LIFETIME OF A COSMIC-RAY SPOT

Understanding the cosmic-ray (CR) transport in the Galactic magnetic field (GMF) is fundamental to tracing their origin. Any source of CRs (Galactic or extragalactic) will be blurred by the GMF to some degree; whether there would be direction–time correlations of any sort in the arrival of the CR depends on the structure and strength of the GMF. In the limit in which CR arriving from a source have traveled much less than one gyroradius, CR arrival directions correspond to the direction of their source in the sky, but it is not obvious that any CR or their sources satisfy this condition. In the opposite limit, all information about the direction and point-like nature of the source may be lost, but not necessarily.³ If most CR sources lie in the latter limit, one would predict an isotropic sky (except possibly anisotropy caused by anisotropy in source density). However, for some CR energies, neither limit is applicable, and the interesting question may be asked of whether the arrival statistics are truly Gaussian or whether there is a correlation between the arrival times and/or directions of CR arriving from a distant point source. A CR hotspot from a certain direction in the sky, for example, may be the result of their origin from a point source even if that source is in a different direction. However, this hotspot may move around the sky on a timescale short compared to the duration of the time bins of the observation (which are limited from below by the requirement of accumulating a statistically viable number of CR) in which case it would be highly blurred and likely go undetected or be highly controversial. Moreover, dispersion in flight times between the source and Earth can blur different transient sources in time so that, even if they are active at different epochs, they overlap inseparably in time during the period of observation.

In this Letter, therefore, we address the following questions: for what sources are CR hotspots possible and what would be the angular width and lifetime of such a hotspot?

To answer these questions, one needs to quantify the angular displacement of the CRs produced by a change of phase of the MHD disturbances in which the CRs propagate and through which the Earth moves.

In the following, we define by $N(E)={\rm{\Delta }}{EdN}(E)/{dE}\equiv$ $[{\rm{\Delta }}E/E]{{dU}}_{\mathrm{CR}}(E)/{dE}\,\simeq$ $[{\rm{\Delta }}E/E]{U}_{\mathrm{CR}}(E)/E$ the number of CRs at some energy E made by the same source, where ${{dU}}_{\mathrm{CR}}(E)/{dE}$ is the total output energy of the source in ultrahigh-energy cosmic rays (UHECRs) per unit particle energy interval of the source, U_CR(E) is the total energy output in UHECRs of energy less than E, and ${\rm{\Delta }}E$ is the width of the energy bin over which the location of the hotspot is more or less in the same region of the sky. If the deflection at a given value of rigidity⁴ ρ is much smaller than the measured hotspot itself, then the size is presumably determined by the variation in rigidity of the trajectories and ${\rm{\Delta }}E$ can be of order E. However, over a distance comparable to or greater than a gyroradius, a small change in rigidity can cause a large change in the directions from which CR from a given source will arrive. When we consider UHECRs in the rigidity range ∼5–10 EV, the gyroradius is of the order of a kiloparsec (assuming that the value of the GMF in the vicinity of the solar system is 6 μG, according to Beck 2008), meaning that Galactic sources of UHECRs could be at distances of order several gyroradii. A situation could therefore arise where a real hotspot could be obtained, but that its position would be highly sensitive to rigidity. Similarly, if the CR composition is dominated by two different ion species at a given energy, then there might be two different hotspots. The prospect of measuring such effects is highly motivated because it would be a very powerful method for diagnosing the GMF. To establish whether or not this is possible, however, requires detailed calculations and will in any case depend on the exact location of the source.

