Conditions for supernovae driven galactic winds
aa r X i v : . [ a s t r o - ph . C O ] S e p Draft version October 5, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
CONDITIONS FOR SUPERNOVAE DRIVEN GALACTIC WINDS
Biman B. Nath , Yuri Shchekinov Raman Research Institute, Sadashiva Nagar, Bangalore 560080, India and Department of Physics, Southern Federal University, Rostov on Don, 344090 Russia
Draft version October 5, 2018
ABSTRACTWe point out that the commonly assumed condition for galactic outflows, that supernovae (SNe)heating is efficient in the central regions of starburst galaxies, suffers from invalid assumptions. Weshow that a large filling factor of hot ( ≥ K) gas is difficult to achieve through SNe heating,irrespective of the initial gas temperature and density, and of its being uniform or clumpy. We insteadsuggest that correlated supernovae from OB associations in molecular clouds in the central region candrive powerful outflows if the molecular surface density is > M ⊙ pc − . Subject headings: galaxies: starburst — galaxies: ISM — ISM: bubbles — ISM: clouds— ISM:supernova remnants INTRODUCTION
Standard models of supernovae driven galactic out-flows assume a central region where the SNe energy inputis thermalized (e.g., Larson (1974); Chevalier & Clegg(1985); Heckman et al. (1990); Suchkov et al. (1994);Strickland & Heckman (2009); Sharma & Nath (2013)).These models posit that SNe explosions shock heat theinterstellar medium (ISM) within this region (of size ∼
200 pc); numerical simulations for winds also imple-ment this assumption (e.g., Suchkov et al. (1994, 1996);Cooper et al. (2009)). The basic assumption is thateven with a small heating efficiency ( ∼ . > ∼ K and launch a galacticwind.There are two assumptions here, one involving the en-ergetics of SNe explosions and another to do with thethermalization of this energy. While the energy bud-get can be met in the case of high supernova rate instarbursts, the process of thermalization assumes thatSNe remnants overlap and reach a porosity larger thanunity. Suchkov et al. (1996); Strickland & Heckman(2009) have discussed this issue in the context of theobserved X-ray emission from the outflowing hot gas inM82. The required SNe heating efficiency of ǫ is con-nected with the mass loading factor, the ratio β betweenthe total mass deposition rate and the mass lost throughSNe and stellar winds. The temperature and bright-ness of the gas depend on different combinations of ǫ and β , and Strickland & Heckman (2009) suggested anoptimum condition of β ∼ ǫ ∼ . .
3. Theysuggested that ǫ could be large in the case of a low den-sity gas ( ∼ . − ). The average density of the dif-fuse ionized medium in starbursts is however ≈
24 cm − (Armus et al. 1989).The question is whether or not SNe remnants canoverlap in these regions and sufficiently heat the gas(Melioli & de Gouveia Dal Pino 2004). We argue in thispaper that it is difficult to achieve high porosity for hot( > ∼ –10 K) gas irrespective of the ambient densitybeing small or large, if SNe remnants occur randomly [email protected]; [email protected] in this region. In order to overcome this difficulty wesuggest that galaxies approaching a galactic wind stageare likely to produce super star clusters with an en-hanced star formation rate. We note here in passingthat other processes have also been invoked to aid galac-tic winds, such as radiation pressure (Nath & Silk 2009;Murray et al. 2011) and turbulence (Scannapieco 2013),and also cosmic rays (Uhlig et al. 2012) which can op-erate outside the central region. POROSITY IN A UNIFORM ISM
Consider the estimation of the porosity of hot ( > ∼ K) gas in a uniform ISM of ambient density n . In thecontext of the three-phase ISM model, the porosity ofthe coronal gas is estimated by the final volume of SNremnants when the shells decelerate to the sound speedof the ambient gas (McKee & Ostriker 1977; Cox 2005).The ‘hot’ interior gas at this stage has a temperature ∼ × K, which was used to infer the three-phasemodel of the ISM. The same expression was, however,used by Heckman et al. (1990) in order to derive a highvalue of porosity of a hotter gas at > ∼ K gas (for n ∼
