Mathematical Principles of Theoretical Physics

7.2. STARS 415

total force consists of four parts: the viscous friction, the pressure gradient, the relativistic
effect, and the thermal expansion force, which are given as follows:

• Theviscous friction forceis caused by the electromagnetic, the weak and the strong
  interactions between the particles and the pressure, and isgiven by

(7.2.5) FνP=ν△P=ν(∆Pθ,∆Pφ,∆Pr),

as defined by (7.1.57).

• Thepressure gradientis defined by:

(7.2.6) −∇p=−

(

1

r

∂p

∂ θ

,

1

rsinθ

∂p

∂ φ

,

1

α

∂p

∂r

)

.

• Therelativistic effectis reflected in the following terms:

(7.2.7) FGP=

(

−

ν

2 α^2 r

dα

dr

∂

∂r

(rPθ),−

ν

2 α^2 r

dα

dr

∂

∂r

(rPφ),

ν

2 α

∂

∂r

(

1

α

dα

dr

Pr

))

,

which can be regarded as the coupling interaction between the gravitational potentialα

and the electromagnetic, the weak and the strong potentialsrepresented by the viscous

coefficientν.

We shall see that the force (7.2.7) is responsible to the supernova’s huge explosion.

• Thethermal expansion force is due to the coupling between the gravity∇ψand the
  heatQ:

(7.2.8) FT=

(

0 , 0 ,

c^2

2 α

dψ

dr

βT

)

,

which is the main driving force for generating stellar interior circulations and nebular

matter spurts of red giants.

The two forces (7.2.7) and (7.2.8) are the main driving forces for the stellar motion, and
hereafter we derive their explicit formulas.

1.Formula for thermal expansion force.The thermal expansion force (7.2.8) is radially
symmetric, which is simply written in ther-component form

fT=

c^2

2 α

dψ

dr

βT.

In its nondimensional form,fTis expressed as

(7.2.9) fT=σ(r)T,

andσ(r)is called the thermal factor given by

(7.2.10) σ(r) =−

c^2 r^40 β

2 κ^2

1

α

dT

dr

dψ

dr

.