## L 4ac S2 T3geff dT 4 2 Lp T geff Sp dMP42

which expresses the radiative transfer in rotating stars.

4.1.2 Conservation of Energy

The net energy outflow dLP from a shell comprised between the isobars W and W + dW is equal to f f dn dLP = eg dn da = dW eg——do , (4.3)

J W=const J W=const dW

where e is the net rate of energy production in the shell. Using (2.36) and the constancy of g(1 - r2 sin2 9 Q a) on an isobar, one can write dLP = dW < — > SPg (1 - r2sin2 9 Q a) . (4.4)

geff

With (2.39), one obtains by decomposing the energy generation rate into its nuclear, gravitational and neutrino components, dLP = < (enucl - ev + egrav) gef > (4 5)

which is the equation for energy production in a rotating star in equilibrium.

### 4.1.3 Structure Equations for Rotating Stars

Because of the non-constancy of the density and temperature on isobars in the case of shellular rotation, the above equations are not as simple as in the conservative case. We shall now examine under which conditions, one can transform equations (2.42), (2.46), (4.2), (4.5) into a usable form as proposed by Kippenhahn & Thomas [283].

First, one sees that if, instead of g, one considers the quantity g (2.46) as a dependent variable, the continuity equation for the mass keeps its usual form. We shall consider also a mean temperature T obtained from the equation of state with as input variables g, the isobaric P and the chemical composition. The chemical composition is supposed to be homogeneous on an isobaric surface due to the strong horizontal turbulence. The equations of energy conservation and energy transport are written with these mean values of density and temperature, in addition one makes the following approximations for energy conservation (4.5)

< (£nucl - £v + %av)ge- > ^ £nucl(- T) _ (g, T) + e^Q, T) , (4.6) < gf >

and for the radiative transfer (4.2), T3geff dT

k dMP

In convective regions, the temperature gradient is the adiabatic gradient (Sect. 5.3) and we approximate the average gradient there by d ln T „ d ln T

With these changes of variables and approximations we recover the set of stellar structure equations, dP

GMP ~4nrP

GMP 4 nrP

dMP 4nrPg

fP min

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