SEARCH FOR ANISOTROPIES IN COSMIC-RAY POSITRONS DETECTED BY THE PAMELA EXPERIMENT

Measurements of cosmic ray positrons address a number of questions in contemporary astrophysics, such as the nature and distribution of particle sources in our Galaxy and the subsequent propagation of cosmic rays through the Galaxy and the solar magnetosphere. Positrons are a natural component of the cosmic radiation, produced in the interaction between cosmic rays and interstellar matter. Observations by the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) experiment (Adriani et al. 2009, 2010, 2013), confirmed by other recent measurements (Ackermann et al. 2012; Accardo et al. 2014), revealed a positron excess above the predictions of commonly used propagation models (e.g., Moskalenko & Strong 1998; Delahaye et al. 2009). Either the standard paradigm of cosmic ray propagation in the galaxy should be revised or additional sources of cosmic rays should be introduced to explain the positron anomaly at high energy. There are two primary hypotheses for these new sources: (1) a source linked to particle physics, e.g., a dark matter decay or annihilation (e.g., Cirelli et al. 2008); or (2) a nearby astrophysical source, e.g., a pulsar or supernova remnant (e.g., Hooper et al. 2009). In the first case, among the various dark matter models proposed, Batell et al. (2010) and Schuster et al. (2010) suggested that a significant fraction of positrons detected at Earth could be produced by dark matter particles annihilating in the neighborhood of the Sun. In this case, the electron/positron pair emission would be strongly correlated with the center of the Sun and would produce visible effects in an anisotropy study.

In the second case, the localization of the astrophysical sources could result in anisotropy in the arrival direction of the cosmic ray positrons. Several authors (e.g., Ptuskin & Ormes 1995; Kobayashi et al. 2004; and Büsching et al. 2008), estimated that the expected anisotropy in cosmic ray positrons from supernova remnants and/or from pulsars should be of the order of a few percent or less.

Directional and timing data from PAMELA are exploited to search for positron anisotropies that can provide unique information on new sources of cosmic rays. Previous searches have already been carried out using the combined flux of electrons and positrons (Ackermann et al. 2012; Campana et al. 2013; Accardo et al. 2014) and the results were compatible with an isotropic distribution of the arrival directions of detected particles.

In this paper the analysis is focused on positrons since they represent a cleaner sample of cosmic rays produced by possible sources with respect to electrons. In most theoretical models, electrons and positrons from new sources are produced in pairs. However, the e⁺ flux consists of only two components, e⁺ from secondary production and from unknown sources, while the e⁻ flux contains, in addition and in a larger amount, also primary e⁻. Since the cosmic ray primary electrons in this energy range represent the 90% of the total e⁻ flux, the search for anisotropies in the pure electron sample will be dominated by an isotropic background.

In the following, an analysis method is presented for studying the anisotropy of positrons, using the back-tracing of particle trajectories in order to identify arrival directions far from the Earth. An evaluation of systematic uncertainties is also provided.

The PAMELA experiment has been operating on board the satellite Resurs-DK1 since 2006 June. Its design is optimized for precision studies of light particles and antiparticles in primary cosmic rays between a few tens of MeV and a few hundreds of GeV.

Since its launch, PAMELA has collected $\sim 8\cdot {10}^{9}$ events, of which $\sim 5\cdot {10}^{6}$ electrons and positrons, whose analysis is described in several publications (Adriani et al. 2009, 2010, 2011).

The PAMELA detector (Picozza et al. 2007) is equipped with a magnetic spectrometer, comprising a permanent magnet hosting a tracking system. The tracking system consists of six double-sided microstrip silicon planes, which allow the determination of the particle charge and rigidity (momentum divided by charge) with high precision. An imaging electromagnetic calorimeter, made of 44 silicon planes interleaved with 22 plates of tungsten absorbers, is mounted below the spectrometer to accurately perform particle identification and to measure the energy of electrons and protons. A Time of Flight system made up of three planes of double layers of plastic scintillators permits measurements of particle velocity and energy loss. It also provides the main trigger for the experiment. Another layer of scintillators and a neutron detector, placed below the calorimeter, give additional information about the shower extent and improves lepton/hadron discrimination. Particles in the PAMELA acceptance due to scattering or interactions are rejected during offline analysis of the anti-coincidence system signals.

