Astr 8000 – Problem Set #3 – Due March 26, 2009 1. The equation ...

Astr 8000 – Problem Set #3 – Due March 26, 2009
1. The equation of hydrostatic equilibrium is
dp
dz = −ρg
where the pressure derivative with respect to height z in a stellar atmosphere depends
on density ρ and gravitational acceleration g. In hot stars, the pressure is the sum of
gas pressure p g and radiation pressure p R which is given by
∫
p R = (4π/c)
K ν dν
where K ν is the second moment of the intensity field. The depth variation of K ν can
be obtained from the first moment of the transfer equation,
dK ν
dz = −χ νH ν
where H ν is the Eddington flux (first moment of the intensity field).
(a) Show that by changing the height variable to column mass dm = −ρ dz, the
equation of hydrostatic equilibrium can be written as
or
∫
dp g /dm = g − (4π/c) (χ ν /ρ)H ν dν
dp g /dm = g(1 − Γ)
where Γ is the ratio of radiative to gravitational acceleration.
(b) Suppose hot stars contain only fully ionized H and He (abundance Y). If electron
scattering alone is taken to provide a lower limit on the opacity, show that
χ ν /ρ = n e σ e /ρ = σ e (1 + 2Y )/(m H (1 + 4Y ))
where σ e is the cross section for electron scattering and m H is the mass of hydrogen.
(c) Show that
Γ = σ e(1 + 2Y ) L
m H (1 + 4Y ) 4πcGM
where L is the luminosity, G is the gravitational constant, and M is the mass.
(d) Use the following data to express Γ in terms of L/L ⊙ amd M/M ⊙ :

σ e = 6.65 × 10 −25 cm 2 electron −1 , Y = 0.1, m H = 1.673 × 10 −24 g, L ⊙ = 3.85 × 10 33
erg s −1 , M ⊙ = 1.989 × 10 33 g, and G = 6.6726 × 10 −8 cm 3 g −1 s −2 .
(e) Howarth & Prinja (1989, ApJS, 69, 527) give a mass and luminosity calibration
for O stars that can be fit by
L/L ⊙ = 317(M/M ⊙ ) 1.82 .
Use this relationship with your expression for Γ to find the maximum possible
stellar mass (i.e., where outward radiative acceleration equals inward gravitational
acceleration). [40]
2. Use the Saha ionization equation to estimate the ratio of the numbers of negative
hydrogen ions to neutral hydrogen atoms in the solar atmosphere. Use the form of
the Saha equation given by
N j /N j+1 = n e
U j (T)
U j+1 (T) C I T −3/2 e χ I/(kT)
where C I = 2.07 × 10 −16 in c.g.s. units. Assume that the partition functions for
neutral hydrogen and the negative hydrogen ion equal 2 and 1, respectively. Further
assume an electron density of 5 × 10 13 cm −3 and a temperature of 6300 ◦ K (typical
values near optical depth unity). [30]
3. Calculate the discrete ordinate solution to the grey atmosphere
I i (τ) = 3 4 F(τ + Q + µ i +
n−1 ∑
α=1
S = J(τ) = 3 n−1
4 F(τ + Q + ∑
α=1
L α e −kατ
1 + µ i k α
)
L α e −kατ )
using the single Gauss quadrature results for n = 4 in Table 10.1 from Collins (1989,
Fundamentals of Stellar Astrophysics [New York: Freeman]; see next page). The
quadrature points are
µ = [.9602899, .7966665, .5255324, .1834346]
(Chandrasekhar 1960, Radiative Transfer [New York: Dover]). Make a plot of the
limb darkening law for the surface intensity (I(µ) at τ = 0) for both the Eddington
approximation and the discrete ordinate method. Determine the degree of radiation
anisotropy
(I(µ = .9602899) − I(µ = .1834346))/I(µ = .9602899)
fro τ = 0, 1, 10, and 100. Plot the temperature versus depth relation, T/T eff versus
log τ, between τ = 0.01 and 100 for both the Eddington approximation result and the
discrete ordinate solution. [30]