Double Power-law Spectrum Research Articles

Double Power-law Formation by Sequential Particle Acceleration

Spectral double power laws are common in solar high-energy phenomena such as flares and interplanetary energetic-electron events. However, the physical mechanism that produces the changes in power-law index within a single spectrum is unclear. We developed a fully analytical method of forming single power-law spectra from sequential acceleration of particles orbiting inside and hopping between simulated large-scale magnetic islands formed by flare reconnection. Here, we extend the analytical method to the formation of double power-law spectra by assuming sequential acceleration in two successive regions with different acceleration and particle-transport rates. The resulting spectral distribution is continuous and smooth, with a flattening at low energies, two power-law regions at mid-energies, and a steep rollover at high energies. The model provides analytical expressions for the spectral indices, all energy breaks, and normalization constants as functions of just three physical parameters of each acceleration region: (1) the energy gain in each accelerator, (2) the percentage of particles transferred between accelerators, and (3) the number of accelerators visited. One of the most salient predictions of our work is that the spectral index at high (low) energies is determined by the parameters of the first “seed” (second) acceleration region. By constructing the spectral distribution through an iterative analytical process, the evolution toward a double power law is easily characterized and explained. Our analytical model provides tools to interpret space- and ground-based observations from RHESSI, FOXSI, NuSTAR, Solar Orbiter/STIX, EOVSA, and future high-energy missions.

Non-universality of the Turbulent Spectra at Sub-ion Scales in the Solar Wind: Dispersive Effects versus the Doppler Shift

Large surveys of the power spectral density of the magnetic fluctuations in the solar wind have reported different slope distributions at MHD, sub-ion and sub-electron scales: the smaller the scale, the broader the distribution. Here, we review briefly some of the most relevant explanations of the broadening of the slopes at sub-ion scales. Then, we present a new one that has been overlooked in the literature, which is based on the relative importance of the dispersive effects with respect to the Doppler shift due to the mean flow speed. We build a toy model based on a dispersion relation of a linear mode that matches at high frequency (ω ≳ ω ci) the Alfvén (respectively whistler) mode at high oblique (respectively quasi-parallel) propagation angles θ kB . Starting with a double power-law spectrum of turbulence in the inertial range and at the sub-ion scales, the transformed spectrum (in frequency f) as it would be measured in the spacecraft reference frame shows a broad range of slopes at the sub-ion scales that depend both on the angle θ kB and the flow speed V. Varying θ kB in the range 4°–106° and V in the range 400−800 km s−1 the resulting distribution of slopes at the sub-ion scales reproduces quite well the observed one in the solar wind. Fluctuations in the solar wind speed and the wavevector anisotropy of the turbulence may explain (or at least contribute to) the variability of the spectral slopes reported from spacecraft observations in the solar wind.

Kinetic Scale Magnetic Reconnection with a Turbulent Forcing: Particle-in-cell Simulations

Turbulent magnetic reconnection has been observed by spacecraft to occur commonly in terrestrial magnetosphere and the solar wind, providing a new scenario of kinetic scale magnetic reconnection. Here by imposing a turbulent forcing on ions in particle-in-cell simulations, we simulate kinetic scale turbulent magnetic reconnection. We find formation of fluctuated electric and magnetic fields and filamentary currents in the diffusion region. Reconnection rate does not change much compared to that in laminar Hall reconnection. At the X-line, the electric and magnetic fields both exhibit a double power-law spectrum with a spectral break near local lower-hybrid frequency. The energy conversion rate is high in turbulent reconnection, leading to significant electron acceleration at the X-line. The accelerated electrons form a power-law spectrum in the high energy range, with a power-law index of about 3.7, much harder than one can obtain in laminar reconnection.

Parametric Evolution of Power-law Energy Spectra of Energetic Electrons in the Coronal Loops

Fast electron beams (FEBs) are one of the main products of various active events and are ubiquitous in solar, space and cosmic plasmas. They reveal themselves in hard X-ray and radio emissions. The observed characteristics of X-ray and radio emissions sensitively depend on the energy distribution of FEBs, which usually have a power-law energy spectrum. As FEBs travel in the solar atmosphere, their energy distribution can considerably vary due to the interaction with ambient plasmas. Tang et al. investigated the evolution of the energy spectrum of the FEBs traveling along a flare loop and discussed the possible effects on associated hard X-ray (HXR) and radio emissions. Considering the ubiquitous coronal loops in active regions, in the present paper, we investigate the parametric evolution of the energy spectra of FEBs when propagating along coronal loops. Here, we take the sunpot atmospheric model as an approximate coronal loop atmosphere model. The results show that the energy loss has an important impact on the cutoff behavior and energy spectra of FEBs when precipitating in a coronal loop with density ratio n b /n e = 0.01. The initially single power-law spectrum with a steepness cutoff can evolve into a more complex double power-law spectrum or two “knees” power-law spectrum with a flattened steepness cutoff behavior or saturation cutoff behavior. Our calculations also demonstrate that the energy spectrum evolution is not obvious if n b /n 0 = 0.001 as Tang et al. asserted. The present results are helpful for a more comprehensive understanding of the dynamic spectra of HXR and radio emissions from FEBs.

