Stellar Structure Section 4: Structure of Stars Lecture 6 – Comparison of M-L-R relation with observation Pure M-L relation How to find T eff A different way of doing homology … … more flexible, but equivalent L-T eff relations How to make realistic stellar models Homology by simple scaling

M-L-R relation – comparison with observation To compare with observation, need to know values for and. Two approximate expressions, already known in 1920s: Electron scattering:  = constant( =1, =3) (4.17) “Kramers’ law”:    T -3.5 ( =2, =6.5) (4.18) Electron scattering gives a pure mass-luminosity relation, while Kramers’ law gives one with very weak dependence on radius (see blackboard). The mass dependence brackets the observed value (see blackboard). Applies to main-sequence stars, but not to giants, which have a different distribution of chemical composition with mass.

Pure M-L relation To eliminate the radius completely, need to know  as well as and. Can then find relation between any pair of variables. In particular, can find how the effective temperature scales with mass, using the definition from Lecture 1 (see blackboard). This demonstrates that the main sequence is the locus of stars with the same composition but different mass. We can also find (see later) the equation for the main sequence in the HR diagram (the luminosity-effective temperature relation).

Alternative method of finding scaling relations – 1 Same approximations as before, but now illustrate method using radius as dependent variable. Equations are: (2.1) (2.2) (4.23) (4.24) (2.8)

Alternative method of finding scaling relations – 2 Consider two main-sequence model stars, A and A* Relate their variables by scaling relations: m = M*(r)/M(r), x = r*/r, l = L*(r)/L(r), p=P*(r)/P(r), z =  *(r)/  (r) and t = T*(r)/T(r). (4.25) Write down full equations for model A* Use scaling relations to replace the variables for model A* by those for model A (see blackboard) Divide resulting equations by the corresponding equations for model A Hence derive 5 relations between the 6 scaling factors (see blackboard)

Application of scaling relations The scaling relations are: p = mz/x, m = x 3 z, t = z l /xt, l = x 3 z 2 t , p = zt (4.27) Solve for 5 scaling factors in terms of the 6 th, chosen according to what scaling we want to find, e.g. m to find scalings with mass, x to find scalings with radius Examples:  eliminate p, z from 1 st and last to find t = m/x => T  M/R (as in Theorem IV)  eliminate z from 1 st two to find p = m 2 /x 4, or P  M 2 /R 4 (as in Theorem I)

Equivalence of two homology arguments Omit equation involving  Eliminate p, t, and z from other 4 equations – find relation equivalent to the M-L-R relation found earlier (see blackboard) General solution in terms of (e.g.) x is very messy (see blackboard), so usually only treat special cases Taking (e.g.)  = 13: Electron scattering:( =1, =3) l = x 4 => L  R 4 Kramers’ law:( =2, =6.5) l = x 8.5 => L  R 8.5 Then use L  R 2 T eff 4 to eliminate R and find slope of main sequence in HR diagram (see blackboard)

Further progress requires numerical integration of full equations Equations numerically difficult – non-linear, and 2-point boundary conditions Starting from centre and integrating outwards requires guesses for central density and temperature – solutions are unstable and diverge near the surface Similar problem integrating in from surface – so need to do both and match the solutions half-way Once one solution found, can look for small changes (in mass or time) to this solution, and therefore linearise the equations Converting them to difference equations, and including the boundary conditions, then allows solution by matrix inversion

Simple scaling (doesn’t prove homology, but gives results) Replace derivatives by ratios of typical values (see blackboard for all that follows) Leave out all constants and just use proportionality Find scalings for P, , T from pressure balance, mass conservation and equation of state – independent of opacity or energy generation laws Solve other two equations to give two alternative scalings for L in terms of M and R Eliminate L to find mass-radius relation Eliminate R to find mass-luminosity relation Use L  R 2 T eff 4 to find relations involving effective temperature