This apparatus permits electrons and positrons to be separated from the proton background over the rigidity range 10 GV $\leqslant$ R $\leqslant$ 200 GV and permits the measuring of their incoming directions with an accuracy of about two degrees. For each particle crossing the detector the arrival direction from space is reconstructed using the trajectory inside the instrument and the satellite position. The satellite orbit, 70° inclination, and 350–610 km altitude allow PAMELA to perform a very detailed measurement of the cosmic radiation in different regions of Earth's magnetosphere. Back-tracing of detected particles in the geomagnetic field is performed in order to obtain their initial spatial distributions, outside of the Earth's magnetosphere.

This data analysis aims to identify the presence of any large-scale pattern in the distribution of arrival directions of cosmic ray positrons detected by PAMELA. The entire data set of PAMELA up to 2010 January was analyzed, constructing sky maps for positrons in Galactic coordinates. In order to perform a search for anisotropies in these data, the experimental particle flux, measured by the instrument for different directions, is compared with background (or coverage) maps derived from an isotropic particle flux.

Coverage maps can be determined through an accurate simulation of particles coming from any direction of the sky, but this would require a very precise knowledge of the instrument exposure, defined as the projection of the acceptance in each sky direction integrated over the full live time of the detector. Since the result depends on the position and orientation of the spacecraft as a function of time, small inaccuracies can lead to the creation of fake anisotropy signals. To overcome this problem, the background map for isotropic expectation was constructed using actual PAMELA data. The flux of protons is highly isotropic, as previously shown with PAMELA data (Giaccari et al. 2013). A set of proton events was selected, as described in Section 4, for the same period of time as used for the positron data. The instrument exposure, dead times, and other detector effects are therefore included when calculating the ratio of positrons over protons. The comparison between the signal and background maps was performed with two independent and complementary techniques: a statistical significance test introduced by Li and Ma (Li & Ma 1983) and a spherical harmonic analysis.

3.1. Back Tracing

3.2. Sky Pixelation

3.3. Significance Maps

The statistical significance of the signal over an isotropic background in each direction of the sky was derived following the technique described in the work of Li & Ma (1983). The significance is

$$\begin{eqnarray}S & = & \pm \sqrt{2}\{{N}_{\mathrm{on}}\mathrm{ln}[\displaystyle \frac{1+\alpha }{\alpha }(\displaystyle \frac{{N}_{\mathrm{on}}}{{N}_{\mathrm{on}}+{N}_{\mathrm{off}}})]\\ & & +{{N}_{\mathrm{off}}\mathrm{ln}[(1+\alpha )(\displaystyle \frac{{N}_{\mathrm{off}}}{{N}_{\mathrm{on}}+{N}_{\mathrm{off}}})]\}}^{1/2},\end{eqnarray} \tag{ 1 }$$

where ${N}_{\mathrm{on}}$ and ${N}_{\mathrm{off}}$ are respectively the observed and the expected number of events in a certain angular window of the sky, after the proton and positron maps are normalized to the same number of events.

In Equation (1) α is the ratio of the exposure in the on-source region divided by the exposure in the off-source region; in this case the on-source and off-source regions coincide so $\alpha =1$. The significance is defined as positive if there is an excess and is otherwise defined as negative in the presence of a deficit of events. If no signal is measured, the distribution of the significance S is a normal function. The results are reported bin per bin, as a number of standard deviations.

