Models Of Gamma-ray Emission Research Articles

Cosmic-Ray Injection from Star-Forming Regions.

At present, all physical models of diffuse Galactic γ-ray emission assume that the distribution of cosmic-ray sources traces the observed populations of either OB stars, pulsars, or supernova remnants. However, since H_{2}-rich regions host significant star formation and numerous supernova remnants, the morphology of observed H_{2} gas (as traced by CO line surveys) should also provide a physically motivated, high-resolution tracer for cosmic-ray injection. We assess the impact of utilizing H_{2} as a tracer for cosmic-ray injection on models of diffuse Galactic γ-ray emission. We employ state-of-the-art 3D particle diffusion and gas density models, along with a physical model for the star-formation rate based on global Schmidt laws. Allowing a fraction, f_{H_{2}}, of cosmic-ray sources to trace the observed H_{2} density, we find that a theoretically well-motivated value f_{H_{2}}∼0.20-0.25 (i)provides a significantly better global fit to the diffuse Galactic γ-ray sky and (ii)highly suppresses the intensity of the residual γ-ray emission from the Galactic center region. Specifically, in models utilizing our best global fit values of f_{H_{2}}∼0.20-0.25, the spectrum of the galactic center γ-ray excess is drastically affected, and the morphology of the excess becomes inconsistent with predictions for dark matter annihilation.

Improved cosmic-ray injection models and the Galactic Center gamma-ray excess

Fermi-LAT observations of the Galactic Center (GC) have revealed a spherically- symmetric excess of GeV gamma rays extending to at least 10 deg from the dynamical center of the Galaxy. A critical uncertainty in extracting the intensity, spectrum, and morphology of this excess concerns the accuracy of astrophysical diffuse gamma-ray emission models near the GC. Recently, it has been noted that many diffuse emission models utilize a cosmic-ray injection rate far below that predicted based on the observed star formation rate in the Central Molecular Zone. In this study, we add a cosmic-ray injection component which non-linearly traces the Galactic H2 density determined in three-dimensions, and find that the associated gamma-ray emission is degenerate with many properties of the GC gamma-ray excess. In models that utilize a large sideband (40x40 deg surrounding the GC) to normalize the best-fitting diffuse emission models, the intensity of the GC excess decreases by a factor of 2, and the morphology of the excess becomes less peaked and less spherically symmetric. In models which utilize a smaller region of interest (15x15 deg) the addition of an excess template instead suppresses the intensity of the best-fit astrophysical diffuse emission, and the GC excess is rather resilient to changes in the details of the astrophysical diffuse modeling. In both analyses, the addition of a GC excess template still provides a statistically significant improvement to the overall fit to the gamma-ray data. We also implement advective winds at the GC, and find that the Fermi-LAT data strongly prefer outflows of order several hundred km/s, whose role is to efficiently advect low-energy cosmic rays from the Galactic center. Finally, we perform numerous tests of our models, and conclude that they significantly improve our understanding of multi-wavelength non-thermal emission from the GC.

Variability of GeV gamma-ray emission in QSO B0218+357 due to microlensing on intermediate size structures

Strong gravitational lensing leads to an occurrence of multiple images, with different magnifications, of a lensed source. Those magnifications can in turn be modified by microlensing on smaller mass scales within the lens. Recently, measurements of the changes in the magnification ratio of the individual images have been proposed as a powerful tool for estimation of the size and velocity of the emission region in the lensed source. The changes of the magnification ratios in blazars PKS1830-211 and QSO B0218+357, if interpreted as caused by a microlensing on individual stars, put strong constraints on those two variables. These constraints are difficult to accommodate with the current models of gamma-ray emission in blazars. In this paper we study if similar changes in the magnification ratio can be caused by microlensing on intermediate size structures in the lensing galaxy. We investigate in details three classes of possible lenses: globular clusters (GC), open clusters (OC) and giant molecular clouds (GMC). We apply this scenario to the case of QSO B0218+357. Our numerical simulations show that changes in magnifi- cations with similar time scales can be obtained for relativistically moving emission regions with sizes up to 0.01 pc in the case of microlensing on the cores of GCs or clumps in GMCs. From the density of such structures in spiral galaxies we estimate however that lensing in giant molecular clouds would be more common.

