Mathematical Principles of Theoretical Physics

438 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

andt=r/v, wherevis the particle velocity. Replacingvby the speed of lightc, we have

ε 0 =

hc ̄

2 r

.

Bym 0 =ε 0 /c^2 , the densityρof (7.3.19) is written as

(7.3.20) ρ=

3 h ̄

8 πcr^4

or equivalently r^4 =

3 h ̄

8 πcρ

.

Insertingr^4 of (7.3.20) into (7.3.18), we derive the interaction potential pressurePin the form

(7.3.21) P=

8 cρg^2

3 ̄h

.

3.Criticalδ-factors.It is known that the central pressure of a star with massmand radius
r 0 can be expressed as

(7.3.22) PM=

Gm^2

r^40

=

2 πc^2

3

ρ δ, δ=

2 mG

c^2 r 0

,

whereδis theδ-factor.
We infer from (7.3.21) and (7.3.22) the criticalδ-factor as

(7.3.23) δc=

4

π

g^2

̄hc

,

wheregis one of the interaction charges in (7.3.17).
The criticalδ-factor in (7.3.23) provides criterions for the three types of astronomical
bodies of (7.3.16).

4.Physical significance ofδc.It is clear that for a star withm> 1. 4 M⊙if

(7.3.24) δ<

4

π

g^2 n

hc ̄

,

then the neutron potential pressurePnin (7.3.21) is greater than the star pressurePMin
(7.3.22):
Pn>PM.

In this case, neutrons in the star cannot be crushed into quarks. Hence, (7.3.24) should be a
criterion to determine if the body is a neutron star. It is known that

g^2 n∼ ̄hc.

Thus, we take

(7.3.25) g^2 n=

π

4

̄hc,

and (7.3.24) is just the black hole criterion.