This technique, developed in γ-astronomy to search for point sources, can be easily arranged to search large anisotropies integrating the signal and the background maps over various angular scales in order to increase the sensitivity up to a large-scale pattern. In this survey, different angular radii for the integration, from 10° to 90°, were used. The center of each bin of the sky pixelization is considered and the content of the bin is given by the cumulative number of events falling within a given angular distance from the center of that bin. Both signal and background maps are smoothed following this process. In this way the obtained integrated maps have correlated neighboring pixels and none of the information at a given angular scale is lost.

The statistical Li & Ma significance test is subsequently applied to study possible deviations from isotropy. In the case of no strong anisotropies, each significance histogram approximates a Gaussian distribution. The significance distributions become narrower with increasing integration radius—this is expected because in the integrated maps the bins are strongly correlated and identical events are included in the calculation.

3.4. Spherical Harmonic Analysis

The orbital parameters of the Resurs-DK1 satellite provide a relatively uniform exposure over the full celestial sphere. This permits the study of the angular power spectrum of the arrival directions of the events, i.e., the cosmic ray intensity over the celestial sphere can be expanded in spherical harmonics. The power spectrum analysis is a powerful tool for studying the anisotropy patterns and provides information on the angular scale of the anisotropy in the map.

The positron (or electron) relative intensity map I, defined as the relative deviation of the observed number of events N from the expected number of events $\langle N\rangle$ in each angular bin of the sky defined by the Galactic coordinates $(\mathrm{glon},\mathrm{glat})$ is considered,

$$\begin{eqnarray}&&I(\mathrm{glon},\mathrm{glat})=\displaystyle \frac{N(\mathrm{glon},\mathrm{glat})-\langle N(\mathrm{glon},\mathrm{glat})\rangle }{\langle N(\mathrm{glon},\mathrm{glat})\rangle }.\end{eqnarray} \tag{ 2 }$$

The relative map I is expanded on the basis of spherical harmonics functions Y_lm

$$\begin{eqnarray}&&I(\mathrm{glon},\mathrm{glat})=\displaystyle \sum _{l=0}^{\infty }\displaystyle \sum _{m=-l}^{m=l}{a}_{{lm}}{Y}_{{lm}}(\mathrm{glon},\mathrm{glat}).\end{eqnarray} \tag{ 3 }$$

The full anisotropy information is encoded into the set of spherical harmonic coefficients, a_lm, which can be used to calculate the angular power spectrum

$$\begin{eqnarray}&&C(l)=\displaystyle \frac{1}{2l+1}\displaystyle \sum _{m=-l}^{m=l}{a}_{{lm}}^{2}.\end{eqnarray} \tag{ 4 }$$

The amplitudes of the power spectrum at some multi-pole order are associated with the presence of structures in the distribution of the arrival directions in the sky. Non-zero amplitudes in the multi-pole moments, C_l, arise from fluctuations in the particle flux on an angular scale near $180/l$ degrees.

3.4.1. Dipole Upper Limit

Although all of the spherical harmonic moments are important for characterizing the anisotropy patterns at any angular scale, the dipole moment is of particular interest (Büsching et al. 2008). In fact, if there is a marked directionality, dipole anisotropy is expected, as could be the case for a single source dominating the positron flux. In this case the overall intensity at an angular distance θ from the maximum of the dipole anisotropy can be written as I(θ) = I₀ + I₁cos(θ), where I₀ represents the isotropic signal and I₁ represents the maximum anisotropy due to the dipole. The degree of anisotropy can be expressed as the fraction $\delta ={I}_{1}/{I}_{0}$.

