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Astron. Astrophys. 355, 365-374 (2000)

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2. Theoretical approach

In this work, we consider the theory of a solar gravitation figure to include the effects of differential rotation. The oblateness can be defined as the difference between the equatorial and polar radii, usually expressed in milliarcsecond, or as a dimensionless coefficient defined by

[EQUATION]

where [FORMULA] and [FORMULA] are the equatorial and the polar radius, respectively. This oblateness is linked to the quadrupole moment, that is to say to the potential describing the distribution of mass and velocity inside the Sun. This potential is deduced from the basic figure for steady rotation around a fixed axis (forming the zero-order approximation), with small corrections that can be successively added. Among several possible definitions of the solar surface, one of the best is to define it by a surface of constant gravitational potential, which permits one to treat the problem with a very high accuracy. This gravitational potential is usually developed into spherical harmonics, i.e. the Legendre polynomials [FORMULA]. The even terms are only kept as the figure shows a symmetry created by the rotation around the minor axis. To this gravitational potential must be added the potential due to the rotation of the Sun, so that the total potential [FORMULA] is:

[EQUATION]

where G is the gravitational constant, [FORMULA] the equatorial radius, [FORMULA] the solar mass, r the solar radial vector (taken here as a variable), [FORMULA] the Legendre polynomials, [FORMULA] the latitude, [FORMULA] the rotation and [FORMULA] the coefficients associated with the dynamical form of the Sun. According to the magnitude of the [FORMULA] coefficients, the distortion from a pure sphere is more or less large. Due to the monotonic decreasing function of the distribution of mass in the solar case, the contribution of the first coefficient is the most important and consequently signs the oblateness. To describe this oblateness, we express the ellipsoid shape in Cartesian coordinates:

[EQUATION]

Using the oblateness definition given by Eq. [FORMULA] and introducing the latitude [FORMULA], the following expression is obtained:

[EQUATION]

Developing Expression 4 up to the third order:

[EQUATION]

Expression 5 is solved to obtain r, and using the binomial expansion, we find:

[EQUATION]

Eqs. 2, 5 and 6 can be combined to express the total potential UT limited to the second order as:

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are functions of the oblateness and the latitude, and [FORMULA]. At the equilibrium on the surface, the coefficients of J2 and J4 must vanish, so that, and after some algebra (see for instance Cole, 1978):

[EQUATION]

[EQUATION]

Assuming that the Sun is symmetric around its axis of rotation and introducing spherical polar coordinates [FORMULA], the hydrostatic equation inside the Sun takes the well known form (see also Roxburgh, 1964, Paternó, 1996 and Pijpers, 1998):

[EQUATION]

[EQUATION]

where P is the pressure, [FORMULA] the density, P[FORMULA] the second Legendre polynomial, [FORMULA] the colatitude, and [FORMULA] the gravitational potential satisfying the Poisson equation:

[EQUATION]

By cross-differentiation of Eqs. 10 and 11, P is eliminated; taking into account Eq. 12, the following dimensionless differential equation is obtained:

[EQUATION]

[EQUATION]

where the quantities [FORMULA] and y [FORMULA] have been introduced dimensionless, as well as [FORMULA], [FORMULA] and [FORMULA]. The quantity [FORMULA] is the reference rotation rate, which is taken equal to the rotation rate of the radiative zone (for our purpose, this rotation rate is taken equal to 435 nHz). The second member of this differential equation contains the term of the rotation. Two approaches can be made. The first one consists in trying to give an analytical form of the solution y, and this have been made by Pijpers (1998). The oblateness is deduced from an inverse problem when the kernels of the integral relation are known. The other approach consists of solving Eq. 13 on successive shells of thickness ([FORMULA]), taking into account at each step the boundary conditions. This method allows us to compute the succesive ellipticities of a stratificated Sun. The helioseismic constraints will play an important role in the adopted model of density and in the internal and differential rotation laws. This last method permits one to have access the local distortion of each small volume element in the Sun.

The differential Eq. 13 can be easily solved to obtain [FORMULA] if two boundary conditions are known. The first one is obvious, as the perturbation must vanish at the center. The second one (bearing in mind that [FORMULA] varies as [FORMULA]) is given by the continuity of [FORMULA] and [FORMULA] when crossing the surface of the Sun. These two conditions give:

[EQUATION]

which can be rewritten in the dimensionless form:

[EQUATION]

Since the centrifugal force is a first order term, the density in the last term in Eq. 10 can be replaced by the zero-order spherically symmetric density. This term contains only zero and second Legendre coefficients. This leads us to expand the gravitational potential [FORMULA] in the form:

[EQUATION]

where [FORMULA] is a first order term. As outside the Sun, the gravitational potential satisfies the Laplace equation:

[EQUATION]

The identification of the same order terms in Eqs. 16 and 17 leads, whatever [FORMULA] is, to:

[EQUATION]

hence,

[EQUATION]

or

[EQUATION]

The heterogeneous composition of the internal layers of the Sun are here represented as a series of successive thin shells, the composition of each shell being homogeneous. The oblateness [FORMULA] can be computed for each of these shells assuming that the matter which is outside a shell of a given radius [FORMULA], is in the outer space and satisfies Eqs. 14.

From Eq. 8 limited to the first order, the oblateness at the distance [FORMULA] is expressed for a shell as:

[EQUATION]

Hence, according to the potential,

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are respectively the radius, the mass and the rotation rate of a shell of thickness [FORMULA]. Note, that when integrating [FORMULA] over the whole radius of the Sun, this would lead to a global figure of the Sun which will change with the latitude. This is one of the main features of this paper: the helioid (just as the geoid, but obviously at a very lower degree) is not perfectly ellipsoidal.

A mass and density model for the Sun is required to solve the differential Eq. 13. We have chosen one of the five solar models of Richard et al. (1996) - Model 3 - including the helioseismological constraints. This model is computed with element segregation and with Grevesse values as initial abundances, iterated so that the final abundances are also those given by Grevesse (1991). Moreover, the element segregation, introduced in Model 3, has shown that it fits very well the seismic data.

As the solar rotation depends both on the radial distance and on the colatitude, we need to define a law to express [FORMULA]. Among several laws available, we have chosen that proposed by (Kosovichev, 1996b), which is based on the observed p-mode rotational frequency splittings. The law is represented in terms of associated Legendre functions of order 1, [FORMULA]:

[EQUATION]

where

[EQUATION]

and Ak(r) is a radial function developed in a parametric form. From the analysis of the Big Bear Solar Observatory data, Kosovichev formulates a simple model of solar rotation based on the first three terms of expansion 23 ([FORMULA]):

[EQUATION]

From a numerical point of view, with the value of Ak given in nHz, Eq. 25 is expressed by:

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

This model is particularly valued owing to its analytical form describing fairly well the differential rotation of the convective zone and the rigid rotation of the radiative zone, and not depending on the density [FORMULA]. For lack of data concerning the g -modes, it is not yet possible to know if the rotation in the core is faster or slower than the rotation of the radiative zone. Thus, we adopt the same rotation rate for the core and for the radiation zone: [FORMULA] nHz.

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© European Southern Observatory (ESO) 2000

Online publication: March 17, 2000
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