Assessing Radiation Pressure as a Feedback Mechanism in Star-Forming Galaxies
aa r X i v : . [ a s t r o - ph . C O ] N ov S UBMITTED TO A P J Preprint typeset using L A TEX style emulateapj v. 11/26/04
ASSESSING RADIATION PRESSURE AS A FEEDBACK MECHANISM IN STAR-FORMING GALAXIES B RETT
H. A
NDREWS & T ODD
A. T
HOMPSON S UBMITTED TO A P J ABSTRACTRadiation pressure from the absorption and scattering of starlight by dust grains may be an important feed-back mechanism in regulating star-forming galaxies. We compile data from the literature on star clusters,star-forming subregions, normal star-forming galaxies, and starbursts to assess the importance of radiationpressure on dust as a feedback mechanism, by comparing the luminosity and flux of these systems to their dustEddington limit. This exercise motivates a novel interpretation of the Schmidt Law, the L IR – L ′ CO correlation,and the L IR – L ′ HCN correlation. In particular, the linear L IR – L ′ HCN correlation is a natural prediction of radiationpressure regulated star formation. Overall, we find that the Eddington limit sets a hard upper bound to the lumi-nosity of any star-forming region. Importantly, however, many normal star-forming galaxies have luminositiessignificantly below the Eddington limit. We explore several explanations for this discrepancy, especially therole of “intermittency” in normal spirals—the tendency for only a small number of subregions within a galaxyto be actively forming stars at any moment because of the time-dependence of the feedback process and theluminosity evolution of the stellar population. If radiation pressure regulates star formation in dense gas, thenthe gas depletion timescale is 6 Myr, in good agreement with observations of the densest starbursts. Finally, wehighlight the importance of observational uncertainties—namely, the dust-to-gas ratio and the CO-to-H andHCN-to-H conversion factors—that must be understood before a definitive assessment of radiation pressureas a feedback mechanism in star-forming galaxies. Subject headings: galaxies: general, evolution, ISM, stellar content, starburst — stars: formation INTRODUCTION
Understanding global star formation is crucial in under-standing galaxy evolution and the assembly of the z = 0 stel-lar population over cosmic time. Observations indicate thatonly a few percent of the available gas reservoir in galaxies isconverted into stars per local free-fall time (Kennicutt 1998;Krumholz & Tan 2007). In addition, models of the interstellarmedium (ISM) suggest that energy and momentum injectedby massive stars could act as a feedback loop by driving su-personic turbulence, which would cause most of the gas to beinsufficiently dense to collapse, rendering star formation inef-ficient (Krumholz & McKee 2005). However, the interactionbetween star formation and the ISM is not well understood,and a mechanism for the regulation of star formation acrossthe large dynamic range of star-forming environments has notyet been conclusively identified. Proposed mechanisms in-clude supernova explosions, expanding HII regions, stellarwinds, cosmic rays, magnetic fields, and radiation pressureon dust (McKee & Ostriker 1977; Matzner 2002; Cunning-ham 2008; Chevalier & Fransson 1984; Socrates et al. 2008;Kim 2003; Scoville et al. 2001; Scoville 2003; Thomp-son, Quataert, & Murray 2005, hereafter TQM; Krumholz &Matzner 2009, hereafter KM09; Murray, Quataert, & Thomp-son 2010, hereafter MQT; Draine 2010; Hopkins et al. 2010). In the case of radiation pressure on dust, UV and opticalradiation from OB stars is absorbed and scattered by dustgrains and subsequently re-radiated as IR radiation. The dustgrains are coupled to the gas of the ISM through collisions Department of Astronomy, The Ohio State University, 140 West 18thAvenue, Columbus, OH 43210, [email protected] Center for Cosmology and Astroparticle Physics, The Ohio State Uni-versity, 191 West Woodruff Avenue, Columbus, OH 43210. Alfred P. Sloan Fellow ISM turbulence driven by non-stellar processes, such as disk instabilities,has also been proposed (Sellwood & Balbus 1999; Wada et al. 2002; Piontek& Ostriker 2004, 2007). and magnetic fields, so radiation pressure on the dust ex-erts a force on the gas as well (O’dell et al. 1967; Ferrara1993; Laor & Draine 1993; Murray, Quataert, & Thomp-son 2005). On galaxy scales, TQM showed that radiationpressure could constitute the majority of the vertical pressuresupport in dense starburst galaxies such as ultra-luminous in-frared galaxies (ULIRGs). Likewise, models of giant molec-ular cloud (GMC) disruption predict that radiation pressure isthe dominant feedback mechanism regulating star formationin the birth of massive star clusters (MQT; KM09). In this pic-ture, gas in a marginally-stable galactic disk collapses to forma GMC and a central compact star cluster. When the stellarmass and luminosity of the cluster exceed the Eddington limitfor dust, the overlying gas reservoir is expelled. Thus, the fi-nal stellar mass in individual star clusters is regulated by thedust Eddington limit. The centers of ULIRGs and GMC coresare optically thick to both UV and the re-radiated IR photons,which make them ideal candidates for radiation pressure sup-port since essentially all of the momentum from the starlightis efficiently transferred to the gas. Recent observations in-dicate that the most luminous GMCs in the Milky Way aredisrupted by radiation pressure (Murray 2010).In this paper, we critically assess the theory of radiationpressure regulated star formation by comparing the picturedeveloped by TQM, MQT, and KM09 with the available ob-servations of star-forming galaxies ranging from dense indi-vidual star clusters and GMCs, to normal spiral galaxies andstarbursts. In §2 we describe the current model of radiationpressure feedback. We emphasize the deviations from thesimplest version of the dust Eddington limit in assessing radi-ation pressure regulated feedback that arise from ambiguitiesin the value of the flux-mean dust opacity, and the tendencyfor low-density galaxies to have highly intermittent knots orhotspots of star formation across their disks. In §3, we com-pare data from the literature to models of radiative feedback.In §4, we discuss our conclusions, the major observationaland theoretical uncertainties in our analysis, and the implica-tions of our results. THEORETICAL ELEMENTS
The statement that radiation pressure may be an importantfeedback mechanism in galaxies is equivalent to the statementthat galaxies as a whole or the star-forming subregions withinthem approach or exceed the Eddington limit for dust, F Edd = 4 π Gc Σ κ F , (1)where F Edd is the Eddington flux, Σ is the surface density ofthe dominant component of gravitational potential in the star-forming region, and κ F is the flux-mean opacity. The overallpicture is that star-forming regions meet the Eddington limitand self-regulate in analogy with an individual massive star(TQM; see also Scoville et al. 2001; Scoville 2003; MQT;KM09). We would thus naively expect to test the theory ofradiation pressure regulated star formation by taking the ratioof the observed flux ( F obs ) to F Edd . However, a direct compar-ison between the simple theoretical expectation F obs / F Edd → Σ in equation (1) for unresolved galaxies or unre-solved star-forming subregions. Below, we consider both COand HCN emission (see §3; Figures 1 & 2), but the conversionfrom the luminosity in either of these molecular gas tracers togas mass where the stars are forming, is highly uncertain. An-other uncertainty is the coupling of the radiation field to thegas, which is complicated due to both the non-gray nature ofthe dust opacity and the clumpiness of the gas on all scales(see §2.1 below). Finally, there is an additional complicationnot readily apparent from the time-independent statement ofequation (2): the star formation rate (SFR) across the face ofa large spiral galaxy is highly intermittent so that only a smallnumber of subregions are bright at any time. As discussed byMQT and in detail below (§2.2), this intermittency can causenormal star-forming galaxies to be appear significantly sub-Eddington ( F obs / F Edd ≪
1) when only their average propertiesare considered, but much closer to Eddington when a modelis used to take this effect into account.