By applying Equation (2) the intensity of the dipole anisotropy becomes

$$\begin{eqnarray}&&I(\theta )=\displaystyle \frac{I(\theta )-\langle I(\theta )\rangle }{\langle I(\theta )\rangle }=\displaystyle \frac{I(\theta )-{I}_{0}}{{I}_{0}}=\displaystyle \frac{{I}_{1}\mathrm{cos}(\theta )}{{I}_{0}}.\end{eqnarray} \tag{ 5 }$$

Since the spherical harmonic ${Y}_{1}^{0}(\theta ,\phi )$ depends on θ, it is possible to derive that ${a}_{1}^{0}=\displaystyle \frac{{I}_{1}}{{I}_{0}}\sqrt{\frac{4\pi }{3}}$ , and consequently

$$\begin{eqnarray}&&{C}_{1}={(\displaystyle \frac{{I}_{1}}{{I}_{0}})}^{2}\displaystyle \frac{4\pi }{9}.\end{eqnarray} \tag{ 6 }$$

Starting from Equation (6), the correlation between the dipole amplitude and the power spectrum coefficient with l = 1 is

$$\begin{eqnarray}&&\delta =3\sqrt{\displaystyle \frac{{C}_{1}}{4\pi }}.\end{eqnarray} \tag{ 7 }$$

Considering $\hat{{C}_{1}}$ as the outcome of the calculated angular power spectrum over the signal map for l = 1, for the observed spectrum

$$\begin{eqnarray}&&\hat{\delta }=3\sqrt{\displaystyle \frac{\hat{{C}_{1}}}{4\pi }}.\end{eqnarray} \tag{ 8 }$$

Since $\hat{{C}_{1}}$ follows a ${\chi }^{2}$ distribution with three degrees of freedom centered on C₁, the probability of measuring $\hat{{C}_{1}}$ given the true C₁ is

$$\begin{eqnarray}&&P(\hat{{C}_{1}};{C}_{1})\displaystyle \frac{3\sqrt{3}}{\sqrt{2\pi }{C}_{1}}\sqrt{\displaystyle \frac{\hat{{C}_{1}}}{{C}_{1}}}\mathrm{exp}(-\displaystyle \frac{3\hat{{C}_{1}}}{2{C}_{1}}).\end{eqnarray} \tag{ 9 }$$

From Equations (8) and (9) it is possible to derive the probability of $\hat{\delta }$ as

$$\begin{eqnarray}&&P(\hat{\delta };\delta )=\displaystyle \frac{3\sqrt{6}}{\sqrt{\pi }}\displaystyle \frac{{\hat{\delta }}^{2}}{{\delta }^{3}}\mathrm{exp}(-\displaystyle \frac{3{\hat{\delta }}^{2}}{2{\delta }^{2}}).\end{eqnarray} \tag{ 10 }$$

The upper limit on δ can be calculated using a frequentist approach where the limit corresponds to the solution of the integral

$$\begin{eqnarray}&&{\displaystyle \int }_{0}^{\hat{\delta }}P(\hat{\delta }| \delta )d\delta =1-{\rm{C}}.{\rm{L}}.\end{eqnarray} \tag{ 11 }$$

for a given confidence level (C.L.) (Neyman 1937; Garthwaite et al. 1995).

The sample of positrons selected for this analysis stems from the period from the satellite launch to the end of the solar minimum phase (2006 June–2010 January). This is the same data set as used for previous publications (Adriani et al. 2010). However, the selection criteria are different because a statistical subtraction of proton contamination in the positron sample does not allow event-by-event positron identification, as required for determining the incoming direction and rigidity of each event. The positron sample was instead selected using a cut-based analysis similar to the one developed for the electron flux determination (Adriani et al. 2011). The same track and event quality selection were used; the hadron contamination was reduced to a negligible amount by using a stronger calorimeter selection (with about 80% selection efficiency) combined with activity requirements on the neutron counter. A very clean sample of positrons was obtained from flight data in the rigidity range from 10 to 200 GV. A total number of 1489 positrons with rigidity R > 10 GV was selected for this work. With the same selection a sample of 20,673 electrons and positrons was selected for the anisotropy search in the direction of the Sun. The quality of the sample was verified by comparing the resulting positron fraction to the published one. In the same period, 2006 June–2010 January, the calorimeter was used to select hadronic showers initiated by positively charged particles in the same rigidity range and with the same requirements on the event and track quality. In this way a proton sample of about $4.5\cdot {10}^{5}$ events, which preserved the angular distribution of particles along the PAMELA orbit, was obtained. Finally, a refined analysis on the detector orientation was carried out by comparing the pointing of the instrument determined by the satellite orbital information with the pointing obtained using data from the Earth remote sensing camera that is installed on board the Resurs-DK1 satellite. This refined analysis minimized the uncertainty on the measured absolute incoming trajectory for all of the selected particles, which is used in the back-tracing and in the anisotropy maps.

