Spectral shapes of the fluxes of electrons and positrons and the average residence time of cosmic rays in the Galaxy

The cosmic ray energy spectra encode very important information about the mechanisms that generate relativistic particles in the Milky Way, and about the properties of the Galaxy that control their propagation. Relativistic electrons and positrons traveling in interstellar space lose energy much more rapidly than more massive particles such as protons and nuclei, with a rate that grows quadratically with the particle energy $E$. One therefore expects that the effects of energy loss should leave observable signatures in the $e^{\mp }$ spectra, in the form of softenings centered at the critical energy $E^{*}$. This quantity is determined by the condition that the total energy loss suffered by particles during their residence time in the Galaxy is of the same order as the initial energy. If the electrons and positrons are accelerated in discrete (quasi) pointlike astrophysical objects, such as supernova explosions or pulsars, the stochastic nature of the sources should also leave observable signatures in the $e^{\mp }$ energy spectra and angular distributions above a second (higher) critical energy $E^{†}$, determined by the condition that particles with $E\gtrsim E^{†}$ can propagate only for a maximum distance shorter than the average separation between sources. In this work we discuss the theoretically expectations for the signatures of the energy-loss effects on the electron and positron spectra, and compare these predictions with the existing observations. Recent measurements of the ($e^{-}+e^{+}$) flux have discovered the existence of a prominent spectral break at $E\simeq 1\mathrm{TeV}$. This spectral feature can perhaps be identified with the critical energy $E^{*}$. An alternative hypothesis is to assume that $E^{*}$ has a much lower value of order of a few GeV. Resolving this ambiguity is of great importance for our understanding of Galactic cosmic rays.

Figure 1

Spectra of $e^{\mp }$ measured by PAMELA [1, 2] (squares) and AMS02 [6] (circles) shown in the form $E^3{\varphi }_{e^{\mp }}(E)$ versus $E$. The energy scale of the PAMELA has been rescaled by a factor $f=0.93$. The lines are fit to the data discussed in the text.

Figure 2

Electron spectra measured by AMS02 [6] and by PAMELA [4] in different time intervals. The energy scale of PAMELA has been rescaled by a factor $f=0.93$. The lines are fit to the data discussed in the main text.

Figure 3

Measurements of the electron (left panel) and positron (right panel) spectra taken during 79 different time intervals (of 27 days) by the AMS02 Collaboration [7]. The thick solid lines are the fits of the time-averaged $e^{\mp }$ spectra discussed in this work. The dashed lines are calculated by distorting the average spectra using the FFA model for solar modulations with different parameters.

Figure 4

Measurements of the ($e^{-}+e^{+}$) spectrum obtained by AMS02 [13], FERMI [14], ATIC [15], CALET [16, 17], DAMPE [18], HESS [19, 20, 21], MAGIC [22], and VERITAS [23]. The line is a fit to the data from [24].

Figure 5

Measurements of the ($e^{-}+e^{+}$) spectrum obtained by AMS02 [13], DAMPE [18], and HESS, and of the $e^{+}$ flux by AMS02. The lines are fits of the DAMPE [18] and HESS [21] data.

Figure 6

Spectral indices of the $e^{-}$, $e^{+}$ and ($e^{-}+e^{+}$) fluxes. The index is calculated from Eq. (2) using analytic expressions to fit the data. The fits to the AMS02 data are discussed in the text. The fits to the DAMPE and HESS data are from [18] and [21]. The shaded regions are estimates of the $1\sigma$ uncertainties.

Figure 7

Spectral feature generated by energy losses calculated in a leaky box model. The source spectrum and the escape time have energy dependence $q(E)\propto E^{-\alpha }$ and $T_{\mathrm{esc}}(E)\propto E^{-\delta }$.

Figure 8

Examples of the function $g(x,x_i,{\sigma }^2)$.

Figure 9

Spectral feature generated by energy losses in a diffusive model. The diffusion coefficient is homogeneous in the finite layer $|z|<H$ and has the energy dependence $D(E)=D_0{E}^{\delta }$. The CR source is confined in a thin layer at $z\simeq 0$ and has the form $q(E)=q_0$; $E^{-\alpha }$.

Figure 10

Fits of the radial distribution of pulsars (Yusifov and Kucuk [34]) and supernova remnants (Case and Bhattacharya [33]) in the Mikly Way ($r$ is the distance from the Galactic center).

Figure 11

Space-time distribution of the CR flux (energy losses are considered negligible). The thick closed lines in and around the shaded area are defined by the equation $d\varphi /(d\ln rd\ln t)=\text{const}$ and chosen so that the integral in the region inside the line corresponds to a fraction 0.1, 0.5, and 0.9 of the total flux. The thinner lines are defined by the equation ${\varphi }_s(r,t)/{\varphi }^{*}=f$, with $f={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$, 1, 10, and $10^2$.

