Are starburst galaxies proton calorimeters?

In star-forming galaxies, gamma rays are mainly produced through the collision of high-energy protons in cosmic rays and protons in the interstellar medium (i.e., cosmic-ray induced p0 g- radiation). For a 'normal' star-forming galaxy like the Milky Way, most cosmic rays escape the galaxy before such collisions, but in starburst galaxies with dense gas and huge star formation rate, it is thought that most cosmic rays 'die' in these interactions. To test this, we construct a 'thicktarget' model for the starburst galaxies, in which cosmic rays are accelerated by supernovae, and escape is neglected. This model gives an upper limit to a galaxy's gamma-ray emission, and tests the calorimetric relation between gamma rays and cosmic rays for starbursts. Only two free parameters are involved in the model: the cosmic-ray proton acceleration energy per supernova ecr, and the proton injection spectral index s. We apply our model to five observed starburst galaxies: M82, NGC 253, NGC 1068, NGC 4945 and the Circinus galaxy, and find most of these starbursts are consistent with being proton calorimeters, with ecr (0:1-0:25)1051erg and s -2:1-2:4. But we confirm that for Circinus, other gamma-ray sources are needed to explain its GeV luminosity.