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Using observational data from the literature, we show that the non‐thermal radio luminosity ( L ) of supernova remnants (SNRs) is a strong function of the average gas surface density  (Σ g )  of the galaxy in which the remnants reside, from normal spirals to dense luminous starbursts. Our result supports the interpretation of the radio sources in M82 and Arp 220 as normal SNRs, and not ‘radio’ supernovae (SNe). We combine a simple theory for electron cooling in SNRs with their observed radio luminosities to estimate the remnant magnetic field strength ( B  SNR ): the correlation between  L  and  Σ g   implies that  B  SNR  also increases with  Σ g  . We explore two interpretations of this correlation: (1)  B  SNR  is generated by post‐shock magnetic field amplification, with   B  2  SNR ∝Σ g   and (2)  B  SNR  results from shock compression of the ambient interstellar medium (ISM) magnetic field ( B  ISM ), with  B  ISM  being larger in denser galaxies. We find that shock compression is, on average, sufficient to produce the observed radio emission from SNRs in the densest starburst galaxies; amplification of post‐shock magnetic fields is not required. By contrast, in normal spirals modest post‐shock field amplification in some remnants (a factor of  ∼ few − 10 ) is consistent with the data; we find tentative evidence that both the Alfvén speed within SNRs and the ratio of   B  2  SNR /8π  to the post‐shock pressure (‘ ε  B   ’) are constant in SNRs from galaxy to galaxy. We discuss observational tests that can be used to more definitively distinguish between these two interpretations of the radio luminosities of SNRs. Regardless of which is correct, the radio emission from SNRs provides an upper limit to  B  ISM  that is independent of the minimum energy assumption. For the densest starbursts, the magnetic energy density in the ISM is below the total ISM pressure required for hydrostatic equilibrium; thus magnetic fields are not dynamically important on the largest scales in starbursts, in contrast with spiral galaxies like our own. This dichotomy may have implications for galactic dynamo theory.

Radio emission from supernova remnants: implications for post‐shock magnetic field amplification & the magnetic fields of galaxies