Consider an extragalactic transient source that puts out a total energy U_CR(E) in UHECRs of energy E. The presence of any correlation between different UHECR events would require detecting more than one CR from the same source. This condition can be expressed as

$$\begin{eqnarray}N(E) & \geqslant & [{\rm{\Delta }}E/E][4\pi {D}^{2}({\rm{\Delta }}{\rm{\Omega }}/4\pi )/A][{t}_{d}/{\rm{\Delta }}t]\\ & = & 16{\pi }^{2}[{\rm{\Delta }}E/E]({\rm{\Delta }}{\rm{\Omega }}/4\pi )\,{t}_{d}{D}^{2}/{ \mathcal E },\end{eqnarray} \tag{ 1 }$$

where ${\rm{\Delta }}t$ is the period of observation, typically 5–10 years for recent CR arrays (Auger, TA), t_d is the dispersion of the flight times of the CR propagating from the source to Earth, A is the time average detector area, D is the distance to the source, and ${ \mathcal E }$ is the exposure of the detector. ${\rm{\Delta }}{\rm{\Omega }}$ is the solid angle in steradians subtended by the beam of CR emerging from the source when it finally encounters the Earth and can be written as $4\pi \cdot {U}_{\mathrm{CR}}/{U}_{\mathrm{iso}}=4\pi f$.⁵ The time dispersion introduced by its propagation from the periphery of the Galaxy to Earth, for a particle of rigidity 10¹⁹ V is typically of the order of 10⁴ years (Globus et al. 2016). Extragalactic propagation, of course, could introduce additional dispersion. For TA, the exposure is ${ \mathcal E }=8600\,{\mathrm{km}}^{2}\,\mathrm{sr}\,\mathrm{year}$ as of 2015 May 11 (Matthews 2016). So, for a transient source of CRs at rigidity 10¹⁹ V and charge Z at 100 D₁₀₀ Mpc to create a hotspot of 20n₂₀ CR events at energy ${10}^{19}Z$ eV, the isotropic equivalent energy would need to be ${U}_{\mathrm{iso}}\geqslant 5.6\cdot {10}^{53}Z({t}_{d}/{10}^{4}\,\mathrm{year}){D}_{100}^{2}{n}_{20}$ erg.

If the extragalactic transient source is a gamma-ray burst (GRB) with a total isotropic equivalent energy of ${10}^{51}{U}_{51}$ erg, it would have to occur within $\sim 4.2\,{U}_{51}^{1/2}{({t}_{d}/{10}^{4}\mathrm{year})}^{-1/2}{n}_{20}^{-1/2}{Z}^{-1/2}$ Mpc of Earth. Now, unless the extragalactic magnetic field B is small enough that it induces UHECR deflections and time delays significantly smaller than those in the GMF (i.e., $B\lt 0.3$ nG; Globus et al. 2016) so that ${U}_{\mathrm{iso}}\gg {10}^{51}$ erg and t_d are not raised to much more than 10⁴ years, this distance is less than the GZK length (including pair production and photodissociation processes if they are relevant). So the question would arise as to how much that one nearby source would stand out over other more numerous, distant sources (as per Olber's paradox). This question has been analyzed recently by Globus et al. (2016). If there is an extragalactic magnetic field B that limits the UHECR magnetic horizon⁶ to much less than the GZK distance (e.g., $B\gt 30$ nG), such a magnetic field would raise t_d to at least D/c, so that the source would have to occur at a distance that obeys $D\leqslant 4{U}_{51}/{(D/{10}^{4}{\rm{l}}\mathrm{year})}^{1/2}{n}_{20}^{1/2}{Z}^{1/2}$ Mpc, implying that $(D/{10}^{4}\,{\rm{l}}\,\mathrm{year})\,\leqslant {U}_{51}^{2/3}\cdot {10}^{2}$ or $D\lesssim 300{U}_{51}^{2/3}$ kpc. This would put a GRB well within the local group, and probably within our own Galaxy.

If the extragalactic source is steady at luminosity L, then the ratio ${t}_{d}/{\rm{\Delta }}t$ should be replaced with unity and the total energy should be set to ${U}_{\mathrm{iso}}=L{\rm{\Delta }}t$. So for an observational interval of 5 years, the time-averaged luminosity needed to provide a hotspot of 20n₂₀ events is $3.6\cdot {10}^{45}{D}_{100}^{2}{n}_{20}$ erg s⁻¹.

