Mathematical Principles of Theoretical Physics

430 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

1. When a very massive red giant completely consumes its central supply of nuclear fuels,
   its core collapses. Its radiusr 0 begins to decrease, and consequently theδ-factor increases:

r 0 decreases⇒δ=

2 mG

c^2 r 0

increases.

2. The huge massmand the rapidly reduced radiusr 0 make theδ-factor approaching one:

δ→1 asr 0 →Rs

whereRs= 2 mG/c^2 is the Schwarzschild radius.

3. By (7.2.58), the shrinking of the star slows down:

Pr∼

√

1 −δ,

and nearly stops asδ→ 1.

4). Then the model (7.2.26) is valid, and the eigenvalue equations of (7.2.26) are given
by

(7.2.60)

Pr∆P+

1

κ

FGP+σT~k−∇p=βP,

̃∆T+Pr=βT,

divP= 0.

The first eigenvalueβdepends on theδ-factor, and by (7.2.14)

(7.2.61) β 1 ∼

(

Prδ^2

1 −δ

) 1 / 2

asδ→ 1.

Based on the transition criterion (7.2.25), the property (7.2.61) implies that the star has con-
vection in the shell layer, i.e., the radial circulation momentum fluxPrsatisfies

Pr>0 in certain regions of the shell layer.

5. The radial force (7.2.15) in the shell layer is

fr≃

2 Prδ^2

1 −δ

Pr→∞ asδ→1 andPr> 0 ,

which provides a very riving force, resulting in the supernova explosion, as shown in Figure
7.2.

6. SincePr=0 atr=r 0 , the radial force of (7.2.14) is zero:

fr=0 at r=r 0.

Herer 0 is the radius of the blackhole core. Hence, the supernova’s huge explosion preserves
an interior core of smaller radius containing the blackholecore, which yields a neutron star.