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

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 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 .

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

Many authors have reported observations on the value of 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 , where B is the field at the radius r and the field at the center, and for the angle between magnetic field lines and the axis of symmetry, at seven values of , where 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.

Fig. 2. a  The ratios as functions of 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 is either the outer radius of the penumbra or the outer radius of the coil. b  The angle 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 was about Wb.

The ohmic power density resulting from the presence of an electric current is . For the reference spot, assuming a uniform J, and mA/m². Here, is orthogonal to . According to one source (Lorrain & Koutchmy, 1993), 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.

© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998
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