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

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5. The magnetic flux tube radius b below -36 Mm

Call z the vertical coordinate normal to the photosphere, with the unit vector [FORMULA] pointing up, and the origin at the photosphere. We now calculate the flux tube radius b as a function of the external pressure [FORMULA], 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.

We saw above that, judging by the S&S data, the electric current density J seems to be about uniform. However, let us be more general and set

[EQUATION]

where [FORMULA] is the azimuthal current density at the periphery. Assume that the exponent n is independent of z. If [FORMULA], then J is uniform.

We have no information on the conductivity [FORMULA], except for the value of [FORMULA] at the surface, as in Sect. 4. If we did know the conductivity at depth, we could estimate the value of [FORMULA] 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 [FORMULA],

[EQUATION]

As with a long solenoid, B outside the flux tube is negligible because the return flux extends over a large region.

The magnetic flux in the tube is

[EQUATION]

If [FORMULA] and n are independent of z, then [FORMULA] is also independent of z.

Call B on the axis of the flux tube [FORMULA], and set the magnetic pressure on the axis equal to a fraction K of the external pressure [FORMULA], where [FORMULA]. We do not assign a value to K for the moment. Then

[EQUATION]

Eliminating [FORMULA] from Eqs. (8) and (9),

[EQUATION]

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 [FORMULA] and [FORMULA] is a fair approximation at this stage.

Also,

[EQUATION]

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 [FORMULA] and [FORMULA], then [FORMULA].

From Eq. (9), for any n,

[EQUATION]

and, from Eq. (6), the azimuthal current per meter of length is

[EQUATION]

So, with increasing pressure [FORMULA], or with increasing depth,

  • a) [FORMULA] increases, [FORMULA], from Eq. (9),

  • b) b decreases, [FORMULA], from Eq. (10),

  • c) [FORMULA] increases, [FORMULA], from Eqs. (8) and (10),

  • d) [FORMULA] increases, [FORMULA], from Eqs. (9) and (13), and

  • e) [FORMULA] decreases, from Fig. 1.

To calculate b as a function of z, we now require the external pressure [FORMULA] as a function of depth below the photosphere. See Bahcall & Ulrich (1988), Bahcall & Pinsonneau (1992), and Hendry (1993).

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

[EQUATION]

The fit is near-perfect but, clearly, the Bahcall-Pinsonneau table cannot be extrapolated upward to the photosphere, where [FORMULA]. 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 [FORMULA] Wb, [FORMULA], and [FORMULA], for the range [FORMULA] Mm to [FORMULA] 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, [FORMULA].

[FIGURE] Fig. 3. Magnetic flux tube radius b as a function of depth for the Stanchfield et al. (1997) sunspot. The z-axis points up. Over this range, the flux tube radius decreases slowly with depth. That sunspot had a radius of 3 Mm at the level of the photosphere.

According to the Bahcall & Pinsonneau (1992) table, the pressure at a depth of 50 Mm is [FORMULA] Pa. Then, at that depth, if [FORMULA] and [FORMULA], from Fig. 4 and from Eqs. (10), (9), (12), and (13), [FORMULA] km, [FORMULA] T, [FORMULA] kA/m2, and [FORMULA] kA/m.

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© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998
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