Cosmic Ray Anisotropy and Spectra as Probes for Nearby Sources

Abstract

Cosmic ray (CR) spectra and anisotropy are closely related to the distribution of CR sources, making them valuable probes for studying nearby sources. There are 12 nearby sources located within 1 kpc of the solar system, and which ones are the optimal candidates? In this work, we have selected the Geminga, Monogem, Vela, Loop I, and Cygnus SNR sources as the focus of our research, aiming to identify the optimal candidate by investigating their contribution to the energy spectra and anisotropy using the Spatially Dependent Propagation (SDP) model. Additionally, the anisotropic diffusion effect of the local regular magnetic field (LRMF) on CR particles is also considered in the SDP model. Our previous work only provided 1D anisotropy along the right ascension; this current work will further present 2D anisotropy maps along the right ascension and declination. When the injection power of different nearby sources is roughly equal, the results show that the Geminga, Momogem, and Loop I SNR sources contribute significantly to the nuclear energy spectra. Under the isotropic diffusion without considering the LRMF, the 2D anisotropy maps indicate that the phase points to the nearby source below 100 TeV. We further adjust the injection power of the Monogem SNR source in accordance with the spin-down energy of the Geminga and Monogem pulsars, and find that the contribution of the corrected Monogem SNR can be disregarded. Because the Loop I SNR source is located in the direction of the Galactic Center (GC), it cannot contribute to the excess of CRs in the anti-GC direction. Under anisotorpic diffusion with the consideration of the LRMF, the 2D anisotropy maps show that only the Geminga SNR can match the anisotropy measurement, while the other sources cannot. Finally, we conclude that the Geminga SNR source is the optimal nearby source.

1. Introduction

CRs less than PeV are generally believed to be produced in the Galaxy, and supernova remnants (SNRs) are considered to be the most important galactic sources [1]. CRs can be accelerated to form power-law spectra through the diffusive shock acceleration mechanism at SNRs [2]. The CR spectrum is an important tool for investigating the origin, acceleration, and propagation of CRs. In recent years, with the improvement of the new generation of CR detection technology, numerous space experiments have uncovered subtle anomalies of CR spectra at about 200 GeV, deviating from the expected power-law spectrum. Experiments such as ATIC-2 [3], CREAM [4,5], PAMELA [6], AMS-02 [7,8], and the calorimeter experiment CALET [9] have revealed nuclear spectra become hard at $\mathcal{R}$∼200 GV, while DAMPE [10,11], CREAM [12], and NUCLEON [13] have found that proton and helium spectra become soft at $\mathcal{R}$∼14 TV. So far, several theoretical models have been proposed to explain these spectral anomalies, including the contribution of nearby sources near the solar system to the “bulge” of the CR spectra [14,15], interaction between CRs and accelerating shock waves [16,17], the effect of CR propagation process [15,18], and the superimposition of multiple acceleration sources [19,20].

In addition to the CR energy spectrum, anisotropy is also an important probe for studying CRs. Most of the charged particles in CRs, during their propagation, are subject to deflection and modulation by the galactic magnetic field, and interact with the interstellar medium. As a result, when they reach Earth, they exhibit overall isotropy. However, some ground-based air shower arrays and underground muon detectors, such as Tibet [21,22,23], Super-Kamiokande [24], Milagro [25,26], IceCube/Ice-Top [27,28,29,30,31], ARGO-YBJ [32,33], EASTOP [34], KASCADE [35,36], and HWAC [37,38] have observed small anisotropy with relative amplitudes of the order of ${10}^{-4}$∼${10}^{-3}$ at energies from 100 GeV to hundreds of PeV. The experimental results indicate a complex energy dependence in the amplitude and phase of anisotropy. As the energy increases, the amplitude increases below 10 TeV, decreases from 10 TeV to 100 TeV, and increases again above 100 TeV. Meanwhile, at less than 100 TeV, the phase points towards ∼3 h, which is consistent with the direction of LRMF observed by the IBEX experiment [39], while it points towards the GC above the 100 TeV. In the Multi-TeV energy region, the 2D anisotropy maps obviously present two large-scale structures, which are “beyond” from the heliospheric magnetic tail direction named “Tail-in” and “missing” from the galactic North Pole direction termed “Loss-cone” [21,23]. Generally, the origin of anisotropy may stem from the following factors: nearby sources near the solar system [15,40], the deflection of a local regular magnetic field [40,41,42], CR propagation [15], and the Compton–Getting effect caused by the relative motion between the Earth’s rotation and CRs [22,43].

