Stellar Structure
Section 3: Energy Balance
Lecture 4 – Energy transport processes
Why does radiation dominate?
Simple derivation of transport equation
Energy conservation equation
Full sets of structure equations …
… including boundary conditions
Simple prescription for time evolution
Energy transport – from central
energy source to surface emission
• Radiation – energy carried by photons
• Conduction – energy carried by electrons (mainly)
• Convection – energy carried by large-scale gas motions
Convection: discussed later (Section 4)
Radiation, conduction similar, so can be treated similarly, using a
conduction coefficient λ (see blackboard)
Which carries more energy?
Normally (see blackboard) gas particles possess more energy
than photons – but photons travel much farther:
ℓphoton ≈ 105 ℓelectron .
Exception: high density stars (e.g. WDs) (Pauli exclusion principle)
Energy carried by radiation:
approximate argument (see blackboard)
ℓ = photon
mean free
path
ℓ = r
(T+T)4
 = opacity
(defined
below)
r
T + T
T
T4
r + r
 = cross-sectional area for absorption (per unit volume)
Energy carried by radiation
• Approximate argument gives: rad 
acT 3

• Precise argument (see Handout) gives:
• Hence (see blackboard):
rad
4 acT 3

3 
dT
3 L

dr
16acr 2T 3
(3.23)
• Finally, energy conservation in a spherical shell (see
blackboard) gives:
dL
 4r 2 
dr
(3.24)
The four differential equations of
stellar structure
Assuming (i) steady state (ii) all energy carried by radiation:
dP
GM
 2
dr
r
dM
 4r 2 
dr
4 equations
dT
3 L

2 3
dr
16acr T
7 variables
dL
 4r 2 
dr
Three relations to close the system
Assuming that conditions in stellar interiors are close to
thermodynamic equilibrium:
P = P(, T, composition)
 = (, T, composition)
ε = ε(, T, composition).
Using previous explicit expression for pressure:
T
1 4
P
 aT

3
we also need  = (, T, composition).
Differential equations in terms of
mass as independent variable
Surface best defined by M = Ms, so use mass as variable:
dP
GM

dM
4 r 4
(dividing (2.1) by (2.2))
(3.29)
dr
1

dM 4 r 2 
(3.30)
dT
3L

dM
64 2 acr 4T 3
(3.31)
dL
 .
dM
(3.32)
Then central boundary conditions are: r = L = 0 at M = 0.
(3.33)
Surface boundary conditions
Vogt-Russell “theorem”
• Surface boundary conditions?
• A useful approximation is:
 = T = 0 at M = Ms .
(3.34)
• Actually, Ts ≠ 0 – and is one of the unknowns. But can find
effective temperature from
Ls  4π Rs2 Teff4
since Ls, Rs emerge from solution, and assume Ts ≈ Teff.
• Vogt-Russell “theorem”: 7 equations of stellar structure, plus 4
boundary conditions, completely determine the structure of a
star of given mass and composition. Unique for main-sequence
stars, but not for evolved stars – may also depend on history.
Approximate treatment of stellar
evolution
• Simple treatment uses sequence of static models, differing only in
composition:
• Build initial model, with composition known as function of mass.
• Assume all energy release from nuclear reactions.
• Calculate resultant changes in composition (schematic):
 (compositio n) 
 f (  , T , compositio n)


t
M
.
• Integrate (3.35) for timestep Δt, starting from composition of
previous model.
• Construct sequence of static models of gradually changing
composition and structure.
(3.35)