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Assignment 3

Semester 1, 2007

1. The total pressure P in a non-degenerate, ionized region of a not-too-luminous star can be considered as that of a simple perfect gas alone (i.e. radiation pressure can be ignored). Let the mean molecular weight of particles in this region be µ so that P = k T /µm H . Similarly, the energy density per unit mass is U = 3 k T /µmH . Using the 1st Law of Thermodynamics, 2 show that in this region of the star, the adiabatic exponents 1 , 2 and 3 are all equal to 5 , 3 where the exponents are defined as 1 = ( d ln P )ad , d ln
ad

d ln T 2 - 1 =( ) 2 d ln P and 3 - 1 = (

d ln T )ad . d ln

2. A low mass asymptotic giant branch star consists of a compact degenerate core, which is essentially a white dwarf around which hydrogen and helium burn in shells, and an extended envelope. The compact core has a mass of about 0.6 M and the envelope mass is about one or two tenths of a solar mass. The radius of the nuclear burning shells is 0.1 R while the radius of the extended envelope is 200 R . Such a star may be approximated as a central point source of mass M and luminosity L and an envelope whose mass is so small that it does not contribute to the gravitational acceleration felt by mass elements in the envelope. The envelope can be assumed to consist of a perfect gas and to have a radius r = R outside which the pressure is zero. (a) Let x = r/R and assume the density in the envelope is given by = 0 x-2 . Show that the pressure in the envelope of such a star is given by P= GM 0 1 ( - 1). 3R x3

(b) Use this expression for the pressure to show that the adiabatic wave equation for the star can be written as x d 3 R3 2 3 d [(x - x4 ) ] + [ x - (31 - 4)] = 0. dx dx 1 GM

(c) Show that near x = 0, the solution to this equation is of the form = Ax , where A is a constant and 2 = 3 (31 - 4). 1


(d) For 0 x 1, use a power series of the form =
n=0

an x

+n

to derive a formula for the eigenvalues 2 . Derive and sketch the eigenfunctions for the fundamental mode and first overtone mode, normalized to 1.0 at the surface. Assume 1 = for your plot.

5 3


3. Write some brief notes on the following three classes of variable stars: RR Lyraes; Classical Cepheids; and Long-period Variables (Miras and Semi-Regular Variables). For each class, your notes could mention: stellar mass range; luminosities and/or period luminosity laws; mode(s) of pulsation; where found (old populations, young populations, star clusters,...); evolutionary state; astronomical "uses"; and connection with, or revelations about, mass loss.