Gustavo Medina Tanco

Instituto Astronômico e Geofísico - Universidade de São Paulo, Brasil

Department of Physics and Astronomy, University of Leeds, United Kingdom

gustavo@iagusp.usp.br

Ultra high energy cosmic rays (UHECR), have challenged our imagination for the last four decades. They pose in some respects the same kind of questions that gamma ray bursts (GRB) do. They are relatively rare events of, very likely, extragalactic origin with no certain optical counterpart, poor angular determination, undetermined distance scale to the sources and unknown powering mechanism. To complicate things further, depending on the energy threshold we use to define UHECR, the detection rate can be as low as 1 event per square kilometer per century; this means 1 event per year at the largest experiment running at present (AGASA). Even the nature of the primary is uncertain.

So, what do we know? Although other particles cannot be disregarded, the ratio of muon density to charge particle density in the observed showers seems to point to a hadronic primary; protons, in particular, seem to be favored (Gaisser 1993). This information alone immediately sets an upper limit to the distance-scale to the sources: photo-pion production via cosmic microwave background radiation (CMBR) interactions will hinder proton propagation farther than some few tens of Mpc (e.g., Berezinsky and Grigor'eva 1988, Aharonian and Cronin 1994). In fact, one way of defining UHECR is by using the energy threshold for this interaction (» 3 ´ 10¹⁹ eV) as a lower limit. The photo-pion interaction with the CMBR should produce a bump followed by a cut-off at the highest energy end of the cosmic ray spectrum. This feature is known as the GZK cut-off (Greisen 1966; Zatsepin and Kuz’min 1966), and its exact position depends on the characteristic distance and distribution of the sources (e.g., Berezinsky and Grigor'eva 1988, Yoshida and Teshima 1993, Medina Tanco 1998c). (The GZK cut-off is a very solid prediction, and should hold as long as Lorentz symmetry is not violated at relativistic factors g ³ 10¹¹ – Gonzalez-Mestres 1997, 1998.)

Furthermore, charged particles interact with the magnetic field that permeates the propagation region, hampering the location of the sources. Unfortunately, relatively few is known about the galactic magnetic field (GMF) structure in the Milky Way, specially as we go into the Halo (Vallée 1997). Our knowledge of the intergalactic magnetic field (IGMF) is even more speculative and observationally poor, limited mostly to upper limits and few punctual determinations (Arp 1988, Kim et al. 1989, Kronberg 1994, 1996, Vallée 1997). However, commonly adopted fiducial values for the GMF and IGMF are B_gal » 10^-6 G and B_IGM » 10^-9 G respectively. Therefore, at 10¹⁹ eV protons have a gyroradius of » 10 kpc in the GMF, which is large enough compared to the thickness of the galactic disk to forbid confinement inside the galaxy, but still large enough as to produce a considerable deflection of the incoming particle. The particles are probably extragalactic, but any information on the location of individual sources is lost. However, at 10²⁰ eV protons have a gyroradius of » 10² kpc in the GMF, i.e., » 10³ times the thickness of the galactic disk. Furthermore, the gyroradius of a 10²⁰ eV proton in the IGMF is » 10² Mpc, i.e., larger than the probable distance to the sources. The latter means that at the highest energies observed, the charged particles are not only very likely extragalactic, but they also point in principle to their sources (Medina Tanco et al. 1997). This opens the remarkable possibility of doing an "UHECR astronomy", with charged particles in a similar way as it is done with photons. A peculiar kind of astronomy, though, as the images of the sources should be strongly coupled with the intervening magnetic field. UHECR represent, in this sense, a potentially powerful tool for the mapping of the GMF in the obscured central regions of our galaxy and the weak and elusive IGMF.

In the following sections we show what should be expected if the sources of the particles are distributed in the same way as the luminous matter in the nearby universe (Medina Tanco 1998c), and analyze the effects of the GMF (Medina Tanco 1997a, Medina Tanco et al. 1998) and different configurations of the IGMF on UHECR propagation (Medina Tanco 1997b, 1998b). Finally, we also study what constrains, if any, can be imposed on the source location problem by the detection of possibly correlated cluster (pairs) of events, suggested by AGASA data (Hayashida et al. 1996, Medina Tanco 1997c, 1998a).

While propagating from the source to the detector, UHECR particles have to cross several different regions from the point of view of the magnetic field: the immediate environment of the source, the intergalactic medium, the galactic halo, the galactic disk, the heliosphere and the magnetosphere. The first region is completely unknown, but mainly irrelevant to us here – one can always redefine a source large enough to enclose any "immediate environment". The last two regions have much larger magnetic fields than the GMF or IGMF, but their size is relatively small and so the deflection they produce is negligible. They can be important, however, as sources of differential deflection which may have relevance for composition determination (e.g., Medina Tanco and Watson 1998, A. Cillis, S. J. Sciutto 1997). Therefore, from the astronomical point of view, only the GMF and IGMF matter.

