Chapter 5: Equilibria of collisionless stellar systems

Problem 5.1

Assuming each star is roughly \(1\,M_\odot\), Fornax has \(N = 10^7\) stars. The relaxation time is approximately

\[ \begin{align} t_{\mathrm{relax}} &\approx \frac{N}{8\ln N}\,t_{\mathrm{cross}} \\ &= \frac{N}{8\ln N}\,\frac{2\pi R}{\sigma} \\ &\approx \frac{10^7}{8 \times 16.1} \times \frac{2\pi \times 1\,\mathrm{kpc}}{10\,\mathrm{km/s}} \\ &\approx 4.7 \times 10^4\,\mathrm{Gyr} \end{align} \]

Problem 5.2

From the virial theorem \(2K + W = 0\), if the mass increases by a factor of \(f\) and all the distances remain the same, then \(W\) increases by a factor of \(f^2\) (since \(W \propto M^2/R\)) and thus \(K\) must also increase by a factor of \(f^2\). Since \(K \propto M \sigma^2\), \(\sigma\) must increase by a factor of \(\sqrt{f}\).

Problem 5.3

Acceleration is dominated by distant bodies

The gravitational force from a single body at distance \(r\) is \(Gm/r^2\). In a roughly homogeneous system the enclosed mass grows as \(M(<r) \propto r^3\), so the cumulative acceleration from all mass inside radius \(r\) is

\[ a(<r) \sim \frac{GM(<r)}{r^2} \propto r \]

which increases with \(r\): distant matter dominates. More concretely, the nearest neighbor sits at \(d \sim R/N^{1/3}\) and contributes \(a_{\mathrm{near}} \sim Gm/d^2\), while the total smooth-field acceleration is \(a_{\mathrm{smooth}} \sim GNm/R^2\). Their ratio is

\[ \frac{a_{\mathrm{near}}}{a_{\mathrm{smooth}}} \sim \frac{1}{N}\left(\frac{R}{d}\right)^2 \sim N^{-1/3} \ll 1 \]

so the nearest neighbor's contribution is negligible.

Jerk is dominated by nearby bodies

The jerk from a body at distance \(r\) involves the tidal tensor \(\partial^2\Phi/\partial x_i\partial x_j \sim Gm/r^3\), giving a jerk that scales as \(Gmv/r^3\). The contribution from particles in a shell of radius \(r\) and thickness \(dr\) to the total stochastic jerk magnitude is

\[ d\dot{a}(r) \sim n\,r^2 dr \cdot \frac{Gmv}{r^3} = Gnmv \frac{dr}{r} \]

which implies \(\dot{a}\) logarithmically diverges as \(r \to 0\): nearby bodies dominate. The nearest neighbor at distance \(d \sim R/N^{1/3}\) contributes

\[ \dot{a}_{\mathrm{near}} \sim \frac{Gmv}{d^3} \sim Gnmv \sim G\bar{\rho}\,v \]

which is of the same order as the jerk from the entire smooth-field tidal tensor (\(\dot{a}_{\mathrm{smooth}} \sim G\bar{\rho}\,v\)). Unlike the acceleration, where the nearest neighbor is suppressed by \(N^{-1/3}\), the nearest neighbor contributes an amount of order \(G \bar{\rho} v\), which is comparable to the smooth-field jerk. This implies that the jerk remains stochastic and "grainy" even as \(N \to \infty\) (at fixed total mass and density), whereas the acceleration becomes perfectly smooth in that limit because \(a_{\mathrm{near}}/a_{\mathrm{smooth}} \to 0\).