ASTR 304 - 2003W [Week 9]

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Week 9 - Stellar Black Holes [8] [10]

Summary

Accretion is ubiquitous in the universe and accounts for much if not most of the radiation produced. Shakura and Sunyaev in 1973 presented "the standard model" for accretion disks. Their results have been applied to understand quasars and protostars and much in between.

Reading List

  • ``Black holes in binary systems. Observational appearance.''
    [ ADS, PDF ]
    REF: Shakura, N. I., Sunyaev, R. A. 1973, A & A, 24, 337-355 . Top

  • ``New Evidence for Black Hole Event Horizons from Chandra''
    [ ADS, PDF ]
    REF: Garcia, M. R., McClintock, J. E., Narayan, R., Callanan, P., Barret, D., Murray, S. S. 2001, Astrophys. J. Lett., 553, L47-L50 . Top

Problem Set

Problem 1 - A Simplified Accretion Disk Top

This is a simplified model for an accretion disk. It is simpler than the model outlined in the Shakura & Sunyaev paper but it will give the right order of magnitude for things. We are also using Newtonian gravity.
  1. Let's divide the accretion disk into a series of rings each of mass dm. What is the total energy of a ring at a distance r from the central black hole of mass M?
  2. Let's say that the ring shrinks by a distance dr. What is the change in the energy of the ring (dE/dr) ?
  3. As the ring shrinks mass is moving toward the black hole. Divide both sides the answer to (2) by dt to get an equation for the energy loss rate per radial interval.
  4. What is the energy loss rate per unit area?
  5. Let's assume that this energy is radiated at the radius where it is liberated. Using the blackbody formula what is the temperature of the surface of the disk?
  6. Let's assume that the disk extends from an outer radius rA to an inner radius r0. What is the total luminosity of the disk if the accretion rate is dm/dt? What and where is the peak temperature of the disk? What and where is the minimum temperature of the disk?
  7. Sketch the spectrum from the accretion disk on a log-log plot. You can use temperature units for the energy axis (i.e. kTmax and kTmin). To do this you will have to think about the peak flux from a blackbody at a particular temperature and the size of the disk that radiates at Tmax and Tmin.
  8. The accretion rate is determined by the evolution of the orbit of the black hole with its companion, so it doesn't know about the Eddington limit of the black hole. What do you suppose happens if the rate that matter falls onto the disk exceeds the Eddington limit?
  9. What major bit of physics has been left out of this analysis?

Problem 2 - Thinking about Instruments Top

You will probably have to surf the net a bit or use things you have learned from other courses to work these out, but the equations will be rather simple once you have them.
  1. The black hole in the center of our Galaxy has a mass of 106 . Let us assume that it is a maximally rotating (a=M) Kerr black hole. How big is its horizon? How big is its ergosphere?
  2. What angle does the horizon of the central black hole subtend in the sky?
  3. I would like to build a telescope that can resolve the central black hole. What is the angular resolution of a telescope as a function of the wavelength of the light and the diameter of telescope. You can look up the formula, use dimensional analysis or the Heisenberg uncertainty principle.
  4. What is the diameter of the telescope if you use 2 GHz radio waves?
  5. What is the diameter of the telescope if you use 1 keV X-ray photons? Scaling from Chandra, what is the focal length of the telescope?
Last modified: Tuesday, 06 April 2004 07:28:16