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Astron. Astrophys. 339, 610-614 (1998)

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4. The radial distribution of the azimuthal current density in the flux tube

Weiss (1990), Jahn (1989), and others proposed that there exists a sharp transition between the magnetic flux tube and the external plasma. Hirayama (1992) proposed a thin current sheet at the periphery of magnetic elements, and Solanki & Schmidt (1993), Jahn (1989), and Pizzo (1990) also proposed thin current sheets for sunspots. If J, and hence [FORMULA] in Eq. (5), are significant only near the periphery of the tube then, a short distance inside the flux tube, the plasma flows down along the lines of [FORMULA].

On the contrary, observations seem to indicate that J is uniform, as shown below.

Many authors have reported observations on the value of [FORMULA] as a function of the radius in the umbra and penumbra. Solanki & Schmidt (1993) compiled observations by Beckers & Schröter (1969), Kawakami (1983), Lites & Skumanich (1990), Solanki et al. (1992), and Wittman (1971). See also Koutchmy & Adjabshirizadeh (1981), Abjabshirizadeh & Koutchmy (1983), Arena et al. (1990), Title et al. (1993), Howard (1996), Sutterlin et al. (1996), Abramov-Maksimov et al. (1996), Keppens & Martinez-Pillet (1996), and Stanchfield et al. (1997).

One would hope to gain information on a sunspot's magnetic flux tube by comparing the field of a sunspot with that of coils of various geometries.

We averaged the Solanki & Schmidt (1993) curves for the ratio [FORMULA], where B is the field at the radius r and [FORMULA] the field at the center, and for the angle [FORMULA] between magnetic field lines and the axis of symmetry, at seven values of [FORMULA], where [FORMULA] is the outer radius of the penumbra. See also Solanki (1990).

We then compared with the corresponding variables for six coils of circular cross-section, including a single loop and a cone with its larger radius on top. The outer boundary of the penumbra corresponds to the outer radius of the coil.

The field that agrees best with the S&S data is the one in a plane at the end of a coil of zero inside radius, whose length is 20 times its radius. Figs. 2a and b show plots of the fields.

[FIGURE] Fig. 2. a  The ratios [FORMULA] as functions of [FORMULA] for the sunspot data compiled by Solanki & Schmidt (1993) (broken line) and for a coil of zero inside radius and whose length is equal to 20 times its radius (smooth line). The radius [FORMULA] is either the outer radius of the penumbra or the outer radius of the coil. b  The angle [FORMULA] between a magnetic field line and the axis of symmetry, for the same data. According to both sets of curves, J is somewhat larger than average near the axis.

Our reference sunspot is the one observed by Stanchfield et al. (1997). Its radius was 3 Mm, and B at the center was 0.25 T. Using an average B of 0.125 T, the magnetic flux [FORMULA] was about [FORMULA] Wb.

The ohmic power density resulting from the presence of an electric current is [FORMULA]. For the reference spot, assuming a uniform J, [FORMULA] and [FORMULA] mA/m2. Here, [FORMULA] is orthogonal to [FORMULA]. According to one source (Lorrain & Koutchmy, 1993), [FORMULA] S/m in the photosphere. Then the ohmic power density at the surface is only about 0.1 W/m3, which is less than would be required to make the umbra bright, by many orders of magnitude.

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© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998