Anisotropies of different mass compositions of cosmic rays

TL;DR

This paper investigates how the anisotropies of different cosmic ray mass groups can reveal the influence of nearby sources, providing a new approach to understanding cosmic ray origins and transportation.

Contribution

It introduces a model analyzing anisotropy features of various cosmic ray mass groups from nearby sources, highlighting their potential to distinguish source contributions.

Findings

Anisotropy features are more detectable than spectral features for different mass groups.

Future measurements by DAMPE, HERD, and LHAASO can test the nearby source scenario.

Nearby sources significantly influence cosmic ray anisotropies across mass groups.

Abstract

The spectral hardenings of cosmic ray nuclei above $\sim 200$ GV followed by softenings around 10 TV, the knee of the all-particle spectrum around PeV energies, as well as the pattern change of the amplitude and phase of the large-scale anisotropies around 100 TeV indicate the complexities of the origin and transportation of Galactic cosmic rays. It has been shown that nearby source(s) are most likely to be the cause of such spectral features of both the spectra and the anisotropies. In this work, we study the anisotropy features of different mass composition (or mass groups) of cosmic rays in this nearby source model. We show that even if the spectral features from the nearby source component is less distinctive compared with the background component from e.g., the population of distant sources, the anisotropy features are more remarkable to be identified. Measurements of the anisotropies of each mass composition (group) of cosmic rays by the space experiments such as DAMPE and HERD and the ground-based experiments such as LHAASO in the near future are expected to be able to critically test this scenario.

Tables

Table 1. Table 1 : Injection parameters of the background and local source.

	Background			Local source
Element	Normalization†	$\nu$	${ℛ}_c$	$q_0$	$\alpha$	${ℛ}_c^{\prime }$
	$\left[{\left(m^2\cdot \mathrm{sr}\cdot s\cdot \mathrm{GeV}\right)}^{-1}\right]$		[PV]	[GeV-1]		[TV]
p	$8.04\times {10}^{-5}$	2.45	7	$2.72\times {10}^{52}$	2.10	28
He	$5.47\times {10}^{-5}$	2.39	7	$2.53\times {10}^{52}$	2.10	28
C	$1.12\times {10}^{-5}$	2.40	7	$4.80\times {10}^{50}$	2.05	28
N	$1.45\times {10}^{-6}$	2.45	7	$8.40\times {10}^{49}$	2.05	28
O	$2.30\times {10}^{-5}$	2.43	7	$5.50\times {10}^{50}$	2.05	28
Ne	$4.46\times {10}^{-6}$	2.38	7	$1.02\times {10}^{50}$	2.05	28
Mg	$9.06\times {10}^{-6}$	2.44	7	$7.80\times {10}^{49}$	2.05	28
Si	$1.03\times {10}^{-5}$	2.44	7	$8.70\times {10}^{49}$	2.05	28
Fe	$2.53\times {10}^{-5}$	2.38	7	$1.01\times {10}^{50}$	2.05	28

Equations

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

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

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

Full text

1 Introduction

It is widely postulated that cosmic rays (CRs) with energies less than $\sim$ PeV are mainly generated by the Galactic supernova remnants (SNRs). Through the well-known diffusive shock acceleration process inside SNRs [1, 2, 3, 4], CRs are accelerated to form non-thermal power-law spectra, ${\rm d}N/{\rm d}{\cal R}\propto\cal R^{-\nu}$, with ${\cal R}$ being the rigidity. After escaping from the acceleration sites, they undergo frequent scatterings with the random magnetic turbulence in the Galaxy, whose behaviours are usually described by a diffusion equation. In the conventional propagation model, the diffusion is assumed to be homogeneous and isotropic, with a rigidity-dependence, namely $D({\cal R})\propto{\cal R}^{\delta}$, with $\delta$ being constrained from $0.3$ to $0.6$ from the Boron-to-Carbon ratio [5]. The propagated CR spectrum should then fall as $\phi\propto\cal R^{-\nu-\delta}$. Aside from the diffusion, CR nuclei may also suffer from the convection, re-acceleration as well as the fragmentation by collision with the interstellar gas. The low-energy nuclei also lose their energies due to the ionization and Coulomb scattering. For a comprehensive introduction to the CR propagation in the Galaxy, one can refer to [6, 7].

