Pulsars as the Sources of High Energy Cosmic Ray Positrons
1 Introduction
Dark matter particles annihilating in the Galactic Halo are predicted to generate a number of potentially observable products, including gamma-rays, electrons, positrons, protons and antiprotons. In contrast to gamma-rays, which travel along straight lines, charged particles move under the influence of the Galactic Magnetic Field, diffusing and losing energy, resulting in a diffuse spectrum at Earth. By studying the cosmic ray antimatter spectra, balloon or satellite-based experiments hope to identify signatures of dark matter.
The PAMELA satellite, which began its three-year mission in June of 2006, is designed to measure the spectra of cosmic ray positrons up to 270GeV and electrons up to 2TeV, each with unprecedented precision. Recent results show that the ratio of positrons to electrons plus positrons (the positron fraction) in the cosmic ray spectrum appears to stop decreasing and begins to climb quite rapidly between 10GeV and 100GeV.
A similar trend was in fact also indicated by earlier experiments, including HEAT and AMS-01, although with lesser statistical significance. The behavior in the positron fraction observed by PAMELA is very different from that predicted for secondary positrons produced in the collisions of cosmic ray nuclides with the interstellar medium. Barring systematics (e.g. problems in the positron/proton separation at high rigidity), the data appear to indicate the existence of additional, primary sources of high energy positrons, such as dark matter annihilations taking place in the halo of the Milky Way. It should be noted, however, that while the observed spectral shape can be easily accommodated with annihilating dark matter, the normalization of this contribution to the PAMELA data requires a somewhat large annihilation rate.
In either case, such scenarios are somewhat constrained by observations of gamma-rays, antiprotons and synchrotron emission. The challenges involved in explaining the PAMELA signal with annihilating dark matter lead us to consider a less exotic and purely astrophysical explanation for the observed positron flux. Energetic electron-positron pairs can be in fact produced in astrophysical sources, the leading candidate sites being pulsars -- rapidly spinning, magnetized neutron stars, which emit pulsed electromagnetic radiation, as observed from Earth.
In this paper, we explore the possibility that the positron fraction reported by PAMELA may be generated by mature pulsars. Gamma-ray pulsars are predicted to produce energetic electron-positron pairs with a harder spectrum than that from secondary cosmic-ray induced origin, leading to the possibility that such sources may dominate the cosmic ray positron spectrum at high energies.
We calculate the spectrum of such particles from known local pulsars (Geminga and B0656+14), and from the sum of all pulsars distributed throughout the Milky Way. As found in earlier studies, we find that both local pulsars and the sum of pulsars distributed throughout the Milky Way can contribute significantly to the observed spectrum. At 10GeV, we estimate that on average only 20% of the cosmic ray positrons originate from pulsars within 500 parsecs from the Solar System.
The remainder of this article is structured as follows: In section 2, we review the known properties of pulsars and consider them as sources of high energy electron-positron pairs. In section 3, we consider the nearby pulsars Geminga and B0656+14 and discuss their potential contributions to the cosmic ray positron spectrum. In section 4, we calculate the expected dipole anisotropy from nearby pulsars and compare this to the sensitivity of the Fermi gamma-ray space telescope. We summarize and draw our conclusions in section 5.
In both models of polar gap and outer gap, electrons can be accelerated in different regions of the pulsar magnetosphere and induce an electromagnetic cascade through the emission of curvature radiation, which in turn results in production of photons which are above threshold for pair production in the strong pulsar magnetic field. This process results in lower energy electrons and positrons that can escape the magnetosphere either through the open field lines or after joining the pulsar wind. In this second case, the electrons and positrons lose part of their energy adiabatically because of the expansion of the wind.
The energy spectrum injected by a single pulsar depends on the environmental parameters of the pulsar, but some attempts to calculate the average spectrum injected by a population of mature pulsars suggest that the spectrum may be relatively hard, having a slope of This spectrum, however, results from a complex interplay of individual pulsar spectra, of the spatial and age distributions of pulsars in the Galaxy, and on the assumption that the chief channel for pulsar spin down is magnetic dipole radiation. Due to the related uncertainties, variations from this injection spectra cannot be ruled out. Typically, one concentrates the attention on pulsars of 105 years because younger pulsars are likely to still be surrounded by their nebulae, which confine electrons and positrons and thus prevent them from being liberated into the interstellar medium until later times.
Still, some energetics considerations can be done with simple analytical models; this will also help the understanding of arguments developed in the next section. The rate of energy injection from a single pulsar in the form of pairs is limited by its spin-down power (the rate of energy loss corresponding to the slowing rate of rotation).
To proceed in a more quantitative way towards the calculation of the overall spectrum from Galactic pulsars, one needs to adopt a model for the e+ ? e? acceleration and escape probability from a single pulsar with a given magnetic field, period, etc. and then integrate over a Monte Carlo distribution of these typical parameters in a Galactic Pulsar population.