100 cm − and T ∼ K; their eqn 2) , andthis argument has been repeated by other authors (e.g.,Suchkov et al. (1994); Strickland & Heckman (2009)).The average temperature of the interior gas at this shellspeed is, however, less than 10 K, and cannot be usedto determine the porosity of gas with T > ∼ K.We recall that the average gas temperature inside aSNe remnant decreases rapidly after the gas cools downto a temperature ∼ K. Cox (1972) showed that theenergy of the remnant scales as E ( t ) ∝ R − , after theshell enters the radiative phase (at shell radius R c ). Onehas, E ( t ) = 0 . E (cid:16) RR c (cid:17) − , R c = 22 . E / n − / , (1)where the initial explosion energy is E = 10 E erg,and n is the ambient particle density (in cm − ). The We note that the Slavin & Cox (1993) prescription would haveyielded a value of porosity smaller by a factor ∼ Nath, Shchekinovcorresponding timescale is t c ≈ × yr E / n − / .Inverting this relation, we have the shell radius at a timewhen the internal energy has decreased to a fraction f of the initial value as R ( f ) = 0 . R c f − / . The cor-responding time scale is t ( f ) = 0 . t c f − / . Denotingthe supernova rate density as ν SN (yr − pc − ), we candefine the porosity when the energy has decreased by afraction f , as, P ( f ) ≈ . × ( f / . − / ν SN E / n − / . (2)We have scaled the expression to f = 0 .
5, since the av-erage interior temperature falls to ∼ K at this stage,according to Cox (1972) (see his Figure 2a). This is borneout by the simulations of Shelton (1998) for n = 0 . K after t c = 5 × yr appropriate for this density, asexpected.The condition for ∼ K gas to overlap (requiredto explain X-ray observations ) is more stringent andtherefore we can use the porosity in eqn 2 as an upperlimit. Note that the shell speed in Cox (1972) at thisstage ( f = 0 .
5) drops down to ∼
100 km s − . We canalso use this velocity criterion to estimate the porosityfor hot gas.The typical supernova rate density in starburst regions ν SN ∼ − yr − pc − , implies a porosity ≈ . n = 1, and can be even lower for the average densityin starbursts (Armus et al. 1989). For f ≡ ǫ = 0 .
1, assuggested by Strickland & Heckman (2009), the poros-ity is ≈ . n − / , still less than unity. Moreover, for f ∼ . ≤ × K, not enough to explain the observed X-rayemission. Furthermore, if the gas reservoir of ∼ × M ⊙ required to explain the mass loading rate in M82(Suchkov et al. 1996), were uniformly distributed in the200 pc central region, the density would be ≈
70 cm − ,precluding the possibility of a large porosity factor forhot gas. And if this gas were uniformly heated to 10 K,then it should emit hard X-rays with 10 erg s − , muchlarger than the observed 3 × erg s − .One could argue that most of the mass is in clouds witha small filling factor, which are destroyed by shock waves.Thornton et al. (1998) showed that gas with T ∼ Koccupies a fraction (1 / ∼ − of the SN remnant vol-ume (their Figs 5,6) until the radiative stage t c ∼ yr,when the average temperature is ∼ K. Therefore evenif the shells overlap with the interior gas at mostly 10 K, the filling factor of 10 K gas is small, of order 0 . K (as required by X-ray observation) is of thesame order. Therefore the clouds should have a large,and not small filling factor in order to explain the X-rayobservations.One could argue that SN remnants propagate mostlythrough a hot, low density medium. For a 10 K gas,with (isothermal) sound speed ∼
100 km s − , the rem-nants will stop expanding after reaching this speed. Theporosity of hot SN bubbles will therefore still be givenby P ≈ .
02 for n = 1 cm − and a SN rate densityof few × − yr − pc − , because it corresponds to thesame shell speed. This is the porosity provided by newSNe remnants. However, a 10 K gas will cool in a time scale of 10 n − yr, and will need new SN remnants withlarge filling factor to maintain its temperature, thoughthe above estimate shows the filling factor to be small. Therefore the porosity of hot gas is less than unity ir-respective of the density and temperature of the gas inwhich SNe explode.
