Cosmic rays are a fundamental source of ionization for molecular and diffuse clouds, influencing their chemical, thermal, and dynamical evolution. The amount of cosmic rays inside a cloud also determines the $γ$-ray flux produced by hadronic collisions between cosmic rays and cloud material. We study the spectrum of cosmic rays inside and outside of a diffuse cloud, by solving the stationary transport equation for cosmic rays including diffusion, advection and energy losses due to ionization of neutral hydrogen atoms. We found that the cosmic ray spectrum inside a diffuse cloud differs from the one in the interstellar medium for energies smaller than $E_{br}\approx 100$ MeV, irrespective of the model details. Below $E_{br}$, the spectrum is harder (softer) than that in the interstellar medium if the latter is a power law $\propto p^{-s}$ with $s$ larger (smaller) than $\sim 0.42$.
A flux of cosmic rays (CRs) propagating through a diffuse ionized gas can excite MHD waves, thus generating magnetic disturbances. We propose a generic model of CR penetration into molecular clouds through their diffuse envelopes, and identify the leading physical processes controlling their transport on the way from a highly ionized interstellar medium to a dense interior of the cloud. The model allows us to describe a transition between a free streaming of CRs and their diffusive propagation, determined by the scattering on the self-generated disturbances. A self consistent set of equations,…
Cosmic rays are a fundamental source of ionization for molecular and diffuse clouds, influencing their chemical, thermal, and dynamical evolution. The amount of cosmic rays inside a cloud also determines the $γ$-ray flux produced by hadronic collisions between cosmic rays and cloud material. We study the spectrum of cosmic rays inside and outside of a diffuse cloud, by solving the stationary transport equation for cosmic rays including diffusion, advection and energy losses due to ionization of neutral hydrogen atoms. We found that the cosmic ray spectrum inside a diffuse cloud differs from th…
Aims. We use Fermi and Planck data to calculate the cross correlation between gamma ray signal and gas distribution in different scales in giant molecular clouds (GMC). Then we investigate the cosmic rays (CRs) penetration in GMCs with these informations. Methods.We use the wavelet technique to decompose both the gamma ray and dust opacity maps in different scales, then we calculate the wavelet cross correlation functions in these scales. We also define wavelet response as an analog to the impulsive response in Fourier transform and calculate that in different scales down to Fermi angular reso…
The Tibet ASgamma experiment just reported their measurement of sub-PeV diffuse gamma ray emission from the Galactic disk, with the highest energy up to 957 TeV. These gamma-rays are most likely the hadronic origin by cosmic ray interaction with interstellar gas in the Galaxy. This measurement provides direct evidence to the hypothesis that the Galactic cosmic rays can be accelerated beyond PeV energies. In this work, we try to explain the sub-PeV diffuse gamma-ray spectrum within cosmic rays diffusive propagation model. We find there is a tension between the sub-PeV diffuse gamma rays and the…
The ANTARES neutrino telescope detects the Cherenkov radiation emitted along the path of charged particles produced in neutrino interactions. ANTARES is sensitive to all flavors even though it is optimized for muon neutrinos. Several algorithms estimating the deposited energy in the active volume of the detector have been developed and applied to the reconstruction of the primary neutrino energy - this allows to improve the search for a diffuse flux of astrophysical neutrinos. The search for a diffuse flux of cosmic neutrinos at very high energies (E_ν> 30 TeV) is updated using 4 years of data…
The origin of the knee in cosmic ray spectrum remains to be an unsolved fundamental problem. There are various kinds of models which predict different break positions and the compositions of the knee. In this work, we suggest to use diffuse $γ$-rays and neutrinos as probes to test these models. Based on several typical types of the composition models, the diffuse $γ$-ray and neutrino spectra are calculated, which show distinctive cutoff behaviours at energies from tens of TeV to multi-PeV. The expected flux will be observable by the newly upgraded Tibet-AS$γ$+MD (muon detector) experiment as w…