The Radiation Pressure Force
The coupling between radiation and gas in star-forming en-vironments is complex primarily because the flux-mean opac-ity κ F in equation (1) has a full range of more than 3 dex,depending on whether the SED of the system considered isdominated by UV or FIR light. However, there are two dis-tinct regimes: optically thick to UV but thin to the re-radiatedFIR and optically thick to FIR. We call these the “single-scattering” and “optically thick” limits, respectively. Single-scattering Limit
Regions in the single-scattering limit are optically thick tothe UV but optically thin to the FIR ( τ FIR ∼ κ FIR Σ g / Σ g . ⊙ pc - κ - f - , , (3)where κ FIR = κ f dg , is the Rosseland-mean dust opacitywith κ = κ/ (2 cm g - ) (see §2.1.2) and f dg , = f dg ×
150 is the dust-to-gas ratio. In the single-scattering limit, UV pho-tons are absorbed once and then re-radiated as FIR photons,which free-stream out of the medium. Since the column-averaged flux-mean optical depth in this limit is always equalto unity, the flux-mean opacity for the single-scattering limitis ∼ / Σ g . The Eddington flux is then F sEdd ∼ L ⊙ kpc - (cid:18) Σ g
10 M ⊙ pc - (cid:19) f - , (4)where f gas = Σ g / Σ tot is the gas fraction and Σ tot ≡ Σ g + . Σ ⋆ (Wong & Blitz 2002). The wide range of column densitiesover which this limit is applicable implies that the averagemedium of most star-forming galaxies, some starbursts, andthe GMCs that constitute them is single-scattering. Optically Thick Limit
Dense starbursts and GMCs can reach the high gas sur-face densities Σ g & ⊙ pc - κ - f - , necessary to be-come optically thick to FIR photons ( τ FIR & P rad ∼ τ FIR F / c , and κ FIR depends on temperature (Bell & Lin1994; Semenov et al. 2003). The functional form of κ FIR nat-urally leads to two regimes: “warm” ( T <
200 K) and “hot”(200 K < T < T sub , where T sub ∼ T ∼ τ T ∼ κ FIR Σ F σ SB ∼ κ FIR M g π R M ⋆ Ψ π R σ SB T ∼
290 K κ / Ψ / M / , M / ⋆, R - , (5)where T eff is the effective temperature, κ = κ/ (10 cm g - ), Ψ = 3000 ergs s - g - is the light-to-mass ratio of a zeroage main sequence stellar population, M g , = M g / (10 M ⊙ ),and M ⋆, = M ⋆ / (10 M ⊙ ). Warm Starbursts — For T <
200 K, the Rosseland meanopacity increases as κ FIR ( T ) ≈ κ o T , where κ o ≈ × - cm g - K - f dg , . In this case, F Edd ≈ π Gc σ SB κ o f , f gas ! / ∼ L ⊙ kpc - f - / f - , . (6)Remarkably, the flux necessary to support the medium is in-dependent of Σ (TQM). Hot Starbursts — Intense, compact starbursts may have cen-tral temperatures greater than 200 K. The correspondingopacity is roughly constant with temperature: κ FIR ( T ) ≈ g - f dg , for temperatures 200 K . T . T sub . Fortypical numbers, F thickEdd ∼ L ⊙ kpc - (cid:18) Σ g M ⊙ pc - (cid:19) f - / f - , . (7)The high surface densities necessary to enter this regime mayonly be attained in the pc-scale star formation thought to at-tend the fueling of bright AGN (Sirko & Goodman 2003;TQM; Levin 2007). Galaxies with surface densities less than ∼ ⊙ pc - will be opticallythin with respect to dust. Below this limit, the ionization of neutral hydro-gen will becomes the dominant source of opacity. The large cross-section( σ HI ≈ . × - cm per H atom) implies an incredibly small surface den-sity ( Σ g & - M ⊙ pc - ) is required for the medium to be optically thick toionizing photons. These ionizing photons transfer momentum directly to thegas on the same order as the momentum transfer due to the single-scatteringlimit for dust. Thus, we encompass this limit and the single-scattering limitfor dust under the same heading. GMC Evolution & Intermittency
In order to gauge the importance of GMC evolution and in-termittency, we adopt the simple picture presented by MQTthat marginally stable ( Q ≈
1) disks fragment into sub-unitson the gas disk scale height ( h ) to form GMCs. An individualstar cluster is born, reaches the critical Eddington luminos-ity threshold, and then expels the overlying gas. Importantly,the timescale for collapse and expansion of the GMC is thedisk dynamical timescale, t dyn , which can be much longer thanthe characteristic timescale for the stellar population to de-crease in total luminosity, the main-sequence lifetime of mas-sive stars, t MS ∼ × yr. In this picture, a low-density star-forming galaxy with radius r should have ∼ ( r / h ) sub-units,but only a small fraction ξ (the “intermittency factor”) shouldbe bright at any one time. If each subregion reaches the Ed-dington luminosity for a time t MS and is then dark, and then ifa large number of subregions are averaged, one expects ξ ≡ N on N tot ∼ L obs L Edd ∼ t MS t dyn + t MS , (8)where t dyn ∼ (cid:18) π (2 h )32 G Σ tot (cid:19) / ∼ . × yr h / f / (cid:18)
10 M ⊙ pc - Σ g (cid:19) / , (9) h = h / (100 pc), N on is the number of sub-units that are“on,” and N tot is the total number of sub-units. For ex-ample, the normal star-forming galaxy M51 has a observedbolometric luminosity ( L obs = 0 . L Edd ) that is a factor of ∼ L intEdd = 0 . L Edd ). Although the approximation that the stellarpopulation is bright for a time t MS and then dark is crude, theparameter ξ gives us a way to judge the importance of inter-mittency in normal star-forming galaxies.Note that for higher densities, t dyn decreases and ξ → Σ crit ∼ π (2 h )32 G (0 . t MS ) ∼ × M ⊙ pc - h , (10)which corresponds with a critical midplane pressure P crit ∼ × - ergs cm - h (see eq. 15). For Σ tot > Σ crit , massivestars live longer than the time required to disrupt the parentsub-unit ( t MS > t dyn ). MQT argue that in this regime the mas-sive stars continue to drive turbulence in the gas and maintainhydrostatic equilibrium in a statistical sense until t MS , whenthe process then repeats until gas exhaustion.Several additional elements of GMC evolution are impor-tant in judging whether or not the star formation of galaxiesis regulated by radiation pressure on dust. First, the GMCscollapse from regions of size h , with total mass Σ g π h , andto a size R GMC = h /φ . This implies that the surface density ofindividual GMCs is Σ GMC ∼ φ Σ g , where φ can be ∼ ǫ GMC = M ⋆ / M GMC ( ∝ Σ g in the single-scattering limit), Ψ , and assuming κ FIR ∼ κ o T (appropriate for T .
