Stellar Structure
Section 3: Energy Balance
Lecture 5 – Where do time derivatives matter?
(part 1)
Time-dependent energy equation
Adiabatic changes
Do we need other time derivatives?
• Simple treatment of evolution – only time dependence is in
nuclear reaction network.
• Don’t expect time dependence in mass conservation or
radiative energy transfer equations
• Time derivative in hydrostatic equation only matters if changes
occurring on dynamical timescale – ignore otherwise
• Will need to include time derivatives in energy production
equation if star changes on thermal or Kelvin-Helmholtz
timescales (comparable because from virial theorem U = -Ω/2
=> Eth ≈ Egrav) – happens between nuclear burning stages
Generalisation of energy equation
• Need to use thermodynamics
• Consider change in entropy of a small mass element, relate it to
heat supplied to the element, and apply to the energy equation
– this gives an expression for the rate of change of entropy at
fixed mass (see blackboard)
• Use first law of thermodynamics to relate entropy to thermal
energy and pressure, and hence derive time-dependent form of
energy equation (see blackboard)
• Original equation is true either if there is no variation with time
or if the changes are adiabatic (see blackboard): dQ/dt = 0.
Stellar Structure
Section 4: Structure of Stars
Lecture 5 – Approximation for pressure
(part 2)
Power laws for opacity, energy generation
Resulting set of structure equations
Homologous solutions – formal treatment
Derivation of M-L(-R) relation
Explicit expressions for state
variables (P, , ε, )
• P – ideal gas, neglect radiation pressure (see equation 2.8)
• Opacity and energy generation – use simple power law
approximations:
   0   1T 3
   0 T 
• 0, ε0 both functions of chemical composition
• Note that for 2-body nuclear reactions:
number of reactions/m3   2 => number/kg  
• Mean molecular weight independent of density and
temperature, so write
 = (composition)
Full set of equations, with these
approximations
dP
GM

dM
4 r 4
dr
1

dM 4 r 2 
3 0   1 L
dT

dM
64 2 acr4T 
dL
  0 T 
dM

P  T

B.c.: r = L = 0 at M = 0;  = T = 0 at M = Ms
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
Homologous solutions
• A set of model stars of different mass are said to be
homologous if the whole set can be derived from a model for
one mass by a simple scaling procedure.
• This requires the composition to be the same function of
fractional mass for all the stars, i.e. (schematically):
composition = c(m), the same for all stars,
where m ≡ M/Ms is the fractional mass.
• In a homologous set of models, the shapes of the various
functions (density, pressure, …), as a function of fractional
mass, are independent of the mass of the star (see sketch).
Formal proof of homology, using
mass-dependent equations - 1
• Introduce scaled variables (with bars) by:
r  M s r (m)
  M s  (m)
L  M s L(m)
(4.10)
T  M s T (m)
P  M s P (m)
where the indices , , , ,  are constants.
• Substitute these expressions into the stellar structure equations, and
eliminate the total mass Ms by choosing the indices (see blackboard)
Formal proof of homology, using
mass-dependent equations - 2
• Eliminating the total mass from the structure equations leads to
5 equations for the 5 indices (see blackboard)
• It also leads to 5 structure equations for the barred variables,
that are functions of m alone (see blackboard)
• The boundary conditions are also independent of the total mass
(see blackboard) – but only because of the simple zero
boundary conditions at the surface. If density and temperature
were non-zero, they would need to scale in the same way with
mass, which would introduce an extra, incompatible, constraint.
• One solution of the equations for the barred variables, plus the
scaling relations, gives a solution for any choice of the total
mass – a big saving in effort.
Use of homologous solutions to find
mass-luminosity(-radius) relation
• Even without knowing the energy generation law (i.e. without
having a value for ), can find one important result.
• Solve 4 equations without  for , , ,  in terms of .
• For luminosity exponent, find:
 = (3-) +  - 1 - .
(4.14)
• Now use scaling relations (4.10) at the stellar surface, m = 1, to
find expressions for the total radius and luminosity of the star in
terms of the total mass (see blackboard).
• Use (4.14) to eliminate  and  and give a relation between the
total luminosity, total mass and total radius (see blackboard).
• How does this compare with observation? Next lecture!