Cosmic rays accelerated to ultra-high energies (E>4x10^19 eV) in electric fields in accretion discs around supermassive black holes are discussed. Particle injection spectra are assumed to be harder than those formed in acceleration on shock fronts. It turned out that cosmic rays with injection spectra supposed contribute a little in the particle flux detected with ground-based arrays. But in the extragalactic space particle generate a noticeable flux of the diffuse gamma-ray emission compared with the data obtained with the Fermi LAT instrument (onboard the Fermi space observatory). The inten…
The diffuse flux of cosmic neutrinos has been measured by the IceCube Observatory from TeV to PeV energies. We show that an improved characterization of this flux at the lower energies, TeV and sub-TeV, reveals important information on the nature of the astrophysical neutrino sources in a model-independent way. Most significantly, it could confirm the present indications that neutrinos originate in cosmic environments that are optically thick to GeV-TeV $γ$-rays. This conclusion will become inevitable if a steeper or even uninterrupted neutrino power law is observed in the TeV region. In such …
Diffuse interstellar clouds show large abundances of H_3^+ which can be maintained only by a high ionization rate of H_2. Cosmic rays are the dominant ionization mechanism in this environment, so the large ionization rate implies a high cosmic-ray flux, and a large amount of energy residing in cosmic rays. In this paper we find that the standard propagated cosmic-ray spectrum predicts an ionization rate much lower than that inferred from H_3^+. Low-energy (~10 MeV) cosmic rays are the most efficient at ionizing hydrogen, but cannot be directly detected; consequently, an otherwise unobservable …
The Milky Way is surrounded by a gravitationally bound gas corona extending up to the Galaxy's virial radius. Interactions of cosmic-ray particles with this gas give rise to energetic secondary gamma rays and neutrinos. We present a quantitative analysis of the neutrino and gamma-ray fluxes from the corona of the Milky Way together with a combined contribution of coronae of other galaxies. The high-energy neutrino flux is insufficient to explain the IceCube results, while the contribution to the FERMI-LAT diffuse gamma-ray flux is not negligible.
Cosmic rays (CRs) generate diffuse emission while interacting with the Galactic magnetic field (B-field), the interstellar gas and the radiation field. This diffuse emission extends from radio, microwaves, through X-rays, to high-energy gamma rays. Diffuse emission has considerably increased the interest of the astrophysical community due to recent detailed observations by Planck, Fermi-LAT, and by very-high-energy Cherenkov telescopes. Observations of this emission and comparison with detailed predictions are used to gain information on the properties of CRs, such as their density, spectra, d…
The extragalactic sources of ultra-high-energy (E > 4x10^19 eV) cosmic rays that make a small contribution to the flux of particles recorded by ground-based arrays are discussed. We show that cosmic rays from such sources can produce a noticeable diffuse gamma-ray flux in intergalactic space compared to the the data obtained with Fermi LAT (onboard the Fermi space observatory). A possible type of active galactic nuclei (AGNs) in which cosmic-ray protons can be accelerated to energies 1021 eV is considered as an illustration of such sources. We conclude that ultra-high-energy cosmic rays from t…
In this work, we revisit the all-sky Galactic diffuse $γ$-ray emission taking into account the new measurements of cosmic ray electron/positron spectrum by PAMELA, ATIC and Fermi, which show excesses of cosmic electrons/positrons beyond the expected fluxes in the conventional model. Since the origins of the extra electrons/positrons are not clear, we consider three different scenarios to account for the excesses: the astrophysical sources such as the Galactic pulsars, dark matter decay and annihilation. Further, new results from Fermi-LAT of the (extra-)Galactic diffuse $γ$-ray are adopted. Th…
We use the recent measurement of the isotropic $γ$-ray background (IGRB) by Fermi LAT and analysis of the contribution of unresolved point $γ$-ray sources to IGRB to build constraints on the models of ultra-high cosmic rays (UHECR) origin. We also calculate the minimal expected diffuse $γ$-ray flux produced by UHECR interactions with an interstellar photon background. Finally, for the subclass of dark matter (DM) models with decaying weakly interacting massive particles (WIMP), we build constraints on the particle decay time using minimal expected contributions to the IGRB from unresolved poin…
The diffuse $γ$-ray spectrum at sub-PeV energy region has been measured for the first time by the Tibet-AS$γ$ experiment. It will shed new light on the understanding of origin and propagation of Galactic cosmic rays at very high energies. It has been pointed out that the traditional cosmic ray propagation model based on low energy measurements undershoot the new data, and modifications of the model with new ingredients or alternative propagation framework is required. In this work, we propose that the hadronic interactions between freshly accelerated cosmic rays and the medium surrounding the …