Figure 1 shows the maps obtained for events with a rigidity greater than 10 GV, for selected positrons and protons, taking into account the geomagnetic field. As the PAMELA experiment has a shorter exposure to the poles, the sky is not observed uniformly, leading to a particular shape in the observed maps. The same overall exposure is achieved for both types of particle. Both the signal and the background maps were binned taking into account the detector angular resolution. The color scale represents the number of detected events for each bin. If a signal can be found in the data, it could spread over multiple adjacent bins, making the anisotropy measurement more difficult. Hence, to highlight a possible anisotropy, the bin size must be similar to the angular scale of the anisotropy itself. To increase sensitivity it is possible to merge the content of neighboring bins. The content of each integrated bin is equal to the total number of events belonging to the bins covered by a circular region of a given radius. The radius chosen for the integration represents the angular scale at which the anisotropy is expected. If the integration radius is too small the signal is distributed among adjacent bins, while if it is too large too much background will be integrated. The events and the coverage maps were integrated over the following angular scales: 10°, 30°, 60°, and 90°. The corresponding results are reported in Figure 2 for positrons and in Figure 3 for protons. The color scale now represents the integrated number of events. The exposure distribution varies depending on the integration radius as expected. To estimate the statistical significance of any excess or deficit, the method detailed in Section 3.3 was used. In this method, likelihood functions are applied both to the null hypothesis and to the signal hypothesis; the ratio between these two functions represents the significance of the excess. Significance maps, constructed by comparing signal and corresponding background maps for the different integration radii, are shown in Figure 4. The significance signals are always less than 4σ, indicating that a point-like source cannot be distinguished from the background. In the case of statistical fluctuations for an isotropic sky the distribution of the significance will exhibit a Gaussian distribution with zero mean. In Figure 5, the obtained significance distributions are shown together with a fitted Gaussian function; no significant deviation from isotropy is observed. It can be seen that the significance distribution becomes narrower when increasing the integration radius since with larger integration radius, the bins are strongly correlated, reducing the variance of the distribution. For the study of the angular power spectrum of the arrival direction distribution, the cosmic ray intensity was expanded in spherical harmonics, using the anafast code provided by Healpix software (Gorski et al. 2005). The power spectrum in modes from l = 1 (dipole) up to l = 20 was studied. Due to statistical and systematic uncertainties higher order modes are irrelevant for this search. In Figure 6 the angular power spectra C(l) as a function of the multi-pole l are shown. The band defined in the figures by the two dotted lines shows the 3σ region of the expected power spectrum from an isotropic sky. The power spectra are compatible with the prediction of the isotropic sky and no significant deviation from the isotropy was observed. The value for the measured dipole amplitude ${\hat{\delta }}_{1}$, as expressed in Section 3.4.1, is calculated with the value of coefficient C₁ reported in Figure 6. Solving the integral in Equation (11) for a confidence level at 95%, the limit on the dipole anisotropy parameter is δ = 0.076.

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Systematic uncertainties were estimated with the following methodology. Variables that have an effect on the anisotropy measurement were independently and artificially varied in the selected data samples to understand the effects of the introduced biases on the anisotropy measurement. For each of the new biased samples of positrons and protons the analysis was performed as explained in the previous sections; the resulting Li and Ma distribution and angular power spectrum were compared to the unbiased one and the difference was taken as an estimation of systematic uncertainties for the given bias. Assuming no correlation between the systematic effects (biases), the total resulting systematic uncertainty was obtained by quadratically summing the single components.

