Expert: Philip A. Stahl - 3/13/2008

Question
I have read that solar flare eruptions can be compared to disruptions of inductive electrical circuits, with release of the energy 1/2 (LI^2) where L is the inductance and I is the current associated with the flare "circuit". so how can this inductance actually be measured?

Answer
Hello,

First, there is no way that the inductance in a flare region can be measured. What is more accurate to say is that we have indirect methods of *estimating* it in certain conditions.

Second, one must bear in mind that there will be differences in these estimate methods, depending on: a) what particular parameters are available for the estimate, and b) whether we are looking at flare occurence in a single coronal loop or in a pair of them. In the latter case one will have to have estimates for the self-inductances of *each* loop (say L1 and L2) as well as the mutual inductance (M).

In the case of a single loop with a radius r, and representative length, x, subject to the tearing mode instability (which you can google) D.S. Spicer ('Solar Physics', V53, 1977) has proposed an estimated inductance, L:

L =  0.002 x[ ln (2x/r) - 1]

Hannes Alfven, meanwhile, has proposed that in  a "circuit" with double layers (think of capacitor-like surfaces along solar loops) for which there exist potential drops V(D) and and a current changing at the rate dI/dt with some initial current I and an effective "emf" V(E) and resistance R for the plasma, one has an inductance:

L ~  [V(e) - V(D) - RI]/ (dI/dt)

The current in this system always tends to a "saturation" value:

I_s =  (V(E) - V(D)/ R

and provided I < I_s the applicable double layer will explode before I_s is attained. At that point, the solar flare is triggered.

Alfven argues (see e.g. 'Cosmic Plasma', p. 35) that if the "circuit" can be re-established there is no reason the same process can't be repeated and so "repeat events" can be observed.  And indeed, when we look at specific flares regions (such as occurred in the vicinity of Mount Wilson sunspot group #21862 over 5-8 Nov. 1980, we actually see this repeat flaring.

In another crude estimate - based on the characteristic timescale of electric current dissipation (t) A. Kruger et al have argued the inductance can be estimated using:

L ~  R t

where R is the resistance applicable to the solar loop.

In the case of two loops, of course, the estimate procedure is much more complex. Usually, the inductance will only be accessible IF one can also estimate the mutual inductance, and the self-inductances for the loops are roughly equal.

It also means one can estimate the flare energy - say from the 1-8 Angstrom soft x-ray flux records.  One takes the meximum power P(t) as a function of time, and then takes: E = P(t) x t where t is the flare duration from the SXR record.

Assuming L1 = L2 and M is known or well estimated, and I1 and I2 can be estimated for the respective loops,  then L1, L2 = L can be found using:

L = 2[E - MI1 I2]/ (I1^2 + I2^2)

It is well to point out in most realistic cases this is improbable since L1 will not equal L2!

Yet another means of estimating self-inductance for a loop pair involves also using the magnetic helicity - or getting an estimate of that to work out the self-inductance.

De Moulin and Berger (2006) have shown, for example, that one can find the mutual helicity for a pair of loops in mutual rotation around each other. The procedure for obtaining the requisite angles is long and complex (involving complex numbers and "principal values" of arguments, etc.)  so I will not go into it here, but if you are up to it you can read more in their paper (Solar Phys., V233, p. 3)

To make a long story short, assuming that one can obtain an estimate for this mutual magnetic helicity, H(M_ij), and one can also estimate the force-free parameters (alpha_i, alpha_j) for each loop, then one can estimate for the inductance:

L(i,j) ~  (alpha_i + alpha_j)H(M_ij)/ mu_o

where mu_o is the magnetic permeability of free space.

The main point to get from all this is that estimating flare-associated inductances - whether self or mutual or plain old L, is not any kind of exact science and also based on some significant simplifying assumptions that may well break down in the actual flare scenario.

Hope this helps!