The convectional CR transport picture has successfully reproduced the observed power-law spectrum, the secondary-to-primary ratio, e.g. B/C ratio, the diffuse gamma-ray distribution and so on. However, growing observations challenge the conventional transport model. In recent years, the hardening of CR spectra above a few hundred GeV/nucleon received much attention. They were initially observed by balloon-borne calorimeter experiments ATIC-2 [8, 9], CREAM [10, 11], and get confirmed by precise measurements of space magnetic spectrometer experiments PAMELA [12] and AMS-02 [13, 14] and calorimeter experiment CALET [15]. The finding of the spectral hardenings brings about various alternatives to the traditional CR paradigm (e.g. [16, 17, 18, 19, 20, 21]). Most recently, the DAMPE observation shows clearly that the proton spectrum further experiences a spectral softening at $\sim 14$ TeV [22]. Hints of such spectral features were also found previously by CREAM [23] and NUCLEON measurements [24]. These new observations indicate that the structures of the energy spectra of CRs are more complicated than expected (see e.g., [25, 26] for discussions).

In addition to the unexpected structures in CR energy spectra, the traditional transport scenario also fails to explain the observed anisotropies. Despite that the arrival directions of Galactic CRs are highly isotropic due to their diffusive propagation in the Galactic magnetic field, a weak dipole-like anisotropy is consistently observed, with intensity differences up to $\sim 10^{-4}-10^{-3}$. So far a large amount of observations of anisotropies ranging from TeV to PeV have been carried out by the ground-based experiments, for example Super-Kamiokande [27], Tibet [28, 29, 30, 31], Milagro [32, 33], IceCube/Ice-Top [34, 35, 36, 37, 38], ARGO-YBJ [39, 40] and HAWC [41]. Besides the large-scale anisotropies, some mediate-scale and small-scale anisotropies have also been measured [41, 38, 42].

One of the well-known origins of the large-scale anisotropy is the so-called Compton-Getting effect [43, 44], which is induced by the motion of solar system with respect to a frame in which CR distribution is isotropic. This effect only relies on the power-law index of CR energy spectrum as well as the velocity of solar system, and does not vary with energy. The diffusion of CRs also predicts a large-scale dipole anisotropy, whose amplitude is expected to be proportional to the diffusion coefficient and the phase is along the density gradient of CRs. However the observations show a more complicated energy dependence. Less than $100$ TeV, the amplitude of anisotropy increases first with energy and then decreases, and is far below the prediction of the standard diffusion scenario [45, 46, 47, 48]. Furthermore, the phase also disagrees with the observations. The conventional propagation model predicts the excess of CR flux toward the direction of the Galactic center. However, the TeV measurements show the excess approaches to the direction of the heliotail, i.e. so-called tail-in region [49]. The puzzle is commonly referred to as the “anisotropy problem”. The anisotropy problem may indicate the effects due to the regulation of local magnetic field and/or nearby sources [50, 51, 52, 53, 54, 55].

It has been noted that there exists a common energy scale between the structures of the energy spectra and the large-scale anisotropies, which may indicate a common origin of them [56]. We proposed in a recent work that these features might be the imprints of local sources [56]. The spectral softenings around 10 TeV are due to a nearby source contribution on top of the background component. The low-energy ($\lesssim 100$ TeV) anisotropies are dominated by the local source, whihe the high-energy anisotropies are due to the background. The transition of the low-energy and high-energy components occur at about 100 TeV, forming a dip in the amplitude and a flip of the phase from nearly anti-Galactic center direction to the Galactic center direction. In [56], only protons and Helium nuclei are considered. In this work, we further extend this model to heavier nuclei. We pay particular attention to the anisotropy features of different mass composition (or mass groups), which may be tested in the near future by e.g., LHAASO [57, 58].

2 Model Description

2.1 Spatially-dependent diffusion

We work in a spatially-dependent propagation (SDP) frame, which is motivated by the HAWC observations of extended haloes around pulsars [59]. The SDP model was proposed to account for the hundred GeV spectral hardenings of CRs [21, 60, 61]. It was then found to be able to explain a series of observations of CR spectra and diffuse $\gamma$-rays [62, 63]. The diffusion volume in the SDP model is separated into two regions. Close to the Galactic disk $(|z|<\xi z_{h})$, where $z_{h}$ is the half thickness of the whole diffusive halo, the level of turbulence is expected to be high due to activities of supernova explosions, and hence the diffusion coefficient is relatively small. In the outer halo $(|z|>\xi z_{h})$, particles diffuse much faster. The parameterized diffusion coefficient we adopt is [62, 63]

[TABLE]

where

[TABLE]

with $g(r,z)=N_{m}/[1+f(r,z)]$, and $f(r,z)$ is the source density distribution. The numerical package DRAGON [64] is used to solve the transport equation. In this work, we adopt the diffusion-reacceleration model.