We demonstrate that the UHECRs produced in the nuclear cascade in the jet of Low-Luminosity Gamma-Ray Bursts (LL-GRBs) can describe the UHECR spectrum and composition and, at the same time, the diffuse neutrino flux at the highest energies. The radiation density in the source simultaneously controls the neutrino production and the development of the nuclear cascade, leading to a flux of nucleons and light nuclei describing even the cosmic-ray ankle at $5 \times 10^{18}$ eV. The derived source parameters are consistent with population studies, indicating a baryonic loading factor of about ten. …
Cosmic rays fill up the entire volume of galaxies, providing an important source of heating and ionisation of the interstellar medium, and may play a significant role in the regulation of star formation and galactic evolution. Diffuse emissions from radio to high-energy gamma rays (> 100 MeV) arising from various interactions between cosmic rays and the interstellar medium, interstellar radiation field, and magnetic field, are currently the best way to trace the intensities and spectra of cosmic rays in the Milky Way and other galaxies. In this contribution, I describe our recent work to model…
We estimate the sensitivity of LHAASO telescope for the large angular scale diffuse gamma-ray flux in multi-TeV - multi-PeV energy range. We discuss possible sources of the signal in this energy range including the guaranteed flux from cosmic ray interactions in the interstellar medium and possible flux from decaying dark matter. We show that LHAASO will be able to detect the diffuse cosmic ray induced gamma-ray flux up to high Galactic latitude regions thus providing firm identification of the Galactic cosmic ray component of the astrophysical neutrino signal detected by IceCube and clarifica…
We discuss ultra-high energy (UHE) cosmic rays (CR) from minor sources and their possible contribution to the extragalactic diffuse gamma-ray emission. As an illustration of minor sources we consider possible specific type of active galactic nuclei (AGN) in which supermassive black hole is surrounded by a super strong magnetic field of 10^(10)-10^(11) Gs. In this model we have calculated CR energy spectra at the Earth and intensity of cascade quanta produced by CRs in extragalactic space. Proceeding from numerical results and the data by Pierre Auger Observatory and Telescope Array it is shown…
We study diffuse gamma-ray emission at intermediate Galactic latitudes measured by the Fermi Large Area Telescope with the aim of searching for a signal from dark matter annihilation or decay. In the absence of a robust dark matter signal, we set conservative dark matter limits requiring that the dark matter signal does not exceed the observed diffuse gamma-ray emission. A second set of more stringent limits is derived based on modeling the foreground astrophysical diffuse emission. Uncertainties in the height of the diffusive cosmic-ray halo, the distribution of the cosmic-ray sources in the …
Significant advances have been made in the understanding of the diffuse Galactic hard X-ray continuum emission using data from the INTEGRAL observatory. The diffuse hard power-law component seen with the INTEGRAL/SPI spectrometer has been identified with inverse-Compton emission from relativistic (GeV) electrons on the cosmic microwave background and Galactic interstellar radiation field. In the present analysis, SPI data from 2003 to 2009, with a total exposure time of ~ 10^8 s, are used to derive the Galactic ridge hard X-ray spatial distribution and spectrum between 20 keV and 2.4 MeV. Both…
The propagation of particles accelerated at supernova remnant shocks and escaping the parent remnants is likely to proceed in a strongly non-linear regime, due to the efficient self-generation of Alfvén waves excited through streaming instability near the sources. Depending on the amount of neutral hydrogen present in the regions around the sites of supernova explosions, cosmic rays may accumulate an appreciable grammage in the same regions and get self-confined for non-negligible times, which in turn results in an enhanced rate of production of secondaries. Here we calculate the contribution …
