Dr. I. Fernini Special Topics:
Astrophysics (Phys 493)
Student Name:_________________ ID Number:____________________
Note:
You must
answer all the questions given below. Make sure to show all the steps of your
calculation. Credits won’t be given to any answer not backed by any
explanation. Care should be given to the units used in each problem.
Problem 1 (20
points):
If the brightestappearing
star, Sirius, were three times its present distance, would it still make the
list of the “25 Brightest Stars” (Appendix four, Table A42)?
What about the second
brightest star,
Problem 2 (20
points) :
(a) Verify that about 6 × 10^{11} kg of hydrogen is converted to helium in
our Sun every second.
(b) If
a star is characterized by a mass M = 3 × 10^{32}
kg and L = 2 × 10^{32}
W, how long can it shine at that luminosity if it is 100% hydrogen and converts
all of the hydrogen to helium.
(c) Do
a similar calculation for a star of mass 10^{26} kg and luminosity 4 × 10^{20} W.
Problem 3 (20
points):
Show
that the anisotropic factor f = K/J = 1/3 for each of the following radiation
fields:
(a) I(m) = I_{0} + m I_{1}
(Eddington approximation)
(b) I(m) = I_{0} d(m  m_{0}) where m_{0} = 1/Ö3
Problem 4 (20
points):
The visible spectral lines of
the Balmer series have n_{b} = 2 and n_{a} ≥ 3. The first hydrogen lines discovered were the Balmer
series and they are designed Hα
for n_{a} = 3, Hβ
for n_{a} = 4, Hg for n_{a} = 5, and so forth.
(a) Compute the wavelengths of the
Hα , Hβ , and Hg lines.
(b) If these lines were emitted by
a galaxy receding at 60,000 km/s, at which wavelengths will they be detected on
Earth? Will they be observed in the visible range of the spectrum?
Problem 5 (20
points):
Although detailed models of stellar structure require the
use of complex computer codes, single scalings can be obtained by making rough
approximations. For a variable x, we can substitute Δx/Δr for dx/dr in order to obtain a crude result. (This
method is a rough version of a numerical technique called ‘finite
differences.’)
(a)
Use the equation of hydrostatic equilibrium to show that the
central pressure scales as P_{c} µ
M^{2}/R^{4}. Substitute ΔP/Δr for dP/dr and take this
difference between r=0 and r=R/2, that is, ΔP/Δr ~ [ P(r=R/2)  P_{c}] / ( R/2 – 0).
You may assume that P(r=R/2) is
negligible compared with P_{c}. Also, substitute the mean
density of the star <ρ> for ρ .
(b)
Using the above
approximation, compute the central pressures in comparison with the Sun’s
central pressure for the following stars:
Spectral type 
Mass (/M_{O}) 
Radius (/R_{O}) 
O5 
40 
18 
B0 
16 
7 
A0 
3.3 
2.5 
F0 
1.7 
1.4 
G0 
1.1 
1.1 
K0 
0.8 
0.8 
M0 
0.4 
0.6 