The magnetic field inside the galactic disk, as far as Faraday rotation measures indicate (see Kronberg 1998 and references therein), appears organized on a grand scale with, possibly, field reversals. There is, however, a random component comparable in intensity to ordered field. Few can be said about the symmetry of the GMF by looking to our own galaxy. Nevertheless, for the more than 20 spiral galaxies with measured large scale magnetic fields (Beck et al 1996), the field usually follows the spiral arms and there is also evidence for either an axisymmetric or a bisymmetric pattern, and even for a combination of both. The GMF seems to be compatible with these topologies but, at the present stage, choosing between one model and another is just a working hypothesis.

Measurements of the galactic magnetic halo are also very difficult, but they suggest a scale height of » 4 kpc and an intensity of » 0.1 m G at and outside the solar circle (Kronberg 1998). Comparable or larger scale heights are observed in a few nearby edge-on spiral galaxies (Hummel, Beck and Dahlem 1991).

Therefore, to described the regular large scale component of the GMF, we adopt below the axisymmetric model as Stanev (1997), including a component perpendicular to the galactic plane, plus a superimposed random component of variable correlation length L_c, and amplitude dB/B » 1 – Abramenkov and Krymkin 1990):

where the correlation length is taken as L_c = 100 pc in the Sun’s vicinity, and scales throughout the galaxy according to L_c(r) µ [B(r)]^-2.

The principle of reversibility is applied to the propagation of protons through the GMF. Approximately 10⁴ antiprotons are uniformly injected at Earth with momentum ‘-p’ over 4p sr, and their trajectories are integrated until the interface between halo and IGM, represented by a spherical galactocentric surface of radius R_H = 20 kpc, is reached. This schematically represented in figure 1. In this way, the arrival directions of the UHECR onto the border of the galactic halo (which, if the IGMF is neglected, point to the true location of the sources) can be mapped onto the celestial sphere seen by the observer. Two maps can be constructed in this way, one with the actual position of the sources on the sky and another with the apparent position of their images in the sky seen by the detector due to GMF deflection (for further details and several examples see Medina Tanco 1997a, Medina Tanco et al. 1998).

In figure 2a-b we show the resulting source location error boxes as a function of galactic coordinates for incoming protons, with energies 4 ´ 10¹⁹ and 10²⁰ eV respectively, due to the GMF. It can be seen that the effects are strongly dependent on the direction on the sky and the particle's arrival energy, ranging from almost negligible at the galactic polar regions to severe towards the galactic center. Consequently, identification of extragalactic sources near the inner galactic plane and bulge should be much hampered unless protons of > 10²⁰ eV are used. On the other hand, this relatively large deflections mean that, once a source or distribution of sources is identified, UHECR could be used map the presently poorly known GMF in the central regions of our galaxy. Probably, even an unexpectedly high magnetic field in the halo (Hillas 1998) should produce a recognizable signature on top of a distribution of sources related with an independently known spatial distribution (like the distribution of luminous matter in the nearby Universe).

The effects are substantially larger if UHECR comprise heavier nuclei (Medina Tanco 1997a).

The main problem lies in our lack of knowledge of the IGMF, with the exception of some few observational determinations and upper limits (e.g., Arp 1988, Kim et al. 1989, Kronberg 1994) and numerical simulations of cosmological structure formation (Biermann 1996 and references there in). It seems, however, that fields » 0.1 m G are generally associated with the presence of intergalactic hot gas and galaxies (Kronberg, 1998), whereas field in the general intergalactic medium and in the interior of voids remains basically undetected. On the other hand, evidence from rotation measure in galaxy clusters suggests that the largest reversal scale is L_c » 1 Mpc. This, together with upper limits in rotation measure out to z » 2.5 indicates an upper limit for the IGMF, B_IGMF £ 10^-9 G (Kronberg, 1994). The latter result is also compatible with Vallée's (1990) determination of B_IGMF < 10^-9 G for a regular cosmic magnetic field (outside clusters of galaxies) and mean particle density 10^-7 cm^-3.

