Discrepancies between empirical and theoretical models of the flaring solar chromosphere and their possible resolution

Models of the solar chromosphere during flaring deduced theoretically or empirically are compared. Marked discrepancies are noted and various reasons are offered to explain their existence. A means is presented for testing theoretical heating models (electron heating) by analyzing the net energy loss rates in (observed) empirical atmospheres and inverting the flare energy equation to deduce the parameters of the supposed heating mechanism

Turbulent pitch-angle scattering and diffusive transport of hard-X-ray producing electrons in flaring coronal loops

Recent observations from {\em RHESSI} have revealed that the number of non-thermal electrons in the coronal part of a flaring loop can exceed the number of electrons required to explain the hard X-ray-emitting footpoints of the same flaring loop. Such sources cannot, therefore, be interpreted on the basis of the standard collisional transport model, in which electrons stream along the loop while losing their energy through collisions with the ambient plasma; additional physical processes, to either trap or scatter the energetic electrons, are required. Motivated by this and other observations that suggest that high energy electrons are confined to the coronal region of the source, we consider turbulent pitch angle scattering of fast electrons off low frequency magnetic fluctuations as a confinement mechanism, modeled as a spatial diffusion parallel to the mean magnetic field. In general, turbulent scattering leads to a reduction of the collisional stopping distance of non-thermal electrons along the loop and hence to an enhancement of the coronal HXR source relative to the footpoints. The variation of source size $L$ with electron energy $E$ becomes weaker than the quadratic behavior pertinent to collisional transport, with the slope of $L(E)$ depending directly on the mean free path $\lambda$ again pitch angle scattering. Comparing the predictions of the model with observations, we find that $\lambda \sim$$(10^8-10^9)$ cm for $\sim30$ keV, less than the length of a typical flaring loop and smaller than, or comparable to, the size of the electron acceleration region.Comment: 25 pages, 5 figures, accepted for publication in Astrophysical Journa

Development of a theory of the spectral reflectance of minerals, part 4

A theory of the spectral reflectance or emittance of particulate minerals was developed. The theory is expected to prove invaluable in the interpretation of the remote infrared spectra of planetary surfaces

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.Comment: accepted for publication in Ap

Implications of solar flare hard X-ray "knee" spectra observed by RHESSI

We analyse the RHESSI photon spectra of four flares that exhibit significant deviations from power laws - i.e. changes in the "local" Hard X-ray spectral index. These spectra are characterised by two regions of constant power law index connected by a region of changing spectral index - the "knee". We develop theoretical and numerical methods of describing such knees in terms of variable photon spectral indices and we study the results of their inversions for source mean thin target and collisional thick target injection electron spectra. We show that a particularly sharp knee can produce unphysical negative values in the electron spectra, and we derive inequalities that can be used to test for this without the need for an inversion to be performed. Such unphysical features would indicate that source model assumptions were being violated, particularly strongly for the collisional thick target model which assumes a specific form for electron energy loss. For all four flares considered here we find that the knees do not correspond to unphysical electron spectra. In the three flares that have downward knees we conclude that the knee can be explained in terms of transport effects through a region of non-uniform ionisation. In the other flare, which has an upward knee, we conclude that it is most likely a feature of the accelerated spectrum

Fast electron slowing-down and diffusion in a high temperature coronal X-ray source

Finite thermal velocity modifications to electron slowing-down rates may be important for the deduction of solar flare total electron energy. Here we treat both slowing-down and velocity diffusion of electrons in the corona at flare temperatures, for the case of a simple, spatially homogeneous source. Including velocity diffusion yields a consistent treatment of both "accelerated" and "thermal" electrons. It also emphasises that one may not invoke finite thermal velocity target effects on electron lifetimes without simultaneously treating the contribution to the observed X-ray spectrum from thermal electrons. We present model calculations of the X-ray spectra resulting from injection of a power-law energy distribution of electrons into a source with finite temperature. Reducing the power-law distribution low-energy cutoff to lower and lower energies only increases the relative magnitude of the thermal component of the spectrum, because the lowest energy electrons simply join the background thermal distribution. Acceptable fits to RHESSI flare data are obtained using this model. These also demonstrate, however, that observed spectra may in consequence be acceptably consistent with rather a wide range of injected electron parameters

A Fokker-Planck Framework for Studying the Diffusion of Radio Burst Waves in the Solar Corona

Electromagnetic wave scattering off density inhomogeneities in the solar corona is an important process which determines both the apparent source size and the time profile of radio bursts observed at 1 AU. Here we model the scattering process using a Fokker-Planck equation and apply this formalism to several regimes of interest. In the first regime the density fluctuations are considered quasi-static and diffusion in wavevector space is dominated by angular diffusion on the surface of a constant energy sphere. In the small-angle ("pencil beam") approximation, this diffusion further occurs over a small solid angle in wavevector space. The second regime corresponds to a much later time, by which scattering has rendered the photon distribution near-isotropic resulting in a spatial diffusion of the radiation. The third regime involves time-dependent fluctuations and, therefore, Fermi acceleration of photons. Combined, these results provide a comprehensive theoretical framework within which to understand several important features of propagation of radio burst waves in the solar corona: emitted photons are accelerated in a relatively small inner region and then diffuse outwards to larger distances. En route, angular diffusion results both in source sizes which are substantially larger than the intrinsic source, and in observed intensity-versus-time profiles that are asymmetric, with a sharp rise and an exponential decay. Both of these features are consistent with observations of solar radio bursts.Comment: 28 pages , 1 figure, submitted to Ap

