Accelerated Electron Spectrum Research Articles

Review and meta-analysis of electron temperatures from high-intensity laser–solid interactions

The accelerated electron spectrum from high-intensity laser–solid interaction is often conveniently described using a Boltzmann distribution, whose temperature is known within the field as the hot-electron temperature. The importance of the electron temperature is highlighted by the sheer number of experimental and simulation studies on the subject over the past three decades. Recently, multi-kJ, multi-ps pulses have yielded electron spectra with temperatures far beyond the expected ponderomotive result. Expressions that predict the electron temperature considering laser parameters beyond intensity and wavelength have been developed, albeit using small datasets. In this review, we present what is, to the best of our knowledge, the largest dataset of electron temperatures gathered from experimental measurements and particle-in-cell simulations. This dataset allows us to compare existing analytical and empirical hot-electron temperature scaling models over a wide parameter range. We also develop new scaling models that incorporate the laser pulse duration of the laser and the plasma scale length. Three models that include pulse-duration and scale length dependence are especially successful at predicting both simulated and experimental data. The dataset will soon be made publicly available to encourage further investigation.

A Remarkably Narrow RHESSI X-Ray Flare on 2011 September 25

Abstract The unusually narrow X-ray source imaged with the Ramaty High Energy Solar Spectroscopic Imager (RHESSI) during an impulsive spike lasting for ∼10 s during the Geostationary Operational Environmental Satellite C7.9 flare on 2011 September 25 (SOL2011-09-25T03:32) was only ∼2″ wide and ∼10″ long. Comparison with Helioseismsic and Magnetic Imager magnetograms and Atmospheric Imaging Assembly images at 1700 Å shows that the X-ray emission was primarily from a long ribbon in the region of positive polarity with little if any emission from the negative polarity ribbon. However, a thermal plasma source density of ∼1012 cm−3 estimated from the RHESSI-derived emission measure and source area showed that this could best be interpreted as a coronal hard X-ray source in which the accelerated electrons with energies less than ∼50 keV were stopped by Coulomb collisions in the corona, thus explaining the lack of the more usual bright X-ray footpoints. Analysis of RHESSI spectra shows greater consistency with a multi-temperature distribution and a low-energy cutoff to the accelerated electron spectrum of 22 keV compared to 12 keV if a single-temperature distribution is assumed. This leads to a change in the lower limit on the total energy in electrons by an order of magnitude, given the steepness of the best-fit electron spectrum with a power-law index of ∼6.

Determination of the Total Accelerated Electron Rate and Power Using Solar Flare Hard X-Ray Spectra

Abstract Solar flare hard X-ray (HXR) spectroscopy serves as a key diagnostic of the accelerated electron spectrum. However, the standard approach using the collisional cold thick-target model poorly constrains the lower-energy part of the accelerated electron spectrum, hence the overall energetics of the accelerated electrons are typically constrained only to within one or two orders of magnitude. Here, we develop and apply a physically self-consistent, warm-target approach that involves the use of both HXR spectroscopy and imaging data. This approach allows an accurate determination of the electron distribution low-energy cutoff, and hence the electron acceleration rate and the contribution of accelerated electrons to the total energy released, by constraining the coronal plasma parameters. Using a solar flare observed in X-rays by RHESSI, we demonstrate that using the standard cold-target methodology, the low-energy cutoff (hence the energy content in electrons) is essentially undetermined. However, the warm-target methodology can determine the low-energy electron cutoff with ∼7% uncertainty at the 3σ level, hence it permits an accurate quantitative study of the importance of accelerated electrons in solar flare energetics.

Two-stage Electron Acceleration by 3D Collisionless Guide-field Magnetic Reconnection

Abstract We report a newly found two-stage mechanism of electron acceleration near X-lines of 3D collisionless guide-field magnetic reconnection in the nonrelativistic regime typical, e.g., for stellar coronae. We found that after electrons are first pre-accelerated during the linear growth of reconnection, they become additionally accelerated in the course of the nonlinear stage of 3D guide-field magnetic reconnection. This additional acceleration is due to the filamentation of electric and magnetic fields caused by streaming instabilities. In addition to enhanced parallel electric fields, the filamentation leads to additional curvature-driven electron acceleration in the guide-field direction. As a result, part of the accelerated electron spectra becomes a power law with a spectral index of ∼−1.6 near the X-line. This second stage of acceleration due to nonlinear reconnection is relevant for the production of energetic electrons in, e.g., thin current sheets of stellar coronae.