Effect of the magnetization parameter on electron acceleration during relativistic magnetic reconnection in ultra-intense laser-produced plasma

Relativistic magnetic reconnection (MR) driven by two ultra-intense lasers with different spot separation distances is simulated by a three-dimensional (3D) kinetic relativistic particle-in-cell (PIC) code. We find that changing the separation distance between two laser spots can lead to different magnetization parameters of the laser plasma environment. As the separation distance becomes larger, the magnetization parameter σ becomes smaller. The electrons are accelerated in these MR processes and their energy spectra can be fitted with double power-law spectra whose index will increase with increasing separation distance. Moreover, the collisionless shocks’ contribution to energetic electrons is close to the magnetic reconnection contribution with σ decreasing, which results in a steeper electron energy spectrum. Basing on the 3D outflow momentum configuration, the energetic electron spectra are recounted and their spectrum index is close to 1 in these three cases because the magnetization parameter σ is very high in the 3D outflow area.

Parametric Evolution of Power-law Energy Spectra of Flare Accelerated Electrons in the Solar Atmosphere

Abstract It is well known that solar hard X-ray bursts (HXRs) and solar radio bursts (SRBs) from solar flares both are produced directly by fast electron beams (FEBs) traveling through the solar atmosphere. The observed characteristics of HXRs and SRBs sensitively depend on the energy distribution of FEBs, which are believed commonly to have a power-law energy spectrum. When FEBs propagating in the solar atmosphere, however, their energy spectra can considerably vary due to the interaction with the atmospheric plasmas and this may significantly influence the observational characteristics of the producing HXRs and SRBs. In the present paper, based on flare atmospheric models, we investigate the parametric evolution of power-law spectra of FEBs due to their energy losses when propagating along flaring loops. The results show that an initially single power-law spectrum with a lower-energy cutoff can evolve into a more complex double power-law spectrum or a broken power-law spectrum with multi-breaking knees because of the dependence of the energy loss on the initial energies. The possible effects of the energy-spectral evolution of FEBs on observational characteristics of their HXR flares and SRBs are discussed. The present results are helpful to understand the physics of dynamical spectra of HXRs and SRBs from solar flares.

Double power-law spectra of energetic electrons in the Earth magnetotail

Abstract. In this paper, we consider electron acceleration in the vicinity of X-line and corresponding formation of energy spectra. We develop an analytical model including the effect of the electron trapping by electrostatic fields and surfing acceleration. Speiser, Fermi and betatron mechanisms of acceleration are also taken into account. Analytical estimates are verified by the numerical integration of electron trajectories. The surfing mechanism and adiabatic heating are responsible for the formation of the double power-law spectrum in agreement with the previous studies. The energy of the spectrum knee is about ~150 keV for typical conditions of the Earth magnetotail. We compare theoretical results with the spacecraft observations of electron double power-law spectra in the magnetotail and demonstrate that the theory is able to describe typical energy of the spectra knee. We also estimate the role of relativistic effects and magnetic field fluctuations on the electron acceleration: the acceleration is more stable for relativistic electrons, while fluctuations of the magnetic field cannot significantly decrease the gained energy for typical magnetospheric conditions.

THE γ-RAY SPECTRUM OF GEMINGA AND THE INVERSE COMPTON MODEL OF PULSAR HIGH-ENERGY EMISSION

We reanalyze the Fermi spectra of the Geminga and Vela pulsars. We find that the spectrum of Geminga above the break is exceptionally well approximated by a simple power law without the exponential cut-off, making Geminga's spectrum similar to that of Crab. Vela's broadband gamma-ray spectrum is equally well fit with both the exponential cut-off and the double power law shapes. In the broadband double power-law fits, for a typical Fermi spectrum of a bright \gamma-ray pulsar, most of the errors accumulate due to the arbitrary parametrization of the spectral roll-off. In addition, a power law with an exponential cut-off gives an acceptable fit for the underlying double power-law spectrum for a very broad range of parameters, making such fitting procedures insensitive to the underlying Fermi photon spectrum. Our results have important implications for the mechanism of pulsar high energy emission. A number of observed properties of \gamma-ray pulsars, i.e., the broken power law spectra without exponential cut-offs and stretching in case of Crab beyond the maximal curvature limit, spectral breaks close to or exceeding the maximal breaks due to curvature emission, a Crab patterns of relative intensities of the leading and trailing pulses repeated in the X-ray and \gamma-ray regions, all point to the inverse Compton origin of the high energy emission from majority of pulsars.

