Mathematical Principles of Theoretical Physics

7.3. BLACK HOLES 441

whereRis the object radius,Rsis the radius of the black hole,ρis the mass density outside
the core, and

(7.3.36) Mr=Mb+

∫r

Rs

4 πr^2 ρdr.

SinceRs≪R, we takeRs=0 in the integrals (7.3.35) and (7.3.36). We assume that the
densityρis a constant. Then, it follows from (7.3.35) and (7.3.36) that

V=− 4 πGρ

∫R

0

[

Mbr+

4 π

3

ρr^4

]

dr=− 4 πGρ

(

MbR^2

2

+

4 π

3 × 5

ρR^5

)

.

Byρ=M 1 /^43 πR^3 andMb=M−M 1 ,we have

(7.3.37) V=−

G

R

(

3

2

MM 1 −

9

10

M^21

)

.

The stability of the object requires that−Vtakes its maximum at someM 1 such that
dV/dM 1 =0. Hence we derive from (7.3.37) that

(7.3.38) M 1 =

5

6

M, Mb=

1

6

M.

The relation (7.3.38) means that a black hole with massMbcan form an astronomical object
with massM= 6 Mb.

3.Relation between radius and temperature.A black hole with massMbdetermines the
massMof the corresponding astronomical system:M= 6 Mb. Then, by the Jeans relation
(7.3.33), the radiusRand average temperatureTsatisfy

(7.3.39) T=

2 × 6 GmMb

5 kR

whereTis expressed as

T=

3

4 πR^3

∫

BR

τ(x)dx,

whereBRis the ball of this system, andτ(x)is the temperature distribution. Letτ=τ(r)
depend only onr, then we have

(7.3.40) T=

3

R^3

∫R

0

r^2 τ(r)dr.

4.Solar system.For the Sun,M= 2 × 1030 kg andR= 7 × 108 m. Hence the mass of the
solar black hole core is about

M⊙b=

1

3

× 1030 kg,

and the average temperature has an upper limit:

T=

4

5

×

6. 7 × 10 −^11 m^3 /kg·s^2 × 1030 kg× 1. 7 × 10 −^27 kg

1. 4 × 10 −^24 kg·m^2 /s^2 ·K× 7 × 108 m

≃ 108 K.