X-rays diagnostics of the hot electron energy distribution in the intense laser interaction with metal targets

A two-temperature hot electron energy distribution has been revealed by modeling of bremsstrahlung emission, measured by the radiation attenuation and half-shade methods, and Kα emission from a massive silver cylinder irradiated by a subpicosecond s-polarized laser pulse with a peak intensity of about 2 × 1019 W/cm2. To deduce parameters of the hot electron spectrum, we have developed semi-analytical models of generation and measurements of the x-rays. The models are based on analytical expressions and tabulated data on electron stopping power as well as cross-sections of generation and absorption of the x-rays. The Kα emission from thin silver foils deposited on low-Z substrates, both conducting and nonconducting, has been used to verify the developed models and obtained hot electron spectrum. The obtained temperatures of the colder and hotter electron components are in agreement with the values predicted by kinetic simulations of the cone-guided approach to fast ignition [Chrisman et al., Phys. Plasmas 15, 056309 (2008)]. The temperature of the low-energy component of the accelerated electron spectrum is well below the ponderomotive scaling and Beg's law. We have obtained relatively low conversion efficiency of laser energy into the energy of hot electrons propagating through the solid target of about 2%. It is demonstrated that the assumption about a single-temperature hot electron energy distribution with the slope temperature described by the ponderomotive scaling relationship, without detailed analysis of the hot electron spectrum, can lead to strong overestimation of the laser-to-electron energy-conversion efficiency, in particular, the conversion efficiency of laser energy into the high-temperature component of the hot electron distribution.

Fermi-LAT Observations of High-energy Behind-the-limb Solar Flares

Abstract We report on the Fermi-LAT detection of high-energy emission from the behind-the-limb (BTL) solar flares that occurred on 2013 October 11, and 2014 January 6 and September 1. The Fermi-LAT observations are associated with flares from active regions originating behind both the eastern and western limbs, as determined by STEREO. All three flares are associated with very fast coronal mass ejections (CMEs) and strong solar energetic particle events. We present updated localizations of the >100 MeV photon emission, hard X-ray (HXR) and EUV images, and broadband spectra from 10 keV to 10 GeV, as well as microwave spectra. We also provide a comparison of the BTL flares detected by Fermi-LAT with three on-disk flares and present a study of some of the significant quantities of these flares as an attempt to better understand the acceleration mechanisms at work during these occulted flares. We interpret the HXR emission to be due to electron bremsstrahlung from a coronal thin-target loop top with the accelerated electron spectra steepening at semirelativistic energies. The >100 MeV gamma-rays are best described by a pion-decay model resulting from the interaction of protons (and other ions) in a thick-target photospheric source. The protons are believed to have been accelerated (to energies >10 GeV) in the CME environment and precipitate down to the photosphere from the downstream side of the CME shock and landed on the front side of the Sun, away from the original flare site and the HXR emission.

The collisional relaxation of electrons in hot flaring plasma and inferring the properties of solar flare accelerated electrons from X-ray observations.

X-ray observations are a direct diagnostic of fast electrons produced in solar flares, energized during the energy release process and directed towards the Sun. Since the properties of accelerated electrons can be substantially changed during their transport and interaction with the background plasma, a model must ultimately be applied to X-ray observations in order to understand the mechanism responsible for their acceleration. A cold thick target model is ubiquitously used for this task, since it provides a simple analytic relationship between the accelerated electron spectrum and the emitting electron spectrum in the X-ray source, with the latter quantity readily obtained from X-ray observations. However, such a model is inappropriate for the majority of solar flares in which the electrons propagate in a hot megaKelvin plasma, because it does not take into account the physics of thermalization of fast electrons. The use of a more realistic model, properly accounting for the properties of the background plasma, and the collisional diffusion and thermalization of electrons, can alleviate or even remove many of the traditional problems associated with the cold thick target model and the deduction of the accelerated electron spectrum from X-ray spectroscopy, such as the number problem and the need to impose an ad hoc low energy cut-off.