200 K), one finds that therequired gas surface density for a GMC to be optically thickto the FIR is Σ τ FIR =1GMC ∼ ǫ - / , . M ⊙ pc - . This GMC gassurface density would correspond to an average gas surfacedensity for the galaxy that is φ times smaller ( Σ g ∼ ⊙ pc - ). Thus, the medium surrounding the central star cluster may be optically thick to the FIR even if the av-erage gas surface density of the galaxy is less than the naiveestimate given in §2.1.2.At surface densities in excess of τ FIR = 1 for the GMCs, themodels of MQT rely on the fact that the medium is in factoptically-thick to FIR radiation. This is in sharp contrast tothe work of KM09, where they argue that the effective opti-cal depth is always ∼ τ FIR by a factor of a few in systemsas dense as the putative GMCs in Arp 220 because of the time-dependence of the GMC disruption process (see §4.3). RESULTS
We compile data of super star clusters, normal star-forminggalaxies, local starburst galaxies, ULIRGs, sub-millimetergalaxies (SMGs), hyper luminous infrared galaxies, and cir-cumnuclear starbursts to assess feedback from radiation pres-sure. Below, we test the hypothesis that radiation pressureis dynamically important in galaxies and star-forming subre-gions by comparing data to the Eddington limit (§2) on a va-riety of physical scales ranging from globally-averaged prop-erties of galaxies to individual star-forming subregions withingalaxies.
IR Luminosity vs. Molecular Line Luminosity
We show the total IR luminosity L IR as a function of molec-ular line luminosity L ′ CO (Figure 1) and L ′ HCN (Figure 2) forour sample of star-forming galaxies. L IR is known to trace thetotal light from massive stars (e.g., Kennicutt 1998), whereas L ′ CO and L ′ HCN provide a measure of the total gas mass anddense gas mass, respectively. Under the assumption that thetotal gravitational potential is dominated by the gas on thephysical scales where the stars are forming, the Eddington lu-minosity is related to L ′ CO by L Edd = 4 π Gc κ X CO L ′ CO , (11)where X CO is the L ′ CO -to- M H conversion factor and κ isthe appropriate flux-mean or Rosseland-mean opacity (either We used the J = 1–0 line unless only higher order lines were available. Aravena et al. 2008; Becklin et al. 1980; Beelen et al. 2006; Benfordet al. 1999; Capak et al. 2008; Carilli et al. 2005; Casoli et al. 1989; Chap-man et al. 2005; Chung et al. 2009; Combes et al. 2010; Coppin et al. 2009;Coppin et al. 2010; Daddi et al. 2007; Daddi et al. 2009; Daddi et al. 2010a;Downes & Solomon 1998; Gao et al. 2007; Gao & Solomon 1999; Gao &Solomon 2004a,b; Genzel et al. 2003; Graciá-Carpio et al. 2008; Greve etal. 2005; Greve et al. 2006; Isaak et al. 2004; Kim & Sanders 1998; Knud-sen et al. 2007; Mauersberger et al. 1996; Mirabel et al. 1990; Momjian etal. 2007; Murphy et al. 2001; Neri et al. 2003; Riechers et al. 2006; Riech-ers et al. 2007; Riechers et al. 2008; Sajina et al. 2008; Sakamoto et al. 2008;Sanders et al. 1991; Schinnerer et al. 2006; Schinnerer et al. 2007; Schinnereret al. 2008; Smith & Harvey 1996; Solomon et al. 1997; Solomon & VandenBout 2005; Walter et al. 2003; Walter et al. 2009; Weiß et al. 2001; Yan etal. 2010; Young & Scoville 1982; Yun et al. 2001. In reality, for normal galaxies a fraction of the UV and optical light es-capes before being reprocessed by dust, and a fraction of the IR is diffuse andlikely not associated with star formation (e.g., Kennicutt et al. 2010, Calzettiet al. 2010). The UV and IR luminosities are roughly equal at a bolometricluminosity of L bol ∼ L ⊙ , but the UV luminosity is an order of magnitudelarger than the IR luminosity at L bol ∼ . L ⊙ (Martin et al. 2005). Thus,we expect that galaxies with L IR . L ⊙ to move closer to the Eddingtonlimit in Figure 1. F IG . 1.— IR luminosity as a function of CO line luminosity. The different symbols correspond to different rotational transitions of CO: solid circles ( J = 1–0),crosses ( J = 2–1), squares ( J = 3–2), and triangles ( J = 4–3). The data are the same in both panels. The lines in the left panel correspond to the single-scatteringEddington limit (solid line; assuming h = 100pc and r = 10kpc; §2.1.1) and the single-scattering Eddington limit accounting for intermittency (dot-dashed line;eq. 8). The lines in the right panel show the optically thick Eddington limit for our preferred value of the Rosseland-mean opacity (shaded region; κ FIR = 5–10 cm g - f dg , ) and for an enhanced dust-to-gas ratio (dashed line; κ FIR = 30 cm g - f dg , ). Note that no galaxies are significantly super- or sub-Eddington.We emphasize that it is not possible to determine which limit is applicable without knowing a surface density, so dense star-forming regions can be opticallythick at L ′ CO . K km s - pc and approach the optically thick Eddington limit (see §2.1). The single-scattering Eddington limit was calculated by adoptingthe Galactic CO-to-H conversion factor X MWCO = 4 . ⊙ (K km s - pc ) - . The optically thick Eddington limit was calculated by adopting the ULIRG CO-to-H conversion factor X ULIRGCO = 0 . ⊙ (K km s - pc ) - . single-scattering or optically thick; see §2.1.1–2.1.2). Al-though counterintuitive, the single-scattering Eddington lumi-nosity lies below the optically thick Eddington limit becausethe dust opacity is column-averaged and highly non-grey (see§2.1). We adopt X MWCO = 4 . ⊙ (K km s - pc ) - for normalgalaxies (including a correction factor of 1.36 to account forHe; Strong & Mattox 1996; Dame et al. 2001), and X ULIRGCO =0 . ⊙ (K km s - pc ) - for galaxies with L IR ≥ L ⊙ (as appropriate for starbursts and ULIRGs; e.g., Downes &Solomon 1998). Similarly, if the majority of the IR luminositycomes from regions where the dense molecular gas dominatesthe potential, then L Edd is related to L ′ HCN by L Edd = 4 π Gc κ FIR X HCN L ′ HCN , (12)where we explicitly write κ = κ FIR because the critical den-sity for HCN emitting gas is large enough that these re-gions should always be optically-thick (§2.1.2). In eq. 12,we take an L ′ HCN -to- M denseH conversion factor of X HCN =3 M ⊙ (K km s - pc ) - , but we caution that X HCN is uncertainto a factor of ∼ h = 100 pc and r = 10 kpc (eq. 11). Theshaded region in the right panel is the optically thick Edding-ton limit (eq. 11 with κ FIR = 5–10 cm g - ) and the dashed Using ρ crit , HCN ∼ - g cm - , τ FIR ∼ κ FIR ρ crit , HCN R is larger thanunity for scales R & pc and κ FIR & few cm g - . line shows the optically thick Eddington limit for an enhancedopacity ( κ FIR = 30 cm g - f dg , ; where f dg , = f dg ×
