Samuel Coradetti

This page summarizes my research during the summer 2002 REU (Research Experience for Undergraduates) program at Montana State University, Bozeman. I'm worked under Dr. Stephane Regnier and Dr. Richard Canfield to develop a method to analyze the complexity of the magnetic potential field above solar active regions. This method, which I call the "dipolarization threshold", was largely a failure. The results it gives do not consistantly agree with qualitative judgements of complexity, and on it's own at least, show no correlation with the probablity of an active region to produce CMEs or flares.


Background

When a flare and its associated CME erupt from the sun, a burst of radiation, followed by a hail of relativistic particles and a cloud plasma propagate out into the solar system. Besides producing pretty aurora when the plasma hits the earth's ionosphere, these geomagnetic storms can destroy satellite circuitry, cause large scale blackouts, disrupt radio communication over large portions of the earth's surface and pose health risks to astronauts. As we become more and more dependent on our satellite fleet and expand human exploration of space, understanding the forces that cause flares and CMEs and learning to accurately predict them is taking on critical importance.

These eruptions originate from areas of intense magnetic fields in the corona above active regions (the area around sunspots). While the process is still not completely understood, it is thought that reconnections of these magnetic fields release the enormous energy for solar eruptions. At several hundred Gauss, these magnetic fields are on the order of a thousand times stronger than the earth's magnetic field. They are so strong that the completely dominate all other forces in the corona. The solar atmosphere is in the plasma state, in which the hydrogen and helium nuclei have been stripped of their electrons. This soup of particles is electrically neutral, but individual nuclei and electrons have electric charge that prevent them from moving perpendicularly to the field lines. So when the field lines move, they pull the plasma along with them. At the site of a magnetic reconnection, there is a rapid shift as contorted field lines snap back like rubber bands, accelerating a giant bubble of plasma and magnetic loops into space at speeds up to 2000 km/s.


Cooling plasma constrained by magnetic field lines.

A simple example of magnetic reconnection. Two field lines of opposing polarity are slowly forced together. When they meet they reconnect and snap back into a lower energy configuration. The upper loop is ejected into space, dragging plasma with it.

The Goal

There are two factors which theorists believe increase the likely hood of active regions to erupt, the twist of the fields and the complexity of their spatial configuration. The former seems to be considered the most important by the most solar physicists at this point. My project, however, focuses on the later. Anyone who spends time watching active regions for solar eruptions will soon note that active regions that seem qualitatively more complex are much more likely to erupt and when they do, the intensity of the eruption seems linked to this complexity. That is, active regions that appear as a clearly separated pair of sunspots will remain relatively quiet, and those that appear as a tangled overlapping mess will likely do something cool.

Quantitatively describing this geometrical complexity has been difficult because there are so many variables to account for. The goal of my project is to describe this complexity by calculating the height (from the photosphere) at which the magnetic field simplifies to approximate a dipolar field. This height, which I call the "dipolar threshold", would provide a rough description that naturally incorporates many aspects of an active region's distribution in a more succinct manner than would be possible with a more direct geometrical analysis.

The Zeeman Effect

To find the dipolar threshold, we need to know something about the magnetic field above active regions. Since we can't exactly fly a Hall probe through the corona, we have to measure it through secondary effects. Field lines can often be deduced by watching plasma flow, but not precisely or reliably. Currently, the best way we have of measuring the solar magnetic field is by taking advantage of a quantum mechanical process known as the "Zeeman effect". Electrons have a magnetic moment determined by their orbital and spin angular momentum. When this moment interacts with an external magnetic field, the energy of an electron orbit is shifted by: (messy constant)(L+2S) dot B. Where L is the orbital angular momentum, S the spin angular momentum and B the external magnetic field. This energy shift changes the energy differences between electron orbits, thus the wavelengths that can be emitted or absorbed by the electron are changed as well. Since this shift depends on the quantum states of the electron, the wavelengths are not uniformly shifted, but split into three components. This spectral splitting can be measured with an interferometer to determine the magnitude of the magnetic field in the line-of-site direction of the instrument.

The direction of the line-of-site field is determined by the circular polarization of the shifted lines. Electrons with angular momentum aligned with the magnetic field will emit and absorb light that is circularly polarized in the same direction. Those with momentum aligned against the magnetic field will emit light that is polarized in the opposite direction. So if the spectral line with the higher frequency (increased energy) is circularly polarized in the positive or counter-clockwise sense (towards the observer), then the magnetic field is pointing towards the observer as well. If it is polarized in the negative or clockwise sense, then it is pointing away from the observer.


