{ "COSMIC RAY SAMPLING OF A CLUMPY INTERSTELLAR MEDIUM ": [ "()arXiv:1311.0006v1 [astro-ph.HE] 31 Oct 2013 " ], "ABSTRACT ": [ "How cosmic rays sample the multi-phase interstellar medium (ISM) in starburst galaxies has important implications for many science goals, including evaluating the cosmic ray calorimeter model for these systems, predicting their neutrino fluxes, and modeling their winds. Here, we use Monte Carlo simulations to study cosmic ray sampling of a simple, two-phase ISM under conditions similar to those of the prototypical starburst galaxy M82. The assumption that cosmic rays sample the mean density of the ISM in the starburst region is assessed over a multi-dimensional parameter space where we vary the number of molecular clouds, the galactic wind speed, the extent to which the magnetic field is tangled, and the cosmic ray injection mechanism. We evaluate the ratio of the emissivity from pion production in molecular clouds to the emissivity that would be observed if the cosmic rays sampled the mean density, and seek areas of parameter space where this ratio di ers significantly from unity. The assumption that cosmic rays sample the mean density holds over much of parameter space; however, this assumption begins to break down for high cloud density, injection close to the clouds, and a very tangled magnetic field. We conclude by evaluating the extent to which our simulated starburst region behaves as a proton calorimeter and constructing the time-dependent spectrum of a burst of cosmic rays. " ], "1. INTRODUCTION ": [ "Starburst galaxies are complex environments with intense star formation, “clumpy,” multi-phase interstellar gas, supernovae-driven winds, and tangled magnetic fields. It is remarkable, then, that despite drastically di erent environments, both quiescent and star-forming disk galaxies have a well-established linear correlation between their far-infrared (FIR) and radio luminosities ((<>)Helou et al. (<>)1985). In these systems, the FIR-radio luminosity correlation suggests a fundamental relationship between the star formation processes, resulting in FIR emission from dust heated by young, massive stars, and the cosmic ray population, producing radio synchrotron emission from relativistic electrons spiraling along magnetic field lines. ", "To explain the FIR-radio luminosity correlation, the cosmic ray calorimeter model for starburst galaxies has been explored (e.g., (<>)Voelk (<>)1989; (<>)Thompson et al. (<>)2006; (<>)Persic et al. (<>)2008; (<>)de Cea del Pozo et al. (<>)2009; (<>)Lacki et al. (<>)2011; (<>)Paglione & Abrahams (<>)2012; (<>)Yoast-Hull et al. (<>)2013). This model suggests that the energy imparted to cosmic rays by supernovae is entirely expended through observable emission within the starburst region. If this model holds for cosmic ray electrons, then the observed radio synchrotron emission should be solely dependent on the supernova rate and therefore on the star formation rate (SFR; there is, of course, also a dependence on magnetic field strength, but this too appears to be tied to the SFR; (<>)Schleicher & Beck ((<>)2013)). Likewise, if the model holds for cosmic ray protons, then we expect a similar relationship between the observed γ-ray emission from pion production and the SFR. ", "In order to test the calorimeter model for starburst galaxies, it is necessary to understand how cosmic rays sample the multi-phase interstellar medium (ISM) as they undergo radiative, collisional, and advective losses within the starburst region. In an ISM consisting of cold, molecular clouds embedded in a hot, low-density medium, cosmic rays will only sample the mean density of the ISM if they are able to e ectively enter the molecular clouds. The leading theory of cosmic ray acceleration suggests that energetic particles undergo first-order Fermi acceleration ((<>)Fermi (<>)1949), or di usive shock acceleration (e.g., (<>)Bell (<>)1978; (<>)Blandford & Ostriker (<>)1978), at supernova shock fronts. Additionally, cosmic rays may undergo second-order Fermi acceleration ((<>)Fermi (<>)1949) in the di use ISM. Both first- and second-order Fermi acceleration can only occur eÿciently in low-density environments, where the acceleration mechanism can impart energies in excess of ionization losses and the small-scale magnetic field fluctuations which scatter the particles can propagate. For these reasons, the majority of cosmic rays are believed to be injected in the hot, low-density medium. ", "The hot medium where cosmic rays are injected has a very large filling factor compared to the molecular gas. Additionally, the hot medium may be actively advected from the region by a galactic wind. In order for the cosmic rays to sample the mean density of the ISM, the magnetic field lines along which they propagate must intersect a suÿciently large number of molecular clouds, and the cosmic rays must remain within the region for a suÿciently long time before they are advected away. Therefore, cosmic ray sampling of the mean density of the ISM is far from a foregone conclusion. ", "In this paper, we use Monte Carlo simulations to study ", "TABLE 1 ", "Properties of M82 ", "cosmic ray sampling of an ISM with properties similar to that of the prototypical starburst galaxy M82. We selected this galaxy for our study because it has a well-studied starburst region with well-observed masses of its multi-phase ISM, galactic wind speed, and supernova rate. In M82, the ISM consists of a hot, di use medium in which dense, warm, ionized