We provide our estimates of the intensity of the gamma-ray emission with an energy near 0.1 TeV generated in inrergalactic space in the interactions of cosmic rays with background emissions. We assume that the cosmic ray sources are point-like and these are active galactic nuclei. The following pjssible types of sources are considered: remote and powerful ones, at redshifts up to z=1.1, with a monoenergetic particle spectrum, E=10^21 eV; the same objects, but with a power-law particle spectrum; and nearby sources with redshifts 0<z<=0.0092, i.e. at distances no larger than 50 Mpc also with a p…
It was recently discovered that in some regions of the Galaxy, the cosmic ray (CR) abundance is several orders of magnitude higher than previously thought. Additionally, there is evidence that in molecular cloud envelopes, the CR ionization may be dominated by electrons. We show that for regions with high, electron-dominated ionization, the penetration of CR electrons into molecular clouds is modulated by the electric field that develops as a result of the charge they deposit. We evaluate the significance of this novel mechanism of self-modulation and show that the CR penetration can be reduce…
Galactic Diffuse Gamma-Ray and Neutrino Emission from Cosmic-Ray Interactions in Stellar Atmospheres
Purple Mountain Observatory, Chinese Academy of Sciences（中国科学院紫金山天文台）；School of Astronomy and Space Science, University of Science and Technology of China（中国科学技术大学天文与空间科学学院）
We investigate the ionization of the diffuse interstellar medium by cosmic rays by modeling their propagation along the wandering magnetic fields using a Monte Carlo method. We study how low-energy cosmic rays propagate in turbulent, translucent molecular clouds, and how they regulate the ionization and both lose and gain energy from the medium. As a test case, we use high spatial resolution (0.03 pc) CO maps of a well-studied high latitude translucent cloud, MBM 3, to model turbulence. The propagation problem is solved with a modified Monte Carlo procedure that includes trapping, energization…
Low energy cosmic rays are the major ionization agents of molecular clouds. However, it has been shown that, if the cosmic ray spectrum measured by Voyager 1 is representative of the whole Galaxy, the predicted ionization rate in diffuse clouds fails to reproduce data by 1-2 orders of magnitude, implying that an additional source of ionization must exist. One of the solutions proposed to explain this discrepancy is based on the existence of an unknown low energy (in the range 1 keV-1 MeV, not probed by Voyager) cosmic ray component, called carrot when first hypothesized by Reeves and collabora…
Synchrotron diffuse radiation (SDR) emission is one of the major Galactic components, in the 100 MHz up to 100 GHz frequency range. Its spectrum and sky map provide valuable measure of the galactic cosmic ray electrons (GCRE) in the relevant energy range, as well as of the strength and structure of the Galactic magnetic fields (GMF), both regular and random ones. This emission is an astrophysical sky foreground for the study of the Cosmic Microwave Background (CMB), and the extragalactic microwave measurements, and it needs to be modelled as better as possible. In this regard, in order to get …
The Picard code for the numerical solution of the Galactic cosmic ray propagation problem allows for high-resolution models that acknowledge the 3D structure of our Galaxy. Picard was used to determine diffuse gamma-ray emission of the Galaxy over the energy range from 100 MeV to 100 TeV. We discuss the impact of a cosmic-ray source distribution aligned with the Galactic spiral arms for a range of such spiral-arm models. As expected, the impact on the gamma-ray emission is most distinct in the inverse-Compton channel, where imprints of the spiral arms are visible and yield predictions that are…
A search for a diffuse flux of astrophysical muon neutrinos, using data collected by the ANTARES neutrino telescope from December 2007 to December 2009 is presented. A $(0.83\times 2π)$ sr sky was monitored for a total of 334 days of equivalent live time. The searched signal corresponds to an excess of events, produced by astrophysical sources, over the expected atmospheric neutrino background without any particular assumption on the source direction. Since the number of detected events is compatible with the number of expected background events, a 90% c.l. upper limit on the diffuse $ν_μ$ flu…
We show that the observed fluxes, spectra and sky distributions of the high energy diffuse backgrounds of astronomical neutrinos, gamma rays and cosmic ray positrons satisfy simple relations expected from their common production in hadronic collisions of high energy cosmic rays (CRs) with diffuse matter. This provides compelling evidence that the high energy neutrino, gamma ray, and positron backgrounds have a common origin - hadronic meson production by cosmic rays in our Galaxy and in external galaxies rather than through the decay/annihilation of dark matter particles.
