# Pulsar interpretation of lepton spectra measured by AMS-02

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## Abstract

Alpha Magnetic Spectrometer (AMS-02) recently published its lepton spectra measurement. The results show that the positron fraction no longer increases above \(\sim \)200 GeV. The aim of this work is to investigate the possibility that the excess of positron fraction is due to pulsars. Nearby known pulsars from the ATNF catalog are considered to be a possible primary positron source of the high energy positrons. We find that the pulsars with age \(T\simeq (0.45{-}4.5)\times 10^{5}\) year and distance \(d<0.5\) kpc can explain the behavior of positron fraction of AMS-02 in the range of high energy. We show that each of the four pulsars—Geminga, J1741-2054, Monogem, and J0942-5552—is able to be a single source satisfying all considered physical requirements. We also discuss the possibility that these high energy \(e{}^{\pm }\) are from multiple pulsars. The multiple pulsar contribution predicts a positron fraction with some structures at higher energies.

### Keywords

Dark Matter Spectral Index Proton Flux Multiple Pulsar Positron Fraction## 1 Introduction

The positron fraction spectrum \({e}^{+}\)/(\({e}^{+}+{e}{}^{-}\)) in the cosmic ray (CR) contains two components: secondary \(e{}^{\pm }\) produced by nuclei collision and primary \({e}^{-}\). It is currently believed that these two components, each of which will produce a diffused power low spectrum, predict a positron fraction which goes down with energy. However, the latest results measured by the Alpha Magnetic Spectrometer (AMS-02) with high accuracy indicate that the positron fraction increases with energy above \(\sim \)8 GeV and does not increase with energy above \(\sim \)200 GeV [1, 2]. This “increasing” behavior, which is also observed by the payload for antimatter matter exploration and light-nuclei astrophysics (PAMELA) [3, 4, 5] and the Fermi large area telescope (Fermi-LAT) [6, 7], is not compatible with only diffused power low components. The “cutoff” behavior above 200 GeV, which can be well described by a common source term with an exponential cutoff parameter in Eq. (1) of [1], indicates that potential sources produce the excess of electron and positron pairs.

AMS-02 [1, 2] is a state-of-the-art astroparticle detector installed on the International Space Station (ISS). It carries a Transition Radiation Detector (TRD) and a Electromagnetic Calorimeter (ECAL). These two sub-detectors provide independent proton/lepton identification, which will achieve a much larger proton rejection power of AMS-02 compared with PAMELA which has only one electromagnetic calorimeter for proton/lepton identification using the 3D shower shape and energy–momentum match (E/P). Compared with Fermi-LAT, AMS-02 has a large magnet which can identify charge sign of the particle. Thus, the contamination of electrons (also called “charge confusion” in [1]) in the positron sample of AMS-02 is much smaller than that of Fermi-LAT. For the reasons given above, there is much less proton or charge confusion contamination in AMS-02 measurement than that in PAMELA or Fermi-LAT. Here, we only interpret the AMS-02 result as due to the lack of knowledge of the contamination control in PAMELA and Fermi-LAT measurements.

AMS-02’s recent measurements of the positron fraction [1], \({e}{}^{+}\) flux, \({e}{}^{-}\) flux [2], and (\({e}{}^{-} + {e}{}^{+}\)) flux [8] were published. The \({e}{}^{-}\) flux contains three components: primary \({e}{}^{-}\), secondary \(e{}^{-}\), and \({e}{}^{-}\) from unknown sources. The \({e}{}^{+}\) flux contains only two components: \({e}{}^{+}\) from secondary production and primary \({e}{}^{+}\) from sources to be identified. To avoid the unnecessary uncertainty of primary \({e}{}^{-}\), the \(e{}^{+}\) flux seems to be an ideal spectrum to study extra sources. However, there is an acceptance uncertainty from the detector itself in the \(e{}^{+}\) and \(e{}^{-}\) fluxes. This uncertainty in \(e{}^{+}\) flux is strongly correlated with that in \(e{}^{-}\) flux [2], especially at high energies. The positron fraction can avoid this systematic uncertainty [1]. For example, one can clearly see a drop at the last point (350–500 GeV) in the positron fraction but cannot tell a drop at the last point (370–500 GeV) in the \(e{}^{+}\) flux due to its larger error bars. Therefore, the positron fraction is used to study extra sources, while the \(e{}^{-}\) flux is used to estimate the primary \(e{}^{-}\), which will affect the denominator of \(e{}^{+}\)/(\(e{}^{+}+e{}^{-}\)).

