Electron and positron spectra in the three-dimensional spatial-dependent propagation model

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Zhen Tian, Wei Liu, Bo Yang, Xue-Dong Fu, Hai-Bo Xu, Yu-Hua Yao and Yi-Qing Guo. Electron and positron spectra in the three-dimensional spatial-dependent propagation model[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/8/085102
Zhen Tian, Wei Liu, Bo Yang, Xue-Dong Fu, Hai-Bo Xu, Yu-Hua Yao and Yi-Qing Guo. Electron and positron spectra in the three-dimensional spatial-dependent propagation model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/8/085102 shu
Received: 2020-02-10
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Electron and positron spectra in the three-dimensional spatial-dependent propagation model

  • 1. Institute of Applied Physics and Computational Mathematics, Chinese Academy of Engineering, Beijing 100094, China
  • 2. Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
  • 3. College of Physics, Sichuan University, Chengdu 610065, China

Abstract: The spatial-dependent propagation (SDP) model has been demonstrated to account for the spectral hardening of both primary and secondary Cosmic Rays (CRs) nuclei above about 200 GV. In this work, we further apply this model to the latest AMS-02 observations of electrons and positrons. To investigate the effect of different propagation models, both homogeneous diffusion and SDP are compared. Different from the homogeneous diffusion, SDP brings about harder spectra of background CRs and thus enhances background electron and positron fluxes above tens of GeV. Thereby the SDP model could better reproduce both electron and positron energy spectra when introducing a local pulsar. The influence of background source distribution is also investigated, in which both axisymmetric and spiral distributions are compared. We find that taking account for the spiral distribution lead to a larger contribution of positrons for energies above multi-GeV than the axisymmetric distribution. In the SDP model, when including a spiral distribution of sources, the all-electron spectrum above TeV energies is thus naturally described. In the meantime, the estimated anisotropies in the all-electrons spectrum show that in contrary to the homogeneous diffusion model, the anisotropy under SDP is well below the observational limits set by the Fermi-LAT experiment, even taking a local source into account.


    1.   Introduction
    • The accurate measurements of cosmic-ray electron and positron fluxes have permitted to unveil abundant unexpected features. A remarkable positron excess above 10 GeV was discovered by the PAMELA experiment [1]. Before long this anomaly was substantiated by the Fermi-LAT observation [2]. The AMS-02 experiment has extended their measurements up to 1 TeV with an unprecedented high precision [3]. Apart from confirming an excess of positron flux above ~25 GeV, AMS-02 also hinted a gradual positron drop-off starting at ~284 GeV. But this feature has to be validated by the future precise measurements. As for electrons, the measurements by the AMS-02 experiment showed that the energy spectrum could not be described by a single power-law [4]. The power index changes at about 30 GeV. Moreover, the analysis of the difference between the electron and positron fluxes [46] suggested that from ~30 GeV, there is still a distinct excess in the propagated electrons, compared with the conventional homogeneous diffusion model. Later the DAMPE experiment [26] found a spectral hardening above ~50 GeV in the all-electron (electron + positron) spectrum. Recently the AMS-02 collaboration updated their measurement of electrons and confirmed the excess starting from ~42 GeV [13]. In addition, the HESS experiment reported a spectral break around 1 TeV in the all-electron spectrum, which resembled the knee region at ~4 PeV in the all-particle spectrum [14, 42]. This feature was also observed by the other experiments such as MAGIC [24], VERITAS [17] and DAMPE [26].

      In the conventional homogeneous diffusion model, the positrons are usually regarded as the secondary CRs and their propagated energy spectrum is expected to be a featureless single power-law. The discovery of positron excess with respect to the conventional model hints the existence of extra primary components and has called a lot of attention. Numerous models have been proposed to explain this phenomenon, which could be attributed to either astrophysical, such as local pulsars [62, 78] and the hadronic interactions inside supernova remnants (SNRs) [35, 48], or more exotic origins like the dark matter self-annihilation or decay [20, 75]. Additionally, the $ \rm e^+/e^- $ ratio is interpreted as a charge-sign dependent particle injection and acceleration [51]. For an extensive introduction of the relevant models, one can refer to the reviews [22, 31] and references therein. For the observed break-off of all-electron spectrum, it is argued to be caused by the radiation cooling of electrons surrounding SNRs [72] or a threshold interaction during the transport of CR electrons [39, 73].

      Due to severe energy losses, the typical propagation distance of energetic electrons/positrons is much shorter compared with the low-energy ones. Thus most of them come from the sources within a few kpc. In this case, both propagation process and source distribution should have great impact on the electron and positron spectra. In recent years, the SDP model has drawn more and more attentions. It was originally introduced by [66] to account for the spectral hardening of CR proton and helium above 200 GV. Another possible interpretation of such remarkable spectral features can be attributed to the correponding break in the injection spectrum of CRs [84, 85]. Different from the homogeneous diffusion, the whole diffusive region is split into two zones characterized by different diffusion properties in the SDP scenario. The Galactic disk and its surrounding areas within a few hundred parsecs are called the inner halo (IH), and the extended regions outside of IH are called outer halo (OH) [34, 38, 50, 66]. Around IH, the diffusion process is much slower than OH. Recent observations of radial gamma-ray profile of Geminga and PSR B0656+14 pulsars by the HAWC experiment indicate that the diffusion around these sources is inhibited by about two orders of magnitude lower than the values inferred by fitting B/C ratio [5, 7, 29]. This provides a support to the SDP scenario.