gas and dense, cold, molecular clouds are found (e.g., (<>)Westmoquette et al. (<>)2009). In particular, CO measurements suggest that the galaxy contains ∼3 ±1 ×108 M⊙of molecular gas that is largely found within clumpy clouds in the starburst region (e.g., (<>)Naylor et al. (<>)2010). Additionally, M82 has a well-observed galactic wind that travels approximately perpendicularly to the galactic plane and is believed to be primarily driven by supernova shock heating ((<>)Chevalier & Clegg (<>)1985). Estimates of the outflow velocity range from ∼500 −600 km s−1 as indicated by optical emission lines of ionized gas ((<>)Shopbell & Bland-Hawthorn (<>)1998, and references therein) to as high as ∼1400 −2200 km s−1 as suggested by X-ray observations ((<>)Strickland & Heckman (<>)2009). Although we use properties suggestive of M82 in this study (see Table 1 for a summary), our analysis is applicable to many other such systems. ", "Here, we evaluate how cosmic ray sampling of the ISM in starburst galaxies is a ected by the number of molecular clouds, the galactic wind speed, the extent to which the magnetic field is tangled, and the cosmic ray injection mechanism. In Section 2, we describe our Monte Carlo simulation parameters and the calculation of an emissivity ratio used to quantify the sampling behavior of the cosmic ray population. Section 3 details the results of our simulations and highlights the region of parameter space in which the emissivity ratio is elevated by a factor of a few. In Section 4, we review our results, compare them to existing models, and discuss their implications for exploring the cosmic ray calorimeter model and constructing the time-dependent spectrum of a burst of cosmic rays. " ], "2. MONTE CARLO SIMULATIONS ": [], "2.1. Simulation Parameters ": [ "The multi-phase ISM in the starburst region is modeled to consist of a hot, low-density medium in which cold molecular clouds are embedded. Observations and numerical simulations suggest that the molecular medium consists of many clumpy, fragmented clouds (e.g., (<>)Blitz (<>)1993; (<>)Inoue & Inutsuka (<>)2012). Although ∼3 ±1 ×108 M⊙of molecular gas has been detected in the star-", "TABLE 2 ", "Model Parameters ", "burst nucleus (e.g., (<>)Naylor et al. (<>)2010), we take a conservative order of magnitude estimate of ∼1 ×108 M⊙of molecular gas within our spherical starburst region of radius R= 100 pc, corresponding to a mean number density n∼486 cm−3 . Assuming that the upper limit on the maximum mass of giant molecular clouds in the Milky Way of ∼104 −106 M⊙is applicable to M82, this implies on the order of Nc ∼103 molecular clouds. We hold the total mass of molecular gas constant while varying the number of clouds from Nc = 200 to Nc = 3000 clouds to explore both the nominal case (Nc ∼3000) as well as the limit of very small cloud numbers and very high cloud densities (Nc ∼200). Each cloud is taken to have a volume of Vc ∼27 pc 3 , and thus the clouds’ volume filling factor varies from ∼0.1% to ∼2%. ", "Note that we are explicitly neglecting the contribution of the hot, low-density medium to the density sampled by cosmic rays. Due to the di usive nature of particle motion along magnetic field lines (which reduces their e ective propagation speed from cto the Alfv´en speed, vA), the contribution of the low-density gas is greater than the simple ratio nh /nc , where nh and nc are the mean densities of the hot gas and the cold molecular gas, respectively. We estimate that this is approximately a 10% e ect for the M82 environment. Likewise, warm (∼104 K) ionized gas, in addition to hot gas and cold molecular gas, is undoubtedly present and would also have a small e ect on the model. We will consider these small contributions in future work. ", "Additionally, a supernova-driven galactic wind is included in the model with a linear profile and a wind speed of vadv at a height z= 100 pc above the mid-plane (v= vadv(z/100 pc)). We vary vadv from 0 to 2000 km s−1 , with 500 km s−1nominally taken as the favored value ((<>)Yoast-Hull et al. (<>)2013). We assume the wind carries away the magnetic field lines embedded in the hot gas, together with the cosmic rays loaded onto the field lines, but that the molecular component remains behind. ", "Since the starburst region of M82 is highly turbulent, we expect the magnetic field to have a significant random component. Cosmic ray propagation along tangled magnetic field lines is approximated as a random walk process governed by a mean free path, λmfp, which parameterizes the magnetic correlation length. As the magnetic geometry is not well known, we vary λmfp from 0.5 pc to 25 pc. A longer value of λmfp (i.e., λmfp = 50 pc) was found to yield comparable results to the λmfp = 25 pc case. Due to scattering by short wavelength Alfv´en waves generated by the cosmic ray streaming instability, cosmic ", "Fig. 1.