CR spectra and anisotropy from GeV to ∼100 TeV have some common anomalous characteristics, suggesting that they may have a common origin. In recent years, a large number of studies have revealed that nearby sources are closely related to these anomalies. The work in ref. [40] shows that the Geminga source and the anisotropic diffusion of CRs induced by the LRMF can explain both nuclear spectra and anisotropy. The study in ref. [44] indicates that the Geminga SNR is the sole optimal candidate, while the status of the Monogem SNR is controversial due to the disparity in anisotropy between the model calculation and the observations. Moreover, the Vela SNR contributes to a new spectral structure beyond TeV energy. The work in ref. [45] demonstrates that only the Geminga SNR could be the proper candidate for the local CR source by a fitting calculation. The work in ref. [46] has found that Monogem can reasonably account for primary electron excess and proton spectrum. The work in ref. [42] presents that an excellent candidate for the local CR source responsible for the dipole anisotropy at 1∼100 TeV is the Vela SNR. We found that the age, location, and injection power of nearby sources are crucial in contributing to the energy spectrum and anisotropy.

In the conventional diffusion propagation model of CRs, the non-uniform distribution of CR background sources predicts large-scale anisotropy. Under isotropic diffusion, the anisotropy amplitude increases with energy, and the phase always points towards the GC, which is clearly inconsistent with experimental observations. For example, at ∼100 TeV, the expected amplitude is two orders of magnitude higher than what is observed in experiments. If a nearby source, such as the Geminga SNR, is introduced in the anti-GC direction in the propagation model, it can counterbalance the CR flux from the GC direction, thus helping to alleviate the discrepancies between the theoretical predictions and experimental observations. Another solution for this issue is the SDP model, which diminishes the anisotropy amplitude because of the lower diffusion coefficient in the inner halo compared to the outer halo, but it does not resolve the low-energy phase issue [15]. By observing neutral particles passing through the heliosphere boundary, the IBEX experiment revealed that the LRMF follows $(l,b={210.5}^{\circ },-{57.1}^{\circ })$, in the range of 20 pc around the solar system [39,47]. The direction of the LRMF is coincident with the phase of anisotropy below 100 TeV. Several studies [42,47,48] have demonstrated that this coincidence is due to anisotropic diffusion, which guides CRs to propagate along the LRMF. Therefore, in this work, we establish a unified model that incorporates the effects of nearby sources and the anisotropic diffusion induced by the LRMF based on the SDP model.

There are 12 nearby sources located within 1 kpc of the solar system, and for this study, we have selected Geminga, Monogem, Vela, Loop I, and Cygnus, as listed in

Table 1. The location and age of the five known SNRs. References: 1 [49], 2 [50], 3 [51], 4 [52], 5 [53].

SNR	l	b	d	${\mathit{T}}_{\mathit{age}}$	$\mathit{Ref}$
				$\left[\mathit{pc}\right]$	$\left[\mathit{Kyr}\right]$
Geminga	$194.3^{\circ }$	$-13.1^{\circ }$	330	345	1
Monogem	$203.0^{\circ }$	$12.0^{\circ }$	288	86	2
Vela	$263.9^{\circ }$	$-3.3^{\circ }$	295	11	3
Loop I	$329.0^{\circ }$	$17.5^{\circ }$	170	200	4
Cygnus Loop	$74.0^{\circ }$	$-8.5^{\circ }$	540	10	5

as the focus of our investigation. The objective of this study is to identify the optimal candidate source by analyzing their contribution to CRs’ spectra and anisotropy using the unified propagation model. Our previous work [40] focused solely on studying the 1D anisotropy phase along the right ascension. This current study will further investigate 2D anisotropy maps along the right ascension and declination. This paper is organized as follows: Section 2 presents the model description and methods; in Section 3, the results of CR spectra and anisotropy are presented and discussed; Section 4 gives the summary.

