Cosmic Ray Spectra and Anisotropy in an Anisotropic Propagation Model with Spiral Galactic Sources

Abstract

In our previous work, we investigated the spectra and anisotropy of galactic cosmic rays (GCRs) under the assumption of an axisymmetric distribution of galactic sources. Currently, much observational evidence indicates that the Milky Way is a typical spiral galaxy. In this work, we further utilize an anisotropic propagation model under the framework of spiral distribution sources to study spectra and anisotropy. During the calculation process, we adopt the spatial-dependent propagation (SDP) model, while incorporating the contribution from the nearby Geminga source and the anisotropic diffusion of cosmic rays (CRs) induced by the local regular magnetic field (LRMF). By comparing the results of background sources with spiral and axisymmetric distribution models, it is found that both of them can well reproduce the CR spectra and anisotropy. However, there exist differences in their propagation parameters. The diffusion coefficient with spiral distribution is larger than that with axisymmetric distribution, and its spectral indices are slightly harder. To investigate the effects of a nearby Geminga source and LRMF on anisotropy, two-dimensional (2D) anisotropy sky maps under various contributing factors are compared. Below 100 TeV, the anisotropy is predominantly influenced by both the nearby Geminga source and the LRMF, causing the phase to align with the direction of the LRMF. Above 100 TeV, the background sources become dominant, resulting in the phase pointing towards the Galactic Center (GC). Future high-precision measurements of CR anisotropy and spectra, such as the LHAASO experiment, will be crucial in evaluating the validity of our proposed model.

1. Introduction

With the improvement of cosmic ray (CR) observation technology, a new generation of experiments have entered the era of high-precision measurement and unveiled a series of unexpected phenomena. In recent years, multiple experiments such as ATIC-2 [1], CREAM [2,3], PAMELA [4], AMS-02 [5,6], DAMPE [7], and the calorimeter experiment CALET [8] have observed that the spectra of proton and helium nuclei become harder at rigidity $\mathcal{R}$ ∼ 200 GV. Furthermore, DAMPE [9], CREAM [10], NUCLEON [11], and CALET [12] found that spectra soften at $\mathcal{R}$ ∼ 14 TV. This subtle anomaly in spectra obviously deviates from the expected CR power law spectrum. The main theoretical explanations for the anomaly are as follows: the sources near to the Solar System contributing to the “bulge” of the CR spectra [13,14]; interaction between CRs and accelerating shock waves [15,16]; effects in the CR propagation process [14,17]; and multiple acceleration sources as superimposed factors [18,19].

CRs, mostly charged particles, become isotropic as they travel through the Milky Way due to deflections in the galactic magnetic field (GMF). However, extensive observations show that there exists a subtle CR anisotropy with relative amplitudes in the order of ${10}^{-4}$∼${10}^{-3}$ at a wide energy range. Tibet [20,21,22], Super-Kamiokande [23], Milagro [24,25], IceCube/IceTop [26,27,28,29,30], ARGO-YBJ [31,32], EAS-TOP [33], KASCADE [34,35], and HWAC [36,37] have revealed that the structure and intensity of anisotropy are obviously energy-dependent. The amplitude of anisotropy increases first and then decreases with energy below 100 TeV, but gradually increases again above 100 TeV. Meanwhile, the phase points toward a value of ∼3 h below 100 TeV, whereas it flips and points towards ∼−6 h above 100 TeV, in the direction of the GC. It is noteworthy that, at tens of TeV, the morphology of anisotropy exhibits an excess named “tail-in” and a deficit named “loss-cone”. Obviously, the above anisotropic features contradict the expectations of the conventional propagation model. In general, CR anisotropy may arise due to the following several causes: the uneven distribution of CR sources, such as sources near the Solar System [14,38]; the modulation effect of large-scale local magnetic fields [38,39,40]; the process of CR propagation in the Galaxy [14]; and the Compton–Getting effect caused by the relative motion between the observer and the CR flux [41].

