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In the following black hole model, electrons and positrons form a neutral gas which is confined by gravitation. The smaller masses are supported against gravity by electron degeneracy pressure. Larger masses are supported by ideal gas and radiation pressure. In each case, the gas is a polytrope which satisfies the Lane-Emden equation. Solutions are found that yield the physical properties of black holes, for the range 1000 to 100 billion solar masses.

The discovery of very large black holes in the early Universe is physical evidence that these black holes formed during the Big Bang. The electron-positron model requires vast numbers of positrons, and these were available only during the lepton epoch of the Big Bang [

It will be shown in the following that the smaller black holes achieve equilibrium as a degenerate quantum gas. The gas is in its ground state and does not radiate. The radius of such objects decreases with increasing mass, but it cannot be smaller than the Schwarzschild radius, R s = 2 G M / c 2 . The gas remains nonrelativistic until that point. Larger masses reach equilibrium as a mixture of ideal gas and radiation. Their radius is equal to R s , so that the radiation is confined. The average density decreases as 1 / M 2 , and as a result, the gas remains nonrelativistic throughout.

The Fermi energy for a completely degenerate ( T = 0 ) gas of N / 2 electrons is given by [

ϵ F = ( 3 π 2 2 ) 2 / 3 ℏ 2 2 m 5 / 3 ρ 2 / 3 (1)

where ρ is the mass density of the neutral lepton gas. The pressure of the degenerate gas is

P = ( 3 π 2 2 ) 2 / 3 ℏ 2 5 m 8 / 3 ρ 5 / 3 = K ρ 1 + 1 n (2)

Therefore, the gas is a polytrope [

K = ( 3 π 2 2 ) 2 / 3 ℏ 2 5 m 8 / 3 = 1.71 × 10 18 ( cgs units ) (3)

A spherically symmetric mass, in hydrostatic equilibrium, satisfies the gravitational field equation

1 r 2 d d r ( r 2 d ψ d r ) = 4 π G ρ (4)

as well as

d P d r = − ρ d ψ d r (5)

Together, they yield the pressure formula

1 r 2 d d r ( r 2 ρ d P d r ) = − 4 π G ρ (6)

If the mass is a polytropic gas, then

( n + 1 ) K 1 r 2 d d r ( r 2 d ρ 1 / n d r ) = − 4 π G ρ (7)

Define ρ = ρ 0 θ n and substitute to find

[ ( n + 1 ) K 4 π G ρ 0 1 − 1 n ] 1 r 2 d d r ( r 2 d θ d r ) = − θ n (8)

Finally, define r = α ξ where the constant

α = [ ( n + 1 ) K 4 π G ρ 0 1 − 1 n ] 1 / 2 (9)

in order to obtain the Lane-Emden equation [

1 ξ 2 d d ξ ( ξ 2 d θ d ξ ) = − θ n (10)

The function θ ( ξ ) and the variable ξ are dimensionless. The initial conditions at the center ( ξ = 0 ) are

θ ( 0 ) = 1 and d θ d ξ | 0 = 0 (11)

so that ρ 0 represents the density at the center.

The Lane-Emden equation has been solved numerically for many values of the polytropic index n. The following is a summary for the case n = 1.5 . The solution θ ( ξ ) decreases monotonically from θ ( 0 ) = 1 to θ ( ξ 1 ) = 0 , where ξ 1 = 3.654 (from the tables). This corresponds to zero density and pressure at the surface, R = α ξ 1 . This would yield the radius, once the value of α is known. However, a more direct approach is provided by the mass-radius relation [

K = N n G M ( n − 1 ) / n R ( 3 − n ) / n = N 1.5 G M 1 / 3 R (12)

The tabulated coefficient N 1.5 = 0.424 , so that

R = K 0.424 G M − 1 / 3 = 6.05 × 10 25 M − 1 / 3 ( cm ) (13)

This shows that the radius decreases with increasing mass. It will continue to do so until the Schwarzschild radius, R s = 2 G M / c 2 , is reached at M = 8 × 10 6 M ⊙ . This defines the largest intermediate-mass black hole.

The average density ρ ¯ = 3 M / 4 π R 3 and the central density ρ 0 = 5.99 ρ ¯ (tables) are found by substituting (13)

ρ 0 = 6. × 10 − 78 M 2 ( g ⋅ cm − 3 ) (14)

while the central pressure is

P 0 = K ρ 0 5 / 3 = 3.9 × 10 − 112 M 10 / 3 ( dyn ⋅ cm − 2 ) (15)

Finally, the Fermi energy at the center is (1)

ϵ F 0 = 1.35 × 10 − 60 M 4 / 3 ( erg ) (16)

These physical properties are tabulated below (

The central density, pressure, and kinetic energy all increase rapidly with mass. In the final line, R = R s .

