Changes between Version 3 and Version 4 of u/Polytropes
- Timestamp:
- 07/11/13 16:00:59 (12 years ago)
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u/Polytropes
v3 v4 46 46 There are only three analytic solutions: n=0. n=1 and n=5. The n=0 case corresponds to an incompressible fluid, i.e when [[latex($\rho=\rho_{c}=const.$)]]. Thus the density is constant through out the star, but the pressure still goes to zero at the surface. This case is considered a crude approximation to the interior of our earth. The n=5 corresponds to the radius of this star being infinite. It can be shown that all indices greater than or equal to five will have infinite radii. The two cases that correspond to real stars are the n=1.5 and n=3 case, whose solutions are found numerically. The n=1.5 case is useful for approximating fully convective stars, and the n=3 case useful for approximation main sequence and relativistic degenerate cores of white dwarfs (though for the white dwarf case there is some special additions that need to made concerning K). 47 47 48 In order to make solutions of the Lane-Emden equation correspond to physical values, two input parameters are needed. These parameters can be K and central density, stellar mass and stellar radius, K and stellar mass etc. In the wiki:LE_Module, we require stellar mass and central density be given. We do not explicitly calculate K 49 48 In order to make solutions of the Lane-Emden equation correspond to physical values, two input parameters are needed. These parameters can be K and central density, stellar mass and stellar radius, K and stellar mass etc. In the wiki:LE_Module, we require stellar mass and central density be given. We do not explicitly calculate K, but rather redefine a few terms as follows: 49 [[latex(\begin{center}$\rho=\rho_{c} \theta^{n}$\end{center})]] 50 [[latex(\begin{center}$P=\frac{4 \pi G \alpha^{2} \rho_{c}^{2} \theta^{n+1}}{n+1}$\end{center})]] 51 where 52 [[latex(\begin{center}$\alpha=[\frac{-M}{4 \pi \rho_{c} \xi_{1}^{2}} (\frac{d\theta}{d\xi})_{\xi_{1}}]^{1/2}$\end{center})]] 50 53 51 54
