Interpretation of the Spectra and Anisotropy of Galactic Cosmic Rays

Abstract

Recent measurements of the spectra and anisotropy of cosmic rays (CRs) show a fine structure that reflects the spectral hardenings of CRs nuclei at the rigidity $\mathcal{R}$ ∼ 200 GV followed by softenings at $\mathcal{R}$ ∼ 10 TV, and reveal complicated energy dependence of the amplitude and phase of anisotropy from 100 GeV to PeV. Numerous studies have shown that the existence of nearby CR sources and a local interstellar magnetic field (LIMF) near the solar system are crucial for such CR spectral and anisotropic patterns. In this work, we analyze the CR spectra of different CR components and the anisotropy considering the nearby Geminga supernova remnants (SNRs) source. In the calculation process, we also introduce the anisotropic diffusion of CRs induced by the LIMF based on the spatial-dependent propagation (SDP) model. As a result, our model can simultaneously account for the CR spectra and the anisotropy from 100 GeV to PeV. Future high-precision measurements of the CR anisotropy, for example, by the LHAASO experiment, would be of the essence in the assessment of our proposed model.

1. Introduction

It is widely believed that cosmic rays (CRs) below PeV energies originate from galactic sources, presumably supernova remnants (SNRs) [1]. CRs can be accelerated to form non-thermal power law spectra, $dN/d\mathcal{R} \propto {\mathcal{R}}^{-v}$, through the diffusive shock acceleration mechanism at SNRs, with $\mathcal{R}$ being the rigidity and ν an injection power index [2,3,4,5]. After CRs are released by their sources, they enter the Milky Way Galaxy and interact with irregular magnetic fields and interstellar gas, which can be described by a diffusion process with the diffusion coefficient of $D\left(\mathcal{R}\right)\propto {\mathcal{R}}^{\delta }$ and $\delta ~ 0.38–0.5$ as inferred from the Boron-to-Carbon (B/C) ratio. After the diffusion transportation, the CR spectrum further softens into the form of $\Phi \propto {\mathcal{R}}^{-v-\delta }$ [6].

Recent observations of the CR energy spectra indicate that the spectral structures are more complicated than ever expected. The spectral hardenings of CR nuclei at $\mathcal{R}$ ~ 200 GV are shown by a large number of experiments, such as ATIC-2 [7], CREAM [8,9], PAMELA [10], AMS-02 [11] and the calorimeter experiment CALET [12]. Most recently, measurements of the proton and helium spectra by the DAMPE [13], CREAM [14] and NUCLEON experiments [15] reveal that the spectra become soft at 14 TeV.

Owing to the deflection of the galactic magnetic field (GMF), galactic cosmic rays (GCRs) lose their original direction and become almost isotropic. However, small CR anisotropy with relative amplitudes at the level of 10⁻⁴~10⁻³ is still observed at a wide energy range from 100 GeV to PeV. Complex energy dependence of the CR anisotropy is unveiled by numerous experiments, such as Tibet [16,17,18], Super-Kamiokande [19], Milagro [20,21], IceCube/IceTop [22,23,24,25,26], ARGO-YBJ [27,28], HAWC [29], EASTOP [30], and KASCADE [31]. Less than 100 TeV, the amplitude of anisotropy increases gradually up to 10 TeV at first with energy and then decreases, but increases again above 100 TeV. The phase reverses from $\mathrm{R}.\mathrm{A}.\simeq 3^h$ to the galactic center (GC) at around 100 TeV.

In general, the CR anisotropy is attributed to the following reasons. One is the nearby sources, such as SNRs, which can influence the anisotropy by changing the spatial gradient of the CR particles in the solar system [32,33]. The second is the local interstellar magnetic field (LIMF) at the level of ~3 µG as inferred from the Interstellar Boundary Explorer (IBEX) [34] experiment along (l, b) = (210.5°, −57.1°). The Larmor radius of PeV CRs is much smaller than their scattering length in the LIMF, which indicates that the LIMF can deflect the PeV CR particles [35,36,37,38] so that the CR diffusion is anisotropic in the LIMF, i.e., stronger in the direction parallel to the LIMF and weaker in the perpendicular direction. Several literatures [36,39,40,41] also illustrate that the observation of the anisotropic phase of CRs is coincident with the LIMF.

Neither the spectral hardenings at ~200 GV nor the complex energy dependence of the anisotropy can be explained by the conventional propagation model. The common features of the CR energy spectra and anisotropy from 10 to 100 TeV indicate a common origin of the CRs. Based on the conventional propagation model, various improvements have been explored to explain the spectra and anisotropy, including those by introducing the nearby source [32,42,43,44], the spatially dependent diffusion process [45,46,47], the LIMF [36,48], and the ensemble fluctuations of CR sources.