      However a comprehensive study of SDP model to both electron and positron spectra is still unavailable. The prediction of the SDP model on the electrons and positrons has been preliminarily investigated by [34, 67]. It can be found that the expected positron flux fails to explain the AMS-02 measurements. As for the electron, above ~ 200 GeV, the theoretical flux starts to be lower than the observed flux, despite that the spectra at low energy can be reproduced. In particular, recently the AMS-02 collaboration published their updated observations of electrons and positrons, which affirmed the excess of electron flux above ~ 30 GeV found by [26, 46]. This feature has not been well studied in the past years. Meanwhile, in the previous studies, the CR sources are regarded as axisymmetric distributed by default. The effect of spiral distribution has been investigated until recent years [19, 36, 60], but it has not been studied in the SDP model. In this work, we investigate the influences of propagation effect and source distribution on both electron and positron spectra and thus different theoretical propagation models are compared in detail: homogeneous diffusion with axisymmetric distribution, SDP with axisymmetric distribution and SDP with spiral distribution respectively. We find that in the homogeneous diffusion with axisymmetric distribution model, the electron and positron spectra are hard to be simultaneously accounted for even if introducing a local pulsar. This is due to the excess of electron flux. With the specific transport parameters, SDP model could prominently enhance the background electron and positron fluxes. Therefore compared with the homogeneous diffusion, the SDP model could better describe the electron and positron spectra. The all-electron spectrum could also be well explained under the SDP with spiral distribution model up to 20 TeV. The spectral break above TeV energies is a superposition of the contributions of the background components and the local source. The anisotropies of electrons are computed, which shows that the expected anisotropy in the SDP model is far below the current observational limit, even a local source is taken into account.

      The rest of the paper is organized as follows. In Sec.2, both spatial-dependent propagation model and spiral distribution of sources are introduced in detail. Sec.3 represents the calculated results and Sec.4 is reserved for the conclusion.

    2.   Model Description

      2.1.   Spatial-dependent propagation

    • After entering the interstellar space, CRs undergo the random walks within the Galactic magnetic field by bouncing off the magnetic waves and magnetohydrodynamic turbulence. This process is usually described by a diffusion equation. The diffusive region, which is called magnetic halo, is approximated as a flat cylinder with its radius $ R = 20 $ kpc, equivalent to the Galactic radius. The half-thickness $ z_h $ is unknown,and is typically constrained by fitting the B/C ratio. The Galactic disk, where both CR sources and interstellar gas are mainly spread across, is located in the middle of the magnetic halo. The width of Galactic disk is is approximated to be invariant spatially and equals to 200 pc. In addition to diffusion, CRs may still go through convection, diffusive-reacceleration, fragmentation, radioactive decay and other energy losses before arriving at the Solar system. CR nuclei lose their energy principally via ionization, Coulomb scattering and adiabatic expansion. For electrons and positrons, their major energy losses are the bremsstrahlung, synchrotron radiation and the inverse Compton scattering. The general convection-diffusion equation is written as:

      $\begin{split} \frac{{\partial \psi ({{r}},p,t)}}{{\partial t}} =& \nabla \cdot ({D_{xx}}\nabla \psi - {V_c}\psi ) + \frac{\partial }{{\partial p}}\left[ {{p^2}{D_{pp}}\frac{\partial }{{\partial p}}\frac{\psi }{{{p^2}}}} \right]\\ &- \frac{\partial }{{\partial p}}\left[ {\dot p\psi - \frac{p}{3}(\nabla \cdot {V_c}\psi )} \right] - \frac{\psi }{{{\tau _f}}} - \frac{\psi }{{{\tau _r}}} + q({{r}},p,t)\;, \end{split}$


      where $ \psi({{r}},p,t) $ is the CR number density per unit of total momentum p at position $ {{r}} $ and time t. At the boundary of magnetic halo, a free escape condition is imposed, i.e. $ \psi(R, z, p) = \psi(r, \pm z_h, p) = 0 $. The CR flux is defined as $ \Phi = v\psi/4\pi $. For a comprehensive introduction to the CR propagation in the Galaxy, one can refer to [54, 64]. $ D_{xx} $ is the spatial diffusion coefficient. In the homogeneous diffusion model, $ D_{xx} $ is only a function of rigidity $ {\cal R} = pc/Ze $, i.e.

      ${D_{xx}}({\cal R}) = {D_0}\beta {\left( {\frac{{\cal R}}{{{{\cal R}_0}}}} \right)^\delta }\;.$


      $ \beta $ is the particle velocity in unit of the speed of light c. But in the SDP model, the homogeneous hypothesis is abandoned. It consists in allowing CRs to experience a spatial dependent diffusion, when they transport in the region near the Galactic plane. As summarized in the Introduction, the classical cylindrical propagation region is divided into IH and OH, in terms of two types of Galactic turbulence. The type of the diffusion process in the IH is determined by the intense turbulence, which are governed by the nature and the scale of magnetic-filed irregularities generated by the SN explosions [81]. While there are no SNRs in the OH, CRs are responsible for generating their own turbulence by CR-induced streaming instability [69]. Compared with the SNR-driven propagation in the IH, the diffusion situation outside of IH approaches to the homogeneous assumption of evident rigidity dependent. Hence the corresponding diffusion coefficient $ D_{xx} $ is parameterized as:

      ${D_{xx}}(r,z,{\cal R}) = {D_0}F(r,z)\beta {\left( {\frac{{\cal R}}{{{{\cal R}_0}}}} \right)^{\delta (r,z)}}\;.$


      Both $ F(r,z) $ and $ \delta(r, z) $ are anti-correlated with the source distribution $ f(r, z) $ [38, 50]. To be more specific,

      $F(r,z) = \left\{ {\begin{array}{*{20}{l}} {g(r,z) + \left[ {1 - g(r,z)} \right]{{\left( {\dfrac{z}{{\xi {z_h}}}} \right)}^n},}&{|z| \leqslant \xi {z_h}}\\ {1,}&{|z| > \xi {z_h}} \end{array},} \right.$



      $\delta (r,z) = \left\{ {\begin{array}{*{20}{l}} {g(r,z) + \left[ {{\delta _0} - g(r,z)} \right]{{\left( {\dfrac{z}{{\xi {z_h}}}} \right)}^n},}&{|z| \leqslant \xi {z_h}}\\ {{\delta _0},}&{|z| > \xi {z_h}} \end{array},} \right.$


      in which $ g(r,z) = N_m/[1+f(r,z)] $. $ \xi $, which characterizes the size of IH, is determined by fitting the hardening of the proton spectrum. So the width of IH is $ 2\xi z_h $, and the corresponding OH is $ 2(1-\xi) z_h $. The higher the hardening energy is, the smaller $ \xi $ is. The diffusive reacceleration is represented as a diffusion in the momentum space. Its diffusion coefficient $ D_{pp} $ is related to $ D_{xx} $ by

      ${D_{pp}}{D_{xx}} = \frac{{4{p^2}v_A^2}}{{3\delta (4 - {\delta ^2})(4 - \delta )}}\;,$


      in which $ v_A $ is the Alfvénic velocity. $ V_{c} $ is the convection velocity and $ \dot{p} $ represents the energy loss rate. $ \tau_{f} $ and $ \tau_{r} $ are the characteristic timescales for the fragmentation and radioactive decay respectively.

