On Estimating the Flux of the Brightest Cosmic Ray Source above 57x10^18 eV

The detection of anisotropic cosmic rays (CR) above 57 ×10 18 eV (Abraham et al. 2008(Abraham et al. , 2007) ) by the newly constructed Pierre Auger Observatory (Abraham et al. 2004) has lead to speculation that CR astronomy may soon be a reality. If this is true, it is expected that the brightest source will be the first to be resolved. Since any meaningful astronomical measurement (e.g., sky position, energy spectrum, or morphology) will require the detection of many particles from the source object, the near-future prospects for CR astronomy depend largely on the flux of the brightest CR source in the sky. Consider that the Auger Observatory detects approximately 23 CR above 57 × 10 18 eV per year (Abraham et al. 2007). If the brightest source contributes 10% of the flux, the Pierre Auger Observatory will detect 23 events from this source in a decade; the era of CR astronomy will begin. If, on the other hand, the flux of the brightest source is far below 1%, the Pierre Auger Observatory may not detect more than one CR from this source during its 20-year operational lifetime.

In this letter, we estimate the flux of the brightest source above 57 × 10 18 eV by postulating two general conditions. The conditions are consistent with a large number of leading source scenarios; e.g., radio loud galaxies (Fraschetti & Melia 2008), active galactic nuclei (Becker & Biermann 2009;Farrar & Gruzinov 2008), and gamma ray bursts (Waxman 1995;Wick et al. 2004). The first condition is that the source objects are associated with galaxies, but the sources are relatively rare or rarely active such that many galaxies, such as our own and our closest neighbors, do not contain a luminous source. This implies a source number density dN/dV < 10 -2 Mpc -3 . The second condition is that the CR particles are protons or heavy nuclei such as iron and the Greisen-Zatsepin-Kuzmin (GZK) energy loss processes (Greisen 1966;Zatsepin & Kuzmin 1966) are occurring. Above 10 20 eV, the energy loss length for protons and iron is tens of Mpc. This effectively limits the propagation distance of the highest energy CR to 100 Mpc, which implies a source number density dN/dV > 10 -6 Mpc -3 . The energy loss lengths of intermediate weight nuclei are less than either protons or iron-like heavy nuclei. Therefore, the second condition is conservative, i.e. it implies the broadest range of source number densities. This range of source densities fully encompasses the ranges recently estimated by Takami & Sato (2009) and Cuoco et al. (2008).

In this work, we only consider the integrated CR flux above a threshold energy E th = 57 × 10 18 eV. Because of GZK energy losses, sources beyond a distance D, the so-called GZK horizon, will be nearly undetectable in this energy range. The GZK horizon is a strong function of the threshold energy, decreasing as E th increases. By an accident of nature, the GZK horizon is approximately the same for both protons and iron nuclei (Harari et al. 2006). Therefore, we will only examine the proton case in detail here, realizing that the iron case will have similar results.

To reproduce the observed energy spectrum we assume that the proton injection spec-trum (i.e., the near-source spectrum) is a power law dn/dE ∝ E -α with α = 2.6, as suggested by Allard et al. (2007). Our estimation of the GZK horizon then proceeds as follows. As in Harari et al. (2006), we define a GZK attenuation factor A(E th , x) as the fraction of particles originally above E th that still have an energy above E th after traversing a distance x. The attenuation factor is then A(E th , x) = ((E th + ∆E)/E th ) 1-α , where ∆E is the average energy loss of a proton traversing a distance x. The energy loss ∆E is calculated by solving dE/dx = -E/λ, where λ is the proton energy loss length in the extragalactic medium (e.g., see Protheroe & Johnson (1996)). This is the so-called continuous energy loss approximation. In Fig. 1, we show the results of this calculation for E th = 57 × 10 18 eV. There is a rather sharp breakpoint at approximately 250 Mpc beyond which the attenuation factor rapidly decreases exponentially. A reasonable choice for the GZK horizon is then 250 Mpc.

Let us examine a simplified scenario where there are N sources of equal luminosity L evenly distributed within a sphere of radius D = 250 Mpc centered at earth. We approximate the attenuation factor as a step function, with a value of one within the sphere and zero outside. The flux received at the detector from a single source inside the sphere is q = ωLr -2 , where ω is the detector’s sky exposure at the source location, and r is the source distance. We assume ω is constant o

…(Full text truncated)…