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Active star formation in galaxies leads to acceleration of protons and electrons via the Fermi-I diffusive shock acceleration mechanism in SN remnants. Under equilibrium conditions in a galaxy, a minimum-energy configuration of the magnetic field and the energetic particles may be attained. Energy densities of particles and magnetic fields may then be in approximate equipartition, implying that the energetic proton energy density, Up, can be deduced from the detected level of synchrotron radio emission. In this radio-based approach Up can be estimated if the source size, distance, radio flux, and radio spectral index are known.

In a gamma-based approach, Up can be obtained from the measured GeV-TeV spectral flux, which is mostly due to p-p interactions, as described the previous section. Only recently have such measurements become possible, at present only for 10 sources.

In the SN method, with an assumed fraction of SN kinetic energy that is channeled into particle acceleration, Up can be estimated if the size of the star-forming region and SN rate are known, as well as an estimate for the proton residence timescale from the presence (or absence) of a galactic wind emanating from the star-forming region. This timescale is largely determined by the advection timescale (~ 105yr) in SBGs, and by the pi0-decay timescale (~ 107yr) in low-SFR (quiescent) galaxies.

Expanding on our previous work (Persic & Rephaeli 2010), we show that the three methods give consistent results for Up for a sample of 10 galaxies with widely varying levels of star formation activity, from very quiescent to extreme SBs. These are the only galaxies of their kind for which gamma-ray data, in addition to radio data and SN rates, are available (see Table 1).

Table 1. Star-forming galaxies: the data.

Object DL[1] rs[2] f1 GHz[3] alphaNT[4] ne, th[5] LTIR[6] SFR[7] nuSN[8] Mgas[9] Lgamma[10] Notes
  (Mpc) (kpc) (Jy)   (cm-3) (erg/s) (Modot/yr) (yr-1) (Modot) (erg/s)  

Arp 220 74.7 0.25 0.3 0.65 300 45.75 253 3.5 9.24-0.11+0.10 < 42.25 SB
M 82 3.4 0.23 10.0 0.71 200 44.26 8.2 0.25 9.37-0.14+0.09 40.21-0.13+0.10 SB
NGC 253 2.5 0.20 5.6 0.75 400 44.23 7.7 0.12 9.20-0.11+0.10 39.76-0.19+0.14 SB
Milky Way 4.4 0.01 43.75 2.5 0.02 9.81-0.16+0.12 38.91-0.15+0.12 quiescent
M 31 0.78 4.5 4.0 0.88 0.01 42.98 0.43 0.01 9.88-0.15+0.11 38.66-0.10+0.09 quiescent
M 33 0.85 2.79 3.30 0.95 0.03 42.68 0.22 0.003 9.35-0.19+0.13 < 38.54 quiescent
LMC 0.049 2.4 285.0 0.84 0.01 42.45 0.13 0.002 8.86-0.18+0.12 37.67-0.05+0.05 quiescent
SMC 0.061 1.53 45.3 0.85 0.01 41.45 0.01 0.001 8.66-0.06+0.03 37.04-0.14+0.11 quiescent
NGC 4945 3.7 0.22 5.5 0.57 300 44.02 4.7 0.1-0.5 9.64-0.40+0.10 40.30-0.16+0.12 SB+Sy2
NGC 1068 16.7 1.18 6.6 0.75 300 45.05 50 0.2-0.4 9.71-0.19+0.11 41.32-0.23+0.15 SB+Sy2

