Nonuniform Target Ionization and Fitting Thick Target Electron Injection Spectra to RHESSI Data

Past analyses of flare hard X-ray (HXR) spectra have largely ignored the effect of nonuniform ionization along the electron paths in the thick-target model, though it is very significant for well-resolved spectra. The inverse problem (photon spectrum to electron injection spectrum F ₀(E ₀)) is disturbingly non-unique. However, we show that it is relatively simple to allow for the effect in forward fitting of parametric models of F ₀(E ₀)) and provide an expression to evaluate it for the usual single power-law form of F ₀(E ₀)).The expression involves the column depth N _* of the transition region in the flare loop as one of the parameters so data fitting can enable derivation of N _* (and its evaporative evolution) as part of the fitting procedure. The fit to RHESSI data on four flares for a single power law F ₀(E ₀)) is much improved when ionization structure is included compared to when the usual fully ionized approximation is used. This removes the need, in these events at least, to invoke broken power laws, or other forms, of the acceleration spectrum F ₀(E ₀)) to explain the observed photon spectrum     

The effect of wave generation on HXR bremsstrahlung spectra from flare thick-target beams

Fast electrons in the solar atmosphere are detected by their hard X-ray bremsstrahlung and by type III radio bursts caused by ‘bump-on-tail’ plasma wave generation. This paper investigates empirically the effect of wave generation on the HXR spectrum. Purely collisional propagation of an electron beam generates a bump in the distribution function, due to stopping of low-velocity electrons. The consequent positive gradient means there is a possibility of wave generation, production of type III radio bursts, and energy redistribution of the electron beam. We have represented this relaxation parametrically and calculated the global bremsstrahlung HXR emission spectrum. We show that for a range of relaxed forms, with different local electron spectral shapes, the bremsstrahlung spectrum integrated over the whole target is identical in shape to the purely collisionally evolved beam. Our results show that spatially integrated HXR spectral measurements would be unable to distinguish between the presence or absence of relaxation effects. Only spatially resolved hard X-ray spectra, such as anticipated from the HESSI mission, will be able to remove this ambiguity in HXR diagnostics of beam relaxation.     

Joint Effects of Compton Backscattering and Low-Energy Cutoff on the Flattening of Solar Hard X-Ray Spectra at Lower Energies

Theoretical calculation has shown that the spectrum of the Compton backscattering component in solar hard X-ray flares has a peak around 30 keV for a primary power-law source. Thus the superposition of the Compton backscattering component could cause a photon spectrum received at the Earth to be flattened below the peak energy and steeper above the peak energy. On the other hand, because a thick-target bremsstrahlung photon with a given energy E only could be produced by a nonthermal electron with an energy larger than E, thus if a power-law electron spectrum is cutoff below an energy E _c, then the produced photon spectrum will become flattened below E _c. In this work we present a calculation of the joint effects of the Compton backscattering and the low-energy cutoff on the spectral characteristics of the received solar hard X-ray in the energy range 10–100 keV. The results show that the flattening caused purely by the Compton backscattering could be comparable with that by the low-energy cutoff for hard spectra. So, it is obvious that the joint effects of the low-energy cutoff and the Compton backscattering could result in the received photon spectra to be much more flattened at lower energies. On the other hand, compared to the primary photon spectrum, the received photon spectral index will increase about 0.15 due to the Compton backscattering at higher energy, which seems independent of the primary spectral index.     

Determination of Electron Flux Spectra in a Solar Flare with an Augmented Regularization Method: Application to Rhessi Data

Kontar et al. (2004) have shown how to recover mean source electron spectra in solar flares through a physical constraint regularization analysis of the bremsstrahlung photon spectra I() that they produce. They emphasize the use of non-square inversion techniques, and preconditioning combined with physical properties of the spectra to achieve the most meaningful solution to the problem. Higher-order regularization techniques may be used to generate forms with certain desirable properties (e.g., higher-order derivatives). They further note that such analysis may be used to infer properties of the electron energy spectra at energies well above the maximum photon energy observed. In this paper we apply these techniques to data from a solar flare observed by RHESSI on 26 February, 2002. Results using different orders of regularization are presented and compared for various time intervals. Clear evidence is presented for a change in the value of the high-energy cutoff in the mean source electron spectrum with time. We also show how the construction of the injected electron spectrum F₀(E₀) (assuming that Coulomb collisions in a cold target dominate the electron transport) is facilitated by the use of higher-order regularization methods.     