<---(absorbtion)--- frequency increasing ---(emition)--->
A schematic of the spectral lines split by the Zeeman effect. Spectral lines are split with the frequency of circularly polarized light increasing when the polarization is aligned with the external magnetic field and decreasing when it is aligned against the magnetic field.

Magnetograms

The Michelson Doppler Imager on board the SOHO spacecraft analyzes the Zeeman effect on the Ni I photospheric absorption line at 6767.8 Å. With this data it produces snapshots of the magnetic field called magnetograms. These magnetograms show only the line-of-site component, which is a reasonable approximation of the vertical field through the photosphere,as long as only areas near the center of the solar disk are considered. To the left is an example of a magnetogram next to an image in visible light from about the same time. The black indicates negative flux, white positive and grey neutral. Notice that the areas of strong magnetic flux are in the vicinity of the sunspots. Sunspots are dark because they are cooler than the surrounding areas. The strong magnetic fields constrain the plasma and keep it from convecting so heat cannot transfer as quickly from deeper layers.

Extrapolating the Magnetic Field

The magnetogram data only describes the vertical component of the field at the photosphere. To find the dipolar threshold, we need information about the fields in the corona high above the photosphere. This must be extrapolated from the magnetograms. This is not a trivial excersize. We know the magnetic fields in the corona follow the equation:


Where alpha is an independent function of x,y and z. Unfortunately, this equation is really hard to solve. So hard, if fact, that it hasn't been solved yet. If we assume that alpha is a constant, however, the solution becomes relatively straitforward. I used a solution devolved by G. Allen Gary (The Astrophysical Journal Supplement Series, February 1989). If we take the components of B in the corona to be given by the inverse Fourier transform of the components of the power spectrum in the corona...

...and we assume that the power spectrum in the corona is related to the power spectrum on the photosphere by...
where

...and the power spectrum on the photoshpere is the Fourier transform of the magnetogram...

...then we get the following equation for the components of B:

This can be plugged into the first equation and solved for the power spectrum components in the corona.
where and
Look closely at that last one. To find alpha we need the x and y components of the power spectrum on the photosphere. To find those, we need the x and y components of B on the photosphere. Uh-oh... we don't have those, do we? No we don't, we only have the z component. So what do we do now? Well, we can simply assume alpha is zero. This isn't as arbitrary as it sounds. The zero alpha solution is the potential field solution, the lowest possible energy configuration given the boundary conditions. Thus, with this solution we can calculate a lower bound estimate of complexity of an active region.

Finding the dipolar threshold

Now that we know how to extrapolate the potential field, we can investigate the evolution of that field at different heights above the photosphere and identify the point at which it becomes dipolar. So when is a magnetic field dipolar enough to be called dipolar? Take a look at the following series of images.

Original Magnetogram
Extrapolated Field ~4500km
~13500km

~22500km

As we look at the field at higher and higher levels, the smaller flux sources are slowly washed out until we are left with two smeared blobs of opposite flux. At which of these qualifies as dipolar is a matter of the criteria applied. The third image appears dipolar, but there is a secondary source of positive flux above the major negative source. Is this source significant? Well... it depends. When it comes down to it, some threshold criteria must simply be chosen and applied consistently to all data that is to be compared. For this project, I set the criteria at 10% of the maximum. That is if there are no secondary flux sources where the field strength exceeds 10% of the maximum strength of the primary sources, then the field qualifies as a dipole. If we apply this criteria to the data above, the fourth image is the dipolar threshold. If the criteria was set at 50%, however, the third height would qualify.

Length Scales

So now we can calculate the dipolar threshold, but before we can make much use of it, we need some length scales to compare it to. Otherwise, comparisons between active regions of different sizes will be meaningless becuase a larger active region will naturally have a higher dipolar threshold. I took two measures of active region size for comparison. The first, the "l" scale, is the differnce between the centroids of positive and negative flux. A centrtiod is a weighted average of position, sort of a "center of mass" for the flux. The second, the "r" scale, is the radius of a circle that includes 90% of the total flux of the active region.