[1]Distance (from Ackermann et al. 2012).
[2]Effective radius of star-forming region. See text. Data are from Persic & Rephaeli 2010 and refs. therein (Arp 220, M 82, NGC 253), Beck & Gräve 1982 (M 31), Tabatabaei et al. 2007 (M 33), Weinberg & Nikolaev 2001 (LMC), Wilke et al. 2003 (SMC), Moorwood & Oliva 1994 ( NGC 4945), Spinoglio et al. 2005 (NGC 1068).
[3]1 GHz flux density. Data are from Persic & Rephaeli 2010 and refs. therein (Arp 220, M 82, NGC 253)), Beck & Gräve 1982 (M 31), Tabatabaei et al. 2007 (M 33), Klein et al. 1989 (LMC), Haynes et al. 1991 (SMC), Elmouttie et al. 1997 ( NGC 4945), Kühr et al. 1981 (NGC 1068).
[4]Non-thermal spectral radio index. Data are from Persic & Rephaeli 2010 and refs. therein (Arp 220, M 82, NGC 253), Beck & Gräve 1982 (M 31), Tabatabaei et al. 2007 (M 33), Klein et al. 1989 (LMC), Haynes et al. 1991 (SMC), Elmouttie et al. 1997 ( NGC 4945), Kühr et al. 1981 (NGC 1068).
[5]Thermal electron density. Data are from Roy et al. 2010 (Arp 220), Petuchowski et al. 1994 (M 82), Kewley et al. 2000 and Corral et al. 1994 (NGC 253), Cox 2005 (Milky Way), Beck 2000 (M 31), Tabatabaei et al. 2008 (M 33), Points et al. 2001 (LMC), Sasaki et al. 2002 (SMC), Spoon et al. 2000 (NGC 4945), Kewley et al. 2000 (NGC 1068).
[6]Total IR [i.e., (8-1000)µm] luminosity, in log (from Ackermann et al. 2012).
[7]Star formation rate, from SFR = LIR / (2.2 × 1043 erg/s ) (Kennicutt 1998).
[8]Core-collapse SN rate. Data are from Persic & Rephaeli 2010 and references therein (Arp 220, M 82, NGC 253), Diehl et al. 2006 (Milky Way), van den Bergh & Tammann 1991 (M 31, M 33, SMC, LMC; see also Pavlidou & Fields 2001), Lenain et al. 2010 and references therein (NGC 4945, NGC 1068). For NGC 1068 we also computed an upper limit to the SN rate (nuSN leq 0.39) using Mannucci et al.'s (2003) formula nuSN = (2.4 ± 0.1) × 10-2 [LFIR / (1010 Lodot)] yr-1, being fFIR = 1.26 × 10-11(2.58 f60 + f100) erg cm-2s-1 (see Helou et al. 1988) with f60 appeq 190 Jy and f100 appeq 277 Jy.
[9]Gas mass (neutral plus molecular hydrogen: MHI + MH2), in log. Data are from: Torres 2004 for Arp 220; Abdo et al. 2010a for M 82, NGC 253, and the Milky Way; Abdo et al. 2010b for M 31 and M 33Abdo et al. 2010c for the LMC; Abdo et al. 2010d for the SMC; and Lenain et al. 2010 for NGC 4945 and NGC 1068.
[10]High-energy (> 100 MeV) gamma-ray luminosity, in log (from Ackermann et al. 2012).

3.1. Particles and magnetic field

The population of NT electrons consists of primary (directly accelerated) and secondary (produced via pi± decays) electrons. While the exact form of the steady-state electron energy spectrum is not a single power law, at high energies the flattening of the spectrum due to Coulomb losses can be ignored, justifying the use of the approximate single power-law form. The combined (primary plus secondary) electron spectral density distribution is then

Equation 1 (1)

where the electron Lorentz factor gamma is in the range gamma1 leq gamma leq gamma2, Ne,0 is a normalization factor of the primary electrons, chi is the secondary-to-primary electron ratio, and q geq 2 is the spectral index. Ignoring the contribution of low-energy electrons with gamma < gamma1, the electron energy density is Ue = Ne,0 (1 + chi) me c2gamma1gamma2 gamma1-q dgamma, where gamma2 is an upper cutoff whose exact value is irrelevant in the limit of interest, gamma2 >> gamma1. For q > 2 and gamma2 >> gamma1,

Equation 2 (2)

For a population of electrons (specified by Eq. 1) traversing a homogeneous magnetic field of strength B in a region with (a spherically equivalent) radius rs located at a distance d, standard synchrotron relations yield

Equation 3 (3)

where the scaled flux is f1 GHz Jy, a(q) is defined and tabulated in, e.g., Tucker (1975), and psi ident (rs / 0.1 kpc)-3 (d / Mpc)2 (f1 GHz / Jy). Use of Eq.(2) then yields

Equation 4 (4)

In order to compute Ue from Eq.(4) we need to specify gamma1 and B. To do so we make the following assumptions:

(i) The low-energy limit of the electron power-law spectrum, gamma1, marks the transition (for decreasing energy) from Coulomb (Rephaeli 1979) to synchrotron losses. For an electron of energy gamma, the synchrotron loss rate is

Equation 5 (5)

whereas the Coulomb loss rate is

Equation 6 (6)

(Rephaeli 1979). We then simply assume that electrons lose energy via Coulomb scattering for gamma < gamma1 and via synchrotron cooling for gamma > gamma1.

