How cosmic ray electron propagation affects radio–far-infrared correlations in M 31 and M 33

We investigate the effect of propagation of cosmic ray electrons (CRE) on the non-thermal (NTH; synchrotron)–far-infrared correlations in M 31 and M 33. The thermal (TH) and NTH emission components of the radio continuum emission at 1.4 GHz and one higher frequency are compared with dust emission from M 31 and M 33 using Spitzer data. In both galaxies the TH emission is linearly correlated with the emission from warm dust (24 μ m, 70 μ m), but the power laws of the NTH–FIR correlations have exponents b < 1 that increase with increasing frequency. Furthermore, the values of b for M 33 are significantly smaller (b ≃ 0.4) than those for M 31 (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 M 31 only has a thin NTH disc with a scale height of h = 0.3–0.4 kpc at 1.4 GHz, whereas M 33 has a similar thin disc as well as a thick disc with scale height hthick ≃ 2 kpc. In the thin discs, the (deprojected) diffusion length at 1.4 GHz is ≃1.5 kpc, yielding a diffusion coefficient of ≃2 × 1028 cm2 s−1. The structure, strength and regularity of the magnetic field in a galaxy as well as the existence of a thick disc determine the diffusion of the CRE, and hence, the power-law exponent of the NTH–FIR correlations.