      $ q({{r}},p,t) $ in Equ. (1) is the source term. The SNRs are generally considered as CR sources. P, He, C, O, e- etc. are regarded as primary CRs. The SNRs are regarded as sources of the so-called primary CRs, and to produce power-law energy spectra through diffusive shock acceleration process [18, 23]. In this work, all of background SNRs are hypothesized to have the same injection spectra. The comprehensive analysis of the diffusion-reacceleration (DR) model [40, 68] have shown that a pure power-law injection spectrum is not enough to well describe the observational data. The observational data could be better described after introducing a broken power-law injection spectrum at low energies. The observations of the nearby molecular clouds and the supernova remnants interacting with molecular clouds confirm the need for a low-energy break in the CR nuclei injection spectrum [8, 58]. Moreover, the fitting to the diffuse synchrotron radiation has also implied a break in the injection electron spectrum [65]. The low-energy break in the spectrum may related to the acceleration process at sources [80] transition from the convective to diffusive regime [15, 57], change in the energy dependence of the diffusion coefficient [57], etc. Hence to fit the low energy spectra, the injection spectra of proton and electron are assumed to have a broken power-law:

      ${q^{\rm{p}}}({\cal R}) = q_0^{\rm{p}}\left\{ {\begin{array}{*{20}{l}} {{{\left( {\dfrac{{\cal R}}{{{\cal R}_{{\rm{br}}}^{\rm{p}}}}} \right)}^{\nu _1^{\rm{p}}}},}&{{\cal R} \leqslant {\cal R}_{{\rm{br}}}^{\rm{p}}}\\ {{{\left( {\dfrac{{\cal R}}{{{\cal R}_{{\rm{br}}}^{\rm{p}}}}} \right)}^{\nu _2^{\rm{p}}}}\exp \left[ { - \dfrac{{\cal R}}{{{\cal R}_{\rm{c}}^{\rm{p}}}}} \right],}&{{\cal R} > {\cal R}_{{\rm{br}}}^{\rm{p}}} \end{array}} \right.$



      ${q^{{{\rm{e}}^ - }}}({\cal R}) = q_0^{{{\rm{e}}^ - }}\left\{ {\begin{array}{*{20}{l}} {{{\left( {\dfrac{{\cal R}}{{{\cal R}_{{\rm{br}}}^{{{\rm{e}}^ - }}}}} \right)}^{\nu _1^{{{\rm{e}}^ - }}}},}&{{\cal R} \leqslant {\cal R}_{{\rm{br}}}^{{{\rm{e}}^ - }}}\\ {{{\left( {\dfrac{{\cal R}}{{{\cal R}_{{\rm{br}}}^{{{\rm{e}}^ - }}}}} \right)}^{\nu _2^{{{\rm{e}}^ - }}}},}&{{\cal R} > {\cal R}_{{\rm{br}}}^{{{\rm{e}}^ - }}} \end{array}} \right.$


      where $ \nu $ and $ q_{0}^{\rm p/e^-} $ are the spectral index and injection proton(electron) number at rigidity $ {\cal R}_{\rm br}^{\rm p} $($ {\cal R}_{\rm br}^{{\rm e}^-} $) respectively, and $ R_{\rm c} $ is the cut-off rigidity.

      Other species, e.g. Li, Be, B and e+, are hardly synthesized during the stellar nucleosynthesis. They are generally considered to bring forth from the fragmentation of parent nuclei throughout the transport. They are usually defined as secondaries. For the production of Li, Be and B, the so-called straight-ahead approximation [54] is adopted, in which the kinetic energy per nucleon is conserved during the spallation process. The production rate is thus

      ${Q_j} = \sum\limits_{i = {\rm{C}},{\rm{N}},{\rm{O}}} {({n_{\rm{H}}}{\sigma _{i + {\rm{H}} \to j}} + {n_{{\rm{He}}}}{\sigma _{i + {\rm{He}} \to j}})} v{\psi _i}\;,$


      where $ n_{{\rm H}/{\rm He}} $ is the number density of hydrogen/helium in the ISM and $ \sigma_{i+{\rm H/He}\rightarrow j} $ is the total cross section of the corresponding hadronic interaction.

      Different from the secondary nuclei, the differential production cross section of the positrons have an energy distribution. The source term of positron is a convolution of the energy spectra of primary nuclei $ \psi(p) $ and the relevant differential cross section $ \mathrm{d} \sigma_{i+{\rm H}/{\rm He}\rightarrow j}/ \mathrm{d} p_j $, i.e.

      $\begin{split} {Q_j} =& \sum\limits_{i = {\rm{p}},{\rm{He}}} {\int d } {p_i}v\left\{ {{n_{\rm{H}}}\frac{{{\rm{d}}{\sigma _{i + {\rm{H}} \to j}}({p_i},{p_j})}}{{{\rm{d}}{p_j}}}} \right.\\ &\left. { + {n_{{\rm{He}}}}\frac{{{\rm{d}}{\sigma _{i + {\rm{He}} \to j}}({p_i},{p_j})}}{{{\rm{d}}{p_j}}}} \right\}{\psi _i}({p_i}). \end{split}$


      As for the energetic electrons and positrons, the energy loss from synchrotron radiation and inverse Compton scattering have to be considered [56]. A full-relativistic treatment of the inverse-Compton losses [27] has been implemented in the DRAGON package [30]. Due to the systematic uncertainties stemming from the p-p interaction models, propagation models, nuclear enhancement factor from heavy elements etc., the evaluated background positron flux may have large uncertainties [28, 34, 77]. To encompass the above uncertainties, the propagated background positron flux is rescaled by a factor $ c^{\rm e^+} $ [47, 77].