COLLISIONAL RELAXATION OF ELECTRONS IN A WARM PLASMA AND ACCELERATED NONTHERMAL ELECTRON SPECTRA IN SOLAR FLARES

Extending previous studies of nonthermal electron transport in solar flares which include the effects of collisional energy diffusion and thermalization of fast electrons, we present an analytic method to infer more accurate estimates of the accelerated electron spectrum in solar flares from observations of the hard X-ray spectrum. Unlike for the standard cold-target model, the spatial characteristics of the flaring region, especially the necessity to consider a finite volume of hot plasma in the source, need to be taken into account in order to correctly obtain the injected electron spectrum from the source-integrated electron flux spectrum (a quantity straightforwardly obtained from hard X-ray observations). We show that the effect of electron thermalization can be significant enough to nullify the need to introduce an {\it ad hoc} low-energy cutoff to the injected electron spectrum in order to keep the injected power in non-thermal electrons at a reasonable value. Rather the suppression of the inferred low-energy end of the injected spectrum compared to that deduced from a cold-target analysis allows the inference from hard X-ray observations of a more realistic energy in injected non-thermal electrons in solar flares.

ON THE SPECTRAL HARDENING AT ≳300 keV IN SOLAR FLARES

It has been noted for a long time that the spectra of observed continuum emissions in many solar flares are consistent with double power laws with a hardening at energies $\sim > $ 300 keV. It is now largely believed that at least in electron-dominated events the hardening in photon spectrum reflects an intrinsic hardening in the source electron spectrum. In this paper, we point out that a power law spectrum of electron with a hardening at high energies can be explained by diffusive shock acceleration of electrons at a termination shock with a finite width. Our suggestion is based on an early analytical work by Drury et al., where the steady state transport equation at a shock with a tanh profile was solved for a $p$-independent diffusion coefficient. Numerical simulations with a $p$-dependent diffusion coefficient show hardenings in the accelerated electron spectrum which are comparable with observations. One necessary condition for our proposed scenario to work is that high energy electrons resonate with the inertial range of the MHD turbulence and low energy electrons resonate with the dissipation range of the MHD turbulence at the acceleration site, and the spectrum of the dissipation range $\sim k^{-2.7}$. A $\sim k^{-2.7}$ dissipation range spectrum is consistent with recent solar wind observations.

PROBING DYNAMICS OF ELECTRON ACCELERATION WITH RADIO AND X-RAY SPECTROSCOPY, IMAGING, AND TIMING IN THE 2002 APRIL 11 SOLAR FLARE

Based on detailed analysis of radio and X-ray observations of a flare on 2002 April 11 augmented by realistic 3D modeling, we have identified a radio emission component produced directly at the flare acceleration region. This acceleration region radio component has distinctly different (i) spectrum, (ii) light curves, (iii) spatial location, and, thus, (iv) physical parameters from those of the separately identified, trapped or precipitating electron components. To derive evolution of physical parameters of the radio sources we apply forward fitting of the radio spectrum time sequence with the gyrosynchrotron source function with 5 to 6 free parameters. At the stage when the contribution from the acceleration region dominates the radio spectrum, the X-ray- and radio- derived electron energy spectral indices agree well with each other. During this time the maximum energy of the accelerated electron spectrum displays a monotonic increase with time from ~ 300 keV to ~ 2 MeV over roughly one minute duration indicative of an acceleration process in the form of growth of the power-law tail; the fast electron residence time in the acceleration region is about 2 - 4 s, which is much longer than the time of flight and so requires a strong diffusion mode there to inhibit free-streaming propagation. The acceleration region has a relatively strong magnetic field, B ~ 120 G, and a low thermal density, less than 2x10^9 cm^-3. These acceleration region properties are consistent with a stochastic acceleration mechanism.