Furthermore, for those spatial scales where measurements are available, the intensity of astrophysical magnetic fields seems to correlate remarkably well with the density of thermal gas in the medium. Figure 3 (an adaptation of figure 1 of Vallée 1997) shows that we have probably two regimes: one at small ( £ galactic) scales, and other at large (extragalactic) scales. It is apparent that B can be reasonably well fitted by a single power law over » 14 orders of magnitude in thermal gas density at sub-galactic scales. Besides, a correlation is also suggested at very large scales from, galactic halos to the environments outside galactic clusters, over » 4 orders of magnitude in thermal gas density.

Thus, the observational clues/constrains given heretofore point to a possible picture in which the IGMF correlates with the distribution of matter as traced, for example, by the distribution of galaxies. A high degree of non-homogeneity should be expected, with relatively high values of B_IGMF over small regions (» 1 Mpc) of high matter density (c.f., Arp’s 1988 determination of B_IGMF » 3x10^-7 G for the Virgo cluster or Kim’s et al. 1989 B_IGMF » 10^-6 G for the Coma cluster). These systems should be immersed in vast low density/low B_IGMF regions with B_IGMF < 10^-9 G. Furthermore, from rotation measure, the topology of the field should be such that it is structured coherently on scales of the order of the correlation length L_c which, in turn, scales with IGMF intensity: L_c µ B_IGMF^-2(r). B_IGMF vectors should be independently oriented at distances > L_c. Therefore, a scenario can be built in which the IGMF presents a cell-like spatial structure, with cell size given by the correlation length, L_c; and such that: L_c µ B_IGMF^-2(r) and B_IGMF µ r _gal^0.35(r), where r _gal is the galaxy density, and the IGMF is uniform inside cells of size L_c and randomly oriented with respect to adjacent cells (Medina Tanco 1997b, Medina Tanco et al 1997). The observed IGMF value at a given point, like the Virgo cluster (» 10^-7 G, Arp 1988), can be used as the normalization condition for the magnetic field intensity.

The density of galaxies, r _gal, is estimated using the latest release (version of Jul 27, 1998) of the CfA Redshift Catalogue (Huchra et al 1992). Figure 4 shows an Aitoff (equal area) projection of the galaxy distribution up to a depth of 5100 km/sec (equivalent to » 100 Mpc for H=50 km/sec/Mpc).

The spatial distribution of the sources of UHECR is tightly linked to the nature of the main acceleration mechanism involved.

Two kinds of acceleration mechanisms can be envisaged for the generation of UHECR: bottom-up and top-down mechanisms. Bottom-up mechanisms consist in the acceleration particles injected at lower energies. Some examples are: particle acceleration in the accretion flows of cosmological structures (e.g., Norman, Melrose and Atcherberg 1995, Kang, Ryu and Jones 1996), galaxy collisions (Cesarsky and Ptuskin 1993, Al-Dargazelli et al. 1997, but see Jones 1998), galactic wind shocks (Jokipii and Morfill 1987), pulsars (Hillas 1984, Shemi 1995), active galactic nuclei (Biermann and Streitmatter 1987), powerful radio galaxies (Rawlings and Saunders 1991, Biermann 1998), gamma ray bursts (Vietri 1995, 1998, Waxmann 1995, but see Stanev, Schaefer and Watson 1996), etc.. Top-down mechanisms, already form the particles at high energies and simply cascade down their energy up to the observed values, avoiding in that way severe problem like energy losses during acceleration. Some examples are: the decay of topological defects into super-heavy gauge and Higgs bosons, which then decay into high energy neutrinos, gamma rays and nucleons with energies up to the GUT scale (» 10²⁵ eV) (e.g., Bhattacharge, Hill and Schramm 1992, Sigl, Schramm and Bhattacharge 1994, Berezinsky, Kachelrie and Vilenkin 1997, Berezinsky 1998), high energy neutrino annihilation on relic neutrinos (Waxmann 1998), etc.

In the case of bottom-up mechanisms the sources of the particles should be related to the distribution of luminous matter in the Universe. For top-down mechanisms, on the other hand, the distribution of the sources may or may not be related to the distribution of luminous matter. However, an isotropic distribution of sources is expected in most of the models.

In what follows, we consider a distribution of sources that closely resembles the distribution of luminous matter in the nearby universe, as represented by the observed r _gal, and assume that the scenario described above is an acceptable representation of the IGMF up to a distance 100 Mpc from our galaxy. Protons are injected at the sources (one source at the location of each galaxy) with a power law injection spectrum, dN_inj/dE µ E^-n , with n = 3 above a threshold of 4 ´ 10¹⁹ eV. Each source is a standard candle, i.e., the luminosity in UHECR, L_UHECR =const. The 3D trajectories of the particles are calculated from the sources to the detector, taking into account adiabatic energy losses due to redshift, pair production and photo-pion production due to interactions with the CMBR. The interaction mean free times [(1/E)dE/dt]^-1 as a function of energy calculated by Berezinsky and Grigor'eva (1988), are used to determine, randomly, the interaction of protons with CMBR photons during the propagation and the subsequent evolution of UHECR particle energy.

