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The Question

(Submitted May 30, 1996)

We understand that in recent years discovery of intense 511-keV annihilation lines from black holes indicate that steady-state thermal annihilation pair plasmas may exist and that there is a fundamental limit to the temperature above which pair creation can no longer be balanced by annihilation and that this limit is referred to as the BKZS limit. Can you let us know the derivation of the letters B, K, Z & S in the naming of this limit? Could you give us a reference to some place where we can get more information on this subject?

The Answer

Thank you very much for your interesting, very high level, question. We could not answer your question ourselves, but we have managed to find an expert, Dr. Charles Dermer of Naval Research Laboratory, who could. His answer is given here.

The questioner is quite right about what he has heard. There is a fundamental limiting temperature above which steady thermal plasmas cannot exist. The limit is named after the authors of the paper which points out this limit, and the complete reference is:
Bisnovatyi-Kogan, G. S., Zel'dovich, Ya. B., and Sunyaev, R. A., 1971, Soviet Astronomy, AJ, vol. 15, p. 17.

The question outlines the essential reason for this limit: at sufficiently high temperatures, there is a competition between two-body processes. On the one hand, collisions of electrons with other particles (such as electrons, positrons, or protons), makes electron-positron pairs through the process:

  1. particle 1 + particle 2 --> particle 1 + particle 2 + electron + positron. The electrons and positrons are made at the expense of the kinetic energy of particles 1 and 2.
  2. On the other hand, pair production is balanced by the pair annihilation process: electron + positron --> two gamma-ray photons.
At sufficiently high temperatures, the addition of electron-positron pairs through process (1) makes additional electron-positron pairs through process (1) and so on, and this cannot be balanced by the pair annihilation rate. The result is unlimited production of pairs if one requires the system to remain at a fixed temperature. In reality, of course, energy cannot be continuously injected and the system cools, so that the runaway pair production is quenched.

BKSZ calculated a maximum temperature of about 20 MeV; subsequent research revised that maximum to about 12 MeV (see, for example, A. A. Zdziarski, 1982, Astronomy and Astrophysics Letters, vol. 110, p. L7). This is for a completely transparent medium, and the maximum temperature is even less if the system is opaque (i.e., has finite optical depth).

The discovery of time-variable sources of 0.511 MeV annihilation radiation was the impetus for this work, although the reality of black hole sources of annihilation radiation is now in dispute (although diffuse annihilation radiation in the galaxy and on the Sun from radioactive beta-emitters is beyond question). A Scientific American article by Gehrels, et al. (December 1993, page 68) discusses cosmic annihilation radiation, though not in great detail.

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