2. Model and Methods

2.1. Spatially Dependent Diffusion

After being accelerated in the “source” region, the primary CRs enter interstellar space and undergo frequent scattering by the interstellar turbulent magnetic field, resulting in random walks within the galaxy. This random process is known as CR diffusion propagation. The region in which CRs diffuse within the Galaxy is known as the magnetic halo, often approximated as a cylinder with a radial boundary equal to the galactic radius, i.e., $R=20$ kpc, and a half thickness $z_h$ of a few kpc. The value of $z_h$ is typically determined by fitting the B/C ratio along with the diffusion coefficient [54]. Both CR sources and the interstellar medium are typically assumed to be concentrated near the galactic disk, with an average thickness $z_s$ of approximately 0.2 kpc. CRs in the magnetic halo go through diffusion, convection, reacceleration, energy loss, nuclear and nucleonic fragmentation, as well as the decay of unstable secondary particles. This comprehensive process can be described by the propagation equation [54]. In the conventional propagation model, the diffusion coefficient is a scalar that depends solely on rigidity. This model has successfully explained the cosmic ray power-law spectrum, the B/C ratio, and the distribution of diffuse gamma rays and so on. However, growing experimental observations have challenged the conventional model.

In the past few years, the SDP model with two halos has attracted much attention and been applied to more and more CR research fields. It was originally introduced to account for the spectral hardenings of proton and helium at ∼200 GeV [18]. Afterwards, it was further used to explain the excess of secondary and heavier components [55,56,57,58], diffuse gamma ray distribution [59], and large-scale anisotropy [15,45]. Recent measurements of the TeV halo around the pulsar have found that CRs diffuse significantly slower than the inferred boron–carbon ratio, which strongly supports the SDP model [60,61].

In the SDP model, the galactic diffusion halo is divided into two regions, i.e., inner halo (IH) and outer halo (OH). The galactic disk and its surrounding region is called the IH, while the diffusion region outside the IH is called the OH. In the IH region, where there are more sources, the activity of supernova explosions will lead to more intense turbulence. Therefore, the diffusion of CRs will be slow, and the diffusion coefficient will be less dependent on the rigidity. Whereas in OH region, the diffusion of CRs is less affected by stellar activity, and the diffusion coefficient is consistent with the traditional propagation model and only depends on rigidity.

In this work, we adopt the SDP model and the diffusion coefficient is parameterized as [58,59]

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

(1)

where r and z are cylindrical coordinates, $\mathcal{R}$ is particle’s rigidity, and $D_0$ is a constant. The total half-thickness of the propagation halo is $z_0$, and the half-thickness of the IH is $\xi z_0$. The parameterization of $F(r,z)$ can be parameterized as

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

(2)

where $g(r,z)=N_m/[1+f(r,z)]$, and $f(r,z)$ is the source density distribution.

In this work, we adopt numerical package DRAGON to solve the transport equation [62].

2.2. Background Sources

SNRs are considered the most likely sites for the acceleration of GCRs, where charged particles are accelerated to a power-law distribution through diffusive shock acceleration. The spatial distribution of the background sources are approximated as axisymmetric, following the distribution of SNRs [63]:

$$f(r,z)\propto {(r/r_{\odot })}^{1.69}exp[-3.33(r-r_{\odot })/r_{\odot }]exp(-|z|/z_s),$$

(3)

where $r_{\odot }=8.5$ kpc represents the distance from the solar system to the GC and $z_s=0.2$ kpc. Formula (3) indicates that the density distribution of the SNRs decreases exponentially along the vertical height from the galactic plane.