By contrast, if the source of the UHECR is a Galactic GRB, then the smaller distances allow much "hotter" spots than extragalactic ones. For example, if a Galactic GRB puts out ${10}^{49}{U}_{i,49}$ erg in UHECRs of species i, at $E\sim {10}^{19}$ eV, at a distance of 10 kpc, Equation (1) implies that the TA would detect up to $\sim {10}^{4}{U}_{i,49}$ in ions of species i. Now it could be the case that for the most recent Galactic GRBs, the fireball jet was directed away from our line of sight and 10¹⁹ eV protons escaped with small deflection, with a disproportionately few ever reaching the Earth, while heavier nuclei were trapped by the GMF for much longer times and scattered in the Earth's direction. In this case, the expected number of CR counts in the hotspot could be much less. While the event rate of observed GRBs is of the order of 1–3 Gpc⁻³ yr⁻¹(Wanderman & Piran 2010), i.e., one per ${10}^{6.5}$ to $\sim {10}^{7}$ Galaxies per year, the true rate, including the majority directed away from the Earth, could be much higher (Guetta & Eichler 2010). Thus, the true rate of GRBs in our own Galaxy could be much less than the lifetime of heavy ion UHECRs in our Galaxy, and there could be many Galactic GRBs contributing to Galactic UHECRs at a given time. If there are, say, N such GRBs distributed in the Galactic plane more or less uniformly, then the "closest" one, i.e., the one contributing most to the anisotropy, would likely contribute on the order of $\sim 1/\mathrm{ln}N$ of the total count rate. This argument is meant to establish that hotspots could comprise only a small fraction of the total UHECRs in a given energy bin.

If there are enough counts to make a hotspot, this is not yet a sufficient condition to observe one. For if the trajectories are deflected enough that they distribute their arrivals all over the observer's sky, they would not be identifiable as having all come from the same source. Yet another condition is that, over the observation period, the apparent position of the hotspot not wander too much beyond its own radius, otherwise the time integrated signal would smear out the hotspot. In this Letter, we concern ourselves with the question of how much a hotspot would wander during a 5 year observational interval.

We use a representative model of the GMF by Jansson & Farrar (2012, hereafter JF12). The large-scale coherent field is the sum of three distinct components: (i) the disk field, which is concentrated in the plane and closely follows the spiral arms of the Galaxy; (ii) the toroidal halo field, which decreases exponentially with scale height, and takes different amplitudes (with different signs) below and above the plane; and (iii) the out-of-plane halo field (see JF12 for details). We add a purely turbulent component following the numerical procedure of Giacalone & Jokipii (1999), assuming a Kolmogorov-like turbulence. The intensity of the turbulent field is 3 times the magnitude of the regular component, and its coherent length is ∼200 pc (Beck et al. 2016). This is a comparable or slightly larger coherence length than used in earlier papers (e.g., Giacinti et al. 2012; Abreu et al. 2013; Keivani et al. 2015). This model also includes an anisotropic turbulent component originating from isotropic turbulent fields by the action of shear. It should be added that the turbulence need not be Kolmogorov and that a correlation length 200 pc is not enough to scatter 10 EeV particles resonantly, so the results are sensitive to assumptions about the magnetic turbulence at somewhat higher length scales.