It is believed that anisotropy is related to the distribution of CR sources and CR propagation. In our previous work [38,42], based on the assumption that CR background sources follow an axisymmetric distribution, we built a hybrid model to calculate the CR spectra and anisotropy. The model successfully reproduced the experimental observations. Current extensive experimental observations reveal that the Milky Way is a typical spiral galaxy. The spiral arms, where high-density gas accumulates, are hotspots for rapid star formation [43,44]. Therefore, there is a high correlation between the distribution pattern of CR sources (especially supernova remnants, SNRs) and the spiral arm structure. In recent years, the impact of the spiral distribution of CR sources has garnered attention in research. Several studies have demonstrated that this spiral distribution of CR sources can significantly influence the positron and electron spectra, providing a more compelling explanation for the observed excess of positrons and electrons at 30 GeV [45,46]. Does the spiral distribution of CR sources affect nucleon spectra and anisotropy? In this work, we will utilize an anisotropic propagation model to investigate the energy spectra and anisotropy under the spiral distribution scenario. This will contribute to a deeper understanding of the origins and propagation mechanisms of CRs.

The rest of this paper is organized as follows: Section 2 introduces the model in detail, including the SDP model, the spiral structure of background sources, nearby sources, anisotropic diffusion, and large-scale anisotropy. In Section 3, the results of CR spectra and anisotropy are presented and thoroughly discussed; Section 4 provides the summary.

2. Model Description

2.1. Spatial-Dependent Propagation Model

In recent years, the SDP model has been proposed and widely applied. It was initially introduced as a two-halo model to account for the excess of primary proton and helium fluxes at $\mathcal{R}\sim$ 200 GV [16]. Afterwards, it was further extended to explain the excess of secondary and heavier components [45,47,48,49], the diffuse gamma-ray distribution [50], and large-scale anisotropy [51,52]. Recent observations of halos surrounding pulsars have revealed that CRs diffuse much slower than previously inferred from the boron-to-carbon (B/C) ratio, providing robust support for the hypothesis that diffusion may be spatially dependent [53,54].

In the SDP model, the whole galactic diffusive halo is clearly divided into two distinct zones characterized by different diffusion properties. The galactic disk and its surrounding area are referred to as the inner halo (IH), while the extensive diffusive region outside the IH is called outer halo (OH). In the IH region, where there are more sources, the activity of supernova explosions will lead to more intense turbulence. Consequently, the diffusion of CRs in the IH region is slowed down, and the diffusion coefficient exhibits a lesser dependence on rigidity. In the OH region, however, the diffusion of CRs is less affected by stellar activity, so CRs diffuse faster. The diffusion coefficient only depends on rigidity and is consistent with the conventional propagation model.

In this work, we adopt the SDP model, and the diffusion coefficient is parameterized as [38,42]

$$\begin{array}{cc}\hfill D_{xx}\left(r,z,\mathcal{R}\right)& =D_{0}F(r,z){\left({\displaystyle \frac{\mathcal{R}}{{\mathcal{R}}_0}}\right)}^{{\delta }_{0}F(r,z)}\hfill \end{array}$$

(1)

where r and z are cylindrical coordinates, $\mathcal{R}$ is a particle’s rigidity and the terms $D_0$ and ${\delta }_0$ are two constants. The reference rigidity ${\mathcal{R}}_0$ is fixed at 4 GV. The parameterization of $F(r,z)$ is given by

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

(2)

with $g(r,z)=N_m/[1+f(r,z)]$, where $f(r,z)$ denotes the source density distribution and $N_m$ is a normalization factor. The total half thickness of the propagation halo is $z_0$, with $\xi z_0$ and $(1-\xi )z_0$ being the half thicknesses of the IN and OH, respectively. In this work, we apply the diffusion–reacceleration model. The diffusive reacceleration is represented as a diffusion in the momentum space. Its diffusion coefficient $D_{pp}$ is related to $D_{xx}$ by

$$\begin{array}{cc}\hfill D_{xx}{D}_{pp}& ={\displaystyle \frac{4p^2{v_A}^2}{3\delta \left(4-{\delta }^2\right)\left(4-\delta \right)}}\hfill \end{array}$$

(3)

where $\delta ={\delta }_{0}F(r,z)$ and $v_A$ is the Alfvénic velocity.

Numerical simulations of CR propagation through the Galaxy help to understand the galactic CR transportation. Recently, a variety of codes have emerged for simulating the CR propagation process, including GALPROP [55], DRAGON [56], and PICARD [57]. In this work, we adopt the numerical package DRAGON to solve the CR transport equation.

