1 [3 p.] First things first. 4 [2p.] Mass to light II. 5 [3p.] Merger remnants

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1 Name and student number: [will be needed in the real exam] ASTC22 (GALACTIC AND EXTRAGALACTIC ASTROPHYSICS) PREPARATION SETFOR FINAL EXAM (#7) - PROBLEMS. SOLVED, SOME NOT. SOME FULLY Points in the square brackets give the relative weight with which the problems count toward the final score. This part of the exam is worth 30% of the total course score. Various physical constants are given at the end of this part. Please write legibly and explain what you are doing. Describe any obstacles you encounter - if you get stuck toward the end or with numerical evaluation of your result, but were going in the right direction, you ll get a sizable credit! 1 [3 p.] First things first What is the estimated tidal (Jacobi) radius of a small black hole of mass m = M found at a distance of one core radius r c = 1 kpc from the center of a spherical stellar system. Observations show radial velocity dispersion σ r = 120 km/s. 2 [2p.] Mass to light I Sove exercise 4.2 from the book. 3 [2p.] Mass to light III Solve exercise [2p.] Mass to light II Solve exercise [3p.] Merger remnants Solve exercise

2 6 [5p.] A visit to Dr. Hubble Sandy Faber and her colleague Robert Jackson happen to visit Edwin Hubble during his daily tea time. Hubble shows them a picture of the Coma cluster. ( In the center there is an elliptical galaxy NGC 4881, which interests Hubble. It is a 13th-magnitude (more precisely, m V = 12.7 mag) galaxy on the outskirts of the Coma Cluster, a great cluster of galaxies more than 5 times farther away than the Virgo Cluster. The radial velocity of NGC 4881, based on the Doppler displacement of lines in its spectrum, is about 7000 km/sec. He has heard that Faber and Jackson could predict luminosty of that galaxy, if he told them the dispersion of velocities of stars along the line of sight, σ. They confirm the rumor and tell Hubble about their empirical law. Hubble reveals that he had measured and found σ to be 300 km/s. He d like to know the galaxy s luminosity to try to find the value of his (Hubble s) constant, by first finiding the distance from distance modulus, or the difference of apparent and absolute magnitude of the galaxy. The astronomers quickly agree to do a calculation before finishing their tea. Scribbling on napkins, they finally obtain H 60 km/s/mpc. They think they did their calculation right, but to be absolutely sure (and to have a nice solution with important steps described in words!) they ask their ASTC22 students to repeat that calculation. Hint: If you ever need the absolute magnitude of the sun or other types of stars, look them up in Tables Chapter 1 summarizes the basic formulae for magnitudo brightness scale, if you need a refresher (Hubble s guests didn t, of course). 7 [5p.] Another visit to Dr. Hubble A week later, Brent Tully and J. Richard Fisher meet Edwin Hubble over tea. They also are interested in the Coma cluster, especially in one spiral galaxy to the right of the giant elliptical NGC ( It s a 16-th magnitude object in the visible, having solartype spectrum and colors at all wavelengths. It moves at a speed 7300 km/s away from us. They would very much like to know how far the galaxy is, so they can compute its luminosity in units of L. Their ultimate goal is to determine the maximum rotation speed V m ax of the galaxy, based on an empirical law they ve discovered and compare it with the results obtained with radio telescopes by their colleagues. They wonder what value of his constant Hubble would recommend. Hubble since the last week he believes that H 60 km/s/mpc (and we know why). The astronomers doodle furiously but, due to the last week s visit, run out of napkins before coming up with the expected rotational speed of the galaxy. Can you help them find V m ax? 2

