A&A 377, 688-690 (2001)

DOI: 10.1051/0004-6361:20011104

helioseismic estimates of the internal structure and rotation

**I. W. Roxburgh**

Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

LESIA, Observatoire de Paris, Place Jules Janssen, 92155 Meudon, France

Received 12 July 2001 / Accepted 31 July 2001

**Abstract**

We determine the gravitational multipole moments
,
of the sun using a model of the interior structure and of
solar rotation obtained from helioseismic inversions. The differential
rotation of
the convective zone and the underlying transition zone make only a small ()
contribution to the quadrupole moment *J*_{2} which
is found to have a value
.

**Key words: **rotation - helioseismology

In the case of axial symmetry the external gravitational potential of the sun,
,
can be expressed in the form

(1) |

where the

The equations governing the equilibrium of a rotating star
(neglecting any circulation currents) are

(2) |

where

Since the ratio of centrifugal force to gravity is small we solve Eqs. (2) by perturbation analysis writing

(3) |

where , with the values in the unperturbed, non rotating, state and the perturbations.

Eliminating the pressure by taking the curl(rot) of the first of Eqs. (2), and retaining only terms of first order in the perturbations gives

(4) |

Eliminating gives the equation for the perturbation

(5) |

To solve this equation we expand in terms of Legendre Polynomials

(6) |

and express the right hand side of Eq. (5) in terms of associated Legendre polynomials

(7) |

where the

(8) |

These are subject to the boundary conditions that the are regular at

(9) |

Given from the unperturbed model, these equations can be solved to determine the , and hence the multipole moments .

If
is a function of radius only, Eq. (8) reduces to that given in Roxburgh (1964b):

(10) |

It is convenient to introduce the dimensionless variables

(11) |

in terms of which Eq. (8) becomes

(12) |

where the

(13) |

Many authors have determined the internal hydrostatic structure of the sun
by inversion techniques using the measured values of the solar p-mode oscillation frequencies; here we use the inverted model determined by Marchenkov et al. (2000) using the full non-linear sucessive Born approximation inversion technique. Likewise several authors have
determined
the solar diferential rotation from the measured rotational splitting of the frequencies; here we use the results of Kosovichev (1998) which are given in
the convenient form:

(14) |

where in units of nHz and

(15) |

where

The coefficients *a*_{2n}(*x*) were determined using this model by evaluating
and hence the left hand side of Eq. (13) on
,
and then inverting to give the
*a*_{2n}(*x*), *n*=1,5. The resulting equations for the *y*_{2n} were then solved subject to the appropriate boundary conditions. The external multipole moments are given by
where
is the solar surface.
Two solar models were used in the calculations:
**bsun**: the inverted model of Marchenkov et al. (2000).
**ssun**: the solar model S of Christensen-Dalsgaard et al. (1996) determined by using a stellar evolution code including diffusion, but with
polynomial smoothing in the very central core (
).
The results are given in Table 1.

For comparison the value of *J*_{2} for uniform rotation was also determined, taking the constant value to be 435 nHz - the value given by Kosovichev for
the solar interior. This gave the values

bsun | |

ssun |

The results for the two models are in good agreement - as was to be expected since there are no major differences between the inverted model and model S (the main differences being in the layers just below the convective zone and in the inner core).
The value of *J*_{2} for non-uniform rotation (
)
is close to the value obtained by Pijpers (1998) (
)
using a seismically determined rotation profile
,
and also in close agreement with the value obtained by Paterno et al. (1996) (
).
They are considerably larger than the value of
found in earlier work (Roxburgh 1964); this is to be expected since the simple model
of the sun used in 1964 was much more centrally condensed that current models determined either by inversion of the observed solar oscillation frequencies, or from current evolutionary models. The value obtained is considerably larger than the value of
obtained by
Godier & Rozelot (1999).

The inclusion of the differential rotation makes only a small difference ()
to the value of *J*_{2}. This is as expected. In the convective envelope and transition layer below the envelope, the departure of the rotation from its uniform value is nowhere large, the departure of the spherically averaged rotation is even smaller, and the mass and inertia of the envelope are small. We do not find the large variation by a factor of 3 in *J*_{2}
reported in Godier & Rozelot (1999).

The value of *J*_{4} obtained here
is comparable with the values obtained by Ulrich and Hawkins (1981) using
a simple theoretical model in which the rotation was assumed constant on cylinders within the convective zone and uniform in the radiative core.

We note that the value of *J*_{2} determined here is sufficiently small so as
not to play any significant role in the advance of the perihelion of Mecury,
therefore supporting the prediction of the General Theory of Relativity.

- Christensen-Dalsgaard, J., Dappen, W., Antia, H. M., et al. 1996, Science, 272, 1296 In the text
- Godier, S., & Rozelot, J.-P. 1999, A&A, 350, 310 In the text NASA ADS
- Kosovichev, A. G. 1998, in Sounding Solar and Stellar Interiors, IAU Symp. 181, Poster Volume, ed. J. Provost, & F.-X. Schmider, Obs. Côte d'Azur, 97 In the text
- Marchenkov, K. I., Roxburgh, I. W., & Vorontsov, S. V. 2000, MNRAS, 312, 39 In the text NASA ADS
- Paterno, L., Sofia, S., & Di Mauro, M. 1996, A&A, 314, 190 In the text
- Pijpers, F. P. 1998, MNRAS, 297, 76 In the text
- Roxburgh, I. W. 1964a, MNRAS, 128, 157 In the text NASA ADS
- Roxburgh, I. W. 1964b, Icarus, 3, 92 In the text NASA ADS
- Ulrich, R. K., & Hawkins, G. W. 1981, ApJ, 249, 831 In the text NASA ADS

Copyright ESO 2001