Intro to Astronomy - 2002 Final Exam Solutions

Problem 1 (13 points) a. A neutron star whose mass is 1.4 Mo is in orbit about a normal star. Matter is Howing from the normal star onto the collapsed star at a rate of 10' grams per second. i. Assume that all of the gravitational potential energy of the infalling matter is converted to radiation. Compute the bolometric luminosity of the neutron star. ii. Suppose that the surface of the neutron star radiates this energy as a blackbody of a single temperature T. Compute T. Also compute the quantity kT' (in eV) and comment on the type of electromagnetic radiation you think is being emitted. (Take the radius of the neutron star to be Rys = 10 km.) b. A new X-ray pulsar was discovered by MIT scientists last week. It has an orbital period of 42 minutes. The neutron star moves in a circular orbit about the center of mass of the binary with a velocity of 11 km s*', while its unseen companion star has an orbital velocity of 770 km 8~'. Find the masses of the unseen companion star and the neutron star (in units of Mo). Problem 2 (12 points) The figures below show H-R diagrams for two globular clusters: M3 and M5. (V is the apparent visual magnitude.) 8-VI O Identify the various stellar evolutionary phases seen on the H R diagram for M3. You may write your answers directly on the plot. b. Find the relative distances to the two clusters, i.e., dus/dus- c. What are the predominant nuclear reactions powering the different evolutionary phases, and where are they taking place within the star (e.g., in a shell or in the core)? d. How many main-sequence stars with V = 20 equal the brightness of a single giant with V = 12.5? Problem 3 (Short Answer Questions - 13 points) a. Use a dimensional analysis of the equation of hydrostatic equilibrium to derive a radius mass relation, M(R), for white dwarfs. Assume that the equation of state for degenerate matter is P = Kp/3, where P is the pressure, p is the mass density, and K is a constant. b. What is 21-cm radiation, and what can be learned by studying it? c. What are the basic ingredients required to compute the "Oort limit" on the local mass density in the galactic plane? (Not to be confused with the Oort A and B coefficients for studying galactic rotation.) Problem 4 (Short Answer Questions - 12 points) a. A globular cluster contains 108 stars. Assume they all have the same mass, m and the same speed, v. Use the virial theorem to estimate a characteristic size of the cluster. b. Explain what the cosmic microwave background (CMB) is, and how it originates. c. Name three effects that dust has on starlight propagating through it. Problem 5 (13 points) a. A reasonable approximation to the My - P relation for Cepheid variables is: (Mv) = -2.3 10510(P/days) - 1.7 A Cepheid variable with mean apparent magnitude (m) = 19 is found in the galaxy M31 with a pulsation period of 30 days. Find the distance to M31 in terms of a distance modulus and in parsecs. to trime in disnes of dient to dad ander can are yed ceil in atre moe including Cepheids). Problem 6 (12 points) A spiral galaxy is observed to have a rotation curve that can be well fitted by the following analytic expression: +(R) = 200 (1 -e-R/Ro) km/sec, where e-R/Ro is an exponential function of R, the radial distance from the center of the galaxy, and R, is a constant whose value is 4 kpc. (Assume that the galaxy is viewed edge on so that the maximum Doppler velocities are observed.) a. Estimate how much mass there is interior to a distance of 16 kpc from the center. (Express your answer in units of Mo-) b. Show that for smali distances from the galaxy center (R < 4 kpc) the angular frequency, S, is approximately constant. [Hint: Use a Taylor's series expansion for u(R).] (If you are not able to do this part analytically, you may demonstrate the answer numerically with your calculator.) c. Find the rotation period (2=/2) of stars near the center, in units of years. d. For large distances, R, the rotation curve approaches a nearly constant value. Assume that the gravitating matter which produces this rotation curve is distributed spherically symmetrically about the center with a mass density that falls off with distance as p(R) a R-°, for R > 4 kpc. Find the value of a that leads to a flat rotation curve. You should consider only large distances in this part of the problem. Problem 7 (12 points) The plots below show three quasar spectra. The most prominent line in each spectrum is due to Ha with a rest wavelength of 6562 A. For the purpose of this problem, assume a value for the Hubble constant, Ho, of 70 km s"' Mpe"'. a. For the most distant of these quasars, find its approximate red shift, z = (1 - lo)/Ao- b. What is the approximate recession velocity of this quasar? c. Estimate the distance to the quasar. d. The energy flux from this quasar is 3 x 10-13 ergs cm-* sec-' (3 × 10-16 Watts m~*), a substantial fraction of which is emitted in the Ha line. Find the approximate luminosity of the quasar. e. A model for the central object in this quasar involves a supermassive black hole of 107 Mo- Is this consistent or not with the concept of an "Eddington limit"? Explain. Problem 8 (13 points) The graph below shows a plot of the recessional velocity of galaxies as a function of their distance. a. Use the diagram to estimate the Hubble constant, Ho (to better than a factor of 2). b. Use the answer to part (a) to estimate the age of the universe. c. For the case of a "flat", Do = Pm = 1 universe, use Newtonian cosmology to derive a more exact relation between the Hubble constant and the current age of the universe.