We use right-handed Cartesian (x, y, z) coordinate system, where the Galactic center (GC) is at the origin, Galactic North is in the positive z-direction, and the Sun is located at x = −8.5 kpc. The method of antiparticle tracing (Thielheim & Langhoff 1968) is applied, using a numerical tool developed by Rouillé d'Orfeuil et al. (2014). We represent a CR spot by a set of 121 antiprotons, with the initial directions of the velocity vectors (pointing to the Galactic coordinates l, b) are uniformly distributed on a grid of 3° square. The trajectories are calculated by integrating numerically the equations of motion of the antiparticles in the GMF, with a precision of 10⁻¹⁰. The antiparticle starts its propagation on the position of the Earth in the Galaxy. At a given CR rigidity ρ, the direction and time coordinates $(\theta ,\phi ,t)$ each map to a prior coordinate $r(\theta ,\phi ,t,{t}^{\prime },\rho )$, ${\theta }^{\prime }(\theta ,\phi ,t,{t}^{\prime },\rho )$, ${\phi }^{\prime }(\theta ,\phi ,t,{t}^{\prime },\rho )$ at an earlier time ${t}^{\prime }$. A necessary and sufficient condition for a CR arriving at Earth from direction $\theta ,\phi$ at time t is that there is a CR source at $r(\theta ,\phi ,t,{t}^{\prime },\rho )$, $\theta (\theta ,\phi ,t,{t}^{\prime },\rho )$, $\phi (\theta ,\phi ,t,{t}^{\prime },\rho )$. Directional anisotropy observed at any time t from Earth results from there being more source strength at coordinates $r(\theta ,\phi ,t,{t}^{\prime },\rho )$, ${\theta }^{\prime }(\theta ,\phi ,t,{t}^{\prime },\rho )$, ${\phi }^{\prime }(\theta ,\phi ,t,{t}^{\prime },\rho )$ for some directions than for others, as illustrated in Figure 1, where CR trajectories are shown. Here, the spot of 121 CRs is back-propagated into two different directions: the GC and the anticenter. The CR have all the same rigidity, 3 EV (left panel), or 10 EV (right panel). These calculations were done assuming two different configurations of the turbulence, figured in orange and blue, respectively.

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Let us determine the width of the energy bin over which the barycenter of the hotspot is more or less in the same region of the sky. At a given energy time t, the angular decorrelation due to a change in energy $\delta E$ is defined by the quantity $\delta \theta (t,\delta E)={\cos }^{-1}[{\boldsymbol{v}}(t,E)\cdot {\boldsymbol{v}}(t,E+\delta E)]$, where ${\boldsymbol{v}}(t,E)$ is the velocity vector of a CR of energy E at time t. The mean value of this angular decorrelation as a function of propagation distance ct, is shown in Figure 2, for $\delta E/E=0.1$ (dotted lines), and $\delta E/E=0.02$ (dashed lines). The spread is figured by the shaded areas. Here, we back-propagated the 121 CRs in the Galactic plane (b = 0°) and in 4 different directions: l = 0° (GC), 60°, 90°, 180° (anticenter). The above exercise demonstrates that 10 EV UHECRs arriving simultaneously from a given 3°  × 3° region of the sky ("hotspot") could have come from a single source located at ∼3 kpc regardless of the direction if their difference in energy (or in the case of multi-species ions, rigidity) is less than 10%. In the anticenter direction there is no limit for the source distance, as the angular decorrelation is $\delta \theta \lesssim 1$°.