      In this work, we adopt the common DR model. The numerical package DRAGON is used to solve the propagation equation to obtain distribution of background CRs. It is based on a Crank-Nicolson second-order implicit scheme [61]. Less than tens of GeV, the CR fluxes are impacted by the solar modulation. The well-known force-field approximation [59] is applied to describe such an effect, with a modulation potential $ \phi $ adjusted to fit the low energy data. The modulated flux is calculated according to

      $\frac{{{\Phi ^{{\rm{TOA}}}}({E^{{\rm{TOA}}}})}}{{{\Phi ^{{\rm{IS}}}}({E^{{\rm{IS}}}})}} = {\left\{ {\frac{{{p^{{\rm{TOA}}}}}}{{{p^{{\rm{IS}}}}}}} \right\}^2}\;,$


      in which $ p^{\rm TOA} $ and $ p^{\rm IS} $ correspond to modulated and interstellar momentum respectively [54]. For a nucleus with charge Z and atomic number A, the modulated and interstellar total energy $ E^{\rm TOA} $ and $ E^{\rm IS} $ are related by

      ${E^{{\rm{TOA}}}}/A = {E^{{\rm{IS}}}}/A - |Z|\phi /A\;.$

    • 2.2.   Spiral distribution of CR sources

    • A great deal of observational evidence (see [45, 74] and references therein) have indicated that the Milky Way is a typical spiral galaxy. The high density gas concentrates in the spiral arms, which triggers a rapid star formation. It turns out that the distribution of SNRs are highly correlated with the spiral arm patterns. In main of the past studies, the CR sources are usually approximated as axisymmetric-distributed, which is parameterized as

      $f(r,z) = {\left( {\frac{r}{{{r_ \odot }}}} \right)^\alpha }\exp \left[ { - \frac{{\beta (r - {r_ \odot })}}{{{r_ \odot }}}} \right]\exp \left( { - \frac{{|z|}}{{{z_s}}}} \right)\;.$


      $ r_\odot $ represents the solar distance to the Galactic center, which is adopted as 8.5 kpc here. The parameters $ \alpha $ and $ \beta $ are taken as 1.09 and 3.87 respectively in this work [37]. Perpendicular to the Galactic plane, the density of CR sources descends as an exponential function [37], with a mean value $ z_{s} = 100 $ pc. This approximation is plausible as the diffusion length of CRs is usually much longer than the characteristic spacing between the adjacent spiral arms. However, subject to the synchrotron radiation and inverse Compton scattering, the transport distance of the energetic electrons is much shorter and the above approximation might become less accurate. Thus the inclusion of a more realistic description of the spiral distribution is expected to have striking impact on the observed spectrum of high energy electrons. Observationally, there are still some uncertainties on the structure of the spiral arms, owing to our position in the Galaxy. The outer part of the Milky Way seems to have four arms, but the number of arms in the inner part is still being debated. The observations for the spiral structure and number of spiral arms are reviewed in [70, 71].

      In this work, a model established by [33] is used to describe the spiral distribution of SNRs. The Galaxy consists of four major arms spiraling outward from the Galactic center. The locus of the i-th arm centroid expresses analytically as a logarithmic curve: $ \theta(r) = k^{i} \ln(r/r^{i}_{0}) + \theta_{0}^i $, where r is the distance to the Galactic center. The values of $ k^{i} $, $ r^{i}_{0} $ and $ \theta_{0}^i $ for each arm are listed in Table 1. Along each spiral arm, there is a spread in the normal direction which follows a Gaussian distribution, i.e.


      Table 1.  The values of the parameters $k^{i}$, $r^{i}_{0}$ and $\theta^{i}_{0}$ for four Galactic arm centroids

      ${f_i} = \frac{1}{{\sqrt {2\pi } {\sigma _i}}}\exp \left[ { - \frac{{{{(r - {r_i})}^2}}}{{2\sigma _i^2}}} \right]\;,\;\;\;i \in [1,2,3,4]\;,$


      where $ r_i $ is the inverse function of the i-th spiral arm's locus and the standard deviation $ \sigma_i $ is taken to be $ 0.07 r_i $. The number density of SNRs at different radii is still consistent with the radial distribution in the axisymmetric case, i.e. Eq (13). Fig. 1 shows the spatial distribution of SNRs in the spiral model. The solar system lies between the Carina-Sagittarius and Perseus spiral centroids.

      Figure 1.  Top view of the density distribution of SNRs in the Galaxy. The Galaxy is assumed to have four spiral arms, with the Sun (denoting as a black point) lying bewteen the Garina-Sagittarius and Perseus arms, about 8.5 kpc away from the Galactic center.

    • 2.3.   Local source

    • Due to the energy loss during the propagation, the characteristic lifetime of energetic electrons and positrons at energy E is less than $ 2.3 \times 10^{5} (E/\rm TeV)^{-1} $ yr, which suggests the corresponding diffusion distance is $ \sim 1 \cdot (E/\rm TeV)^{-1/3} $ kpc [41]. Therefore at about TeV energies, the CR electrons and positrons originate from sources within 1 kpc around the solar system. Within 1 kpc, the number of generated SNRs is $ \sim 2.5 $ less than $ 10^5 $ yrs, assuming 1 supernova explosion per century. The continuity hypothesis is no longer valid. Studies show that the discrete effect of nearby CR sources could induce large statistical fluctuations [21, 55], especially at high energies. The role of nearby sources on the electrons and positrons has been studied in recent works (see [7, 32, 48] and references therein). In this work, we assume a nearby pulsar to account for the excess of positrons above $ \sim 20 $ GeV. The diffusion process of electrons/positrons injected instantaneously from a point source is described by a time-dependent propagation equation,

      $\frac{{\partial \varphi }}{{\partial t}} - \nabla ({D_{xx}}\nabla \varphi ) + \frac{\partial }{{\partial E}}(\dot E\varphi ) = Q(E,t)\delta ({{r}} - {{{r}}^\prime })\;.$


      The corresponding Green's function $ G({{r}}-{{r}}^{\prime}, t- t^{\prime}, E) $ can be found in [44]. The energy dependence is taken as a single power-law distribution, namely

      $Q(E,t) = {Q_0}(t){\left( {\frac{{\cal R}}{{1\;{\rm{GV}}}}} \right)^{ - \gamma }}\exp \left[ { - \frac{{\cal R}}{{{\cal R}_{\rm{c}}^{{{\rm{e}}_ \pm }}}}} \right]\;,$


      in which $ {\cal R}^{\rm e_{\pm}}_{\rm c} $ is the cutoff rigidity. For the local pulsar, a continuous injection process of electron-positron pairs is considered. The injection rate is time-dependent, which decays as pulsar-type [41], i.e.

