We investigate the effect of propagation of cosmic- ray electrons ( CRE ) on the nonthermal ( synchrotron ) – far-infrared correlations in M 31 and M 33 .
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 M 31 and M 33 using Spitzer data .
In both galaxies the TH emission is linearly correlated with the emission from warm dust ( 24 \mu \mathrm { m } , 70 \mu \mathrm { 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 \simeq 0.4 ) than those for M 31 ( b \simeq 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 disk with a scale height of h = 0.3 - 0.4 \mathrm { kpc } at 1.4 GHz , whereas M 33 has a similar thin disk as well as a thick disk with scale height h _ { \mathrm { thick } } \simeq 2 \mathrm { kpc } .
In the thin disks , the ( deprojected ) diffusion length at 1.4 GHz is \simeq 1.5 \mathrm { kpc } , yielding a diffusion coefficient of \simeq 2 10 ^ { 28 } \mathrm { cm } ^ { 2 } / \mathrm { 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 .