Recent studies have proposed some interpretations, such as dark matter annihilation or decay [9, 10, 11, 12, 13, 14, 15, 16, 17], supernova remnants (SNRs) [18, 19, 20, 21, 22, 23], secondary production in the interstellar medium (ISM) [24], and pulsars [16, 17, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Cosmic-ray flux data can also be taken together with other observations (like the dark matter relic density and the direct detection experimental results etc.) to give a combined constraint on dark matter models [40, 41]. Besides a dark matter scenario, the others can provide astrophysical explanations which do not require the existence of new particles. The SNRs model, for instance in [22, 42], introduces some new mechanisms for the propagation model or special distributions of the primary sources. The “model-independent” approach from [24] sets an upper limit of the positron fraction by neglecting radiative losses of electrons and positrons but does not indicate any obvious cutoff in the spectrum. Among them, the pulsar interpretation is one of the scenarios which predicts a cutoff at a few 100 GeV in the positron fraction spectrum and does not contradict other cosmic-ray spectra (e.g. boron-to-carbon). The pioneering work on a pulsar interpretation of the positron fraction has been performed [29, 31, 38] a few years ago. Combined analyses of the recent AMS-02 lepton data have been performed by [16, 23], with a global fit on the positron fraction [1], \(e{}^{+}\) flux, \(e{}^{-}\) flux and (\(e{}^{-}+e{}^{+}\)) flux. To avoid the over-estimation of the \(\chi {}^{2}\), however, only two out of four spectra should be used in the fit. As for the reasons given by the previous paragraph, we only study the positron fraction and \(e{}^{-}\) flux in this paper.

A pulsar is widely regarded as a rotating neutron star with a strong magnetosphere, which can accelerate electrons, which will induce an electromagnetic cascade through the emission of curvature radiation [43, 44, 45, 46]. This leads to the production of high energy photons, which eventually induces \(e{}^{+} e{}^{-}\) pair production. This process produces the same amount of high energy \(e{}^{+}\) and \(e{}^{-}\), which can escape from the magnetosphere and propagate to the earth. There is a cutoff energy of the photons produced in a pulsar, which leads to a cutoff in the positron fraction.

In this paper, DRAGON [47, 48, 49, 50, 51] is used as a numerical tool to model the propagation environment, to tune the related parameters and to estimate the \(e{}^{\pm }\) background. The authors of [49, 50, 51, 52] did a very complete work on three-dimensional cosmic-ray modeling. In the 3-D models, they pointed out that the spiral arms have an effect on the propagation parameters. A 2-D model is used in this paper because we focus on the lepton spectra implications. Due to the energy loss of the leptons, the effect of spiral arms on the high energy leptons is less important than that of the additional nearby sources contribution. ROOT is used to minimize \(\chi {}^{2}\) to get the best fit results. We consider six nearby pulsars from the ATNF catalog [53, 54] as the possible extra single sources of the high energy positrons. We find only four, which are Geminga, J1741-2054, Monogem, and J0942-5552, that can survive from all considered physical requirements. We then discuss the possibility that these high energy \(e{}^{\pm }\) are from multiple pulsars. The multiple pulsar contribution predicts a positron fraction with some structures at higher energies.