Three-dimensional analytical description of magnetized winds from oblique pulsars

Rotating neutron stars, or pulsars, are plausibly the source of power behind many astrophysical systems, such as gamma-ray bursts, supernovae, pulsar wind nebulae and supernova remnants. In the past several years, 3D numerical simulations made it possible to compute pulsar spindown luminosity from first principles and revealed that oblique pulsar winds are more powerful than aligned ones. However, what causes this enhanced power output of oblique pulsars is not understood. In this work, using time-dependent 3D magnetohydrodynamic (MHD) and force-free simulations, we show that, contrary to the standard paradigm, the open magnetic flux, which carries the energy away from the pulsar, is laterally non-uniform. We argue that this non-uniformity is the primary reason for the increased luminosity of oblique pulsars. To demonstrate this, we construct simple analytic descriptions of aligned and orthogonal pulsar winds and combine them to obtain an accurate 3D description of the pulsar wind for any obliquity. Our approach describes both the warped magnetospheric current sheet and the smooth variation of pulsar wind properties outside of it. We find that generically the magnetospheric current sheet separates plasmas that move at mildly relativistic velocities relative to each other. This suggests that the magnetospheric reconnection is a type of driven, rather than free, reconnection. The jump in magnetic field components across the current sheet decreases with increasing obliquity, which could be a mechanism that reduces dissipation in near-orthogonal pulsars. Our analytical description of the pulsar wind can be used for constructing models of pulsar gamma-ray emission, pulsar wind nebulae, and magnetar-powered core-collapse gamma-ray bursts and supernovae.

NEW LIMITS ON GAMMA-RAY EMISSION FROM GALAXY CLUSTERS

Galaxy clusters are predicted to produce gamma-rays through cosmic ray interactions and/or dark matter annihilation, potentially detectable by the Fermi Large Area Telescope (Fermi-LAT). We present a new, independent stacking analysis of Fermi-LAT photon count maps using the 78 richest nearby clusters (z<0.12) from the Two Micron All-Sky Survey (2MASS) cluster catalog. We obtain the lowest limit on the photon flux to date, 2.3e-11 ph/s/cm^2 (95% confidence) per cluster in the 0.8-100 GeV band, which corresponds to a luminosity limit of 3.5e44 ph/s. We also constrain the emission limits in a range of narrower energy bands. Scaling to recent cosmic ray acceleration and gamma-ray emission models, we find that cosmic rays represent a negligible contribution to the intra-cluster energy density and gas pressure.

CONSTRAINTS ON VERY HIGH ENERGY EMISSION FROM GRB 130427A

Prompt emission from the very fluent and nearby (z=0.34) gamma-ray burst GRB 130427A was detected by several orbiting telescopes and by ground-based, wide-field-of-view optical transient monitors. Apart from the intensity and proximity of this GRB, it is exceptional due to the extremely long-lived high-energy (100 MeV to 100 GeV) gamma-ray emission, which was detected by the Large Area Telescope on the Fermi Gamma-ray Space Telescope for ~70 ks after the initial burst. The persistent, hard-spectrum, high-energy emission suggests that the highest-energy gamma rays may have been produced via synchrotron self-Compton processes though there is also evidence that the high-energy emission may instead be an extension of the synchrotron spectrum. VERITAS, a ground-based imaging atmospheric Cherenkov telescope array, began follow-up observations of GRB 130427A ~71 ks (~20 hr) after the onset of the burst. The GRB was not detected with VERITAS; however, the high elevation of the observations, coupled with the low redshift of the GRB, make VERITAS a very sensitive probe of the emission from GRB 130427A for E > 100 GeV. The non-detection and consequent upper limit derived place constraints on the synchrotron self-Compton model of high-energy gamma-ray emission from this burst.

Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance

The quantum electrodynamical (QED) process of Compton scattering in strong magnetic fields is commonly invoked in atmospheric and inner magnetospheric models of x-ray and soft gamma-ray emission in high-field pulsars and magnetars. A major influence of the field is to introduce resonances at the cyclotron frequency and its harmonics, where the incoming photon accesses thresholds for the creation of virtual electrons or positrons in intermediate states with excited Landau levels. At these resonances, the effective cross section typically exceeds the classical Thomson value by over 2 orders of magnitude. Near and above the quantum critical magnetic field of 44.13 TeraGauss, relativistic corrections must be incorporated when computing this cross section. This paper presents formalism for the QED magnetic Compton differential cross section valid for both subcritical and supercritical fields, yet restricted to scattered photons that are below pair creation threshold. Calculations are developed for the particular case of photons initially propagating along the field, mathematically simple specializations that are germane to interactions involving relativistic electrons frequently found in neutron star magnetospheres. This exposition of relativistic, quantum, magnetic Compton cross sections treats electron spin dependence fully, since this is a critical feature for describing the finite decay lifetimes of the intermediate states. The formalism employs both the Johnson and Lippmann (JL) wave functions and the Sokolov and Ternov (ST) electron eigenfunctions of the magnetic Dirac equation. The ST states are formally correct for self-consistently treating spin-dependent effects that are so important in the resonances. Relatively compact analytic forms for the cross sections are presented that will prove useful for astrophysical modelers.

