Download Ay123 Fall 2011 STELLAR STRUCTURE AND EVOLUTION Problem Set 5

Ay123
Fall 2011
STELLAR STRUCTURE AND EVOLUTION
Problem Set 5
Solutions due Thursday November 13th 2011
1. Use the Saha equation for hydrogen to plot the ionization fraction y as function of temperature T for
a density of ρ = 10−10 g cm−3 . Calculate the adiabatic gradient Γ1 = (d ln P/d ln ρ)ad as function of
temperature for the same value of ρ. Make sure the temperature range of your plot covers both ionized
and non-ionized states.
2. Assume the Sun is fully ionized and fully convective all the way to its surface.
(a) Show that the sound speed close to the surface is given by c2 = (γ − 1) g z where z = R − r << R
is the distance from the surface.
(b) For a chosen horizontal wavelength ky (see HKT p410) find the associated spherical harmonic l in
terms of R and, for a given frequency ω, the maximum distance zmax an acoustic wave can penetrate
into the star.
(c) For a standing wave, show that
Z
zmax
kr dr = n π
0
where n is the number of nodes. Hence find a relation between n, l and ω.
3. Consider the restoring force from buoyancy for an element displaced upward in a star. Show that
F = −N 2 ∆x = −
g ∆ρ
δρ
[
−
] dx,
ρ ∆x ∆x
where ∆ρ is the difference in atmospheric density from the initial to final (upward) position while δρ is
the same for gas inside the displaced gas, and N is the bouyancy or Brunt–Vaisala frequency.
[NB: In class, the Brunt-Vaisala frequency was given the symbol ωBV ]
a. Show that for adiabatic motions
N 2 = −g [
1 dT
γ − 1 1 dP
−
]
γ P dx
T dx
b. Show that N becomes 0 when the temperature gradient becomes steep enough to satisfy the condition
for convection. This confirms that gravity waves due to buoyancy oscillations cannot propagate in a
convective region of a star.
c. Estimate the Brunt–Vaisala frequency and corresponding period for the atmospheres of the Earth
and Mars. Indicate what assumptions you are making. Estimate the acoustic wave cutoff frequency
ωc , and corresponding period, for the Earth as well. What happens to a wave in the frequency interval
between ωc and N ?
4. Consider a thin spherical shell of radius r and thickness ∆ r in the envelope of a star. Assuming that
the mass density of an element of volume (∆ r)3 = ∆ m / ρ where ∆ m is the element mass, and that
the fractional change in temperature across the element is large, show that the shell luminosity is given
by
L∝
T4
κ ρ4/3