The "l" length scale

The "r" length scale

Results

So what happens when you apply this method? Below is a table of active regions I used my programs on. For each there is a snapshot of the region's MDI data, the h/l and h/r for this snapshot and a link to a plot showing how these values evolve over a 5 day period. There are two groups of active regions I studied. The first group consists of 7 active regions I picked out to give a range of complexity and eruptive activity. Looking at just these 7, the method seems to match fairly well with qualitative judgements of complexity. Notice, however that h/l and h/r seem too low for AR9682. The second group consists of 21 active regions. These were chosen by picking 5 time periods of 10 days and analyzing any active region which produced sunspots and transited the center of the solar disk (30E to 30W) during that peroid. So this group represents a somewhat random selection of active regions - although I did pick periods that included at least some CMEs to insure I had at least some data from CME producing regions. At the time I began analyzing this second group, I thought the method summarized complexity fairly well. I wanted a more random sample to investigate the connection between h/l and h/r and the likelyhood for CME production. However, with the larger sampling of active regions, it became clear that the method has some real problems. The most striking example of the inconsistancy with qualitative judgements is the comparison of AR9733 and AR9737. AR9733 is one of the most complex regions analyzed, yet has among the lower h/l and h/r, while AR9333 is one of the more simple regions anlayzed yet has the highest h/l and h/r of all.

Hand Picked Regions

AR9077 - 7/13/00
h/l = 2.452
h/r = 0.798
Time Series Plot
AR9358 - 2/21/01
h/l = 0.412
h/r = 0.304
Time Series Plot
AR9522 - 7/2/01
h/l = 0.272
h/r = 0.206
Time Series Plot
AR9682 - 10/31/01
h/l = 0.267
h/r = 0.186
Time Series Plot
AR9798 - 1/27/01
h/l = 0.228
h/r = 0.206
Time Series Plot
AR9916 - 4/26/02
h/l = 0.246
h/r = 0.193
Time Series Plot
AR8027 - 4/8/1997
h/l = 0.524
h/r = 0.218
Time Series Plot

Regions From 10 Day Windows

AR8099 - 10/30/1997
h/l = 0.387
h/r = 0.284
Time Series Plot
AR8100 - 11/2/98
h/l = 0.255
h/r = 0.191
Time Series Plot
AR8210 - 5/2/98
h/l = 0.892
h/r = 0.339
Time Series Plot
AR8214 - 5/4/98
h/l = 0.274
h/r = 0.297
Time Series Plot
AR8216 - 5/5/98h/l = 1.342
h/r = 0.481
Time Series Plot
AR8217 - 5/8/98
h/l = 0.519
h/r = 0.269
Time Series Plot
AR9231 - 11/18/00
h/l = 0.440
h/r = 0.240
Time Series Plot
AR9232 - 11/19/00
h/l = 0.412
h/r = 0.297
Time Series Plot
AR9235 - 11/20/00
h/l = 0.493
h/r = 0.312
Time Series Plot
AR9236 - 11/24/00
h/l = 0.148
h/r = 0.120
Time Series Plot
AR9415 - 4/9/01
h/l = 4.684
h/r = 1.319
Time Series Plot
AR9417 - 4/7/01
h/l = 0.226
h/r = 0.158
Time Series Plot
AR9418 - 4/10/01
h/l = 0.382
h/r = 0.315
Time Series Plot
AR9420 - 4/11/01
h/l = 0.673
h/r = 0.440
Time Series Plot
AR9422 - 4/13/01
h/l = 1.905
h/r = 0.498
Time Series Plot
AR9426 - 4/15/01
h/l = 1.897
h/r = 0.570
Time Series Plot
AR9732 - 12/12/01
h/l = 0.229
h/r = 0.206
Time Series Plot
AR9733 - 12/14/01
h/l = 0.257
h/r = 0.223
Time Series Plot
AR9737 - 12/15/01
h/l = 3.707
h/r = 1.227
Time Series Plot
AR9738 - 12/17/01
h/l = 0.412
h/r = 0.344
Time Series Plot
AR9741 - 12/18/01
h/l = 0.503
h/r = 0.368
Time Series Plot

Not surprisingly, these measures that don't effectively measure complexity, also don't have any relavance for CME production. Below are plots of the average h/l and h/r for each active region versus the number of CMEs they produced during the 5 days they were analyzed (it takes about 5 days for an active region to go from 30E to 30W). Theree are no discernable trends. Are regions with higher h/l and h/r more likely to erupt? No.

The relative values of h/l and h/r an not significant either. Below is a histogram of the values of h/l and h/r at the time of CMEs divided by the average h/l and h/r for the active region during the 5 day period. Are regions more likely to erupt when h/l and h/r are higher relative to their average values for that region? No.