The aim of this work is to explore a scenario that can simultaneously explain the CR anisotropy and energy spectra from 100 GeV to PeV. A large number of works have shown that the CR energy spectra and anisotropy are sensitive to the age, position, power, and cut-off rigidity of the nearby source [48,49,50]. Because the Geminga SNR is located close to the anti- GC direction and far from the galactic disk, it is regarded as a prime candidate [51]. Our method is based on the spatial-dependent propagation (SDP) model, with introducing the anisotropic diffusion induced by magnetic fields and considering the contribution from the nearby Geiminga SNR source. We calculated the energy spectra of different CR components and CR anisotropy. Ultimately, our model can simultaneously explain CR anisotropy and energy spectra. For example, the high-precision measurements of CR anisotropy by the LHAASO experiment could help test this scenario”.

2. The Model of Cosmic Rays Propagation

2.1. Spatially Dependent Diffusion

The recent HAWC observations [29] have shown that the diffusion coefficient of GCRs near the galactic disk is at least two orders of magnitude smaller than the conventional one [51]. Therefore, in this work, we adopt the SDP model [46,47,52], whose diffusion coefficients differ in the inner halo and the outer halo. The SDP model was first proposed to explain the hundred GeV spectral hardenings of CRs [10] and further applied to account for the secondary and heavier components [47,53,54,55,56], the diffuse gamma-ray distribution [57] and the large-scale anisotropy [49,50].

The diffusion volume of the SDP model is separated into two regions, i.e., the inner halo (|z| < ξz_h) and the outer halo (|z| > ξz_h). In the inner halo, particles diffuse slowly because the level of turbulence is expected to be high due to activities of supernova explosions; whereas in the outer halo, the diffusion coefficient is much larger.

The parameterized diffusion coefficient which we adopt is [55,57],

$$D_{xx}\left(r,z,\mathcal{R}\right)=D_{0}F\left(r,z\right){\left(\frac{\mathcal{R}}{{\mathcal{R}}_0}\right)}^{{\delta }_{0}F\left(r,z\right)},$$

(1)

F(r, z) is parameterized as,

$$F\left(r,z\right)=\{\begin{array}{cc}g\left(r,z\right)+\left[1-g\left(r,z\right)\right]{\left(\frac{z}{\xi z_0}\right)}^n,\hfill & \left|z\right|\le \xi z_0\hfill \\ 1,\hfill & \left|z\right|>\xi z_0\hfill \end{array}$$

(2)

where g(r, z) = N_m/[1 + f(r, z)], and f(r, z) is the source density distribution. The spatial distribution of sources takes the form of the SNR distribution [58], and f(r, z) $\propto$ (r/r_⊙)^1.69exp[−3.33(r − r_⊙)/r_⊙] exp(−|z|/z_s), where r_⊙ = 8.5 kpc and z_s = 0.2 kpc. In this work, we adopt the numerical package DRAGON to solve the transport equation [59].

The injection spectrum of background sources is assumed to be a power-law of rigidity with a high-energy exponential cutoff, $q(\mathcal{R})$ $\propto {\mathcal{R}}^{-v}$exp(−$\mathcal{R}$/$\mathcal{R}$_c).

2.2. Nearby Source

The Green’s function method is adopted to calculate the time-dependent propagation of CR particles from the nearby source, assuming a spherical geometry with infinite boundary conditions [42,43]. The GCR density of the nearby source as a function of the injection spectrum is described as,

$$\Phi \left(r,\mathcal{R},t\right)=\frac{q_{inj}\left(\mathcal{R}\right)}{{\left(\sqrt{2\pi }\sigma \right)}^3}\exp \left(-\frac{r^2}{2{\sigma }^2}\right),$$

(3)

where $q_{inj}\left(\mathcal{R}\right)\delta \left(t\right)\delta \left(r\right)$ is the instantaneous injection spectrum of a point source, σ($\mathcal{R}$, t) = $\sqrt{2D\left(\mathcal{R}\right)t}$ the effective diffusion length within time t, and D($\mathcal{R}$) the diffusion coefficient which is adopted as the value near the solar system. The injection spectrum is also parameterized as a cutoff power-law form, $q_{inj}\left(\mathcal{R}\right)=q_0{\mathcal{R}}^{-\alpha }\exp \left(-\mathcal{R}/{{\mathcal{R}}^{\prime }}_c\right)$. The normalization $q_0$ is obtained through fitting the GCR energy spectra.

In this work, we select the Geminga SNR as the contribution source to the spectral anomaly above 200 GeV and the complex evolution of anisotropy at lower than 100 TeV [49,50]. The Geminga SNR is located in the direction of l = 194.3°, b = −13° [60] and its distance to the solar system is d ~ 330 pc. Its explosion time was about τ = 3.4 × 10⁵ years ago [51].

