The existence of an equilibrium solution to the CBE does not assure its stability. Real stellar systems are subject to perturbations, and if these grow they may completely transform the initial equilibrium. Thus it is critical to check that our galaxy models are actually stable.

The Jeans (1929) instability is probably the most basic instability in gravitating systems. It was originally derived to discuss a since-abandoned model for the formation of the Solar System, but it is now recognized as of fundamental importance for cosmology.

To understand the Jeans instability, consider a nearly-uniform
distribution of stars containing a *slightly* overdense spherical
region with radius `L` and density `rho`. *The
overdense region will collapse if the random velocities of stars are
not large enough to carry them out of the region before the collapse
can occur* (e.g. Jeans 1929, Toomre 1964).

The collapse timescale can be estimated by considering a star at
rest on the edge of the sphere. The gravitational acceleration of
this star is just `GM/L^2`, where the mass of the sphere is

4pi 3 (1) M = --- L rho . 3Then the time for this star to reach the center is just half the Keplerian period for an orbit with semimajor axis

1 (L/2)^3/2 3 pi 1/2 (2) t_coll = - 2pi --------- = (--------) . 2 (G M)^1/2 32 G rhoThe timescale for stars to escape the overdense region is of order

(3) t_esc = L / v_rms .Notice that

3 pi v_rms^2 1/2 (4) L_J = (------------) . 32 G rhoOnly those overdense regions with

To form a physical understanding of disk instabilities, we can first consider the Jeans instability in a non-rotating disk, and then consider separately the effects of rotation (Toomre 1964, GKvdK89, Ch. 9.5).

A 2-D version of the Jeans instability serves as a model for local
gravitational instability in a stellar disk. Let `Sigma` be
the surface density of the disk, and suppose that there is a region,
of radius `L` and mass

2 (5) M = pi L Sigma ,which is slightly overdense. Approximating the collapse time by the same Keplerian formula as above, we have

1 (L/2)^3/2 pi L 1/2 (6) t_coll = - 2pi --------- = (---------) , 2 (G M)^1/2 8 G Sigmawhile the escape time is again given by Eq. (3). Notice that

pi v_rms^2 (7) L_J = -- ------- . 8 G SigmaAs before, only overdense regions with

In a differentially rotating disk the local angular velocity is
Oort's constant `B`. A circular region collapsing from radius
`L` to radius `L_1` will conserve its angular momentum, so
its angular velocity is

2 2 (8) Omega_1 = B L / L_1 .If we analyze the motion of the region in a frame of reference rotating with angular velocity

2 2 4 3 (9) a_r = L_1 Omega_1 = B L / L_1 .There is also an inward acceleration due to gravity:

2 (10) a_g = - G M / L_1 ,where once again a point-mass approximation has been used. Now the key idea is that

d a_r | d a_g | (11) ----- | = - ----- | d L_1 | d L_1 | |L_1 = L |L_1 = Land solving for

2pi G Sigma (12) L_rot = --- ------- . 3 B^2Rotation prevents collapse on scales

Combining the above results, we conclude that only regions with radii
satisfying `L_J < L < L_rot` can collapse; smaller scales
are stabilized by random motion, while larger scales are stabilized by
rotation. Thus *a disk is locally stable if* `L_J >
L_rot` (Toomre 1964). Setting `L_J = L_rot` and solving for
the *r.m.s.* stellar velocity yields

4 G Sigma (13) v_rms,min = ----- ------- . 3^1/2 BIf the local velocity dispersion is greater than

In general, the procedure for analyzing the stability of a stellar system is:

**1.** Start with an equilibrium solution to the CBE and Poisson
Equation:

(14) f = f_0(r,v) , Phi = Phi_0(r) .

**2.** Introduce perturbations scaled by `epsilon <<
1`:

f = f_0(r,v) + epsilon f_1(r,v,t) , (15) Phi = Phi_0(r) + epsilon Phi_1(r,t) .

