Question

Simple Model of White Dwarf Consider a simple model of a white dwarf by assuming that the mass density is uniform. Then its gravitational potential energy is EG = ?3GM² / 5R. Further assume that the nuclei are carbon and the electrons are degenerate and ultra-relativistic. (a) Derive the electron density in terms of M, mH and R, where M and R are the mass and radius of the white dwarf respectively. (b) Derive the Fermi energy ?F in terms of M, mH and other constants. (c) The density of states of ultra-relativistic electrons is g(?) ? ?². Derive the average electronic energy in terms of ?F. (d) By considering the R dependence of the total energy, show that when M exceeds a threshold, the radius R converges to 0. Use this to estimate the Chandrasekhar mass (in MSun). ? = 1.055 × 10?³? Js, c = 3 × 10? ms?¹, G = 6.67 × 10?¹¹ Nm²kg?², mH = 1.66 × 10?²? kg, MSun = 1.99 × 10³? kg ? = pc p = ?k kF³ = 3?²n

          Simple Model of White Dwarf
Consider a simple model of a white dwarf by assuming that the
mass density is uniform. Then its gravitational potential energy is
EG = ?3GM² / 5R. Further assume that the nuclei are carbon and the
electrons are degenerate and ultra-relativistic.
(a) Derive the electron density in terms of M, mH and R, where M
and R are the mass and radius of the white dwarf respectively.
(b) Derive the Fermi energy ?F in terms of M, mH and other
constants.
(c) The density of states of ultra-relativistic electrons is g(?) ? ?².
Derive the average electronic energy in terms of ?F.
(d) By considering the R dependence of the total energy, show
that when M exceeds a threshold, the radius R converges to 0.
Use this to estimate the Chandrasekhar mass (in MSun).
? = 1.055 × 10?³? Js, c = 3 × 10? ms?¹, G = 6.67 × 10?¹¹ Nm²kg?², mH
= 1.66 × 10?²? kg, MSun = 1.99 × 10³? kg ? = pc p = ?k kF³ = 3?²n

Simple Model of White Dwarf
Consider a simple model of a white dwarf by assuming that the
mass density is uniform. Then its gravitational potential energy is
EG = ?3GM² / 5R. Further assume that the nuclei are carbon and the
electrons are degenerate and ultra-relativistic.
(a) Derive the electron density in terms of M, mH and R, where M
and R are the mass and radius of the white dwarf respectively.
(b) Derive the Fermi energy ?F in terms of M, mH and other
constants.
(c) The density of states of ultra-relativistic electrons is g(?) ? ?².
Derive the average electronic energy in terms of ?F.
(d) By considering the R dependence of the total energy, show
that when M exceeds a threshold, the radius R converges to 0.
Use this to estimate the Chandrasekhar mass (in MSun).
? = 1.055 × 10?³? Js, c = 3 × 10? ms?¹, G = 6.67 × 10?¹¹ Nm²kg?², mH
= 1.66 × 10?²? kg, MSun = 1.99 × 10³? kg ? = pc p = ?k kF³ = 3?²n

Added by Richard N.

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