Figure 5 shows an all sky Aitoff projection, in galactic coordinates with the galactic anticenter at the coordinate origin, of the arrival probability distribution of UHECR protons with energy E_arrival > 4 ´ 10¹⁹ eV. The available data on UHECR for the same energy interval is also displayed in the same figure for the experiments of AGASA, Haverah Park, Yakutsk and Volcano Ranch. The thick strip at zero galactic latitude covers the position of the galactic plane in the figure. Obscuration by dust in this region severely hinders the observation of galaxies, and so our approximation does not apply.

In figure 6, the median of the deflection (defined as the angle between the arrival direction of a particle and the true direction to its source) as a function of energy is shown, together with 63% and 95% confidence levels. It can be seen that, despite de fact that the median decreases strongly with increasing energy, there is considerable dispersion and an appreciable fraction of particles can arrive with large angular deflections (> 20° ) at 10²⁰ eV.

At the present low level of statistics, it is difficult to say something conclusive about any possible correlation; however, the angular distribution of observed events seems very much isotropic, in contrast to the inferred arrival probability. It is premature to say whether this is telling something about the distribution of the sources or the structure of the IGMF (c.f., Hillas 1998).

Undoubtedly, if a point-like UHECR source exist (i.e., spatially confined to less than the angular resolution of the current experiments, » 1° ) at a distance of no more than a few photo-interaction mean-free paths, it will eventually emerge observationally as the accumulation of events in a relatively small solid angle. How small this solid angle should be to be statistically significant is still open to debate. However, it is very exciting that some few "hot spot" may be already appearing in the data, as reported some time ago by the AGASA collaboration (Hayashida et al., 1996). Table I reproduce the three published pairs of events. Actually, there may be as much as 8 pairs of events and at least two triples if data from other experiments are combined. Nevertheless, based on past experiences of a long history of possible GRB repeaters (e.g., Fishman 1996, Meegan et al. 1995, Hartmann et al. 1995, Brainerd et al., 1995), caution must be exercised when associating two events only by their angular proximity on the sky, specially when the data is so scarce.

If UHECR are charged particles, protons as it is more likely, and the components of the pairs have a common origin, then the observed clustering impose severe constraints on the characteristics of the propagation region and/or their sources (e.g., Cronin 1996, Sigl et al 1996). Catastrophic extragalactic events, like GRB or the decay of topological defects (TD), which are able to produce the particles over a very short period of time, should only be consistent with the data for a suitable combination of low intergalactic magnetic field (IGMF) and distance to the source. Nevertheless, the stirring of the intergalactic medium (IGM) by large agglomerates of galaxies, shocks excited in binary collisions of galaxies or the bow shocks preceding fast moving galaxies in dense environments are examples of quiescent sources that could produce chance pairings of UHECR events on the sky. If these quiescent sources are traced by the distribution of luminous matter in the nearby universe, then the probability of the corresponding chance pairing can be estimated and compared with the observations.

In order to analyze the pairs, the propagation problem can be divided into two different calculations: one involving the IGMF, from the source to the border of the galactic halo and the other the GMF, through the galactic halo and galactic plane to detector (see Medina Tanco 1998a for details).

First, the trajectories of the individual particles through the GMF are calculated using the model described above (Medina Tanco et al. 1998, Medina Tanco 1997a). If the particles are simultaneously released at the source, this constrains the amount of time delay due to intergalactic propagation alone and, consequently, the range of IGMF values and source distances allowed. The separation angle between the momenta of the particles at their arrival at the border of the halo, q_halo ,can also be estimated. The latter can be seen as a matching condition to be met by the particle trajectories at the border of the halo.

Second, the same numerical scheme of Medina Tanco et al (1997) is used to estimate the arrival relative-deflection distribution function for some allowed combinations of IGMF and distance to the source. The comparison of this distribution function with the previously calculated q_halo , gives a quantitative idea of the likelihood of the observed events being the result of point-like sources.

Pair 1 is the only one in which the high energy event arrives first. Therefore, it could be produced, in principle, in a burst. Figure 7.a shows the distribution function of arrival time delays, D t_arr, at the border of the halo for protons of the observed energies. It is assumed that the particles originated with a power law energy spectrum at the source, dN_inj/dE µ E^-2. Two possible combinations of IGM and source distance are shown which would be able to reproduce the observed D t_arr » 1.3 yr at the border of the halo.

