity distribution. Then Eq. 8 becomes
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 beta(r). One option is to assume that the velocity distribution is isotropic, so that beta = 0 for all r. This allows us to calculate mass profiles, but the results are uncertain because there is no good reason why velocity distributions should be isotropic. Another option is to assume that the galaxy has a constant mass-to-light ratio, and calculate the anisotropy profile beta(r) needed to satisfy Eq. 13 (Binney & Mamon 1982).
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 k is Boltzmann's constant, m_p is the proton mass, and mu is the mean molecular weight (e.g. Sarazin 1987). The parallels between Eq. 13 and Eq. 14 are obvious.
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.
Last modified: January 31, 1995