(ii) The particle energy density is in equipartition with that of the magnetic field, Up + Ue = B2 / 8 pi. In terms of the p/e energy density ratio, kappa, the equipartition condition is Up [1+ (1 + chi) / kappa] = B2 / 8pi, so that

Equation 7 (7)

Inserting Eq.(7) into Eq.(5) we get (dgamma / dt)syn propto gamma12(9-q) / (5 + q). Once the value of ne,th is specified (see Table 1), by equating Eqs.(5, 6) we deduce gamma1.

The secondary-to-primary electron ratio chi, which appears in Eq.(7), depends on the injection p/e number ratio, rp/e = (mp / me)(qinj-1)/2, and on the gas optical thickness to p-p interactions. Given the branching ratios in p-p collisions, only a third of these collisions produce electrons. The mean free path of CR protons due to p-p interactions in a medium of density np is lambdapp = (sigmapp np)-1; for protons with kinetic energy T ~ few TeV the cross section is sigmapp appeq 50 mb = 5 × 10-26 cm2 (Baltrusaitis et al. 1984). For a typical SB ambient gas density np appeq 150 cm-3, lambdapp ~ 43 kpc. The probability for a single CR proton to undergo a pp interaction in its 3D random walk through a region of radius rs ~ 0.25 kpc (also typical of SB nuclei) is then √3 rs / lambdapp appeq 0.01. Thus, in a typical SB environment, characterized by relatively strong non-relativistic shocks (qinj = 2.2), the secondary to primary electron ratio is chi = chi03 (rs / lambdapp) appeq 0.3. In a more quiescent environment, with typical values np appeq 1 cm-3 and rs ~ 2.5 kpc, chi appeq 0.03. The higher value found in SBs is in approximate agreement with results of detailed numerical starburst models for energies gtapprox 10 MeV (plotted in, e.g., Paglione et al. 1996, Torres 2004, De Cea et al. 2009, Rephaeli et al. 2010).

To compute the p/e energy density, kappa, we assume power-law spectra: (i) The electron spectral index qe is deduced from the measured radio index alpha, generally qe = 2 alpha + 1. (ii) The proton spectral index is assumed to be close to the injection value, qp ~ qinj appeq 2.1-2.2, for the dense SB environments hosted in the central regions of some galaxies, and equal to the leaky-box value, qp = qinj + delta appeq 2.7 (where delta appeq 0.5 is the diffusion index) for more quietly star forming galaxies.

Finally, we obtain an explicit expression for Up:

Equation 8 (8)

Using Eq.(8), values of Up can be obtained from the relevant observational quantities for our sample galaxies; these values are listed in Table 2. The quantities in Eq.(8) are usually well determined for our sample galaxies, except for the p/e energy density ratio kappa, for which a spectral index, qp, must be assumed. Given its possible values (i.e., qp appeq 2.1-2.2 in SB regions, and qp appeq 2.1-2.2 for quiescent galaxies), the uncertainty in the spectral index, delta qp appeq 0.1, translates to a factor of ~ 2 uncertainty on kappa, i.e. typically an uncertainty of ~ 50% on Up as deduced from Eq.(8).

3.2. Energetic particles and gamma-ray emission

Based on the calculation of gamma-ray emission from SFGs outlined in Section 1, Up can be estimated directly from recent measurements of the nearby galaxies. In SBGs, such as M 82 and NGC 253, the central SB region (referred to also as the source region) with a radius of ~ 200-300 pc is identified as the main site of particle acceleration. The injected particle spectrum is assumed to have an index q = 2, the theoretically predicted Np / Ne ratio is adopted, and equipartition is assumed. A measured radio index alpha appeq 0.7 in the source region implies q = 2 alpha + 1 appeq 2.4 there, indicating a substantial steepening due to diffusion (D propto gamma-delta), that cause the steady-state particle spectral index to be q0+ delta above some break energy. The procedure is similar when star formation does not (largely) occur in a burst in the nuclear region, but proceeds more more uniformly across the disk.

For a source with ambient gas number density ngas, proton energy density Up, and volume V, the integrated hadronic emission from pp-induced pi0 decay is

Equation 9 (9)

with the integral emissivity ggeq epsilon[eta] in units of photon s-1(H-atom)-1(eV/cm3)-1 (Drury et al. 1994). Thus, Up can be determined from measurements of Lgeq epsilon and ngas(r), and the particles steady-state energy distributions can be numerically calculated in the context of the convection-diffusion model.

