Propagation of Ultra-high-energy Cosmic Rays in Galactic Magnetic Field
aa r X i v : . [ a s t r o - ph . H E ] A p r Propagation of Ultra-high-energy Cosmic Rays in GalacticMagnetic Field
Hajime Takami
Max Planck Institute for Physics, F ¨ ohringer Ring 6, 80805 Munich, Germany Abstract.
The propagation trajectories of ultra-high-energy cosmic rays (UHECRs) are inevitably affected by Galacticmagnetic field (GMF). Because of the inevitability, the importance of the studies of the propagation in GMF have increasedto interpret the results of recent UHECR experiments. This article reviews the effects of GMF to the propagation and arrivaldirections of UHECRs and introduces recent studies to constrain UHECR sources.
Keywords:
Ultra-high-energy cosmic rays, Galactic magnetic field
PACS:
INTRODUCTION
The origin of ultra-high-energy cosmic rays (UHECRs)has been an open problem in astrophysics for about 50years, despite both theoretical and experimental efforts.A main difficulty to identify UHECR sources originatesfrom the charge of UHECRs, which leads to the deflec-tion of their propagation trajectories in Galactic magneticfield (GMF) and intergalactic magnetic field (IGMF).These fields make UHECRs arriving at the Earth losedirectional information on their sources contrary to neu-tral particles, e.g., photons and neutrinos. However, thedeflection is expected to be small (several degree) at thehighest energies ( ∼ eV) even at an upper limit valueof averaged IGMF, B IGMF l c , IGMF1 / < ( )( ) / [1], if UHECRs are protons. Recent g -ray observationsimply ∼ − G in voids [2], although this estimationis controversial [3], and give conservative lower limitsof IGMF in voids 10 − G [4, 5]. Although lower val-ues of these constraints seems to be positive to identifyUHECR sources directly, structured IGMF, i.e., magneticfield in clusters of galaxies and filamentary structure, cansignificantly contribute to the deflections, which includeslarge uncertainty in the modelling [6, 7, 8, 9, 10].The propagation of UHECRs in GMF has also beenstudied from early days [11, 12]. In 1990s, the impor-tance of GMF was revisited [13, 14], which was stim-ulated by potential anisotropy in the arrival distributionof UHECRs [15]. In addition, since UHECRs detectedat the Earth are inevitably affected by GMF, the effectof GMF is essential to consider what UHECR sourcesare. Moreover, after the identification of several UHECRsources, UHECRs can be used as a background source toconstrain local GMF structures.The importance of considering the effect of GMF hasincreased recently. In order to interpret the correlationbetween UHECRs and extragalactic astrophysical ob- jects reported by the Pierre Auger Observatory (PAO)[16] and to constrain UHECR sources, the propagation ofUHECRs in Galactic space should be taken into accountbecause of its inevitability. The properties of the propa-gation highly depends on the composition of UHECRs,which is still controversial between the PAO [17] andHigh-Resolution Fly’s Eye (HiRes) [18] [and TelescopeArray (TA) (see Dr. Tameda’s talk in this workshop)],since the deflections of trajectories of UHECRs are sen-sitive to their charge (e.g., Ref. [14]). Also, in the case oftransient sources, GMF produces significant time-delayof UHECRs compared to neutral particles emitted at thesame time [19].This article reviews the effects of GMF to the propa-gation and arrival directions of UHECRs and introducesrecent studies to constrain UHECR sources.
GMF MODELING
Here, we give readers a minimal summary on GMF mod-elling. See Dr. Han’s talk in this workshop for recentprogress on GMF observations and more details of mod-elling.GMF is modelled by two components; magnetic fieldsin the Galactic disk and in the Galactic halo. Note thatthe halo field usually means magnetic field far from theGalactic disk. The disk field has been modelled by abisymmetric spiral structure (BS) [20], an axisymmetricspiral structure (AS) [13], and concentric rings with fieldreversals (e.g. Refs. [21, 22]). Which model can repro-duce observations globally better is still a controversialissue [22, 23], but the existence of (at least one) fieldreversals has been reported (e.g. [24]). There is not evi-dence that the directions of the field above and below theGalactic plane is different. z [ k p c ] x [kpc] BS-SBS-AAS-SAS-A -20-15-10-5 0 5 10 15 20-20 -15 -10 -5 0 5 10 15 20 z [ k p c ] x [kpc]BS-S + Dipole + Turb.BS-A + Dipole + Turb.AS-S + Dipole + Turb.AS-A + Dipole + Turb. FIGURE 1.
