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

Abstract

Modern X-ray and $\gamma$-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 $\gamma$-rays but not in GeV $\gamma$-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.