In figure 1, we show the spectrum of positrons and the positron fraction resulting from the sum of all pulsars throughout the Milky Way. In the upper panels, we show results for different rates of pulsar birth (one per 10, 25, or 100 years). The dashed line represents the baseline result neglecting the contribution from pulsars, including only the positrons produced as secondaries in the hadronic interaction of cosmic rays. In the right frames, the positron ratio is obtained considering, besides secondary leptons, also the primary electrons accounted as in, to ease the comparison with previous literature. In the right frames, the measurements of HEAT (light green and magenta) and the measurements of PAMELA (dark red) are also shown.
The cutoff in the positron spectrum derived in our calculations is solely the result of the corresponding cutoff in the injection spectrum shown in eq. (2.7). This cutoff is determined by the details of the development of the electromagnetic cascade in the pulsar magnetosphere and, even more importantly, by the distribution of periods, magnetic fields, and radii of mature pulsars. The exact value of the cutoff energy should, therefore, not be considered to be a robust prediction of the theory, although it represents a good estimate of the order of magnitude of the cutoff energy. For instance, in ref. it is argued that the typical energy of electrons and positrons in the cascade associated with the polar gap is at energies lower than 10GeV, pulsars are not expected to contribute any appreciable flux because of the very hard spectrum, compared to the spectrum of secondary positrons produced in hadronic interactions of cosmic rays diffusing throughout the Galaxy. These secondary electrons approximately reproduce the steep spectrum of the parent nuclei and at low energies dominate the observed positron fraction. Since the spectrum of positrons from pulsars is important only at relatively high energies, we have neglected here the role of solar modulation.
Nearby pulsars as a source of high energy cosmic ray electrons and positrons In this section, following earlier studies, we re-explore the possibility that an individual or small number of nearby pulsars dominate the cosmic ray positron spectrum within the energy range studied by PAMELA. As argued in the previous section, in order to contribute significantly such a pulsar can be neither too young nor too old.
The spectrum of electron positron pairs at Earth is again calculated by solving the transport equation, but this time for a single source. Moreover we consider the case of a bursting source, namely one in which the duration of the emission is much shorter than the travel time from the source. For the pulsars discussed above this seems indeed to be appropriate. For the simple case of a power-law spectrum of electrons and positrons. The Fermi sensitivity shown is for the spectrum integrated above a given energy.
In figure 5 we plot the level of anisotropy expected for a Geminga-like and a B0656+14-like pulsar if they are responsible for the majority of the observed positron excess. The two dashed lines show the sensitivities of Fermi to anisotropy at 95 confidence level and at 5&sigma; confidence level, after five years of observation (integrated above a given energy). We find that Fermi should be capable of identifying a single local source (or multiple sources in the same direction of the sky) if that source injected the bulk of its electrons/positrons within the last few hundred thousand years (the B0656+14-like and Geminga-like cases correspond to injection 110,000 and 370,000 years ago, respectively). If only a fraction of the high energy positrons observed by PAMELA originate from a given nearby pulsar, the corresponding solid lines shown in figure 5 should be multiplied (reduced) by this factor.
Alternatively, if dark matter annihilations throughout the Milky Way's halo are primarily responsible for the excess in the high energy cosmic ray positron spectrum, a small dipole anisotropy in the direction of the Galactic Center could also be generated. Fortunately, both B0656+14 and Geminga are in approximately the opposite direction, allowing for a potentially unambiguous discrimination between these possibilities. In the special and relatively unlikely case that a nearby dark matter subhalo in the direction of B0656+14/Geminga is responsible for the observed flux, it would be difficult to distinguish between pulsar and dark matter origins using this technique.
If anisotropy studies should prove inconclusive in resolving this issue, other information could be inferred from the shape of the positron fraction and of the overall electron/positron spectrum. Peculiar shapes can result from the superposition of the overall pulsar spectrum plus local contributions (see, for example, figure 4 or ref). Future studies of the electron and positron spectra at higher energies will be especially important, as the spectral cutoff in the pulsar case is typically expected to be smoother and less sudden than that predicted from annihilating dark matter.
Furthermore, combining electron/positron measurements with those of antiprotons, antideuterons and diffuse gamma-rays may prove useful in distinguishing between these possibilities. Population studies of pulsars in the gamma-ray band by Fermi are also expected to refine theoretical predictions and shed light on this issue.
5 Conclusions
The results recently reported by PAMELA strongly indicate the existence of a primary source or sources of high energy cosmic ray positrons. This result is very interesting, even if of purely astrophysical origin. Several papers have appeared recently which discuss this signal within the context of dark matter annihilations. In this article, we have instead explored the possibility that the observed flux of high energy positrons is the result of electron-positron pairs being produced in nearby and galactic pulsars. We find that pulsars throughout the Milky Way, and a small number of nearby mature pulsars, such as B0656+14 and Geminga, could each plausibly generate the observed flux of positrons. To normalize the overall flux, on the order of a few percent of the pulsars' spin down power is required to be transferred into the production of electron-positron pairs. The prediction in the case of the sum of all pulsars in the Galaxy appears somewhat more robust in that it relies on the average statistical properties of these astrophysical objects rather than on the specific characteristics of nearby pulsars.
It is remarkable that in this case, reasonable values for the parameters can lead to a positron spectrum consistent with the observations. Also, a pulsar origin would naturally fit the absence of an excess in the anti-proton data, since differently from dark matter scenarios no hadronic cascades are associated with the production of pairs in the magnetospheres.