      ${Q_0}(t) = \frac{{{q_0}}}{{{{[1 + (t - {t_i})/{\tau _0}]}^2}}}\;,$


      where $ t_i $ is the initial time of releasing electron-positron pairs. The characteristic duration $ \tau_0 $ is taken as $ 10^4 $ years as typically done in past literature, even if the value is uncertain and might vary for each specific pulsar, see e.g. [52]. The propagated spectrum is a convolution of Green function and time-dependent injection rate $ Q_0(t) $ [41, 43], i.e.

      $\varphi ({{r}},E,t) = \int_{{t_i}}^t G ({{r}} - {{{r}}^\prime },t - {t^\prime },E){Q_0}({t^\prime }){\rm{d}}{t^\prime }\;.$

    3.   Results
    • To demonstrate the effects of SDP and spiral distribution of background sources, the exploration of different propagation models is categorized as SDP with axisymmetric distribution (denoted as SDP+axis), SDP with spiral distribution (denoted as SDP+spiral) and homogeneous diffusion with axisymmetric distribution respectively. It is worth to be mentioned, the situation of homogeneous diffusion It is worth to be mentioned, the situation of homogeneous diffusion considers two kinds of injection spectra for nuclei: one is single power-law (denoted as HD + axis) and another has a spectral break above ~ 200GV (denoted as HDB + axis). Under the homogeneous diffusion, the free parameters that are fixed in order to accommodate the data are $ D_{0} $, $ \delta_{0} $, $ z_{h} $ and $ v_{A} $. As for the SDP, two additional parameters are included, $ N_{\rm m} $ and $ \xi $ respectively. The injection parameters of the background protons(heliums, electrons) are $ A^{\rm p/e^-} $, a, $ \nu^{\rm p/He/e^-}_1 $, $ \nu^{\rm p/He/e^-}_2 $, $ \nu^{\rm p/He/e^-}_3 $, $ {\cal R}^{\rm p/He/e^-}_{\rm br} $, $ {\cal R}^{\rm p/He}_{\rm brh} $ and $ {\cal R}_{\rm c} $. Here $ A^{\rm p/e^-} $ is normalization flux for the background protons(electrons). The parameter of a represents the elemental abundance of primary helium compared with proton. The secondary electron and positron fluxes are rescaled by the factor $ c^{e^+} $. The injection parameters of the local pulsar are $ q_{0} $, $ \gamma $ and $ {\cal R}_{\rm c}^{\rm e_\pm} $. In addition, for proton, helium, electron and positron, each has individual modulation potential $ \phi $.

      In this section, first of all, the transport parameters of four models are fixed by fitting the B/C ratio and proton spectrum. Then the energy spectra of electrons and positrons under four models are compared by the fixed transport parameters. Finally, the anisotropies of all-electrons are calculated.

    • 3.1.   B/C ratio and proton spectrum

    • The benchmark values of the transport parameters for the four propagation models are fixed in order to explain the B/C ratio, as illustrated in the Fig. 2. The orange, green, blue and black solid lines represent HD+axis, HDB+axis, SDP+axis and SDP+spiral respectively. The corresponding transport parameters are listed in table 2. Compared with the HD picture, both SDP scenarios favor a larger $ \delta_0 $ and halo size $ z_h $. This could also be inferred from the Bayesian analysis of the two-halo model [34]. In contrast to the HD model, the required alfvénic velocity is smaller, namely $ \sim 6 $ km/s. This could be understood as follows. In SDP picture, both $ \delta $ and $ D_0 $ vary spatially and are smaller nearby the Galactic disk. A smaller $ D_{xx} $ corresponds to a larger $ D_{pp} $ according to Equ. (6). Hence even if the alfvénic velocity is smaller, the diffusion-reacceleration still have significant effect in SDP model, compared with that in the HD model.

      Figure 2.  Fitting to the B/C ratio under three propagation models, which are HD+axis (orange), SDP+axis (blue) and SDP+spiral (black) respectively. The data are taken from AMS-02 experiment [10] In this work, the benchmark values of the transport parameters for these propagation models are fixed via fitting the B/C ratio and the proton spectrum.

      $N_{\rm m}$0.1410.24

      $A^{\rm p}\; [\rm m^{-2}sr^{-1}s^{-1} GeV^{-1}]$$^\dagger$0.0450.04350.04360.0436
      $\nu_{1}^{\rm p}$
      ${\cal R}_{\rm br}^{\rm p}$[GV]9.911.05.55.5
      $\nu^{\rm p}_{2}$2.352.352.402.3
      ${\cal R}_{\rm brh}^{\rm p}$[GV]750
      $\nu^{\rm p}_{3}$2.11
      $\phi^{\rm p}$[MV]500800800800
      ${\cal R}_{\rm c}$[TV]1458682

      $\nu_{1}^{\rm He}$
      ${\cal R}_{\rm br}^{\rm He}$[GV]9.911.05.55.5
      $\nu^{\rm He}_{2}$2.352.432.512.41
      ${\cal R}_{\rm brh}^{\rm He}$[GV]370
      $\nu^{\rm He}_{3}$2.19
      $\phi^{\rm He}$[MV]250250500650
      $^\dagger$ The normalization is set at kinetic energy per nucleon $E_{\rm n} = 100$ GeV.