ON THE ORIGIN OF THE 1/ f SPECTRUM IN THE SOLAR WIND MAGNETIC FIELD

We present a mechanism for the formation of the low frequency 1/f magnetic spectrum based on numerical solutions of a shell reduced-MHD model of the turbulent dynamics inside the sub-Alfv\'enic solar wind. We assign reasonably realistic profiles to the wind speed and the density along the radial direction, and a radial magnetic field. Alfv\'en waves of short periodicity (600 s) are injected at the base of the chromosphere, penetrate into the corona and are partially reflected, thus triggering a turbulent cascade. The cascade is strong for the reflected wave while it is weak for the outward propagating waves. Reflection at the transition region recycles the strong turbulent spectrum into the outward weak spectrum, which is advected beyond the Alfv\'enic critical point without substantial evolution. There, the magnetic field has a perpendicular power-law spectrum with slope close to the Kolmogorov -5/3. The parallel spectrum is inherited from the frequency spectrum of large (perpendicular) eddies. The shape is a double power-law with slopes of -1 and -2 at low and high frequencies respectively, the position of the break depending on the injected spectrum. We suggest that the double power-law spectrum measured by Helios at 0.3 AU, where the average magnetic field is not aligned with the radial (contrary to our assumptions) results from the combination of such different spectral slopes. At low frequency the parallel spectrum dominates with its characteristic 1/f shape, while at higher frequencies its steep spectral slope (-2) is masked by the more energetic perpendicular spectrum (slope -5/3).

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.

Proton acceleration in a single-loop disrupted during collision of two moving solitary magnetic kinks

We investigate a new model of single-loop flare based on the fact that a magnetic flux tube with axial current can be explosively disrupted during collision of two moving solitary magnetic kinks (MoSMaK), particularly paying attention to an acceleration mechanism of high energy protons. By using three-dimensional electromagnetic fields obtained from a resistive three-dimensional MHD equations during the single-loop flare, we investigate the orbit of many protons to obtain their energy spectra. We found that the protons can be accelerated to γ-ray-emitting energies (>1 MeV) with double power-law spectra up to about 25 MeV. There appears a breaking point of the index of the power-law spectra from 1.8 to 2.1 near 10 MeV. The protons are accelerated mainly in one direction along the loop by the electric field produced near the three-dimensional localized current associated with the magnetic reconnection process in the disrupted loop. We study the loop disruption for two different plasma beta in the center of the loop; and . In a high beta case (), the disruption of the loop is more violent than that of . We found that the proton acceleration for the case of occurs mainly at two different regions; one is inside the loop and the other is in the upper part of the disrupting loop. However, the energy spectra of protons are very similar for both cases.

1∕f noise in hydrogenated amorphous silicon–germanium alloys

Measurements were made of conductance noise of a-Si:H and a-Si1−xGex:H in two different geometries: one where the current flow is transverse to the surface and the other where it is longitudinal to the surface. Because of the large change in sample resistance between the two geometries, it was not possible to measure both geometries at the same temperature. For both geometries, alloying with up to 40% Ge reduces the noise magnitude by several orders of magnitude over that found in a-Si:H. The decrease is incompatible with several popular noise models. Extrapolating the temperature trends for each geometry shows that it is possible that the noise observed in the transverse samples has the same origin as the higher frequency part of the double power law spectra observed in the longitudinal samples.

Alfven waves in regions of magnetic reconnection and the acceleration of newborn ions

We use a 2-D hybrid numerical simulation to study velocity-driven magnetic reconnection in low-beta plasmas. Our results show that Alfven waves can be produced in the process. Under their action newborn ions undergo pitch-angle scattering and assume a spherical shell distribution. A part of the ions are accelerated to a maximum energy of about 4( m iV 2 A0 /2). The acceleration lasts over a time scale of 100/Ω i , and is extremely rapid. The energy spectrum of the accelerated ions is a double power-law spectrum.

On the Lower Energy Cutoff of Nonthermal Electrons in Solar Flares

A quantitative method to determine the lower energy cutoff (E,) of power-law electron beams is established. We apply this method to the 54 hard X-ray events observed with BATSE/Compton Gamma Ray Observatory (CGRO). The results show that about 75% of the observed broken double power-law spectra of hard X-rays can be explained by a lower energy cutoff in the power-law electron beams. The values of E,, varying among the flares, are all greater than the usually accepted 20 keV! On average, E-c is about 69 keV. So high a lower energy cutoff of nonthermal electrons implies that nonthermal electrons might not be as important in powering solar flares as was previously thought. Further significance of this finding is discussed.