Mathematical Principles of Theoretical Physics

440 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

Due to the closedness of black holes, planets, stars and galaxy nuclei are stable, which
cannot be absorbed into the inner cores of black holes and vanish.

1.Jeans theory on the origin of stars and galaxies.In the beginning of the twentieth cen-
tury, J. Jeans presented a general theory for the formation of galaxies and stars. He thought
that the Universe in the beginning was filled with chaotic gas, and various astronomical ob-
jects were formed in succession by a process of gas decomposition into bulks of clouds,
consequently forming galaxies, stars, and planets.
According to the Jeans theory, a ball of clouds with homogeneous densityρcan be held
together only if

(7.3.30) V+K≤ 0 ,

whereVis the total gravitational potential energy, andKis the total kinetic energy of all
particles. The potential energyVis

(7.3.31) V=−

∫R

0

GMr

r

× 4 πr^2 ρdr=−

3 GM^2

5 R

,

whereMis the mass of the cloud,Mr= 4 πr^3 ρ/3, andRis the radius. The kinetic energyK
is expressed as the sum of thermal kinetic energies of all particles:

K=

3

2

NkT,

whereNis the particle number,Tis the temperature, andkis the Boltzmann constant. As-
sume that all particles have the same massm, thenN=M/m, and we have

(7.3.32) K=

3 M

2 m

kT.

Thus, by (7.3.30)-(7.3.32) we obtain that

(7.3.33)

GM

R

≥

5

2 m

kT.

The inequality (7.3.33) is called the Jeans condition.

2. Masses of astronomical objects. The Jeans condition (7.3.33) guarantees only the
   gaseous clouds being held together, and does not imply that the gas clouds can contract to
   form an astronomical object. However, a black hole must attract the nebulae around it to form
   a compact body.
   We consider the mass relation between an astronomical object and its black hole core.
   The massMof the object is

(7.3.34) M=Mb+M 1 ,

whereMbis the mass of the black hole, andM 1 is the mass of the material attracted by this
black hole. The total binding potential energyVof this object is given by

(7.3.35) V=−

∫R

Rs

GMr

r

× 4 πr^2 ρdr,