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Stellar Structure

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

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Stellar Structure

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

  2. 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, butnotto giants, which have a different distribution of chemical composition with mass.

  3. 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).

  4. 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)

  5. 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)

  6. Application of scaling relations • The scaling relations are: p = mz/x, m = x3z, t = zl/xt, l = x3z2t, p = zt (4.27) • Solve for 5 scaling factors in terms of the 6th, 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 1st and last to find t = m/x => T  M/R (as in Theorem IV) • eliminate z from 1st two to find p = m2/x4, or P  M2/R4 (as in Theorem I)

  7. 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 = x4 => L R4 Kramers’ law: (=2, =6.5) l = x8.5 => L  R8.5 • Then use L R2Teff4 to eliminate R and find slope of main sequence in HR diagram (see blackboard)

  8. 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

  9. 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 R2Teff4 to find relations involving effective temperature

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