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As seen in Figure 1, two different configurations of the magnetic turbulence could lead to different positions of the CR source in the observer sky. Now, we know that our Galactic environment changes with time due to the motion of the solar system through space. The Local Standard of Rest solar velocity is 13–19 km s⁻¹ (Frisch & Slavin 2006). The velocity of the ISM flow into the heliosphere measured by Ulysses and IBEX experiments is 23–26 km s⁻¹ (Gry & Jenkins 2014). At a velocity of 20 km s⁻¹, which is the value we adopt, the solar system traverses ∼21 au in approximately 5 years, and thus potential variations in the parameters of the magnetic turbulence on that observation time should affect the spot image. To quantify these variations, we compared the back-propagated trajectories of the 121 antiprotons before and after changing the position of the Earth by 21 au. At a given energy E, the angular decorrelation due to a change in Earth's position in an interval time $\delta t$ is $\delta \theta (\delta t,E)={\cos }^{-1}[{\boldsymbol{v}}(t,E)\cdot {\boldsymbol{v}}(t+\delta t,E)]$, and its value for $\delta t=5\,\mathrm{year}$ as a function of the propagation time is shown in Figure 3. We see that the motion of the Earth through the GMF may cause one region of the observer's sky to be occupied by different source directions at different times. Because the total number of steradians in the sky is a constant, we may confidently conclude that the previous source direction "occupying" that sky direction moved to a different sky direction and that the CR signal produced by a sufficiently distant source moves around in the observer's sky. We see that the angular decorrelation of 10 EV CRs from point sources starts to be significant only for propagation distances greater than 10 kpc, with $\delta \theta \sim 15$° for $l=0^\circ$ (GC). We found no angular decorrelation for $| l| \geqslant 90$°, as well as in the direction of the TA hotspot ($b\sim 49^\circ$, $l\sim 177^\circ$) (Abbasi et al. 2014). We predict $\delta \theta \sim 2$° from Cen A ($b\sim 20^\circ$, $l\sim 310^\circ$) should a hotspot be reported in that direction. However, there is no significant evidence for the existence of correlated multiplets in the Auger data (Abreu et al. 2012), and the largest departure from isotropy in the Cen A direction was found to have a post-trial probability of ∼1.4% (Aab et al. 2015a). Keivani et al. (2015) showed that the arrival direction distribution of any given 10 EV CRs from Cen A may change from one realization to another of the Galactic turbulent field, but the difference appears to be less than the overall spread.

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We have shown that changing the rigidity of a UHECR from a given source changes the expected arrival direction, as does motion of the solar system through the ISM. For a realistic energy bin, typically at least 10% or so variation in energy, the uncertainty in arrival direction for a 10 EV CR can be as high as 40° if it arrives from the direction of the GC and has propagated more than 3 kpc. From the anticenter, on the other hand, a 10% spread in rigidity leads to a directional uncertainty of only of order 1° regardless of propagation distance. The motion of the Earth over 5 years causes changes in arrival direction that can be as large as 20° from the direction of the GC, but the change is found to always be less than the width caused by a 10% spread in rigidity. We conclude that

(a) Hotspots, if they are found to exist at UHECR energies, should be stable to within their instantaneous width, over a 5 year observation period (unless the detector's energy bins can be made nearly monoenergetic). The hotspot that the observer detects probably integrates over all the individual positions of the source, but, in principle, if enough UHECRs are detected, there could be time–position correlations within the hotspot.

(b) The anticenter direction is much better connected to the solar system than the GC direction. This means that if UHECR sources are distributed in the Galactic plane, they are much better represented at Earth from the anticenter direction, and, as already noted in Kumar & Eichler (2014), this can cause a large-scale anisotropy from the anticenter direction. This could explain the anisotropy above 8 EeV that has recently been reported (Aab et al. 2015b).

At the same rigidities as the protons, heavier ion species emanating from the same source would appear at significantly higher energies (Lemoine & Waxman 2009). Thus, the energy spectrum of a hotspot, should one be revealed, could be a powerful tool for diagnosing the composition of UHECRs. For example, if a hotspot is detected at one energy but not at another energy in the same place, then the source composition at a given rigidity is pure enough that no other species makes a detectable hotspot at the same rigidity. Similarly, at a given energy, the location and angular size of a hotspot could vary with ion species and a double or multiple hotspot could provide a similar diagnostic of composition.

We thank Denis Allard and Rainer Beck for helpful discussions. N.G. is indebted to Benjamin Rouillé d'Orfeuil for important contributions to the initial version of the code. N.G. acknowledges the support of the I-CORE Program, The Israel Science Foundation (grant 1829/12), the Israel Space Agency (grant 3-10417), and the Lady Davis foundation. D.E. acknowledges support from the ISF, including an ISF-UGC grant, the Israel-US Binational Science Foundation, and the Joan and Robert Arnow Chair of Theoretical Astrophysics.