Three main sources of systematic errors were identified:

• 1.  
  Particle direction reconstruction. Experimentally, the particle direction reconstruction in celestial coordinates results from the combination of the particle direction in the PAMELA reference frame measured by the tracking system and the pointing direction of the satellite (hence of PAMELA which is solidly attached to the Resurs-DK1). The inclination information of the satellite is provided to PAMELA by the satellite every 1.5–180 s. The higher rate is used during the satellite movements that are performed to observe the Earth with the optical camera mounted on the satellite. Hence, satellite inclination information derives from an interpolation of points as a function of time and has a resolution that is usually better than about two degrees. To estimate the effects of a possible systematic in the particle direction reconstruction on the anisotropy measurements, the inclination measurement was changed for each event by choosing random values according to a Gaussian distribution, with a variance equal to the experimental angular resolution.
• 2.  
  Particle rigidity determination. Particle rigidity is measured by the tracker by determining the deflection of charged particles in the magnetic field. The maximum detectable rigidity (MDR) in PAMELA is about 1.2 TV, resembling the accuracy in the cluster position determination of about 3 μm in the bending view and the spectrometer magnetic field of about 0.4 Tesla. Residual coherent misalignment of the tracker silicon sensors after detector calibration could bring a systematic shift in the rigidity measurement of about 15% at the MDR, corresponding to a maximum shift in the deflection measurement of about 0.0001 GV⁻¹. A systematic error in the rigidity measurement could have an impact in the anisotropy measurement, since the applied back-tracing procedure relies on the particle rigidity to perform the integration of motion in the Earth's magnetic field. To estimate the effects of these systematic uncertainties, a deflection uncertainty ±10⁻⁴ GV⁻¹ was added and subtracted to the deflection measurement of selected particles
• 3.  
  Event time determination. Absolute UTC time is saved by the PAMELA CPU with a precision of one millisecond. The absolute time of the PAMELA CPU is synchronized with the satellite with a precision of one second at least once per orbit. Different clock frequencies between the two clocks and the absolute time at ground are negligible. Time determination is used to calculate the satellite position in latitude, longitude, and altitude given the orbital elements. Hence, the uncertainty in time corresponds to an uncertainty in the orbital positioning and influences the back-tracing procedure and the anisotropy estimation. Systematic effects were estimated by biasing the sample adding and subtracting one second to the measured time on an event-by-event basis.

The resulting estimated systematic uncertainty on the anisotropy measurement is of the order of about 12% and is shown as gray bands in Figures 5 and 6.

An excess of electrons and positrons was also searched for in the Sun direction. The reference frame used in this analysis is the ecliptic coordinate system centered on the Sun and referred to the J2000 epoch (Murray 1989).

The number of electron and positron events within annuli centered on the Sun in the range 0°–90°, in 5° steps, is calculated. The results are reported in Figure 7 as a function of the angular distance from the Sun direction and are compared to the expectations from an isotropic sky. The gray boxes represent the 3σ bounds for an isotropic sky calculated with an elliptical reference frame on subsets of coverage map. Data are consistent with the isotropic expectation within a 3σ interval. Similar results were obtained by the Fermi LAT collaboration (Ajello et al. 2011) without selecting electrons and positrons separately.

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The detection of an anisotropy signal might be used to distinguish between models that could explain the cosmic ray positron excess (e.g., Adriani et al. 2009), measurements above the predictions of commonly used propagation models. A variety of models, invoking either dark matter or astrophysical objects as positron sources, e.g., see review papers by Boezio et al. (2009) and Serpico (2012), have been put forward to explain this excess. As a result of the different localization of such sources, an anisotropy in the arrival particle distribution cannot be excluded. Various authors, e.g., Ptuskin & Ormes (1995); Kobayashi et al. (2004); Büsching et al. (2008); Ackermann et al. (2010); and Di Bernardo et al. (2011), studied the expected anisotropy in cosmic ray positrons from supernova remnants and/or from pulsars. For example, Ackermann et al. (2010) and Di Bernardo et al. (2011) estimated an anisotropy of the order of about 1% for 100 GeV positrons accelerated by a Vela supernova remnant, while Büsching et al. (2008) estimated an anisotropy of about 3% and about 0.3% for positrons with energies between 10 and 100 GeV accelerated by Monogem and Geminga pulsars, respectively. In the case of a dark matter origin, Ackermann et al. (2010) estimated an anisotropy of the order of per-mil.