Hard X-rays from solar flares — time characteristics in correlation with energy and angular distributions of electrons

Fine time variation of hard X-rays has been explained in terms of a spread in the angle of incidence of the source electrons in non-thermal thick-target model for bremsstrahlung generation. The electron energy and angular distributions have been calculated by combining small angle scatterings using analytical treatment with a large angle collision using Monte Carlo calculations as a function of column density. The incidence angles of electrons are taken as 0, 30, and 60°. Using the Bethe-Heitler cross section and the above calculated electron distributions, the bremsstrahlung flux for different photon energies as a function of column density has been studied. The computed X-ray pulse as a function of column density has been converted into time profile. It corresponds well with the observed fine time structure. The calculated spectra of X-rays at the peak and valley are also consistent with the observations. The variation of photon flux with time has also been computed for photon energies 20, 50, and 100 keV for 90 and 180° observation angles together with the changes in spectral shapes of photon energy spectrum at different times for 90 and 180° observation angles.     

Study of hard X-ray characteristics of solar flares - support for non-thermal processes

The spatial and angular distributions and also the energy spectrum of hard X-rays from solar flares have been studied in terms of the energy and angular distributions of the accelerated electron beam. The incident electron distributions as functions of column density have been computed by combining the analytical treatment of small-angle scattering with the Monte-Carlo calculations for large angle scattering. To start with monoenergetic electrons at 0°, 30°, and 60° incidence angles have been taken. Using the Bethe-Heitler total cross section and the Sauter differential cross section along with the calculated electron distributions, the bremsstrahlung flux and its angular distribution for different photon energies > 10 keV have been studied as function of column density. The shape of the calculated curves agrees with the observations of PVO/ISEE-3 lending support to the beamed thick-target model for X-ray generation with continuous injection.Physics Department, Vishwa Bharti Institution, Rainawari, Srinagar, India.     

Beam-driven return current instability and anomalous plasma heating in solar flares

We consider the problem of ion-acoustic wave generation, and resultant anomalous Joule heating, by a return current driven unstable by a small-area thick-target electron beam in solar flares. With a prescribed beam current evolution, j _b (t) (and, therefore, a prescribed return current j _p (t) = –j _b (t)), and using an approximate local treatment with a two component Maxwellian plasma, and neglecting energy losses, we demonstrate the existence of two quite distinct types of ion-acoustic unstable heating regimes. First, marginally stable heating occurs when the onset of instability occurs at electron-ion temperature ratios T _e /T _i > 4.8. Secondly, there exists a catastrophic heating regime for which marginally stable evolution is impossible, when the onset of instability occurs at T _e /T _i < 4.8.For the marginally stable case, we solve the electron and ion heating equations numerically and find that rapid anomalous Ohmic heating occurs in a substantial plasma volume. This large hot plasma emits thermal bremsstrahlung hard X-rays ( 20 keV) comparable to, or exceeding, the nonthermal bremsstrahlung which would have been emitted by the beam in a conventional thick target, large area, collisional scenario without anomalous effects. This means that, contrary to the usual assumption, onset of return current instability need not turn off hard X-ray production by a beam, though changing its source from direct to indirect. Indeed with small beam areas, this indirect mechanism can result in a higher hard X-ray bremsstrahlung efficiency than in a conventional collisional thick target.The catastrophic heating regime, for which we expect much larger wave levels, is discussed qualitatively, and preliminary results cited of an alternative approach, incorporating an equation directly describing the electrostatic wave energy level. Which of these two regimes will pertain in any particular case depends (discontinuously) on the beam and atmospheric parameters and we suggest that this effect may manifest itself in the distinctive temporal behaviour of X-ray flares.     

Thick target X-ray bremsstrahlung from partially ionised targets in solar flares

The effect of partial ionisation of a thick target bremsstrahlung source on the emitted X-ray intensity is analysed. It is shown that a totally ionised target produces an X-ray burst only about one third as intense as that from an unionised target.In the case of a solar flare plasma target, the ionisation decreases with increasing depth in the flare. Thus, in an X-ray flare model in which electrons are continuously accelerated down into the chromosphere, high energy photons are produced with increased efficiency in the deeper layers of the flare plasma with consequent hardening of the X-ray spectrum. As a result, the spectra of nonthermal electrons in flares, inferred from X-ray spectra, are steepened and their total energy correspondingly increased.     

