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Test of the Superdiffusion Model in the Interstellar Medium Around the Geminga Pulsar

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Test of the Superdiffusion Model in the Interstellar Medium Around the Geminga Pulsar Test of the superdiffusion model in the interstellar medium around the Geminga pulsar Sheng-Hao Wanga;b, Kun Fanga,∗ Xiao-Jun Bia;b,y and Peng-Fei Yina aKey Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, Beijing 100049, China bUniversity of Chinese Academy of Sciences, Beijing 100049, China (Dated: January 6, 2021) Abstract The TeV γ-ray halo around the Geminga pulsar is an important indicator of cosmic-ray (CR) propagation in the local zone of the Galaxy as it reveals the spatial distribution of the electrons and positrons escaping from the pulsar. Considering the intricate magnetic field in the interstellar medium (ISM), it is proposed that superdiffusion model could be more realistic to describe the CR propagation than the commonly used normal diffusion model. In this work, we test the superdiffu- sion model in the ISM around the Geminga pulsar by fitting to the surface brightness profile of the Geminga halo measured by HAWC. Our results show that the chi-square statistic monotonously increases as α decreases from 2 to 1, where α is the characteristic index of superdiffusion describ- ing the degree of fractality of the ISM and α = 2 corresponds to the normal diffusion model. We find that model with α < 1:32 (or < 1:4, depending on the data used in fit) is disfavored at 95% confidence level. Superdiffusion model with α close to 2 can well explain the morphology of the arXiv:2101.01438v1 [astro-ph.HE] 5 Jan 2021 Geminga halo, while it predicts much higher positron flux on the Earth than the normal diffusion model. This has important implication for the interpretation of the CR positron excess. ∗ [email protected] y [email protected] 1 I. INTRODUCTION Several middle-aged pulsars, such as the Geminga pulsar, are reported to be surrounded by TeV γ-ray halos with scales larger than 20 pc [1]. These γ-ray halos are generated by free electrons and positrons1 diffusing out from the corresponding pulsar wind nebulae (PWNe), rather than being interpreted by γ-ray PWNe [2]. According to the evolution model of PWN, the original PWNe of the middle-aged pulsars were broken long time ago [3]. These pulsars are currently traveling in the interstellar medium2 (ISM) and driving bow-shock PWNe with scales . 1 pc. This is consistent with the observations of the X-ray PWN of the Geminga pulsar [5{7]. The TeV γ-ray halos are generated mainly through the inverse Compton scattering (ICS) of the homogeneous cosmic microwave background, so the morphologies of the halos unambiguously indicate the electron propagation in the ISM. Cosmic-ray (CR) propagation in the ISM is usually modeled by the diffusion process considering the turbulent nature of the ISM [8]. The diffusion approximation is based on the assumption that the inhomogeneity of the chaotic magnetic field has small-scale character and is negligible in terms of the scale of interest. However, multi-scale inhomogeneities could exist in the ISM. The ISM is more likely to be a fractal type and the normal diffusion can be generalized to superdiffusion [9{12], where the CR propagation is simulated by L´evy flights instead of the Brownian motion. This model has been applied in the Galactic-scale propagation of CRs to explain features of the CR energy spectra [9, 13]. In this work, we test the superdiffusion model in a local region of the Galaxy by explaining the morphology of the Geminga halo. As the diffusion packets are different for the normal diffusion and superdiffusion models, the γ-ray morphologies predicted by them could also be distinct. This paper is organized as follows. In Section II, we introduce the superdiffusion model and the solution of the propagation equation. Then we briefly introduce the information of Geminga in Section III. We fit the γ-ray surface brightness profile (SBP) of the Geminga halo measured by HAWC with different propagation models and discuss the fitting results in Section IV. In Section V we discuss the impact on the interpretation of the positron excess according to the results of Section IV. Finally, we conclude in Section VI. 1 Electrons will denote both electrons and positrons hereafter. 2 The pulsars could also be inside their old host SNRs if the SNRs are large enough [4]. 