— Summary of the cosmic ray sampling behavior during a representative run of the Monte Carlo code for each  mfp considered. From top to bottom, the figures show the distance traveled by particles before leaving the starburst region, the distance traveled inside of molecular clouds, and the average density sampled normalized to the mean density of the ISM, neglecting any contribution from the hot or warm ionized gas. The particles show a significant range of sampling behaviors; all density distributions show a peak at ˆP /ˆSB = 0, where the particles sample no clouds, as well as varying degrees of a tail at ˆP /ˆSB> 5, where they∼sample average densities greater than the mean density of the ISM by as much as an order of magnitude. These results were found for injection at the center of the starburst region, Nc = 3000 clouds, vadv = 500 km s−1 , and NP = 106 particles, although the results are broadly consistent with the full parameter space. ", "rays are taken to travel along the field lines at the Alfv´en speed, vA ((<>)Kulsrud & Pearce (<>)1969). Given a magnetic field strength B= 275 µG and an average density of the hot medium n= 0.33 cm−3 ((<>)Yoast-Hull et al. (<>)2013), the Alfv´en speed is found to be vA = 960 km s−1 . ", "We use three methods of injecting the cosmic rays into the starburst region. First, we consider the simple scenario where all cosmic rays are injected at the center of the region. Second, in accordance with models of distributed cosmic ray acceleration such as second-order Fermi acceleration by interstellar turbulence, we inject the particles randomly throughout the region. Finally, in agreement with models of point source cosmic ray acceleration such as first-order Fermi acceleration by supernova shocks, we inject the particles at randomly chosen supernova shock sites. We choose 30 injection sites due to observational evidence from radio interferometry for ∼30 active supernova remnants (SNRs) in M82 (e.g., (<>)Fenech et al. (<>)2010). One might expect SNRs and star-forming molecular clouds to spatially coincide, as young, massive stars are not likely to travel far from their place of birth over the course of their lifetimes. Additionally, recent observations of γ-ray emission from molecular clouds associated with SNRs (e.g., (<>)Aharonian et al. (<>)2008b,(<>)a; (<>)Abdo et al. (<>)2010, (<>)2011; (<>)Ajello et al. (<>)2012) provide evidence for a spatial correlation between SNRs and molecular clouds. Thus, for our final injection method, our injection sites are chosen randomly on spheres of radius r= 3 pc centered on clouds. For all cosmic ray injection methods, we disallow injection inside of clouds due to the reduced acceleration eÿciency expected in cold, dense environments. ", "The Monte Carlo simulations are run with NP = 104 particles for a given choice of Nc, vadv, λmfp, and cosmic ray injection mechanism (see the Appendix for a discussion of the number of particles necessary to achieve convergence). At the beginning of a run, we select the cosmic rays’ initial injection sites as well as a random molecular cloud distribution, and we map the state of the ISM onto a three-dimensional grid with a resolution of 0.5 pc. Starting from its injection site, a given particle executes a random walk governed by a mean free path λmfp broken down into steps of length l= 0.5 pc. At each step, the contribution of advection to the particle motion, the component of the ISM being sampled (in/out of cloud), and the particle’s presence in the starburst region (in/out of region) are assessed. If a particle is inside of a cloud at the beginning of a step, the galactic wind does not act on the particle for that step, and the particle is taken to travel at the speed of light instead of at the Alfv´en speed. The particles are taken to free-stream inside of molecular clouds due to the destruction by ion-neutral damping of Alfv´en waves resulting from the streaming instability ((<>)Kulsrud & Pearce (<>)1969). The particle is scattered when it has traveled a distance equal to its mean free path, and the process repeats until the particle has traveled out of the region. ", "Note that we do not explicitly model energy losses along the particle trajectories, deferring a discussion of it to Section (<>)4. The cloud column densities (∼5 ×1023 cm−2 , Nc = 3000 clouds; ∼7 ×1024 cm−2 , Nc = 200 clouds) are well below the characteristic column densities for energy loss (NE ∼2 ×1026 cm−2 , E= 1 GeV; ", "NE ∼3 ×1025 cm−2 , E= 1 TeV) except for very small Nc and/or very high proton energies. This suggests that cosmic rays make several passes though clouds before undergoing collisional energy losses. See Table 2 for a summary of our model parameters. ", "Figure 1 summarizes the sampling behavior of the cosmic rays for a single representative run of the Monte Carlo code. For each λmfp, we show distributions of the total distance traveled, the total distance traveled in molecular clouds, and the average density sampled ( ρP ) normalized to the mean density of the ISM in the starburst region ( ρSB ). It is clear from the density distributions that the particles display a diverse range of sampling behaviors. The extremes of this behavior are seen in the peaks at ρP /ρSB = 0, where no clouds are sampled, as well as in the tails at ρP /ρSB ∼>5, where the particles sample densities as high as an order of magnitude greater than the mean density of the ISM. Note that for λmfp >1, a peak appears near ρP /ρSB = 1. These trends hint that as we proceed to more precisely quantify the sampling behavior, we may reasonably expect the particles to roughly sample the mean density of the ISM over much of parameter space. " ], "2.2. Emissivity Calculations ": [ "To quantitatively assess the assumption that cosmic rays sample the mean density of the ISM in the starburst region and evaluate its e ect on γ-ray emission, we use our simulation results to construct a function ϕ(n/n)d(n/n), which gives the fraction of cosmic rays that sample gas density nrelative to the mean density n. The emissivity due to pion production by cosmic rays in molecular clouds is then given by " ], "(1) ": [ "where nand