      Table 2.  Parameters of injection spectrum of proton, helium and propagation under HD + axis, HDB + axis, SDP + axis and SDP +spiral models. Under the HD and HDB models, the free propagation parameters are $D_{0}$, $\delta_{0}$ and $z_{h}$. In SDP models, two additional parameters, namely $N_{m}$ and $\xi$, are included. The parameters of injection spectrum of proton are the normalization $A^{\rm p}$, spectral indexes $\nu^{\rm p}_{1}$ below $R_{\rm br}^{\rm p}$, $\nu_{2}^{\rm p}$ between $R_{\rm br}^{\rm p}$ and $R_{\rm brh}^{\rm p}$, $\nu^{\rm p}_{3}$ above $R_{\rm brh}^{\rm p}$ and cutoff-rigidity $R_{c}$. Besides that, the parameters of injection spectrum of helium contain $\nu^{\rm He}_{1}$, $\nu^{\rm He}_{2}$, $\nu^{\rm He}_{3}$, $R_{\rm br}^{\rm He}$ and $R_{\rm brh}^{\rm He}$. The parameter of a represents the elemental abundance of helium compared with proton. $\phi^{\rm p}$ and $\phi^{\rm He}$ represent the modulation potential for proton and helium.

      It is worth to note that the SDP scenarios predicts a flattening of the B/C ratio above 1 TeV. Compared with primaries, the hardening of secondaries principally has two sources: the hardening of propagated primary spectrum and spatial-dependent propagation. Therefore the propagated secondary spectrum is harder than that of primary, which has been substantiated by the AMS-02 observations of secondary nuclei [12]. Because of it, an excess in the secondary-to-primary ratios is naturally generated under the SDP model [34, 50, 66]. Future precise measurements of B/C ratio above TeV could verify this picture.

      Fig. 3 shows the propagated proton spectra of four different propagation models. The red squares, gray inverted triangles, black crosses and violet circles represent AMS-02, CREAM, DAMPE and CALET measurements respectively. The parameters of proton injection spectrum are listed in table 2. The HD model predicts that the propagated spectrum falls off as a featureless power-law starting from tens of GeV (shown as orange line). This is visibly at variance with the current observations. Compared with that in HD model, the propagated proton spectrum under HDB + axis assumption (shown as green line) is in a good agreement with observations, in which the spectral index changes $ \rm \nu^{p}_{2} = 2.35 $ to $ \rm \nu^{p}_{3} = 2.11 $ at rigidity $ \rm R_{brh}^{p} $ of 750GV. Under SDP model, the high-energy CRs mainly originate from the neighboring regions around solar system and their transport mainly occurs in the IH region, in which the rigidity dependence of the diffusion coefficient is weaker. Thus the propagated proton spectrum has a significant hardening above $ \sim 200 $ GeV when $ \xi = 0.1 $, which well reproduces the observations of AMS-02, CREAM and CALET. Though DAMPE has released a similar hardening structure, its data is a little discrepant from the others. Therefore, our results do not fit with DAMPE well. Moreover, as can be seen that the distribution of CR sources make no difference to the proton flux less than hundreds of GeV. Since the diffusion length of low energy protons is much longer than the characteristic interval of neighbouring spiral arms, the distribution of CR sources does not prominently modify the spectrum. But for high-energy protons, the major diffusion region is at the IH region, whose thickness is comparable to the characteristic interval of neighbouring spiral arms. In this case, the source distribution could not be neglected. Above a few TeV, the spiral distribution enhances the proton flux, compared with axisymmetric case.

      Figure 3.  Corresponding propagated proton and helium spectra. The experimental data are taken from AMS-02 [10, 11] (red square), CREAM [76] (grey inverted triangle), DAMPE [82](black cross) and CALET [83](violet solid circles).

      In addition that, the predicted helium spectra are demonstrated at the bottom of Fig. 3 as well. Table 2 displays the parameters of injection spectrum of helium under four theoretical models. Observationally, the fitted injection spectrum of helium is 0.08 harder than proton under HDB picture, while 0.11 in the framework of SDP models. This may originate from the acceleration process at source region. To reproduce the flattening of proton and helium spectra above ~ 200 GeV, in HDB model the spectral break of injection for proton and helium is tuned at rigidity $ \rm R_{brh}^{p} $ = 750 GV and $ \rm R_{brh}^{He} $ = 370GV respectively. The cut-off rigidity $ {\cal R}_{\rm c} $ under HDB and SDP models are adjusted to 145 TV and around 85TV.

      It is worth to further evaluate the total energy of CRs injected by a single SNR in the SDP model, according to the normalization $ A^{\rm p} $. By approximating the SDP model as an one-dimensional two-halo model [66], the total energy of CRs above 1 GeV released by each SNR is roughly $ 4.5\times 10^{49} $ erg, assuming one supernova explosion per century. This is in compatible with the transferred shock kinetic energy of the typical core-collaspe supernova.

    • 3.2.   electron and positron spectra

    • Fig. 4 demonstrates the calculated positron spectra under different propagation models respectively: HD+axis (upper left), HDB+axis (upper right), SDP+axis (lower left) and SDP+spiral (lower right). The blue and green lines are the components contributed by the background and local sources respectively, in which the background positron fluxes before and after rescaling are shown as blue dash and solid lines respectively. The black line is the sum of rescaled background and local fluxes. Table 3 lists the distance and age of local source as well as the parameters for the injection spectrum of background and local sources.

      $A^{\rm e^-}\; [\rm m^{-2}sr^{-1}s^{-1} GeV^{-1}]$$^\dagger$
      $\nu_{1}^{\rm e^-}$1.71.691.531.14
      ${\cal R}_{\rm br}^{\rm e^-}$[GV]445.15.1
      $c^{\rm e^+}$

      t[yr]$3.0 \times 10^{5}$$2.4 \times 10^{5}$$2.2 \times 10^{5}$$2.9 \times 10^{5}$
      $q_{0}$[ GeV$^{-1}$]$3.56 \times 10^{50}$$3.0 \times 10^{50}$$ 1.29 \times 10^{50}$$2.25 \times 10^{50}$
      ${\cal R}^{\rm e^{\pm}}_{\rm c}$[TV]8888
      $\phi^{\rm e^{+}}$[MV]14001420650580
      $\phi^{\rm e^{-}}$[MV]12101210660620
      $^\dagger$ The normalization is set at kinetic energy $E = 10$ GeV.

