# Polytrope

If we choose the polytropic equation of stateC.1,

 (C.1)

the hydrostatic balance is expressed by
 (C.2)

where we used equations (4.1) and (4.2). A hydrostatic gaeous star composed with a polytropic gas is called polytrope. Normalizing the density, pressure and radius as
 (C.3) (C.4) (C.5)

we obtain a normalized equation as
 (C.6)

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

 (C.9)

where represents the zero point of or the surface radius normalized by . For or , equation (C.9) reduces to
 (C.10)

Thus, the mass does not depend on the central density for polytrope. For or , is written down as
 (C.11)

where and . Polytrope with gas, the mass-density relation becomes . While, for of , is written down as
 (C.12)

where and .

Kohji Tomisaka 2012-10-03