Examples of the projected trajectories of UHECRs ( left : protons, right : irons) with 10 . eV with the same arrivaldirections in different GMF models. This figure is originally from Ref.[25]. In the halo field, exponential decay models of the spi-ral disk field in the direction perpendicular to the Galac-tic plane have been used [13, 26]. Faraday rotation mea-surements of extragalactic radio sources indicates the di-rections of magnetic fields parallel to the Galactic planeabove and below the Galactic plane are opposite [27].This anti-symmetric structure is theoretically supportedby so-called A0 mode in dynamo theories. So, somemodels treat GMF in the disk and GMF in the halo sep-arately [28]. Recent observations have confirmed thistreatment and have revealed the scale height of electrondensity in the disk higher than in previous understanding[22, 23].A dipole field, as a GMF component parallel to theGalactic disk, is also often taken into account [26, 29].The dipole field is motivated by (i) a vertical field of0.2-0.3 m G observed in the vicinity of the solar system[30], (ii) strong filamentary magnetic field perpendicularto the Galactic plane in the vicinity of the Galactic center[31] and (iii) a theoretical result of A0 dynamo, but thereis no clear evidence of the dipole field. In extragalacticedge-on galaxies, X-shaped structures of magnetic fieldabove and below the disk have been observed, whichimplies galactic wind [32, 33]. The shape of the verticalcomponent of GMF is still a controversial issue.In addition to these coherent components, the turbu-lent component of GMF, whose strength is 0.5-2.0 timesas strong as the coherent components, is implied [34].Only upper limits of the correlation length of the turbu-lent component ( ∼
100 pc) have been estimated due toa limited angular resolution of radio observations [34].Although the turbulent component is not important forprotons, it plays an significant role for heavy nuclei be-cause the Larmor radius of heavy nuclei in the m G fieldapproaches to the correlation length.
PROPAGATION OF UHECRS IN GMF
In order to investigate the propagation of UHECRs inGMF, a backtracking method has been often adopted.In this method, particles with the charge opposite tothat of particles which we are interested in are injectedfrom the Earth and their trajectories are calculated. Then,we can regard the trajectories of these anti- particles asthose of particles arriving from extragalactic space. Inmany cases, all the energy-loss processes of UHECRsare negligible because of the size of the Galaxy smallerthan the energy-loss lengths of UHECRs. This methodallows us to save much CPU time, since only the particlesreaching the Earth are taken into account. This methodcan be applied to simulate the arrival distribution ofUHECRs taking GMF and even structured IGMF intoaccount [29, 8].Since the solar system is embedded in the Galacticdisk 8.5 kpc away from the Galactic center, magneticfield in the vicinity of the solar system mainly contributesto the total deflections of UHECRs in Galactic space.Fig. 1 shows the examples of the trajectories of UHE pro-tons ( left ) and irons ( right ) with the energy of 10 . eVwith the same arrival directions ( ℓ , b ) = (0 . ◦ , − . ◦ ).The GMF models adopted in this figure are BS and ASmodels for spiral field in the disk with two exponentialdecay scale in the direction perpendicular to the Galacticdisk. The symbols S and A means symmetric and anti-symmetric field below and above the Galactic plane. Thetrajectories of protons seems almost to be straight linesby eye, but are deflected several degrees mainly in theGalactic disk. The field reversals of the disk field af-fect the total deflection of protons. In the BS models, thenearest field reversal is located at 0.5 kpc inside the solarsystem. When a charged particle passes through the fieldreversal, the direction of its deflection becomes opposite,and the total deflection angle is suppressed [26]. There-ore, BS models, in general, predict smaller deflectionangles of UHECRs than AS models. We can observe thiseffect even for irons (green line), but sometimes heavynuclei are confined in the Galactic disk even in the high-est energy range. The propagation trajectory of a parti-cle shown by the red line is very complex and is con-fined in the Galactic disk for a long time, and thereforedirectional information on its source is completely lost.The distribution of the deflection angles of UHE ironswas well studied by Refs. [14, 37], similarly to protoncases [38], demonstrating that irons with the energy of6 × eV are deflected more than 25 ◦ in more than80% region of the sky assuming a GMF model proposedby Ref. [28] plus weak dipole field. Ref. [25] calcu-lated the back-tracked positions of UHECRs observed bythe Pierre Auger Observatory (PAO) onto the surface ofMilky Way sphere on the assumption of pure iron com-position and demonstrated the correlation between UHE-CRs and nearby galaxies [16] is strongly disturbed. Fig. 2is the arrival directions of 27 UHECRs above 5 . × eV in the first public data of the PAO [35] ( red ), the back-tracked directions of them on the assumption of protons( green ) and irons ( blue ). A BS-S model is adopted. Theblack points are galaxies in the IRAS catalog within 75Mpc [36]. While the back-tracked directions are close tothe original arrival directions in the case of protons, theback-tracked directions of irons are far away from theoriginal directions and no longer correlate with the dis-tribution of galaxies. Pure iron composition at the Earthis an extreme and unrealistic case even in the case whenall the UHECRs emitted from sources are irons becausephotodisintegration with cosmic microwave backgroundduring propagation in intergalactic space. Nevertheless,it is an intriguing problem whether the correlation can bereproduced under heavy-nuclei dominated compositionif the correlation is not a statistical fake.As mentioned above, the spiral field in the vicinityof the solar system mainly contribute to the total de-flections. Thus, reflecting the direction of local magneticfield, UHECRs from extragalactic sources in the north-ern Galactic hemisphere are deflected to the directionof Galactic south. This tendency can be seen in the fig-ure. In the case of anti-symmetry above and below theGalactic plane, the deflection directions are opposite. Inboth cases, since the Larmor radius of a charged parti-cle is proportional to its energy, the arrival directions ofUHECRs from a source are arranged in the order of theirenergy if the coherent component of GMF is dominated[26, 29, 25].The Liouville’s theorem leads the isotropic distribu-tion of UHECRs at the Earth if UHECR flux is isotropicoutside GMF. However, the distribution of the momen-tum directions of back-tracked UHECRs injected fromthe Earth isotropically is not isotropic because there arethe trajectories of UHECRs which cannot reach the Earth FIGURE 2.
Arrival directions of the 27 PAO events [35]( red ) and their back-tracked directions assuming protons( green ) and irons ( blue ) in galactic coordinates. A BS modelwith symmetry above and below the Galactic plane is assumed.Black points are galaxies in the IRAS catalog within 75 Mpc[36]. This figure is originally from Ref.[25]. [26, 39, 38, 37]. In other words, there are directions inwhich sources can provide the Earth with UHECRs ef-ficiently (or inefficiently). Such an effect is called mag-netic lensing (de-lensing) [40, 41]. Interestingly, this ef-fect can occur even in turbulent magnetic field in a few% of the whole sky.
SEARCHING FOR UHECR SOURCES
The correlation between UHECRs and the positions ofsource candidates has been expected to give a hint toidentify UHECR sources, and therefore a lot of effortshas been dedicated. Although the modifications of thearrival directions by GMF are not taken into account toavoid model dependence in many studies, several testsconsidered plausible GMF models and obtained positiveresults. In early days, Ref. [42] tested the correlation be-tween observed UHECR events and BL Lac objects tak-ing a BS model into account. The authors treated thecharge of the UHECRs as a free parameter, and obtaineda positive correlation signal in the case of Q = + Q = −
1, where Q is thecharge of UHECRs in the unit of the absolute value ofthe electron charge. This implies that the composition ofUHECRs is mainly protons. Recently, Ref. [43] exam-ined the correlation between the PAO events and g -raysources detected by Fermi Large Area Telescope [44].Although strong ( ∼ s ) correlation was found, the mod-ification by GMF did not largely change the result be-cause a Cen A region is dominated in the signal.Ref. [45] tested the positional correlation between ob-served UHECR events and galaxies with extended ra-dio jets taking modifications of their arrival directionsby several GMF models. The authors claimed that onethird of UHECRs correlates with extended radio jets ifthey are light nuclei and proposed the idea that the re- F r equen cy Separation Angle: q [deg]n s = 10 -4 Mpc -3 N p = 200, E > 60 EeVd < 75 MpcBS-S + Dipole + Turb. (South)BS-S + Dipole + Turb. (North)AS-A + Dipole + Turb. (South)AS-A + Dipole + Turb. (North) FIGURE 3.
