Evidence of a thick halo for the spatial-dependent propagation model with Cosmic Ray anisotropy
Bing-Qiang Qiao, Yu-Hua Yao, Wei Liu, Qiang Yuan, Xiao-Jun Bi, Hong-Bo Hu, Yi-Qing Guo
DDraft version March 1, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Evidence of a thick halo for the spatial-dependent propagation model with Cosmic Ray anisotropy
Bing-Qiang Qiao, Yu-Hua Yao,
2, 1
Wei Liu, Qiang Yuan,
3, 4
Xiao-Jun Bi,
1, 5
Hong-Bo Hu,
1, 5 and Yi-Qing Guo
1, 5 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China College of Physics, Sichuan University, Chengdu 610064, P.R. China Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,China Center for High Energy Physics, Peking University, Beijing 100871, China University of Chinese Academy of Sciences, 19 A Yuquan Rd, Shijingshan District, Beijing 100049, P.R.China
ABSTRACTThe spatial-dependent propagation (SDP) model with a nearby source works well to reproduce the co-evolving features of both cosmic ray (CR) nuclei spectra and anisotropy. However, it is well known that theSun is offset from the Galactic plane. This will lead to a dominating anisotropy in perpendicular direction,which is discrepant with observations. Thus it is necessary to reboot further investigation into the effect ofthe solar offset. In this work, for the first time the combined studies of the solar offset, nuclei spectra andanisotropy are performed based on the SDP model. As a result, to reproduce CR spectra and anisotropy, thethickness of the halo ( ξ z h ) increases linearly with the displacement of the Sun. When the offset is ∼ ξ z h is about 0.9 kpc, which is a much thicker halo than usually.Moreover, the PeV anisotropy could estimate the value of diffusion coefficient, thus breaking the degeneracy ofdiffusion coefficient and halo thickness. Therefore it is a good messenger to constrain the halo thickness. Onthe other hand, the anisotropy in PeV energy region, as a new probe, might also shed new light to constrainthe solar offset. We hope that the anisotropy at the energies of ∼ TeV to PeV can be finely measured byLHAASO experiment, leading to a better understanding about the thick halo. INTRODUCTIONRecent years great progress has been made on thespectral measurement of Cosmic Rays (CRs) withballoon-borne and space-borne experiments. The finestructure of spectral hardening of nuclei at 200 GVwas observed by ATIC-2 (Panov et al. 2006), CREAM(Ahn et al. 2010) and PAMELA (Adriani et al. 2011).AMS-02 also confirmed the hardening with unprece-dented precision, although the spectral line-shape hasslight discrepancy with each other (Giesen et al. 2015).At higher energies, the DAMPE observation clearly re-vealed that the proton spectrum further experiencesa spectral softening at ∼ ∼ ∼ Corresponding author: Yu-Hua Yao, Yi-Qing [email protected], [email protected] from nearby SNRs (Thoudam & H¨orandel 2012), there-acceleration mechanism of old SNRs sources (Bier-mann et al. 2010; Thoudam & H¨orandel 2014), thecombination effects from different group sources (Zat-sepin & Sokolskaya 2006; Yuan et al. 2011) and theSpatial-Dependent Propagation (SDP) of CRs (Tomas-setti 2012; Gaggero et al. 2015; Jin et al. 2016).Due to diffusive propagation of particles in the Galac-tic magnetic field and the distribution of sources in theGalaxy, a small degree of anisotropy in the arrival di-rection of CRs is predicted. Indeed, various CR, γ -ray,and neutrino observatories could identify anisotropy inthe arrival directions of CRs at the level of 10 − − − ,see Ref. (Mollerach & Roulet 2018; Deligny 2019) fora recent review. However, the data from studies ofARGO-YBJ (Bartoli et al. 2013, 2015), EAS-TOP (Agli-etta et al. 1996; Abbasi et al. 2012), IceCube/IceTop(Aartsen et al. 2013), and Tibet-AS γ (Amenomori et al.2017), indicate that the TeV–PeV dipole anisotropy isnot described by a simple power law and undergoes arapid phase flip at an energy of 0.1–0.3 PeV (Ahlers2016). These