ncr are the number densities of the medium and the cosmic rays, respectively, σcoll is the cross section for pion production, vis the velocity of the particles, and Eint is the energy produced by the interaction. If the cosmic rays do indeed sample the mean density of the ISM, the emissivity is given by ", "Thus, the ratio of emissivities αis given by " ], "(2) ": [ "(3) ", "Although σcoll is a function of energy, αis independent of energy for all cosmic rays that satisfy the propagation assumptions given at the beginning of this section (i.e., the streaming instability results in propagation at the Alfv´en speed). αis easily obtained by appropriately normalizing, binning, and summing over the distributions of average densities sampled weighted by the densities themselves (see, e.g., Figure 1(c)). Thus, we determine αfor the range of parameters discussed in Section 2.1 and seek areas of parameter space where αdeparts significantly from unity. See the Appendix for a discussion of our determination of αfor a given choice of Nc, vadv, λmfp, and cosmic ray injection mechanism. The Appendix also details our estimation of the errors on αthat ", "originate from both the finite number of particles per run as well as the changes in the random cloud distributions and injection sites from run to run. " ], "3. RESULTS ": [], "3.1. Emissivity Ratios for all Parameters ": [ "Fig. 2.— Mean emissivities for all combinations of cosmic ray mean free path, injection mechanism (indicated by color), three wind speeds (shape), and two cloud numbers (fill). The lower panel is a closer view of a portion of the upper. It is clear that over the vast majority of parameter space, the emissivity ratios are clustered around = 1 and thus the cosmic rays roughly sample the mean density of the ISM. However, as discussed in Section 3.2, cosmic ray injection near the clouds in the limit of small cloud number and thus high cloud density results in emissivity ratios that are elevated by a factor of a few and the assumption that the cosmic rays sample the mean density breaks down. Additionally, cosmic ray injection randomly throughout the region results in emissivities that are suppressed by as much as a factor of ∼2; however, the suppression of the emissivity at low values of is not as dramatic as its elevation at high . ", "We begin by examining the densities sampled over the entire range of parameters considered in this study (i.e., all combinations of cosmic ray mean free path, injection mechanism, two molecular cloud numbers (Nc = 200, 3000 clouds), and three wind speeds (vadv = 0, 500, and 2000 km s−1)). In Figure 2, we see that the cosmic rays roughly sample the mean density of the ISM over much of parameter space. Overall, we do not achieve emissivities suppressed below α= 1 by a factor of more than ∼2; we ", "do achieve emissivities elevated above α= 1 by a factor of a few; and these modestly elevated emissivities occur only for cosmic ray injection near clouds in the limit of small cloud number and thus high cloud density. This case will be considered separately in Section 3.2. ", "Although much ofover parameter space, and cosmic ray injection mechanism. We note that αthere are still clear trends relating αto Nc, λmfp, vadv, tends to increase as λmfp decreases, suggesting that cosmic rays with short λmfp generally travel along more convoluted trajectories and thus spend more time sampling the starburst region as well as the clouds that they encounter. We also note a somewhat greater spread in αat shorter λmfp that may be due to more frequent scattering allowing for more varied paths through the starburst region and thus more varied emissivity outcomes. ", "For cosmic ray injection near clouds and randomly throughout the region, αtends to decrease as vadv increases. When injection occurs far from the midplane, the particle’s trajectory is immediately a ected by the wind, and advective losses dominate. However, for injection at the center of the region and short λmfp, αinstead increases with vadv. For central injection, the particle initially experiences very weak wind speeds, and advective losses are negligible. Additionally, for higher vadv, these particles may travel along more eÿcient trajectories and thus sample more clouds than for lower vadv, where they travel more convoluted trajectories and are more likely to circumvent clouds. ", "The cosmic ray injection mechanism also has other effects on the value of α. At a given λmfp, emissivities from injection at the center of the starburst region exceed those of injection randomly throughout the region, while injection near clouds in the case of Nc = 3000 clouds tends to fall between these two limiting cases. It is expected that cosmic rays injected at the center have the opportunity to sample greater densities than those injected randomly because they generally spend more time within the starburst region. Additionally, cosmic rays injected near clouds in the Nc = 3000 clouds case are expected to sample greater densities than those injected randomly because of their initial proximity to clouds. However, as these cosmic rays also sample lesser densities than those injected at the center of the region, it appears that this e ect is outweighed by the relatively shorter time spent within the region. ", "Finally, Nc has a clear e ect on the value of α, albeit one moderated by injection mechanism. When cosmic rays are injected near clouds and thus have ample opportunity to encounter dense gas, a smaller number of denser clouds produces a higher emissivity than a larger number of less dense clouds. However, when cosmic rays are randomly injected, they have a decreased likelihood of encountering dense gas; thus, despite the increase in cloud density, the decrease in cloud number may suppress the emissivity such that the results are comparable for both a low and high number of clouds. " ], "3.2. Emissivity Ratios for Injection Near Nc = 200 Clouds ": [ "We now discuss the region of parameter space where the cosmic rays sample the highest densities. As shown in Figure 3, the cosmic rays achieve the highest αvalues when injected the molecular clouds innear ", "Fig. 3.