      Table 3.  Parameters of injection spectra of background electrons and local electron-positron pairs under HD + axis, HDB + axis, SDP + axis and SDP +spiral models. The parameters of background electrons contain normalization factor $A^{e^{-}}$, spectral index $\nu^{\rm e^{-}}_{1}$($\nu^{e^{-}}_{2}$) below(above) $R^{e^{-}}_{\rm br}$. $c^{e^{+}}$ represents the scale factor of the background positron flux. The corresponding parameters of local electron-positron pairs are injection rate of electron-positron pairs $q_{0}$, spectral index $\gamma$ and cut-off rigidity $R^{e^{\pm}}_{c}$. $\phi^{e^{+}}$ and $\phi^{e^{-}}$ are the modulation potentials for electrons and positrons respectively.

      Figure 4.  The positron spectra computed under four propagation models, i.e. HD+axis (upper left), HDB +axis (upper right), SDP + axis (lower left), SDP + spiral (lower right). To take the uncertainties from p-p collision cross section, propagation etc. into account, the secondary positrons under HD + axis, HDB + axis and SDP + axis models are multiplied by a scale factor $ c^{e^{+}} $. The red squares are the measurements from the AMS-02 experiment [3]. The blue dash and solid lines are the background flux and the rescaled one (multiplied by $ c^{e^{+}} $) respectively, while the green solid lines represents the fluxes from the local source. The black solid line is the sum of rescaled background and local fluxes. To describe the drop of positron above several hundreds of GeV recently unveiled by AMS-02, the cutoff rigidity of local source $ {\cal R}^{\rm e_{\pm}}_{\rm c} $ is 8 TV.

      In the HD model, the scale factor $ c^{e^+} $ is up to 2.0, and the studies of [47] and [48] come to a similar conclusion. Influenced by a noticeable growth of proton fluxes above tens of GeV, the HDB scenario would undoubtedly contribute more background component of positrons than that of HD model. Thus, its corresponding scale factor naturally decreases down to 1.8. But the SDP model brings about a broken power-law in the propagated positron spectrum so that above tens of GeV the positron spectrum becomes flatter. This appreciably enhances the secondary positron flux at higher energy. Compared with nuclei, the broken energy shifts from $ \sim 200 $ GeV to $ \sim 20 $ GeV, due to the energy loss of positrons during the transport. As can be seen, $ c^{e^+} $ in the SDP with axisymmetric distribution drops to 1.4 dramatically. According to [28], [77] and [34], the uncertainties of production cross section models differ the secondary positron flux up to tens of percentages. The uncertainties from the propagation and injection are ~30% at ~100 GeV, and gradually increase with energy [34]. In the model of spiral distribution, $ c^{e^+} $ further reduces to 1.0 for the propagated parameters in Table 3.

      In all four models, the injection spectrum of local source is harder than that of background. But in the HD and HDB models, the difference of both is larger. The power index of local source is around 2.00, while it is 2.8 for the background SNRs. But in the SDP models, the obtained power index of local pulsar increases and their difference becomes smaller. Moreover, it is noteworthy that in the HD models, the propagated positron spectrum show a pronounced high-energy tail, which is obviously different from the high-energy sharp fall-off under the assumption of transient injection [48]. This is caused by the continuous injection process of pulsar [41, 63]. Likewise similar phenomenon is also visible in the corresponding electron spectra, as shown in Fig. 5. But this feature disappears in the SDP models. Instead, the energy spectra above TeV energies slowly decline. Since around the Galactic disk, diffusion process is much slower, the energetic electrons from the local source suffer more energy loss so that most of them have exhausted before arriving at solar system. To fit the high-energy fall-off, the cutoff rigidity of local injection electron/positron spectrum is 8 TV.

      Figure 5.  The electron spectra computed under four transport models, i.e. HD+axis (upper left), HDB +axis (upper right), SDP + axis (lower left), SDP + spiral (lower right). The red squares are the measurements from the AMS-02 experiment [13]. The blue and violet lines represent the background primary and secondary electron fluxes respectively, while the green line refers to the electron flux from the local source.

      The observations confirm that there is a hardening above ~40 GeV in the electron spectrum, which indicates a primary component. Under the model of local pulsar, the contribution of electrons from the local source is determined by the positron data. It can be seen that, above ~100 GeV, the electron data could not be described both in the HD model, even with a local source. This is due to the excess of primary electron flux. In the observed electron flux, it contains both primary and secondary components, despite the latter are far less than the former. On the other hand, for either the propagation process or the local pulsar, almost same amount of electrons and positrons are generated. Thereupon, the difference between electrons and positrons are expected to be the primary components and described by a single power-law spectrum in the conventional diffusion model. However the analysis of [46] revealed that there is also a significant excess in this component. One of the origins for the extra electrons is the local SNRs. In a recent work by [48], to well describe both electron and positron spectra, a nearby supernova remnant surrounding by a giant molecular cloud is introduced, in which the additional electron-positron pairs, apart from the primary electrons, are produced inside the SNR through interactions between CRs and the molecular clouds. For HDB model, its flux of propagated secondary electrons is enhanced slightly, which is attributed to the hardening of proton spectrum. In comparison with primary component, this contribution of secondary electrons is small. Likewise, it is obviously seen that above ~ 100 GeV, the electron spectrum under HDB model could not reproduce the AMS-02 observations. In the SDP model, the calculated electron flux turns to be flatter so that the background electrons significantly augment at high energy, as shown in Fig. 5. The overabundant primary electrons are attributed to the effect of SDP. So the observed electron spectrum could be better described in the SDP model. Besides, the modulation potential $ \phi $ in the conventional DR model is larger [47], that is, 1200 MeV for electron and 1400 MeV for positron respectively. But in the SDP model, its value is greatly reduced, which is around ~650 MeV.