Figure 12

Space and time distribution of the CR sources (energy losses are considered to be negligible). Left panel: Plot of $1/\varphi d\varphi /d\rho$ versus $\rho =r/H$. Right panel: Plot of $1/\varphi d\varphi /d\tau$ versus $\tau =t/T_{\mathrm{esc}}(E)$.

Figure 13

Space-time distribution of the CR flux (energy losses are considered to be dominant). See Fig. 11 for the meaning of the different lines. The calculation is performed by assuming that the source spectrum is a power law with exponent $\alpha =2.4$, and the diffusion coefficient has the energy dependence $D_0{E}^{\delta }$ with $\delta =0.4$.

Figure 14

Space and time distribution of the CR sources (energy losses are dominant). Left panel: Plot of $1/\varphi d\varphi /d{\rho }^{\prime }$ versus ${\rho }^{\prime }=r/T_{\max }(E)$. Right panel: Plot of $1/\varphi d\varphi /d{\tau }^{\prime }$ versus ${\tau }^{\prime }=t/T_{\text{loss}}(E)$.

Figure 15

The top (bottom) panel shows the shape of the distribution $d{N}_s/d{\varphi }_s$ ($d\varphi /d{\varphi }_s$) as a function of ${\varphi }_s$, calculated assuming that energy losses are negligible. The distributions are shown in the energy-independent scaling form, plotting the quantity $1/N^{*}d{N}_s/d\ln x$ ($1/\varphi d\varphi /d\ln x$ in the bottom panel) where $x={\varphi }_s/{\varphi }^{*}$ and the quantities $N^{*}$ and ${\varphi }^{*}$ are given in Eqs. (76) and (68), with $R=H$ and $T=T_{\mathrm{esc}}(E)$.

Figure 16

The top (bottom) panel shows the shape of the distribution $d{N}_s/d{\varphi }_s$ ($d\varphi /d{\varphi }_s$) as a function of ${\varphi }_s$ calculated by assuming that energy losses are dominant for CR propagation (escape is negligible). The distributions are shown in the energy-independent scaling form, plotting the quantity $1/N^{*}d{N}_s/d\ln x$ ($1/\varphi d\varphi /d\ln x$ in the bottom panel) where $x={\varphi }_s/{\varphi }^{*}$ and the quantities $N^{*}$ and ${\varphi }^{*}$ are given in Eqs. (76) and (68) with $R=R_{\text{loss}}(E)$ and $T=T_{\text{loss}}(E)$. The shape of the distributions depends on the exponents $\alpha$ and $\delta$, and the curves describe the cases $\{\alpha =2.6,\delta =0.4\}$ and $\{\alpha =2.6,\delta =0.0\}$.

Figure 17

The curves show, as a function of the kinetic energy $E_k$, the quantities $N_p^{*}$ and $N_e^{*}$, which give the (approximate) number of sources that contribute to the observed flux. The curves are calculated using a generalization of Eqs. (82) and (84), assuming that the frequency of sources in the entire Milky Way is $(50\mathrm{years}{)}^{-1}$ with $\delta =0.4$, and for two different values of the critical energy $E^{*}$: $E^{*}=3\mathrm{GeV}$ and $E^{*}=900\mathrm{GeV}$. The two points along the curves for $N_e^{*}$ indicate the energy $E_e^{†}$ [see Eq. (93)] where one expects that granularity effects become dominant.

Figure 18

Cumulative flux ${\varphi }_{\mathrm{cum}}$ as a function of the number of source events $N_s$ calculated by assuming that energy losses are negligible. The relation is shown in the energy-independent form ${\varphi }_{\mathrm{cum}}(N_s)/\varphi$ versus $N_s/N^{*}$ where $N^{*}$ is given in Eq. (76) with $R=H$ and $T=T_{\mathrm{esc}}(E)$.

Figure 19

As in Fig. 18, but with the flux calculated by assuming that energy losses are the dominant effect in propagation. The quantities $N^{*}$ are defined in Eq. (76) with $R=R_{\text{loss}}(E)$ and $T=T_{\text{loss}}(E)$. The shape of the curve depends on the exponents $\alpha$ and $\delta$. The two lines in the figure describe the cases $\{\alpha =2.6,\delta =0.4\}$ and $\{\alpha =2.6,\delta =0.0\}$.

Figure 20

Spectra of $p$, $e^{-}$, $e^{+}$, ($e^{-}+e^{+}$), and $\overline{p}$. The high energy data points for protons are from CREAM [39]; the data points for ($e^{-}+e^{+}$) are from DAMPE and HESS. All other points are from AMS02. The lines superimposed to the $e^{-}$, $e^{+}$, and $\overline{p}$ data points are simple power-law fits for $E>30\mathrm{GeV}$. The line for the proton data is a broken power-law fit taken from [9].