High-energy emission from star-forming galaxies
Adopting the convection-diffusion model for energetic electron and proton propagation, and accounting for all the relevant hadronic and leptonic processes, the steady-state energy distributions of these particles in the starburst galaxies M 82 and NGC 253 can be determined with a detailed numerical treatment. The electron distribution is directly normalized by the measured synchrotron radio emission from the central starburst region; a commonly expected theoretical relation is then used to normalize the proton spectrum in this region, and a radial profile is assumed for the magnetic field. The resulting radiative yields of electrons and protons are calculated: the predicted 100 MeV and 100 GeV fluxes are in agreement with the corresponding quantities measured with the orbiting Fermi telescope and the ground-based VERITAS and HESS Cherenkov telescopes. The cosmic-ray energy densities in central regions of starburst galaxies, as inferred from the radio and -ray measurements of (respectively) non-thermal synchrotron and -decay emission, are eV cm, i.e. at least an order of magnitude larger than near the Galactic center and in other non-very-actively star-forming galaxies. These very different energy density levels reflect a similar disparity in the respective supernova rates in the two environments. A relationship is then predicted, in agreement with preliminary observational evidence.
INAF and INFN, Trieste Tel-Aviv University and University of California, San Diego \PACSes \PACSit95.85.Pw: -ray sources \PACSit98.54.Ep: starburst galaxies and infrared excess galaxies \PACSit98.56.Ne: spiral galaxies (M 31 and M 33) \PACSit98.56.Si: Magellanic clouds and other irregular galaxies \PACSit98.70.Sa: Cosmic rays
In the nuclear regions of starburst (SB) galaxies, active star formation (SF) powers emission of radiation directly by supernova (SN) explosions and indirectly by SN-shock heating of interstellar gas and dust, as well as from radiative processes involving SNR-accelerated cosmic-ray electrons (CRe) and protons (CRp).
Some basic considerations suggest that the timescales required for CRp to be accelerated (by SN shocks) and lose energy (via pion decay into photons and pairs, or via advection) are shorter than timescales of SB activity in galaxies, that are themselves comparable to galactic dynamical timescales. A consequence is that in a SB region a balance can roughly be achieved between energy gains and losses for galactic CRs during a typical burst of SF . Under basic hydrostatic and virial equilibrium conditions in a galaxy, a minimum-energy configuration of the field and the CRs may be attained. This implies that energy densities of particles and magnetic fields can be in approximate equipartition .
The equipartition assumption enables deduction of the CRp energy density, , from the measured synchrotron radio emission (which can be observed relatively easily) and a theoretically motivated injection p/e ratio. Alternatively, can be estimated also from SN rates and the fraction of SN energy that is channeled into particle acceleration. Knowledge of enables prediction of -ray emission [either at high energies (HE: 100 MeV) or at very high energies (VHE: 100 GeV)], which is mostly due to CRp interactions with ambient gas protons, via decay.
2 HE emission from star-forming galaxies
In this section we will review some basic features of SB modeling, notably applied to the local galaxies M 82 and NGC 253, and the status of observations of star-forming galaxies in the HE/VHE -ray domain.
In both nearby SB galaxies, M 82 and NGC 253, the central SB region (which will also referred to as the source region) with a radius of 300 pc and height of 300 pc is identified as the main site of particle acceleration. Here, the injection particle spectrum is assumed to have a non-relativistic strong-shock index . A theoretical ratio, predicted from charge neutrality of the injected CRs, is likely to hold in this source region – as is also the assumption of equipartition.
Due to the implicit dependences in the expression for the synchrotron flux, CR/field equipartition is implemented iteratively to solve for (primaries plus secondaries), , and . For both M 82 and NGC 253, central values G have been obtained in detailed models (M 82: [3, 4]; NGC 253: [5, 6, 7]).