Black holes of mass greater than M = 8 × 10 6 M ⊙ are supported against gravity by ideal gas and radiation pressure. In all cases, R = R s so that the lepton gas and radiation are confined. The pressure is given by

M | R > R s | ρ 0 | P 0 | ϵ F 0 |
---|---|---|---|---|

( M ⊙ ) | (cm) | (g·cm^{−3}) | (Pa) | (eV) |

10^{3} | 4.8 (10^{13} | 2.6 (10^{−5}) | 3.9 (10^{9}) | 2.1 |

10^{4} | 2.25 (10^{13} | 2.6 (10^{−3}) | 8.4 (10^{12}) | 4.5 (10) |

10^{5} | 1.05 (10^{13}) | 2.6 (10^{−1}) | 1.8 (10^{16}) | 9.6 (10^{2}) |

10^{6} | 4.8 (10^{12}) | 2.6 (10) | 3.9 (10^{19}) | 2.1 (10^{4}) |

8 (10^{6}) | 2.4 (10^{12}) | 1.7 (10^{3}) | 3.9 (10^{22}) | 3.3 (10^{5}) |

P = P gas + P rad = ρ m k T + a 3 ( k T ) 4 (17)

where a = π 2 / 15 ( ℏ c ) 3 . The mixture may be treated analytically by adopting the following device from the standard stellar model [

β P = ρ m k T and ( 1 − β ) P = a 3 ( k T ) 4 (18)

Eliminate k T from these equations to find

P = [ 3 a 1 − β m 4 β 4 ] 1 / 3 ρ 4 / 3 = K ρ 4 / 3 (19)

If β is assumed to be a constant, then the mixture is a polytrope of index n = 3 , which satisfies the Lane-Emden Equation (10). The solution for n = 3 yields the mass-radius relation (12)

K = 0.364 G M 2 / 3 = 2.43 × 10 − 8 M 2 / 3 ( cgsunits ) (20)

Therefore, the value of β is determined by the mass alone

1 − β β 4 = a 3 K 3 m 4 = 6.9 × 10 − 83 M 2 (21)

The average density of the supermassive black hole ρ ¯ = 3 M / 4 π R s 3 yields the central density (tables)

ρ 0 = 6.93 ρ ¯ = 5.1 × 10 83 M − 2 ( g ⋅ cm − 3 ) (22)

The central pressure is

P 0 = K ρ 0 4 / 3 = 1.0 × 10 104 M − 2 ( dyn ⋅ cm − 2 ) (23)

Finally, the temperature is found from (18) k T = β m K ρ 1 / 3 . At the center,

k T 0 = 1.8 × 10 − 7 β ( erg ) (24)

Physical properties are tabulated below (

The central density, pressure, and temperature all decrease with increasing mass. The values of β show that radiation pressure dominates in the larger masses.

M | R = R s | ρ 0 | P 0 | k T 0 | β |
---|---|---|---|---|---|

( M ⊙ ) | (cm) | (g·cm^{−3}) | (Pa) | (eV) | |

8 (10^{6}) | 2.4 (10^{12}) | 2 (10^{3}) | 3.9 (10^{22}) | 1.1 (10^{5}) | 0.99 |

10^{7} | 3 (10^{12}) | 1.3 (10^{3}) | 2.5 (10^{22}) | 1.1 (10^{5}) | 0.98 |

10^{8} | 3 (10^{13}) | 1.3 (10) | 2.5 (10^{20}) | 6.2 (10^{4}) | 0.62 |

10^{9} | 3 (10^{14}) | 1.3 (10^{−1}) | 2.5 (10^{18}) | 2.3 (10^{4}) | 0.23 |

10^{10} | 3 (10^{15}) | 1.3 (10^{−3}) | 2.5 (10^{16}) | 8 (10^{3}) | 0.08 |

10^{11} | 3 (10^{16}) | 1.3 (10^{−5}) | 2.5 (10^{14}) | 2 (10^{3}) | 0.02 |

The model presented here is a great improvement over the previous work [

It is evident from

Supermassive black holes must be stable, in order to endure for more than 10 billion years. The lepton model is stable, in part, because there are no thermonuclear reactions. For this to remain the case, it is important that the black hole stays free of contamination by baryons. In particular, there can be no flow of matter into the black hole. A recent observation of the Milky Way’s black hole shows that this is, indeed, the case [

The author declares no conflicts of interest regarding the publication of this paper.

Dalton, K. (2019) Supermassive Black Holes. Journal of High Energy Physics, Gravitation and Cosmology, 5, 984-988. https://doi.org/10.4236/jhepgc.2019.53052