      The electron and positron spectra are add up together to further compare with the observed all-electron spectrum, which is shown in Fig. 6. The experimental data of AMS-02, CALET, DAMPE and H.E.S.S are represented by red squares, violet circles, black crosses and green inverted triangles, respectively. As can be seen from the figure, from ~50 GeV to ~1 TeV, the measurement by DAMPE experiment apparently disagrees with AMS-02 and CALET. Furthermore, the H.E.S.S experiment extended its measurement of all-electron spectrum up to 20 TeV and the data has a good agreement with both AMS-02 and CALET at around TeV energies. Therefore we do not fit the DAMPE data in this work. We compare the calculated all-electron spectra of the four models with the observations of AMS-02, CALET and H.E.S.S experiments. As can be seen, the all-electron spectra could be naturally reproduced by the SDP + spiral distribution scenario within the whole energy range. The break around TeV is caused by the local source, while the power-law fall-off above that energy comes from the background components.

      Figure 6.  All-electron (electron + positron) spectra calculated under four models. The red squares, violet circles and green inverted triangles are the data of AMS-02 [13], CALET [9] and HESS [42] respectively. The black crosses correspond to the measurements of DAMPE experiment [26].

    • 3.3.   All-electron anisotropy

    • The electrons and positrons above TeV chiefly originate from the local sources within 1 kpc around the solar system. Like CR nuclei [16, 49], the local source may also leave imprint on the all-electron anisotropy as well as the energy spectra. The dipole anisotropy is defined as

      $\delta = \frac{{{\psi _{{\rm{max}}}} - {\psi _{{\rm{min}}}}}}{{{\psi _{{\rm{max}}}} - {\psi _{{\rm{min}}}}}} = \frac{{3{D_{xx}}}}{v}\frac{{|\nabla \psi |}}{\psi }\;.$


      The anisotropies of electron plus positron from the local sources have been studied in the recent works based on the homogeneous diffusion model (see [32, 48, 63] and references therein). In this work, both background sources and local pulsar could contribute to the anisotropy. Since the Galactic SNRs concentrate at the inner disk, there is a radial CR streaming coming from the Galactic center. Meanwhile under spiral distribution, the SNRs are distributed along the spiral arms, there is another CR streaming perpendicular to the direction of Galactic center. Thereupon, the background anisotropy is a combined effect of two directions.

      Fig. 7 illustrates the energy dependence of anisotropy of electron plus positron when the local pulsar is at the direction of Galactic center. The blue solid lines denote the anisotropy from the background and the green lines show the anisotropy generated by the local pulsar when neglecting the background anisotropy, while the black lines are sum of background and local. In the HD and HDB models, the local source dominates the anisotropy and total anisotropy is very large, which is close to the observational upper limits. This is caused by the larger diffusion coefficient nearby the solar system. But in the SDP models, around Galactic disk, the diffusion coefficient is much smaller. Therefore in contrary to the HD and HDB models, the SDP models predict a very small background anisotropy. Even taking the local source into account, the total anisotropy is still well below the current upper limits, compared with the conventional propagation models [63].

      Figure 7.  Anisotropies of electron with the local source at the direction of Galactic center under the models of the HD+axis (upper left), HDB + axis (upper right), SDP+axis (lower left) and SDP+spiral (lower right)

      Fig. 8 shows the same electron anisotropies but with the local pulsar at the direction of anti-Galactic center. The CR streaming from local pulsar opposes to that of background, which could further lower the total anisotropy. When the CR streaming from local pulsar is comparable to that of background, the total anisotropy reaches a minimum. In the HD and HDB models, from tens of GeV to ~1 TeV, the local source always dominates, except for the lower energy, where the background streaming takes effect. Similarly for the case of SDP + axis distribution. But for SDP plus spiral distribution model, the background and local streaming are comparable at ~100 GeV and ~2 TeV respectively, there are two minimums in the total anisotropy. With this energy range, the anisotropy is dominated by the local pulsar, while background sources play major part in the rest of energy.

      Figure 8.  Anisotropies of electron with the local source at the direction of anit-Galactic center.

    4.   Conclusion
    • With the improvements of energy resolution, particle identification and available observation range of both space-based and ground-based experimental instruments, the measurements of CR electrons and positrons become more and more precise in the recent years. The current measurements unveil more unexpected features and raise new challenges to the conventional propagation model. In this work, we study the effects of SDP and spiral distribution by explaining the latest electron and positron observations. To demonstrate the both effects, four kinds of propagation models are compared. Compared with the homogeneous diffusion model, SDP could prominently enhance the background electron and positron flux at high energy, so that the excess of primary electron could be accounted for by the SDP effect without introducing other local SNRs. Both electron and positron spectra could be explained by the SDP model. In the meantime, the all-electron spectrum up to 25 TeV could also be naturally described in the SDP plus spiral distribution scenario. The break around TeV is generated by the local source and the high-energy power-law fall-off is principally attributed to the background components.

      In the local source model, both the distance and age are evaluated via fitting the AMS-02 observations. The local pulsar, for example $ t = 2.9\times10^{5} $ years and $ r = 0.25 $ kpc in SDP + spiral model, is very close to the Geminga pulsar ($ t = 342 $ kyr and $ r = 0.25 $ kpc). We use some properties of Geminga listed in the ATNF catalog [53] to further investigate the converted efficiency to electron-positron pairs of the local pulsar. According to the parameters given in Tab. 3, the total rotational energy loss is estimated as $ 9.83\times10^{48} $ ergs. The total injection energy of electron-positron pairs from this pulsar is $ 1.97\times10^{48} $ ergs, and the corresponding converted efficiency is nearly 20%. These values are consistent with the general estimation. For the local pulsars assumed in the other models, the corresponding estimations are also compatible with the expected estimation of a mature pulsar.

      The anisotropy of electron under four models are calculated. In the conventional model, the total anisotropy is dominated by the local source, and very close to the upper limits set by the Fermi-LAT experiment due to the larger diffusion coefficient. But in the SDP model, thanks to the smaller diffusion coefficient around the Galactic disk, the level of the background anisotropy is much lower. In this case, even including a local source, the total anisotropy is still very small. The future precise measurements of electron anisotropy by the DAMPE, LHAASO [25] and HERD [79] experiments could test our model.

      DRAGON [30] available at https://github.com/cosmicrays

Reference (85)



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