3. Anisotropic Diffusion and Large-Scale Anisotropy

3.1. Anisotropic Diffusion

According to the introduction, it seems clear that the LIMF (l, b = 210.5°, −57.1°) observed by IBEX should be a key factor in the explanation of the anisotropy of CRs. The diffusion of CRs in magnetic field is anisotropic, and the diffusion tensor D_ij associated with the magnetic field is written as,

$$D_{ij}\equiv D_{\perp }{\delta }_{ij}+\left(D_{\parallel }-D_{\perp }\right)b_i{b}_j,b_i=\frac{B_i}{\left|\mathit{B}\right|},$$

(4)

where b_i is the i-th component of the unit vector of the ordered magnetic field B in the chosen coordinate system [61], and $D_{\parallel }$ and $D_{\perp }$ are the diffusion coefficients aligned parallel and perpendicular to the ordered magnetic field, respectively.

In this work, two different rigidity dependents $D_{\parallel }$ and $D_{\perp }$ are shown as follows [62],

$$D_{\parallel }=D_{0\parallel }{\left(\frac{\mathcal{R}}{{\mathcal{R}}_0}\right)}^{{\delta }_{\parallel }}$$

(5)

$$D_{\perp }=D_{0\perp }{\left(\frac{\mathcal{R}}{{\mathcal{R}}_0}\right)}^{{\delta }_{\perp }}=\epsilon D_{0\parallel }{\left(\frac{\mathcal{R}}{{\mathcal{R}}_0}\right)}^{{\delta }_{\parallel }}$$

(6)

where ε = $D_{0\perp }/D_{0\parallel }$ is the ratio between the perpendicular and the parallel diffusion coefficients at the reference rigidity $\mathcal{R}$₀, which has been studied by numerous of literatures [38,42,63,64,65,66,67].

CR particles usually propagate in a magnetic field consisting of two parts, i.e., a regular part, B₀, and a turbulent part, δB. The total magnetic field is their vector sum, B = δB + B₀. The Larmor radius of a TeV CR particle in the LIMF is smaller than its mean free path. According to the quasi-linear theory [63,64], the ratio between $D_{\parallel }$ and $D_{\perp }$ is given by,

$$\frac{D_{\perp }}{D_{\parallel }}~F\left(k\right)~\frac{\delta B{\left(k\right)}^2}{B_0^2}\ll 1$$

(7)

where F(k) is the normalized power of the turbulent modes with wave number k [40,62,68], and δB(k)² = ${{\displaystyle \int }}^{ }{d}^3\delta B{\left(k\right)}^2\ll {\left|B_0\right|}^2$. If δB/B₀ ∼ 1, $D_{\perp }\simeq D_{\parallel }$, then the diffusion of CRs with higher energies tends to be isotropic.

3.2. Large-Scale Anisotropy

It is known that the amplitude of the dipole anisotropy is proportional to the spatial gradient of the CR density and the diffusion coefficient. In an anisotropic diffusion model, its vector form can be written as [36,41],

$$\delta =\frac{3}{v\psi }{D}_{ij}\frac{\partial \psi }{\partial x_j}$$

(8)

The rigidity dependence of the dipole amplitude results from both the diffusion tensor D_ij and ∇ψ/ψ.

4. Results and Discussion

4.1. The Energy Spectra of Different Primary CR Components

It is known that the observed spectra are inversely proportional to the diffusion coefficient, which is a reflection of the average interstellar environment. The range of the LIMF measured by the IBEX experiment is ~20 pc [34,62], however, the path length of the GCR propagation is much longer than this distance. Therefore, the LIMF is believed to have little influence on the energy spectra. Thus, the CR flux intensities from both the background and the nearby sources are approximately calculated under the isotropic diffusion model in this work.

The propagation parameters are determined by fitting the B/C ratio. Figure 1 illustrates the fitting of the B/C ratio, which indicates that the relevant parameters are reasonable. The corresponding transport parameters are shown in

Table 1. Transport parameters of SDP model.

D0 [cm2·s−1]	δ0	Nm	ξ	n	vA [km·s−1]	zh [kpc]
4.87 × 1028	0.58	0.62	0.1	4	6	5

.

First, we calculate the energy spectra of primary CR components, including proton, He, C, N, O, Ne, Mg, Si and Fe nuclei. The injection spectra of background sources with a high-energy exponential cutoff are based on Z-dependent cutoff. The cutoff rigidity, $\mathcal{R}$_c ~ 7 PV, is obtained by fitting the proton and helium spectra observed by KASCADE [70]. The injection power of each element roughly follows the rule that element abundance decreases from light to heavy nuclei. Similarly, the injection spectra of the nearby source are assumed to be Z-dependent cutoff with ${{\mathcal{R}}^{\prime }}_c$ ~ 25 TV. The corresponding injection parameters are given by fitting the energy spectra, which are shown in

Table 2. Injection parameters of the background and the nearby source (The normalization is set at total energy E = 100 GeV).