THE SPECIFIC ACCELERATION RATE IN LOOP-STRUCTURED SOLAR FLARES—IMPLICATIONS FOR ELECTRON ACCELERATION MODELS

We analyze electron flux maps based on RHESSI hard X-ray imaging spectroscopy data for a number of extended coronal loop flare events. For each event, we determine the variation of the characteristic loop length $L$ with electron energy $E$, and we fit this observed behavior with models that incorporate an extended acceleration region and an exterior "propagation" region, and which may include collisional modification of the accelerated electron spectrum inside the acceleration region. The models are characterized by two parameters: the plasma density $n$ in, and the longitudinal extent $L_0$ of, the acceleration region. Determination of the best-fit values of these parameters permits inference of the volume that encompasses the acceleration region and of the total number of particles within it. It is then straightforward to compute values for the emission filling factor and for the {\it specific acceleration rate} (electrons s$^{-1}$ per ambient electron above a chosen reference energy). For the 24 events studied, the range of inferred filling factors is consistent with a value of unity. The inferred mean value of the specific acceleration rate above $E_0=20$ keV is $\sim10^{-2}$ s$^{-1}$, with a 1$\sigma$ spread of about a half-order-of-magnitude above and below this value. We compare these values with the predictions of several models, including acceleration by large-scale, weak (sub-Dreicer) fields, by strong (super-Dreicer) electric fields in a reconnecting current sheet, and by stochastic acceleration processes.

BAYESIAN CONFIDENCE LIMITS OF ELECTRON SPECTRA OBTAINED THROUGH REGULARIZED INVERSION OF SOLAR HARD X-RAY SPECTRA

Many astrophysical observations are characterized by a single, non-repeatable measurement of a source brightness or intensity, from which we are to construct estimates for the true intensity and its uncertainty. For example, the hard X-ray count spectrum from transient events such as solar flares can be observed only once, and from this single spectrum one must determine the best estimate of the underlying source spectrum I({epsilon}), and hence the form of the responsible electron spectrum F(E). Including statistical uncertainties on the measured count spectrum yields a 'confidence strip' that delineates the boundaries of electron spectra that are consistent with the observed photon spectrum. In this short article, we point out that the expectation values of the source brightness and its variance in a given photon energy bin are in general not (as has been assumed in prior works) equal to n, the number of counts observed in that energy bin. Rather, they depend both on n and on prior knowledge of the overall photon spectrum. Using Bayesian statistics, we provide an explicit procedure and formulas for determining the 'confidence strip' (Bayesian credible region) for F(E), thus providing rigorous bounds on the intensity and shape of the accelerated electron spectrum.

RELATION BETWEEN THE CORONAL MASS EJECTION ACCELERATION AND THE NON-THERMAL FLARE CHARACTERISTICS

We investigate the relationship between the main acceleration phase of coronal mass ejections (CMEs) and the particle acceleration in the associated flares as evidenced in Reuven Ramaty High Energy Solar Spectroscopic Imager non-thermal X-rays for a set of 37 impulsive flare-CME events. Both the CME peak velocity and peak acceleration yield distinct correlations with various parameters characterizing the flare-accelerated electron spectra. The highest correlation coefficient is obtained for the relation of the CME peak velocity and the total energy in accelerated electrons (c = 0.85), supporting the idea that the acceleration of the CME and the particle acceleration in the associated flare draw their energy from a common source, probably magnetic reconnection in the current sheet behind the erupting structure. In general, the CME peak velocity shows somewhat higher correlations with the non-thermal flare parameters than the CME peak acceleration, except for the spectral index of the accelerated electron spectrum, which yields a higher correlation with the CME peak acceleration (c ?0.6), indicating that the hardness of the flare-accelerated electron spectrum is tightly coupled to the impulsive acceleration process of the rising CME structure. We also obtained high correlations between the CME initiation height h 0 and the non-thermal flare parameters, with the highest correlation of h 0 to the spectral index ? of flare-accelerated electrons (c 0.8). This means that CMEs erupting at low coronal heights, i.e., in regions of stronger magnetic fields, are accompanied by flares that are more efficient at accelerating electrons to high energies. In the majority of events (~80%), the non-thermal flare emission starts after the CME acceleration, on average delayed by 6 minutes, in line with the standard flare model where the rising flux rope stretches the field lines underneath until magnetic reconnection sets in. We find that the current sheet length at the onset of magnetic reconnection is 21 ? 7?Mm. The flare hard X-ray peaks are well synchronized with the peak of the CME acceleration profile, and in 75% of the cases they occur within ?5?minutes. Our findings provide strong evidence for the tight coupling between the CME dynamics and the particle acceleration in the associated flare in impulsive events, with the total energy in accelerated electrons being closely correlated with the peak velocity (and thus the kinetic energy) of the CME, whereas the number of electrons accelerated to high energies is decisively related to the CME peak acceleration and the height of the pre-eruptive structure.