— Mean emissivities and 68% confidence intervals for cosmic ray injection near clouds in the limit of small cloud number (Nc = 200 clouds) and high cloud density. In this region of parameter space, we achieve emissivities elevated above = 1 by a factor of a few, with the highest emissivities achieved with large vadv and small mfp. ", "the limit of small cloud number (Nc = 200 clouds) and thus high cloud density. This result is not surprising, as cosmic ray injection near high-density molecular gas has the greatest likelihood of interaction between the particles and the molecular medium. As noted in Section 3.1, the value of αis highest for short λmfp (λmfp 5 pc),<∼where αvaries from ∼2 -7.5, than for longer λmfp , ", "where αis ∼2 -3. In addition to spending more time sampling the starburst region, particles with short λmfp may also be more likely to enter the clouds accompanying their injection sites than those that are able to travel more eÿciently away. ", "As is clear in Figure 3, the case of no galactic wind results in αvalues lower than that of high wind (vadv = 2000 km s−1) by a factor of ∼2, while the case of moderate wind (vadv = 500 km s−1) results in intermediate emissivities. At short λmfp, the cases of high and moderate vadv become comparable. Particles that are injected between the galactic plane and their accompanying cloud are likely to be advected into the cloud in the presence of a wind; the likelihood of this occurring may increase with vadv. Particles with short λmfp also have an increased likelihood of entering the clouds at their injection sites due to their inability to eÿciently travel away from these sites, rendering the emissivity less sensitive to vadv. ", "In summary, cosmic ray injection near Nc = 200 clouds for short λmfp and moderate to high vadv results in αvalues that are modestly elevated above α= 1 by a factor of a few. Note that the uncertainty associated with these emissivities increases dramatically in the limit of short λmfp and high vadv. The former dependence suggests that particles that experience more frequent scattering travel along more diverse trajectories and thus have more diverse emissivity outcomes. Additionally, the latter dependence implies that a higher vadv results in greater populations of particles that are rapidly advected away (resulting in low emissivities) as well as particles that enter the clouds near their injection sites (high emissivities). Therefore, although this region of parameter space results in elevated emissivities, the nature of the random cloud and injection point distributions is important in determining the observed emissivity. ", "We have now seen that cosmic rays injected near Nc = 200 high-density clouds sample densities in excess of the mean density of the ISM, while those injected near Nc = 3000 lower density clouds do not. Thus, for the case of injection near clouds, we seek an upper limit on the number of clouds necessary to achieve elevated emissivities. In Figure 4, we consider αfor vadv = 500 km s−1and a range of Nc values for which 200 ≤Nc ≤3000 clouds. It is clear that very low values of Nc are required to achieve elevated emissivities. For the shortest λmfp, decreasing Nc from 3000 to 1000 clouds increases αby only a factor of ∼2. It is only when Nc is decreased to ∼400 clouds that αbecomes elevated by a factor of a few. Thus, in the case of cosmic ray injection near clouds, a very small cloud number and thus very high cloud density is required to achieve elevated emissivities. " ], "3.3. Cosmic Rays that Do Not Sample Clouds ": [ "We now consider the fraction of cosmic rays that escape from the starburst region without sampling molecular clouds at all. This sheds light on the relationship between the sampling behavior of individual particles and the sampling of the particle population as a whole. In Figure 5, it is clear that over the full parameter space considered, the fraction fof cosmic rays that escape from the starburst region without sampling clouds ranges from near complete sampling (f∼2×10−5 , Nc = 3000 clouds, vadv = 0 km s−1 , injection at the center) to near com-", "Fig. 4.— Mean emissivities and 68% confidence intervals are strongly dependent on the cloud number and thus the cloud density for cosmic ray injection near clouds. Though the emissivity is significantly elevated for short mfp in the limit of small Nc, these emissivities decrease very rapidly as Nc increases. At Nc values greater than several hundred, approaches unity for all mfp. ", "Fig. 5.