50) dueto an assumed higher dust-to-gas ratio in dense star-formingenvironments. We plot the single-scattering (left panel) andoptically thick (right panel) Eddington limits separately forclarity, but the data are the same in both panels. The differentsymbols indicate various rotational transitions of CO: solidcircles ( J = 1–0), crosses ( J = 2–1), squares ( J = 3–2), andtriangles ( J = 4–3).Note that no galaxies exceed the optically thick Edding-ton limit and most galaxies are neither significantly super- orsub-Eddington with respect to the single-scattering Eddingtonlimit. We caution that the applicable Eddington limit for anyindividual galaxy cannot be determined in this plot due to thelack of surface density measurements, which dictate the op-tical depth to the FIR and the relevant Eddington limit. Forexample, high surface density star-forming regions can be op-tically thick at L ′ CO . K km s - pc and lie to the left of thesingle-scattering Eddington limit (solid line in left panel) butbelow the optically thick Eddington limit (shaded region inright panel). For the single-scattering limit, our assumption of r = 10 kpc is accurate to a factor of ∼ r - , so it is only ac-curate to a factor of ∼
25. Some compact starbursts have radiimuch smaller than our assumed radius, so they are opticallythick, even at low L ′ CO and this explains why they exceed thesingle-scattering limit in the left panel but are below the opti-cally thick limit in the the right panel. In addition, note thatthe optically-thick Eddington limit is a hard upper bound toa galaxy’s IR luminosity, which suggests that radiation pres-sure feedback may set the maximum SFR of a galaxy. In the F IG . 2.— IR luminosity as a function of HCN line luminosity. The differentsymbols correspond to different rotational transitions of HCN: solid circles( J = 1–0) and crosses ( J = 2–1). We show the optically thick Eddingtonlimit for our preferred value of the Rosseland-mean opacity (shaded region; κ FIR = 5–10 cm g - f dg , ). The dashed line shows the effect of a factorof 3 increase in the dust-to-gas ratio for the optically thick Eddington limit( κ FIR = 30 cm g - f dg , ). Note that all galaxies are within ∼ conversion factor X HCN = 3 M ⊙ (K km s - pc ) - . Eddington-limited model, the scatter in the L IR – L ′ CO relationmay be due to variations in h , X CO (see §4.5), the dust-to-gasratio/metallicity (see §4.4), the effective radii, and the depthof the stellar potential. The intermittency of star formation will likely affect theEddington limit for CO-emitting gas (dot-dashed line in leftpanel). L ′ CO traces the total molecular gas reservoir includ-ing the molecular gas that is not actively participating instar formation, such as GMC envelopes and diffuse inter-cloud gas. The gas mass relevant for the Eddington limitmay be overestimated for galaxies in the single-scatteringlimit. To account for this, we multiply the Eddington lu-minosity by the intermittency factor for CO-emitting gas ξ ∼ .
06 for the Milky Way value of X CO , h = 100 pc, r =10 kpc, and L ′ CO = 10 K km s - pc (see eq. 8 and §2.2).The intermittency factor approaches unity when L ′ CO ∼ × K km s - pc h r / X MWCO . Thus, compact star-formingregions, such as the nuclear starbursts of ULIRGs, have ξ ∼ L ′ CO due to their very small radii (e.g., Downes &Solomon 1998).Figure 2 shows the L IR – L ′ HCN relation for our sample of star-forming galaxies. The shaded region represents the optically-thick Eddington limit (eq. 12 with κ FIR = 5–10 cm g - ) andthe dashed line shows the optically thick Eddington limit foran enhanced opacity ( κ FIR = 30 cm g - f dg , ; where f dg , = f dg × Note that previous work by Krumholz & Thompson (2007) andNarayanan et al. (2008) explains the slopes of the L IR – L ′ CO and L IR – L ′ HCN relations by comparing the critical density of the gas tracer to the mediandensity of the ISM. in dense star-forming environments. The circles and crossescorrespond to the J = 1–0 and the J = 2–1 rotational transitionsof HCN, respectively.The L IR – L ′ HCN relation (Figure 2) is tight and linear overseveral orders of magnitude, implying that stars form out ofdense gas (Gao & Solomon 2004a,b; Wu et al. 2005). Thedense gas fraction ( L ′ HCN / L ′ CO ) is nearly constant for galax-ies with L IR . L ⊙ (Gao & Solomon 2004b), so L ′ CO canbe used to indirectly trace the dense gas mass M denseH . How-ever, the dense gas fraction increases dramatically in LIRGsand ULIRGs ( L IR & L ⊙ ), so CO does not trace dense gasmass in these galaxies (Gao & Solomon 2004b). HCN, on theother hand, has a critical density for excitation that is ∼ t HCNdyn ∼ × yr ρ - / , HCN ≪ t MS → ξ ≈ , (13)where ρ crit , HCN ∼ - g cm - .If the picture of radiation pressure feedback is correct, thenit should determine the L IR – L ′ HCN correlation directly. In fact,both the Eddington limit and the data show a linear relationbetween L IR and L ′ HCN . The galaxies closely follow but donot exceed the Eddington limit for our preferred value of theRosseland-mean opacity ( κ FIR = 5–10 cm g - f dg , ). If theopacity is higher ( κ FIR = 30 cm g - f dg , ), then many galax-ies are consistent with Eddington and a number are super-Eddington. For any of the values of the opacity that we as-sume, the general agreement between L IR and L ′ HCN suggeststhat radiation pressure may play an important role in regulat-ing star formation (Scoville et al. 2003). However, a numberof important factors remain uncertain, which we discuss in §4.