The injection spectrum of background sources is assumed to be a power-law of rigidity with a high-energy exponential cutoff, $q({\cal R})\propto{\cal R}^{-\nu}\exp(-{\cal R}/{\cal R}_{\rm c})$. The cutoff rigidity of each element could be either $Z$- or $A$-dependent. The spatial distribution of sources takes the form of SNR distribution [65], $f(r,z)\propto(r/r_{\odot})^{1.69}\exp[-3.33(r-r_{\odot})/r_{\odot}]\exp(-|z|/z_{s})$, where $r_{\odot}=8.5$ kpc and $z_{s}=0.2$ kpc.

2.2 Local source

The time-dependent propagation of CRs from the local source is obtained using the Green’s function method, assuming a spherical geometry with infinite boundary conditions. The solution is

[TABLE]

where $q_{\rm inj}({\cal R})\delta(t)\delta(\vec{r})$ is the instantaneous injection spectrum of a point source, $\sigma({\cal R},t)=\sqrt{2D({\cal R})t}$ is the effective diffusion length within time $t$. The diffusion coefficient $D({\cal R})$ is taken the value nearby the solar system. The injection spectrum is again parameterized as a cutoff power-law form, $q_{\rm inj}({\cal R})=q_{0}{\cal R}^{-\alpha}\exp(-{\cal R}/{\cal R}^{\prime}_{\rm c})$. The normalization $q_{0}$ is determined through fitting to the GCR energy spectra. The distance and age of the local source are set to be $d=330$ pc and $\tau=3.2\times 10^{5}$ years [56], respectively. The direction of the local source is obtained through fitting to the data of the anisotropies. We find that for $l=161^{\circ}$ and $b=9^{\circ}$, both the amplitudes and phases of the large-scale anisotropies can be reproduced (see below). We further assume that the local source contributes only to primary nuclei (such as p, He, C, O, Fe), rather than secondary nuclei (such as B and Be).

2.3 Sun’s vertical location

It should be noted that usually the solar system is assumed to be located at the mid-plane of the Galactic disk, and the source distribution is symmetric above and below the disk. However, it has long been known that the Sun locates slightly above the Galactic plane (towards the north Galactic pole). The inferred distance above the mid-plane is from several parsecs to $\sim 20$ pc [66, 67, 68]. The offset may induce a net vertical flow outwards from the Galactic plane, which generates a corresponding component of anisotropy. The total large-scale anisotropies thus include three components, the radial component, the vertical component, and the local source component. The sum of these three components give the total anisotropies which can be used to compare with the data. The vertical location of the Sun is assumed to be 10 pc above the Galactic mid-plane in this work.

3 Results

The model parameters are tuned according to the B/C and 10Be/9Be ratios, the energy spectra of various nuclei species, the all-particle spectra, and the amplitudes and phases of the anisotropies. The diffusion coefficient parameters are $D_{0}=8.34\times 10^{28}$ cm2 s*-1*, $\delta_{0}=0.65$, $N_{m}=0.39$, $n=3.5$, $\xi=0.1$, and $\eta=0.05$. The thickness of the propagation halo is $z_{h}=10$ kpc, and the Alfvénic velocity is $v_{A}=6$ km s*-1*. Note that a larger value of $z_{h}$ is adopted in this work compared with Ref. [62]. This is to suppress the vertical component of the anisotropies which seems to be lack in the observations at high energies ($>100$ TeV). It is interesting to note that previous studies under the simple one-zone propagation framework also give a relatively large value of the halo height [69, 70]. The comparison of the B/C and 10Be/9Be ratios between the model predictions and the data is given in Fig. 1.

Fig. 2 shows the propagated spectra of primary CR components, including protons, He, C, N, O, Ne, Mg, Si, and Fe nuclei. In each panel the blue and red lines are the contributions from the background and the local source respectively, and the black solid line is the sum of them. The corresponding injection parameters are given in Table 1. The spectral indices of the local source component are assumed to be slightly harder than that of the background component, which helps fit the data better. This is reasonable due to the diversity of CR sources, as can be inferred from the $\gamma$-ray observations of SNRs [87]. We can see that the addition of the local source component can simultaneously account for the spectral hardening features at $\sim 200$ GV, and the softening features at $\sim 10$ TV. The SDP model can also give a concave shape of the propagated CR spectra, which was previously proposed to account for the spectral hardenings [62, 63]. However, in this work the SDP-induced spectral hardenings do not specifically correspond to the measured hardenings. Nevertheless, the SDP model is still necessary in suppressing the anisotropies as will be shown below.

