Cosmic ray anisotropies from transient extragalactic sources

We study the spectrum and anisotropies of ultrahigh energy cosmic ray transient sources, accounting for the effects of their propagation through the turbulent extragalactic magnetic fields. We consider either bursting sources or sources emitting since a given initial time. We analyze in detail the transition between the diffusive and the quasirectilinear regimes, describing some new features that could be present.

Figure 1

Density distribution as a function of R for different values of $d$. For the illustrative cases $d=2$, 8, and 32 we also show the curves that correspond to the distributions in Eqs. (5) and (7).

Figure 2

Density distribution as a function of $R$ for different values of $d\le 4$. Also shown are the profiles expected according to Eq. (12) and with dotted lines the individual contribution from the prompt component, while with dashed lines that of the diffusive component.

Figure 3

Fraction of the prompt contribution $f(d)$ as a function of $d$, together with the fit (see text).

Figure 4

Normalized CR density as a function of $E/E_{\mathrm{c}}$ for a source at $r_{\mathrm{s}}=4\mathrm{Mpc}$. For the smaller values of $ct=4.5$ and 5 Mpc, the long dashed lines indicate the contribution from the prompt component, and short dashes that from the diffusive component. For the larger values of $ct$ shown the prompt component becomes negligible. Left panel is for $l_{\mathrm{c}}=30\mathrm{kpc}$, right panel for $l_{\mathrm{c}}=100\mathrm{kpc}$.

Figure 5

Distribution of CR arrival directions as a function of $\cos \theta$ for values of the normalized source distance $R_{\mathrm{s}}\equiv r_{\mathrm{s}}/l_D=0.5$ and 2. The lines correspond to different values of $d$, which is the distance traveled by the particles since the emission of the burst, in units of $l_D$. When $d/R$ is larger than a few, the distribution approaches that of a dipole with amplitude $R/2d$.

Figure 6

Distribution of CR arrival directions as a function of $\cos \theta$ for a source at a distance $r_{\mathrm{s}}=4\mathrm{Mpc}$ that had a burst at a time such that $ct=4.4$, 6, 8, and 16 Mpc, as labeled. The lines correspond to different values of $E/E_{\mathrm{c}}$ within an interesting range in which the propagation is transitioning away from rectilinear.

Figure 7

Values of $⟨\cos \theta ⟩$ (left) and $⟨{\cos }^2\theta ⟩$ (right), as a function of the normalized source distance $R_{\mathrm{s}}$, for different values of the time of emission of the burst as indicated. The black dashed lines in the left plot correspond to the function $(R_{\mathrm{s}}/2d)(1+(R_{\mathrm{s}}/d{)}^{2.5})$, which is a good fit for $d>8$ for all values of $R_{\mathrm{s}}$. For $d<8$ it is a good fit for $R_{\mathrm{s}}>d/2$ only. In the right plot the black dashed lines correspond to the fit by the function $(1+(R_{\mathrm{s}}/d{)}^4(1+(R_{\mathrm{s}}/d{)}^8))/3$.

Figure 8

Enhancement factor of cosmic ray density as a function of $R$ for sources emitting since different times, parametrized by the values of $d_i$ quoted.

Figure 9

Distribution of CR arrival directions as a function of $\cos \theta$ for values of the normalized source distance $R_{\mathrm{s}}=0.5$ and 2 for sources emitting since different times, $d_i$ as quoted. The dashed lines correspond to the distribution in Eq. (13) with values of $\kappa$ and $i$ as given by Eqs. (27), (18) using (25) or (26) depending on the value of $d_i$.

Figure 10

Left: Mean value of $⟨\cos \theta ⟩$ obtained through numerical simulations (points) for different values of $d_i$ as labeled in the plot. The solid lines corresponds to the fits from Eq. (25) for $d_i>2$, and from Eq. (26) for $d_i\le 2$. Right: Concentration parameter $\kappa$ obtained from simulations using Eq. (17) (points) and fit from Eq. (27) (lines).

Figure 11

Enhancement factor $\xi$ for a source at a distance $r_{\mathrm{s}}=4\mathrm{Mpc}$, and for different extension of the emission period as a function of $E/E_{\mathrm{c}}$. A coherence length of 30 kpc was adopted for the turbulent magnetic field.

Figure 12

Values of $⟨\cos \theta ⟩$ and $\kappa$ as a function of $E/E_{\mathrm{c}}$ for the same values of the source distance and coherence length adopted in Fig. 11.