The aim of my project is to anlyse the output from Professor Longcope's PREFT program and adapt the flux tube parameters in order to best match the observational data corresponding to the X-ray loop top source from flar 2004 February 26th. This should give us insight into the validity of magnetic reconnection and subsequent tube retraction in explaining the energy release and emission as observed by RHESSI.
My initial efforts have been strongly based in understanding the theoretical concepts underpinning the PREFT model (see background theory). This has involved several meetings with Dana and lots of reading. The relevant papers are:
The below animation demonstrates the general features of flux tube retraction as modelled by PREFT. The tube is accelerated downwards by the tension forces and as centrally orientated plasma flows are generated, a density and temperature spike forms at the loop top.
Having grasped a working knowledge of the theory, my initial task was to calculate the retraction velocity of the tube bends - or the so called rotational discontinuies. The RD shock position was determined by calculating where the velocity gradient was steepest. The length of the tube to this point was calculated using the summation of pythagorean length elements. Then the change in length was dvided by the time step to deduce the velocity. The results are shown below:
Note the velocity remains approxmately of the order of the Alfevn speed as should be expected of a shock propagting via the restoring force of magnetic tension.
In order to further develop the visualisation options for flare analysis I have developed a widget which can take user input and scroll temporally through evolution. This helps quickly identify features and allows for easy usability.
Having familiarised myself with the nature of the physical evolution and the means through which to access the data through structures contained within save files, I will now move on to running some PREFT simulations of my own.
A solar flare will likely be powered by several retracting loops all with slightly different loop parameters. In the PREFT model the first step is initialising these parameters:
The 171 Å TRACE observations of coronal loops are able to constrain the final contracted loop lengths and the Magnetic field strength. This is demonstrated in the figures below taken from Longcope, Qiu annd Brewer 2016.
The other variables can be varied in order to omptimise fitting of the synthesised light curve to the observed RHESSI emission spectra. This has previously been done for an intermediate length single loop. My first task was to run additional simulations where the initial and final loop lengths are scaled linearly.
The simulations can be quite lengthy so this took some time. I was able to achieve 4 different parameter runs which cover the range of final lengths indicated by the figure above. The other parameters were held constant:
Analysis:
Here we see that as the length increases the apex temperature rises higher. However the overall profile has a similar shape - each peaking at approximately 15 seconds. As the loop lengths are scaled linearly the overall retraction length also scales linearly - hence there is more magnetic energy released and we should expect the a greater temperature increase. However, why should the heating effects have approximately the same temporal profile?
This can be explained via the initialisation of the flux tube parameters in accordance with the Rosner, Tucker and Vaiana (RTV) equilibrium model [6]. This imposes energy equilibrium during the initialisation and uses scaling laws to constrain the tube. See a simplistic discussion of such Coronal Loop Equilibria. The main result is the scaling law: $$T_{apex}^{3}\propto P L$$ Thus as we increase the length for a given apex temperature the pressure will decrease. This in turn means that the denisty will decrease through the ideal gas equation of state. Finally the Alfven velocity must increase as the density decreases but the field strength remains the same: $$v_{A}=\frac{B}{\sqrt{\mu_{0}\rho}}$$ Hence as L increases the propagation speed of the rotational discontinuities increases also. Thus the points at which the flux tube is straightened occurs at roughly the same time despite different lengths of retraction - this explains the coincident kinks in the temperature evolution above. This does not change the physics of the loop overly hence we decided to run some more simulations such that the initial temperature is varied instead with all other quantities fixed.
I have spent some time developing my PREFT widget to allow for the visualisation of other physical quantities. This required effective management of widget event structures and has certainly challenged me to improve my IDL capabilities. This should allow for quick and user friendly evaluation of PREFT data runs in the future.
Over the weekend I also ran some more simulations varying the initial apex temperature of the loop. This will allow us to examine a different physical regime. Now the alfven speeds change whilst the loop retraction lengths remain unchanged - thus the timescales will vary and should exhibit different heating effects. Some initial insight can be achieved by plotting the apex temperature vs time for each of the runs as shown below:
Here we see a somewhat surprising trend as the lower initial temperatures result in a larger maximal temperature and a quicker time evolution of the tubes (as indicated by the tube straightening kink position). Referring back to the RTV setup as before, now the temperature is raised whilst the length remains fixed. This implies that pressure and density increase with temperature which in turn relates to a lower alfven speed. Thus cooler tubes intially retract quicker and hence accelerate the plasma flows to greater velocities. These faster flows result in a more energetic collision at the loop top - raising temperature higher.
To further explore these runs, Dana suggested we move on to synthesising the photon emission spectra. This is done by loading in the PREFT tarr data into a program called rh_spec_thist.pro which performs several theoretical steps [6].
With a working understanding of the spectral synthesis process I have set about devloping a useful widget to help visualise the light curves and compare them for the different temperature runs - I have termed this tool spectralVIEW!
This allows me to simultaneously compare the light curves of the 5 temperature runs over each energy bin. I can also switch to plotting the energy-intensity spectra and scroll through for all times in the flare. The save button allows me to save any images I desire.
The light curves maintain the same temporal profile across all energies since the flaring process is modelled with the convolution of only one loop type. Hence each energy bin simply reflects the rise and fall in flaring as the reconnection rate varies.
The sharp fall off in the low temperature spectra indicates a softer x-ray emission when the initial apex temperature is reduced. This is interesting as we previously noted that the initially low T apex tubes actually achieve a higher maximal apex temperature. However, this does not seem to translate into inreased emission. Again this comes down to denisty. Let us examine the density evolution of the loop centre with time.
Indeed emission is not soley determined by temperature but also the mass denisty - as more particles are confined to a smaller region the collisional mean free path is decreased as free-free interactions occur more often. Since the density of the high temperature loops is greater the $n_{e}^2$ term will dominate in the intensity integrals. Note that the difference beween initial apex density and maximum apex density increases also with the temperature increase.
Now that I am familiar with the methods of spectral generation and the output plots I am ready to begin comparing the PREFT synthesised spectra with the RHESSI observational results. I have extracted the 2004 data and plotted as shown below:
Here we see the RHESSI spectrum in white and the PREFT spectrum in red. Note that between the dashed orange lines marking 10-30keV, the PREFT run parameters have been adapted to achieve an optimal fit. Below 10keV emission features are important - notably the FeXXVI emission line at approx 6.9keV. Furthermore above 30keV the photon count becomes too small and begins to become dominated by noise. Thus we limit our range of interest within the interval where Bremsstrahlung radiation dominates. This allows us to fairly compare with the PREFT model which synthesises the spectra based on such free-free interactions.
As we see in the paper the PREFT model departs from such a good fit at earlier and later times of the flare:
Thus I need to combine both trial and error and logical guessing to determine how to achieve a better fit at these times by changing the PREFT run parameters. Having accumulated a wide selection of runs already I first write a script to help compare the fits to the RHESSI spectra at different times. This script, rh_compare performs several tasks:
Implementing this procedure allowed for quick and insightful analysis of my initial runs. I have initially looked at 3 times:
By comparing the length varying runs and the temperature varying runs I will try to see if there are consistent trends in the spectral response. Then I can try adjust run parameters accordingly to provide a better fit.
Results:
I have chosen to adopt the RMSD_log as my primary error measure. This places equal emphasis on factor deviations from the observed spectra as opposed to absoulte deviations in which the low energy dominates due to the logarithmic nature of the plot. This means a lower RMSD_log corresponds to a better fit visually on the log plot.
In recreating the optimal fit from Longcope 2016 [6] at time 2:00:50 we find an RMSD_log of 0.0648595. This acts as a threshold level for suitably good fits which we will try and achieve at other times.
Temperature Varying Runs:
Here we see that although neither curve fits the RHESSI data satisfactorily we do see a clear response as the PREFT plots pivot with temperature. As T increases the spectra becomes softer and steeper. This is again explained with the RTV initialisation. We previously saw that higher initial temperatures evolve slower and reach reduced maximal apex temperatures. Hence there is reduced emission in the high energy regime in accordance with brehmsstrahlung theory. Such a pivot action can be exploited to customise future runs.
Length Varying Runs:
In this case all other parameters are fixed and only the final and initial loop lengths are scaled. Again this plot doesn't display a perfect fit, but suggests a clear trend as larger loop retractions exhibit increased emission across the whole spectra. The factor enhancement is somewhat greater in the hard regime hence reducing the steepness of the curve.
We do however see a good fit for the time 1:58:50 with the following run. This has a RMSD_log below that of the reference run and hence is suitable for convolution into the flare evolution.
Scaling Law:
Discussing these results with Dana he explained that through some useful approximations we can in fact derive a comprehensive scaling law. This wraps up all the free run parameters and relates them to the maximal apex temperature achieved - thus theoretically justifying the hardening or softening of the spectra.
Longcope showed that a retracting fluxtube releases magnetic energy/Mx at a rate:
$$\dot{W}_{M}=\frac{B}{4 \pi}\frac{dL}{dt}=\frac{B}{4\pi}4v_{a}\sin{(\Delta\theta/4)}^{2}$$We are only interested in the energy which is thermalised (via viscous forces) at the loop top gas dynamic shock. Thus we consider the kinetic energy of horizontal motion - exracted using the fraction $v_{x}^{2}/v^{2}$. Therefore the flare energy flux (multiply through by the magnetic field):
$$F=\frac{B^{2}}{\pi}\sin{(\Delta\theta/4)}^{2}v_{a}=\frac{B^{2}}{\pi}\sin{(\Delta\theta/4)}\frac{B}{\sqrt{\mu_{0} \rho}}$$Assuming an approximately steady state energy density within the loop top, then the energy flux must be balanced by thermal conduction to the foot points.
$$\frac{F}{L}=\frac{\partial}{\partial l}\left(\kappa \frac{\partial T}{\partial l}\right)$$Now we insert characteristic quantities and use the classical Spitzer-Harm form for the conduction coefficient - this should be valid when we reach an approximate equilibria whereby the thermal conduction fronts have reached the footpoints.
$$\frac{B^{3}}{\pi L\sqrt{\mu_{0}\rho}}\sin\left(\Delta\theta/4\right)^4 = \frac{\kappa_{0}T^{7/2}}{L^{2}}$$Now inserting the RTV relationship $T_{\mathsf{0}}^{2}=\rho L_{0}$, cancelling terms, rearranging and dropping constants we find the scaling law:
$$T_{\mathsf{apex\_max}} \propto L_{f}^{2/7}L_{0}^{1/7}B_{0}^{6/7}T_{0}^{-2/7}\sin(\Delta\theta/4)^{8/7}$$This is a powerful relationship which suggests how the loop evolution may respond to a change of parameters. Note it again expresses why a larger initial apex temperature leads to a smaller final apex temperature as we have previously found and explained - unsurprising since this scaling law incorporates the RTV result. Furthermore it suggests that as the final relaxed length increases the maximal temperature increases and hence the spectrum should be harder. This pleasingly agrees with our findings above also.
This scaling law seems to be a useful guide for adjusting our run parameters and predicting the repsonse. I will next try different tube bends $\theta$ and test the response. The scaling law motivates us to hypothesise that a greater bend will produce a higher maximal apex temperature and a harder spectrum -we will see in the coming week!
I started the week by attempting to verify the scaling law temperature dependence. With many different temperature runs, I was able to plot the maximal apex temperature vs the initial apex temperature. See below:
The predicted -2/7 law, when fitted using the leftmost point, deviates considerably for larger initialisation T. However we do note that a -4/7 power law does fit much better.
To begin the week I wanted to perform a check on my runs which I have been meaning to get round to! Indeed, I want to check that each PREFT run has been continued long enough such that the tube is sufficiently relaxed and emission has dropped back to background. If this were not the case then the individual tube bremsstrahlung intensity being convoluted with the magnetic reconnection rate would not be a true representation of the emission profile - we would be missing out on the end chunk of emission!
In order to investigate this I examined a particular lengthy PREFT run and synthesised several spectra curtailing the tarr file at different times before passing to 'tube_brem_band'. I could then examine at what point the tail of the single tube light curve [10-30keV bin] had a negligible effect on the final convolved spectra.
Below we have a plot of spectra produced corresponding to a flare time of 2hr 0min 50s. Each colour is produced by convolving different lengths of the initial tarr file as indicated by the dashed lines in the lower plot.
The above figure shows that at high energies the tail of the single tube light curve seems to have little effect irrespective of the temporal extension of the tarr file. However, we do notice the spectra is underestimated for low energies if we perform runs for a shorter time. A useful measure of relaxation is the ratio of the final emission to the peak emission over the light curve. From the above figure we reason that once this reaches 0.04 then there is little appreciable change for the spectra - the tube has sufficiently relaxed and any further tail emission will not be important.
This threshold was then incorporated into my 'rh_compare' program and each PREFT run was re-examined to check if it met this condition. 3 runs lay just outside this relaxation threshold so the tarr files were extended further via 'flare_cont.pro'. Once this is done, the spectral synthesis can be more accurately constructed.
The runs which initially failed this test corresponded to larger loops or lower initial apex temperatures. Longer retractions mean it takes more time to relax to equilibrium - loops take longer to snap into their final stable configuration. As we have found, low initial apex T leads to higher apex T. This higher tempaerature may persist for longer before relaxing to the background. Hence care should be taken in future to allow for longer runs in such parameter regimes.
Once this revision to my code was implemented I went about a variety of theta runs and examined the spectral response. This threw up some surprising results. As $\Delta\theta$ was initially decreased and the tube became less bent, the spectra became softer as expected - the tube retracts slower and less heating should be present. However, after a certain point the spectra suddenly became harder and the maximal temperatures achieved at the loop top shot up. See the figure below:
This trend initially puzzled me. Clearly something in the model is producing an articfial enhancement of the temperature if the angle parameter is set to around 103° or less. Use of my visualisation widget allowed me to reexamine the physics of the situation and see what went wrong. Essentially the tube is never able to quite relax to the final length given these smaller input angles. It gets close but then begins to bend and overshoots the footpoints in an unphysical manner! The flaw is comparable to defining a triangle with two sides and an angle but a baseline which is too small.
This motivated me to geometrically re-examine the situation to help derive suitable length and angle parameter ranges for the future. For a given bend $\Delta\theta$, a final relaxed loop length $L_{f}$ and a height of the retracted loop $h$ we find the initial loop length should be set to:
$$L_{0} = \frac{2 h}{\sin(\Delta\theta/2)}+\frac{L_{f}-2h/\sin(\Delta\theta/2)}{\cos(\Delta\theta/2)}$$By introducing the free variable $h$, we constrain the final length - simply recasting the problem in terms of different independent variables. Maximal retraction is achieved for $h=0$ and minimal retraction for $h=L_{f}\sin(\Delta\theta)$ which implies the initial loop length is equal to the final loop length, so no retraction. Estimates for h are suggested by the LFFF extrapolation loop heights found in the Longcope 2016 [6] and are used to inform our runs.
With this improved understanding of the relationship between angles and lengths I now begin to combine the varaition of both properties to try and achieve a softer spectrum.
I have tried many different runs of late. However, the relationship between lengths and angle discussed above restricts the flexibility of these paramteters. Indeed as the angle used decreases, the difference in length also decreases by the above formula (for a given h). Thus creating a softer spectrum also drops the entire spectrum too low. Thus the counterproductive combination keeps us away from achieving the desired spectral profile.
I will continue to investigate different runs to try and find an effective compromise. However, in the mean time I have suggested a couple of alternative routes to achieve a softer fit.
Simultaneous Retraction of Different Loops:
I suggest the idea of combing two different loops. The spectra are simply combined using a weighted sum to produce a resultant curve. By looping through weightings we can find the balance which minimises the RMSD_log error and hence provides a better fit. Using this idea some preliminary results do show we can get closer fits:
Clearly this is a much improved fit. We are still off the desired error value but the above plot demonstrates how future runs can be combined to achieve an optimal fit. However, we should note that in the above plot the initial pre reconnection apex temperature is raised beyond what thought initially feasible. This leads us onto the next route for further exploration.
Enhanced Initial Loop Temperatures:
Despite flares capable of reaching around 50MK post retraction, in the pre-retraction loop top the temperatures are typically believed to be limited to around 2-3MK. Theoretically, since thermal conduction can only occur along field lines there can be no heating of the separate pre-reconnection loop tops. However, we see that if we ignore this and raise the temperature, we do achieve the softer spectra for later times of the flare. Possible mechanisms could be enhanced radiative heating of the loop tops from already reconnected loops below? This requires more thought but as common practice in physics - our models and observations should inform us as to the theory.
The initial apex temperature is boosted to 13MK. Due to the RTV initialisation, the progress of loop retraction is much slower in this case. Furthermore, the increased densities lead to higher pressures at the chromospeheric footpoints which can trigger a 'firehose' instability at later times. Care is taken to curtail the run before this yet still achieve sufficient relaxation of the flux tube.
The result is a spectral curve which has a very pleasing steep profile indicating softer emission. However the curve has been translated upwards. Thus motivating a run with a similar temperature but reduced angle and length of retraction. Nonetheless we observe a highly improved fit when averaged with another high temperature run as indicated by the yellow dashed line. This encourages us to explore such abnormally high temperature runs further.
Week 6 was a shorter working week due to the long Independence Weekend break. Coming back on the Wednesday, I was able to examine some of the runs I have been fine tuning to get a good single loop fit for the later/softer times of the flare. This was achieved and the results are shown below:
Note the visual fit is very good with the RMSE_log measure - only increased by the unavoidable noise at the highest energies. This again relies on us exploiting very high temperatures and densities in the loop initialisation. Indeed this is somewhat unexpected since pre-flare conditions are typically only 2-3MK. Thus 11.5MK is controversially high!
However, models, theory and observation should all complement one another and hence such a result should make us go back and re-examine all aspects. Does there exist some mechanism whereby the looptop density and temperature is enhanced over the impulsive phase of the flare? Since thermal conduction only occurs along field lines it seems difficult to drive such enhancement over separate loops but this question should be examined further.
Having settled the time extremes of the flare, I now turn my attention to try finding good fits at 2 more intermediate times. I was ableto find a good fit for time 01:59:50 by reexamining my previous data runs as shown below:
This is suitably below the error threshold and can be taken forward immediately to the final convolution. However, as per the historical record of this project, the later time of 02:03:30 has proved more difficult to find. I have performed many runs and am slowly honing in on a suitable fit. My latest attempts are able to achieve a good match at the energy limits but curves slightly above the RHESSI data at intermediate energies:
I am hopeful that in the coming week I can find an improved fit for this time.
The main drive of this week was preparation for the Midterm Presentations. Each member of the Solar REU program is given 15 minutes to share with the other students and academics the progress we have made. This was a valuable opportunity to practice our presentation skills and with many wise words of advice from Prof Priest, I have certainly learnt a lot from this experience.
It was fascinating to hear about the diverse range of projects. There is some incredibly interesting work being done by all students. It was also neat to see how such different projects all link together to form a larger solar story - transitioning from active regions, to energetic events such as solar flares and CMEs and then onto effects at the Earth and the observational instrumentation being developed.
For those interested, my presentation can be viewed here: Midterm Presentation
Over the weekend I have continued to experiment with runs to achieve a better fit at the time 02:03:30. This had led to a much better comparison which is suitable for the staggered convolution process.
Combined with the other 4 times we now have a suitable collection of tube retractions to form the staggered convolution. The first step is to distribute the flux transfer rate between each of the five loop retractions. This is done using a simple multiplicative triangular switch on-off structure which always sums to one. See below:
Multiplication through by these flux share profiles gives the $\dot{\phi}_{i}(t)$ corresponding to each loop type. This is shown below:
We now simply perform the RHESSI synthesis for each tube but take the different $\dot{\phi}_i(t)$ as the input flux transfer rates. The resulting spectral responses are summed for all times to produces the net emission behaviour. This can be compared with the observed RHESSI data across all time as we have done before and we can see if such a multi-convolution has a good fit. The following movie allows us to appreciate the results:
Multi Convolution Movie 1: Download 10/07/2017
The upper plot compares the RHESSI spectra with the PREFT simulation spectra - the same comparison I have been performing over the last few weeks. The middle plot shows the light curve for a particular energy bin made up of the contributions from each separate tube type. The bottom plot compares the RMSD_log error for the multiple convolution method (red) against the single loop method (black). The single loop compared against in black is the loop corresponding to the middle time indicated by the yellow dotted line. Note the comparison is curtailed at 7500s. Beyond this time there are gaps in the RHESSI data which prevent a well defined error measure being calculated.
Overall we see that for the middle impulsive phase we do achieve a very good fit throughout the flare using the multiple loop convolution method. This has a particular advantage over the single loop for later times where the RMSD_log errors diverge and the red line is much more favorable. For further improvement we should also look just before the peak of the flare - around 7000s. Furthermore note the slight red hump around 7300s coincides with a gap in the loop time being fitted. Another loop here could help combat this and achieve yet a better result. Both these times should be investigated further and incorporated into the convolution.
Another route of analysis is now being investigated - comparison with the GOES x-ray channels. These are in fact US weather satellites - Geostationary Operational Environmental Satellites - which are equipped with various instrumentation to perform solar meaurements as a side project. We are interested in the X-ray sensor which measures the net flux over two passbands - 0.5 to 4Å and 1 to 8Å. Similar to RHESSI spectral synthesis we can synthesise the emission over these two GOES channels and compare with the observational data. Using just a single loop (specifically that chosen to fit at time 02:00:50) we see the following fit:
Here we see the fit does roughly resemble the two channels but there is a large error. Instead we hope that the multiple loop method will be more effective. Again by summing the GOES signals from a multi convolution method we get the net result which compares favorably over the impulsive phase of the flare. See the video below:
Multi Convolution Movie 2: Download 11/07/2017
Here we see that the GOES data is a very good fit across both channels. The black solid line represents the observational 1-8Å whist the correpsonding black dashed line is the synthesised data. Similarly for the red lines in the higher energy- shorter wavelength passband. Also notice how the largest deviations in the GOES synthesis from observation coincide with the largest differences in the RHESSI errors. This again emphaises the importance of finding a better RHESSI fit at the earlier and intermediate-late times of the flare.
Although I don't have perfect fits at these other times I have made a preliminary analysis to see if the idea is promising. Note that the $\dot{\phi}_{2}(t)$ flux share is shifted to an ealier centred time in order to reduce the effect of the $\dot{\phi}_{1}(t)$ contribution quickly. Why do I do this? The small initial temperatures of this new earliest loop $\rightarrow$ high maximal temperatures and a longer time to decay to negligible emission. Hence, even when these loops are no longer being triggered their effect lingers on. Thus I minimise this by ensuring they only dominate the flux transfer rate for a very short time at the beginning. The results of this 7 loop convolution are shown in the video below:
Multi Convolution Movie 3: Download 13/07/2017
Here we see that the fit is drastically improved at earlier times and also around 7300s where we have also inserted another loop. This lends strong support to the multi-convolution method!
Now that I have had some notable success with these runs it is worth examining if there are trends in the parameter variations as we progress through the time evolution of the flare. Below are plotted the parameters for each loop used against the time that loop was found to be an effective fit.
Time (s) | Run Index | L0 (Mm) | L0_f (Mm) | $\Delta L$ (Mm) | T (MK) | $\Delta\theta$ (°) |
---|---|---|---|---|---|---|
6900 | pre2 | 110 | 80 | 30 | 7 | 115 |
7130 | run3 | 74 | 50 | 24 | 7.5 | 110 |
7190 | runt1 | 67 | 45 | 22 | 8 | 110 |
7250 | run1 | 67 | 45 | 22 | 7.5 | 110 |
7310 | frun20 | 65 | 45 | 20 | 10.5 | 109 |
7410 | frun20 | 65 | 45 | 20 | 10.5 | 109 |
7550 | frun13 | 67 | 47 | 20 | 11.5 | 102 |
We observe three parameter trends. As time moves forwards the collapsing loops seem to be retracting less, of higher temperature (and thus density) and the bend decreases. The increasing temperature again suggests the possibility of a density enhancement mechanism at the loop top with time. The decreasing length of retraction and initial bend is consistent with the geometry of a growing post flare arcade where the later loops retract less from the diffusion region to the top of the forming arcade. See the figure below taken from Forbes & Acton (1996):
However, to be truly consistent with this model I should also try make final lengths smaller for earlier times as these form the lower level arcade loops. The initial lengths should then correspond to similar retraction $\Delta L$ in order to achieve good spectral fits as before. Trying to achieve these runs is the next task to tackle over the remainder of the project.
Length:
I have started the week trying to find PREFT runs which follow the same trends but with adjusted final lengths to match the basic idea that as time goes on the final loop length should increase. Indeed loops retract onto the ever growing post flare arcade. The following figure illustrates a possible simple 2D non-erupting flare setup.
As time moves forward the post flare arcade grows and the current sheet shrinks. Thus, assuming the typical point of reconnection moves upwards with the centre of this current sheet and the top of the equilibrium loop settles at the bottom, then overall there will be a decrease in retraction length in accordance with my parameter findings. Such a growing trend in later loops is supported by the observed final loop lengths corresponding to the 2004 flare event:
Here we have the plane of sight (POS) loop lengths. Thus the actual final lengths accounting for projection effects are roughly 50% larger. Hence the straight line fit gradient when scaled implies the final lengths grow at roughly 30Mm/hr. Over the impulsive phase (~1:55-2:10) of the flare this only corresponds to an increase of 7.5Mm. From my efforts of spectral fitting such changes themselves will not massively affect the spectral results. Indeed the overall change in length will be more important in this case. I believe that adjustment of the other parameters with such a final length variation would still allow us to achieve a good fit.
However as we see from the plot there is a large distribution in loop lengths present at all times hence the idea of time ordered lengths should be scrutinised further. Indeed this plot images the loops in 171 Â when they have cooled to approximately 1MK. Hence the time they appear on the diagram is actually after they have originally formed, retracted and cooled into the bandpass, which can vary depending on parameter choices. Furthermore, as the flare ribbons grow and the flare spatially evolves, triggering neighbouring reconnection events, then shorter loops can again form over the new free ends of the arcade. All of this information needs to be disentangled to fully undertsand the length variation with time. Ultimately mutliple loops lengths will be retracting at any time so a full model of loop formation and arcade growth would need to be developed to get a complete picture. For the purposes of this initial proof-of-concept study a simple length ordering is sufficient. Although PREFT runs have proved difficult this week with a variety of crashes slowing me down I have managed a rough demonstration that the lengths can be ordered temporally and give suitable fits. The following parameters were used in producing the run below:
Time (s) | Run Index | L0 (Mm) | L0_f (Mm) | $\Delta L$ (Mm) | T (MK) | $\Delta\theta$ (°) |
---|---|---|---|---|---|---|
6900 | time1_2 | 80 | 50 | 30 | 6.5 | 115 |
7130 | run3 | 74 | 50 | 24 | 7.5 | 110 |
7190 | time3_2 | 77 | 55 | 22 | 8.3 | 110 |
7250 | time4_2 | 79 | 57 | 22 | 8.5 | 110 |
7310 | time5 | 80 | 60 | 20 | 10.5 | 109 |
7410 | time5 | 80 | 60 | 20 | 10.5 | 109 |
7550 | time7 | 85 | 65 | 20 | 12 | 102 |
Multi Convolution Movie 4: Download 19/07/2017
Here we see the fit is again improved in comparison to the single loop run. With more work to achieve better runs at the different times and inclusion of more intermediate time loops this can be improved. Note the difficulty in fitting the earlier times is in fact due to non-thermal effects which is indicated by the observed straight line power-law behaviour on the log-log plot. Since PREFT is not trying to emulate these effects, we should not worry so much about the fit at this time.
Angle:
The shear angle $\Delta\theta$ in question needs to be defined carefully with respect to the figure above. It is the angle between the field lines either side of the current sheet as seen looking onto the current sheet plane from the side. Hence not necessarily related to the footpoint shear or the ribbon shear. See the figure below:
As the flare evolves and the current sheet retracts to a null point the magnetic field strength should decrease. Thus if we assume that the guide field component strength remains constant this implies the vertical field must decrease. This has the effect of shallowing the tube bend and decreasing $\Delta\theta$ in accordance with my findings.
All these discussions emphasise that such parameter trends are compatible with theoretical models. However, the flexibility of models and complexity of such events does not say this has to be the case. Hence these trends are interesting but by no means a general statement of flare behaviour. Thus, the multi convolution flare loop method is not overly restricted in terms of lengths and $\Delta\theta$ parameter choice. However I do believe that the increasing temperature initialisation is fundamnetal to achieving the desired fits. I suggest this strongly points towards a firm theoretical mechanism which facilitates a clear density enhancement.
This week I have tied up the project and started formalising my findings in a formal report. This should allow my work to be carried forward in the future and also be incorporated into a publication. This report can be downloaded below.
Formal Report: Download 28/07/2017
This final week has allowed me to finish collating my results and prepare the handover material which should provide enough information to replicate my work in the future. After editing the final report, I have also prepared a simple text file which documents many of the relevant programs I used/wrote and how they work together within the programming framework of my project.
Programming Index: Download 01/08/2017
To complement this file I have also done a screen capture tutorial where I talk through the rough programming structure of the project and highlight useful procedures.
Programming Tutorial: Download 01/08/2017
The project was formally brought to a conclusion on Wednesday with our final presentations. This was a fantastic opportunity to hear all the amazing work performed by the REU students and I'm proud with what we have all achieved this summer. My final presentation is available for download below:
Final Presentation: Download 02/08/2017
All that remains is for me to thank the REU program and the various staff who have contributed to making this an incredible summer. Many thanks in particular to my supervisor Prof Dana Longcope for his continued mentorship and guidance throughout the project. I had an amazing time working with him and he has truly instilled within me an enthusiasm for solar physics which I look forward to exploring further as I undertake my disseration in 3D magnetic reconnection this coming year.
Ⓒ Callum Fairbairn 2017 |