An analysis of Fermi-LAT observations of the outskirts of the Coma cluster of galaxies

We analyse the data from the Fermi-Large Area Telescope in order to search for a ring-like gamma-ray structure around the Coma cluster. The ring-like structure has recently been suggested to be detected with VERITAS at energies higher than 220 GeV and could possibly be associated with an accretion shock. Our analysis of the Fermi data is performed at energies > 100 MeV and we find no detection of this structure in the Fermi data. We derive the 95 per cent upper limit on the flux from the region covering the proposed ring-like structure. The derived upper limit on the flux at > 100 MeV cannot be incorporated with the detection of an accretion shock wave around Coma at the significance of 4.5 sigma by VERITAS at very high energies, if the production mechanism of the gamma-ray emission generates a photon spectrum with a power index of 2 in the broad energy band. The model of gamma-ray emission induced by ultrahigh-energy protons can reconcile the results of the VERITAS and Fermi observations.

DIFFUSE PIONIC GAMMA-RAY EMISSION FROM LARGE-SCALE STRUCTURES IN THEFERMIERA

For more than a decade now the complete origin of the diffuse gamma-ray emission background (EGRB) has been unknown. Major components like unresolved star-forming galaxies (making <50% of the EGRB) and blazars (<23%), have failed to explain the entire background observed by \emph{Fermi}. Another, though subdominant, contribution is expected to come from the process of large-scale structure formation. The growth of structures is accompanied by accretion and merger shocks, which would, with at least some magnetic field present, give rise to a population of structure-formation cosmic rays (SFCRs). Though expected, this cosmic-ray population is still hypothetical and only very weak limits have been placed to their contribution to the EGRB. The most promising insight into SFCRs was expected to come from \emph{Fermi}-LAT observations of clusters of galaxies, however, only upper limits and no detection have been placed. Here, we build a model of gamma-ray emission from large-scale accretion shocks implementing a source evolution calibrated with the \emph{Fermi}-LAT cluster observation limits. Though our limits to the SFCR gamma-ray emission are weak (above the observed EGRB) in some case, in others, some of our models can provide a good fit to the observed EGRB. More importantly, we show that these large-scale shocks could still give an important contribution to the EGRB, especially at high energies. Future detections of cluster gamma-ray emission would help place tighter constraints on our models and give us a better insight into large-scale shocks forming around them.

Gamma-ray emission from proton–proton interactions in hot accretion flows

We present a model of gamma-ray emission through neutral pion production and decay in two-temperature accretion flows around supermassive black holes. We refine previous studies of such a hadronic gamma-ray emission by taking into account (1) relativistic effects in the photon transfer and (2) absorption of gamma-ray photons in the radiation field of the flow. We use a fully general relativistic description of both the radiative and hydrodynamic processes, which allows us to study the dependence on the black hole spin. The spin value strongly affects the gamma-ray emissivity within ~10 gravitational radii. The central regions of flows with the total luminosities L < 0.001 of the Eddington luminosity are mostly transparent to photons with energies below 10 GeV, permitting investigation of the effects of space-time metric. For such L, an observational upper limit on the gamma-ray (0.1-10 GeV) to X-ray (2-10 keV) luminosity ratio of L_{0.1-10 GeV}/L_{2-10 keV} << 0.1 can rule out rapid rotation of the black hole; on the other hand, a measurement of L_{0.1-10 GeV}/L_{2-10 keV} ~ 0.1 cannot be regarded as the evidence of rapid rotation, as such a ratio can also result from a flat radial profile of gamma-ray emissivity (which would occur for nonthermal acceleration of protons in the whole body of the flow). At L > 0.01 of the Eddington luminosity, the gamma-ray emission from the innermost region is strongly absorbed and the observed gamma-rays do not carry information on the value of the black hole spin. We note that if the X-ray emission observed in Centaurus A comes from an accretion flow, the hadronic gamma-ray emission from the flow should contribute significantly to the MeV/GeV emission observed from the core of this object, unless it contains a slowly rotating black hole and protons in the flow are thermal.

