## Properties of the Isobars

In the case of shellular rotation, the centrifugal force cannot be derived from a potential and thus (2.3) does not apply. Let us consider the surface of constant W (2.9),

As in Sect. 1.2.1, the gravitational potential is defined by d0/dr = GMr/r2 and 0 = -GMr/r in the Roche approximation. The components of the gradient of W are in polar coordinates (r, $)

dW d0 9 9 9 9 dQ — = — - Q2 r sin2 $ - r sin2 $Q —, (2.28)

dr dr dr

1 S = 1 d0 - Q2 r sin $ cos $ - r2sin2 tfQ1 ^ , (2.29) r d$ r d$ r d$

The first two components of the gravity gef = (-geff,r, geff,$, 0) are according to (2.11) in the Roche model, d0 2 2

geff r --Q2r sin2 $ and dr geff,$ = Q2r sin $ cos $ . (2.30)

Thus, by comparing these terms and the derivatives of W, one can write geff = -VW - r2 sin2 $QVQ. (2.31)

The equation of hydrostatic equilibrium VP = g geff is thus

Since Q is constant on isobars, the vector VQ is parallel to VP. The hydrostatic equation (2.32) implies the parallelism of VP and VW. Thus, in this non-conservative case the surfaces defined by W = const. (2.27) are isobaric surfaces [408], but they are not equipotential and the star is said to be baroclinic. In the case of solid body rotation, isobars and equipotentials coincide and the star is barotropic.

In literature, W and & are often defined with different signs, care has to be given because this may lead to different expressions.

Thus, for shellular rotation one may choose to write the equations of the stellar structure on the isobars and use, with little changes, a method devised for the conservative case [283], with the advantage to keep the equations for stellar structure one-dimensional. The main change concerns the expression of the average density between isobars, as given below (cf. 2.47). Some further properties of baroclinic stars are studied in Sect. 10.5.3.

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