Problem set #5 (the last one). Due .....................................

The gas responds strongly (non-linearly) to the force associated with a spiral wave in the stellar disk of a galaxy. Your objective will be to calculate how strongly, and to formulate an approximate criterion for non-linear response in terms of the strength and wavelength of the spiral wave.

We shall neglect the self-gravity of gas, the modifications of stellar spiral wave due to the presence of gas, and all the non-linear effects in general. We will treat the ISM as a perfect gas with soundspeed v_s, subject locally around some radius of interest, r, to external force described by potential

Phi_1(r,theta,t)=F Omega^2 r |k|^(-1) exp i(k r + m  theta -  omega t)

where k is the radial wavenumber of a tightly wrapped wave (kr>>m), and F is the amplitude factor of the force.

Starting from the Eulerian equations of hydrodynamics for a vertically averaged (2-D) gas disk, i.e. two momentum equations and one continuity equation, perform the linearization of the equations around the unperturbed circular orbits (cf. BT), assuming that F<<1 and that the gas response has the same (r,theta,t) dependence as Phi_1.
Solve the momentum eqs. to obtain the v_r and v_theta components of perturbed velocity. Then plug the v's into the continuity equation, neglecting m/r compared with k in the derivations, in the spirit of WKB (or tight-winding) approximation. Justify your assumption by considering the solar neighborhood with pitch angle of m=2 arms equal to 15 degrees. Derive an equation connecting the density contrast Sigma_1/Sigma_0 with the F factor and v_s.

What happens if we set v_s=0 at that point, i.e. neglect gas pressure and talk about free (test) particles? What is the density contrast, and when does it exceed unity (the criterion for non-linear response)? (compare result with BT, pp.386-91)

Is the pressure-related term negligible in general? For instance, in the solar neighborhood, where  kappa^2 - (m Omega - omega)^2 is comparable with kappa^2 ? If not, what is the density contrast in the gas? How does it scale with v_s? What is the non-linearity criterion for the gas? Imagine that the gas was suddenly heated to v_s corresponding to the stelar disk dispersion of velocities = 40 km/s, instead of the actual 7 km/s. How would the  Sigma_1/Sigma_0 change?

Estimate the minimum forcing F required for non-linear gas response in the solar neighborhood. Can F be determined observationally?

1. Rederive the formulae (BT eqs. 7-10) for the perturbation of the velocity of a test particle of mass M, in an encounter with particle m, using the impulse approximation rather than the full 2-body trajectories (hyperbolae) as in BT. Did you get the same or a different result?
2. Study the motion of a test particle representing a merging satellite system moving in the potential of a disk galaxy NGC666 (cf. problem 6), subject to additional dynamical friction force dv_M/dt given by the Chandrasekhar formula. Assume NGC666 does not change in time, and its stellar disk has velocity dispersion sigma_*=40 km/s (on top of the circular velocity).
3. Chose a method for determining the disk (and halo) potential - may be numerical or analytical. Starting position of the satellite galaxy is in a plane inclined to the disk of NGC666 by 40 degrees, on a circular orbit of radius r=15 kpc.
4. Analyze the 3-D trajectory of the satellite system, and the time of the merger.
5. What is responsible for the merging process, disk halo, or both, and does the answer vary in time?
6. Could some results be predicted by a rough estimate using the Chandra formula even without detailed calculations?
7. Which assumptions and features of your calculation are least reliable or realistic?

A massive black hole is suspected in the nucleus of galaxy NGC777, which at r>100 pc has luminosity profile of a King sphere with core radius r0=100 pc and sigma_0=200 km/s. Inside r0, at r=25 pc the velocity dispersion increases to sigma(25 pc)=280 km/s. The hole grew adiabatically in time shorter than the relaxation time of the nucleus (BT, pp. 545-546). Knowing that sigma(r)^2=sigma_0^2+GM_bh/r, calculate:

• The central mass density rho0 in the initial isothermal King sphere (before the black hole growth).
• The radius of influence of the hole, or the cusp radius r_bh.
• The mass of the black hole, M_bh
• Density profile of a stellar cusp. Sketch on a log-log plot the density profile inside and outside the nucleus.
• M_cusp (mass of the cusp).
• The Schwarzschild radius of the black hole. (r_g=2GM_bh/c^2)
• Show that an average star collides with another star in the cusp on time scale t_coll >> t_relax
• Consider a disk of gas surrounding the black hole out to radius r_bh, of uniform density Sigma. Such a profile characterizes, e.g., a stationary accretion disk with a constant kinematic viscosity coefficient. Let the disk mass M_d be maximum, i.e. the disk is marginally stable against axisymmetric perturbations (Toomre number Q=1, cf. GT). Assuming that disk gas temperature T~10,000 K (and thus, soundspeed v_s~10 km/s) everywhere in the disk, estimate M_d, and compare it with M_cusp, initial King core mass, and the M_bh. Can you estimate the disk thickness (vertical scale height)?