406 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
̃∆is the Laplace operator for scalar fields given by
(7.1.58) ̃∆T=
1
r^2 sinθ∂
∂ θ(sinθ∂T
∂ θ)+
1
r^2 sin^2 θ∂^2 T
∂ φ^2+
1
αr^2∂
∂r(r^2∂T
∂r),
the nonlinear term(u·∇)vis
(7.1.59)
(v·∇)vθ=vθ
r∂vθ
∂ θ+
vφ
rsinθ∂vθ
∂ φ+vr∂vθ
∂r+
vθvr
r−
cosθv^2 φ
rsinθ,
(v·∇)vφ=vθ
r∂vφ
∂ θ+
vφ
rsinθ∂vφ
∂ φ
+vr∂vφ
∂r+
vφvr
r+
cosθvφvθ
rsinθ,
(v·∇)vr=vθ
r∂vr
∂ θ+
vφ
rsinθ∂vr
∂ φ+vr∂vr
∂r−
1
αr(v^2 θ+v^2 φ)+1
2 αdα
drv^2 r,and the divergent term divvreads
(7.1.60) divv=
1
rsinθ∂(sinθvθ)
∂ θ+
1
rsinθ∂vφ
∂ φ+
1
r^2√
α∂(r^2√
αvr)
∂r.
7.1.4 Momentum representation
The Universe, galaxies and galactic clusters are composed of stars and interstellar nebulae.
Their velocity fields are not continuous. Hence it is not appropriate that we model cosmic
objects using continuous velocity fieldv(x,t)as in the Navier-Stoks equations or by discrete
position variablesxk(t)as in theN-body problem.
The idea is that we use the momentum density fieldP(x,t)to replace the velocity field
v(x,t)as the state function of cosmic objects. The main reason is that the momentum density
fieldPis the energy flux containing the mass, the heat, and all interaction energy flux, and
can be regarded as a continuous field. The aim of this section is to establish the momentum
form of astrophysical fluid dynamics model.
The physical laws governing the dynamics of cosmic objects are as follows
(7.1.61)
Theory of General Relativity,
Newtonian Second Law,
Heat Conduction Law,
Energy-Momentum Conservation,
Equation of State.The mathematical expressions of these laws are given respectively in the following:
1.Gravitational field equations.(7.1.62)
Rμ ν−1
2
gμ νR=−8 πG
c^4Tμ ν+1
2
(D ̃μΦν+D ̃νΦμ),D ̃μ=Dμ+e
̄hcAμ,