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Prospects for Detection of Synchrotron Emission from Secondary Electrons and Positrons in Starless Cores: Application to G0.216+0.016

We investigate the diffusion of cosmic rays into molecular cloud complexes. Using the cosmic-ray diffusion formalism of Protheroe, et al. (2008), we examine how cosmic rays diffuse into clouds exhibiting different density structures, including a smoothed step-function, as well as Gaussian and inverse-$r$ density distributions, which are well known to trace the structure of star-forming regions. These density distributions were modelled as an approximation to the Galactic centre cloud G0.216+0.016, a recently-discovered massive dust clump that exhibits limited signs of massive star formation and thus may be the best region in the Galaxy to observe synchrotron emission from secondary electrons and positrons. Examination of the resulting synchrotron emission, produced by the interaction of cosmic ray protons interacting with ambient molecular matter producing secondary electrons and positrons reveals that, due to projection effects, limb-brightened morphology results in all cases. However, we find that the Gaussian and inverse-$r$ density distributions show much broader flux density distributions than step-function distributions. Significantly, some of the compact (compared to the $2.2''$ resolution, 5.3 GHz JVLA observations) sources show non-thermal emission, which may potentially be explained by the density structure and the lack of diffusion of cosmic rays into the cloud. We find that we can match the 5.3 and 20 GHz flux densities of the non-thermal source JVLA~1 and 6 from Rodr\'ıguez & Zapata (2014) with a local cosmic ray flux density, a diffusion coefficient suppression factor of $χ=0.1-0.01$ for a coefficient of $3\times10^{27}$ cm$^2$ s$^{-1}$, and a magnetic field strength of 470 $μ$G.