825: Ast Radiative Gas Dynamics Problem Set 1 Due Thursday, January 22
1) Lets start simply, with a hydrostatics problem, rather than a hydrodynamics problem. The Earths atmosphere may be approximated as a static, plane-parallel atmosphere, with a uniform gravitational acceleration g . To make things easier still, lets assume that the Earths atmosphere is made of an ideal gas. a) If P (z ) and T (z ) are the pressure and temperature at a height z above sea level, show that 1 dP gm = , P dz kT where g = 980 cm s2 is the magnitude of the gravitational acceleration and m = 4.8 1023 g is the mean molecular mass of the atmosphere. b) Assume that the decrease in pressure with height is due to an adiabatic expansion of the gas; in this case, the atmosphere follows a polytropic equation of state, with P , where is the adiabatic index. What are the resulting pressure, density, and temperature as a function of height, given their values P0 , 0 , and T0 at sea level? Give numerical results, adopting reasonable values for the sea-level temperature and pressure. c) The results of section (1b) are nonsense in the limit of large z . [Hint: if your answer to section (1b) is not nonsensical in this limit, youve done something wrong.] What false assumptions did we make that led to this nonsensical result? d) Now assume that the atmosphere is not adiabatic, but isothermal (that is, the heating and cooling processes in the atmosphere keep it at a uniform temperature T = T0 .) What are P (z ) and (z ) in this case? Again, give numerical results.
2 2) Lets now look at a steady-state hydrodynamics problem. (Steady-state means that u, , P , and so forth have no explicit time dependence.) a) A star of mass M and radius R emits a spherically symmetric wind at a constant rate M . The wind is expanding freely into a surrounding vacuum. Write down the Euler equations (equations 2.3, 2.4, and 2.5 of the notes) for this spherically symmetric, nonrotating, system. steady-state b) Assume that the pressure gradient and self-gravity of the gas are negligible compared to the gravitational acceleration g = GM /r2 of the central star. Find the radial velocity u(r) of the wind in terms of M , R , and u u(R ). c) Using the continuity equation and the wind velocity, u(r), nd the density (r) of the stellar wind. d) At what radius does the winds self-gravity become comparable to the gravity of the central star? (An approximate estimate is all right.)
3 3) Now lets start looking at simple shocks traveling through dierent media. a) A normal plane-parallel nonradiative shock moves with Mach number M1 = 1.5 through the earths atmosphere (1 = 1.2 103 g cm3 , P1 = 1.0 106 dyne cm2 , = 7/5). What are 2 , P2 , and T2 immediately after passage of the shock? What is the relative velocity, u1 u2 , of the pre-shock and post-shock gas? b) A normal plane-parallel nonradiative shock moves with Mach number M1 = 1.5 through the warm neutral ISM (1 1024 g cm3 , P1 4 1013 dyne cm2 , = 5/3). What are 2 , P2 , and T2 immediately after passage of the shock? What is the relative velocity, u1 u2 , of the pre-shock and post-sh...