Note: Some results of exercises in Chapter 4, as indicated, are used in solving the following exercise.
A quasar (contraction of the name quasi-stellar object) is an astronomical object that pours out a prodigious amount of energy. Because they are so bright, quasars are the most distant visible objects, some of them as much as 14 x 109 light-years distant. A single bright quasar can give off energy at a rate greater than that of all the stars in our galaxy, though most quasars emit energy at a more modest rate than that. What is the source of this energy? We do not know. The energy emission rate can sometimes change significantly in a short time, implying that the emitting structure is small. (Otherwise the limiting speed of light would prevent different parts of the structure from "cooperating" to change the emission rate.) Evidence is accumulating that the energy comes from stars torn apart and spiraling into a black hole, material that is heated to extreme temperatures in the process and emits radiation copiously prior to disappearing across the horizon of the black hole. In essence, some fraction of the energy at infinity of the in-falling material is converted into light.
Most theories of quasar energy production assume that the black hole involved is rotating rapidly. The rotating black hole is the subject of Project F, The Spinning Black Hole. Here we analyze a simpler (and less realistic) model of energy production around a nonrotating black hole.
To show the order of magnitude of gravitational energy that is available, consider the following simplified encounter: A stone of mass m is in the stable orbit of smallest possible radius around a black hole. A second stone of equal mass m, initially at rest at a great distance, plunges radially toward the black hole and collides with the stone in the circular orbit. Assume that the entire kinetic energy of the pair, as observed by a local shell observer, is converted into light that travels outward. (Do not worry about conservation of momentum for the shell observer.)
A. What is the total kinetic energy of the two stones measured by the shell observer? Assume that all of this kinetic energy is converted into a light flash directed radially outward.
B. What is the total energy of the resulting light flash received by a remote observer?
C. What fraction of the total rest energy m + m = 2m has been turned into light at infinity? Compare this with an energy-conversion fraction of 0.1% or less for nuclear reactions on Earth.
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