Astron. Astrophys. 321, 465-476 (1997)

Appendix A

Although the Kippenhahn & Thomas (1970) method is applicable only in conservative cases (see section 2), most authors use it to treat the case of a "shellular rotation law" (clearly a non conservative case) without checking the conditions under which such a procedure is possible. Here we bring some clarifications on the implicit assumptions made when the the Kippenhahn & Thomas method is used in the case of a "shellular rotation law".

A.1 The interior stellar structure equations

Let us show that for a "shellular rotation law" the surfaces given by the equation

are isobars. , where V is the gravitational potential, r is the radius, the angular rotation rate and the colatitude.

The hydrostatic equilibrium implies that

where is the effective gravity. In spherical coordinates, its components write

The components of the gradient of in polar coordinates r, are

Comparing (A.4) with (A.2) and (A.5) with (A.3), one can write

Thus the expression for the hydrostatic equilibrium may be written

Since is constant on the isobars, the vector is parallel to the vector . The hydrostatic equation (A.7) implies then that is parallel to . Therefore the surfaces defined by constant correspond to isobaric surfaces and this ends the demonstration. It is interesting to note that the shape of the isobars in the case of a "shellular rotation law" are identical to the shape of the equipotentials in a conservative situation.

Once the shape of the isobars is known, one can write the stellar structure equations on these isobars. In the following we shall drop the subscript P attached to and the symbol g will be used instead of . Following the method of Kippenhahn & Thomas we define by

where is the volume inside an isobar. For any quantity, q, which is not constant over an isobaric surface, a mean value is defined by

where is the surface of the isobar and is an element of that surface.

A.1.1 Hydrostatic equilibrium equation

The effective gravity can no longer be defined by ( and ), since is not a potential (dn is the distance between two neighboring isobaric surfaces =const. and =const.). Let us use the fact that is parallel to and thus can be expressed by with a scalar which depends only on . Replacing by this expression as a function of in A.7 leads to the following expression for the effective gravity

which implies that

The hydrostatic equation then writes

or

From this equation, one can immediately deduce that the quantity is constant on an isobar.

In order to have , i.e. the mass inside the isobar, as independant variable, one has to find the expression for . One can write, using (A.8)

The fact that is constant on an isobar leads to

Thus, using (A.9) and (A.11),

Setting

G being the gravitational constant, the hydrostatic equilibrium equation writes

Thus expressed in terms of the lagrangian variable , the hydrostatic equation keeps the same form as in the conservative case.

A.1.2 Conservation of the mass

The volume of a shell comprised between two isobars is by definition of

It can also be expressed by

which finally gives

From (A.14) and (A.15), one can deduce and, using (A.11), one obtains

where

Let us note that is not equal to . Indeed is obtained by averaging the density over the volume between two isobars, instead is an average performed on an isobaric surface.

A.1.3 Conservation of the energy

One can write that the net energy outflow from a shell comprised between the isobars and is equal to

where is the net rate of energy production in the shell. Using (A.8) and the constancy of on an isobar, one can write

Using expression (A.11), one finally obtains decomposing into its nuclear, gravitational and neutrino components,

A.1.4 Radiative equilibrium

Locally, the equation of radiative transfer writes

where F is the radiative flux at a given point on the isobar. Using the fact that and the expressions (A.8) and (A.11), one has that

Integrating over an isobar, one obtains

A.1.5 Convective transport

In a convective region one has that locally, i.e. at a given point on an isobar,

Taking the averages on an isobar of the both sides of this equality, implies that

A.1.6 Simplifying assumptions

From the equations presented above, one can see that because of the non constancy of the density and temperature on isobars in the case of a "shellular rotation law", it is not possible to write as simple equations as in the conservative case. However let us see under which conditions one can transform the equations (A.13), (A.16), (A.19), (A.21) and (A.22) into the form proposed by Kippenhahn & Thomas (1970).

First, from equation (A.16), one immediately sees that if, instead of , one considers as a dependant variable the quantity , the continuity equation for the mass keeps its usual form. We shall consider also a mean temperature obtained from the equation of state with as input variables , p and the chemical composition. The chemical composition is supposed to be homogeneous on an isobaric surface due to the strong horizontal turbulence. The energy conservation equation and the energy transport equation are written using these mean values of the density and the temperature, making the following approximations:

and

With these changes of variables and these approximations we recover the set of stellar structure equations given by Kippenhahn & Thomas (1970), i.e.:

where

Partial derivatives have replaced total derivatives to allow for the fact that the quantities depend not only on but also on time. Let us note here that the simplifying assumptions (A.23) to (A.26) do not appear too severe. First we have shown that the equations describing the hydrostatic equilibrium and the conservation of mass are strictly valid in the case of a "shellular rotation law", provided that is considered as the dependant variable for the density. Moreover the strong horizontal turbulence responsible for the constancy of on isobars will likely homogenize the chemical composition and reduce the constrasts in densities and temperatures on isobars, making the above approximations justified. In the present computation we used the Roche model for the computation of the gravitational potential.

A.2 The equations for the stellar envelope

We call envelope (cf. Kippenhahn et al. 1967) the layers connecting the inner solutions to the atmosphere. In the envelope, convection is treated non adiabatically, partial ionisation is treated in details and the 's are considered to be zero. The envelopes of massive hot stars contain only a few thousands of the total mass, thus we may consider that the envelope rotates with a uniform angular velocity equal to that of the first (outermost) interior shell. In that case we have locally a solid rotation law. The independant variable , used in the interior is advantageously replaced by the pressure which in the outer parts of the star covers a much wider range of values. Using the pressure as the independant variable, the equations of stellar structure become:

A.3 The equations for the atmosphere

In the atmosphere, the mass, the radius and the luminosity are supposed to keep constant values. Only the hydrostatic equilibrium equation and the radiative transfer equations must be solved. We shall suppose that is constant as a function of the depth and take the same value as in the envelope. In this case, the effective gravity can be derived from a potential and one can write (see eq. A.9)

Using the fact that (see expressions (A.14) and (A.15), with ), one has that

Defining the optical depth by , one obtains

which can be transformed into

The equation of radiative equilibrium may be transformed into a relation linking the temperature to the optical depth. Let us call the radiation pressure. One can write

Since , one obtains

On the other hand, in the diffusive approximation, one can write locally at

Together with the expression (A.37), one obtains

Using the expression for the optical depth given above, one has that

Integrating from to the surface

In the case the specific intensity can be considered as isotropic and as having non zero value only in the outward direction, one has that

Thus one can write

Now the theorem of von Zeipel (1924) enables ones to write

Thus

Let us define by the expression . Then

With , and introducing the expression A.32 for one finally obtains

A.4 The equation for the local conservation of the angular momentum

Actually, one should resolve an equation of transfer for the angular momentum. However in the case of "no wind" and for asymptotic regime, the advection of angular momentum by the meridional currents are compensated by the diffusion of angular momentum through shear flow (cf. Zahn 1992). Thus one can consider that the angular momentum per unit mass in a given shell remains constant with time.

The momentum of inertia dI of a shell comprised between two isobars is

Using (A.8) and the constancy of on an isobar, one can write

With (A.11) this gives

Since is constant on an isobar, the angular momentum, dG, can be written

Thus, the angular momentum per unit of mass for a given shell comprised between two isobars is given by

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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