2.2. Spiral Distribution of CR Sources

In view of the fact that the diffusion length of CRs is usually much longer than the characteristic spacing between the adjacent spiral arms, CR sources are generally approximated as axisymmetric. In our previous work, the source density distribution is parameterized as

$$f(r,z)={(r/r_{\odot })}^{\alpha }exp[-\beta (r-r_{\odot })/r_{\odot }]exp(-|z|/z_s),$$

(4)

where $r_{\odot }=8.5$ kpc represents the solar distance to the GC. The parameters $\alpha$ and $\beta$ are taken as 1.69 and 3.33, respectively [58]. Perpendicular to the galactic plane, the density of CR sources descends as an exponential function, with a mean value $z_s=0.2$ kpc.

However, a large number of observations have indicated that the Milky Way is a typical spiral galaxy [43,44]. Due to the position of the Solar System within the Milky Way, we still have uncertainties regarding the structure of the spiral arms. Currently, it is generally believed that the Milky Way possesses four spiral arms, yet its internal structure remains relatively unclear. The spiral arms where high-density gas accumulates are regions of rapid star formation. Consequently, the distribution of SNRs is highly correlated with the spiral arm structure. In this work, we adopt a model established by Faucher-Giguere and Kaspi to describe the spiral distribution [59]. The Galaxy consists of the four major spiral arms extending outward from the GC: Norma, Carina–Sagittarius, Perseus, and Crux–Scutum. The locus of the i-th arm centroid is expressed as a logarithmic curve: ${\theta }^i\left(r\right)=k^{i}ln(r/r_0^i)+{\theta }_0^i$, with $i\in \left\{1,2,3,4\right\}$, where r is the distance to the GC and the values of $k^i$, $r_0^i$, and ${\theta }_0^i$ have been fixed for each arm [46]. Along each spiral arm, there is a spread in the normal direction which follows a Gaussian distribution, i.e.,

$$f_i={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_i}}exp\left[-{\displaystyle \frac{{(r-r_i)}^2}{2{\sigma }_i^2}}\right]$$

(5)

where $r_i$ is the inverse function of the i-th spiral arm’s locus and the standard deviation ${\sigma }_i$ is $0.07r_i$. The number density of SNRs at different radii is still consistent with the radial distribution in the axisymmetric case, i.e., Equation (4).

The injection spectrum of background sources is assumed to be a power-law of rigidity with a high-energy exponential cutoff, i.e.,

$$q\left(\mathcal{R}\right)=q_0{\mathcal{R}}^{-\nu }exp(-\mathcal{R}/{\mathcal{R}}_{\mathrm{c}})$$

(6)

where $q_0$ is a normalization factor, $\nu$ is the power index, and ${\mathcal{R}}_{\mathrm{c}}$ is the rigidity cutoff. The cutoff rigidity of each element could be either Z- or A-dependent.

2.3. Nearby Source

In this work, the CR flux from the nearby source can be computed by solving the time-dependent diffusion equation using the Green’s function method, assuming a boundary at infinity [60,61]. Regarding the scenario of instantaneous and point-like injection, the CR density is calculated using a function of time, location, and rigidity, i.e.,

$$\psi (r,\mathcal{R},t)={\displaystyle \frac{q_{\mathrm{inj}}\left(\mathcal{R}\right)}{{\left(\sqrt{2\pi }\sigma \right)}^3}}exp\left[-{\displaystyle \frac{{(r-r^{\prime })}^2}{2{\sigma }^2}}\right],$$

(7)

where $q_{\mathrm{inj}}\left(\mathcal{R}\right)$ is parameterized as a power-law function of rigidity with an exponential cutoff, i.e.,

$$q_{\mathrm{inj}}\left(\mathcal{R}\right)={q^{\prime }}_0{\mathcal{R}}^{-\alpha }exp(-\mathcal{R}/{\mathcal{R}}_{\mathrm{c}}^{\prime }),$$

(8)

where ${q^{\prime }}_0$ is a normalization factor, $\alpha$ is the power index, ${\mathcal{R}}_{\mathrm{c}}^{\prime }$ is the rigidity cutoff, $\sigma (\mathcal{R},t)=\sqrt{2D\left(\mathcal{R}\right)t}$ is the effective diffusion length within time t, and $D\left(\mathcal{R}\right)$ is the diffusion coefficient, which is adopted as the value near the Solar System.

