Astron. Astrophys. 356, 301-307 (2000)

1. Introduction

The extrapolation of solar magnetic fields from measurements of the line-of-sight component using the force-free assumption

or equivalently

holds fairly well in the corona, where the plasma is much smaller than unity. Note that modern magnetographs (THEMIS: Mein & Rayrole 1985, Rayrole 1992; LEST: Stenflo 1985) will provide all of the three components of , at least at one level in the photosphere, and therefore from the set of Eqs. (2) written as

will enable us to determine only from the vertical component of the density current vector (3d) if (1) holds , but not from (3a) or (3b) since the vertical derivatives and cannot be easily estimated from observations. On the opposite, system (3) can be used to extrapolate to the lower corona (Wu et al. 1985, 1990; Cuperman et al. 1989, 1990), but lots of difficulties arise because of the ill-posedness of the problem (Amari et al. 1997). Other methods, existing or presently under development, are based on iterations or MHD codes (Amari & Démoulin 1992; Amari et al. 1997; Démoulin et al. 1997; Mc Clymont et al. 1997) and they are expected to produce stable solutions.

Nevertheless, at the photosphere holds, and the force-free balance (1a) must be replaced by the magnetostatic balance (4a) on the horizontal

as far as inertial terms of acceleration Jm and advection Jm are negligible with respect to the pressure force Jm and the magnetic force Jm^-3), estimated on the basis of typical values ms^-1; T; m; s; Pa; kgm^-3). This is a starting assumption, since we know that velocity and magnetic field are well correlated in sunspots (Berton 1986). Moreover, there is serious observational evidence that magnetic forces are important up to 400 km above the photosphere and become negligible beyond (Metcalf et al. 1995).

If horizontal pressure gradients are by some means measured, one may think to solve Eqs. (4) with standard boundary conditions ( known at the photosphere). In a first step brightness fluctuations I currently observed across sunspots may be related to horizontal temperature gradients through the blackbody assumption of radiation transfer

where h and k denote respectively Planck and Boltzmann constants, and the absolute temperature T is connected to the thermal pressure p by the perfect gas law

where the density is assumed to vary only with height ( m²s^-2K^-1 in the photosphere). Actually, the relative brightness is measured at the photospheric level

with respect to the continuum value far from the line centre.

With this background, the relationship between I, p and T will be discussed in Sect. 2, the calculation algorithms will be explained with demonstrations of the well-posedness in Sect. 3, and prospects will be proposed in the concluding Sect. 4.

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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