Polytrope

If we choose the polytropic equation of state^{C.1},

(C.2) |

(C.3) | |||

(C.4) | |||

(C.5) |

we obtain a normalized equation as

which is called Lane-Emden equation of index . The boundary condition at the center of polytrope should be

(C.7) | |||

(C.8) |

at . Solution of this equation is plotted for several in Figure C.1.

Mass of the polytrope is written down as

where represents the zero point of or the surface radius normalized by . For or , equation (C.9) reduces to

Thus, the mass does not depend on the central density for polytrope. For or , is written down as

where and . Polytrope with gas, the mass-density relation becomes . While, for of , is written down as

where and .