In our previous works, the spectral anomaly at 200 GeV and dipole anisotropy below 100 TeV are attributed to the Geminga SNR. The Geminga SNR is located in the direction of $(l,b)=(194.3^{\circ },-{13}^{\circ })$, where l and b denote the galactic longtitude and latitude and its distance to the Solar System is $d\sim 330$ pc [62]. Its explosion time was about $\tau =3.4\times {10}^5$ years ago, which is inferred from the spin-down luminosity of the Geminga pulsar [63]. In this work, we also select the Geminga SNR as the optimal source.

2.4. Anisotropic Diffusion and Large-Scale Anisotropy

By observing neutral particles traversing through the heliosphere boundary, the IBEX experiment has unveiled that the LRMF aligns with coordinates $(l,b)=(210.5^{\circ },-57.1^{\circ })$, within a 20 pc radius [40,64]. We have discovered that the direction of the LRMF is generally consistent with the CR anisotropy observed below 100 TeV. Furthermore, some studies have also revealed that the TeV CR anisotropy is associated with the LRMF [38,39,40,42].

When CRs are deflected by magnetic fields, they diffuse anisotropically. It is generally believed that CRs diffuse faster along the direction of the magnetic field than perpendicular to it. The corresponding dipole anisotropy is expected to be modified by the LRMF. In this scenario, the diffusion coefficient D is replaced by a tensor $D_{ij}$. The $D_{ij}$ associated with the magnetic field is written as

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

(9)

where $D_{\Vert }$ and $D_{\perp }$ are the diffusion coefficients aligned parallel and perpendicular to the ordered magnetic field B, and $b_i$ is the i-th component of its unit vector [62], respectively. In this work, the values $D_{\Vert }$ and $D_{\perp }$ are parameterized as a power-law function of rigidity [41,65],

$$\begin{array}{cc}\hfill D_{\Vert }& =D_{0\Vert }{\left({\displaystyle \frac{\mathcal{R}}{{\mathcal{R}}_0}}\right)}^{{\delta }_{\parallel }},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill D_{\perp }& =D_{0\perp }{\left({\displaystyle \frac{\mathcal{R}}{{\mathcal{R}}_0}}\right)}^{{\delta }_{\perp }}=\epsilon D_{0\Vert }{\left({\displaystyle \frac{\mathcal{R}}{{\mathcal{R}}_0}}\right)}^{{\delta }_{\perp }}.\hfill \end{array}$$

(11)

where $\epsilon ={\displaystyle \frac{D_{0\perp }}{D_{0\Vert }}}$ is the ratio between the perpendicular and parallel diffusion coefficients at the reference rigidity ${\mathcal{R}}_0$.

Under the anisotropic diffusion model, the dipole anisotropy can be written as

$$\delta ={\displaystyle \frac{3\mathit{D}}{v}}.{\displaystyle \frac{\nabla \psi }{\psi }}={\displaystyle \frac{3}{v\psi }}{D}_{ij}{\displaystyle \frac{\partial \psi }{\partial x_j}}.$$

(12)

where v is the velocity of the CRs.

3. Results and Discussion

3.1. Proton and Helium Spectra of Nearby Sources

Since the spatial scale of the LRMF is significantly smaller than the average propagation length of CRs deduced from the B/C ratio, the LRMF does not have a remarkable impact on the energy spectra. Consequently, in this work, the isotropic diffusion SDP model within the framework of the spiral distribution of background sources is employed to calculate the nucleon spectra.

Firstly, the propagation parameters for the SDP model can be determined by fitting the B/C ratio. Figure 1 displays the fitting results of the B/C ratio, which align well with the experimental data from AMS-02 [66,67]. The corresponding propagation parameters are as follows: $D_0=9.0\times {10}^{28}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, ${\mathcal{R}}_0=4\mathrm{GV}$, ${\delta }_0=0.58$, $N_{\mathrm{m}}=0.5$, $\xi =0.12$, $v_A=6\mathrm{km}/\mathrm{s}$, and $z_0=5\mathrm{kpc}$.

The normalization, power index, and cutoff rigidity for each element of the background source injection spectra are determined by fitting to the energy spectra derived from experimental observations. The cutoff rigidities of different compositions are regarded as the limits of acceleration in the sources and are presumed to be Z-dependent, with a high-energy exponential cutoff. Similarly, the injection spectrum of the nearby source is also set using the same methodology.