features indicate that a new component ofCRs is required to understand the complicated energydependence anisotropy. Some claimed that these puzzle a r X i v : . [ a s t r o - ph . H E ] F e b was caused by the effect of the regulation of local mag-netic field and/or nearby sources(Schwadron et al. 2014;Savchenko et al. 2015; Mertsch & Funk 2015; Ahlers2016; Sveshnikova et al. 2013; Liu et al. 2017). Worksin (Liu et al. 2019) considered the common evolving ori-gin of energy spectra and the large-scale anisotropy forexplanation. In the later scenario, the spectral softensaround 10 TeV are due to a nearby source contribu-tion on top of the background component. The low-energy ( (cid:46)
100 TeV) anisotropy are dominated by thelocal source, while the high-energy anisotropy are dueto the background. The transition of the low-energyand high-energy components occur at about 100 TeV,forming a dip in the amplitude and a flip of the phasefrom nearly anti-Galactic center direction to the Galac-tic center direction.It seems that the structures of the spectra andanisotropy have been reproduced under these situa-tions. However, one critical factor has been ignored inprevious works, i.e. the Sun’s offset. Usually the solarsystem is assumed to be located at the mid-plane of theGalactic disk, and the source distribution is symmetricabove and below the disk. Yet it has long been knownthat the Sun locates slightly above the Galactic plane(towards the north Galactic pole). The inferred distanceabove the mid-plane is from several to ∼
30 pc (Joshi2007; Bobylev & Bajkova 2016; Yao et al. 2017). Theoffset would induce a net vertical flow outwards fromthe Galactic plane, which generates a correspondingcomponent of anisotropy. This is totally inconsistentwith experimental observations. On the other hand,the estimated solar offset has a significant spread ofvalues employed with a variety of different methods,although recent analyses have tended to have smalleruncertainties. The combined study of nuclei spectra andanisotropy maybe shed light on this topic. In this work,we perform further studies aimed at the anisotropyproblem. The paper is organized in the following way,section 2 describes the model description, section 3presents the results of the calculation compared withthe observation. Finally, section 4 gives the conclusion. MODEL DESCRIPTIONIt is generally believed that supernova remnants(SNRs) are the main suspects as sources of GalacticCRs. They can accelerate the CRs to very high energywith the expanding diffusive shocks generated duringtheir active period (Bell 1978b,a; Blandford & Ostriker1978). Before arriving at earth, those CRs have trav-eled in the galaxy for ∼ years after they diffuse awayfrom the acceleration sites (Garcia-Munoz et al. 1977).During the journey, the impacts due to the fragmenta- tion and radioactive decay in the InterStellar Medium(ISM) result in the production of secondary particles.Meanwhile, the electron suffers energy loss in the Inter-Stellar Radiation Field (ISRF) and magnetic field. Thisjourney can be described by the propagation equationas: ∂ψ ( r ,p,t ) ∂t = q ( r , p, t ) + ∇ · ( D xx ∇ ψ − V c ψ )+ ∂∂p p D pp ∂∂p p ψ − ∂∂p (cid:2) ˙ pψ − p ( ∇ · V c ψ ) (cid:3) − ψτ f − ψτ r (1)where q( r , p , t) is the acceleration sources, ψ ( r , p , t) isthe density of CR particles per unit momentum p atposition r , V c is the convection velocity, ˙p ≡ dp / dt ismomentum loss rate, τ f and τ r are the characteristictime scales for fragmentation and radioactive decay re-spectively; D xx and D pp are the diffusion coefficients incoordinate and momentum space respectively. In fact,the processes of convection ( V c ) is ignored in this work.Then, the value of ψ ( r , p , t) is dependent on the D xx ,q( r , p , t) and position r .2.1. Spatial-dependent diffusion