The injection spectrum of background sources is assumed to be a power-law of rigidity with a high-energy exponential cutoff, $q\left(\mathcal{R}\right)\propto {\mathcal{R}}^{-\nu }exp(-\mathcal{R}/{\mathcal{R}}_{\mathrm{c}})$. The cutoff rigidity of each element could be either Z- or A-dependent.

2.3. Nearby Source

We solve the time-varying propagation equation of CRs from nearby sources assuming a spherical geometry with infinite boundary conditions and using Green’s function method [64,65].

The CR density of nearby sources as a function of the location, time, and rigity is described by

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

(4)

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

2.4. Anisotropic Diffusion and Large-Scale Anisotropy

The amplitude of the dipole anisotropy is proportional to the spatial gradient of the CR density and the diffusion coefficient. In the conventional propagation model, the anisotropy can be written as [42,48]

$$\delta ={\displaystyle \frac{3D}{v}}{\displaystyle \frac{\nabla \psi }{\psi }}.$$

(5)

The Larmor radius of PeV CRs is much smaller than their scattering length in the LRMF, which indicates that the LRMF can deflect the PeV CR particles. CRs diffuse anisotropically in the local interstellar space under the influence of the LRMF, and the diffusion tensor $D_{ij}$ associated with the magnetic field is written as

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

(6)

where $D_{\Vert }$ and $D_{\perp }$ are the diffusion coefficients aligned parallel and perpendicular to the ordered magnetic field, $b_i$ is the i-th component of the unit vector [66], respectively. In this work, the values of $D_{\Vert }$ and $D_{\perp }$ are parameterized as power-law function of rigidity, and are shown as follows [40,67]:

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

(7)

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

(8)

where $\epsilon ={\displaystyle \frac{D_{0\perp }}{D_{0\Vert }}}$ is the ratio between the perpendicular and parallel diffusion coefficient at the reference rigidity ${\mathcal{R}}_0$. Some works have studied the value of $\epsilon$, ${\delta }_{\parallel }$, and ${\delta }_{\perp }$ [64,68,69]. When $D_{\perp }/D_{\Vert }\ll 1$, the perpendicular diffusion coefficient is much smaller than the parallel one, and CRs are more likely to diffuse along the magnetic field. When $D_{\perp }/D_{\Vert }\approx 1$, the perpendicular diffusion is close to the parallel one, and CRs diffuse almost isotropically.

Under the anisotropic diffusion model, the form of Formula (5) can be written as

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

(9)

3. Results and Discussion

3.1. Proton and Helium Spectra of Five Nearby Sources

The spatial scale of the LRMF is 20 pc, which is much smaller than the average propagation length of CRs, so the LRMF has almost no effect on the energy spectrum [47]. Therefore, the SDP model under isotropic diffusion can be used to calculate the energy spectra of background sources and nearby sources.

We obtained the propagation parameters by fitting the B/C ratio. Figure 1 presents the comparison of the B/C ratio between the model prediction and the observation data of AMS-02. The corresponding propagation parameters are, respectively, $D_0=4.87\times {10}^{28}{\mathrm{cm}}^2$, ${\delta }_0=0.58$, $N_{\mathrm{m}}=0.62$, $\xi =0.1$, $n=4$. The Alfvénic velocity is $v_A=6\mathrm{km}·{\mathrm{s}}^{-1}$, and the half-thickness of the propagation halo is $z_h=5$ kpc.

First, we calculate the proton and helium spectra with the contribution from five different nearby sources, as listed in Table 1. The Z-dependent cutoff is applied to the injection spectra of the background and nearby sources with a high-energy exponential cutoff. The normalization, power index, and cutoff rigidity are obtained by fitting the energy spectra assuming that the injection power of each source is roughly equal. The corresponding injection parameters of different nuclei in the background and nearby sources are shown in

Table 2. Injection parameters of the background and nearby sources.

