Astrophysics Assignment #7, Spring 2012 “Something unknown is doing we don’t know what — that is what our theory amounts to.” ∼Sir Arthur Stanley Eddington 1: Hubble and the Distance to the Triangulum Nebula (M33) Hubble documented his measurement of Cepheid variables in M33 in 1926 (E. Hubble, ApJ 63, 236 [1926]). Figure 1 of that paper, showing the light curves of apparent magnitude m vs. period P for 4 different Cepheids, is shown below. Recall that for a period Pd measured in days, the period-luminosity relationship is: Mv = −2.81 log Pd − 1.43. a ◮ For each of the 4 Cepheids, use the graph to measure the period P and the peak apparent magnitude m. b ◮ For each of the 4 Cepheids, use the period-luminosity relationship to compute the absolute magnitude M. c ◮ Use the distance modulus to compute the distance to the Triangulum Galaxy for each of the 4 Cepheids. Take the average of the 4 results as your estimate of the distance to the galaxy. 2: Tidal Disruption by a Black Hole Consider a star of radius R⋆ and mass M⋆, in an orbit around a galactic black hole of mass M•. The star makes a distance of closest approach to the black hole when it is a distance rp from the center of the black hole. a ◮ Calculate the tidal force Ftidal across the diameter of the star, where Ftidal = Fnear −Ffar. For this exercise, imagine the star as the total mass divided in half and concentrated in small mass points m located on the side nearest and farthest from the black hole respectively. Then Fnear and Ffar are the gravitational force of the black hole on a small parcels of the star m. b ◮ Set your expression for the tidal force equal to the the gravitational attraction of the two parcels m; this is the case where the gravitational pull of the black hole is strong enough to pull the star apart (tidally disrupt the star). Solve for rp, which is your estimate of rtidal, the tidal disruption radius. 6 c ◮ Assuming the Milky Way black hole has a mass of M• =4 × 10 M⊙, how close could the Sun get before it was ripped apart? Astrophysics — Assignment 07 1 3: Escaping a Globular Cluster 5 The Great Globular Cluster in Hercules (M13) has a total mass of 6 × 10 M⊙, and a radius of 84 lyr. a ◮ Compute the escape velocity from the edge of the globular cluster. b ◮ What is the ratio of the escape velocity from M13 to the escape velocity from the surface of the Earth? Comment on whether this result is surprising to you or not and why. 4: Galactic Rotation Curves Vera Rubin and her collaborators documented the rotation curves of many spiral galaxies. The rotation curves of 7 different galaxies is shown below (Figure 3 from V. Rubin et al., ApJ 225, L107 [1978].). Let’s focus on the curve for NGC 3145, which is an SBc galaxy (just like the Milky Way) in Hydra, 167 million lyr away. a ◮ For at least 15-20 points along the curve for NGC 3145, measure the speed v and the radius r. b ◮ Enter your data into a spreadsheet. Create a new column for the mass as a function of radius, M(r), calculated from your data. Create a fourth column for the escape speed as a function of radius, vesc(r), calculated from your data. Create graphs of your results. Print out your graphs and your spreadsheet and attach it to this assignment. Astrophysics — Assignment 07 2 5: Accreting Black Holes? [∼BOB 24.31] −3 If the accretion rate at the center of the Milky Way is 10 M⊙/yr, and if it has remained constant over the past 5 billion years, how much mass has fallen into the center over that period of time? Compare your answer with 6 the estimated mass of the black hole at Sgr A*, M• =4 × 10 M⊙. 6: Massing the Black Hole from Stellar Orbits [∼BOB 24.32] a ◮ Compute the lowest possible density of Sgr A* based on the data obtained from the orbit of S2 (P = 15.2 yr, rp = 120 AU, e =0.87). Assume a spherically symmetric mass distribution. 6 b ◮ Assuming a mass of 4 × 10 M⊙ and a radius of 1 AU (roughly the current limit of resolution at the center 3 3 3 of the Milky Way), estimate the density of Sgr A*. Express your answer in kg/m , M⊙/AU , and M⊙/pc . Astrophysics — Assignment 07 3.