Dr. I. Fernini                              Special Topics: Astrophysics (Phys 493)

Dec. 21, 2002                                                  Fall 2002 ‏

 

                                                            FINAL  EXAM

 

 

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 brightest-appearing star, Sirius, were three times its present distance, would it still make the list of the “25 Brightest Stars” (Appendix four, Table A4-2)?

 

What about the second brightest star, Canopus?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem 2 (20 points) :

 

(a)   Verify that about 6 × 10¹¹ kg of hydrogen is converted to helium in our Sun every second.

(b)   If a star is characterized by a mass M = 3 × 10³² kg and L = 2 × 10³² 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 10²⁶ kg and luminosity 4 × 10²⁰ 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) = I₀ + m I₁    (Eddington approximation)

            (b) I(m) = I₀ d(m - m₀)   where m₀ = 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 M²/R⁴. 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: