RHESSI/GOES Xray Analysis using Multitemperature plus Power law Spectra. J.McTiernan (SSL/UCB). ABSTRACT:
RHESSI/GOES Xray Analysis using Multitemperature plus Power law Spectra.
We present spectral fits for RHESSI and GOES solar flare data that include both a Differential Emission Measure for thermal emission and a power law fit for nonthermal emission. This improvement over the traditional isothermal approximation for thermal flare emission is intended to help to resolve the ambiguity in the range where the thermal and nonthermal components may have similar photon fluxes. This "crossover" range can be anywhere from 10 to 30 keV for medium to large solar flares. It is also expected that the low energy cutoff of the nonthermal electron spectrum lurks in this energy range, or below, but is obscured by thermal emission.
In this work we demonstrate the fitting process using simulated data, and then apply the process (initially presented at the 2007 SPD/AAS meeting for two solar flares and simulated data) to a large sample of flares. Our results indicate that it is often difficult to distinguish between thermal and nonthermal emission in a single spectrum, more so than in the isothermal approximation. This creates large uncertainties on the calculation of quantities such as the energy in the thermal plasma, the low energy cutoff of the nonthermal spectrum and the energy in nonthermal electrons. This research is supported by NASA contract NAS5-98033 and NASA grant NNX08AJ18G.
INTRODUCTION: This work has two goals: 1) to demonstrate the ability to obtain a differential emission measure (DEM) for solar flares in the temperature range above 3 MK using RHESSI and GOES data, and 2) to use it to help in quantitative estimates for the low-energy cutoff of nonthermal emission.
RHESSI and GOES are used because of the availability of the data for a large number of flares. In previous work, we have used this combination of instruments to get DEM estimates for solar active region emission (McTiernan, 2008, submitted to Apj). Solar flare emission often has a substantial nonthermal component to the emission, which can extend out to γ-ray energies. This contaminates a DEM calculation, typically a large fraction of the DEM piles up at the highest temperature allowed in the calculation. To avoid this problem here we estimate the nonthermal emission from the hard Xray spectrum and allow for the presence of nonthermal emission at lower energy when obtaining the DEM.
For this calculation, the DEM is estimated by an arbitrary N-element power law in the temperature range from 3 to 60 MK. The fitting procedure for the DEM is as follows: First a single power law is fit to the whole temperature range. Next, this range is split into two bands and fit again. A reduced χ2 is calculated,
which is the standard χ2 divided by the number of data points minus the number of fit parameters. Next the range is split into 3 bands and fit again. If the value of reduced χ2 decreases, then this fit is retained and a 4 element power law is tried. This process of adding power law components is repeated until a minimum value for the reduced χ2 is found.
TEMPERATURE RESPONSES: Fig. 1 shows temperature responses for the two GOES channels and selected RHESSI energies. GOES is included in the calculation to obtain the low energy part of the DEM estimate; RHESSI does not have much response to temperatures less than about 10 MK. For an Xray spectrometer, the response is a monotonically increasing function of temperature. This can cause difficulties in DEM calculations.
TEST DEM CALCULATIONS: In order to test the method of fitting the DEM, we create a trial DEM function and integrate over the response function. The resulting data is input into the DEM calculation and the output is compared to the initial function.
Tests are shown in Fig. 2. The red line is the test DEM, and the black line is the calculated DEM. The DEM is fit using a simulated annealing technique. Error bars are calculated via a Monte Carlo process in which the count rates are varied, depending on their uncertainties, for multiple trials. The final uncertainty is obtained from the standard deviation of the fit values for the different trials. (If the error bar runs to the bottom of the plot, the uncertainty is larger than the DEM itself, so the DEM for those regions is consistent with zero.) Uncertainties for RHESSI count rates are obtained using Poisson statistics. Uncertainties in the GOES data are estimated to be 10%. (Garcia, (1994 Solar Physics, 154, 275) estimates GOES uncertainties to be 16% in the long wavelength channel and 14% in the short wavelength channel.)