On the bremsstrahlung efficiency of nonthermal hard X-ray source models

It has often been stated, but never rigorously proven, that interpreting observed hard X-ray emission in terms of a thick-target source gives a lower limit to the flux of electrons which have to be injected into the source. The truth of this statement, for theburst-integrated emission, is rigorously established here. Also an explicit inversion for the injected electron flux in terms of the photon spectrum is given, for the case where all electrons traverse a single value of column density. This generalises the previous results for the thick- and thin-target limits.The use of the standard thick-target formalism for the interpretation of instantaneous (as opposed to burst-integrated) photon fluxes is also discussed. By considering the specific case of the thick-target trap model, it is shown that use of this formalism can either under-estimate or over-estimate the injected electron flux, at different times in the same event, but that integration of the inferred electron fluxes over the event nevertheless yields the true, burst-integrated electron flux.     

Multi-Wavelength Analysis of the Flare on 2 October 1993

By use of Yohkoh hard X-ray flux and soft X-ray images, and of vector magnetograms and 2D spectral observations, a 1N/C6.5 flare observed on 2 October 1993 is analysed in detail. Evidence is provided not only morphologically but also quantitatively that the dynamics at kernels A and C of the flare in the impulsive phase were controlled mainly by electron beam bombardment, while the heating of kernel B is mainly due to heat conduction. By plotting the energy gradient of the electron energy flux as a function of energy for the various spectral indexes observed during the flare, the acceleration mechanism is found to be such that there is a constant energy E₀, close to 20 keV, for which the electron flux d F₁/dE is constant. It is shown that such a conclusion can be reached more directly by using the photon flux, which in that case must be constant for E=E₀, whatever the value of the power index. This result implies also that the electron spectrum is represented by a power law and that the X-ray photons are produced in a thick target. Instantaneous momentum balance is shown to exist between the upflowing soft X-ray-emitting and the downflowing Hα- emitting plasma at the kernels of the flare. The observed Hα red asymmetry is well reproduced by the non-LTE computation, with the down-moving condensation included. The observation of the magnetic field suggests that the flare was triggered probably by magnetic flux emergence.     

Empirical correction of RHESSI spectra for photospheric albedo and its effect on inferred electron spectra

Photospheric Compton backscatter (albedo) makes a significant contribution to observed hard X-ray (HXR) spectral fluxes over the RHESSI energy range and should be allowed for in HXR spectral interpretation. The full correction problem is nonlinear and messy but we offer a simple approximate first-order correction procedure for global HXR spectra based upon empirical fits to published albedo simulations. We also illustrate the impact of this correction on inferred electron spectra for the thin- and thick-target models.     

A Study of the Low Energy Cutoff of Nonthermal Electrons in RHESSI Flares

The flattening at the low energy end of the hard X-ray (HXR) photon spectrum of solar flares was generally thought to be due to a cutoff of nonthermal electrons in flares. However, some authors have suggested that inverse Compton scattering (i.e., the albedo effect) or certain other reaction of flare photons with the lower atmosphere can also lead to the flattening. This paper adopts the method of deriving the cutoff proposed by Gan et al. ^[12–14], and makes a statistical analysis on 100 flares observed by the satellite Ramaty High Energy Solar Spectroscopy Imager (RHESSI) in 2002–2005. We found that after the albedo correction, the HXR photon spectra of 18 flares can be fitted with single powerlaw spectra, and those of 80 flares, with double power-law spectra. Besides, 21 flares can be directly interpreted with a single power-law electron spectrum plus a low energy cutoff. The range of the low energy cutoff is 20–50 keV and the mean value is approximately 30 keV. Some other possible interpretations are also investigated.     

Observations of solar gamma ray continuum between 360 keV and 7 MeV on August 4, 1972

Measurements were made of the time-averaged gamma ray energy loss spectrum in the energy range 360 keV to 7 MeV by the gamma ray detector on the OSO-7 satellite during the 3B flare on August 4, 1972. The differential photon spectrum unfolded from this spectrum after subtracting the background spectrum and contributions from gamma ray lines is best described by a power law with spectral index of 3.4±0.3 between 360–700 keV and by an exponential law of the form exp (-E/E ₀) with E ₀ = 1.0±0.1 MeV above 700 keV. It is suggested that this spectrum is due to nonthermal electron bremsstrahlung from a population of electrons, with a strong break in the spectrum at 2 MeV. Since the observational data indicates that the matter number density must be n _H ? 5 × 10¹⁰ cm^-3 in the production region, the number of electrons above 100 keV required to explain the results is ?2 × 10³⁴.     