Adopting the convection-diffusion model for energetic electron and proton propagation, and accounting for all the relevant hadronic and leptonic processes, the steady-state energy distributions of these particles in the galaxies M 82 and NGC 253, in both the SB nucleus and the disk, can be determined with a detailed numerical treatment. In particular, to numerically follow particle energy losses and propagation outside the source region, one needs to know of the HI and HII densities and their profiles, as well as the spatial variation of the mean strength of the magnetic field. (E.g., assuming magnetic flux freezing in the ionized gas, then .)
A measured radio index of 0.7 in the central disk implies there. This implies a substantial steepening of the CRe spectrum from the injection value, . The steady-state electron and proton spectra in the SB region of NGC 253 are shown in fig. 1. At low energies both spectra are flat, whereas at GeV the stronger electron losses result in steeper electron (than proton) spectra.
Electron emissions by bremsstrahlung and Compton scattering are shown in fig. 2; also shown is -ray emission from decay (following pp collisions). As expected, the losses due to bremsstrahlung dominate the lower energy regime, whereas losses due to decay dominate at higher energies. (While synchrotron emission extends to the X-ray region, it is negligible at much higher energies.) Our main interest here is the VHE emission, which – as anticipated – is mainly from the latter process.
It is interesting to note that the numerically predicted VHE -ray flux is a factor 6 lower than that obtained in an approximate treatment where the impact of radial energy losses and propagation mode of the CRp is ignored . This simplified approach results in an unrealistically high contribution of the main disk (exterior to the SB region) to the total TeV emission: in fact even the slight steepening of the CRp spectrum, occurring in the disk because of energy losses, significantly lowers the TeV emissivity  and hence the predicted source flux.
We note that the related neutrino flux ( eventually produce ) at energies higher than 100 GeV is about a third of the corresponding photon flux .
2.2 -ray detections
The two local SB galaxies M 82 and NGC 253 are the only non-AGN extragalactic sources that, up to now, have been detected in both the GeV  and TeV [11, 12] domains. The measured fluxes and spectra of both galaxies in the two bands agree with predictions of recent numerical models (M 82: [3, 4]; NGC 253: [5, 6, 7]). The highest-SFR galaxy in the nearby universe, Arp 220, was undetected by MAGIC .
HE -ray detections were obtained for a number of low SFR galaxies: the Large Magellanic Cloud (LMC: ), the Small Magellanic Cloud (SMC: ), the Andromeda galaxy (M 31: ), and the composite Sy2/SB galaxies NGC 1068 and NGC 4945 . The scenario of mostly hadronic HE -ray emission is generally confirmed for these galaxies, except for NGC 1068 where emission from the active nucleus may be dominant. Only flux upper limits exist for the Triangulum galaxy (M 33: ).
3 CRs in star-forming galaxies
The CRp energy density in galaxies can be either 1) measured directly if the GeV-TeV spectral flux is known, or 2) evaluated indirectly if source size, distance, and radio spectral index and flux are known, and particles/field equipartition and a p/e ratio are assumed, or 3) estimated if the SN rate, the CRp residence timescale, the energy per SN going into CRs, and the size of the SF region are known.
1) CRs and GeV-TeV emission. The detection of M 82 and NGC 253 confirmed values eV cm, resulting from accurate numerical treatments based on the solution of the diffusion-loss equation for the accelerated particles (see refs. above). In the 200 pc region of the Galactic center eV cm based on HESS observations ). For the comparable environment of the Andromeda galaxy, the Fermi/LAT detection implies an average at a level of the average Galactic value . For the even more quiet environments of the LMC and the SMC), actual Fermi/LAT GeV detections imply, respectively, and eV cm [14, 15].
2) CRs and radio emission. Based on method 2) above, for the central SB regions of NGC 253, M 82, and Arp 220, respectively,  evaluate , 97, and 520 eV cm. These radio-based estimates match, to within a factor of 2, those derived from GeV/TeV-based measurements.
