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Measurements and Modeling of Total Solar Irradiance in X-Class Solar Flares

The Total Irradiance Monitor (TIM) from NASA's SOlar Radiation and Climate Experiment (SORCE) can detect changes in the Total Solar Irradiance (TSI) to a precision of 2 ppm, allowing observations of variations due to the largest X-Class solar ares for the first time. Presented here is a robust algorithm for determining the radiative output in the TIM TSI measurements, in both the impulsive and gradual phases, for the four solar ares presented in Woods et al. (2006), as well as an additional are measured on 2006 December 6. The radiative outputs for both phases of these five ares are then compared to the Vacuum Ultraviolet (VUV) irradiance output from the Flare Irradiance Spectral Model (FISM) in order to derive an empirical relationship between the FISM VUV model and the TIM TSI data output to estimate the TSI radiative output for eight other X-Class ares. This model provides the basis for the bolometric energy estimates for the solar ares analyzed in the Emslie et al. (2012) study.


INTRODUCTION
The Total Solar Irradiance (TSI) is the energy released by the Sun across the entire electromagnetic spectrum. TSI contributions due to solar flares are normally masked by other solar variations, such as acoustic oscillations and granulation, which occur on time scales similar to the initial rapid bolometric enhancement from solar flares. In the last decade, large solar flares have been detected from the TSI signal directly from the Total Irradiance Monitor (Kopp et al. 2005b;Kopp & Lean 2011) and through an epoch superposition technique using the Variability of solar IRradiance and Gravity Oscillations (VIRGO) experiment (Kretzschmar et al. 2010). Solar flares that exhibit measurable visible light enhancements have been deemed "white light flares" (Hudson et al. 1992).
It is possible that these white light emissions contribute large amounts of energy to the overall flare energy budget. Their contribution to the TSI energy budget is still an ongoing investigation. This paper aims to help constrain the variation in the TSI due to flares. Solar flares are the dominant cause of short time scale Vacuum Ultraviolet (VUV, 0.1 -200 nm) emission. VUV wavelengths normally contribute a small portion to the TSI, but can vary by a large factor during solar activity. These emissions are one of the major drivers and energy input source of the Earth's upper atmosphere. The increase in VUV irradiance and charged particles from the Sun can effect Earth's thermosphere, mesosphere and ionosphere (Qian et al. 2011). Radio transmissions can be interrupted and Global Positioning Systems (GPS) can be disturbed. An increase in knowledge of the flare VUV and TSI energy release is desired for studies of comparing various energies in Solar Eruptive Events (SEEs), such as was done in Emslie et al. (2012).
Physically, a solar flare is defined as an event where stored magnetic energy is released via an instantaneous (less than a second) rearrangement of stressed magnetic field lines, presumably magnetic reconnection, that accelerates charged particles both toward the solar suface and outward into space (Hudson 2011). The radiative enhancement stems from an increase in particle flux in the solar atmosphere and can be partitioned into two phases, an impulsive and a gradual phase. The initial flare energy release accelerates charged particles inward towards the more dense plasma by the solar surface. This energy input to the lower solar atmosphere heats the plasma and drives an increase of the hard X-ray flux, via Bremsstrahlung. The heated particles confined within the magnetic loops subsequently radiate thermally in soft X-rays and in some portions of the VUV.
The initial hard X-ray temporal enhancement is called the impulsive phase and the gradual phase is the radiative cooling of the heated plasma in soft X-rays and VUV. The impulsive phase generally lasts for 5 -10 minutes and the gradual phase peaks shortly after the impulsive phase and can last for several hours. This observed relation between the hard and soft X-ray components, deemed the Neupert Effect (Dennis & Zarro 1993;Veronig et al. 2002;Benz 2008;Hudson 2011) originates from the work of Neupert (1968), who noticed that the integral of the centimeter radio flux roughly matched the soft X-ray flux. The acceleration of electrons outward to relativistic speeds generating radio emission is linked to the hard X-ray emission fom the inward penetrating electrons. Investigations of white light flares in spectral lines and continuum bands with multiple satellites demonstrate their temporal and spatial coherence with hard X-ray emission (Martínez Oliveros et al. 2011;Watanabe et al. 2013). This supports the case of white light emission occurring primarily during the impulsive phase.
This study is focused on the radiative aspect of large X-Class solar flares. An X-Class flare is a flare that has a peak irradiance greater than 10 −4 W m −2 in the 0.1 -0.8 nm channel of the Geostationary Operational Enviromental Satelites (GOES) X-Ray Sensor (XRS) (García et al. 1994). In this paper we present results on decomposing flare components into each of the two phases in the VUV and in the TSI for the five flares obsereved by TIM. We also use a VUV empirical model as a proxy for un detected solar flare TSI variations. Our updated algorithm is used to revise values in the study by Emslie et al. (2012). In Section 2 we discuss the data and empirical model used for this study. The time series fitting procedure used to fit the empirical model to the the observed TSI data is and described in Section 3. Section 4 explains how these results are used in estimating the total X-Class solar flare radiative output during TIM eclipse periods. We discuss results and future work in Sections 5 and 6 respectively.