In figure 7.b, the distribution functions of the angle between the momenta of the pair of paricles at the border of the halo, q _pp, are shown for the same cases of figures 7.a. It can be seen that if both, D t_arr and q _pp are to be satisfied simultaneously, then a very rear event was observed.

On the other hand, in pairs 2 and 3 the lower energy particle arrives first. This means that a point source cannot have emitted both UHECR simultaneously. Therefore, if the point source hypothesis is to be maintained, we must assume either that the source is quiescent or, if bursting, that a finite acceleration time is involved which delays the emission of the high energy component. In this case, the sum of the arrival time delay, the time delay due to the propagation through the galactic disc and halo, and the time delay due to intergalactic portion of the trajectories, is a lower limit to the lifetime of the source. Again, the galactic and intergalactic trajectories must verify the matching of q_HALO at the border of the galactic halo. Applying the same procedure as for pair 1, it is found that q_HALO(pair 2) » 2°and q_HALO(pair 3) » 2°-5.5°, depending on the model adopted for the galactic magnetic field. Numerical simulations for the IGM propagation of the proton components of pairs 2 and 3 are also shown in figures 7.a and 7.b. A distance of 30 Mpc to the source and two different values of the IGMF, B_IGM = 10^-12 and 10^-9 Gauss, were used. The lower value of the IGMF is the one imposed by a bursting pair 1, and the second is the current upper limit for the IGMF. It can be seen from figures 7.a and 7.b that, as for pair 1, the assumption of a 10^-12 Gauss IGMF leads to a very low probability for an event with q_HALO on the order of a few degrees. Taking into account the galactic propagation, the lower limits for the lifetime of single sources for pairs 2 and 3, with B_IGM » 10^-12 Gauss, are »10 and »100 yr respectively. On the other hand, from the point of view of q_HALO, a consistent picture can be obtained for a higher value of the IGMF, say near 10^-9 Gauss. However, the lower limit for the lifetime of a single source scales up to few times 10⁵ yr and so the source should be quiescent. Furthermore, the sources should probably extend over a large volume of space, perhaps enclosing more than one galaxy, in order to be able to confine » 10²⁰ eV particles.

The previous results seem to point to a chance clustering of the three proposed pairs of events. However, the problem remains that the chance probability for the pairs quoted by Hayashida et al (1996) is only 2.9%. We note, however, that this chance probability was derived under the assumption that the arrival direction distribution is uniform over the sky, which is arguable.

In fact, under the assumption that the sources are distributed following r _gal , the arrival probability is by no means isotropic. A clearer picture can be obtained if maps similar to figure 5 are produced for sources at different distance intervals (Medina Tanco 1998a). It is possible to see then that pair 2 is on top of a maximum of the arrival probability for sources located between 20 and 50 Mpc, while pair 1 is also located on a high arrival probability region for sources at more than 50 Mpc. This is in contrast with the chance probability estimated by Hayashida et al. (1996), and points to either different non-correlated sources of the components of each pair, or to very extended quiescent sources involving several galaxies or clusters.

The third pair, on the other hand, comes from a region of space where no large clustering of galaxies exist up to the depths considered. As the components cannot have originated simultaneously at the same extragalactic source because of galactic propagation constraints, they must have come from isolated sources. This could be interpreted as an indicative that very large agglomerates of galaxies working co-operatively in the acceleration process are not needed in order to accelerate UHECR.

UHECR are obviously an extremely interesting object of research in itself. They are also, undoubtedly, a promising tool for probing the GMF and the IGMF. They can be especially powerful in helping to understand the large scale of the magnetic field in the central regions of our own galaxy and in the galactic halo. Furthermore, they can be the only way to measure the average intensity and to study the spatial structure of the IGMF inside walls and voids at scales of tens of Mpc. Unfortunately, the available data at present seems insufficient to reach any definite conclusion about the UHECR origin. More data is badly needed, and just some few more years at the present detection rate will not do. The more attractive applications will require a change of orders of magnitude in our capacity of gathering data. Hopefully, the Southern site of the Auger experiment should be operative in few years time, and OWL is on the run for the next century (Streitmatter, 1998).

Finally, it must be stressed that all the previous analysis must be severely compromised if the IGMF were coherently structured on large scales (» 10-20 Mpc) with an IGMF on the order of » m G inside cosmological walls and almost negligible in the surrounding voids (see Medina Tanco 1998b) as recently proposed by Ryu, Kang and Biermann (1998).

Acknowledgments: I wish to thank the kind hospitality of the cosmic rays group at the University of Leeds (UK), and to Prof. A. A. Watson so many discussions and valuable comments.

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