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We investigate the effect of propagation of cosmic-ray electrons (CRE) on thenonthermal (synchrotron) - far-infrared correlations in M31 and M33. Thethermal (TH) and nonthermal (NTH) emission components of the radio continuumemission at 1.4 GHz and one higher frequency are compared with dust emissionfrom M31 and M33 using Spitzer data. In both galaxies the TH emission islinearly correlated with the emission from warm dust (24 \mu m, 70 \mu m), butthe power laws of the NTH-FIR correlations have exponents b < 1 that increasewith increasing frequency. Furthermore, the values of b for M33 aresignificantly smaller (b ~ 0.4) than those for M31 (b ~ 0.6). We interpret thedifferences in b as differences in the diffusion length of the CRE. We estimatethe diffusion length in two ways: (1) by smoothing the NTH emission at thehigher 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 NTHemission at the same frequency has b = 1, assuming that the TH emissionrepresents the source distribution of the CRE. Our smoothing experiments showthat M31 only has a thin NTH disk with a scale height of h = 0.3-0.4 kpc at 1.4GHz, whereas M33 has a similar thin disk as well as a thick disk with scaleheight h_thick ~ 2 kpc. In the thin disks, the (deprojected) diffusion lengthat 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 aswell as the existence of a thick disk determine the diffusion of the CRE, andhence, the power-law exponent of the NTH-FIR correlations.

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