Download Radiative Energy Transport

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Radiative energy transport in stellar interior
In a normal star supporting itself through nuclear fusion, energy is generted in
the deep interior, transported through the bulk of the stellar material and then
radiated away from the surface. In a steady state the rate of radiation from the
surface (i.e. the Luminosity) exactly equals the rate of energy generation in the
interior. There are three main processes responsible for energy transport: radiative
diffusion, conduction and convection, all driven by the radial temperature gradient
from the centre to the surface. Existence of this temperature gradient makes the
radiation field in the interior anisotropic, but this departure from isotropy is so
small (about 1 part in 1010 ), that the radiation field is still very well approximated
by a blackbody at the local temperature, which is a function of radius.
Radiative diffusion
In the interior of the star the mean free path of photons is very small compared to
the distance to travel to the surface. In this regime the transport of energy can be
treated as a diffusion process. We recall that the diffusive flux j of particles in the
presence of a gradient of number density n is given by
j = −D∇n
(1)
where D is the coefficient of diffusion,
1
D = vlp
3
(2)
v being the mean velocity and lp the mean free path of the particles.
Similarly, the diffusion of radiative energy can be expressed as
Fν = −Dν ∇uν
(3)
where Fν is the diffusive energy flux at a frequency ν, uν is the radiative energy
density, and the diffusion coefficient
c
c
1
=
.
Dν = clν =
3
3αν 3κν ρ
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Here lν is the photon mean free path at frequency ν, αν the absorption coefficient
and κν is the absorption coefficient per unit mass, known as the opacity. The
energy density
8πhν3
4π
1
Bν (T ) =
uν =
3
c
c exp(hν/kT ) − 1
Then
4π ∂Bν
∇T
∇uν =
c ∂T
and the total frequency integrated flux
" Z ∞
#
Z ∞
4π
1 ∂Bν
F=
Fν dν = −
dν ∇T.
3ρ 0 κν ∂T
0
We define a Rosseland mean opacity κrad by
R ∞ 1 ∂B
Z ∞
ν
dν
1 ∂Bν
π
1
0 κν ∂T
= R ∞ ∂B
dν
=
3
ν
κrad
acT 0 κν ∂T
dν
0
(4)
∂T
where a is the radiation constant. Inserting this, we can write the flux as
F=−
4acT 3
∇T = −krad ∇T
3κrad ρ
(5)
which defines the coefficient of radiative transport krad . In a spherically symmetric situation, at a radius r the local luminosity L = 4πr 2 |F| and the temperature
gradient is then
3 κrad ρL
∂T
=−
(6)
∂r
16πac r2 T 3
in Eulerian variables and
∂T
3 κrad L
=−
(7)
∂m
64π2 ac r4 T 3
in Lagrangian variables.
Conduction
Conductive energy transport proceeds by the collision between gas particles. In
ordinary stars the conductive energy transport is negligible compared to the radiative diffusion, because the mean free path of particles is much smaller than that of
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photons, and the mean speed of particles is much less than c. In degenerate cores
of massive stars, however, conductive transport can play a major role.
Conductive transport can be written in a form similar to radiative diffusion
Fcd = −kcd ∇T
where kcd is the conductivity. Defining a “conductive opacity” κcd by
kcd =
we can write the total flux
4ac T 3
,
3 κcd ρ
!
4ac T 3 1
1
F=−
∇T
+
3 ρ κrad κcd
If we now define an effective opacity κ through
1
1
1
=
+
κ κrad κcd
then the temperature gradient can still be written as in equations (6) and (7) by
replacing κrad with κ.
In hydrostatic equilibrium
From eq. (7) we can then write
∂P
Gm
=−
∂m
4πr4
κL
3
(∂T/∂m)
=
(∂P/∂m) 16πacG mT 3
A quantity often used in stellar physics is the logarithmic variation of temperature
with pressure
d ln T
∇=
(8)
d ln P
For combined radiative and conductive transport this then works out to be
κLP
3
16πacG mT 4
This can be rearranged to write the flux F = L/4πr 2 as
∇rad =
Frad + Fcon =
4acG T 4 m
∇rad
3 κPr2
(9)
(10)