Molecular Schmidt Law and Radiation Pressure
The Schmidt law is a tight power law relation between thesurface density of star formation rate ˙ Σ ⋆ and the gas surfacedensity ( ˙ Σ ⋆ ∝ Σ . ; Kennicutt 1998). Furthermore, Bigiel etal. (2008) found that the Schmidt law for molecular gas is lin-ear within local star-forming galaxies ( ˙ Σ ⋆ ∝ Σ . ). In the leftpanel of Figure 3, we plot ˙ Σ ⋆ vs. Σ H for individual aperturesof THINGS galaxies (small dots; Bigiel et al. 2008; Leroyet al. 2008), THINGS galaxies with H detections (open cir-cles), starburst galaxies (solid circles), M82 super star clus-ters (stars; McCrady et al. 2003; McCrady & Graham 2007),and the Galactic Center star cluster (diamond; Paumard etal. 2006). We compare the data with the Eddington limit us-ing ˙ Σ ⋆ and Σ H as proxies for the radiation and gravitationalforces, ˙ Σ Edd ⋆ = 4 π G Σ H ǫ c κ , (14) Aravena et al. 2008; Becklin et al. 1980; Benford et al. 1999; Capaket al. 2008; Casoli et al. 1989; Chapman et al. 2005; Coppin et al. 2009;Coppin et al. 2010; Daddi et al. 2009; Downes & Eckart 2007; Downes &Solomon 1998; Greve et al. 2005; Knudsen et al. 2007; Mauersberger etal. 1996; Momjian et al. 2007; Neri et al. 2003; Paumard et al. 2006; Riecherset al. 2007; Riechers et al. 2008; Sajina et al. 2008; Sakamoto et al. 2008;Schinnerer et al. 2006; Schinnerer et al. 2007; Schinnerer et al. 2008; Smith& Harvey 1996; Walter et al. 2003; Walter et al. 2009; Weiß et al. 2001; Yanet al. 2010; Young & Scoville 1982; Yun et al. 2001 F IG . 3.— Star formation rate surface density ( ˙ Σ ⋆ ) as a function of the molecular gas surface density ( Σ H ) (left panel) and radiation pressure as a functionof midplane pressure ( P mid ; eq. 15) (right panel). The different symbols represent 750 pc apertures of THINGS galaxies (small dots), THINGS galaxies (opencircles), starburst galaxies (solid circles), M82 super star clusters (stars), and the Galactic Center star cluster (diamond). The solid line in the ˙ Σ ⋆ – Σ H plot isthe Eddington limit for the single-scattering ( κ = κ s ) limit. The shaded region corresponds to the optically thick Eddington limit for our preferred value of theRosseland-mean opacity ( κ FIR = 5–10 cm g - f dg , ). The dashed line shows the effect of a factor of 3 increase in the dust-to-gas ratio for the optically thickEddington limit ( κ FIR = 30 cm g - f dg , ). In the P rad – P mid plot, the solid line shows the Eddington limit adopting κ FIR = 10 cm g - f dg , for optically thickgas. The dot-dashed lines (both panels) are the intermittent Eddington limit (eq. 8). The hatched regions (both panels) are the critical surface density or pressurefor h = 30–100 pc (eq. 10) where t MS ∼ t dyn and τ FIR ∼ Σ H was calculated from L ′ CO using X MWCO if L IR < L ⊙ and X ULIRGCO if L IR > L ⊙ . Overall,the Eddington limit suggests that radiation pressure sets an upper bound to the ˙ Σ ⋆ or the radiation pressure of a star-forming region or galaxy. Most star-formingregions or galaxies are sub-Eddington, but a few THINGS apertures and optically thick starbursts are super-Eddington (for κ = 10 cm g - f dg , ). Several moreoptically thick starbursts will be consistent with or even exceed the Eddington limit for κ = 30 cm g - f dg , . The rough agreement between starburst galaxiesand the intermittent Eddington limit reinforces the likely importance of intermittency. However, the intermittent Eddington limit mildly under-predicts ˙ Σ ⋆ and P rad at for Σ H .
10 M ⊙ pc - and P mid . - ergs cm - , indicating that the effect of intermittency may be overestimated. where ǫ ≈ × - is the efficiency of the conversion of massinto luminosity during the star formation process assuminga Kroupa (2001) broken power law IMF that extends up to120 M ⊙ . For star clusters we assume a light-to-mass ratioappropriate for a zero age main sequence stellar population( Ψ = 3000 ergs s - g - ) and that the stellar mass is a lower limiton the gas mass of the parent GMC. We use the same X CO values as in Figure 1 and again caution that X CO is uncertainto a factor of a few and may vary systematically from normalgalaxies to starbursts (see §4.5).For the molecular Schmidt law, we find that the Edding-ton limit is an upper bound to ˙ Σ ⋆ . Most star-forming regionsand galaxies follow the Eddington limit (solid line) and arewithin ∼ ˙ Σ Edd ⋆ ∝ Σ H /κ FIR ∝ Σ . ) pro-vides a firm upper bound to ˙ Σ ⋆ for our preferred value of thedust opacity ( κ FIR = 5–10 cm g - f dg , ). If the dust opacity( κ FIR = 30 cm g - f dg , ) is enhanced due to a higher assumeddust-to-gas ratio, then some galaxies reach the optically thickEddington limit and a few galaxies exceed it.In the right panel of Figure 3, we plot the radiation pres- sure from UV and FIR photons versus the midplane pressure.These pressures will balance each other at Eddington (solidline): P rad ∼ (1 + τ FIR ) Fc ∼ P mid = π G Σ g Σ tot , (15)where we take Σ tot ≡ Σ g + . Σ ⋆ (Wong & Blitz 2002). The P rad – P mid plot shows that radiation pressure correlates stronglywith midplane pressure over 10 orders of magnitude. TheEddington limit serves as a rough upper limit to P rad , andmost galaxies are within 2 dex of the Eddington limit. Wenote that some of the THINGS apertures and some dense star-bursts reach or exceed the Eddington limit. For galaxies with P mid < P crit , the critical midplane pressure (hatched region;see §2.2 & 4.2), we expect that the effects of intermittencyare important; however, the intermittency adjusted Eddingtonlimit (dot-dashed line) under-predicts P rad for star-forming re-gions with P mid . - . ergs cm - . The intermittency factormay overestimate the importance of intermittency because ofthe simplifying assumption that subregions are “on” or “off”(see §2.2). We also see that galaxies and star-forming re-gions with 10 - ergs cm - . P mid . P crit tend to fall signifi-cantly below the Eddington limit, possibly because our simpleparametrization of X CO (see §4.5) is overestimating M H (and P mid ) for these systems. Radiation pressure becomes increas-ingly more important in the optically thick limit ( P rad & P crit )as some galaxies and star-forming regions meet and exceedEddington. As expected from Figures 1 & 2, if we assume F IG . 