SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

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 [FORMULA] given by the equation

[EQUATION]

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

The hydrostatic equilibrium implies that

[EQUATION]

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

[EQUATION]

[EQUATION]

The components of the gradient of [FORMULA] in polar coordinates r, [FORMULA] are

[EQUATION]

[EQUATION]

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

[EQUATION]

Thus the expression for the hydrostatic equilibrium may be written

[EQUATION]

Since [FORMULA] is constant on the isobars, the vector [FORMULA] is parallel to the vector [FORMULA]. The hydrostatic equation (A.7) implies then that [FORMULA] is parallel to [FORMULA]. Therefore the surfaces defined by [FORMULA] 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 [FORMULA] and the symbol g will be used instead of [FORMULA]. Following the method of Kippenhahn & Thomas we define [FORMULA] by

[EQUATION]

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

[EQUATION]

where [FORMULA] is the surface of the isobar and [FORMULA] is an element of that surface.

A.1.1 Hydrostatic equilibrium equation

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

[EQUATION]

which implies that

[EQUATION]

The hydrostatic equation then writes

[EQUATION]

or

[EQUATION]

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

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

[EQUATION]

[EQUATION]

The fact that [FORMULA] is constant on an isobar leads to

[EQUATION]

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

[EQUATION]

Setting

[EQUATION]

G being the gravitational constant, the hydrostatic equilibrium equation writes

[EQUATION]

Thus expressed in terms of the lagrangian variable [FORMULA], 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 [FORMULA]

[EQUATION]

It can also be expressed by

[EQUATION]

[EQUATION]

which finally gives

[EQUATION]

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

[EQUATION]

where

[EQUATION]

Let us note that [FORMULA] is not equal to [FORMULA]. Indeed [FORMULA] is obtained by averaging the density over the volume between two isobars, instead [FORMULA] 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 [FORMULA] and [FORMULA] is equal to

[EQUATION]

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

[EQUATION]

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

[EQUATION]

A.1.4 Radiative equilibrium

Locally, the equation of radiative transfer writes

[EQUATION]

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

[EQUATION]

Integrating over an isobar, one obtains

[EQUATION]

A.1.5 Convective transport

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

[EQUATION]

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

[EQUATION]

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 [FORMULA], one considers as a dependant variable the quantity [FORMULA], the continuity equation for the mass keeps its usual form. We shall consider also a mean temperature [FORMULA] obtained from the equation of state with as input variables [FORMULA], 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:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

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

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

Partial derivatives have replaced total derivatives to allow for the fact that the quantities depend not only on [FORMULA] 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 [FORMULA] is considered as the dependant variable for the density. Moreover the strong horizontal turbulence responsible for the constancy of [FORMULA] 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 [FORMULA] '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 [FORMULA], 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:

[EQUATION]

[EQUATION]

[EQUATION]

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 [FORMULA] 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)

[EQUATION]

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

[EQUATION]

Defining the optical depth by [FORMULA], one obtains

[EQUATION]

which can be transformed into

[EQUATION]

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

[EQUATION]

Since [FORMULA], one obtains

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

Integrating from [FORMULA] to the surface

[EQUATION]

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

[EQUATION]

Thus one can write

[EQUATION]

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

[EQUATION]

Thus

[EQUATION]

Let us define [FORMULA] by the expression [FORMULA]. Then

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

With (A.11) this gives

[EQUATION]

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

[EQUATION]

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

[EQUATION]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
helpdesk.link@springer.de