Are starburst galaxies proton calorimeters?

Several starburst galaxies have been observed in the GeV and TeV bands. In these dense environments, gamma-ray emission should be dominated by cosmic ray (CR) interactions with the interstellar medium (p_crp_ism → π⁰ → γγ). Indeed, starbursts may act as proton 'calorimeters' where a substantial fraction of CR energy input is emitted in gamma-rays. Here, we build a one-zone, 'thick-target' model implementing calorimetry and placing a firm upper bound on gamma-ray emission from CR interactions. The model assumes that CRs are accelerated by supernovae (SNe), and all suffer nuclear interactions rather than escape. Our model has only two free parameters: the CR proton acceleration energy per SN ∈_cr, and the proton injection spectral index s. We calculate the pionic gamma-ray emission from 10 MeV to 10 TeV, and derive thick-target parameters for six galaxies with Fermi, H.E.S.S., and/or VERITAS data. Our model provides good fits for the M82 and NGC 253, and yields ∈_cr and s values suggesting that SN CR acceleration is similar in starbursts and in our Galaxy. We find that these starbursts are indeed nearly if not fully proton calorimeters. For NGC 4945 and NGC 1068, the models are consistent with calorimetry but are less well-constrained due to the lack of TeV data. However, the Circinus galaxy and the ultra-luminous infrared galaxy Arp 220 exceed our pionic upper-limit; possible explanations are discussed.