The_Cambridge_Handbook_of_Physics_Formulas

9.5 Stellar evolution
183
Compact objects and black holes
Schwarzschild
radius
Gravitational
redshift
Gravitational
wave radiation a L g = 32 5
Rate of change of
orbital period
r s = 2GM
c 2 ≃ 3 M M ⊙
km (9.73)
(
ν ∞
= 1− 2GM ) 1/2
ν r rc 2 (9.74)
G 4
c 5 m 2 1 m2 2 (m 1 +m 2 )
a 5 (9.75)
Ṗ = − 96
5 (4π2 4/3 G5/3 m 1 m 2 P −5/3
)
c 5 (m 1 +m 2 ) 1/3 (9.76)
r s
G
M
c
M ⊙
r
ν ∞
ν r
m i
a
L g
P
Schwarzschild radius
constant of gravitation
mass of body
speed of light
solar mass
distance from mass centre
frequency at infinity
frequency at r
orbiting masses
mass separation
gravitational luminosity
orbital period
Neutron star
p pressure
degeneracy
pressure
p = (3π2 ) 2/3 ¯h 2 ( ) 5/3 ρ
= 2 5 m n m n 3 u (9.77) ¯h (Planck constant)/(2π)
m n neutron mass
(nonrelativistic)
ρ density
Relativistic b p = ¯hc(3π2 ) 1/3 ( ) 4/3 ρ
= 1 4 m n 3 u (9.78) u energy density
Chandrasekhar
mass c M Ch ≃ 1.46M ⊙ (9.79) M Ch Chandrasekhar mass
Maximum black
hole angular
momentum
Black hole
evaporation time
Black hole
temperature
J m = GM2
c
(9.80)
J m
maximum angular
momentum
τ e ∼ M3
M⊙
3 ×10 66 yr (9.81) τ e evaporation time
T =
¯hc3
8πGMk ≃ M 10−7 ⊙
M K (9.82) T temperature
k Boltzmann constant
a From two bodies, m 1 and m 2 , in circular orbits about their centre of mass. Note that the frequency of the radiation
is twice the orbital frequency.
b Particle velocities ∼ c.
c Upper limit to mass of a white dwarf.
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