In this work, the spatial-dependent propagation frameis adopted, which is motivated by the HAWC obser-vations of extended haloes around pulsars Abeysekaraet al. (2017). Here the diffusion volume in the SDPmodel is divided into two regions as inner halo (IH) andouter halo (OH). Close to the Galactic disk ( | z | < ξ z h )is labelled IH region, where z h is the half thickness ofthe whole diffusive halo and ξ is the thickness fractionof inner halo. In the IH region, the level of turbulenceis expected to be high due to activities of supernova ex-plosions, and hence the diffusion coefficient is relativelysmall. Contrast to in the IH region, particles diffusemuch faster in the OH ( | z | > ξ z h ). According to Guo& Yuan (2018); Liu et al. (2018), ξ is settled as 0.1 andthe parameterized diffusion coefficient we adopt is D xx ( r, z, R ) = D F ( r, z ) β η (cid:18) RR (cid:19) δ F ( r,z ) , (2)where F ( r, z ) = g ( r, z ) + [1 − g ( r, z )] (cid:18) zξz h (cid:19) n , | z | ≤ ξz h , | z | > ξz h , (3)with g ( r, z ) = N m / [1 + f ( r, z )], and f ( r, z ) is the sourcedensity distribution. The numerical package DRAGONEvoli et al. (2008) is used to solve the transport equa-tion. In this work, we adopt the diffusion-reaccelerationmodel. The injection spectrum of background sourcesis assumed to be a power-law of rigidity with a high-energy exponential cutoff, i.e. q( R ) ∝ R − ν e ( −R / R c ) .The cutoff rigidity of each element could be either Z -or A -dependent. The spatial distribution of sourcestakes the form of SNR distribution Case & Bhattacharya(1996), f ( r, z ) ∝ ( r/r (cid:12) ) . e [ − . r − r (cid:12) ) /r (cid:12) ] e ( −| z | /z s ) ,where r (cid:12) = 8 . s = 0 . Local source
The time-dependent propagation of CRs from thelocal source is obtained using the Green’s functionmethod, assuming a spherical geometry with infiniteboundary conditions. The solution is φ ( r, R , t ) = q inj ( R )( √ πσ ) exp (cid:18) − r σ (cid:19) , (4)where q inj ( R ) δ (t) δ ( r ) is the instantaneous injectionspectrum of a point source, σ ( R , t) = (cid:112) R )t isthe effective diffusion length within time t . The diffu-sion coefficient D( R ) is taken the value nearby the solarsystem. The injection spectrum is again parameterizedas a cutoff power-law form, q inj ( R ) = q R − α e ( −R / R (cid:48) c ) .The normalization q is determined through fitting tothe GCR energy spectra. The direction of the localsource is obtained through fitting to the data of theanisotropy, and other detailed parameters of the source.2.3. Solar Offset
Several methods have been adopted to measured thesolar offset from the local mean Galactic plane, whichcan be roughly classified in two categories: the matter-based (H II , molecular and methanol masers, open clus-ters) and stars-based (pulsars, Optical stars, Wolf-Rayetstars et al.) methods. FIG.1 presents these estimatedoffset distances summarized in the table 1 of litera-ture(Yao et al. 2017), which span in a large range from5 to ∼
30 pc. It is worth making an effort to calculatethe average offset for these two categories, which are8 . ± . . ± . RESULTSBase on above description, firstly we discuss the influ-ence of the halo thickness on the CRs flux density with - OB starts (2007)Open cluster(2007)Open cluster (2014)Various stars(1942)Wolf-Rayet stars(1990)Optical stars(1995)Optical stars(2001)OB stars(2001)Cepheid variables(2009)Magnetars(2014)Pulsars(2017)Open cluster (2016)Methanol mesers(2016)HII regions(2016)Giant molecular clouds(2016)
Fig. 1.
Displacement of the Sun from the galactic planefrom previous works summarized in the (Yao et al. 2017) andreferences therein. The red and blue points are classified asstars-based and matter-borne methods, respectively. Twovertical dotted lines represent their mean values, the shadowareas are the corresponding errors of mean value. Duringthe procedure, data from open clusters (light orange) areomitted due to their inconsistencies. Detailed calculationssee the main text. - - - - Z-Z - · - · ] - s r - s - m - [ G e V F - - - - R-R
Fig. 2.
