MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.282 May 23, 2002 Final Exam Name Last First (please print) 1. Work all problems. 2. Closed book exam; you may use two pages of notes. 3. Wherever possible, try to solve the problems using general analytic expressions. Plug in numbers only as a last step. Problem 1 2 3 4 5 Grade Grader 6 7 8 Total Constant c (speed of light) G (gravitation constant) k (Boltzmann's constant) (Planck's constant) Wiproton eV (electron Volt) Mo (solar mass) USEFUL CONSTANTS 3 x 10 cm/sec cgs units 7x10- dyne-cm²/g² erg/K 1.4 x 10-16 6.6 x 10-27 erg-sec 1.6 x 10-24 g 1.6 x 10-12 erg 2 x 1039 g mks units 3 x 108 m/sec 7 x 10-11 N-m³/kg² 1.4 x 10-20 J/K 6.6 x 10-34 J-sec 1.6 x 10-27 kg 1.6 x 10-10 J 2 x 10 kg 4 x 102% J/sec (solar luminosity) 4 x 1033 erg/sec Re (solar radius) 7 x 1010 em 7 x 10 m a (Stefan-Boltzmann cons) 6 x 105 erg/cm²-sec-K4 6 x 10 J/m²-sec-K4 A(Angstrom) 10- 10-10 m kım (kilometer) pe (parsec) 105 10 m 3x10 cm 3 x 1016 m kpe (kiloparsec) 3 x 1021 CITI 3 x 1019 m Mpc (megaparsec) 3 x 1024 cm 3 x 10 m year 3 x 107 sec day 86400 sec AU 1.5 x 1043 1' (arc minute) 1/3400 rad 1" (arc second) 1/200,000 rad 3 x 107 sec 86400 sec 1.5 x 10" 1/3400 rad 1/200,000 rad Problem 1 (13 points) a. A neutron star whose mass is 1.4 M_☉ is in orbit about a normal star. Matter is flowing from the normal star onto the collapsed star at a rate of 10⁷ 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. [L = \frac{GMm}{R}] [= 1.9 \times 10^{37} erg/s] 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 R_ns = 10 km.) [\sigma T^4 \cdot 4\pi R^2 = L \Rightarrow T = (\frac{L}{4\pi R^4 \sigma})^{1/4}] [kT_{eff} = 18.7 \times 10^6 K \approx 1 keV (X-ray band)] 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⁻¹, while its unseen companion star has an orbital velocity of 770 km s⁻¹. Find the masses of the unseen companion star and the neutron star (in units of M_☉). [a_x = \frac{v_P \cdot P}{2\pi} = \frac{770 \cdot 42 \cdot 60}{2\pi} = 309,000 km] [M_c = \frac{a_x^3}{GP^2} = 0.02 M_☉] Kepler's 3rd law: [\frac{M_T^2}{a^3} = \frac{4\pi^2}{G}] [M_T = 1.435 M_☉] [M_c + M_n = M_T] [M_c + M_n = M_c(1 + \frac{M_n}{M_c}) = 1.435 M_☉] [M_n = M_c(\frac{M_T}{M_c} - 1) = 1.435 M_☉ - 0.02 M_☉ = 1.415 M_☉] 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., d_M5/d_M3. 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? Solution: b. The horizontal branch for M3 is at ~ V = 15.65, that for M5 is at 15.25. Thus, M3 has a distance modulus that is ~ 0.4 magnitudes greater than that for M5. Each 5 magnitudes is a factor of 10 farther away. Therefore, d_M5/d_M3 = 10^(0.4/5) = 10^(0.08) = 1.2. c. On the main sequence, H → He in the core. Giant branch: H → He in a shell around the core of He. AGB: H → He and He → C in shells around the core. Horizontal branch: He → C in the core. d. V = -2.5 log(L/L₀) = 20 12.5 = -2.5 log(N*L_giant/L₀) = -2.5 log N - 2.5 log(L_giant/L₀) 12.5 = -2.5 log N + 20 log N = -7.5/-2.5 = 3 N = 1000

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