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How cosmic-ray electron propagation affects radio-far-infrared correlations in M31 and M33

We investigate the effect of propagation of cosmic-ray electrons (CRE) on the nonthermal (synchrotron) - far-infrared correlations in M31 and M33. The thermal (TH) and nonthermal (NTH) emission components of the radio continuum emission at 1.4 GHz and one higher frequency are compared with dust emission from M31 and M33 using Spitzer data. In both galaxies the TH emission is linearly correlated with the emission from warm dust (24 μm, 70 μm), but the power laws of the NTH-FIR correlations have exponents b < 1 that increase with increasing frequency. Furthermore, the values of b for M33 are significantly smaller (b ~ 0.4) than those for M31 (b ~ 0.6). We interpret the differences in b as differences in the diffusion length of the CRE. We estimate the diffusion length in two ways: (1) by smoothing the NTH emission at the higher frequency until the correlation with NTH emission at 1.4 GHz has b = 1, and (2) by smoothing the TH emission until the correlation with the NTH emission at the same frequency has b = 1, assuming that the TH emission represents the source distribution of the CRE. Our smoothing experiments show that M31 only has a thin NTH disk with a scale height of h = 0.3-0.4 kpc at 1.4 GHz, whereas M33 has a similar thin disk as well as a thick disk with scale height h_thick ~ 2 kpc. In the thin disks, the (deprojected) diffusion length at 1.4 GHz is ~ 1.5 kpc, yielding a diffusion coefficient of ~ 2 10^28 cm^2/s. The structure, strength and regularity of the magnetic field in a galaxy as well as the existence of a thick disk determine the diffusion of the CRE, and hence, the power-law exponent of the NTH-FIR correlations.