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 due to near stars?

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$}\vert$. In such a time-independent potential both the angular momentum, ${\bf L}=m\,{\bf r}\times{\bf v}$, and energy, $E=v^2/2+\Phi$, 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), and then sum over encounters. This derivation was first done by Chandrasekhar, the derivation here is slightly different from BT (p. 188).

Let's compute the change in velocity $\delta\hbox{$\bf v$}$ of this star. Because we want to treat the case where the effect of encounters is small, we'll approximate the orbit as a straight line, traversed with constant velocity $v$. 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. Combined with Newton's law

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

yields the net change $\vert\delta\bfv_\perp\vert$ over a single encounter:

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

Thus the change is roughly equal to the force at closest approach, $Gm/b^2$, times the time this force acts, $\Delta t\sim b/v$. Because of symmetry, encounters are equally likely to change the velocity in the positive direction, as in the negative direction. Therefore the mean change $\langle \delta\bfv_\perp\rangle=0$. We will therefore consider the rate of change of $\delta\bfv_\perp^2$.

To compute the rate of change of $\delta\bfv_\perp^2$, due to many encounters, proceed as follows. Suppose the test star moves with velocity $v$ through a stellar system with radius $R$, and (uniform) number density of stars, $n$. When travelling for a time $\delta t$, the number of encounters with impact parameter between $b$ and $b+db$, is simply the volume of the cylindrical shell, $dV=2\pi\,b\,db\,\times v\delta t$, times the number density of stars, $n$. Here, $v\delta t$ is the length of the cylinder, $2\pi\,b\,db$ is the surface area of the shell. Since each of these encounters leads to a change in velocity squared by an amount $\delta\bfv_\perp^2$, we find for the rate of change

$${d^2 \delta\bfv_\perp^2\over dt\,db} = n\,v\,2\pi\,b\,\times ({2Gm\over b\,v})^2\,.$$ (8.4)

This is the product of the rate of encounters, $n\,v\,2\pi\,b db$, with the velocity change per encounter. Now we can integrate over impact parameter to obtain for the rate of change in velocity

$${d\delta\bfv_\perp^2\over dt} = \int_0^\infty n\,v\,2\pi\,b\,({2Gm\over b\,v})^2\,db=2\pi\,nv\,({2Gm\over v})^2\,\int_0^\infty {db\over b}\,.$$ (8.5)

Unfortunately this integral diverges, both at small impact parameters (which are few in number, but cause large velocity changes), and at large impact parameters (which cause small velocity changes yet are very numerous). How to overcome this divergence?

For a real stellar system, there cannot be encounters with impact parameter larger than the size $R$ of the system. For the smallest impact parameter $b$, 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 in mind we replace the previous equation by

$${d\delta\bfv_\perp^2\over dt} \approx 2\pi\,nv\,({2Gm\over v})^2\... ...rm min}}^R {db\over b}=2\pi\,nv\,({2Gm\over v})^2\,\ln({R\over b_{\rm min}})\,.$$ (8.6)

We can simplify this by assuming that the stellar system is in virial equilibrium, in which case the kinetic energy of all stars combined, $K\sim N\,m\,v^2/2$ is half of the potential energy, $U\sim G\,(N\,m)^2/R$, where $m$ is the mass of a star. Therefore

$$v^2\approx {GNm\over R}\,.$$ (8.7)

Combining this with the expression for $b_{\rm min}$ it is easy to show that $R/b_{\rm min}=N$, the total number of stars.

So far we have computed the rate of change of the velocity. To judge whether the change of velocity is large or small, we can multiply the rate with the crossing time of the system, $t_{\rm cr}\equiv R/v$. It is a measure of how long it takes to cross the stellar system, on average. The velocity change during a single crossing of the stellar system is therefore

$$\delta\bfv_\perp^2\sim {d\delta\bfv_\perp^2\over dt}\,t_{\rm cr} = 2\pi\,nv\,({2Gm\over v})^2\,\ln(N)\times {R\over v}$$

$$= {6\ln(N)\over N}\, v^2\,.$$ (8.8)

The first line combines Eq.(8.6) with our finding $R/b_{\rm min}=N$ and the definition of the crossing time. The second step assumes virial equilibrium, Eq. (8.7). We are finally in a position to define the relaxation time $T_{\rm relax}$, as the time it takes for encounters to change the velocity by order of itself, hence

$${v^2\over T_{\rm relax}}\equiv {d\delta\bfv_\perp^2\over dt} = {6\ln(N)\over N}\, {v^2\over t_{\rm cr}}\,$$ (8.9)

hence we get as our final result

$$T_{\rm relax} = {N\over 6\ln(N)}\,t_{\rm cr}\,.$$ (8.10)

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 with $N=10^5$ say, $r=10\pc$, and $v=10{\hbox{\rm km s$^{-1}$}}$, $T_{\rm cross}\approx 1{\hbox{\rm Myr}}$. 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, and derive fluid-like equations for the behaviour of a galaxy. The stars will be like the gas particles in the usual gas equations. This is what we'll do next.