Mathematical Principles of Theoretical Physics

408 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

ρ= mass + electromagnetism + potential + heat.

5.Equation of state:

(7.1.68) p=f(ρ,T).

Remark 7.5.Both physical laws (7.1.63) and (7.1.67) are the more general form than the
classical ones:

(7.1.69)

m

dv

dτ

=Force the Newton Second Law,

∂m

∂ τ

+div(mv) =0 the continuity equation,

wheremis the mass density. Hence the momentum representation equations (7.1.65)-(7.1.67)
can be applicable in general. The momentumPrepresents the energy density flux, consisting
essentially of
P=mv+radiation flux+heat flux.

Hence in astrophysics, the momentum densityPis a better candidate than the velocity field
v, to serve as the continuous-media type of state function.

7.1.5 Astrophysical Fluid Dynamics Equations

Dynamic equations of stellar atmosphere

Different from planets, stars are fluid spheres. Like the Sun, most of stars possess at-
mospheric layers. The atmospheric dynamics of stars is an important topic, and we are now
ready to present the stellar atmospheric model.
The spatial domain is a spherical shell:

M={x∈R^3 |r 0 <r<r 1 }.

The stellar atmosphere consists of rarefied gas. For example, the solar corona has mass
density aboutρm= 10 −^9 ρ 0 whereρ 0 is the density of the earth atmosphere. Hence we use
the Schwarzschild solution in (7.1.44) as the metric:

(7.1.70) α(r) =

(

1 −

2 mG

c^2 r

)− 1

, r 0 >

2 mG

c^2

.

wheremis the total mass of the star, and the conditionr 0 > 2 mG/c^2 ensures that the star is
not a black hole.
The stellar atmospheric model is the momentum form of the astrophysical fluid dynamical
equations defined on the spherical shellM:

(7.1.71)

∂P

∂ τ

+

1

ρ

(P·∇)P=ν∆P+μ∇(divP)−∇p−

mGρ

r^2

( 1 −βT)~k,

∂T

∂ τ

+

1

ρ

(P·∇)T=κ∆ ̃T,

∂ ρ

∂ τ

+divP= 0 ,