Through adding different compositions together, we get the all-particle spectrum as shown in Fig. 3, compared with the weighted data [88]. The knee structure of the all-particle spectrum can be properly reproduced by the background component assuming a $Z$-dependent cutoff with ${\cal R}_{c}\sim 7$ PV. In this case we find that the knee of the all-particle spectrum is mainly due to the suppression of the light components (protons and He nuclei). This is because we try to fit the KASCADE spectra of protons and He [76]. If alternatively the light component spectra from the Tibet experiments [89] are used, a smaller cutoff rigidity would be obtained [90].

As suggested in Ref. [56], the softening features in the energy spectra and the energy-dependent anisotropies might have a common origin. In Ref. [56], only the light components of protons and Helium nuclei were considered. Here we add all the major compositions as shown in Fig. 2 together. The corresponding amplitudes and phases of the dipole anisotropies are given in Fig. 4. The energy dependences of both the amplitudes and phases can be well reproduced in this model. Compared with Ref. [56], the dip of the amplitudes becomes wider, which matches better with the data. The transition of the phase also becomes smoother, and can be tested by improved measurements in future. Note that the direction of the local source is different from that assumed in Ref. [56], due to the inclusion of the vertical anisotropoies in this work.

We further calculate the anisotropies of different compositions. Considering the limited particle identification capability of the ground-based experiments, the primary components are divided into four mass groups, i.e. p+He, C+N+O, Ne+Mg+Si, and Fe, respectively. The resulting anisotropies are shown in Fig. 5. Dip structures of the amplitudes and phase flippings are visible for each mass group. We expect that the observations of anisotropies of different mass groups by LHAASO would be promising in revealing these structures, and give a critical test of this model.

The above discussion is based on the assumption of a $Z$-dependent cutoff energy of the local source spectra. We further investigate the effect due to an $A$-dependent cutoff. The comparision of the anisotropy amplitudes and phases of protons and Helium nuclei for $Z$- and $A$-dependent cutoff are shown in Fig. 6. For both cases, the model parameters are tuned to fit the energy spectra of different compositions and the total anisotropies. It is clearly shown that the energies of the dip of protons and Helium nuclei can effectly distinguish these two assumptions. A clear identification of protons and Helium individually is a little bit challenging for ground-based experiments [91]. The measurements of anisotropies of protons and p+He are possible for the LHAASO experiment [91], which can also be very important in probing the $Z$- and $A$-dependent cutoff assumptions of the model.

4 Discussion

Measurements of CRs enter a precise era thanks to fast development of space and groundbased experiments in recent years. Based on the new features of the CR spectra, including the spectral hardenings at $\sim 200$ GV and softenings at $\sim 10$ TV, together with the inhomogeneous diffusion inferred by the HAWC observations of pulsars and the long time puzzle of the energy-dependent evolution of the dipole anisotropy features, an SDP frame with contributions from a local CR source was established and could explain most of these new observational facts [56].

In this work, we extend this model to study the anisotropies of heavier nuclei. We find that the dip structure (phase flipping) of the total amplitudes (phases) of the anisotropies becomes smoother after adding heavier nuclei. This is because the dip features and phase changes for different species depend on energy, and the measurements of all species of CRs give an average effect of them. The anisotropies of different mass groups are also investigated. Similar dip features and phase changes are predicted for all of these compositions, with different characteristic energies. We further explore the differences of the large-scale anisotropy features between $Z$- and $A$-dependent assumptions of cutoff of the local source spectra. It is expected that future precise measurements of the anisotropies of different compositions or mass groups by e.g., DAMPE [113], HERD [114], and LHAASO [58].

Finally we comment that the spectral features of the electrons and positrons, particularly the remarkable positron excess (e.g., [115, 116]), can also be properly reproduced under the same framework of the source and propagation models as discussed in this work [117]. It is thus very encouraging to establish a unified scenario of GCR origin and propagation based on new precise observations of CRs and $\gamma$-rays.

Acknowledgments

This work is supported by the National Key Research and Development Program of China (No. 2018YFA0404203), the National Natural Science Foundation of China (Nos. 11875264, 11635011, 11761141001, 11663006, 11722328, 11851305).