Relaxation in stellar systems

As a star moves through a stellar system, it will feel the gravitational force due to all other stars. Is the motion of this star mainly determined by the average gravitational force of all other stars combined, or is it mostly sensitive to the force of a few of the stars nearest to it?

To see that this would make a huge difference, consider the case where a single star (a test particle) moves through a smooth density distribution with spherically symmetric density distribution $\rho(r)$, where $r=\vert\hbox{$\bf r$}\vert$. As you know, in such a potential both the angular momentum ${\bf L}$ and energy $E$ of the star will be conserved along its orbit. Now suppose such a density distribution to be represented by a finite number of stars. As our test star moves along its orbit, it will be deflected by encounters with these other stars. In such a situation, neither ${\bf L}$ nor $E$ will be conserved, and so the properties of its orbit will change in time. Clearly, the longer the star orbits in this density distribution, the more $E$ will differ from its initial value. The relaxation time $T_E$ is a measure of how long the star remembers its initial energy. We will see that the more stars are there in the galaxy, the longer is the relaxation time. You might have expected this, since for an infinite number of stars, the density is smooth, ${\bf L}$ and $E$ are constant, and hence $T_E\rightarrow \infty$.

To estimate $T_E$, we will compute the change in energy $\Delta E$ of a star due to a single encounter with another star, as function of the impact parameter $b$ and impact velocity $v$ (the velocity at infinity, before the encounter started). You'll notice that this derivation is very similar to what we did to obtain the expression for thermal bremsstrahlung. This derivation follows BT (p. 188).

Let's computed the change in velocity $\delta\hbox{$\bf v$}$ of our star. To make things simple, we'll approximate the orbit as a straight line, traversed with constant velocity $v$. Now the force perpendicular to the orbit is

$$\hbox{$\bf F$}_\perp = {Gm^2\over b^2+(vt)^2} \cos(\theta)\approx {Gm^2\over b^2} \left[1+({vt\over b})^2\right]^{-3/2} ,$$ (8.1)

where I've assumed equal mass stars. The approximation $\cos(\theta)=b/r\approx b/vt$ is valid for $b$ small. Now Newton's law tells us that

$$m{\bf {\dot v}}_\perp = \hbox{$\bf F$}_\perp .$$ (8.2)

Combining these two, we find the net change $\vert\delta\hbox{$\bf v$}_\perp\vert$ over a full encounter:

$$\vert\delta\hbox{$\bf v$}_\perp\vert \approx {Gm\over bv} \int_{-\infty}^{\infty} (1+s^2)^{-3/2} ds ={2Gm\over b v} .$$ (8.3)

Thus the change is roughly equal to the force at closest approach, $Gm/b^2$, times the duration of this force, $\Delta t=b/v$.

Suppose our test star moves with velocity $v$ through a stellar system with $N$ stars, and radius $R$. When it traverses the system once, the number $\delta n$ of encounters with other stars, with impact parameter between $b$ and $b+db$ is

$$\delta n = {N\over \pi R^2} 2\pi b db = {2N\over R^2} b db .$$ (8.4)

Because of symmetry, the mean change $\delta\hbox{$\bf v$}_\perp$ is of course zero. So to characterise the effect of encounters, let's compute

$$\delta v^2_\perp = \left({2G m\over bv}\right)^2 \delta n .$$ (8.5)

To find the net change, we need to integrate over the impact parameter $b$. A recurring problem in calculations of this kind, is that the integral $\int_0^\infty db/b$ diverges, both for small and for large impact parameters. Now for the largest $b$ we can simply use $R$, the radius of the system. For the smallest $b$, we can take $b_{\rm min}=Gm/v^2$, for which the change in velocity is comparable to the initial velocity (cfr.Eq. 8.3). Recall that our derivation assumed small changes anyway.

With this caveat, we can integrate Eq. (8.5) over $b$ to find the net change in $v_\perp^2$ when the star moves through the system once, as

$$\Delta v_\perp^2 \equiv \int_{b_{\rm min}}^R \delta v^2_\perp \approx 8N\left({Gm\over Rv}\right)^2 \ln\Lambda ,$$ (8.6)

where

$$\ln\Lambda \equiv \ln\left({R\over b_{\rm min}}\right)$$ (8.7)

is the (logarithm of the) ratio of the largest over the smallest impact parameter. Now if the system is in virial equilibrium, then the kinetic energy $K=(1/2)Nmv^2$ of its stars, should be of order its potential energy, $U=GM^2/R$, or neglecting factors of order unity, and using $M=Nm$

$$v^2\approx {GNm\over R} .$$ (8.8)

This is the typical velocity of a star. Substituting this in Eq. (8.7), using $b_{\rm min}\approx Gm/v^2$ gives

$$\ln\Lambda\approx \ln N .$$ (8.9)

Finally eliminating $R$ between Eqs. (8.6+8.8) gives

$${\Delta v_\perp^2\over v^2} = {8\ln(N)\over N} .$$ (8.10)

If the star makes many crossing of the galaxy, $v_\perp^2$ will change by of order $\Delta v_\perp^2$ at each crossing, so that the number of crossings $n_{\rm relax}$ that are required for its velocity to change by of order itself is given by

$$n_{\rm relax} = {N\over 8\ln(N)} .$$ (8.11)

Thus, the relaxation time may be defined as $T_{\rm relax}=n_{\rm relax}\times T_{\rm cross}$, where $T_{\rm cross}\equiv R/v$ is the crossing time, i.e. the time it takes the average star to cross the system. So we find

$$T_{\rm relax} = {N\over 8\ln(N)} T_{\rm cross} .$$ (8.12)

For systems of the same mass and size, and hence a given crossing time, we find that the relaxation time increases with $N$ as $T_{\rm relax}\propto N/\ln(N)$. Put differently, the effect of encounters decreases with increasing particle numbers as $N/\ln(N)$.

For a Globular Cluster, $N=10^5$ say, $r=10\pc$, and $v=10{\hbox{\rm km s$^{-1}$}}$, hence $T_{\rm cross}\approx 1{\hbox{\rm Myr}}$. Putting in the numbers^8.1, we find $T_{\rm relax}\approx 10^9$ years, much smaller than the ages of the stars in the GC. Hence we expect stellar encounters to be important in shaping the structure of GCs.

However, for a big elliptical galaxy, $N=10^{11}$, say. The crossing time is of order $20{\hbox{\rm kpc}}/200{\hbox{\rm km s$^{-1}$}}\approx 10^2 M{\hbox{\rm yr}}$, and $T_{\rm relax}\sim 10^{13}M{\hbox{\rm yr}}$ which is much larger than the age of the Universe ( $\sim 10^{10}{\hbox{\rm yr}}$). Therefore we can completely neglect encounters between stars, when describing the equation of stellar dynamics in a galaxy. This is what we'll do next.