5. The magnetic flux tube radius b below -36 Mm
Call z the vertical coordinate normal to the photosphere, with the unit vector pointing up, and the origin at the photosphere. We now calculate the flux tube radius b as a function of the external pressure , and then use the Standard Solar Model (Bahcall & Pinsonneau 1992) to deduce the flux tube radius b as a function of the depth z, where z is negative.
where is the azimuthal current density at the periphery. Assume that the exponent n is independent of z. If , then J is uniform.
We have no information on the conductivity , except for the value of at the surface, as in Sect. 4. If we did know the conductivity at depth, we could estimate the value of from the values of J and of B.
Below -36 Mm, b varies slowly with z and we can use the analysis of Lorrain & Salingaros (1993). At the radius ,
As with a long solenoid, B outside the flux tube is negligible because the return flux extends over a large region.
If and n are independent of z, then is also independent of z.
The magnetic flux tube radius is not very sensitive to the value of either n or K: if n increases from 0 to 5, b decreases by a factor of 2/3 while, if K decreases from 1 to 0.1, b increases by a factor of 1.8. In view of the coarseness of our model, setting and is a fair approximation at this stage.
With the z-axis pointing up as above, the last derivative is negative, and the flux tube radius b increases with increasing z, because U, defined in Eq. (10), is positive.
For the Stanchfield et al. (1997) sunspot, if and , then .
So, with increasing pressure , or with increasing depth,
The Bahcall & Pinsonneau (1992) table extends from the center of the Sun up to -36 Mm, but the range -100 Mm to -36 Mm is sufficient for our purposes. A third-order regression analysis for this range yields
The fit is near-perfect but, clearly, the Bahcall-Pinsonneau table cannot be extrapolated upward to the photosphere, where . The table given in Bahcall & Ulrich (1988) does not extend above -56 Mm.
Fig. 3 shows the magnetic flux tube radius b for the Stanchfield et al. (1997) sunspot as a function of z, using Eq. (10) with Wb, , and , for the range Mm to Mm. Note that the radius scale covers about 50 km, while the depth scale covers 60 Mm. So b varies slowly with z at these depths. Indeed, at -50 Mm, .
According to the Bahcall & Pinsonneau (1992) table, the pressure at a depth of 50 Mm is Pa. Then, at that depth, if and , from Fig. 4 and from Eqs. (10), (9), (12), and (13), km, T, kA/m2, and kA/m.
© European Southern Observatory (ESO) 1998
Online publication: October 21, 1998