Figure 2 shows the proton (left) and helium (right) spectra, with the red dash-dotted, blue dashed, and black solid lines representing the contributions from the nearby Geminga source, background sources, and sum of all, respectively. It is evident that the Geminga source significantly contributes to the hardening of the proton and helium nucleus energy spectra at 200 GeV. The corresponding injection parameters for the background and Geminga sources are listed in

Table 1. Injection parameters of the background and nearby Geminga source.

Element	Background			Geminga Source
	${\mathit{q}}_{\mathbf{0}}$	$\mathsf{\nu }$	${\mathcal{R}}_{\mathit{c}}$	${\mathit{q}}_{\mathbf{0}}^{\prime }$	$\mathsf{\alpha }$	${\mathcal{R}}_{\mathit{c}}^{\prime }$
	${\mathbf{GeV}}^{-\mathbf{1}}{\mathbf{m}}^{-\mathbf{2}}{\mathbf{s}}^{-\mathbf{1}}{\mathbf{sr}}^{-\mathbf{1}}$		PV	GeV−1		TV
p	$4.36\times {10}^{-2}$	2.30	5	$7.74\times {10}^{52}$	2.16	22
He	$2.27\times {10}^{-3}$	2.21	5	$2.35\times {10}^{52}$	2.10	22
C	$1.0\times {10}^{-4}$	2.24	5	$7.80\times {10}^{50}$	2.13	22
N	$1.16\times {10}^{-5}$	2.20	5	$1.03\times {10}^{50}$	2.13	22
O	$1.24\times {10}^{-4}$	2.25	5	$9.0\times {10}^{50}$	2.13	22
Ne	$1.22\times {10}^{-5}$	2.20	5	$1.10\times {10}^{50}$	2.13	22
Mg	$1.83\times {10}^{-5}$	2.23	5	$1.02\times {10}^{50}$	2.13	22
Si	$2.35\times {10}^{-5}$	2.29	5	$1.02\times {10}^{50}$	2.13	22
Fe	$2.47\times {10}^{-5}$	2.26	5	$2.75\times {10}^{50}$	2.13	22

The normalization is set at total energy $E=100$ GeV.

. The spectral indices of the nearby source component are slightly harder than those of the background component, enhancing the fit to the observed data. To explain the softening observed at tens of TeV in the proton and helium spectra, the cutoff rigidity of the nearby source has been determined to be 22 TV. Furthermore, to accurately depict the all-particle spectra and the cutoffs of proton and helium at PeV energies, the cutoff rigidity of the background sources is established at 5 PV. It is apparent that the contribution by the nearby Geminga SNR source can simultaneously account for both the spectral hardening features at $\mathcal{R}\sim 200$ GV and the softening features at $\mathcal{R}\sim 10$ TV.

We also present the all-particle spectrum of CRs, as shown in Figure 3. These results are in good agreement with the Hörandel experiment and successfully reproduce the “knee” structure.

3.2. Anisotropy

Contrary to the energy spectra of CRs, the LRMF can significantly deflect the propagation direction of CRs below 100 TeV, subsequently influencing the dipole anisotropy. Therefore, when calculating the anisotropy, we incorporate the anisotropic diffusion effect of CRs induced by the LRMF. The parameters of the parallel diffusion coefficient $D_{\Vert }$ are set to be the same as those of the diffusion coefficient in the isotropic propagation model, as referenced in Section 2.1. CRs included in an energy range of a few GeV to several hundred GeV are believed to propagate faster parallel to the LRMF than perpendicular to it. Therefore, we set $D_{\Vert }>D_{\perp }$, with $\epsilon =0.01$ and $\mathsf{∆}\delta ={\delta }_{\perp }-{\delta }_{\Vert }=0.32$.

Figure 4 illustrates the evolution of anisotropic amplitude $A_1$ and phase ${\alpha }_1$ with energy, within the context of the spiral distribution of background sources, incorporating contributions from the Geminga SNR and LRMF. It is evident that both the phase and amplitude agree well with experimental data, validating the reasonableness of our model. Below 100 TeV, the phase points in the direction of the LRMF. The results suggest that, although the nearby flux is subdominant, the Geminga source and the deflection caused by the LRMF are the primary factors influencing the anisotropic phase. Above 100 TeV, the phase points to the GC, indicating background sources dominate, since galactic CR sources are more abundant in the inner galaxy.