— Fraction of cosmic rays f that escape the starburst region without sampling molecular clouds as a function of mfp. Though the cosmic rays approximately sample the mean density of the ISM over much of parameter space, they clearly undergo a diverse range of sampling behaviors, from near complete sampling of clouds (f ∼2 ×10−5 ) to near absence of sampling (f ∼0.98). These fractions are determined from single NP = 106 particle runs, and will change slightly for di erent distributions of molecular clouds and cosmic ray injection sites. ", "plete escape (f∼0.98, Nc = 200, vadv = 2000 km s−1 , random injection). As we found that cosmic rays approximately sample the mean density of the ISM over much of parameter space, we see that emissivity values of α∼1 are achieved over the full range in f. Therefore, we may find α∼1 both when the vast majority of particles sample roughly the mean density (occurring mainly for high Nc values), as well as when only a small minority of particles encounter the molecular medium (mainly low Nc values). ", "When we are interested in the mean density sampled by the cosmic ray population as a whole, such as when evaluating the cosmic ray calorimeter model, these two broad sampling behaviors can be considered comparable. However, if large numbers of cosmic rays escape from the star-", "burst region without losing energy to collisional processes in the molecular medium, this may have important implications for the galactic environment. It has been shown that when the cosmic rays are self-confined, meaning that they generate the waves that trap them, they may contribute to driving a galactic wind ((<>)Breitschwerdt et al. (<>)1991; (<>)Everett et al. (<>)2008). Thus, we may be more likely to observe a galactic wind under conditions for which cosmic rays are able to travel through the hot medium to escape the starburst region without undergoing energy losses in molecular clouds. " ], "4. DISCUSSION AND APPLICATIONS ": [], "4.1. Comparison to a “Back of the Envelope” Model ": [ "We now discuss our Monte Carlo simulations in the context of a simple, “back of the envelope” model of cosmic ray sampling of a clumpy ISM. We refer to the model in Section 4.1 of (<>)Yoast-Hull et al. ((<>)2013) adapted to account for di usive cosmic ray propagation. This model suggests that cosmic rays will sample the mean density of the ISM if two general conditions are met. First, the magnetic field lines along which they propagate must pass through a representative sample of the varied components of the ISM. Second, they must be able to travel along or di use across field lines to encounter these components before they are advected from the region. To determine whether these conditions are met, we compare the timescale τdiff for cosmic rays to di use between clouds to the timescale τadv for them to be advected from the region. We define the former as ", "(4) ", "where lc is the mean distance between clouds and the factor of lc/λmfp accounts for cosmic ray di usion with a mean free path λmfp <lc. lc can be approximated as lc ∼R/Nc1/3 , where Ris the radius of the starburst region. We take lc to be the mean minimum distance that cosmic rays must travel to reach a cloud assuming that there is no spatial correlation between clouds and cosmic ray injection sites. For this reason, this model is applicable only to cosmic rays traveling along suÿciently tangled magnetic field lines (λmfp <lc) and to cosmic rays that have been injected either centrally or randomly in the starburst region. ", "We define the condition under which cosmic rays will encounter molecular clouds to be ", "(5) ", "where τadv = 2R/vadv is the advection timescale for cosmic rays that experience a mean advecting wind speed of ∼vadv/2. Taking vA = 960 km s−1 , vadv = 500 km s−1 , and R= 100 pc, we arrive at a relationship between Nc and λmfp that predicts whether or not cosmic rays are able to encounter clouds before being advected from the starburst region: ", "(6) ", "We now compare this prediction with the results of our ", "Monte Carlo simulations. Considering broadly the predictions of Equation (6) for central or random cosmic ray injection, vadv = 500 km s−1 , and λmfp <lc, this prediction suggests that for all but the shortest λmfp, cosmic rays should encounter dense gas for both Nc = 200 and 3000 clouds. For central injection, there is very good agreement between this prediction and our simulation results, as our αvalues are indeed close to unity. For random injection, however, there is only very broad agreement, as αis instead close to ∼0.6. This discrepancy is not surprising for the case of random injection, as our simple model does not account for the e ective decrease in the advection timescale for particles injected considerably closer than R= 100 pc to the edge of the region. ", "Additionally, for λmfp = 0.5 pc, Equation (6) suggests that we require Nc ∼>350 clouds for the cosmic rays to encounter dense gas. For central cosmic ray injection, however, we find from our simulations that the cosmic rays do indeed interact with dense gas for both Nc = 200 and 3000 clouds. For random injection, we again find that the value of αis relatively insensitive to cloud number. This discrepancy is due to our simple model’s inability to account for the e ect of increasing cloud density with decreasing cloud number. In our simulations, we find that despite fewer cosmic rays sampling clouds in the limit of small cloud number, we still observe emissivities comparable to those found for larger cloud numbers due to the increase in cloud density. ", "Thus, our simple model is useful for broadly considering constraints on the conditions for cosmic ray sampling of molecular clouds. However, this model solely suggests whether or not cosmic rays encounter dense gas. We have seen that the observed emissivity is dependent not only on getting