Histograms of the separation angles between200 simulated protons above 6 × eV and their sources with10 − Mpc − within 75 Mpc which gives maximal significanceagainst random event distribution. The peak positions dependon the field reversals of GMF. This figure is originally fromRef.[25]. maining two thirds are heavy nuclei, i.e., a mixed com-position scenario. If radio galaxies are responsible for theobserved UHECRs, a BS-S or BS-A model is supported.Interestingly, several trajectories of UHECRs pass closeto Galactic magnetars and microquasars under the BS-S model. The propagation of UHECRs from Galacticsources was studied by Ref. [46]. UHECRs with 10 -10 eV from Galactic sources inside the solar systemarrive from the northern Galactic sky and from the south-ern Galactic sky for a AS-S and BS-S models, respec-tively. In the case that the A-type parity of GMF is real-ized, the difference between the two models is not clear.These results can easily scale to the case of heavy nucleiat the highest energy by using the rigidity scaling. Thus,large-scale anisotropy is expected at the highest energyif heavy nuclei are dominated and the S-type parity ofGMF is realized.Another approach is to estimate typical angular scalewithin which positive correlation between UHECRs andtheir sources appears by simulations. This angular scalecan be used for the cross-check of the results from cor-relation studies between observed UHECRs and the dis-tribution of source candidates (galaxies), because we donot know what objects are UHECR sources. In addi-tion, since the typical angular scale depends on composi-tion, the estimation also gives an important informationto constrain composition indirectly. Ref. [25] simulatedthe arrival distribution of protons taking GMF into ac-count and estimated typical angular scale within whichpositive correlation between the simulated protons andnearby sources used in the simulations. Fig. 3 shows oneof the histograms of the angular scale within which thecorrelation signal is maximized in the case of the number density of UHECR sources of n s = − Mpc − . Resultsa bit depend on GMF models and the number densityof UHECR sources n s . The typical angular scale is ∼ ◦ for n s = − Mpc − and ∼ ◦ for n s = − Mpc − inthe cases of BS models. AS models predict angular scalelarger than those in BS models due to larger deflectionsin AS models. The smaller angular scale is predicted inthe larger number density because the cross-correlationfunction includes correlation between simulated UHE-CRs and sources not emitting the simulated UHECRs lo-cated in the vicinity of the sources contributing to theobserved (simulated) events.Motivated by ordered deflections by the regular com-ponent, the reconstruction of GMF structure along theline of sight and of the positions of sources has been stud-ied by simulations [47, 48]. A recent study concludedthat about ten events from a source above 3 × eV al-lowed to reconstruct the source position with an accuracyof 0 . ◦ and the orthogonal component of the magneticfield with respect to the line of sight with an accuracy of0.6 kpc Z − [48].Candidates of the arranged events were found in thedata recently published by the PAO in a Cen A region[49]. The source position reconstructed by the arrangedevents is located at 8 . ◦ from M87, a nearby powerful ra-dio galaxy. According to a statistical test, the probabilitythat such a set of events are realized in random back-ground is ∼ × − . This scenario offers the possibilitythat the excess of UHECR events around Cen A is notproduced by Cen A, and gave us not only indirect evi-dence that M87 is a nearby source of UHECRs but alsoa possible solution to the fact that highest energy eventsdo not seem to correlate with the position of the Virgocluster [50, 51].GMF produces the time-delay of UHECRs comparedto neutral particles emitted at the same time. The time-delay plays an important role if the production of UHE-CRs is transient. For transient sources, this time-delaymakes the apparent duration of UHECR bursts and weobserve the sources as apparently steady sources in theobservation time-scale of human beings [52, 19]. Com-bined by the estimation of the number density of UHECRsources on the assumption of steady sources [53, 54], thetime-delay by GMF leads an upper limit of the rate ofUHECR bursts as (60-3000) Gpc − yr − [19]. Althoughthe range means that this upper limit depends on themodelling of GMF, this limit rules out several transientphenomena for UHECR sources. SUMMARY
We have reviewed the effects of GMF to the arrival direc-tions of UHECRs. The importance of GMF is in the factthat all the UHECRs arriving at the Earth are affected byMF. A remarkable feature of GMF is its regular com-ponent. The regular component deflects the trajectoriesof UHECRs effectively and therefore significantly con-tributes to the arrival directions of UHECRs despite thesmaller size of the Galaxy than the propagation distanceof UHECRs in intergalactic space. The dominance of theregular component in UHECR deflections also allows usto use UHECRs as a spectrograph of GMF. Combinedwith increasing number of detected UHECR events andthe improvement of the understanding of UHECR com-position, the studies of UHECR propagation in GMF willhelp us understand the origin of UHECR sources.
ACKNOWLEDGMENTS
H.T. thanks to the organizers of this workshop for invit-ing me. H.T. is partially supported by Grants-in-Aid forScientific Research from the Ministry of Education, Clu-ture, Sports, Science and Technology of Japan throughNo.19104006 (through Katsuhiko Sato).
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