4.— The Eddington ratio ( Γ = P rad / P mid ) as a function of radius for THINGS galaxies with H detections. Γ H (= P rad / ( π G Σ ); solid triangles) and Γ tot (= P rad / (0 . π G Σ g Σ tot ); open circles) represent two ways to calculate the midplane pressure. The open squares ( Γ inttot ) show the effect of adjusting Γ tot forintermittency (see eq. 8). Γ tot tends to be sub-Eddington, rising to a peak at r ∼ Γ inttot is super-Eddington for r & ξ likely overestimates the importance of intermittency. Γ H generally follows the trend of Γ tot in the innerregions of galaxies but increases to super-Eddington values as Σ H nears the detection threshold. a larger dust-to-gas ratio and opacity ( κ = 30 cm g - f dg , )potentially appropriate for dusty galaxies, then more of theoptically thick starbursts would be super-Eddington. Radiation Pressure on Sub-galactic Scales
So far we have evaluated radiation pressure on a galaxy-wide scale; however, the distribution of gas and star forma-tion in galaxies is inhomogeneous. Consequently, the Ed-dington ratio ( Γ = P rad / P mid ) will likely vary on sub-galacticscales. We use observations from the THINGS survey (Wal-ter et al. 2008; Leroy et al. 2008; Bigiel et al. 2008) to calcu- late the Eddington ratio as a function of radius in azimuthallyaveraged radial bins and for semi-resolved (750 pc) aper-tures. Since the THINGS galaxies are generally in the single-scattering limit (see eq. 3), we conservatively adopt the radia-tion pressure to be P IRrad = F IR / c (see eq. 15). For the midplanepressure given in eq. 15, the corresponding Eddington ratiois Γ tot = P IRrad / (0 . π G Σ g Σ tot ). Stars and atomic gas may notcontribute significantly to the surface density in regions of ac-tive star formation, so we also calculate the Eddington ratioassuming that the midplane pressure depends only on the to-tal gas surface density Γ g = P IRrad / ( π G Σ ) or the molecular gas F IG . 5.— The molecular gas Eddington ratio Γ H = P rad / ( π G Σ ) as afunction of radius for NGC 6946. The line styles show the uncorrected Ed-dington ratio (triangles and thin solid line) and Γ H corrected for an X CO gradient (dashed line), for a dust-to-gas ratio gradient plus a factor of φ ( φ = h / R GMC ; see §2.2; dotted line), and for X CO and dust-to-gas ratio gradi-ents plus the intermittency factor and the φ factor (thick solid line). After allof these factors are accounted for, the Eddington ratio is ∼ X CO and dust-to-gasratio gradients on the Eddington ratio in NGC 6946, but the profiles from theother THINGS galaxies shown in Figure 4 are qualitatively similar. surface density Γ H = P IRrad / ( π G Σ ). Intermittency may beimportant because the THINGS observations cannot resolveindividual star-forming regions. We calculate the Eddingtonratio corrected for intermittency ( Γ inttot = Γ tot /ξ , see eq. 8). InFigure 4, we plot Γ tot (open circles), Γ H (solid triangles), and Γ inttot (open squares) as a function of radius for azimuthally av-eraged rings. We find that Γ g is similar to Γ tot , so we omit Γ g for clarity.At intermediate radii ( r ∼ → several kpc), Γ tot and Γ H generally increase from sub-Eddington ( Γ ∼ .
1) to approach-ing or exceeding the Eddington limit ( Γ ∼ Γ tot reaches amaximum Eddington ratio at r ∼ detection threshold, Γ H increases rapidly due to a small Σ H with large error bars(see, e.g., Figure 40 from Leroy et al. 2008), and thus the largevalue of Γ H at large r is consistent with Eddington to withinthe errors on Σ H . For r > Γ tot <
1, the intermit-tency factor can boost Γ inttot to the Eddington limit, suggestingthat intermittency is important.The Γ < X CO (Sodroski et al. 1995;Arimoto et al. 1996) and an increasing gradient in the dust-to-gas ratio (Muñoz-Mateos et al. 2009). We adopt the X CO gra-dient given by eq. 10 of Arimoto et al. (1996) for data from the F IG . 6.— Histogram of the Eddington ratio ( Γ = P rad / P mid ) for 750 pcapertures of THINGS galaxies. The midplane pressure was calculated us-ing two different methods: Γ H = P rad / ( π G Σ ) (dashed line) and Γ tot = P rad / (0 . π G Σ g Σ tot ) (dotted line). The solid line ( Γ inttot ) shows the effect ofadjusting the Eddington ratio by the intermittency factor (eq. 8). Γ tot wascalculated for all apertures with either an H or HI detection, but Γ H couldbe calculated only for apertures with H detections. The Γ tot distribution ismostly sub-Eddington, but intermittency pushes the majority of the Γ inttot dis-tribution above the Eddington limit, implying that ξ may overestimate theimportance of intermittency. The Γ H distribution is less peaked and shiftedto higher Eddington ratios than the Γ tot distribution. Some apertures in the Γ H distribution are at or above the Eddington limit in spite of the observa-tions not being able to resolve individual star-forming regions. Milky Way, M31, and M51 (log X / X e = 0 . r / r e - r e is the effective radius, which we assume to be 7 kpc and X e is the value of X CO at the effective radius). To account forthe dust-to-gas ratio gradient, we use a power law interpola-tion between f dg = 1 /
30 at 0.1 kpc and f dg = 1 /
150 at 10 kpc,motivated by Figure 15 of Muñoz-Mateos et al. (2009). Inaddition, collapsing GMCs enhance the surface density by afactor of φ ( φ = h / R GMC ; see 2.2), making some regions opti-cally thick to FIR radiation. In Figure 5, we show the molec-ular Eddington ratio as a function of radius for NGC 6946since this galaxy is well below ( ∼ X CO gradient, a dust-to-gas ratio gradient, and a sur-face density enhancement in the GMCs, we find that Γ H ∼ X CO , and dust-to-gas ratio gra-dients are similar to those adopted for NGC 6946. Thus, itis at least in principle possible to explain the nominally sub-Eddington inner regions of local star-forming galaxies usinga combination of these effects.In Figure 6, we plot the distributions of the individual Ed-dington ratios for the THINGS apertures (750 pc resolution): Γ tot (dotted line), Γ H (dashed line), and Γ inttot (solid line). Thedistribution of Γ tot is peaked around Γ tot ∼ .