In addition to the high-energy detections of the two local SBGs M 82 and NGC 253 (Abdo et al. 2010a, Acciari et al. 2009; Acero et al. 2009), several galaxies with low SFR were also detected by the Fermi telescope. (i) the Andromeda galaxy M 31 (Abdo et al. 2010b), with Up appeq 0.35 eV cm-3; (ii) the Large Magellanic Cloud (LMC) whose average spectrum, either including or excluding the bright star-forming region of 30 Doradus, suggests Up appeq 0.2-0.3 eV cm-3 (Abdo et al. 2010c); (iii) SMC for which Up appeq 0.15 eV cm-3 was deduced (Abdo et al. 2010d). For the Milky Way, the modeling of the Galactic diffuse HE emission along the lines outlined above requires an average Up appeq 1 eV cm-3 (Strong et al. 2010; Ackermann et al. 2011). Our values for Up determined from the measured GeV-TeV fluxes are listed in Table 2.

3.3. Energetic particles and Supernovae

The SN origin of energetic particles suggested early on; as a test of this hypothesis, we estimate of Up by combining the SN rate with the proton residence time, taures, assuming a fiducial value for the fraction of SN kinetic energy that is channeled to particle acceleration. The residence time is determined from the p-p interaction time, and the two propagation timescales of advection and diffusion:

(i) The energy-loss timescale for pp interactions, taupp = (sigmapp c np)-1; for protons with kinetic energy E gtapprox 10 TeV for which sigmapp appeq 50 mb, this timescale is

Equation 10 (10)

(ii) Particle advection out of the disk mid-plane region in a fast SB-driven wind occurs on a timescale tauadv determined from the advection velocity for which we adopt (except where noted otherwise) the nominal value vadv ~ 1000 km s-1, deduced from measurements of the terminal outflow velocity of ~ 1600-2200 km s-1 in M 82 (Strickland & Heckman 2009; see also Chevalier & Clegg 1985 and Seaquist & Odegard 1991). For a homogeneous distribution of SNe within the SB nucleus of radius rs, the advection timescale is

Equation 11 (11)

(iii) As noted in the previous section, diffusion is likely to be random walk against magnetic field inhomogeneities, with an estimated central diffusion coefficient, D ~ 3 × 1028 cm2/s, assuming a magnetic coherence scale of lambda ~ 1 pc. Thus, diffusion out of the central 0.5 kpc is estimated to occur on a timescale

Equation 12 (12)

Now, since the weighted residence time is

Equation 13 (13)

it is expected that - under typical conditions in central SB regions - p-p collisions and advection, more so than diffusion, effectively determine the survival there of energetic protons.

During taures, the number of SN is nuSN taures; the kinetic energy deposited by each of these into the ISM is Eej = 1051 erg (Woosley & Weaver 1995). Arguments based on the energetic particle energy budget in the Galaxy and SN statistics suggest that a fraction eta ~ 0.05-0.1 of this energy is available for accelerating particles (e.g., Higdon et al. 1998). Thus, the proton energy density can be expressed as

Equation 14 (14)

The resulting values of Up in the sample galaxies are listed in Table 2.

Table 2. Star-forming galaxies: Proton energy densities+.

Object gamma-ray radio SN other   rs taures
  meth. meth. meth. meth. (kpc) (yr)

Arp 220    – 1027 1142    –  0.25 2.0E+4
M 82 250a,c  250  234    –  0.23 4.5E+4
NGC 253 220b,c  230  213    –  0.20 6.7E+4
Milky Way   1d   –  1.2    1j  4.4 2.0E+7
    6e   –   5    –  0.2 2.5E+6
M 31 0.36f  0.22 0.7    –  4.5 2.5E+7
M 33 <0.43f  0.38 0.7    –  2.8 2.0E+7
LMC 0.21-0.31g  0.22 0.4    –  2.5 4.4E+7
SMC 0.15h  0.39 1.1    –  1.5 1.4E+7
NGC 4945 200i  201 215    –  0.22 4.5E+4
NGC 1068    –   65  61    –  1.2 1.0E+6

+ Values are in eV cm-3.

(a) Acciari et al. (2009; see also Persic et al. 2008, De Cea et al. 2009). (b) Acero et al. (2009). (c) Abdo et al. (2010a). (d) Strong et al. (2010). (e) Aharonian et al. (2006). (f) Abdo et al. (2010b), and Drury et al. (1994) for M 33. (g) Abdo et al. (2010c). (h) Abdo et al. (2010d). (i) Lenain et al. (2010). (j) Webber (1987).

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