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

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Presentation transcript:

Stellar Structure: TCD 2006: building models I

Stellar Structure: TCD 2006: 3.2 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: 3.3 homologous stars

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

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

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

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

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

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

Stellar Structure: TCD 2006: 3.10 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 P s =0, T s =T eff. Estimate R and L. Integrate inwards to a point m f, to obtain P if, T if, l if, and r if. Outward solution. At m=0 we have r c =0, l c =0. Estimate P c and T c. Integrate outwards to m f, to obtain P of, T of, l of and r of. In general:  P f =P if –P of  0,  T f =T if –T of  0,  l f =l if –l of  0,  r f =r if –r of  0. Repeat the inward solution with R+  R, L and with R, L+  L. Repeat the outward solution with P c +  P c, T c, and with P c, T c +  T c.

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

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

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

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

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

Stellar Structure: TCD 2006: 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.