Chapter 26

Write a general expression for the force of dynamical friction as $$f_{d} \simeq C(G M)^{a}\left(v_{M}\right)^{b} \rho^{c},$$ where \(C\) is dimensionless, and \(a, b,\) and \(c\) are constants. Set up a system of three linear equations for \(a, b,\) and \(c\) such that \(f_{d}\) has units of force. Solve the system and show that Eq. ( 1 ) is obtained. $$f_{d} \simeq C \frac{G^{2} M^{2} \rho}{v_{M}^{2}},$$

Suppose that the original density distribution of the proto-Galactic nebula envisioned by Eggen, Lynden-Bell, and Sandage had a radial functional dependence similar to the dark-matter halo. Beginning directly with the radial equation of motion, $$\frac{d^{2} r}{d t^{2}}=-\frac{G M_{r}}{r^{2}},$$ show that the free-fall time would be proportional to radius. Hints: Multiply both sides of the equation by \(v=d r / d t\) and use the relation $$v\left(\frac{d v}{d t}\right)=\frac{1}{2} \frac{d v^{2}}{d t}.$$ You may also find the following definite integral helpful: $$\int_{0}^{1} \frac{d u}{\sqrt{\ln \left(\frac{1}{u}\right)}}=\sqrt{\pi}.$$

Estimate the scale height associated with a Galactic disk having a temperature of \(10^{4} \mathrm{K}\). With which component of our Galaxy might such a disk correspond today?