EQUATIONS OF STELLAR STRUCTURE
General Equations
We shall consider a spherically symmetric, self-gravitating star. All the physical quantities will
depend on two independent variables:
radius
and
time
, (
r, t
). First, we shall derive all the equations
of stellar structure in a general, non spherical case, but very quickly we shall restrict ourselves to
the spherically symmetric case.
The microscopic properties of matter at any given point may be described with density
ρ
, tem-
perature
T
, and chemical composition, i.e. the abundances of various elements
X
i
, with
i
= 1
,
2
,
3
....
having as many values as there are elements. All thermodynamic properties and transport coeffi-
cients are functions of (
ρ, T, X
i
). In particular we have: pressure
P
(
ρ, T, X
i
), internal energy per
unit volume
U
(
ρ, T, X
i
), entropy per unit mass
S
(
ρ, T, X
i
), coefficient of thermal conductivity per
unit volume
λ
(
ρ, T, X
i
), and heat source or heat sink per unit mass
ǫ
(
ρ, T, X
i
).
All the partial
derivatives of
P
,
U
,
λ
, and
ǫ
are also functions of
ρ
,
T
, and
X
i
. Using these quantities the first law
of thermodynamics may be written as
T dS
=
d
parenleftbigg
U
ρ
parenrightbigg
−
P
ρ
2
dρ,
(1
.
1)
If there are sources of heat,
ǫ
, and a non-vanishing heat flux
vector
F
, then the heat balance equation may
be written as
ρT
dS
dt
=
ρǫ
−
div
vector
F.
(1
.
2)
The heat flux is directly proportional to the temperature gradient:
vector
F
=
−
λ
∇
T.
(1
.
3)
The equation of motion (the Navier-Stockes equation of hydrodynamics) may be written as
d
2
vector
r
dt
2
+
1
ρ
∇
P
+
∇
V
= 0
,
(1
.
4)
where the gravitational potential satisfies the Poisson equation
∇
2
V
= 4
πGρ,
(1
.
5)
with
V
−→
0 when
r
−→ ∞
.
In spherical symmetry these equations may be written as
1
ρ
∂P
∂r
+
∂V
∂r
+
d
2
r
dt
2
= 0
,
(1
.
6a)
1
r
2
∂
∂r
parenleftbigg
r
2
∂V
∂r
parenrightbigg
= 4
πGρ,
(1
.
6b)
F
=
−
λ
∂T
∂r
,
(1
.
6c)
1 — 1