Not Where You Left Them: Displaced $γ$-Rays and X-Rays Reveal the Cosmic Ray Scattering Rate

Abstract

Modern X-ray and $γ$-ray instruments are revealing a growing class of Galactic non-thermal sources whose emission centroids are measurably offset from the nearest plausible sites of cosmic ray (CR) acceleration. Such "displaced" sources are seen in keV X-rays and TeV-PeV $γ$-rays but not in GeV $γ$-rays, have hard spectra, and are not associated with gas clumps, suggesting a leptonic origin. We develop a general framework for understanding displacement, whereby relativistic CR electrons (CRe) injected into the interstellar medium (ISM) with a strongly anisotropic pitch-angle distribution propagate a finite distance from their acceleration site before scattering processes isotropise their directions sufficiently for the emission to become visible. We use CR transport simulations to investigate under what circumstances displacement is likely, finding that it requires an initial pitch angle distribution $\lesssim 45^\circ$ wide, a line of sight broadly edge-on to the magnetic field, and that the source be measured in a waveband where emission is dominated by CRe for which the radiative-loss and pitch-angle scattering timescales are comparable. For typical Galactic conditions the latter condition is satisfied only for CRe energies $\gtrsim$ 10 TeV, explaining why displaced sources appear at X-ray and TeV but not GeV energies. We further show that, when displacement is detected, it allows a direct inference of the CRe pitch-angle scattering rate.

Figures (11)

• Figure 1: Steady-state solutions to the CRe transport equation, showing the distribution function $f(\zeta, q, \mu)$ for our fiducial case ($k_p = -1.51$, $\log q_\mathrm{cut} = 1.6$, and $\mu_{\rm inj}=0.86$) as a function of position $\zeta$ and pitch angle $\mu$ for three different momenta $q$, as indicated in the labels of each panel. To facilitate comparisons across different values of $q$, the distribution functions in each panel have been normalized to set the maximum to unity.
• Figure 2: CRe distributions for our fiducial case ($k_p = -1.51$, $\log q_\mathrm{cut} = 1.6$, and $\mu_{\rm inj}=0.86$) $f(\mu, q, \zeta)$, normalized by its maximum, shown for three pitch angles $\mu = -0.98, 0, 0.98$ (different line styles) at two momenta. Low-$q$ distributions are broad and nearly isotropic, whereas high-$q$ distributions are narrower and exhibit a clear $\mu$ dependence, with the largest peak displacement for $\mu = 0.0$. The values of our metrics of interest are given in the following \ref{['t:table']}, with further discussion provided in \ref{['SS:case_study']}.
• Figure 3: Four metrics of interest for detectable displaced CRe signals for our fiducial case ($k_p = -1.51$, $\log q_\mathrm{cut} = 1.6$, and $\mu_{\rm inj}=0.86$) as a function of dimensionless CRe momentum $q$ and the cosine of the angle between the magnetic field and the observer $\mu$. Top Panel: The dimensionless position of the maximum of the signal, $\zeta_\mathrm{max}$. Second Panel: Sharpness parameter $S$ measuring the ratio of the distance from source to the peak of the emission $\zeta_\mathrm{max}$ to the FWHM of the emission. Third Panel:$C_\mathrm{on}$, defined as the ratio of the signal brightness at its (displaced) maximum to the signal brightness at the position of the CRe source. Bottom Panel: Ratio of the brightness at the emission peak to the brightness that would be produced by a source emitting cosmic rays with an isotropic distribution of pitch angles, $C_\mathrm{iso}$. Note that the dynamic range on the colour bar is the same for all panels. Each panel is centered on its characteristic mid-value -- 0.1 for $\zeta_\mathrm{max}$, 0.5 for $S$, 5 for $C_{\rm on}$, and 1 for $C_{\rm iso}$ -- with logarithmic scaling spanning one decade in dynamic range ($0.1\times v_{\mathrm{max}}$ to $v_{\mathrm{max}}$), where $v_{\mathrm{max}} = [0.2,\, 1.0,\, 10,\, 2.0]$.
• Figure 4: Sharpness parameter $S$ as a function of $q_\mathrm{cut}$ for $\mu_\mathrm{inj} = 0.86$, $k_p = -1.51$ (as used in the case study in \ref{['SS:case_study']}), and for an observer at $\mu=0.5$ and three CRe energies $q = 0.01, 0.1$, and 1. The figure illustrates that $q_\mathrm{cut}$ has no effect on $S$ as long as $q_\mathrm{cut} \gg q$.
• Figure 5: Same as \ref{['f:color_map_2']}, but now showing how the metrics of interest vary with $k_p$ and $\mu_{inj}$ for $\mu=0.5$ and $q=1$. The white contours in the middle two panels indicate $S = 0.3$ and $C_\mathrm{on} = 5$, our conditions for displaced emission.