**3.** Plug these perturbed solutions into the CBE and Poisson
Equation, and keep only terms of `O(epsilon)`. This yields
*linearized* forms of these equations (see BT87, Ch. 5).

**4.** Solve the linearized equations to find the time-development
of an initial `f_1(r,v,0)`. If *any* initial perturbation
can be shown to grow with time, the system is unstable. To prove
stability one must, in principle, consider all possible perturbations,
and show that *none* lead to growing solutions.

**Local analysis:** If the equilibrium solution is spatially
homogeneous, *or* if the characteristic length-scale of the
perturbations is much smaller than the characteristic length-scale of
the system (WKB approximation), the imposed perturbations can be
Fourier-analyzed in space and time into components of the form

i(k.r - omega t) (16) f_1(r,v,t) = f_a(v) e ,where

-i omega t gamma t (17) f_1 ~ e = e .

The WKB analysis of a differentially-rotating disk galaxy is covered in BT87, Ch. 6.2. Here I will only quote results for axisymmetric perturbations, which locally have the form

i(k R - omega t) (18) f_1 ~ e ;it turns out that such perturbations are sufficiently general to expose the most important physical effects. The dispersion relations resulting from such perturbations involve a quantity not yet mentioned: the radial or

d Omega^2 1/2 (19) kappa = (R --------- + 4 Omega^2) , dRwhere

For a *gas* disk, the dispersion relation is

2 2 2 (20) omega = kappa - 2 pi G Sigma |k| + k v_s ,where

For a *stellar* disk, the dispersion relation depends on the
detailed form of the distribution function. If the random stellar
velocities in the disk are assumed to have a gaussian distribution, the
dispersion relation is

2 2 Omega k^2 sigma_R^2 (21) omega = kappa - 2 pi G Sigma |k| F(-----, -------------) , kappa kappa^2where

In either case, local stability against axisymmetric perturbations is
assured if `omega^2 > 0` for all values of `k`. This
condition implies that

kappa v_s (22) Q_gas = ---------- > 1 , pi G Sigma kappa sigma_R (23) Q_stars = ------------- > 1 3.36 G Sigmafor locally stable gaseous and stellar disks, respectively (Toomre 1964).

An estimate of `Q` for the solar neighborhood is given in
BT87, Ch. 6.2. For the solar neighborhood, the surface density and
epicyclic period are roughly

2 (24) Sigma = 75 M_solar / pc , (25) kappa = 36 km / s / kpc ,and the radial velocity dispersion, averaged over the vertical extent of the disk, is

(26) sigma_r = 45 km / s .To account for the finite thickness and gas content of the galactic disk, the coefficient of

(27) Q = ~1.7 ,so it appears that the Milky Way is locally stable.

For other disk galaxies, the radial dispersion profile may be
estimated by comparing gaseous and stellar rotation velocities; the
latter lag the former by an amount proportional to `sigma_R^2`
due to asymmetric drift. The few galaxies which have been studied so
far yield `Q = 1.5` to `2`; moreover, `Q` appears
to be fairly independent of the radius `R` (GKvdK89, Ch.
10.2).

It is easy to understand why `Q > 1`; if galactic disks
were locally unstable to gravitational collapse then massive clumps of
stars would form and scatter other stars, increasing the velocity
dispersion until `Q = 1` was reached. But the actual
mechanism(s) responsible for randomizing the velocities of disk stars
are not completely understood. Scattering by giant molecular clouds
(which may represent gravitationally-collapsed clumps in the
*gaseous* disk) can explain part of the velocity increase, but
apparently not all of it (*e.g.* Wielen & Fuchs 1990).

- Jeans, J.H. 1929,
*Astronomy & Cosmogony*. - Toomre, A. 1964,
*Ap. J.***139**, 1217. - Toomre, A. 1974, in
*Highlights of Astronomy*, ed. G. Contopoulos, p. 457. - Wielen, R. & Fuchs, B. 1990, in
*Dynamics and Interactions of Galaxies*, ed. R. Wielen, p. 318.

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

Last modified: March 24, 1997