A Fokker-Planck Framework for Studying the Diffusion of Radio Burst Waves in the Solar Corona

Electromagnetic wave scattering off density inhomogeneities in the solar corona is an important process which determines both the apparent source size and the time profile of radio bursts observed at 1 AU. Here we model the scattering process using a Fokker-Planck equation and apply this formalism to several regimes of interest. In the first regime the density fluctuations are considered quasi-static and diffusion in wavevector space is dominated by angular diffusion on the surface of a constant energy sphere. In the small-angle ("pencil beam") approximation, this diffusion further occurs over a small solid angle in wavevector space. The second regime corresponds to a much later time, by which scattering has rendered the photon distribution near-isotropic resulting in a spatial diffusion of the radiation. The third regime involves time-dependent fluctuations and, therefore, Fermi acceleration of photons. Combined, these results provide a comprehensive theoretical framework within which to understand several important features of propagation of radio burst waves in the solar corona: emitted photons are accelerated in a relatively small inner region and then diffuse outwards to larger distances. En route, angular diffusion results both in source sizes which are substantially larger than the intrinsic source, and in observed intensity-versus-time profiles that are asymmetric, with a sharp rise and an exponential decay. Both of these features are consistent with observations of solar radio bursts.Comment: 28 pages , 1 figure, submitted to Ap

Electron-Electron Bremsstrahlung Emission and the Inference of Electron Flux Spectra in Solar Flares

Although both electron-ion and electron-electron bremsstrahlung contribute to the hard X-ray emission from solar flares, the latter is normally ignored. Such an omission is not justified at electron (and photon) energies above $\sim 300$ keV, and inclusion of the additional electron-electron bremsstrahlung in general makes the electron spectrum required to produce a given hard X-ray spectrum steeper at high energies. Unlike electron-ion bremsstrahlung, electron-electron bremsstrahlung cannot produce photons of all energies up to the maximum electron energy involved. The maximum possible photon energy depends on the angle between the direction of the emitting electron and the emitted photon, and this suggests a diagnostic for an upper cutoff energy and/or for the degree of beaming of the accelerated electrons. We analyze the large event of January 17, 2005 observed by RHESSI and show that the upward break around 400 keV in the observed hard X-ray spectrum is naturally accounted for by the inclusion of electron-electron bremsstrahlung. Indeed, the mean source electron spectrum recovered through a regularized inversion of the hard X-ray spectrum, using a cross-section that includes both electron-ion and electron-electron terms, has a relatively constant spectral index $\delta$ over the range from electron kinetic energy $E = 200$ keV to $E = 1$ MeV. However, the level of detail discernible in the recovered electron spectrum is not sufficient to determine whether or not any upper cutoff energy exists.Comment: 7 pages, 5 figures, submitted to Astrophysical Journa

Determination of the Acceleration Region Size in a Loop-structured Solar Flare

In order to study the acceleration and propagation of bremsstrahlung-producing electrons in solar flares, we analyze the evolution of the flare loop size with respect to energy at a variety of times. A GOES M3.7 loop-structured flare starting around 23:55 on 2002 April 14 is studied in detail using \textit{Ramaty High Energy Solar Spectroscopic Imager} (\textit{RHESSI}) observations. We construct photon and mean-electron-flux maps in 2-keV energy bins by processing observationally-deduced photon and electron visibilities, respectively, through several image-processing methods: a visibility-based forward-fit (FWD) algorithm, a maximum entropy (MEM) procedure and the uv-smooth (UVS) approach. We estimate the sizes of elongated flares (i.e., the length and width of flaring loops) by calculating the second normalized moments of the intensity in any given map. Employing a collisional model with an extended acceleration region, we fit the loop lengths as a function of energy in both the photon and electron domains. The resulting fitting parameters allow us to estimate the extent of the acceleration region which is between $\sim 13 \rm{arcsec}$ and $\sim 19 \rm{arcsec}$. Both forward-fit and uv-smooth algorithms provide substantially similar results with a systematically better fit in the electron domain.The consistency of the estimates from these methods provides strong support that the model can reliably determine geometric parameters of the acceleration region. The acceleration region is estimated to be a substantial fraction ($\sim 1/2$) of the loop extent, indicating that this dense flaring loop incorporates both acceleration and transport of electrons, with concurrent thick-target bremsstrahlung emission.Comment: 8 pages, 5 figures, accepted to Astronomy and Astrophysics journa