2 II. THE SUPERDIFFUSION MODEL After escaping from the source, CRs are continuously scattered by the chaotic magnetic field in the ISM. If the chaotic magnetic field is uniformly distributed, the particle trans- portation can be simulated by the Brownian motion [8]. However, the realistic magnetic field in the ISM may consist of turbulent and relatively regular components, where very long jumps for CRs are permitted. L´evyflight, which is characterized by the occasionally very long steps, should be the more proper description in this case [9{12]. For one-dimensional L´evyflight, the probability density function (PDF) of the individual step length x has the heavy-tailed form of P (x) / jxj−1−α for x ! 1, where 0 < α < 2. Obviously, the variance of the step length is infinite, which is different from that of the Brownian motion. The widening of the diffusion packet with time is also faster for L´evyflight (/ t1/α) than the normal diffusion case (/ t1=2), so this scenario is named after superdiffusion or accelerated diffusion [11]. As the PDF of L´evyflight can be described by the fractional Laplacian equation, the propagation equation of CR electrons should be @N(E; r; t) − α @[b(E)N(E; r; t)] = −D(E; α)(−∆) 2 N(E; r; t) + + Q(E; r; t) ; (1) @t @E where N is the differential number density of electrons, D is the diffusion coefficient, and Q is the source function for which we will present the details in Section III. The index α represents the degree of fractality of the ISM and for α = 2 the equation degenerates to the 2 normal diffusion case. The radiative energy-loss rate b(E) = b0E induced by synchrotron radiation and ICS must be considered for high-energy electrons. The magnetic field strength is assumed to be 3 µG to obtain the synchrotron term. For the ICS process, we adopt the seed photon field in Ref. [1] and the parameterization given by Ref. [14] to calculate the energy-loss rate, where the Klein-Nishina correction is accurately considered. We solve Eq. (1) with the Green function method. The Green function takes the form of ρ(α)(jr − r jλ−1/α) G(E; r; t; E ; r ; t ) = 3 0 δ(t − t − τ)H(τ) ; (2) 0 0 0 b(E)λ3/α 0 where Z E0 0 Z E0 0 dE D(α; E ) 0 τ = 0 ; λ = 0 dE ; (3) E b(E ) E b(E ) 3 α=2.0 10-1 α=1.8 α=1.6 α=1.4 α=1.2 10-2 α=1.0 10-3 ) α (r; 3 ρ 10-4 10-5 10-6 0 5 10 15 20 r FIG. 1. Probability density functions of three-dimentional spherically symmetric stable distribu- tions for different α. (α) and H is the Heaviside step function. In Eq. (2) ρ3 (r) is the PDF of a three-dimensional (α) spherically-symmetrical stable distribution with index α. The exact expression of ρ3 (r) is Z 1 (α) 1 kα ρ3 (r) = 2 e sin(kr)kdk ; (4) 2π r 0 while in practice it can be expressed with convergent and asymptotic series [15]: 1 n (α) 1 X (−1) 2n + 3 ρ (r) = Γ( )r2n ; (5) 3 2π2α (2n + 1)! α n=0 1 n−1 (α) 1 X (−1) nαπ ρ (r) = Γ(nα + 2) sin( )r−nα−2 : (6) 3 2π2r n! 2 n=1 (α) We show ρ3 (r) in Fig. 1, where we can clearly see the difference between the heavy-tailed distributions for α < 2 and the Gaussian distribution for α = 2. The solution of Eq. (1) can be then expressed as Z Z t Z +1 3 N(E; r; t) = d r0 dt0 dE0 G(E; r; t; E0; r0; t0) Q(E0; r0; t0) R3 −∞ −∞ (7) Z Z t (α) −1/α 3 b(E?) ρ3 (jr − r0jλ ) = d r0 dt0 3/α Q(E?; r0; t0) ; 3 R t−1=(b0E) b(E) λ where E? ' E=[1 − b0E(t − t0)]. 4 From Eq. (7), we can get the electron number density at arbitrary distance from a pulsar. To calculate the γ-ray SBP around the pulsar, we integrate the electron number density along the line-of-sight from the Earth to the vicinity of the pulsar and get the electron surface density at an arbitrary angular distance θ from the pulsar: Z +1 F (θ) = N(lθ)dlθ ; (8) 0 where lθ is the length in that line-of-sight, and N(lθ) is the electron number density at a p 2 2 distance of d + lθ − 2dlθ cos θ from the pulsar, where d is the distance between the pulsar and the Earth. With F (θ) and the standard calculation of ICS [16], we can finally obtain the γ-ray SBP around the pulsar. III. THE SOURCE GEMINGA The Geminga halo is so far the best-studied TeV halo. The γ-ray SBP is precisely measured by HAWC [1] and is an ideal object to investigate the propagation of CR electrons. In the normal diffusion scenario, the derived diffusion coefficient around the Geminga pulsar is more than two orders of magnitude smaller than the average value in the Galaxy indicated by the boron-to-carbon ratio [17].
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