More than 90% of the Galactic gas-related gamma-ray emissivity above 1 GeV is attributed to the decay of neutral pions formed in collisions between cosmic rays and interstellar matter, with lepton-induced processes becoming increasingly important below 1 GeV. Given the high-quality measurements of the gamma-ray emissivity of local interstellar gas between ~50 MeV and ~4 GeV obtained with the Large Area Telescope on board the Fermi space observatory, it is timely to re-investigate this topic in detail, including the hadronic production mechanisms. The emissivity spectrum will allow the interste…
Particle acceleration in the inner ~ 200 pc of the Galaxy is discussed, where diffuse TeV gamma-rays have been detected by the High Energy Stereoscopic System (HESS) observation. The diffuse gamma-ray emission has a strong correlation with molecular clouds with large velocity dispersion, indicating the presence of turbulence. It is argued that the turbulence may contribute to the acceleration of cosmic rays via stochastic acceleration. The stochastic acceleration may energize cosmic-ray protons up to ~ 100 TeV and electrons to ~ 1 TeV in a relatively tenuous medium. The difference in the effic…
It is well known that the diffuse gamma-rays are produced by the collisions between the galactic cosmic rays with the stellar matter (Hayakawa-Morrison Hypothesis). In this paper the author tries to estimate the contribution of star flares in the low energy component of the diffuse component of the galactic gamma-rays. It is expected that high energy part of the disuse component of the galactic gamma-rays reflects the power spectrum of galactic cosmic rays with gamma= -2.75 (in differential). In the galactic disk and bulge, there are more than 100 billion stars. They might often make flare and…
Using three-dimensional magnetohydrodynamics simulations, we show that the efficiency of cosmic-ray (CR) production at supernova remnants (SNRs) is over-predicted if it could be estimated based on proper motion measurements of H$α$ filaments in combination with shock-jump conditions. Density fluctuations of upstream medium make shock waves rippled and oblique almost everywhere. The kinetic energy of the shock wave is transferred into that of downstream turbulence as well as thermal energy which is related to the shock velocity component normal to the shock surface. Our synthetic observation sh…
The Tibet AS$γ$ experiment provided the first measurement of the total diffuse gamma-ray emission from the Galactic disk in the sub-PeV energy range. Based on the analysis of TeV sources included in the HGPS catalog, we predict the expected contribution of unresolved sources in the two angular windows of the Galactic plane observed by Tibet AS$γ$. We show that the sum of this additional diffuse component due to unresolved sources and the truly diffuse emission, due to cosmic ray interaction with the interstellar medium, well saturates the Tibet data, without the need to introduce a progressive…
The interaction of cosmic rays with the gas contained in our Galaxy is a guaranteed source of diffuse high energy neutrinos. We provide expectations for this component by considering different assumptions for the cosmic ray distribution in the Galaxy which are intended to cover the large uncertainty in cosmic ray propagation models. We calculate the angular dependence of the diffuse galactic neutrino flux and the corresponding rate of High Energy Starting Events in IceCube by including the effect of detector angular resolution. Moreover we discuss the possibility to discriminate the galactic c…
Ultra-high cosmic rays (UHECRs) with energies >10^19 eV emitted at cosmological distances will be attenuated by cosmic microwave and infrared background radiation through photohadronic processes. Lower energy extra-galactic cosmic rays (~10^18-10^19 eV) can only travel a linear distance smaller than ~Gpc in a Hubble time due to the diffusion if the extra-galactic magnetic fields are as strong as nano Gauss. These prevent us from directly observing most of the UHECRs in the universe, and thus the observed UHECR intensity reflects only the emissivity in the nearby universe within hundreds of Mpc…
The diffuse $γ$-ray emission was measured up to $957$ TeV by the Tibet-AS$γ$ experiment recently. Assuming that it is produced by the hadronic interaction between cosmic ray nuclei and the interstellar medium, it requires that the cosmic ray nuclei should be accelerated well beyond PeV energies. Measurements of the cosmic ray spectra for different species show diverse results at present. The Tibet experiments showed that the spectrum of proton plus helium has an early knee below PeV. If this is correct, the diffuse $γ$-ray emission would suggest an additional component of Galactic cosmic rays …
In this contribution the current observational results for the diffuse flux of high-energy astrophysical neutrinos are reviewed. In order to understand the science implications, the measurements in different detection channels are discussed and results are compared. The discussion focuses is the energy spectrum, the flavor ratio and large scale anisotropy.