Observation of SN2011fe with INTEGRAL

SN2011fe was detected by the Palomar Transient Factory on August 24th 2011 in M101 a few hours after the explosion. From the early optical spectra it was immediately realized that it was a Type Ia supernova thus making this event the brightest one discovered in the last twenty years. The distance of the event offered the rare opportunity to perform a detailed observation with the instruments on board of INTEGRAL to detect the gamma-ray emission expected from the decay chains of $^{56}$Ni. The observations were performed in two runs, one before and around the optical maximum, aimed to detect the early emission from the decay of $^{56}$Ni and another after this maximum aimed to detect the emission of $^{56}$Co. The observations performed with the instruments on board of INTEGRAL (SPI, IBIS/ISGRI, JEMX and OMC) have been analyzed and compared with the existing models of gamma-ray emission from such kind of supernovae. In this paper, the analysis of the gamma-ray emission has been restricted to the first epoch. Both, SPI and IBIS/ISGRI, only provide upper-limits to the expected emission due to the decay of $^{56}$Ni. These upper-limits on the gamma-ray flux are of 7.1 $\times$ 10$^{-5}$ ph/s/cm$^2$ for the 158 keV line and of 2.3 $\times$ 10$^{-4}$ ph/s/cm$^2$ for the 812 keV line. These bounds allow to reject at the $2\sigma$ level explosions involving a massive white dwarf, $\sim 1$ M$\odot$ in the sub--Chandrasekhar scenario and specifically all models that would have substantial amounts of radioactive $^{56}$Ni in the outer layers of the exploding star responsible of the SN2011fe event. The optical light curve obtained with the OMC camera also suggests that SN2011fe was the outcome of the explosion, possibly a delayed detonation although other models are possible, of a CO white dwarf that synthesized $\sim 0.55$ M$_\odot$ of $^{56}$Ni. For this specific model.

COOLING RATES FOR RELATIVISTIC ELECTRONS UNDERGOING COMPTON SCATTERING IN STRONG MAGNETIC FIELDS

For inner magnetospheric models of hard X-ray and gamma-ray emission in high-field pulsars and magnetars, resonant Compton upscattering is anticipated to be the most efficient process for generating continuum radiation. This is due in part to the proximity of a hot soft photon bath from the stellar surface to putative radiation dissipation regions in the inner magnetosphere. Moreover, because the scattering process becomes resonant at the cyclotron frequency, the effective cross section exceeds the classical Thomson value by over two orders of magnitude, thereby enhancing the efficiency of continuum production and the cooling of relativistic electrons. This paper presents computations of the electron cooling rates for this process, which are needed for resonant Compton models of non-thermal radiation from such highly-magnetized pulsars. The computed rates extend previous calculations of magnetic Thomson cooling to the domain of relativistic quantum effects, sampled near and above the quantum critical magnetic field of 44.13 TeraGauss. This is the first exposition of fully relativistic, quantum magnetic Compton cooling rates for electrons, and it employs both the traditional Johnson and Lippman cross section, and a newer Sokolov and Ternov (ST) formulation of Compton scattering in strong magnetic fields. Such ST formalism is formally correct for treating spin-dependent effects that are important in the cyclotron resonance, and has not been addressed before in the context of cooling by Compton scattering. The QED effects are observed to profoundly lower the rates below extrapolations of the familiar magnetic Thomson results, as expected, when recoil and Klein-Nishina reductions become important.