X-Ray and Gamma-Ray Emission of the Shell-Type Supernova Remnant RX J1713.7−3946

To study possible origins of high-energy gamma-ray emission from shell-type supernova remnants, RX J1713.7$-$3946, we model their nonthermal photon spectra from X-ray to very high energy (VHE) gamma-ray bands in the frame of non-linear diffusive shock acceleration, respectively. In our modeling, the accelerated proton spectrum is calculated based on a semi-analytical solution of the problem of particle acceleration at non-linear shock waves with a free escape boundary at some location upstream, and the accelerated electron spectrum is similar to the proton's spectrum when the electron's momentum is much less than its maximum momentum, which is determined by equating the synchrotron loss time with the acceleration time, and the cutoff shapes are assumed to be an super-exponential. In the one-zone model, we calculated the non-thermal photon spectra of the supernova remnants (SNR) for both leptonic and hadronic cases, and found out that the spectra calculated for both cases can reproduce the observed data by Suzaku, Fermi LAT, and HESS under reasonable model parameters. However, high-energy gamma-ray emission would have a hadronic origin if the observed thickness of the X-ray filament of the SNR is taken into account.

G-BAND AND HARD X-RAY EMISSIONS OF THE 2006 DECEMBER 14 FLARE OBSERVED BYHINODE/SOT ANDRHESSI

We report on G-band emission observed by the Solar Optical Telescope onboard the Hinode satellite in association with the X1.5-class flare on 2006 December 14. The G-band enhancements originate from the footpoints of flaring coronal magnetic loops, coinciding with non-thermal hard X-ray bremsstrahlung sources observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager. At the available 2 minute cadence, the G-band and hard X-ray intensities are furthermore well correlated in time. Assuming that the G-band enhancements are continuum emission from a blackbody, we derived the total radiative losses of the white-light flare (white-light power). If the G-band enhancements additionally have a contribution from lines, the derived values are overestimates. We compare the white-light power with the power in hard X-ray producing electrons using the thick target assumption. Independent of the cutoff energy of the accelerated electron spectrum, the white-light power and the power of accelerated electrons are roughly proportional. Using the observed upper limit of ~30 keV for the cutoff energy, the hard X-ray producing electrons provide at least a factor of 2 more power than needed to produce the white-light emission. For electrons above 40 keV, the powers roughly match for all four of the time intervals available during the impulsive phase. Hence, the flare-accelerated electrons contain enough energy to produce the white-light flare emissions. The observed correlation in time, space, and power strongly suggests that electron acceleration and white-light production in solar flares are closely related. However, the results also call attention to the inconsistency in apparent source heights of the hard X-ray (chromosphere) and white-light (upper photosphere) sources.