A blind search for positron anisotropies was performed using PAMELA data, taking into account the effects of the Earth's geomagnetic field. The analysis of PAMELA data was carried out by using two techniques: the significance maps, which allowed the study of the directions of possible overdensities at any angular scale, not just the dipole contribution, and the angular power spectrum, which is a method capable of determining the anisotropy magnitude. Results were consistent with isotropy at all angular scales considered. A limit on the dipole anisotropy parameter, δ = 0.076 at the 95% C.L., was obtained. This limit is close but higher than the expected anisotropy for astrophysical sources and about an order of magnitude higher than values found in dark matter models that therefore cannot be excluded. Because of the limited statistics of the positron data set it was not possible to investigate the energy dependence of the dipole amplitude.

A few models (Batell et al. 2010; Schuster et al. 2010) predict that large astrophysical bodies within the Solar System may significantly contribute to cosmic ray positrons, capturing dark matter particles that may first annihilate into metastable mediators. If the mean free path of these mediators is in excess of the solar (or planetary) radius, they can escape and annihilate again, producing gamma-rays and charged particles such as positrons that arrive from directions correlated with the centers of the astrophysical source. The analysis of PAMELA all-electron data in the direction of the Sun is consistent with the isotropic expectation at a 99.7% C.L.

The results of the analysis reported in this paper are in agreement with those published by AMS and Fermi LAT. It should be noted, however, that special care is needed when comparing the results of these three different experiments. AMS results (Accardo et al. 2014) refer to positrons but reporting only the dipole upper limit, while in this paper a multi-pole analysis is presented. Fermi LAT results (Ackermann et al. 2012) refer to the sum of electrons and positrons. In the energy range between tens of GeV and hundreds of GeV, electrons from standard cosmic rays sources are the dominant component. Hence, the study of the pure positron signal is more sensitive, in this energy range, to anisotropies. Finally, in this analysis the effects of the Earth's magnetic field, which could affect a weak anisotropy, were taken into account, while both the AMS and Fermi LAT results refer to the anisotropy as measured in a low Earth orbit.

It should be noted that the local (a few tens of parsecs) turbulent realization of the interstellar magnetic field (Giacinti & Sigl 2012), as well as magnetic reconnection in the heliotail (Lazarian & Desiati 2010) may have an effect on the cosmic ray arrival directions. Such processes have been proposed to explain the anisotopy observed mostly in the multi-TeV energy region (Nagashima et al. 1998; Amenomori et al. 2006; Giullian et al. 2007; Abdo et al. 2009; Aglietta et al. 2009; Abbasi et al. 2010; Iuppa et al. 2013). However, there are no quantitative studies on the effects of these processes on the arrival directions of tens of GeV positrons measured at Earth.

The arrival direction of cosmic ray positrons in the rigidity range from 10 to 200 GV for about 10³ positrons collected in the first 4 years of operation of PAMELA was studied. The effects of the Earth's geomagnetic field were accounted for. The results are consistent with isotropy at all angular scales considered. Moreover, no enhancement of the all-electron flux was observed in the direction toward the Sun.

We would like to acknowledge contributions and support from our universities and institutes, the Italian Space Agency (ASI), Deutsches Zentrum für Luft—und Raumfahrt (DLR), the Swedish National Space Board, the Swedish Research Council, the Russian Space Agency (Roscosmos), and the Russian Scientific Foundation.