1, and the major-ity of apertures are sub-Eddington for Γ tot . The high Γ tot tailof the distribution extends above the Eddington limit, withsuper-Eddington apertures comprising 5% of the total aper-tures and containing 5% of the total flux. Star-forming regionsare unresolved on 750 pc scales, so Γ tot should be adjustedto account for intermittency ( Γ inttot ). However, most apertureslie above the intermittency adjusted Eddington limit. As inFigures 1–5, this shift suggests that intermittency is impor-tant for radiation pressure supported star formation in normalspirals; however, ξ appears to overestimate the importance ofintermittency, possibly due to the simplifying assumption thatsubregions are either “on” or “off” (see §2.2 & 4.2). The dis-tributions of Γ tot and Γ g are similar, so we do not plot Γ g forclarity.The distribution of Γ H is less peaked and shifted to sys-tematically higher values than the distribution of Γ tot with10% of these apertures radiating at or above the Eddingtonlimit. This is not surprising (given Figure 4) because radia-tion pressure will likely be more important in H -dominatedstar-forming regions. The detection limit for H is higher thanthat for HI, so the distribution of Γ H contains fewer aper-tures than Γ tot . In addition, the apertures with H detectionstend to be within 0.4 R (Bigiel et al. 2008), so they mighthave increased metallicities and dust-to-gas ratios with de-pressed X CO values, which would increase the Eddington ratio(see Figure 5, §4.4 and 4.5). Super-Eddington apertures con-tain 6% of the total flux in H -detected apertures across thewhole sample; in NGC 6946, for example, super-Eddingtonapertures contain 10% of the total flux. The super-Eddingtonapertures indicate that radiation pressure can be dynamicallydominant even when individual star-forming regions remainunresolved, suggesting that radiation pressure may be moreimportant on the scale of GMCs and massive star clusters. Fi-nally, we calculated Γ H assuming that UV photons contributeto the radiation pressure P rad = ( F UV + F IR ) / c . The distributionof Γ H remains nearly the same because star-forming regionshave F IR / F UV ≫
1, so we refrain from plotting it in Figure 6. DISCUSSION
We have compared globally-averaged and resolved obser-vations of star-forming galaxies with theoretical expectationsbased on the theory of radiation pressure supported starformation (see §2). Although the uncertainties are large (seebelow), our primary findings are as follows.1. Figures 1–3 show that star-forming galaxies meet, butdo not dramatically exceed, nominal expectations for thedust Eddington limit. When some subregions do seem toexceed the Eddington limit (as in the outer regions of galaxiesin Figure 4 & 5), we consider this to be consistent withEddington since trends in the dust-to-gas ratio and CO-to-H conversion factor, as well as the large-scale stellar potentialand intermittency of the star-formation process ( ξ ; eq. 8)affect the Eddington ratio at order unity.2. The L IR – L ′ HCN plot (Figure 2) provides the strongestevidence for the importance of radiation pressure feedbacksince L ′ HCN is expected to directly trace the dense activelystar-forming gas and L IR traces the total star formation rate.If radiation pressure in fact dominates feedback, we wouldexpect a one-to-one correspondence between these twoquantities, and such a relation is observed (see also Scovilleet al. 2001; Scoville 2003). Nevertheless, for typical valuesof both κ in the optically thick limit and the HCN-to-H conversion factor, the Eddington limit overpredicts L IR bya factor of ∼ conversion factor is smaller (see§4.5 below). If radiation pressure feedback regulates starformation, then this relation is in a sense more fundamentalthan the Schmidt Law because HCN-emitting gas is moreclosely connected with star formation than CO-emitting gas,in which case we would expect ˙ Σ Edd ⋆ = 4 π G Σ denseH / ( ǫ c κ FIR ).3. The central regions of all galaxies in Figure 4 are primafacie substantially sub-Eddington when a constant dust-to-gasratio and CO-to-H conversion factor are applied to all sub-regions without regard to their radial location. If radiationpressure is in fact the dominant feedback mechanism in theseregions, a much higher central dust-to-gas ratio and a lowerCO-to-H conversion factor are required (see Figures 4 & 5).It would be particularly useful for testing radiation pressurefeedback to produce the same profiles in HCN.4. The “break” in the observed Schmidt Law at Σ g ∼ ⊙ pc - (see Figure 3; Daddi et al. 2010b) may bedue to the transition from the single-scattering limit to theoptically thick limit in the GMCs that collapse to form stars,as in MQT.5. If radiation pressure is the primary feedback mechanismfor regulating star formation, then we predict that the SchmidtLaw will follow the form of eq. 14 (for discussion of κ and ξ as well as uncertainties see §2 & 4.2–4.5).6. A testable prediction of radiation pressure feedback is thatall else being equal the star formation rate should dependlinearly on the dust-to-gas ratio in the optically thick limit.In addition to these points, below we note an implication ofradiation pressure feedback that has so far not been stated inthe literature (§4.1). Finally, in the remaining subsections wehighlight the dominant uncertainties in our work as a guidefor future research on the importance of radiation pressurefeedback in star-forming galaxies. Gas Depletion Timescale
The gas depletion timescale, the time required to consumea galaxy’s gas reservoir at the current SFR, is observed to be ∼ t gas = M g ˙ M ⋆ = M g ǫ c ξ L Edd = ǫ c κ π G ξ . (16)Using typical numbers for a spiral galaxy in the single-scattering limit, the gas depletion timescale is t gas ∼ . Σ - / h / , (17)where Σ = Σ /
30 M ⊙ pc - and h = h /
100 pc. This nor-malization of t gas is in good agreement with the observed gasdepletion timescale, but eq. 17 predicts that the gas depletiontimescale should have a strong dependence on Σ g , in contrastto the observations of Leroy et al. (2008) (see their Figure15). For completeness, we note that variations in the dust-to-gas ratio will not affect the dependence of t gas on Σ g in thesingle-scattering limit, but uncertainties in X CO , ξ , and φ (see§4.4–4.5) might impact the gas depletion timescale.Hot starbursts and optically thick subregions (see §2.1.2)have intermittency factors that approach unity and nearly con-stant opacities, so the gas depletion time is approximately0constant, t gas ≈ . κ , (18)where κ = κ FIR /
