Mass-Radius Relation of Degenerate Configurations

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6and diverges as x → ∞. At some point, the second boundarycondition will be satisfied.The pressure at the outer boundary is that of free particlesP =8π15h 3 m ep 5 F (R c) = 110π Z2 e 2 ( 128Z9π 2 ) 4/3 (me e 2¯h 2 ) 4 [ φ (xo )x o] 5/2.The density is the total mass divided by the volume:ρ = 3Am B4π(128Z me e 2 ) 39π 2 ¯h 2 = 4Am BZx o 3( 2me e 2π¯h 2 x o) 3.For low densities, the solution approaches the unique solutionφ(x o ) → 144x −3 o . Thus, P ∝ ρ 10/3 , with K given by Eq. (3).Mass-Radius Relation for Degenerate ObjectsThe mass-radius diagram for cold compact objects is shownin the figure: the solid lines are the limiting expressions Eqs(1–3), the dashed line is the full result, for 12 C. The maximumradius configuration has the properties, approximately, of theplanet Jupiter.In the relativistic limit, for radii much smaller than 5000km, the equation of state will deviate from that of a γ = 4/3gas. Electron capture will reduce Y e and the value of the Chandrasekharmass. Therefore, a regime where dM/dρ c < 0 willexist. Such a regime is dynamically unstable. At sufficientlyhigh density, where nuclear forces become important, the effectivevalue of γ will increase, the mass will reach a minimumvalue (M min ≃ 0.01 M ⊙ , where R ≃ 300 km), and stability isrestored. As the central density increases further, dM/dρ c > 0.This is the neutron star regime. As the mass increases, and theradius shrinks, general relativity, which we have heretofore ignored,becomes important. The most important feature thatgeneral relativity introduces is that at densities well in excess of
7the nuclear saturation density, ρ s = 2.7 · 10 14 g cm −3 , the massreaches a maximum value, in the range 1.5-3 M ⊙ . Larger densityconfigurations are once again dynamically unstable. Themaximum mass is discussed in a subsequent lecture.Cooling of white dwarfsThe interior of a white dwarf has energy transport dominatedby conduction. The electrons are extremely degenerate,nowever, so they must have very large mean free paths. Thethermal conductivity is very high. The temperature gradientmust be rather small. The interior is roughly isothermal. Nearthe surface, isothermality breaks down as the opacity increases.The surface regions are diffusive, with a temperature gradientdTdr = − 3 κρ L4ac T 3 4πr 2.At the high densities, Kramer’s opacity is dominant: κ =κ o ρT −3.5 , with κ o ≃ 4.3 × 10 24 Z(1 + X) cm 2 g −1 . With hydrostaticequilibrium,dPdT = 4ac34πGm (r) T 6/5κ o L ρ .
Page 1 and 2: 1Compact Stars - White Dwarfs, Plan
Page 3 and 4: 3The maximum baryon number for equi
Page 5: 5The electron Fermi energy, modifie
Page 9: 9The cooling rate is L = −dU/dt.

Mass-Radius Relation of Degenerate Configurations

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