Download Ay123 Fall 2011 STELLAR STRUCTURE AND EVOLUTION Problem Set 3

Ay123
Fall 2011
STELLAR STRUCTURE AND EVOLUTION
Problem Set 3
Due Friday, October 28, 2011
1. (10 pts) The aim is to calculate the Rosseland mean opacity in the case of free-free absorption in pure hydrogen. The frequency-dependent opacity is given by the expression:
κν ρ = 1.32 × 1056
ρ2 gf f
(1 − e−hν/kT ) cm−1
ν 3 T 1/2
where gf f is a constant quantum mechanical correction factor called the Gaunt factor.
(i) First derive an expression for ∂Bν /∂T
(ii) Next introduce a dimensionless variable x = −hν/kT .
(iii) Derive an expression for ρκ1ν ∂Bν /∂T and plot the resulting function. Use the plot
to argue that the Rosseland mean opacity is largely determined by κν when the frequency ν
is a few times kT /h.
(iv) Thus show that the Rosseland mean opacity for free-free absorption obeys Kramers
law where κ ∝ ρT −3.5 .
2. (10 pts) The material in the envelope of a star is an ideal gas with γ = 4/3 and the star
is sufficiently centrally concentrated that the mass in the envelope is negligible compared
to that in the core, M. If the envelope is just marginally convectively unstable, show
that the temperature within the envelope varies with radius as
T =
1
GMµmH 1
( −
) + Ts
4k
r
rs
where rs and Rs refer to surface values.
3. (20 pts) Limb Darkening in the Sun
We are going to use the Solar limb darkening to study the properties of the atmosphere of
the Sun, including the variation of opacity with frequency for the Solar atmosphere. Recall
that for the Sun we can resolve the surface. Thus we can measure I(µ, τ = 0), not just
the flux (which is the integral of I(µ, τ = 0), over the surface). Recall that we derived an
approximate solution for T (τ ) for a plane parallel atmosphere, which should be used for this
problem. We also derived in class an appropriate mean optical depth to evaluate I(µ, τ = 0).
Assume the source function is linear in depth, Sν = Cν + Dν τ , where Cν and Dν are
constants for each ν.
a) Use the Solar limb darkening curves given in table 14.16 of the 4th edition of Allen’s
Astrophysical Quantities to find Cν /Dν for λ = 0.2, 0.5, 1.0, 2.0, and 5.0 microns. How
well do the observed limb-darkening curves resemble those of a grey body ? What is the
approximate wavelength of the Rosseland mean opacity for the Sun as judged by the limb
darkening curves ?
b) Assume LTE. The emergent specific intensity at the center of the Sun’s disk in the
continuum is (units are ergs/cm2 /sec/steradian/micron) is 0.014×1010 at 0.2 µm, 3.63×1010
at 0.5 µm, 1.21 × 1010 at 1.0 µm, 0.18 × 1010 at 2.0 µm, and 0.0057 × 1010 at 5.0 µm. Use
the emergent intensity as a function of wavelength at the center of the solar disc to find
temperatures at τν = 1 for each of the five wavelengths listed above.
c) Find Cν and Dν for the 5 wavelengths.
d) Assume τν,i = constant x τν,j for each pair (i,j) of wavelengths from the five above.
At the physical depth z such that τ1µ = 1, what is τ at each of the other 4 wavelengths ?
e) Draw the five points on the T − τ1µ curve.
f) Comment on the variation of the opacity as a function of wavelength in the solar
atmosphere for 0.2 ≤ λ ≤ 5.0 µm based on these values for limb-darkening and intensity at
the center of the Solar disk.