Dissipation and stability of return currents in solar flares

The dynamics of an electron beam, under the effects of Coulomb collisions and classical Ohmic dissipation of the return current, is analysed for a background plasma with a temperature which is time dependent due to the heating effect of beam dissipation offset by thermal conductive cooling. It is shown that the plasma is heated toward a steady state, in time scales short compared to typical flare beam switch on times, and that in this steady state only two regimes of beam dynamics arise.For moderate values of the ratio of beam flux to plasma density (F ₀/n), beam dynamics is dominated by direct Coulomb collisions as in the usual thick target treatment. With increase of F ₀/n, before return current (classical) Ohmic losses can exceed collisions, the return current becomes unstable. In this latter regime beam dynamics is presumably dominated by wave generation and anomalous Ohmic dissipation of the return current, but no detailed treatment is attempted here.     

Photon Spectra of Electron–Electron Bremsstrahlung

A formula is given for the cross section of electron–electron bremsstrahlung in the rest system of the target electron. It is in particular appropriate for the calculation of photon spectra in astrophysical applications, e.g., if the primary electrons have energy distributions in the form of a power law. For the long-wavelength limit of the spectrum the cross section is specified in closed form.     

Comparison of thermal and nonthermal hard X-ray emission in electron-heated solar flares

Using the results of numerical simulations of the solar atmospheric response to heating by nonthermal electron beams during solar flares, we have calculated the spatial and temporal evolution of both (i) the direct (beam-target) nonthermal bremsstrahlung and (ii) the thermal bremsstrahlung arising from the hot plasma energized by the electron beam. Typically, we find that below a certain cross-over energy E ^*, the emission is dominated by the thermal component, while at higher energies the direct bremsstrahlung component becomes more important. This cross-over energy is dependent on the position within the loop, generally increasing with height.We have also investigated the dependence of the cross-over energy E ^* on the parameters of the electron energy input. At the time of peak electron flux injection the cross-over energy E ^* can, for plausible parameters, be as high as 52 keV at the top 1 pixel, and as low as 16 keV at the bottom 1 pixel. We conclude that a possible reassessment of SMM HXIS data as an indicator of the thermal or nonthermal character of the primary energy release (based primarily on the geometric properties of the hard X-ray source) is required. Our results also point to the minimum photon energy that future instruments should observe (where practical, giving due consideration to detector sensitivity) in order to be sure that, in the context of the thick-target interpretation, the nonthermal component is not swamped by the self-consistent thermal counterpart created by the beam heating.     

Flare fragmentation and type III productivity in the 1980 June 27 flare

We present observations of the solar flare on 1980 June 27, 16:14–16:33 UT, which was observed by a balloon-borne 300 cm² phoswich hard X-ray detector and by the IKARUS radio spectrometer. This flare shows intense hard X-ray (HXR) emission and an extreme productivity of (at least 754) type III bursts at 200–400 MHz. A linear correlation was found between the type III burst rate and the HXR fluence, with a coefficient of 7.6 × 10²⁷ photons keV^–1 per type III burst at 20 keV. The occurrence of 10 type III bursts per second, and also the even higher rate of millisecond spikes, suggests a high degree of fragmentation in the acceleration region. This high quantization of injected beams, assuming the thick-target model, shows up in a linear relationship between hard X-ray fluence and the type III rate, but not as fine structures in the HXR time profile.The generation of a superhot isothermal HXR component in the decay phase of the flare coincides with the fade-out of type III production.Universities Space Research Associates.ST Systems Corporation.     

The derivation of parent electron spectra from bremsstrahlung hard X-ray spectra

We formulate a numerical method to derive the spectrum of the parent electrons from the hard X-ray spectrum produced in optically thin bremsstrahlung. The method can utilize any form for the bremsstrahlung cross sections, and it provides accurate estimates of uncertainties in the derived electron spectrum based on uncertainties in the photon measurements. This method is applied to test photon spectra, as well as to hard X-ray spectra of the 27 June, 1980 solar flare which was observed by high spectral resolution detectors. Future measurements with much more sensitive detectors will enable this method to be used to derive detailed, accurate flare electron spectra.     

Flares From the Giant ar 9433 on 24 April 2001

We present H CCD observations of three small-to-medium-size two-ribbon flares observed in the giant AR 9433 on 24 April 2001. Flare observations at other associated wavelengths (e.g., soft X-rays (SXR), hard X-rays (HXR), microwaves (MW)) obtained from archives are also presented and compared. We have tested the Neupert effect for the most energetic third flare. The flare observations are in agreement with the thick-target model. In the case of this flare the HXR emitting electrons appears to be the heating source of SXR and H emissions. The flares are also studied in EUV and UV emissions using TRACE data. We discuss the complexity of the magnetic field using SOHO/MDI magnetograms. The flares are observed to occur in both (f/p) polarity regions in highly sheared magnetic field with emerging flux regions and MMFs.