Part III: Galaxies Lent Term 2010 Examples Sheet # 1: Dynamical Timescales 1. CROSSING TIME: Consider a spherical system with constant density p and radius R. (a) Show that the circular orbital period in such a system is independent of radius, and can be expressed as a function of the density alone. (b) Derive the equivalent expression for the crossing time (R/V) in terms of the total mass and radius of the system. 2. A more realistic approximation for a stellar system is an isothermal sphere, with constant orbital velocity v and radius R. Derive for this case an expression for the crossing time at outer radius R, in terms of R and the mass of the system, and compare to the result from the constant density case above. Explain the origin of any difference in timescales. 3. RELAXATION TIME: The crossing time represents a practical min- imum time for altering the structure of a dynamical system. At the other end of the spectrum is the relaxation time, the time required for a system to reach dynamical equilibrium via 2-body interactions. (a) The relaxation time can be estimated by calculating the mean frac- tional change in the orbital kinetic energy of a star during a single crossing of the stellar system, and then scaling by the crossing time of the system. Begin by calculating the gravitational deflection of a star by passing by another star of equal mass, at impact parameter b. The force acting on the star perpendicular to its motion can be expressed as a function of time: F1 = Gm2 62 1 + ] vt 27-3/2 1
Show that over the entire encounter this introduces a transverse change in velocity: bv 2Gm (b) Next consider a system comprised of N stars of equal mass m, and radius R. Derive an estimate for the total number of encounters that a star makes in one crossing of the cluster (with impact parameters be- tween b and 8b), and show that the total change in the square velocity is: Sv} = ( bv 2Gm ) 2 2N R2 bdb (c) To calculate the total change in v2 we need to integrate over all impact parameters from bmin = Gm/v2 to R (collisions closer than bmin result in captures and are not relevant to this problem). Integrating gives: R AV? = ovi ~ 8N Gm ) (Rm) In R bmin 6min Show the the fractional deviation per orbit crossing: Av2 = 8ln (R/bmin) N v2 (d) Finally show that the relaxation time can be approximately ex- pressed in terms of the crossing time as: trelax N N R tcross 8lnN BlnN v 2
4. Use these results to estimate the values of the crossing and relaxation times for the following systems. For which cases are relaxation effects important? (a) the Galactic Center: (b) elliptical galaxy M87: N = 1012 M = 1012 Mo R = 30kpc (c) Coma cluster: N = 1000 M = 1014 Mo R = 1Mpc N =