10 cm g - . For comparison, Sakamoto etal. (2008) find that the optically thick western nucleus of Arp220 has a gas depletion time of ∼ ∼ π G ξ f gas / ( ǫ c κ ) for small f gas . Intermittency
The intermittency factor ξ (see §2.2; MQT) relates theproperties of radiation pressure dominated star-forming sub-regions to the global properties of a galaxy. However, ξ mayoverestimate the effect of intermittency in some galaxies (seeFigures 3, 4, & 6). We expect the determination of ξ to becomplicated by uncertainty in the timescale for the centralcluster of a sub-unit to be bright ( t MS ∼ t MS as the time a cluster will be bright, since the cluster lumi-nosity drops rapidly after the most massive stars in the clusterexplode as supernovae. However, a cluster will continue toemit after t MS . Indeed, models of cluster luminosity indicatethat a similar amount of momentum will transferred to thegas during the time t → t MS and during the time t MS → t MS ,where the cluster luminosity has dropped by 1 dex after ∼ t MS (Leitherer et al. 1999). Further uncertainty in ξ is due to ambi-guities in calculating the lifetime of a sub-unit ( ∼ t dyn + t MS ),especially the disk dynamical time (see eq. 9). The dynami-cal time likely varies with galactocentric radius because Σ isa strong function of radius (Leroy et al. 2008). Thus, tryingto determine an effective ξ applicable to a galaxy as a wholemay be difficult if ξ changes locally. Overall, we expect theuncertainty in ξ to be a factor of a few to several. The FIR Optical Depth
A key theoretical uncertainty in calculating the Eddingtonlimit for dense starbursts is the effective optical depth ( τ eff ) forsurface densities where τ FIR >
1. In order for radiation pres-sure to be dynamically important in optically thick GMCs, τ eff must exceed unity. Based on the high Mach number tur-bulence simulations of Ostriker et al. (2001), MQT concludethat if the ISM is optically thick on average, then the vastmajority of sight lines will be optically thick. For compari-son, KM09 argue that instabilities, such as Rayleigh-Taylorand photon-bubble instabilities, will reduce the effective opti-cal depth of the dense ISM to ∼
1. However, MQT note thatboth the midplane pressure from gravity P mid ∼ π G Σ and op-tically thick radiation pressure P rad ∼ τ F / c ∝ Σ scale as Σ ,a feature unique to radiation pressure among stellar feedbackprocesses. Thus, if radiation pressure cannot regulate star for-mation in dense, optically thick gas, then no known stellarfeedback process can. Dust-to-Gas Ratio and Metallicity
The coupling between radiation and gas directly dependson the dust-to-gas ratio ( κ ∝ f dg ). In this analysis, we as-sume the Galactic value for the dust-to-gas ratio ( f dg = 1 / f dg and metallicity change with environment. The dust-to-gas ratio has been shown to correlate with metallicity andradius (Issa et al. 1990; Lisenfeld & Ferrara 1998; Draine et al. 2007; Muñoz-Mateos et al. 2009). Muñoz-Mateos etal. (2009) find that the dust-to-gas ratio can climb to values ashigh as f dg ∼ /
10 in the centers of spiral galaxies. This in-crease in metallicity and dust-to-gas ratio is necessary for thecenters of star-forming spirals to be at Eddington (see Figures4 & 5 and §3.3). The average dust-to-gas ratio of local spiralsalso varies by a factor of a few (e.g., M51 has a f dg ∼ / f dg ∼ /
50; Kovács et al. 2006; Michałowski et al. 2010)and sub-mm faint ULIRGs ( f dg ∼ /
20; Casey et al. 2009).Importantly, if we adopt a dust-to-gas ratio potentially appro-priate for dusty starbursts (short dashed line in Figure 1, 2,and the left panel of Figure 3), then a substantial fraction ofoptically thick galaxies would be at or above the Eddingtonlimit (Figure 5 illustrates this for NGC 6946).
Molecular Gas Tracers
The Eddington limit depends strongly on the CO-to-H ( X CO ) and the HCN-to-H ( X HCN ) conversion factors (see eqs.11 & 12). These conversion factors are two of the largestsources of observational uncertainty in our calculations be-cause they vary with excitation conditions ( X ∝ √ n H / T b ,where T b is the brightness temperature) and metallicity. Sev-eral lines of evidence suggest that X MWCO overestimates M H instarburst galaxies. For example, X CO is a factor of ∼ ∼ X CO in normal spirals and ex-treme starbursts, we apply the Milky Way X CO value to galax-ies with L IR < L ⊙ and the ULIRG X CO value to galaxieswith L IR > L ⊙ . Because this prescription is somewhatsimplistic, it probably overestimates M H in moderate lumi-nosity starbursts ( L IR < L ⊙ ), such as M82, and in thecenters of star-forming spirals (see Figures 4 & 5 and §3.3).Additionally, it likely underestimates M H in ultra-luminous( L IR ∼ L ⊙ ) high redshift disk (BzK) galaxies, for whichDaddi et al. (2010a) find a value of X CO that is consistent withthe Galactic value. As a result, moderate luminosity starburstsand the centers of star-forming spirals may be closer to Ed-dington and BzK galaxies might be further below the opticallythick Eddington limit than Figure 3 would suggest.Unfortunately, X HCN is more uncertain than X CO becausethere is no direct calibration of X HCN from Milky Way GMCs.For normal spirals, Gao & Solomon (2004a,b) find X HCN ∼
10 M ⊙ (K km s - pc ) - for virialized cloud cores with h n i =3 × cm - and T b = 35 K. They caution that X HCN could belower in regions of massive star formation due to significantlyhigher brightness temperatures T b ∼ few × K (Boonmanet al. 2001). Ultra-luminous starbursts exhibit widespread in-tense massive star formation, so one might expect that X HCN is lower in more luminous galaxies. For example, Graciá-Carpio et al. (2008) estimate that X HCN should be ∼ L FIR ∼ L ⊙ than at L FIR ∼ L ⊙ .We note that if X HCN is smaller than the assumed value of3 M ⊙ (K km s - pc ) - , then more galaxies will approach orexceed the Eddington limit (see Figure 2). For example, de-creasing X HCN by a factor of ∼ κ FIR ( ∼ REFERENCESAravena, M., et al. 2008, A&A, 491, 173Arimoto, N., Sofue, Y., & Tsujimoto, T. 1996, PASJ, 48, 275Becklin, E. E., Gatley, I., Matthews, K., Neugebauer, G., Sellgren, K.,Werner, M. W., & Wynn-Williams, C. G. 1980, ApJ, 236, 441Beelen, A., Cox, P., Benford, D. 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