A wide range of self-gravitating systems may be idealized as configurations of point masses interacting through gravity. But in galaxies, the timescale over which the cumulative effects of interactions between individual stars cause significant deflections is so long that galaxies can be accurately approximated as continuous systems. This produces a simpler description, the collisionless Boltzmann equation (BT87, Chapter 4).
Any system in which stellar collisions are rare may be idealized as a collection of N point-sized bodies, each with mass m_i, position r_i, and velocity v_i. The hamiltonian for such a system is
-- -- 1 \ 2 G \ m_i m_j (1) H(r,v) = - | m_i v_i + - | ----------- , ~ ~ 2 / 2 / |r_i - r_j| -- -- i i, j#iwhere H depends on all body positions and velocities, the first sum runs over all N bodies, the second runs over all pairs of bodies (twice, hence the factor of 1/2), and G is the gravitational constant. Then the equations of motion are
-- d(r_i) d(v_i) \ m_j (r_j - r_i) (2) ------ = v_i , ------ = G | --------------- , dt dt / |r_j - r_i|^3 -- j#iwhere the sum runs over all bodies except body i.
N-body systems obey several basic conservation laws. In BT87 (Appendix 1.D.2) these laws are derived by manipulating Eq. 2. However, they may also be recognized directly from the form of the hamiltonian; Noether's theorem states that each symmetry of H gives rise to a corresponding conservation law. A symmetry of the hamiltonian is a transformation which leaves the physical system unchanged. For example, translation in time, t -> t + Delta t, is a symmetry of Eq. 1 because H is not an explicit function of time; consequently the total system energy E = T + U = H is conserved. Likewise, symmetry with respect to translation in space, r -> r + Delta r, implies conservation of total linear momentum, and symmetry with respect to rotation gives rise to conservation of total angular momentum.
Another general result shown by manipulating Eq. 2 is the scalar virial theorem (BT87, Chapter 8.1.1), which states that for a system in equilibrium,
(3) 2<T> + <U> = 0 ,where T and U are the total kinetic and potential energy, respectively, and the angle-brackets indicate time-averages. Since E = T+U, the time-averaged kinetic and potential energies are related to the conserved total energy by
(4) <T> = - E , <U> = 2 E .
The total mass M and total energy E of an N-body system thus define characteristic velocity and length scales
2 2 2 <T> |E| M M (5) V = 2 --- = 2 --- , R = - G --- = G ---- . M M <U> 2|E|These are sometimes known as the virial velocity and radius, respectively.
The quantity t_c = R/V is an estimate of the time a typical star takes to cross the system. This timescale may be expressed in several different ways; for example, in terms of the total mass M and energy E, it is
1/2 (6) t_c = G (M^5 / 8 |E|^3) .Note that M and E are conserved, so t_c is a constant even for systems which are far from dynamical equilibrium. In such cases t_c approximates the time-scale over which the system evolves toward equilibrium.
Another expression for t_c follows from the substitution V^2 = G M/R valid for systems near equilibrium:
-1/2 (7) t_c = (G M / R^3) .Here the quantity M/R^3, which has units of density, appears. In systems with galaxy-like density profiles, the virial radius is approximately proportional to the half-mass radius: R = 2.5 R_h. Using this relationship, it follows that
-1/2 (8) t_c = ~1.36 (G rho_h) ,where rho_h is the mean density within R_h. Since the crossing time is just supposed to indicate a typical time-scale for orbital motion, it is usual to drop the numerical constant, and define
-1/2 (9) t_c = (G rho_h) .
Consider an encounter with impact parameter b and velocity v between two stars of mass m. Using the impulse approximation, the transverse velocity acquired is
2 G m (10) v_t = ----- . b vThis approximation is well-justified in systems with large N because large-angle deflections are very rare; the impact parameter leading to a large deflection is
2 (11) b_min = G m / v = ~ R / N ,where the second equality follows from the virial theorem. In the entire system, approximately one close encounter occurs per crossing time, regardless of N.
