The 617 MHz - $\lambda$ 850 $\mu$m Correlation (Cosmic Rays and Cold Dust) in NGC 3044 and NGC 4157

We present the first maps of NGC 3044 and NGC 4157 at $\lambda\,450 \mu$m and $\lambda\,850 \mu$m from the JCMT as well as the first maps at 617 MHz from the GMRT. High latitude emission has been detected in both the radio continuum and sub-mm for NGC 3044 and in the radio continuum for NGC 4157, including several new features. The dust spectrum at long wavelengths required fitting with a two-temperature model for both galaxies, implying the presence of cold dust. Dust masses are $M_d\,=\,1.6\,\times\,10^8 M_\odot$ and $M_d\,=\,2.1\,\times\,10^7 M_\odot$ for NGC 3044 and NGC 4157, respectively, and are dominated by the cold component. There is a clear correlation between the 617 MHz and $\lambda\,850 \mu$m emission in the two galaxies. In the case of NGC 3044, this implies a relation between the non-thermal synchrotron emission and cold dust. The 617 MHz component represents an integration of massive star formation over the past $10^{7-8}$ yr and the $\lambda \,850 \mu$m emission represents heating from the diffuse interstellar radiation field (ISRF). The 617 MHz -- $\lambda\,850 \mu$m correlation improves when a smoothing kernel is applied to the $\lambda\,850 \mu$m data to account for differences between the CR electron diffusion scale and the mean free path of an ISRF photon to dust. The best-fit relation is $L_{617_{\rm MHz}}\,\propto\,{L_{850 \mu{\rm m}}}^{2.1\,\pm\,0.2}$ for NGC 3044. If variations in the cold dust emissivity are dominated by variations in dust density, and the synchrotron emission depends on magnetic field strength (a function of gas density) as well as CR electron generation (a function of massive star formation rate and therefore density via the Schmidt law) then the expected correlation for NGC 3044 is $L_{617_{\rm MHz}}\,\propto\,{L_{850 \mu{\rm m}}}^{2.2}$, in agreement with the observed correlation.