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 => E th ≈ E grav ) – 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, ε 0 both functions of chemical composition Note that for 2-body nuclear reactions: number of reactions/m 3   2 => number/kg   Mean molecular weight independent of density and temperature, so write  =  (composition)

Full set of equations, with these approximations (4.3) (4.4) (4.5) (4.6) (4.7) B.c.: r = L = 0 at M = 0;  = T = 0 at M = M s

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/M s 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: (4.10) where the indices , , , ,  are constants. Substitute these expressions into the stellar structure equations, and eliminate the total mass M s 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 - )  (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!