The first test, (a), is a sanity check, a power law in T. The second test, (b), is a half-gaussian, with a width of 3 MK. The fit is good below about 15 MK, but the error bars get large when the DEM is small. The third test, (c), adds a gaussian component at 20 MK. Here the fit is better for the high T part. The fourth test, (d), combines a power law with a narrow spike at 20 MK, for a test of the temperature resolution. It looks at if the temperature resolution is a few MK at 20 MK. Test (e) combines the narrow spike with the half-gaussian function. The final test, (f), only has the source at 20 MK. The fitting procedure fits this well, but puts some emission measure at low T (albeit with very large error bars).
From the tests we conclude that:
1) Power Laws are easy
2) Gaussians are harder, but fittable.
3) Narrow spikes will have finite width.
4) High T features are fit better.
TESTS OF 2-TEMPERATURE MODEL: For test (g) the input trial DEM is a two-temperature model with sharp spikes at 8 and 20 MK. The power-law model does not do too bad a job in recovering this distribution as long as the two spikes are more than a few MK apart. This is illustrated by test (h), for which the spikes are only separated by 5 MK. Here they are not resolved.
USING REAL DATA: Fig.4 is a plot of a typical RHESSI spectrum as seen in a large solar flare. The emission above 50 keV is fit to a power law spectrum. Below 50 keV the emission is assumed to be a sum of thermal and nonthermal components.
The model for nonthermal emission is very simple. The nonthermal spectrum is fit above 50 keV and extended down to a cutoff energy. Below this energy the spectral index is set to -1.4, which would be the spectrum for a sharp cutoff in the electron spectrum. We hope to put constraints on the cutoff value by seeing which combination of DEM plus cutoff fits the data best.
The photon counts expected from the nonthermal component are subtracted from the total observed photon counts at low energy and the remainder is fit to a DEM.
FLARE SELECTION: Since we require that there be emission above 50 keV, relatively large flares are needed. A sample of 85 flares was identified from the RHESSI flare list. Flares were chosen which had enough emission in the 50 to 100 keV energy range to obtain a good power law fit for a 12 second time interval near the flare peak. Time intervals with data gaps, attenuator state changes or particle events were rejected. RHESSI detectors 1, 3, 4, 6 and 9 were used for the RHESSI spectra, and these were accumulated for the energy range from 6 to 100 keV. Background subtraction was done using the automated procedure used for RHESSI Quicklook spectra, which accesses a database of background spectra accumulated over the whole mission and calculates the background based on the spacecraft position and time.
ANALYSIS: The first step is to fit the spectrum above 50 keV to a power law. Then the DEM calculation is done with the power law cutoff set to 50 keV. Next, the nonthermal spectrum is extended down to 49 keV, then 48 keV, etc…. For each value of the cutoff energy, we then have a DEM model. The “best” cutoff energy is determined by finding the minimum value for the reduced χ2 . In Fig.5 we have plotted the best-fit DEM (for which the cutoff is 26 keV) for a flare which occurred on 20-feb-2002, the first flare in the sample. This is typical for small flares with relatively low GOES temperatures. The DEM is a decreasing function of T.
The bottom panel shows the variation of the reduced χ2 as a function of low energy cutoff. The minimum in the curve is fairly broad, and in the range between 22 and 39 keV the value of χ2 is within 3σ of the minimum value. (For this case, the minimum value of χ2 is 0.4. The uncertainty σ, calculated by varying count rates in a Monte Carlo process, is 0.3.)
20-Feb-2002 11:06:16 to 11:06:28, C7.5 flare. The blue line in the top panel is the best-fit DEM, the red lines show the range of uncertainty. The dashed line on the bottom panel shows the range of “good” cutoff energies.
Here are 20 selected DEMs. The DEM functions do not all look alike. For flares with relatively small GOES size and T, the DEM shown in Fig.5 is typical. For big flares, and flares with high GOES T, above 20 MK there is often a peak in the DEM.
Cutoffs?The range of acceptable cutoff energies varies from flare to flare also. Here the range is plotted versus the ratio of nonthermal emission to thermal emission in the short wavelength GOES channel. There is a weak correlation, which suggests that time intervals with more nonthermal emission relative to thermal emission give more reliable results. This is more likely to be the case in the HXR rise phase of flares, rather than the peak, and this will be investigated in future work.