| | 36 | == Usage of the Lane-Emden Equation: The physical Polytrope == |
| | 37 | |
| | 38 | The Lane-Emden equation has two boundary conditions, which are located at the center of the polytrope or at [[latex($\xi =0$)]]: |
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| | 40 | [[latex(\begin{center}$\theta =1 , \frac{d\theta}{d\xi}=0.$\end{center})]] |
| | 41 | |
| | 42 | Since these requirements occur at the same point, the boundaries are in fact initial conditions. Thus, for every polytropic index value, there can only be one solution to the Lane-Emden equation. |
| | 43 | |
| | 44 | The surface of a polytropic star is the value of [[latex($\xi$)]]when [[latex($\theta = 0$)]], this will be defined as when [[latex($\xi = \xi_{1}$)]]. Thus the surface is defined when the physical pressure and density go to zero. |
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| | 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 | |
| | 48 | In order to make solutions of the Lane-Emden equation correspond to physical values, two scaling parameters are needed. |