In general it is difficult to simultaneously satisfy theconditions for vigorous star formation rate (SFR) andalso fill the central region with hot gas. We can usethe Schmidt-Kennicutt law (Kennicutt 1998) Σ
SFR ≈ . × − Σ . , where Σ SFR is in the units of M ⊙ yr − kpc − and Σ in units of M ⊙ pc − . For a initial massfunction (IMF) of Kroupa (2001); Chabrier (2003), thecorresponding supernova rate is ∼ .
01 yr − (SFR/M ⊙ yr − ). For a scale height H pc, the supernova rate den-sity is ν SN ≈ . × − Σ . H − . The gas density for auniform medium is n = Σ / (2 Hµm H ) ∼
15 Σ H − cm − .Therefore the porosity for the 10 K gas is given by (fromeqn 2), P ≈ × − Σ − . H . E / . (3)The requirement that P ≥ ≤ − H . . (4)Using the Silk-Elmegreen law of star formation (Silk1997, Elmegreen 1987 ), Σ SFR ≈ .
017 Σ Ω, where Ω isthe angular velocity in the units of Myr − , one gets asimilar condition: Σ ≤ − H Ω / . . Since a largedensity is required for high SFR, while a low density isrequired to achieve high porosity of hot gas, the combina-tion of these two requirement leads to a condition that isdifficult to meet. COHERENCY OF SUPERNOVAE
The problem boils down to maintaining a large poros-ity for high temperature gas with a given star formation,or equivalently, SNe rate. This can be achieved by allow-ing SNe remnants to explode in a coherent manner so asto avoid excessive radiative cooling and act collectively.One has to invoke spatial and temporal clustering of SNeevents in order to increase the efficiency of SN heating,even if the heating is confined to small region We can de-fine the coherency of SNe by requiring the four-volumeat t c to be of order unity (4 πR c t c ν SN / > ∼ K.Massive stars (the progenitors of SNe) are likely toform in OB associations in molecular clouds (MCs).These sites can provide the spatial and temporal co-herency needed for this scenario. In order for these SNeto emerge from the parent MC and then heat up the in-tercloud medium, it is necessary for the SNe explosionsto destroy the MC.Consider a MC of mass M MC = 10 M MC , M ⊙ andradius R MC = 5 R MC , pc. One fiducial example ofsuch a MC is provided by G0.253+0.016 (called the‘Brick’) in the central region of Milky Way, with a mass ∼ . × M ⊙ and radius ∼ . µ = 2 . n = 5 × cm − M MC , R − , . The correspond-ing free-fall time scale is t ff ≈ × n − / s ≈ onditions for SN driven galactic wind 31 Myr M − / , R / , . In this environment, a SN remnantshell speed will become v c = 100 km s − at a lengthscale R c ≈ ( E/ρ ) / (2 / v c ) / = 23 pc ( E /n ) / , at t c = (2 / R c /v c = 0 . E /n ) / , consistent withthe results of Chevalier (1999) for SNe inside MCs.One can show that for a star formation rate ǫ ∗ M MC /t ff ,where ǫ ∗ ≈ .
