Astro 405/505: Introduction to Astrophysics, Fall 2002

Assignment #3 Due: Friday, September 27, 2002
BUT
bonus 0.5 points if handed in on Wednesday, September 25
 

  1. The general form for the pressure of an ensemble of particles is:
            P= 1/3 ∫0 ∞n(p)pv dp

    Starting with this general form, derive the equation for the

    1. pressure of photons (the radiation pressure) 

    2. ideal gas law for nonrelativistic fermions (i.e. P=nkT)

    Hints: For a), use the momentum distribution for photons deriveable from the Planck function Bnu(T). To save some time, that gives n(p)=2/h3 (epc/kT-1)-1. For Fermions, you can use the Maxwell-Boltzman number density from Equation 8.1 in your text, suitably transformed to a momentum distribution in the nonrelativistic limit.

     

  2. The first two stellar structure equations relate the radius, mass and pressure within an equilibrium spherically symmetric self-gravitating star.  The kicker, of course, is that the density also appears in these two equations. Assume that the density is approximately constant and equal to the mean density of the star (i.e. 3M/4πR3).

    1. Integrate the equations of continuity and hydrostatic equilibrium. Assume that P=0 at r=R as a boundary condition and evaluate P as a function of radius.

    2. Determine Pc, the central pressure, in terms of M and R. Find the value of Pc in cgs units in terms of the radius in solar radii and mass in solar masses.

    3. Assume that the center of the sun is an ideal, fully ionized gas, with a typical mixture of hydrogen and helium (say, 35 percent H, 63 percent helium, and the rest nitrogen - typical of a middle-age main sequence star) and use your value of Pc and density to compute Tc, in K, again preserving dependence on M and R.

    4. What is the ratio of radiation pressure to gas pressure at the density and temperature you found? What does this say about the role of radiation pressure with increasing mass (assume radius is proportional to mass)?

    5. Clearly, assuming constant density is a rather poor approximation. Does this assumption lead to an overestimate or underestimate of the central pressure? Justify your answer. Estimate how far off your estimate is, quantitatively.

     

  3. Try Problem 10.15 of Carroll and Ostlie...