Mass-Radius Relation for a White Dwarf Star

Abstract

THE peculiar difficulty pointed out by Eddington regarding the ultimate fate of a white dwarf star led Fowler¹ to make the first application of Fermi-Dirac statistics to astrophysics in a fundamental paper which initiated a long series of investigations, particularly by Milne², in recent years. The starting point is the expression which represents (neglecting the effect of relativistic mechanics) the number of wave functions corresponding to eigen-values lying in the energy-range ɛ to ɛ + dt for a free electron confined in a field-free region of volume V; this is assumed to hold for the stellar interiors though gravitational and electrical fields are present. For degenerate matter equation (1) leads to the well-known expression for the pressure which reduces the equation for mechanical equilibrium for a star to Emden's equation of index 3/2 giving the relation (“Emden-solution”) between the radius R and the mass M of a white dwarf. L_o is a length characteristic of the white dwarf theory where ⊙ is the solar mass, m the mass of the electron, m_H the mass of the proton, G the gravitational constant, and ω^o_3/2 = 132·4 is a number defining the Emden-solution of index 3/2. μ is the mean molecular weight per free electron, μ = 1 for ionized hydrogen and μ = A/Z? 2 for any other completely ionized element of atomic weight A and atomic number Z. (The material in the interior of a white dwarf star is completely ionized due to pressure ionization.)