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 L/2 about a point-mass M, or
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 L divided by the r.m.s. stellar velocity, or
(3) t_esc = L / v_rms .Notice that t_coll is independent of L, while t_esc increases linearly with L. Thus small regions have t_esc < t_col and are stable, while large regions have t_esc > t_col and are unstable. The critical radius where collapse is just possible can be estimated by setting t_esc = t_col; the result is the Jeans length,
3 pi v_rms^2 1/2 (4) L_J = (------------) . 32 G rhoOnly those overdense regions with L > L_J are subject to the Jeans instability.
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 t_coll is now proportional to L^1/2; because this is still less than the linear proportionality of t_esc, the reasoning used in the 3-D case applies here too. Thus by setting t_esc = t_coll we obtain the Jeans length in 2-D
pi v_rms^2 (7) L_J = -- ------- . 8 G SigmaAs before, only overdense regions with L > L_J can collapse before they are erased by random motions of stars.
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 = B L / L_1 .If we analyze the motion of the region in a frame of reference rotating with angular velocity Omega we must include an outward-directed pseudo-acceleration (centrifugal force), which at the edge of the disk is
2 2 4 3 (9) a_r = L_1 Omega = 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 collapse will not occur if a_r initially increases faster than a_g (Toomre 1964). This establishes a maximum radius for collapse, because rotation is more important on larger scales. Setting
da_r | da_g | (11) ---- | = - ---- | dL_1 | dL_1 | |L_1 = L |L_1 = Land solving for L yields the maximum unstable radius
2pi G Sigma (12) L_rot = --- ------- . 3 B^2Rotation prevents collapse on scales L > L_rot.
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 v_rms,min then the disk is locally stable.
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(x,v) , Phi = Phi_0(x) .
2. Introduce perturbations scaled by epsilon < < 1:
f = f_0(x,v) + epsilon f_1(x,v,t) , (15) Phi = Phi_0(x) + epsilon Phi_1(x,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(x,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.x - omega t) (16) f_1(x,v,t) = f_a(v) e ,where k is the wave-number and omega is the frequency of the perturbation. If any growing solutions of the linearized CBE exist then there must be solutions of the form in Eq. (16) which also grow, since any solution can be expressed as a sum of these Fourier components. When Eq. (16) is inserted into the linearized CBE and Poisson Equations, the result is a dispersion relation between omega^2 and k. If omega^2 < 0 for any value of k then perturbations with that wave-number are unstable because then omega = i gamma for some real gamma, and the corresponding Fourier component grows like
-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 epicyclic period of a star on a nearly circular orbit,
d Omega^2 1/2 (19) kappa = (R ---------+ 4 Omega^2) , dRwhere Omega(R) is the angular velocity of the circular orbit at radius R.
For a gas disk, the dispersion relation is
2 2 2 (20) omega = kappa - 2 pi G Sigma |k| + k v_s ,where v_s is the speed of sound in the gas.
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 sigma_R is the radial velocity dispersion and the reduction factor F(s,chi) is given in Eq. (6-45) of BT87. Note that F(s,0) = 1; thus Eqs. (20) and (21) are identical in the limiting case where v_s = sigma_R = 0. This is reasonable since the dynamical stability of a perfectly `cold' disk should not depend on its make-up.
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 3.36 in Eq. (23) should be reduced to ~2.9 (Toomre 1974). The result is that for the solar neighborhood
(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).
Last modified: March 6, 1995