We investigate how the extragalactic proton component derived within the "escape model" can be explained by astrophysical sources. We consider as possible cosmic ray (CR) sources normal/starburst galaxies and radio-loud active galactic nuclei (AGN). We find that the contribution to the total extragalactic proton flux from normal and starburst galaxies is only subdominant and does not fit the spectral shape deduced in the escape model. In the case of radio-loud AGN, we show that the complete extragalactic proton spectrum can be explained by a single source population, BL Lac/FR I, for any of th…
Measuring the diffuse Galactic gamma-ray flux in the TeV range is difficult for ground-based gamma-ray telescopes because of the residual cosmic-ray background, which is higher than the gamma-ray flux by several orders of magnitude. Its detection is also challenging for space-based telescopes because of low signal statistics. We characterize the diffuse TeV flux from the Galaxy using decade-long exposures of the Fermi Large Area Telescope. Considering that the level of diffuse Galactic emission in the TeV band approaches the level of residual cosmic-ray background, we estimated the level of re…
Recently, AMS-02 reported their observed results of cosmic rays(CRs). In addition to the AMS-02 data, we add HESS data to estimate the spectra of CR electrons and the diffuse gamma rays above TeV. In the conventional diffusion model, a global analysis is performed on the spectral features of CR electrons and the diffuse gamma rays by GALRPOP package. The results show that the spectrum structure of the primary component of CR electrons can not be fully reproduced by a simple power law and the relevant break is around hundred GeV. At 99\% C.L., the injection indices above the break decrease from…
The observation of the diffuse Galactic gamma ray flux is the most powerful tool to study cosmic rays in different regions of the Galaxy, because the energy and angular distributions of the photons encode information about the density and spectral shape of relativistic particles in the entire Milky Way. An open problem of fundamental importance is whether cosmic rays in distant regions of the Milky Way have the same spectral shape observed at the Earth or not. If the spectral shape of protons and nuclei is equal in all the Galaxy, the dominant, hadronic component of the diffuse gamma ray flux …
Recent multifrequency observations contribute to derive a comprehensive picture of the origin and evolution of relativistic particles in galaxy clusters. In this review I briefly discuss theoretical aspects and open problems of this picture.
A new calculation of nuclear enhancement factor, used in estimation of Galactic diffuse gamma-ray flux from proton-proton interaction in order to take account of heavy nuclei included in cosmic-rays and interstellar matter, is presented by use of a Monte Carlo simulator, DPMJET-3. A new value of 1.8-2.0 in the energy range of 6-1000 GeV/nucleon, slightly increasing with kinetic energy of projectile cosmic rays, is about 20% larger than previous estimates.
Astrophysical neutrinos are expected to be produced in the interactions of ultra-high energy cosmic-rays with surrounding photons. The fluxes of the astrophysical neutrinos are highly dependent on the characteristics of the cosmic-ray sources, such as their cosmological distributions. We study possible constraints on the properties of cosmic-ray sources in a model-independent way using experimentally obtained diffuse neutrino flux above 100 PeV. The semi-analytic formula is derived to estimate the cosmogenic neutrino fluxes as functions of source evolution parameter and source extension in red…
Isotropic diffuse gamma-ray flux in the PeV energy band is an important tool for multimessenger tests of models of the origin of high-energy astrophysical neutrinos and for new-physics searches. So far, this flux has not yet been observed. Carpet-2 is an air-shower experiment capable of detecting astrophysical gamma rays with energies above 0.1 PeV. Here we report the upper limits on the isotropic gamma-ray flux from Carpet-2 data obtained in 1999-2011 and 2018-2022. These results, obtained with the new statistical method based on the shape of the muon-number distribution, summarize Carpet-2 o…
Galactic diffuse TeV gamma-ray emission is produced by the interaction of high-energy cosmic-ray particles with matter and radiation in our Galaxy. The measurement of Galactic diffuse TeV gamma-ray emission would provide strong constraints on the acceleration and propagation of Galactic cosmic rays. The High Altitude Water Cherenkov (HAWC) Observatory, located in central Mexico at 4100 m above sea level, is sensitive to gamma rays between a few hundreds GeV and about 100 TeV. Thanks to its large field of view of 2 steradians and excellent background estimation, HAWC has a unique capability to …
Most of the diffuse Galactic GeV gamma-ray emission is produced via collisions of cosmic ray (CR) protons with ISM protons. As such the observed spectra of the gamma-rays and the CRs should be strongly linked. Recent observations of Fermi-LAT exhibit a hardening of the gamma-ray spectrum at around a hundred GeV, between the Sagittarius and Carina tangents, and a further hardening at a few degrees above and below the Galactic plane. However, standard CR propagation models that assume a time independent source distribution and a location independent diffusion cannot give rise to a spatially depe…