The paper is organized as follows. Section 2 shows the way how the \(e{}^{\pm }\) background is estimated. In Sect. 3, the properties of pulsars are described and the profile of the \(e{}^{\pm }\) fluxes produced by a pulsar is derived. The interpretation of the positron fraction with one single pulsar is discussed in Sect. 4 and the hypothesis as regards the multiple pulsar interpretation is tested in Sect. 5. The conclusions are drawn in Sect. 6. In Appendix A, the diffusion energy-loss equation for a burst-like source is solved in the spherically symmetric approximation.

## 2 Propagation parameters and \(e^{\pm }\) background

*i*per unit momentum interval; \(\mathbf {v}_c\) is the convection velocity, \(\beta \equiv v/c\) is the ratio of the velocity to the speed of light; \(\sigma _{\mathrm {in}}\) is the total inelastic cross section onto the ISM gas, whose density is \(n_{\mathrm {gas}}\); \(\sigma _{ji}\) is the production cross section of the species

*i*by the fragmentation of the species

*j*(with \(j>i\)); and \(Q_i(\mathbf {x},t,p)\) is the source term of species

*i*, which can be thought to be steady \(Q_i=Q_i(\mathbf {x},p)\) for background CR particles.

*D*in the cylindrical coordinate system (

*r*,

*z*) may be parameterized as [48, 56, 57]

*L*is the halo size, and \(\delta \) is the index of the power-law dependence of the diffusion coefficient on the rigidity. \(D_0\) is the normalization of the diffusion coefficient at the reference rigidity \(\rho _0=4\) GV. Previous DRAGON papers [49, 50, 51] tested a few models with the exponential profile, i.e., the left formula in (2), which is more physical than the constant one, i.e., the right one. The effect of choosing different profiles on the electron and positron background is small if the parameters are properly set. In this paper, the constant profile is used in order to compute the pulsar profile in an analytical way, i.e., Eq. (17) in Sect. 3. The function

*f*(

*r*) describes a possible radial dependence of

*D*, and it can be taken to be unity for simplicity.

*D*by [14, 57, 58, 59]

*w*is the ratio of the magnetohydrodynamic wave energy density to the magnetic field energy density, and it is usually taken to be 1.

*B*/

*C*ratio, which is sensitive to the parameters. The Markov chain Monte Carlo algorithm (MCMC, Ref. [60]) is used to determine \(D_{0}\) and \(\delta \). The priors are shown in Table 1. The posterior distributions are shown in a contour in the \(D_{0}\) and \(\delta \) plane in Fig. 1.

The priors of \(D_{0}\) and \(\delta \)

Start value | Minimum value | Maximum value | |
---|---|---|---|

\( D_{0}\, (\times 10^{28}~ \mathrm {cm}^{2}\,\mathrm {s}^{-1}) \) | 3 | 2.5 | 7.5 |

\(\delta \) | 0.40 | 0.20 | 0.65 |

*L*is taken from the MED model of [61], and the \(v_{A}\) is fixed. These parameters are consistent with what the authors of Ref. [16] has got in the reacceleration propagation model.

## 3 e\(^{\pm }\) from a single pulsar

*P*being the period) is a function of time as follows [29, 31, 38]:

*I*is the moment of inertia, which is related to the mass and the radius of the pulsar and can be regarded as a time independent value. The magnetic dipole radiation energy is equal to the energy-loss rate,

*v*of the particle, the speed

*c*of light, \(D_{0}\) and \(\delta \) the same as the parameters used to calculate the background in Sect. 2; and \(b(E)\equiv -\mathrm{d}E/\mathrm{d}t=b_0E^{2}\) with \(b_{0}=1.4\times 10^{-16}~\mathrm {GeV}^{-1}\,\mathrm {s}^{-1}\) is the rate of energy loss due to inverse Compton scattering and synchrotron radiation [6, 31, 38].