Total Irradiance Monitor
Total Solar Irradiance (TSI) data was obtained from NASA's SOlar Radiation and Climate Experiment (SORCE) mission by the Total Irradiance Monitor (TIM). TIM has been taking solar data since SORCE was launched in January of 2003 (Kopp et al. 2005a).
This instrument sets the two criteria in determining which flares are to be studied for this analysis: 1) flares that have occurred since January of 2003 and 2) flares that have an observed signal response in the TSI. Table 1 lists the five flares that met this criteria.
The SORCE spacecraft is in Low Earth Orbit (LEO), which takes about 90 minutes to complete. This orbit leaves TIM with roughly 45 minute windows of solar observing when not inside of Earth's shadow (non-eclipse periods). The TIM has a time cadence of 50 seconds. The SORCE TIM is the first instrument to have the ability to observe solar flares in the TSI (Kopp et al. 2005b;Kopp & Lean 2011), partly due to this time cadence. The TIM rigorous calibration and metrology of all it's individual components are designed to attain a 100 ppm (0.010%) combined standard uncertainty in the TSI (Kopp et al. 2005a).
This absolute precision gives TIM the sensitivity necessary to observe solar flares. Each TIM data point has an uncertainty of roughly σ T SI = 2.72 mW m −2 , corresponding to 2 ppm (1 ppm = 1,360/10 6 W m −2 ).

Flare Irradiance Spectral Model
The Flare Irradiance Spectral Model (FISM) (Chamberlin et al. 2007(Chamberlin et al. , 2008 This relation holds when the magnetic loop plasma is heated by the influx of accelerated particles and cooling by radiation and conduction are negligible (Benz 2008).

DATA ANALYSIS
The .
Where t start and t end are the start and end times of each flare phase respectively.
The low and high wavlength bounds are λ min and λ max respectively. The irradiance contributions computed here are solely from the flare enhancement and not from the rest of the Sun. After the fluence is obtained, Equation 3 gives the conversion, f , in units of ergs m 2 joule −1 , that must be applied to convert from irradiance units of W m −2 at the Earth to ergs at the sun by multiplying where AU is the astronomical unit in meters. This conversion assumes a uniform angular distribution of the flare energy release. A similar calculation was performed in Woods et al. (2006). This algorithm gives an approximation of the energy release in the VUV wavelengths during each phase of the flare with similar accuracies to those of FISM flare estimates. The VUV radiative energies for all flares in this paper are given in Table 1 and Table 2. The subscript, t, in the definitions refer to time in the following expression. The analytical algorithm solves for the scaling coefficients of the impulsive phase I t , with coefficient a, and gradual phase G t , with coefficient b, of the time profiles exactly by a least squares calculation (Bevington & Robinson 2003;Press et al. 2007) so that the difference between the observed data and model function, Z t is minimized.
The values obtained, by mathematical definition, will be the best coefficients if the errors in the data are gaussian distributed and the errors are relatively small compared to the data value. Our error bars (solely from instrument precision and background subtraction), as displayed in Figure 2 and Figure 3) is applied to obtain the TSI contribution for each flare phase at the Sun. The radiative energies for five flares with an observable TIM TSI signature are in Table 1.

RECONSTRUCTED TSI FLARE ENERGY RELEASE
The radiative output from X-class solar flares have large contributions from the VUV wavelength emissions, ∼ 41% for the gradual phase of center flares, ∼ 23% for the gradual phase for limb flares. We take limb flares to have an East-West location > 70 • . Using the 5 flares seen in the TIM TSI and estimated VUV emission simultaneously obtained from TIMED SEE and GOES respectively, the TSI flare irradiances during TIM eclipse periods can be reconstructed. This method can be used to estimate the position dependent total radiative output from flares, which is described in this section.
The average position dependent fractional energy contribution to the flare from the VUV to the total radiative output is found for each flare. The inverse of these values are used to scale the VUV energy output for flares not observed due to TIM eclipse periods (when TIM is behind the Earth with respect to the Sun), to obtain an estimation of the total radiative output. This model is still limited by low statistics (5 solar flares), but as more large flares occur, more robust algorithms can be developed. The impulsive phase, A Imp , and gradual phase B Grad , scale factors for the time integrated VUV irradiance used to approximate the time integrated TSI flare output are given in Equation 5 for the impulsive phase and Equation 6 for the gradual phase. The total number of limb or center flares is given by n, the index for each individual flare in each category is i, V I is the VUV impulsive phase energy, T I is the TSI impulsive phase energy, V G the VUV gradual phase energy and T G the TSI gradual phase energy.

