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• Problem Set
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Week 10 - Supermassive Black Holes

Problem Set

Problem 1 - Accretion-Disk Efficiency

1. Using Newtonian gravity, how much energy is released per unit mass as material spirals to the inner edge of the disk?
2. Now using general relativity, redo the calculation. Assume that the central object is a non-rotating black hole. The energy released per unit mass is given by 1-u_t where u^α is the four-velocity of material in the disk and you are using the Schwarzschild metric.
3. In general relativity, an accretion disk can only extend down to R=6M around a non-rotating black hole. What is the efficiency of accretion onto such a black hole?

Answer for Problem 1

1. In the Newtonian limit we have

   E m

   =

   G M 2 rA

2. We use Kepler's third law Ω² r³ = M to get the four velocity,
   u^t = (1 - 3M/r)^-1/2
   u^φ = Ω u^t
   u_t = (1-2M/r) (1 - 3M/r)^-1/2
   
3. Substituting r=6 M yields,

   1-ut=1-

   2√ 2 3

   c2 = 0.057c2

The Newtonian result is c²/12 ≈ 0.083 c².

Problem 2 - Holes v. Stars

It turns out that the masses of black holes in the centers of galaxies is well correlated with the mass of the bulge of the galaxy (if it is a spiral galaxy) or the entire galaxy if it is an elliptical: M_BH ≈ 0.016 M_bulge.

1. Let's take a bulge of 10⁸ . If the black hole was built up by accretion over the age of the universe, what would its average luminosity be? Let's assume that it is a Schwarzschild hole.
2. The mass-to-light ratio of the bulges of galaxies is given by

   Mb Lb

   = 0.0776

   / | \

   Lb

   \0.18 | /

   What is the luminosity of the stars in bulge?

Answer for Problem 2

1. The mass of the black hole would be 1.6 x 10⁶, so the total energy released in building it up would be 1.6 x 10⁵⁹ ergs. Let's take the age of the universe to be 13.7 billion years or 4.3 x 10¹⁷ seconds, yielding a mean luminosity of 3.8 x 10⁴¹ ergs/s or 9.5 x 10⁷ .
2. Rearranging the mass-to-light ratio equation gives.

   Lb

   = 8.73

   / | \

   Mb

   \0.847 | /

   or 5.2 x 10⁷ .

Problem 3 - Our Very Own Supermassive Black Hole

Andrea Ghez's group at UCLA constructed this beautiful movie of the centralmost arcsecond of our Galaxy. The edge of the box measures on arcsecond on the sky.

1. Use the movie to estimate the mass of the black hole at the center of our Galaxy.
2. How many Schwarzschild radii does the closest star approach the black hole?
3. How big would the black hole look on the sky to the hapless inhabitants on a planet orbiting this star? Would it be as big as the moon, Jupiter, Mars?
4. You have probably assumed something about the orbit of one of the stars. What did you assume? How does the mass of the black hole change if you vary this assumption? What could Andrea's team do to tighten the estimate of the black hole mass?

Answer for Problem 3

1. The image is 400 pixels and the orbit of SO-2 has a semimajor axis of 40 pixels, so its semi-major axis is 0.1 arcseconds. If we remember that one AU subtends one arcsecond at one parsec and take the distance to the Galactic center to be 8.5 kpc, we have a=850 AU. One orbit lasts from 1995.50 to 2010.40 or 14.9 years. Using Kepler's third law we get M=850³ 14.9^-2 =2.7 x 10⁶
2. The star SO-16 really gets closer the the black hole. I would guess about 4 pixels away, so 0.01 arcseconds or 85 AU. The Schwarzschild radius of the black hole is 8 x 10⁶ km or 0.05 AU, so the closest approach is about 1700 Schwarzschild radii.
3. The black hole will subtend 2/1700 radians or about 4 arcminutes. Jupiter's maximum size is a bit less than 1 arcminute, 41 arcseconds. Mars tops out at 25". The moon is about 30'. Venus at its largest is around 57", so the black hole would be a rather impressive sight in the sky a bit larger than the planets. I used this webpage at the U.S. Naval Observatory to figure this out.
4. We assumed that the orbits were in the plane of the sky. If they are inclined relative to the sky, the semimajor axis would increase as 1/(cos i), so the mass estimate would increase as well as 1/(cos³ i). This would mean that to the enperiled inhabitants of a planet orbiting SO-16 the horizon would appear even larger.

   Astrometry on its own can be used to determine the inclination of the orbit, so if we combine this information with radial velocity measurements we would not only get a better handle on the mass of the black hole but also on the distance to the Galactic center.