*d*is the distance between the earth and the source, the diffusion distance \(r_{\mathrm {dif}}\) is given by

Parameters of six nearby single pulsars from the best fit results. The \(\chi ^{2}/ndf\) from the fits of Geminga, J1741-2054, Monogem, J0942-5552, and J1001-5507 are smaller than 1, which show good agreement between those single pulsar models and the experiment

Pulsar name | | | \(\log {}_{10}\left( \frac{Q_{0}}{\mathrm {GeV}}\right) \) | \(\alpha \) | \(A_{\mathrm{prim},e-}\) | \(\chi ^{2}/ndf\) |
---|---|---|---|---|---|---|

Geminga | 0.25 | 3.42 | 50.5 | 2.04 | 0.50 | 26.8/40 |

J1741-2054 | 0.25 | 3.86 | 50.6 | 2.03 | 0.50 | 26.8/40 |

Monogem | 0.28 | 1.11 | 50.1 | 2.15 | 0.50 | 27.3/40 |

J0942-5552 | 0.30 | 4.61 | 50.6 | 2.01 | 0.49 | 27.7/40 |

J1001-5507 | 0.30 | 4.43 | 50.1 | 2.34 | 0.47 | 27.6/40 |

J1825-0935 | 0.30 | 2.32 | 50.5 | 2.61 | 0.44 | 28.8/40 |

The positron fraction from AMS-02 implies a primary positron source with a cutoff energy \(1/E_{s} = 1.84 \pm 0.58\,\mathrm{TeV}^{-1}\) in their “minimal” model [1], which corresponds to \(E_{s} \in \left[ 490,790\right] \). Due to the limitation of the statistics of the high energy \(e^-\) and \(e^+\) measured by AMS-02, the upper bound 790 GeV is not a strict limit. Thus, we consider a primary \(e^+\) and \(e^-\) source contribution with a cutoff energy \(E_{\mathrm {cutoff}}\simeq \) (500–5000) GeV, which corresponds to a pulsar with an age \(T\simeq (0.45{-}4.5)\times 10^{5}\) year according to (19). The term \(\exp \left[ -\frac{d^2}{r_{\mathrm {dif}}^2}\right] \) in (17) tells us that a pulsar with \(d>r_{\mathrm {dif}}\) requires a larger normalization \(Q_0\), which hints at a larger \(E_{\mathrm {out}}\), a larger \(\eta \) in (10), or both. \(r_{\mathrm {dif}}>d\) is required in our study, whose physical interpretation is that the distance a particle travels in the ISM should be larger than the distance between the earth and the source. Equation (18) tells us that \(r_{\mathrm {dif}}\) is a function of the diffusion time \(t_{\mathrm {dif}}\) and the lepton energy *E*, as is shown by Fig. 3 where the color scale indicates \(r_{\mathrm {dif}}\). For \(T\simeq (0.45\sim 4.5)\times 10^{5}\) year and the lepton energy \(E=1000 \) GeV, \(r_{\mathrm {dif}}\) is always greater than 0.5 kpc. Selecting pulsars with \(d<0.5 \) kpc and \(T\simeq (0.45\sim 4.5)\times 10^{5}\) year, the high energy leptons they produced can reach the earth.

Thus, the pulsars with ages \(T\simeq (0.45\sim 4.5)\times 10^{5}\) year and distance \(d<0.5\) kpc can explain the behavior of the positron fraction of AMS-02 at a high energy range.

## 4 Single pulsar interpretation

A few simple examples using a single pulsar are given to explain the high energy positron fraction of AMS-02 [1]. The background electrons and positrons are described in Sect. 2. The primary electron flux is scaled by a normalization factor \(A{}_{\mathrm{prim},e^{-}}\) since it is not possible to constrain the electron flux contribution from SNRs. The age *T* and the distance *d* are taken from the ATNF catalog and the positron fraction is fitted to obtain the free parameters in (17), the spectral index \(\alpha \), and the normalization \(Q_0\). \(Q_0\) is fixed by the relation [23, 32, 37] \(E_{\mathrm {out}}=\int _{E_{\mathrm {min}}}^{E_{\mathrm {max}}} dEEQ(E)\simeq \int _0^\infty dEEQ(E)\), which approximately yields \(Q_{0}\simeq E_{\mathrm {out}}\) for \(\alpha \simeq 2\). The cutoff energy \(E_{\mathrm {cut}}\) is set to be 5000 GeV, which is large enough, as it does not change the shape of the pulsar contribution. Since we are interested in the positron excess at high energies, the fit is started from 10 GeV where the effect of solar modulation is negligible.