02 is the star formation efficiency per free-fall time (Lada et al. 2010), the porosity is given by P =(4 π/ R c t c ν SN ≈ . n / , (similar to estimates in § R c ≫ R MC . Thiseffect is expedited by the ionization of the MC by massivestars prior to SNe events (Monaco 2004). Walch et al.(2012) have found from simulations of a single SN in aMC thatthe ensuing ionization front disperses the cloud on atime scale comparable to the sound crossing time of theionized gas, of order a Myr; moreover radiation pressuremay help in this regard (Murray et al. 2011). INITIATING A GALACTIC WIND
Although star formation in MCs help to concentratethe effects of SN remnants, it remains to be seen whetherthe energetics are sufficient for galactic wind. In the caseof correlated multiple SNe, superbubbles can be ener-getic enough to break out of the disc and reach a suffi-cient height above the disc with enough momentum soas to seed a galactic wind (Mac Low & McCray 1988;Roy et al. 2013) (and references therein). The thresh-old mechanical luminosity for a superbubble to breakoutwith a Mach number of 5–10 depends on the ambientdensity ( ρ ) and sound speed ( c s ), and the scale height( H ). Roy et al. (2013) has determined a threshold lu-minosity of L cr ≈ ρ H c s ≈ . × n H erg s − ,for an ambient gas at 10 K and n = 1 cm − . For 10 K gas, the requirement is 10 times larger. If we use n = 10 cm − , H pc = 50, and T = 10 K, the thresholdmechanical luminosity for breakout is L cr ≈ × ergs − .For a Kroupa/Chabrier IMF, this translates to athreshold SFR of 0 . ⊙ yr − in the central region. Thisalso implies a SNe rate of 10 − yr − , which can be com-pared with the SNe rate 2 . × − yr − in the nuclear (35pc) region of M82 (Table 1 of F¨orster Schreiber et al.(2003)). If we considered the whole region of radius 200pc and total height 100 pc, this would imply a low su-pernova rate density and consequently a small porosity,but that would neglect the effect of coherent SNe insidethe small size of a OB association (few tens of pc). Su-perbubbles breaking out of a stratified disc ultimatelyassume an oval shape, with an extent in the plane of thedisc ≈ π × the scale height, in the Kompaneets approxi-mation. Therefore such a superbubble would ultimatelyengulf the central region, with radial length scale ∼ ∼
50 pc and the gas inside this regionwould be shock heated.Assuming continuous star formation for the time scaleof the wind ( ∼
100 Myr) and a typical SNe explo-sion energy of 10 erg, this critical luminosity im-plies a total number of OB stars of N OB ≈ . × ( H pc / ( n /
10 cm − ). For a Kroupa/ChabrierIMF, the corresponding stellar mass is ≈ . × M ⊙ ( H pc / ( n /
10 cm − ). Matzner & McKee (2000)have estimated that a fraction 0 . .
75 of the molecu-lar cloud mass is ultimately converted into stars. Usingthe upper limit of 0 .
75, one then finds a conservative esti-mate for the required total mass of the parent molecularcloud(s) to be ∼ M ⊙ ( H pc / ( n /
10 cm − ).Another way of estimating this is to use the empiricalSFR in molecular clouds as determined by (Lada et al.2010). They found the SFR to be 4 . ± . × − M th M ⊙ yr − , where M th is the mass of the cloud above a den-sity threshold of n th ≈ cm − . The above SFR thenimplies M th ≈ × M ⊙ . According to Fig 3 of Ladaet al. (2010), such dense clumps amount to a fraction0 . .
15 of the total mass of molecular clouds. Usingthe upper value of 0 .
15 yields a conservative estimate forthe total molecular mass M th ∼ M ⊙ .For molecular clouds in the central regions of galax-ies, the requirement may be stronger than this. In thecentral regions of galaxies, as in our Galaxy, the tur-bulent speed is likely to be large. Krumholz & McKee(2005); Padoan & Nordlund (2011) have shown that dueto turbulence, the threshold gas density for star forma-tion increases to n cr ∼ A x α vir M n , where A x is a con-stant close to unity, α vir ≈ . n is the ambient density, and M is theMach number. Typically M ∼
50, as in the case ofthe ‘Brick’ cloud in the Galactic CMZ (Kruijssen et al.2013). For our fiducial MC, with average density n ≈ × cm − M MC , R − , , the critical density for starformation then becomes n cr ∼ . × cm − . Wecan estimate the mass in clumps with density greaterthan this by using the fact the probability distributionfunction of density in a turbulent ISM at high densityend is described as a power law, dp/dn ∝ n − γ , with γ ≈ . .
75 (Kritsuk et al. 2011). If the cumulativemass (above a certain density) for n th (the thresholdwithout turbulence, 10 cm − ) and n cr (the thresholdwith turbulence) are denoted as M th and M cr , then onehas M cr /M th ≈ ( n cr /n th ) − γ . For γ ∼ .