cosmic rays to the gas, but on additional properties such as the gas density as well. Thus, for accurately predicting observed emissivities, our full Monte Carlo simulations are required to account for all relevant subtleties. " ], "4.2. Implications for Starburst Calorimeter Models ": [ "The starburst calorimeter model suggests that there is a direct relationship between the energy imparted to cosmic rays by supernovae and the energy lost by cosmic rays within the starburst region. Thus, there is a relationship between the supernova rate, and therefore the SFR, and observable emission in the radio (cosmic ray electrons) and γ-ray (cosmic ray protons) regimes. Here, we evaluate the extent to which our simulated starburst region behaves as a cosmic ray calorimeter under a range of physical conditions by comparing the particles’ confinement timescales τC to their energy loss timescales τE. Note that as starburst galaxies with properties like those of M82 are well established to be e ective cosmic ray electron calorimeters (e.g., (<>)Yoast-Hull et al. (<>)2013), we will consider only cosmic ray protons for the remainder of our analysis. ", "The cosmic ray transport equation gives rise to the functional form of τE (e.g. (<>)Yoast-Hull et al. (<>)2013) and is given by ((<>)Longair (<>)2011): ", "(7) ", "Fig. 6.— To assess whether our simulated starburst region is an e ective cosmic ray proton calorimeter, we examine the ratio of the confinement timescale ˝C to the energy loss timescale ˝E. Here, we consider ˝C /˝E for E = 1 GeV and E = 1 TeV protons for a range of conditions on Nc, mfp, vadv, and the cosmic ray injection mechanism. Although the starburst region is an e ective proton calorimeter at low energies (˝C /˝E > 1, E ∼0.1 GeV), it may be at best a partial proton calorimeter for the energies considered here, where ˝C /˝E ∼< 1 over non-trivial portions of parameter space. ", "Here, N(E,t)dEis the cosmic ray number density at time twith energies between Eand E+ dE, dE/dtis the rate at which radiative and collisional losses decrease a particle’s energy, and Q(E,t)dEis the rate at which particles are injected per unit volume with energies between Eand E+dE. Here, τC accounts for both di usive and advective losses and is assumed to be independent of energy. We may assume that Q(E) and N(E) are of the form Q(E) = AE−and N(E) ≈Q(E)τ(E), where τ(E)−1 ≡τC−1 + τE−1 is the total energy loss rate. For the steady state, the energy loss rate τE is then found to be ", "(8) ", "The energy loss rates for cosmic ray protons are due primarily to ionization and pion production, the latter of which is dominant above ∼1 GeV ((<>)Schlickeiser (<>)2002). Note that we do not account for the (negligible) contribution from Coulomb e ects. The energy loss rates from both ionization and pion production are directly proportional to the average density sampled by the particles, ", "defined as the mean density of the ISM modified by a multiplicative factor of α, nP = αnSB . We define the confinement timescale τC for a given vadv, λmfp, and cosmic ray injection mechanism as the median time that NP = 106 particles take to escape from the starburst region. We calculate the ratio of the confinement to energy loss timescales τC/τE and seek physical conditions where the calorimeter model fails to hold (τC/τE <<1). ", "In Figure 6, we show τC/τE for E= 1 GeV and E= 1 TeV protons over the full parameter space considered. At a given energy, the value of τC/τE ranges over more than three orders of magnitude and is strongly dependent on vadv, λmfp, and the cosmic ray injection mechanism. For the lowest energy protons (E∼0.1 GeV), τC/τE ∼>1 for all parameters considered, and the starburst region is an e ective proton calorimeter. At moderate proton energies (E∼1 GeV), however, τC/τE only exceeds unity for all λmfp in the absence of a galactic wind. Additionally, τC/τE does not exceed unity for the longest λmfp for any parameters considered. At higher energies (E∼1 TeV), τC/τE again exceeds unity in all cases except that of large Nc, high vadv, and injection throughout the region. Thus, for moderate to high proton energies, our simulated starburst region can only be deemed a partial proton calorimeter without further knowledge of the physical conditions in the region. Note that the starburst region would be calorimetric over larger portions of parameter space if we had made a more generous estimate of the molecular gas mass contained within the region. " ], "4.3. The Spectrum of a Cosmic Ray Burst ": [ "We conclude by constructing the spectrum of a cosmic ray burst from a single supernova explosion and evaluating how the time evolution of the spectrum is a ected by the density sampled by the particles. Although the steady state solution is generally preferred for the cosmic ray populations of starburst galaxies, the behavior of a burst of cosmic rays may inform our understanding of how smaller scale cosmic ray populations evolve over short timescales. Here, we again consider only cosmic ray protons, as only a small minority of a supernova’s energy is believed to be imparted to cosmic ray electrons ((<>)Blandford & Eichler (<>)1987). ", "To construct the spectrum of a burst of cosmic rays, we begin by dropping both the source and the advec-tion terms from the cosmic ray transport equation; the advection term will be added later. Thus, we have ", "where we define the energy loss rate to be ", "(9) ", "(10) ", "For initial condition N(E,0) = No(E), one can show that N(E,t) takes the