Six nearby single pulsars, whose ages \(T\simeq (0.45\)–\(4.5)\times 10^{5}\) year and distance \(d<0.5\) kpc, are used to fit the positron fraction. The Minuit package in ROOT is used to determine the parameters to minimize \(\chi ^{2}\). The best results of the single pulsars are listed in Table 2.

Electron injection efficiency \(\eta \) of the six nearby pulsars. For a single pulsar interpretation of the positron fraction, the results of Geminga, J1741-2054, Monogem, and J0942-5552 are thought to be reasonable while the possibilities of J1001-5507 and J1825-0935 as the high energy positron sources can be excluded

Pulsar name | \(|\dot{E}|\, (10^{33}~\mathrm {erg})\) | \(E{}_{\mathrm {tot}}\, (10^{49}~\mathrm {erg})\) | \(E{}_{\mathrm {out}}\, (10^{47}~\mathrm {erg})\) | \(\eta \, (\%)\) |
---|---|---|---|---|

Geminga | 32 | 1.2 | 5.19 | 8.7 |

J1741-2054 | 9.5 | 0.46 | 5.83 | 25 |

Monogem | 38 | 0.16 | 1.90 | 24 |

J0942-5552 | 3.1 | 0.21 | 7.02 | 67 |

J1001-5507 | 0.68 | 0.043 | 1.89 | 88 |

J1825-0935 | 4.6 | 0.082 | 5.09 | 120 |

Using the parameters of the best fit results, the positron fraction can be well reproduced by these single pulsar’s contributions. Table 2 tells us that the normalizations \(A{}_{\mathrm{prim},e^{-}}\) are around 0.5 and the spectral indices \(\alpha \) of different pulsars are around 2.

We can estimate the injection efficiency \(\eta \) from the pulsar. Taking Geminga as an example, the spin-down energy-loss rate of Geminga \(|\dot{E}(T)|=3.2\times 10^{34}~\mathrm {erg}/\mathrm {s}\). The total radiation energy of the magnetic dipole can be derived from Eq. (13) as \(E_{\mathrm {tot}}(T)\simeq |\dot{E}(T)|T^2/\tau _0=1.2\times 10^{49}~\mathrm {erg}\). From the fit, we get the injection energy \(E_{\mathrm {out}}/2=10{}^{50.5}~\mathrm {GeV}\simeq 5.19\times 10^{47}\) erg. From (14), we get \(\eta \sim 8.7\,\%\). This efficiency is consistent with the previous studies by [29, 38]. We can perform similar studies on the other five pulsars, whose results are listed in Table 3.

^{1}

## 5 Multiple pulsars interpretation

The extra high energy positrons may come from several pulsars. We perform a similar study for multiple pulsars as we do for a single pulsar. Benefiting from the study in Sect. 4, we can assume that the spectral indices \(\alpha \) of all the pulsars are the same. Considering the physical models of the pulsars to be similar, we make another assumption: that the electron injection efficiencies \(\eta \) are the same. These two assumptions help us reduce the number of free parameters. The discussion of \(\eta \) from a single pulsar in Sect. 4 tells us that Geminga, J1741-2054, and Monogem will give a much larger contribution to the high energy positron than J0942-5552. In other words, the \(\eta \) of J0942-5552 in Table 3 is much larger than that of Geminga, which implies that the contribution from J0942-5552 in the multiple pulsar interpretation can be negligible compared with that from Geminga.

