One of the largest and most complex sunspot groups to transit the Sun occurred in November, 1980 - a white light photo of which is shown above- taken by me using a Celestron 200mm catadioptric telescope. The group was also special in its complex magnetic topology which led to numerous studies including several by me, which featured efforts to arrive at a flare trigger. In this post I describe some of the aspects of the research and also the difficulties encountered. As I pointed out in previously peer-reviewed work (J. Roy. Astron. Soc. Can. Vol. 77, No. 4, p. 203):
"The need for a flare trigger becomes apparent when one appreciates that we often find pre-flare indications, i.e. rapidly growing or rotating spots east-west oriented magnetic inversion line , but no flare occurs. It is as if the safety switch is off, but for some reason the bullet can't be fired."
In the case of the complex spot group shown, it occurred in a very productive active region or AR, designated AR 2776. The key morphological analyses had to trace the existence of a complex loop in the active region, referred to by numerous authors as Loop 'BC'. The estimated positions of the endpoints of this loop are shown in the contour magnetic map shown below, based on a similar one estimated by Marteins (1985).
The key
point to bear in mind is that footpoints B and C form the ends of a primary
loop (‘BC’) whose dimensions, properties
are known. Footpoint A is a possibly emergent footpoint that may or may not have
anything to do with the subsequent motion of B, C. What we seek is a formalism
to describe the net change in magnetic helicity for this system. Ideally this
can also help explain how the region evolved from an isotropic plasma condition
to a flare or pre-flare magnetic configuration during which the Bernstein-mode
hydrodynamic loss-cone instability occurred, as well as beam instability.
As the
geometry indicates, the axis BA transforms to B’A’ while BC transforms to B’C’.
The magnetic inversion line (B çç= 0) is always situated such that B
and C are on opposite sides of it, so B can be regarded as the positive (+)
polarity, and C the negative (-). To
account for this more in conformance with the actual contour images, I looked
at the rotation of the axes (XY ®
X’Y’) and superposed positions B’ and C’ onto the X’Y’ axis, rotating
about respective angles a1 and a2 relative to a fixed A (which is
assumed to change polarities and can therefore be used to arrive at the
magnetic helicity based on interior angle calculations (cf. DeMoulin, Pariat
and Berger, 2006)
Two
de facto flux tubes: BC and B-A-C, (acknowledging A as an emergent flux that
alters polarity) arose from this analysis. To
simplify the treatment, I assessed changes made from the polarities (B+) and
(C-) relative to the fixed flux at (±A). The next diagram resolves this action into the two angles a1 and a2.
a(q )= ò oq [cos q, sin q] / k(q) dq
and:
E = a (2p) - a(0)
with Dq =
q CA - q ¥
Given the error magnitude here was too large, and moreover the model posed the risk of a monopole, I revised the geometry to include a “pseudo-pole” for footpoint A across the magnetic inversion line. The revised model is depicted below
The approach for analysis of the footpoint motions in the model essentially conforms to the prescription proposed by DeMoulin et al, 2006[2] The magnetic helicity (H) is then:
H = 1/ 2p òF òF £ BC, AA’ dFBC dFAA’
The change of the mutual helicity with time relative to the flux tubes chosen is:
d(£ BC, A)/
dt =
dq/ dt [x A- - x B+] + dq/ dt [x A+ - x C-] - dq/ dt [x A+
- x B+] –
dq/ dt [x A- - x C-]
It is important to note that the time integration of the preceding corresponds to the buildup of free magnetic energy in the magnetic configuration. Note that the position of the base footpoints – either B(+) or C(-) with respect to (±A) - can be described by:
x =rA-A, B-C(cos q, sin q).
Following the lead of DeMoulin et al, 2006 (ibid.) in terms of method, one therefore needs a continuous function – without any branch cuts- denoted as a multiple valued function, q m . The integration of the mutual helicity –MFE buildup equation in time then yields:
£ BC, A = { q m A+C- - q m A+B+ + q m A-B+ - q m A-C- }
In terms of interior angles (cf. DeMoulin et al, op. cit.) we have:
£ARCH BC, A = 1/ 2p (a1 + a2)
where:
a1
= a A+ = [q m A+B+ - q m A+C- ]
a2 = a A- = [q m A-C- - q m A-B+ ]
I reiterate here that a1, 2 defines the angular extension of the segments as seen from the emerging flux points ±A[3]. Using the principles of complex numbers, for which – by way of example:
q m A+C- = Im log m (C - - A+) and d log m ( z)/ dz = 1/z
We can also express the result for the mutual helicity of the complex in the revised map:
£ BC, A
= 1/ 2p Im log m [ (A - - B+) (C - - A+) / (B + - A+) (C - - A-)
]
where the bracket makes use of the cross-ratio such that:
K(A,B,C,D) = (A – C) (B – D)/ (A – D) (B – C)
so we obtain:
£ BC, A
= 1/ 2p Im log m K{ A -, B + , C - , A+ }
Using 'rotation' diagrams (e.g. Fig. 2) similar to those from my previously cited paper, e.g.
http://adsabs.harvard.edu/full/1983JRASC..77..203S
I determined a stochastic component emerged with variations in the multiple-valued functions: q m A+B+ , q m A+C- etc. which affected the magnitude of the angles a A+, a A- etc. and hence the magnitude of free energy storage at a given time. The most likely source for the particular energy stochasticism, and hence triggering stochasticism is the emerging flux also inciting mutual polarity intrusion in the region. Thus, the hypothetical trigger should operate in accord with the key condition that (cf. Stahl, 1986a[4]):
¶ / ¶ t {òv B2/2m dV} = 1/m òv div[(v X B) X B] dV - òv {han | Jms |2 }dV
where
the first term on the right hand side embodies the footpoint motions relative
to the photosphere, and the second term, joule heat dissipation in an
anomalously resistive domain at marginal stability. Clearly, only a 10% change
in the magnitudes of q m
A+B+ , q m
A+C- etc can allow for
a significant change in the magnitude of d(£ BC,
A)/ dt and ¶Em/ ¶t.
It is hypothesized here that the conformance of flare incidence to Poisson statistics, i.e.
P(t) = exp (- l) lt / t!
is in large measure determined by the variability in the multiple valued functions, q m. This in turn determines how and when the flare trigger activates. Of course, this variability can arise from either translational motions within the local field, e.g. [x A- - x B+] , [x A+ - x C-] etc. or rotational ones.
In addition, one must be cognizant that the quantitative measurement of a moving magnetic field on a closed surface in the solar atmosphere is extremely difficult (cf. Zhang, 2006[5]). Part of this has to do with the dramatically increased complexity of dealing with moving magnetic field lines.
How specific variations in multiple-valued functions (q m) lead to recognized forms of variation in P(t) and particular triggers will be the topic of a future post.[1] A.L.
MacKinnon, J.C. Brown, and J. Hayward.:1985, Solar Phys. 99, 231.
2] P.DeMoulin,
[3] Note again the revised geometric model for A-A’ assures that the sequence preserves DIV B = 0 so there are no monopoles.
[4] P.A. Stahl: 1986b, Meudon Solar-Terrestrial Predictions Proceedings, (P. Simon, G. Heckman and M.A. Shea, Eds.),
[5] H.Q. Zhang,.: 2006, Chinese Journal of Astronomy and Astrophysics, Vol. 6, p. 6.
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