Schematic diagram of cosmic ray fluxes along withdistances from (R , Z ) in the vertical (blue) and radial (red)directions. Full (dashed) lines correspond with ξz h = 0 . , Z ) = (8 . ,
0) kpc. the Sun’s offset. Then the propagation parameters un-der various scenarios are tuned with B/C, the CR spec-
Tab. 1.
Propagation parameters † . ξ z h (kpc) Z (cid:12) (pc) D ( × cm s − ) δ † N m , n , andv A are adopted as 0.39, 3.5, and 6 km s − , re-spectively. tra are obtained with DRAGON package. Note that thelocal source metioned above is also included in the com-putation. At last, the anisotropy amplitudes and theircorrelation with the solar vertical displacement from theGalactic plane are given.3.1. Effect of halo thickness
As shown in FIG. 2, compared with a consistent of theradial anisotropy ( slope of CR flux ), the anisotropy inthe vertical direction changes significantly with increas-ing vertical distance from the point (R , Z ). Besides,the thickness of the halo has strong (little) influence onthe vertical (radial) anisotropy. A thicker halo wouldcounteract the vertical CR fluxes caused by the solaroffset. In addition, we also study the impact of the localsources and find that they mainly affect the anisotropyat energies lower than hundreds of TeV. -2 -1 B / C E [GeV]
AMS-02 ξ z h = 0.75kpc ξ z h = 1.0kpc ξ z h = 1.2kpc ξ z h = 1.5kpc ξ z h = 2.0kpc Fig. 3.
Model predictions of the B/C ratio compared withAMS-02 measurement(Aguilar et al. 2015, 2017).
B/C and CR spectra
The transport parameters for the SDP model withdifferent values of ξ z h are determined by the B/C ra- E . Φ ( E ) [ G e V . m - s - s r - ] E [GeV]
Proton
AMS-02CREAMKASCADE ξ z h = 0.75kpc ξ z h = 1.0kpc ξ z h = 1.2kpc ξ z h = 1.5kpc ξ z h = 2.0kpc Fig. 4.
Model predictions of the proton at different ξz h , compared with observations from AMS-02(Aguilar et al.2015, 2017), CREAMAhn et al. (2010), KASCADE(Antoniet al. 2005). The dot-dashed and dashed lines are the fluxesfrom the background and local sources, and the solid linesare their sum. Different colors indicate various ξz h scenarios. E . Φ ( E ) [ G e V . m - s - s r - ] E [GeV]
Horandel ξ z h = 0.75kpc ξ z h = 1.0kpc ξ z h = 1.2kpc ξ z h = 1.5kpc ξ z h = 2.0kpc Fig. 5.
Model predictions of the all-particle spectra atdifferent ξz h , the data points are taken from (H¨orandel 2003). tio, as given in FIG. 3. Detailed parameters are pre-sented in TABLE 1. After the determination of basicparameters, the propagated spectra of primary CR pro-tons is shown in FIG. 4. We can see that the addition ofthe local source component can simultaneously accountfor the spectral hardening features at ∼
200 GV, andthe softening features at ∼
10 TV. Note that there areslight differences of the local source under various ξ z h scenarios in order to fit the data well. The all-particle -4 -3 -2 A E [GeV]
Region I Region II Region III
Fig. 6.
The energy dependence of the amplitudes of thedipole anisotropy when adding all of the major CR elementstogether. Observational data is taken from ref. (Ahlers &Mertsch 2017) and references therein. See FIG. 3 for legendsof lines. [ kp c ] h z x [ k p c ] h z T h i s W o r k M a tt e r - ba s e d m e t hod s t a rs - ba s e d m e t hod Fig. 7.
Relationship between the displacement of the Sunfrom the (Galactic plane) point (R , Z ) and the thickness ofhalo. Since ξ = 0 .
1, in order for convenience, the right axisalso shows the total thickness of the halo. The red (blue)vertical line and shaded area are the mean value and errorof mean calculated with previous measurement, respectively.The pink dot shaded area shows the allowable range of thesolar location and the halo thickness estimated with currentanisotropy observations based on SDP model.