	Background			Geminga Source			Monogem Source
Element	Normalization †	$\mathit{\nu }$	${\mathcal{R}}_{\mathit{c}}$	${\mathit{q}}_{\mathbf{0}}$	$\mathit{\alpha }$	${\mathcal{R}}_{\mathit{c}}^{\prime }$	${\mathit{q}}_{\mathbf{0}}$	$\mathit{\alpha }$	${\mathcal{R}}_{\mathit{c}}^{\prime }$
	${\mathbf{\left(}{\mathbf{m}}^{\mathbf{2}}\mathbf{srsGeV}\mathbf{\right)}}^{-\mathbf{1}}$		PV	GeV −1		TV	GeV −1		TV
p	$1.91\times {10}^{-2}$	2.34	7	$8.28\times {10}^{52}$	2.16	25	$2.94\times {10}^{52}$	2.20	22
He	$1.43\times {10}^{-3}$	2.27	7	$2.35\times {10}^{52}$	2.08	25	$1.80\times {10}^{52}$	2.18	22
C	$6.15\times {10}^{-5}$	2.31	7	$7.2\times {10}^{50}$	2.13	25	$6.00\times {10}^{50}$	2.13	22
N	$7.67\times {10}^{-6}$	2.34	7	$1.13\times {10}^{50}$	2.13	25	$7.50\times {10}^{49}$	2.13	22
O	$8.20\times {10}^{-5}$	2.36	7	$1.11\times {10}^{51}$	2.13	25	$1.11\times {10}^{51}$	2.13	22
Ne	$8.05\times {10}^{-6}$	2.28	7	$1.13\times {10}^{51}$	2.13	25	$1.13\times {10}^{50}$	2.13	22
Mg	$1.62\times {10}^{-5}$	2.39	7	$1.08\times {10}^{50}$	2.13	25	$1.08\times {10}^{50}$	2.13	22
Si	$1.28\times {10}^{-5}$	2.37	7	$1.05\times {10}^{50}$	2.13	25	$1.05\times {10}^{50}$	2.13	22
Fe	$1.23\times {10}^{-5}$	2.29	7	$2.20\times {10}^{50}$	2.13	25	$2.20\times {10}^{50}$	2.13	22

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

. Only the injection spectra for the Geminga and Monogem sources are listed here. Figure 2 presents the spectral results of proton (left) and helium (right), where the solid gray line is the contribution of the background sources, the dashed lines in different colors represent the contributions from different single nearby sources, and the solid lines in corresponding colors display the sum of the single nearby sources and background sources. It can be seen that the contribution of the Geminga, Monogem, and Loop I SNR can account for the spectral hardening at ∼200 GeV and softening features at ∼10 TeV, but Vela and Cygnus cannot. This is because the Vela and Cygnus sources are younger than the others, and the low-energy CRs produced by them reach the solar system with difficulty.

3.2. Anisotropy of Geminga, Monogem, and Loop I

Given only the energy spectra of the Geminga, Monogem, and Loop I SNR sources are consistent with the experimental data, we will only analyze the anisotropy of these three sources next.

Unlike the energy spectrum, LRMFs clearly deflect CR particles and affect the anisotropy within the concerned energy region, so the LRMF must be considered in the calculation of anisotropy. The parameters of the parallel diffusion coefficient $D_{\Vert }$ are set as those in Section 2.1. After conducting numerous trial studies on the diffusion coefficient, we set $D_{\Vert }>D_{\perp }$, $\epsilon =0.01$ and the difference between ${\delta }_{\perp }$ and ${\delta }_{\Vert }$ is 0.32, which, due to the CRs from the TeV to PeV energy region are thought to travel faster parallel to the magnetic field than perpendicular to it [40].

Figure 3 illustrates the evolution of the amplitude and phase of anisotropy with energy, taking into account the contribution from the Geminga SNR source. It is evident that the computational results align well with the experimental data. These results further support our previous conclusion regarding the spectra and anisotropy with the contribution from the Geminga SNR [40].