3 8 [4 p.] Maximize superluminal illusion Solve problem 9.6 from the Sparke and Gallagher textbook in section [4 p.] A dissolving dwarf satellite galaxy A dwarf spheroidal galaxy has been found at the distance of only r = 25 kpc from the center of our Galaxy. The angular diameter of the galaxy is 1.14 degrees. The dwarf galaxy contains an estimated mass m M. Calculate the Jacobi tidal radius (r J ) of that systems. Is the dwarf galaxy stable or does it lose stars because of tidal forces? (Assume rotation curve V c = 200 km/s = const. for our Galaxy) The dynamical friction is bringing the dwarf closer toward the center of the Galaxy, toward an eventual tidal destruction. At which distance d will it start losing stars very quickly (catastrophically)? 10 [6 p.] Size of the invisible halo Suppose that our Galaxy has a flat rotation curve with V c = 200 km/s = const only out to some radius r, outside of which the rotation is Keplerian. We are located at about r 0 = 8 kpc distance from the center, and observe in our neighborhood no stars bound to our Galaxy and having galactocentric speed larger than Vc max = 700 km/s. Estimate the radial extent (i.e., radius r ) of the singular isothermal halo of dark matter (density law ρ(r) = ρ 0 (r 0 /r) 2 ), where ρ 0 is a constant, and the above data. Compute also the total mass inside radii r = r 0 and r = r. system. Hint: use the expressions for the escape speed and for the potential of a spherically symmetric 11 [3 p.] Resonances in NGC1604 NGC1604 is a spiral galaxy, where an m = 2 pattern of arms is visible between about 4 kpc and 10.2 kpc from the center. Morphology of gas rings in the galaxy suggests that at 3 kpc there is an Inner Lindblad Resonance, and that the 10.2 radius is very close to the Corotation with the pattern. Astronomers suspect that the rotation curve is flat throughout the observed region. Does the dynamical theory support such a presumption? If yes, where would the Outer Lindblad Resonance be located (so that observers might try to obtain direct confirmation of the arrangement of resonances)? 3

4 12 [5 p.] Cluster M4 M4 is a globular cluster in our Galaxy. Measurement of radial velocities of 29 stars in the cluster gave the following root-mean square (dispersion) of radial velocities relative to the mean velocity of the whole cluster: < v 2 r > = 70.7 ± 0.8 km/s (an actual measurement published in 1991 by Rastorguev and Samus). The light and thus the density distribution (assuming constant M/L ratio) are known in this cluster. Using fig. 3.7 from the 2007 edition of Sparke and Gallagher placed in the powerpoint lecture 9, as well as virial theorem or an appropriate theoretical potential-density pair you choose (with justification!), please estimate the mass of the core of the cluster M4 (an order of magnitude estimate is ok, but any additonal precision would be appreciated. However, don t strive for precision!) Estimate the mean distance between the stars in the core of M4. Estimate the crossing (dynamical) time scale t cross, and the relaxation time t relax in M4. Assuming that it takes 20 relaxation time for a cluster to approach the core collapse stage, is M4 a plausible candidate for a star cluster with a massive black hole inside? 13 [5 p.] NGC 7331 Please solve the problem 5.3 from the textbook, noticing that errata replaces number 3.3 with 5 in the text. 14 [3 points] Does a Supermassive Black Hole chew stars? We re interested in predicting whether or not a SMBH can swallow a star whole. Tidal forces around a black hole are very large but depend on the mass M of a black hole. We d like to know for which masses M sun-like stars are torn apart by tidal forces before reaching a distance equal to one Schwarzschild radius from the center. If a star bursts under the tidal forces of a SMBH, but misses its surface and a gas cloud is flung outwards, what is the highest speed of gas you except to detect spectroscopically? HINT: Assume that Newtonian dynamics is valid down to one Schwarzschild radius from the black hole. Use the formula for the Roche lobe size in a binary system consisting of a SMBH and sun. 4

5 15 The mass of the black hole in the Milky Way s center Determine the mass of a black hole residing in the object called Sagittarius A* (in Galactic Center) from the following observation: star SO-20 was observed from 1995 to 2004 and traversed approvimately 1/4 of its orbit around the Galactic Center. Please notice the angular scale indicated in the figure, make reasonable assumptions about the true orientation (inclination) of the orbit, and assume that Sgr A* is at a distance 8 kpc from us. 16 Some possibly useful constants If you don t have a calculator, you must state that in your solution and provide your calculation rounded off to 2 significant figures (numerical error less than 10% will not lower your score.) Otherwise, at least three significant figures should be carried. c = m/s, = cm/s (speed of light) G = m 3 kg 1 s 2 = cm 3 g 1 s 2 (gravitational const) G = in calculations where units of speed are km/s, mass 1 M, and distance 1 pc. k = J/K = erg/k (Boltzmann) 5

6 m H = kg = g (hydrogen mass) M = kg = g (solar mass) R = m = cm (solar radius) L = J/s = erg/s (solar luminosity) 1 AU = m = cm 1 pc = AU = cm = m 1 yr = s 1 km/s 1 pc/myr distance to the center of the Galaxy (old IAU recomm.) = 8.5 kpc, newer value 8 kpc (use that one!) 6