GAMMA-RAY EMISSION OF ACCELERATED PARTICLES ESCAPING A SUPERNOVA REMNANT IN A MOLECULAR CLOUD

We present a model of gamma-ray emission from core-collapse supernovae originating from the explosions of massive young stars. The fast forward shock of the supernova remnant (SNR) can accelerate particles by diffusive shock acceleration (DSA) in a cavern blown by a strong, pre-supernova stellar wind. As a fundamental part of nonlinear DSA, some fraction of the accelerated particles escape the shock and interact with a surrounding massive dense shell producing hard photon emission. To calculate this emission, we have developed a new Monte Carlo technique for propagating the cosmic rays (CRs) produced by the forward shock of the SNR, into the dense, external material. This technique is incorporated in a hydrodynamic model of an evolving SNR which includes the nonlinear feedback of CRs on the SNR evolution, the production of escaping CRs along with those that remain trapped within the remnant, and the broad-band emission of radiation from trapped and escaping CRs. While our combined CR-hydro-escape model is quite general and applies to both core collapse and thermonuclear supernovae, the parameters we choose for our discussion here are more typical of SNRs from very massive stars whose emission spectra differ somewhat from those produced by lower mass progenitors directly interacting with a molecular cloud.

A unified polar cap/striped wind model for pulsed radio and gamma-ray emission in pulsars

(abridged) Thanks to the recent discovery by Fermi of about fifty new gamma-ray pulsars, it becomes possible to look for statistical properties of their pulsed high-energy emission, especially their light-curves and phase-resolved spectra. These pulsars emit by definition mostly gamma-ray photons but some of them are also detected in the radio band. For those seen in these two extreme energies, the relation between time lag of radio/gamma-ray pulses and gamma-ray peak separation, in case both high-energy pulses are seen, helps to put some constrain on the magnetospheric emission mechanisms and location. This idea is analyzed in detail in this paper, assuming a polar cap model for the radio pulses and the striped wind geometry for the pulsed high-energy counterpart. Combining the time-dependent emissivity in the wind, supposed to be inverse Compton radiation, with a simple polar cap emission model along and around the magnetic axis, we compute the radio and gamma-ray light-curves, summarizing the results in several phase plots. The phase lag as well as the gamma-ray peak separation dependence on the pulsar inclination angle and on the viewing angle are studied. Using the gamma-ray pulsar catalog compiled from the Fermi data, we are able to predict the radio lag/peak separation relation and compare it with available observations taken from this catalog.

Signatures of photon and axion-like particle mixing in the gamma-ray burst jet

Photons couple to Axion-Like Particles (ALPs) or more generally to any pseudo Nambu-Goldstone boson in the presence of an external electromagnetic field. Mixing between photons and ALPs in the strong magnetic field of a Gamma-Ray Burst (GRB) jet during the prompt emission phase can leave observable imprints on the gamma-ray polarization and spectrum. Mixing in the intergalactic medium is not expected to modify these signatures for ALP mass > 10−14 eV and/or for < nG magnetic field. We show that the depletion of photons due to conversion to ALPs changes the linear degree of polarization from the values predicted by the synchrotron model of gamma ray emission. We also show that when the magnetic field orientation in the propagation region is perpendicular to the field orientation in the production region, the observed synchrotron spectrum becomes steeper than the theoretical prediction and as detected in a sizable fraction of GRB sample. Detection of the correlated polarization and spectral signatures from these steep-spectrum GRBs by gamma-ray polarimeters can be a very powerful probe to discover ALPs. Measurement of gamma-ray polarization from GRBs in general, with high statistics, can also be useful to search for ALPs.

Constraining dark matter in galactic substructure

Detecting the dark matter annihilation signal from Galactic substructure, or subhalos, is an important challenge for high-energy gamma-ray experiments. In this paper we discuss detection prospects by combining two different aspects of the gamma-ray signal: the angular distribution and the photon counts probability distribution function (PDF). The true PDF from subhalos has been shown recently (by Lee et al.) to deviate from Poisson; we extend this analysis and derive the signal PDF from a detailed Lambda-CDM-based model for the properties of subhalos. We combine our PDF with a model for Galactic and extra-Galactic diffuse gamma-ray emission to obtain an estimator and projected error on dark matter particle properties (mass and annihilation cross section) using the Fermi Gamma-Ray Space Telescope. We compare the estimator obtained from the true PDF to that obtained from the simpler Poisson analysis. We find that, although both estimators are unbaised in the presence of backgrounds, the error on dark matter properties derived from the true PDF is ~50% smaller than when utilizing the Poisson-based analysis.