In order to understand the phase along the declination, we further calculate 2D anisotropy maps at 10 TeV and 3 PeV, and the results are shown in Figure 4. We compare the 2D anisotropy maps under isotropic diffusion with those under anisotropic diffusion. The phase points to the Geiminga SNR source under isotropic diffusion, while it points in the direction of the LRMF under anisotropic diffusion at 10 TeV, which is consistent with the experimental observation. This indicates that the LRMF can deflect CR particles below 100 TeV. At 3 PeV, the phase is always directed towards the GC under any diffusion, which is attributed to the fact that the background sources are dominant, and the LRMF cannot deflect the CR particles in this energy region.

The solid black line in Figure 5 (left) shows the amplitude of anisotropy with the contribution from the Monogem SNR source. We found that the amplitude is greater than the experimental value. It is known that the spin-down energy of the Monogem pulsar is much lower than that of the Geminga pulsar with the value of $1.8\times {10}^{48}/1.25\times {10}^{49}$ erg, which suggests the injection power of the Monogem SNR is about one-tenth that of the Geminga SNR. Additionally, the gamma emission of the Monogem SNR is lower than that of the Geminga SNR, which also indicates that the injection power of the Monogem SNR is lower [60]. Therefore, we correct the injection power of Monogem according to the spin-down energy of pulsars. The dashed black line in Figure 5 (left) indicates the corrected amplitude of anisotropy, which is lower than the experimental observation. We calculated the corrected proton spectrum again, as shown by the dashed line in Figure 5 (right). The corrected proton spectrum cannot explain the spectrum hardening at 200 GeV. Therefore, after considering the correction, the contribution of the Monogem SNR to energy spectra and anisotropy can be ignored.

Figure 6 shows the anisotropy with the contribution from the Loop I SNR source. It is clear that neither the amplitude nor phase agree with the measurements of the experiment. The Loop I source, located in the direction of the GC, is in the same direction as the background sources. So, their amplitude of synthesis increases as the energy increases; meanwhile, the phase always points toward the GC.

4. Summary

In recent years, a large number of detectors have detected the anomalous structures of CR spectra at 200 GeV and the complex energy dependence of anisotropy from 100 GeV to hundreds of PeV, which have been revealed to be related to nearby sources. The aim of this work is to explore the optimal nearby source using energy spectra and anisotropy as probes. Five nearby sources including Geminga, Monogem, Loop I, Vela, and Cygnus are used as research objects, and the SDP model, which introduces anisotropic diffusion caused by LRMF, is used in the calculation process. Since our previous work only calculated 1D anisotropy along the right ascension and the anisotropy of the declination direction was unknown, this work further provides 2D anisotropy maps along the right ascension and declination.

The CR spectra results indicate that only the older nearby sources such as Geminga, Monogem, and Loop I can explain the nuclear spectral hardening at ∼200 GeV, assuming a common injection power. Meanwhile, younger ones such as Vela and Cygnus cannot explain this phenomenon, mainly because the lower-energy CRs produced by them reach the solar system with difficulty.

The 2D anisotropy maps reveal that the LRMF significantly deflects CR particles below 100 TeV, while it has no effect on particles above the 100 TeV energy region. The combined influence of the nearby source and LRMF dominates the phase of anisotropy below 100 TeV. The Geminga SNR source is located at the anti-GC, below the galactic disk and near the LRMF direction, and its contributions to the energy spectrum and anisotropy agree well with the experimental observations. The anisotropy from the Monogem SNR is obviously higher than the observations. After correcting the injection power of the Monogem SNR in accordance with the spin-down energy of the Geminga and the Monogem pulsars, the contribution from the Monogem SNR can be ignored. The anisotropy of the Loop I source is clearly inconsistent with the experiment, because it is located in the direction of the GC. Therefore, the Geminga SNR source, which can simultaneously explain the proton and helium spectral hardening at ∼200 GeV and the anisotropy from 100 GeV to PeV energy region well, is the optimal nearby source, while the other sources cannot.