	Background			Nearby Source
Element	Normalization	v	${\mathcal{R}}_c$	q0	α	${{\mathcal{R}}^{\prime }}_c$
	(m2·sr·s·GeV)−1		PV	GeV−1		TV
P	1.91 × 10−2	2.34	7	8.28 × 1052	2.16	25
He	1.43 × 10−3	2.27	7	2.35 × 1052	2.08	25
C	6.15 × 10−5	2.31	7	7.20 × 1050	2.13	25
N	7.67 × 10−6	2.34	7	1.13 × 1050	2.13	25
O	8.20 × 10−5	2.36	7	1.11 × 1051	2.13	25
Ne	8.05 × 10−6	2.28	7	1.13 × 1050	2.13	25
Mg	1.62 × 10−5	2.39	7	1.08 × 1050	2.13	25
Si	1.28 × 10−5	2.37	7	1.05 × 1050	2.13	25
Fe	1.23 × 10−5	2.29	7	2.20 × 1050	2.13	25

. The spectral indices of the nearby source component are slightly harder than that of the background, which fits the experimental data better. The spectral index of the helium nuclei is slightly harder than that of the proton. Figure 2 shows the spectral results, where the red and blue lines represent the contributions from the nearby source and the background fluxes, respectively, and each of the black lines is the sum of them. It can be seen that the contribution by the nearby Geminga SNR source can simultaneously account for both the spectral hardening features at $\mathcal{R}$ ∼ 200 GV and the softening features at $\mathcal{R}$ ∼ 10 TV.

We further calculate the all-particle spectra of GCRs, as shown in Figure 3, which are consistent with the observational data.

4.2. Anisotropy

Compared with the energy spectra, the LIMF can radically affect the anisotropy by altering the direction of CR flux entering into the solar system. Therefore, in the process of calculating the anisotropy, we introduce the anisotropic diffusion effect of CRs induced by the LIMF. Despite the uncertainties in the existing measurements, the LIMF is assumed to be in the direction as reported by the IBEX experiment [36,39].

The parameters of the parallel diffusion coefficient $D_{\parallel }$ are set as those shown in Table 1. The CRs in the GeV–PeV energy region require $D_{\parallel }\ge D_{\perp }$, therefore, we set ε = 0.01 and the difference between ${\delta }_{\perp }$ and ${\delta }_{\parallel }$ is 0.3.

Taking into account the contribution of the Geminga SNR, the amplitude and phase of the CR anisotropy are shown in Figure 4. It is obvious that both the phase and the amplitude agree well with the observations. For E < 100 TeV, the nearby source dominates the anisotropic phase, although the nearby flux is sub-dominant. The phase of anisotropy clearly points in the direction of the LIMF (~R.A. = 3^h). It is demonstrated that, compared with the model without considering the magnetic field, the anisotropic phase calculated with considering the LIMF fits better with the experimental observation [49]. The phase is flipped at 100 TeV from ~R.A. = 3^h to the GC. For E ≥ 100 TeV, the contribution from the nearby source and LIMF decreases significantly, and the background becomes dominant instead. The phase of the anisotropy turns to the GC, since galactic CR sources are more abundant in the inner Galaxy.

We further calculate the anisotropies of different compositions. The primary CR components are divided into four mass groups, i.e., P + He, C + N + O, Ne + Mg + Si, and Fe, and the calculated results are shown in Figure 5. The anisotropic amplitudes and phases of each mass group have a similar structure. However, as the nucleon number increases, the position of phase inversion moves towards a higher energy. We hope the coming high-precision measurements of the CR anisotropy will give a critical test on this point predicted by this model, such as the LHAASO experiment.

5. Conclusions

In this work, in view of the CR spectra hardenings at $\mathcal{R}$ ~ 200 GV and softenings at $\mathcal{R}$ ~ 10 TV, as well as the variation of the CR anisotropy with energy from GeV to PeV, we developed a model to explain both the cosmic ray spectra and anisotropy based on the SDP model with consideration of the anisotropic diffusion of CRs induced by the LIMF near the solar system and the contribution by the nearby Geminga SNR source. We first analyzed the energy spectra of different components and the CR anisotropy. The results show that our model can simultaneously account for both the CR spectra hardenings at $\mathcal{R}$ ~ 200 GV and softenings at $\mathcal{R}$ ~ 10 TV, as well as the CR anisotropy. We further explored the anisotropies of different mass groups which have the similar trend of change in amplitude and phase but with different characteristic energies. We hope that future high-precision CR spectral and anisotropic measurements, such as the LHAASO experiment, will help to test our model.