3) CRs and supernova rates. The CRp residence timescale is given by , where the pp interaction timescale is a function of the ambient gas density , and the advection timescale is a function of the speed of the outflowing gas () and of the size (radius) of the SB region (). Typically, yr and yr for the central SB regions of Arp 220 and, respectively, the M 82 and NGC 253 . If however a fast SB-driven wind advects the energetic particles out of the disk plane, then possibly . For M 82, km s . Assuming a homogeneous distribution of SNe within the SB nucleus of radius , the outflow timescale is then yr. So in some SB galaxies . During a number of SN explode and deposit the kinetic energy of their ejecta, erg, into the interstellar medium. The Galactic CR energy budget and SN statistics suggest that of this energy may go into accelerating particles. The CRp energy density in the central SB region is then . A substantial agreement with the equipartition estimates is reached for NGC 253, M 82, Arp 220, Milky Way, LMC: eV cm, respectively . Summarizing, the CR energy densities estimated in five galactic nuclei of similar size (three SB galaxies, the central Galactic region, and the LMC), appear to be largely correlated with key features of the ongoing SF: SN rate and CRp residence time (the latter being, in turn, a function of the local gas density and of the galactic superwind speed). In fact, in these environments, for same (by assumption) and similar (from observations), the CRp energy density seems to be well described just as a function of the number of SN explosions during the CRp residence timescale,
4 He/vhe -ray emission and SFR
The Schmidt-Kennicutt (SK) law of SF, (where gas comprises both HI and H), states that projected SFR varies as a power law the projected gas density. This is true piecewise: both locally and disk-averaged, it is for and for higher densities . If disk thicknesses do not vary much among galaxies, then the SK law can be written in deprojected units with the same index .
For a source with gas number density , proton energy density , and volume , the integrated hadronic -ray photon luminosity above some photon energy is
with the integral emissivity in units of photon s[H-atom][eV/cm] . Using volume-averaged quantities and setting MeV, from Eqs.(1),(2) we can write
This is in agreement with the observational luminosity vs. (gas mass)(SN rate) correlation , which evidently describes the -ray luminosity arising from decay.
Owing to the SK law, the above equation transforms into : if as appropriate for our sample galaxies with , then
Within limited statistics, this prediction agrees with observations .
So the observational non-linear –SFR correlation  stems from the GeV luminosity being (mostly) hadronic in origin, from CRs being linked with SF (through SN explosions), and from the Fermi/LAT-detected galaxies being located in the steep wing (i.e., low- regime) of the SK law of SF.
The link between SF and CR particles was suggested long ago . It is based on the recognition that the CRe’s, responsible for diffuse non-thermal synchrotron emission, are produced in the sites of SN explosions. The rough agreement between the relatively short lifetime ( yr) of massive stars, and the similarly short synchrotron energy loss time of high-energy electrons, led to the expectation that the non-thermal radio emission of a galaxy is a measure of its SF activity on scales much shorter than the Hubble time.
HE/VHE -ray detections (with, respectively, the orbiting Fermi telescope and ground-based IACTs) of some nearby star-forming (SB and normal) galaxies, that span a large range of SFR, have provided direct measurements. These are consistent with theoretical predictions based on radio measurements, and with estimates based on SN rates and local CRp residence times. Should the match between measured and predicted be confirmed, some immediate implications would be:
(i) star-forming galaxies can be powerful particle accelerators, able to achieve CRp energy densities orders of magnitude higher than the Galactic value;
(ii) SNe, both in quietly star-forming galaxies and in very actively star-forming galaxies, probably have a common universal CR acceleration efficiency;
(iii) CR energy densities and equipartition magnetic fields derived from radio measurements can be used as proxies for the quantities characterizing the full particle energy distributions (derived from accurate spectral fits of the GeV-TeV emission): this could be particularly useful in the case of galaxies that are too far away for their (umbeamed) -ray emission to be measured.
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