RESULTS AND CONCLUSION
The TSI energy estimations are the same order of magnitude as previously published values for X-class solar flares (10 31 to 10 32 ergs) and have accuracies consistent to the order given by the uncertainties. TSI radiated energies using the previously described algorithm vary somewhat from the values given the TIM analysis by Woods et al. (2006) by up to a factor of 1.5, where an additional flare on 2006 December 6 is included here that occurred after publication of the aforementioned reference. This is believed to be due to a more robust fitting routine for both the impulsive and gradual phase scaling, but more importantly to the improved background fitting routine that can easily give a factor of 2 in the final results due to the long tail in the decay phase that flares have. The later is most apparent in the 2005 September 7 flare that is dominated by gradual phase emission.
Our results are also consistent within the 1σ uncertainties of the TSI flare contributions derived in the analysis of Kretzschmar et al. (2010) using the PMO and DIRAD radiometers from the Variability of solar IRradiance and Gravity Oscillations (VIRGO) experiment on the SOlar and Heliospheric Observatory (SOHO) mission (Frohlich et al. 1997). As anticipated, the gradual phase contains the bulk of the energy released in solar flares as compared to the impulsive phase (an average of 7.1 times larger), due to the gradual phase lasting much longer. In Table 1 and Table 2 This model can be extended for use on flares lower than X-Class, which was done in an earlier version of this analysis in the investigsation by Emslie et al. (2012). In that study, the reconstructed TSI was computed for the majority of X-class flares of Solar Cycle 23 that were not observed by TIM, due to eclipse periods or flares that occurred before the launch of TIM in 2003. This matured version provides a more accurate estimate of the radiated energy for the energetic comparisons of solar eruptive events. The revised values given here are in general larger by roughly a factor of 2 than those in from the Emslie et al.
(2012) project, due to the modified processes to subtract the highly variable background, also as equally important, the time interval in which the fit is performed and the selection of adequate start and end times for temporal integration of the TSI data.
These newer values are consistent with the conclusions derived in the comprehensive study of Emslie et al. (2012). The bolomertic radiated energy is still over an order of magnitude less than the stored non-potential (free) magnetic energy in the studied active regions. This still agrees with the belief that the available magnetic energy is more than sufficient to power the flare components. The bolometric radiated energy is still less than the Coronal Mass Ejection (CME) kinetic energy. Moreover, the bolometric radiated energy is comparable to the lower limits of the flare-accelerated particles, where the latter could be up to an order of magnitude greater due to the lower energy cut-off being masked by the thermal plasma radiating in soft X-rays. The conclusion persists that there is sufficient energy in accelerated particles to power the radiative energy release. Finally, the correlation between the soft X-ray plasma and bolometric energy is preserved.

DISCUSSION AND FUTURE WORK
A full in-depth knowledge of the energy distribution of solar flares across the entire electromagnetic spectrum is difficult to obtain due to lack of simultaneous observations by the instruments at the time of analysis. These modeled results are the best estimations currently achievable, but are limited to the current TIM TSI data and spectrally to the VUV region. We have broken down the radiative contribution of the VUV (0.1 -190 nm) wavelengths to the total radiative energy release for the largest solar flares, partitioned them into two separate phases and have implemented an empirical model for bolometric flare energy reconstruction. In the future, more steps will be taken to pursue increased knowledge of the spectral energy distribution of flares along with impulsive and gradual phase composition.
Acknowlegments. The authors would like to thank the Laboratory for Atmospheric angood Space Physics (LASP) for their funding and support, as well as support from the NSF REU program.

A. APPENDIX
For completeness, our analytic fitting method is described here. The subscript, t, in the definitions refer to time in the following expressions. The analytical algorithm solves for the scaling coefficients of the impulsive phase I t , with coefficient a, and gradual phase G t , with coefficient b, of the time profiles exactly by a least squares calculation similar to the one in Press et al. (2007). The algorithm is as follows. For a model function, Z t , that is linear in the parameters a and b Z t = aI t + bG t (A1) the coefficients that give the optimal approximation to the actual observed TSI data, D t , with N discrete data points, and propogated uncertainties σ t , can be found by composing a chi-square merit function, differentiating this function with respect to the parameters a and b seperately, setting them equal to zero and then solving these two coupled equations for the coefficients a and b (Equation A2 -Equation A11).
Partial differentiate with respect to a and b, set equal to zero for two seperate equations we use the substitutions, This gives the impulsive and gradual phase scale factors, a and b, respectively.