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3 building models I

3 building models I. solving the system. In time-independent form, the interior of a star can be described by a set of four ordinary differential equations, together with four boundary conditions.

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3 building models I

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  1. 3 building models I Stellar Structure: TCD 2006: 3.1

  2. solving the system • In time-independent form, the interior of a star can be described by a set of four ordinary differential equations, together with four boundary conditions. • If all four boundary conditions were given, say, at the centre, this would require a straightforward numerical integration from the the centre to the surface. Since they are not, some clever techniques are required. • In general, there is no analytic solution and it is necessary to find a numerical solution. • However, by making certain approximations, we can obtain some restricted but very informative solutions. • First, we look at how families of solutions might behave. • Second, we introduce methods for computing full solutions. • Subsequently, we will look at some approximate solutions. Stellar Structure: TCD 2006: 3.2

  3. homologous stars Stellar Structure: TCD 2006: 3.3

  4. homologous stars (2) Stellar Structure: TCD 2006: 3.4

  5. homologous stars (3) Stellar Structure: TCD 2006: 3.5

  6. homology relations: radiative stars Stellar Structure: TCD 2006: 3.6

  7. homology relations: radiative stars (2) Stellar Structure: TCD 2006: 3.7

  8. homology relations: convective stars Stellar Structure: TCD 2006: 3.8

  9. homology relations: upper main-sequence stars Stellar Structure: TCD 2006: 3.9

  10. solving ode’s: shooting • This method divides the star into an inner and outer part.Separate solutions are obtained by estimating a set of additional boundary conditions. These are adjusted until thee two solutions match. • Inward solution.At m=M we have Ps=0, Ts=Teff. Estimate R and L. Integrate inwards to a point mf, to obtain Pif, Tif, lif, and rif. • Outward solution.At m=0 we have rc=0, lc=0. Estimate Pc and Tc. Integrate outwards to mf, to obtain Pof, Tof, lof and rof. • In general: Pf=Pif–Pof0, Tf=Tif–Tof0, lf=lif–lof0, rf=rif–rof0. • Repeat the inward solution with R+R, L and with R, L+L. • Repeat the outward solution with Pc+Pc, Tc, and with Pc, Tc+Tc. Stellar Structure: TCD 2006: 3.10

  11. solving ode’s: shooting (2) Stellar Structure: TCD 2006: 3.11

  12. solving ode’s: difference methods (1) Stellar Structure: TCD 2006: 3.12

  13. solving ode’s: difference methods (2) Stellar Structure: TCD 2006: 3.13

  14. solving ode’s: difference methods (3) Stellar Structure: TCD 2006: 3.14

  15. solving ode’s: difference methods (4) Stellar Structure: TCD 2006: 3.15

  16. 3 building models -- review • We have obtained simple relations for properties of main sequence stars, and looked at methods for solving the full system of stellar structure equations • Homologous stars - families of models • Radiative stars • Convective stars • Solving the full system of 4 odes + 4bcs • Shooting method • Difference Equations • Henyey method • These provide an accurate numerical solution, but not much insight into how stars work. Stellar Structure: TCD 2006: 3.16

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