form: ", "(11) ", "where Eo(E,t) is the energy at time t= 0 of a particle with energy Eat time t. ", "We now restore the advection term to the cosmic ray transport equation: ", "(12) ", "We consider the case where N(E,t) assumes the form N(E,t) = f(E,t)e−t/˝C . By substituting this expression into the transport equation with advective losses added, we discover that f(E,t) is a solution of this equation when advective losses are neglected. Thus, we obtain an expression for N(E,t) accounting for advective losses by multiplying our previous expression for N(E,t) by a factor of e−t/˝C . Additionally, we assume that the initial injected spectrum is a power law No(E) = AE−with spectral index γ∼>2. Therefore, the spectrum of a burst of cosmic rays has the functional form: ", "(13) ", "To find the initial energy Eo given the energy Eat time t, we solve for the value of Eo that satisfies ", "(14) ", "This is done by selecting a value for Eo, numerically integrating backward, and evaluating the consistency of the resulting t′with the desired time t. ", "Finally, we determine the value of Ain order to satisfy: ", "(15) ", "where ESN is the energy released by a supernova explosion, ηis the fraction of a supernova’s energy transferred to cosmic rays, and VSB is the volume of the starburst region. Therefore, Ahas the functional form: ", "(16) ", "where we take γ= 2.1, Emin = 0.1 GeV, ESN = 1051 erg, η= 0.1, and VSB ∼1062 cm3 . Note that we take the cosmic ray acceleration timescale in a supernova shock to be τaccel ∼<104 yr <<τC, and thus τaccel can be taken to be e ectively instantaneous. ", "The time evolution of the spectrum is shown in the upper panel of Figure 7 for α= 1 and τC = 106 yr. This choice of αand τC is consistent with much of the parameter space considered. As in Section 4.2, the energy loss rates for cosmic ray protons are calculated according to (<>)Schlickeiser ((<>)2002). It is evident that the spectrum evolves rapidly with time; on timescales comparable to the confinement time, the number density decreases by several orders of magnitude at all energies. This decline is most dramatic at high energies (E∼>10 GeV), where the spectrum steepens sharply by t∼106 yr due to extreme energy loss rates as well as the initial lack of high-energy particles. ", "In the lower panel of Figure 7, we show the spectrum at time t∼2×105 yr for a range of densities sampled. It is clear that increasing the value of αby a factor of a few rapidly decreases the lifetime of the particle population. ", "Fig. 7.— In the upper panel, we show the time evolution of the spectrum of a burst of cosmic ray protons from a single supernova explosion under the physical conditions of M82. Here, we take = 1 and ˝C = 106 yr, values consistent with much of the parameter space considered. Other confinement times can be easily accounted for by appropriately adjusting the factor of e−t/˝C in Equation (13). The rapid evolution of the spectrum with time suggests that cosmic ray bursts are short lived in these environments. In the lower panel, we show the spectrum of a cosmic ray burst at time t ∼˝accel (dashed line) as well as at t ∼2 ×105 yr (solid lines) for a range of values of . It is clear that cosmic ray bursts that sample densities greater than the mean density of the ISM by a factor of a few have dramatically shortened lifetimes. ", "For example, by comparing the top and bottom panels of Figure 7, we see that the spectrum of a cosmic ray burst with α= 1 that has evolved for t∼1.6 ×106 yr is identical to the spectrum of a burst with α= 8 that has evolved for only t∼2 ×105 yr. ", "Overall, the spectrum of a burst of cosmic rays evolves rapidly under the physical conditions of a starburst galaxy like M82, and thus the cosmic ray population produced by a single supernova explosion under these conditions is a short-lived phenomenon. However, a conservative estimate of the supernova rate in M82 is ∼0.07 yr−1 ((<>)Fenech et al. (<>)2008), and thus we can expect that a supernova explosion will occur in M82 every ∼15 yr. The evolved cosmic ray spectra shown in Figure 7 will therefore never be made manifest, and instead will be regularly replenished by newly injected particles. However, in other star-forming environments where the supernova rate may be very low, such as extreme dwarf ", "galaxies or OB associations, these spectra may indeed be manifested due to a lack of replenishing particles. " ], "5. CONCLUSIONS ": [ "In the interest of understanding how cosmic rays sample the clumpy ISM in a starburst environment, we have undertaken Monte Carlo simulations of cosmic ray sampling of molecular clouds under the physical conditions of the archetypal starburst galaxy M82. Here, we briefly review the results of our study: ", "Though we have illustrated several applications here, cosmic ray sampling of a clumpy ISM may be applied to a wide range of science goals seeking to understand the relationship between the state of the multi-phase ISM, the star formation processes, and the cosmic ray populations of starburst galaxies. ", "We gratefully acknowledge the support of NSF AST-0907837 and NSF PHY-0821899 (to the Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas). We thank Benjamin Brown, Sebastian Heinz, Dan McCammon, and Joshua Wiener for helpful discussions, as well as Masataka Okabe and Kei Ito for supplying the colorblind-friendly color palette used in this paper (see ()fly.iam.u-tokyo.ac.jp/color/index.html). ", "This work has made use of NASA’s Astrophysics Data System. " ] }