5, and for theabove values of two threshold densities, we then have M cr ≈ (1 / M th .Given the uncertainties, we can conclude that the to-tal mass requirement is increased by a factor ≥
10, to ∼ M ⊙ in order to mitigate the effect of turbulenceand create energetic superbubble(s). We can comparethis with the observations of molecular mass in the cen-tral regions of nearby starburst galaxies. Plenisas et al.(1997) estimated a molecular mass of 10 –10 M ⊙ in thecentral regions of nuclear starburst galaxies NGC 2903,NGC 3351, NGC 3504.This threshold molecular mass inside a region of ra-dius 200 pc in radius implies a surface density ≥ M ⊙ pc − , which can be considered as a precondition for pro-ducing coherent SNe in order to initiate a galactic wind.Since the Kennicutt-Schmidt SFR is somewhat lower atsub-kpc scale (e.g. Momose et al. (2013)), this surfacedensity implies a SFR of ∼ ⊙ yr − kpc − in thenuclear 200 pc region.For comparison, most of the nearby galaxies observedin the BIMA SONG survey have central CO surface den- Nath, Shchekinov TABLE 1Observed central molecular surface density in starburstswith wind [Refs: (1) Aalto et al. (2012), (2)Sakamoto et al. (2011), (3) Sargent & Scoville (1991), (4)Scoville et al. (1997), (5) Sofue et al. (2001)]
Name Size of central region (pc) Σ H (10 M ⊙ pc − )NGC 1377 - 5 (1) NGC 253 300 10 (2)
Arp 299 - 62 (3)
Arp 220 100 58 (4)
NGC 3079 125 100 (5) sities less than this and also do not show signs of wind(Helfer et al. 2003). In contrast, CO observations ofULIRGs show the existence in the central few hundredparsecs region a molecular gas of mass (0 . . × M ⊙ , with a surface density of ∼ (0 . × M ⊙ pc − (Solomon et al. 1997). We compare the observed molec-ular surface density of a few nearby starburst with windsin Table 1. DISCUSSION
The nature of the central region of galaxies can affectthe mass loading in outflows, especially the amount ofcold material in the outflow. In the case of SNe rem-nants producing a high porosity of hot gas, molecularclouds are likely to be destroyed, whereas in the presentscenario, the amount of cold/molecular gas in the out-flow could be large. This is because the time scale ofsuperbubble(s) destroying the parent molecular cloud(s) is a few ×
10 Myr (lifetime of massive stars), is also thetime scale of galactic outflows, and the dynamical effectsof starbursts on the surrounding medium. It is thereforereasonable to expect that remnants of molecular cloudswill be advected into the outflows in this case, as hasbeen observed in NGC 253 (Bolatto et al. 2013) with
ALMA , where molecular gas is seen to envelope the X-ray emitting region and superbubbles can identified.It is not necessary that all SNe explode within oneOB association in this scenario. The initial trigger canbe provided by a super star cluster ( M ∼ M ⊙ ;e.g. Walcher et al. 2005), after which the resultingsuperbubble can enhance star formation in the vicin-ity. On one hand the shock wave from superbubble cantrigger star formation in a nearby cloud, and on theother hand, the increased gas pressure can enhance theSFR (Blitz & Rosolowsky 2006). Also, the loss of massthrough the superbubble creates pockets of low densitygas, which can be heated with high efficiency by lattergeneration SN remnants and create a reservoir of hot gas.To conclude, we have argued that it is difficult to ther-malize the energy input from SNe in the central regions( ∼
200 pc) of starbursts and create a large filling factorfor hot ( > ∼ K) gas as is commonly held. We suggestedthat coherency of SNe is need to create superbubbles thatare energetic enough to initiate a galactic wind, and de-termined a threshold molecular surface density > ∼ M ⊙ pc − in the central region for this scenario.We thank the anonymous referee for useful comments.This work is partly supported by an Indo-Russian project(RFBR grant 12-02-92704-IND, DST-India grant INT-RFBR-P121).in the central region for this scenario.We thank the anonymous referee for useful comments.This work is partly supported by an Indo-Russian project(RFBR grant 12-02-92704-IND, DST-India grant INT-RFBR-P121).