A key assumption made in considering the effects of interactions between individual stars is that encounters are not correlated with one another; thus collective effects are neglected. This assumption works well in many cases, though examples of collective relaxation may come up later in this course. If each encounter is uncorrelated with the last, the cumulative effect of many encounters is a random walk in velocity; perturbations add in quadrature. During a single passage through the system, a typical star has
(12) dn = (N / pi R^2) 2 pi b dbencounters with impact parameters between b and b+db. Here the first factor is just the surface density of stars, and the second factor is the area of an annulus with radius b and width db. Adding velocity perturbations in quadrature, the deflection due to these dn encounters is
2 2 2 (13) d v = v_t dn = 8 N (G m / R v) db/b ,and the total velocity perturbation acquired in one crossing time is
/ R 2 | 2 2 (14) Delta v = | d v = 8 N (G m / R v) ln(R/b_min) . | / b_minHere the logarithmic factor arises from the integration over impact parameter from b_min to R; each decade between b_min and R contributes equally to the total deflection. Finally, estimating the encounter velocity v from the virial velocity V = sqrt(G N m / R) gives
2 8 ln N 2 (15) Delta v = ------ V Nfor the total change in a typical star's velocity per crossing time t_c.
The relaxation time is the time over which the cumulative effect of stellar encounters becomes comparable to a star's initial velocity. From Eq. 15 this is
V^2 N (16) t_r = --------- t_c = ------ t_c . Delta v^2 8 ln NIn stellar systems with reasonably large N this time is much longer than the crossing time; the evolution of such systems proceeds on two widely-separated timescales. Relaxation due to stellar encounters plays an important role in the evolution of star clusters; galaxies, however, are generally immune due to their vast numbers of stars.
A typical galaxy has 10^11 stars but is only 100 crossing times old, so the cumulative effects of encounters between stars are pretty insignificant. This justifies the next step, which is to idealize a galaxy as a continuous mass distribution, effectively taking the limit t_r -> infinity.
In this limit, each star moves in the smooth gravitational field Phi(x,t) of the galaxy. Thus instead of thinking about motion in a phase space of 6N dimensions, we can think about motion in a phase space of just 6 dimensions. This is a vast simplification!
Rather than keeping track of individual stars, a galaxy may be described by the one-body distribution function; let
(17) f(r,v,t) dr dvbe the mass of stars in the phase-space volume dr dv at (r,v) and time t. This provides a complete description if stars are uncorrelated, as assumed above.
The motion of matter in phase space is governed by the phase-flow,
. . (18) (r, v) = (v, - grad Phi) .How does this affect the total amount of mass in the phase space volume dr dv? The rate of change of the mass is just the inflow minus the outflow; that is, the flow obeys a continuity equation in 6 dimensions:
df d . d . (19) -- + -- (f r) + -- (f v) = 0 , dt dr dvwhere the derivatives with respect to t, r, and v are understood to be partial derivatives. Using the expression for the phase-flow yields the collisionless Boltzmann equation:
df df df (20) -- + v . -- - grad Phi . -- = 0 . dt dr dvThe collisionless Boltzmann equation or CBE describes the evolution of the distribution function f(r,v,t). It serves as the fundamental equation of galactic dynamics.
The gravitational field Phi(x,t) is given by Poisson's equation,
/ | (21) div grad Phi = 4 pi G | dv f(r,v,t) . | /Eqs. 20 & 21 may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.
Let (r,v) = (r(t),v(t)) be the orbit of a star. What is the rate of change of f(r,v,t) along the star's orbit? The answer is
Df df . df . df (22) -- = -- + r . -- + v . -- Dt dt dr dv df df df = -- + v . -- - grad Phi . -- = 0 , dt dr dvwhere the first equality is just the definition of the convective derivative in phase-space, the second equality follows on substituting the phase-flow (Eq. 18), and the last equality follows from the CBE (Eq. 20). Thus, phase-space density is conserved along every orbit.
This fundamental and completely general result shows that the CBE has a much greater level of symmetry than the N-body equations of motion; whereas the latter conserves a fairly small set of parameters, the CBE conserves f(r,v,t) along an infinite number of stellar orbits. We can take advantage of this infinite array of conservation laws to obtain some important results even when we can't explicitly solve the CBE.
Due date: 2/18/97
8. Assuming an encounter velocity v = 300 km/s, compute the large-angle impact parameter b_min for a solar-mass star, and compare it to the radius of the Sun. Some useful numbers are given in BT87, Appendix 1.A.
9. A physically meaningful phase-space distribution function must never be negative. Prove that if f(r,v,t_0) is not negative at time t_0 then it is not negative at all later times.
Last modified: February 12, 1997