DERIVATION OF STOCHASTIC ACCELERATION MODEL CHARACTERISTICS FOR SOLAR FLARES FROM RHESSI HARD X-RAY OBSERVATIONS

The model of stochastic acceleration of particles by turbulence has been successful in explaining many observed features of solar flares. Here, we demonstrate a new method to obtain the accelerated electron spectrum and important acceleration model parameters from the high-resolution hard X-ray (HXR) observations provided by RHESSI. In our model, electrons accelerated at or very near the loop top (LT) produce thin target bremsstrahlung emission there and then escape downward producing thick target emission at the loop footpoints (FPs). Based on the electron flux spectral images obtained by the regularized spectral inversion of the RHESSI count visibilities, we derive several important parameters for the acceleration model. We apply this procedure to the 2003 November 3 solar flare, which shows an LT source up to 100–150 keV in HXR with a relatively flat spectrum in addition to two FP sources. The results imply the presence of strong scattering and a high density of turbulence energy with a steep spectrum in the acceleration region.

Model calculations of electron precipitation induced ionization patches on the nightside of Mars

Using an electron transport model, we investigate the effect of electron precipitation on the electron density and total electron content in the nightside ionosphere of Mars. As input we use Mars Global Surveyor observations of a typical tail electron spectrum and an auroral‐like electron spectrum. The accelerated electron spectrum increases the maximum number density and total electron content by a factor of 3 over that produced by the typical tail spectrum. Our calculations show a secondary electron density peak due to precipitation of several keV electrons not seen in previous modeling efforts. Regions of enhanced ionization are expected to be localized in space, corresponding to magnetic cusps formed by the interaction of crustal sources with the interplanetary magnetic field. Radio and radar measurements from both Mars Global Surveyor and Mars Express agree with this expectation. The horizontally inhomogeneous regions of ionization can affect signals used for subsurface sounding from orbit.

RHESSI and SOHO CDS Observations of Explosive Chromospheric Evaporation

Simultaneous observations of explosive chromospheric evaporation are presented using data from the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) and the Coronal Diagnostic Spectrometer (CDS) on board the Solar and Heliospheric Observatory. For the first time, cospatial imaging and spectroscopy have been used to observe explosive evaporation within a hard X-ray emitting region. RHESSI X-ray images and spectra were used to determine the flux of nonthermal electrons accelerated during the impulsive phase of an M2.2 flare. When we assumed a thick-target model, the injected electron spectrum was found to have a spectral index of ~7.3, a low-energy cutoff of ~20 keV, and a resulting flux of ≥4 × 1010 ergs cm-2 s-1. The dynamic response of the atmosphere was determined using CDS spectra; we found a mean upflow velocity of 230 ± 38 km s-1 in Fe XIX (592.23 A) and associated downflows of 36 ± 16 and 43 ± 22 km s-1 at chromospheric and transition region temperatures, respectively, relative to an averaged quiet-Sun spectra. The errors represent a 1 σ dispersion. The properties of the accelerated electron spectrum and the corresponding evaporative velocities were found to be consistent with the predictions of theory.

Images of Gradual Millimeter Emission and Multi-Wavelength Observations of the 17 august 1994 Solar Flare

We present a comprehensive analysis of the 17 August 1994 flare, the first flare imaged at millimeter (86 GHz) wavelengths. The temporal evolution of this flare displays a prominent impulsive peak shortly after 01:02 UT, observed in hard X-rays and at microwave frequencies, followed by a gradual decay phase. The gradual phase was also detected at 86 GHz. Soft X-ray images show a compact emitting region (≲20′′), which is resolved into two sources: a footpoint and a loop top source. Nonthermal emissions at microwave and hard X-ray wavelengths are analyzed and the accelerated electron spectrum is calculated. This energy spectrum derived from the microwave and hard X-ray observations suggests that these emissions were created by the same electron population. The millimeter emission during the gradual phase is thermal bremsstrahlung originating mostly from the top of the flaring loop. The soft X-rays and the millimeter flux density from the footpoint source are only consistent with the presence of a multi-temperature plasma at the footpoint.

Studying flare accelerated electrons with BATSE

The Burst and Transient Source Experiment (BATSE) on the Compton Gamma Ray Observatory has opened a hard X-ray window of unprecedented sensitivity from 10 to several hundreds of keV which allows a fresh examination of the details of the accelerated electron spectrum during solar flares. Using the BATSE Spectroscopy detectors we have examined the low energy hard X-ray spectrum for thermal components and evidence of cutoffs in the accelerated electron spectrum.