HIGH-ENERGY EMISSION COMPONENTS IN THE SHORT GRB 090510

We investigate the origin of the prompt and delayed emission observed in the short GRB 090510. We use the broad-band data to test whether the most popular theoretical models for gamma-ray burst emission can accommodate the observations for this burst. We first attempt to explain the soft-to-hard spectral evolution associated with the delayed onset of a GeV tail with the hypothesis that the prompt burst and the high energy tail both originate from a single process, namely synchrotron emission from internal shocks. Considerations on the compactness of the source imply that the high-energy tail should be produced in a late-emitted shell, characterized by a Lorentz factor greater than the one generating the prompt burst. However, in this hypothesis, the predicted evolution of the synchrotron peak frequency does not agree with the observed soft-to-hard evolution. Given the difficulties of a single-mechanism hypothesis, we test two alternative double-component scenarios. In the first, the prompt burst is explained as synchrotron radiation from internal shocks, and the high energy emission (up to about 1 s following the trigger) as internal shock synchrotron-self-Compton. In the second scenario, in view of its long duration (\sim 100 s), the high energy tail is decoupled from the prompt burst and has an external shock origin. In this case, we show that a reasonable choice of parameters does indeed exist to accommodate the optical-to-GeV data, provided the Lorentz factor of the shocked shell is sufficiently high. Finally, we attempt to explain the chromatic break observed around \sim 1e3 s with a structured jet model. We find that this might be a viable explanation, and that it lowers the high value of the burst energy derived assuming isotropy, \sim 1e53 erg, below \sim 1e49 erg, more compatible with the energetics from a binary merger progenitor.

UNCERTAINTIES OF MODELING GAMMA-RAY PULSAR LIGHT CURVES USING VACUUM DIPOLE MAGNETIC FIELD

Current models of pulsar gamma-ray emission use the magnetic field of a rotating dipole in vacuum as a first approximation to the shape of a plasma-filled pulsar magnetosphere. In this paper, we revisit the question of gamma-ray light curve formation in pulsars in order to ascertain the robustness of the "two-pole caustic (TPC)" and "outer gap (OG)" models based on the vacuum magnetic field. We point out an inconsistency in the literature on the use of the relativistic aberration formula, where in several works the shape of the vacuum field was treated as known in the instantaneous corotating frame, rather than in the laboratory frame. With the corrected formula, we find that the peaks in the light curves predicted from the TPC model using the vacuum field are less sharp. The sharpness of the peaks in the OG model is less affected by this change, but the range of magnetic inclination angles and viewing geometries resulting in double-peaked light curves is reduced. In a realistic magnetosphere, the modification of field structure near the light cylinder (LC) due to plasma effects may change the shape of the polar cap and the location of the emission zones. We study the sensitivity of the light curves to different shapes of the polar cap for static and retarded vacuum dipole fields. In particular, we consider polar caps traced by the last open field lines and compare them to circular polar caps. We find that the TPC model is very sensitive to the shape of the polar cap, and a circular polar cap can lead to four peaks of emission. The OG model is less affected by different polar cap shapes, but is subject to big uncertainties of applying the vacuum field near the LC. We conclude that deviations from the vacuum field can lead to large uncertainties in pulse shapes, and a more realistic force-free field should be applied to the study of pulsar high-energy emission.

Scaling laws in high-energy inverse compton scattering. II. Effect of bulk motions

We study the inverse Compton scattering of the CMB photons off high-energy nonthermal electrons. We extend the formalism obtained by the previous paper to the case where the electrons have non-zero bulk motions with respect to the CMB frame. Assuming the power-law electron distribution, we find the same scaling law for the probability distribution function P_{1,K}(s) as P_{1}(s) which corresponds to the zero bulk motions, where the peak height and peak position depend only on the power-index parameter. We solved the rate equation analytically. It is found that the spectral intensity function also has the same scaling law. The effect of the bulk motions to the spectral intensity function is found to be small. The present study will be applicable to the analysis of the X-ray and gamma-ray emission models from various astrophysical objects with non-zero bulk motions such as radio galaxies and astrophysical jets.

Scaling laws in high-energy inverse Compton scattering

Based upon the rate equations for the photon distribution function obtained in the previous paper, we study the inverse Compton scattering process for high-energy nonthermal electrons. Assuming the power-law electron distribution, we find a scaling law in the probability distribution function P_1(s), where the peak height and peak position depend only on the power index parameter. We solved the rate equation analytically. It is found that the spectral intensity function also has the scaling law, where the peak height and peak position depend only on the power index parameter. The present study will be particularly important to the analysis of the X-ray and gamma-ray emission models from various astrophysical objects such as radio galaxies and supernova remnants.