We conducted a comparison of the results obtained from the spiral and axisymmetric source distribution, with the latter referencing our previous work [38,42]. It was found that the energy spectra and anisotropies under both source distribution models align well with experimental observations. However, the propagation parameters and spectral indices of the sources differ. The diffusion coefficient for the spiral distribution is larger than that for the axisymmetric distribution, and the spectral index of the former is slightly harder than that of the latter. These differences may be attributed to the fact that the source distribution affects the propagation of CRs.

In order to further understand the impacts of background sources, nearby sources, and the LRMF on anisotropy, we present 2D anisotropy sky maps as a function of right ascension RA and declination DEC, illustrating the contribution of each factor. Figure 5 displays the 2D anisotropic sky maps at 10 TeV (top) and 3 PeV (bottom), where the left, middle, and right are the results of background sources (BK), background sources plus the nearby Geminga source (BK + Geminga), and background sources plus the nearby Geminga source plus the LRMF (BK + Geminga + LRMF), respectively. In the low-energy region, when only the background sources are considered, the anisotropy points to the GC, as illustrated in Figure 5 (10 TeV BK). The result is clearly inconsistent with experimental observations. When the contribution of the nearby Geminga source is introduced, the phase points towards Geminga, as shown in Figure 5 (10 TeV BK + Geminga). This variation is attributed to the fact that the nearby source significantly alters the gradient of CR intensity in its direction. While the BK + Geminga model results are closer to those of the experiments, there are still some deviations between them. When the contribution of the LRMF is further introduced, the anisotropy points in the direction of the LRMF. Figure 5 (10 TeV BK + Geminga +LRMF) is completely consistent with the experimental observations. This good agreement is due to the anisotropic diffusion effect of the LRMF on CR particles. Figure 5 (bottom) depicts the 2D anisotropic sky maps in the high-energy region, influenced by various factors. Above 100 TeV, the anisotropic sky maps exhibit a relatively straightforward pattern, with the phase consistently directed towards the GC. This indicates that the contribution of background sources is predominant, whereas the contribution of the nearby source is nearly negligible. Additionally, it suggests that the LRMF is unable to deflect CRs at energies exceeding 100 TeV.

4. Summary

Recently, a large number of scientific observations have confirmed that the Milky Way possesses a spiral arm structure. In our previous work, we developed a self-consistent propagation model to analyze energy spectra and anisotropy, assuming an axisymmetric galactic source distribution. The comprehensive model builds upon the SDP model, while also accounting for the contribution from the nearby Geminga source and the anisotropic diffusion effects of the LRMF on CRs. In this work, we further utilize the model to investigate the energy spectra and anisotropy under the framework of the spiral arm source distribution.

Our model can simultaneously explain spectral hardening at 200 GeV and the amplitude and phase variation of anisotropy with energy from 100 GeV to PeV. We found that the results of the spiral distribution are similar to those of the axisymmetric distribution. However, it is worth noting that their propagation parameters differ. Specifically, the diffusion coefficient of the spiral distribution is larger than that of the axisymmetric distribution, and the spectral indices for the spiral distribution are slightly harder. These discrepancies may be attributed to the influence of the source distribution on the propagation of CRs.

To demonstrate the effects of the nearby Geminga source and LRMF, we compare two-dimensional anisotropy sky maps under various contributing factors. Below 100 TeV, it is clear that the nearby Geminga source contributes to the spectral hardening at 200 GeV. Although the contribution of the nearby source to the CR flux is less than that of the background sources, its impact on the anisotropy is predominant. Under the isotropic diffusion model, the anisotropic phase is approximately oriented towards the direction of the nearby Geminga source. If the anisotropic diffusion effect of the LRMF on CRs is further taken into account, the anisotropic phase shifts to align with the direction of the LRMF. Above 100 TeV, the contribution of background sources becomes dominant, and the anisotropic phase is consistently directed towards the GC. High-precision experiments, such as the LHAASO experiment [74], will provide valuable measurements of CR spectra and anisotropies in the future, which will verify our model.