CRs energy spectrum predicted from the SDP model isexhibited in FIG. 5, along with the one from Horan-del spectrum which is extracted from many experimentsdata. 3.3.
Anisotropy and solar offset
The literature proposed that the energy-dependentanisotropy and the softening features in the energy spec-tra might have a common origin(Liu et al. 2019). FIG. 6demonstrates the CR anisotropy amplitudes, the energydependence of anisotropy amplitudes can be well repro-duced in this model. Combined with the local sourcedemonstrated in FIG. 4, the anisotropy of all compo-nents can be divided into three regions by energies asshown in FIG. 6. In the low energy (region I), theanisotropy is influenced by the propagation coefficientand local sources, the former dominates although thelatter is becoming important as energy increases. Thecontribution of the local source to the anisotropy reachesits maximum value at tens of TeV and then goes down,thus it is essential for the reproduction of the dip fea-tures in region II. At the highest energy range (regionIII), the thicker the halo is, the larger the anisotropy is.In this region, the effect of local source becomes negli-gible (this also can be seen from FIG. 4) and only thepropagation effect works. Because the dipole anisotropyestimation is proportional to D xx c (Evoli et al. 2012) andthe ratio of the parameterized diffusion coefficient andhalo thickness, i.e. D xx /ξ z h , is a constant and fixed bythe B / C, the thickness of halo dominates the anisotropy.As we can see, the model curves in FIG. 6 have a tensionwith the observation at energies of PeV as ξ z h is largerthan 1 . D xx , also, ξ z h and solar offset. Owing to limited contri-bution of nearby sources, the anisotropy in PeV energyregion is a good messenger to constrain the halo thick-ness and also solar offset. The halo thickness ξ z h shouldbe less than 1.2 kpc and corresponding to the solar off-set 14 pc with current CR anisotropy measurements.Compared with the two average values inferred in Fig.1, the value interval of anisotropy-estimated solar offsetprefers the measurement from matter-borne method tothe stars-based one. CONCLUSIONMeasurements of CRs enter a precise era thanks tofast development of space-borne and ground-based ex-periments in recent years. Based on the new featuresof the CR spectra, including the spectral hardenings at ∼
200 GV and softenings at ∼
10 TV, together with thelong time puzzle of the energy-dependent evolution ofthe dipole anisotropy features, an SDP frame with con-tributions from a local CR source was established andcould explain most of these new observational facts Liuet al. (2019).In this work, we extend the SDP model to simultane-ously study the CR spectra, anisotropy of CRs, and theSun’s offset from the Galactic plane for the first time.We find that the primary CR protons and all-particlespectra, and the dip structure of the amplitudes of thetotal anisotropy can be well reproduced after consideringthe Sun’s offset. The thickness of the halo increases lin-early with the displacement of the Sun, so a much thickerhalo is required to counteract the affected anisotropy inour model. Recent measurements, i.e. Fermi-bubbles(Su et al. 2010) and large-scale X-ray bubbles (Predehlet al. 2020), support the point of thick halo. More-over, owing to limited contribution of nearby sources,the dipole anisotropy in the PeV energy region can beused to estimate the value of D xx , thus breaking thedegeneracy of D xx /ξ z h , which is tuned by B/C ratio.Therefore it is a good messenger to constrain the halothickness ξ z h . On the other hand, with present observa- tions of CR anisotropy, our model shows that the solaroffset prefers to be less than 14 pc. It is worth notingthat this might be a new (model-dependent) method toconstrain the solar displacement, although in this workwhat we give is an error-free wide range, which largelybecause of current large-uncertainties anisotropy obser-vations. It is expected that future precise measurementsof the anisotropy in the energy range from ∼ TeV to PeVby e.g. LHAASO Bai et al. (2019) could give a finelydetermination of the thickness of the halo, and also, testour model and constrain the Sun’s vertical location.ACKNOWLEDGEMENTSThis work is supported by the National Key R & D Program of China grant No. 2018YFA0404202 ,theNational Natural Science Foundation of China (Nos.11635011, 11875264, 11722328, 11851305, U1738205,U2031110).REFERENCES