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Electron degeneracy pressure in a white dwarf star, of uniform density p, in the nonrela- tivistic case is given by Pwd = ħ² 3memp 25/305/3 where symbols have their usual meanings. Using the result that the central pressure in a star, of radius R and uniform density, under gravitational attraction is given by Pc = Gp² R², derive an expression for the radius Rwd of a white dwarf in terms of its mass M, in the case of nonrelativistic electron degeneracy. Using your result, briefly discuss the limitations of your expression for the radius, in the context of white dwarfs of increasing mass.

Electron degeneracy pressure in a white dwarf star, of uniform density p, in the nonrela- tivistic case is given by Pwd = ħ² 3memp 25/305/3 where symbols have their usual meanings. Using the result that the central pressure in a star, of radius R and uniform density, under gravitational attraction is given by Pc = Gp² R², derive an expression for the radius Rwd of a white dwarf in terms of its mass M, in the case of nonrelativistic electron degeneracy. Using your result, briefly discuss the limitations of your expression for the radius, in the context of white dwarfs of increasing mass.

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a) b) Electron degeneracy pressure in a white dwarf star, of uniform density p, in the nonrela- tivistic case is given by Pwd ħ² 3memp 25/305/3 where symbols have their usual meanings. Using the result that the central pressure in a star, of radius R and uniform density, under gravitational attraction is given by Pc = Gp² R², derive an expression for the radius Rwd of a white dwarf in terms of its mass M, in the case of nonrelativistic electron degeneracy. Using your result, briefly discuss the limitations of your expression for the radius, in the context of white dwarfs of increasing mass. Consider a white dwarf, mass M, radius Rwd and temperature T, consisting entirely of helium nuclei and electrons. Show that the internal thermal energy of the ions alone is given by 3 M Eth= -kT, 8 mp where mp is the proton mass. White dwarfs initially have a very high temperature when they form, and then cool by radi- ation. Derive a differential equation for the rate of change of the temperature of the white dwarf by assuming that the luminosity is entirely accounted for by the change in internal energy. By solving this differential equation derive an estimate of the cooling time of a white dwarf Tcool to a temperature T, and show that Tcool 1 Mk 1 1 32 m₂ TRT wd You may assume that the initial temperature T; of the white dwarf is sufficiently large that T7³ >> T-³. By using your expression found in part (a), find whether the cooling time increases or decreases with the mass of the white dwarf. Briefly describe and explain the evolutionary track in the H-R diagram of a white dwarf as it cools.
Transcribed Image Text:a) b) Electron degeneracy pressure in a white dwarf star, of uniform density p, in the nonrela- tivistic case is given by Pwd ħ² 3memp 25/305/3 where symbols have their usual meanings. Using the result that the central pressure in a star, of radius R and uniform density, under gravitational attraction is given by Pc = Gp² R², derive an expression for the radius Rwd of a white dwarf in terms of its mass M, in the case of nonrelativistic electron degeneracy. Using your result, briefly discuss the limitations of your expression for the radius, in the context of white dwarfs of increasing mass. Consider a white dwarf, mass M, radius Rwd and temperature T, consisting entirely of helium nuclei and electrons. Show that the internal thermal energy of the ions alone is given by 3 M Eth= -kT, 8 mp where mp is the proton mass. White dwarfs initially have a very high temperature when they form, and then cool by radi- ation. Derive a differential equation for the rate of change of the temperature of the white dwarf by assuming that the luminosity is entirely accounted for by the change in internal energy. By solving this differential equation derive an estimate of the cooling time of a white dwarf Tcool to a temperature T, and show that Tcool 1 Mk 1 1 32 m₂ TRT wd You may assume that the initial temperature T; of the white dwarf is sufficiently large that T7³ >> T-³. By using your expression found in part (a), find whether the cooling time increases or decreases with the mass of the white dwarf. Briefly describe and explain the evolutionary track in the H-R diagram of a white dwarf as it cools.
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