We choose three from the four “surviving” pulsars in the multiple pulsars discussion. The input parameters are the age *T*, the distance *d*, and the energy-loss rate \(\dot{E}\) of each pulsar, while the parameters we get from the fit is the normalization factor of primary electron \(A_{\mathrm{prim},e^{-}}\), the spectral index \(\alpha \), and the electron injection efficiency \(\eta \). As shown in Fig. 5a, we obtain a good result from the multiple pulsar fit where \(\chi {}^{2}/ndf=26.9/40\). The parameters we get are \(A{}_{\mathrm{prim},e^{-}}=0.50\), \(\alpha =2.07\), and \(\eta =2.58\,\%\).

The multiple pulsar interpretation predicts a positron fraction with a decrease up to 600 GeV and after that a bump up to 2000 GeV, which is possible to observe with more accumulating AMS-02 data.

Using the parameters from the fit, we can reproduce the electron flux measured by AMS-02 [2] in Fig. 5b. It shows that our electron background estimation in Sect. 2 \(+\) pulsar contribution matches the experimental data especially at high energies. Figure 5 also shows that the effect of the uncertainty due to the propagation model is small at high energy. The solar modulation potential is taken as 550 MV in the best fit result. The solar modulation potential is varied between 400 and 800 MV, showing that its effect on low energy is quite large. To reproduce the low energy electron flux more accurately, we need monthly low energy electron fluxes, which may be published by AMS collaboration to model the solar modulation.

## 6 Discussion and conclusion

In this work, we investigate the possibility that the rise of the positron fraction measured by AMS-02 can be explained by pulsars. The propagation parameters and the injection spectra of nuclei and electrons are tuned according to the boron-to-carbon ratio and the proton flux. It will be better to tune those parameters with boron-to-carbon ratio and proton flux measured by AMS-02, since they are in the same data taking period as the lepton fluxes. We find that both the single pulsar model and the multiple pulsar model can explain the AMS-02 data very well. Six nearby pulsars are investigated as the single pulsar sources of the high energy positrons and finally four survive from all the conditions. The \(\chi {}^{2}\)s of these single pulsars in this work are much smaller that those in [17], mainly because we set the cutoff energy equal to 5000 GeV, while the authors of [17] set it to 1000 GeV. With three mostly contributing pulsars, the multiple pulsar model predicts a positron fraction with a decrease up to 600 GeV and a bump up to 2000 GeV. For the low energy, a simple solar modulation potential cannot explain the measurement well. Thus, we need the monthly electron fluxes which can describe solar activity during the whole period.

It is shown that the positron excess measured by AMS-02 can be explained by the pulsar scenario. Since the multiple pulsars can explain the experimental data well, it will be difficult to exclude the pulsar scenario by isotropy. With accumulating AMS-02 data and future experiments, we can see the positron fraction behavior up to higher energy, which will either confirm or reject the multiple pulsars scenario. If we consider other scenarios, such as dark matter, we have to look into other productions, antiproton production for instance, which have no contribution from pulsars.

## Footnotes

- 1.
Considering that the uncertainty of \(\log {}_{10}(\frac{Q_{0}}{\mathrm {GeV}})\) from the fit is \(\pm 0.1\), the \(E{}_{\mathrm {out}}\) for J1001-5507 is \(1.89^{+0.44}_{-0.36}\times 10^{47}\,\mathrm{erg}\). Thus, \(\eta = 88^{+21}_{-17} \) % for J1001-5507. There is no strong enough evidence that this \(\eta \) is smaller than 1. One should also note that \(\eta = 67^{+14}_{-12} \) % for J0942-5552, which is 2\(\sigma \) smaller than 1.

## Notes

### Acknowledgments

We would like to thank M. Boudaud, S. Caroff, S. J. Lin, A. Putze, S. Rosier-Lees and P. Salati for helpful discussions. This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11375277, 11410301005 and 11005163, the Fundamental Research Funds for the Central Universities, and Sun Yat-Sen University Science Foundation.

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