HD/H2 as a probe of the roles of gas, dust, light, metallicity and cosmic rays in promoting the growth of molecular hydrogen in the diffuse interstellar medium
aa r X i v : . [ a s t r o - ph . GA ] N ov HD / H as a probe of the roles of gas, dust, light, metallicity and cosmic raysin promoting the growth of molecular hydrogen in the di ff use interstellarmedium H. S. Liszt
National Radio Astronomy Observatory520 Edgemont Road, Charlottesville, VA, 22903-2475 [email protected]
ABSTRACT
We modelled recent observations of UV absorption of HD and H in the Milky Way and towarddamped / sub-damped Lyman alpha systems at z = > / N(H ) ratios reflect the separateself-shieldings of HD and H and the coupling introduced by deuteration chemistry. Locally, observationsare explained by di ff use molecular gas with 16 cm − . n(H) .
128 cm − if the cosmic-ray ionization rateper H-nucleus ζ H = × − s − as inferred from H + and OH + . The dominant influence on N(HD) / N(H )is the cosmic-ray ionization rate with a much weaker downward dependence on n(H) at Solar metallicity,but dust-extinction can drive N(HD) higher as with N(H ). At z > ) and somewhat smaller N(H ) / N(H I). Comparison of our Galaxyand the Magellanic Clouds shows that smaller H / H is expected at sub-Solar metallicity and we show bymodelling that HD / H increases with density at low metallicity, opposite to the Milky Way. Observationsof HD would be explained with higher n(H) at low metallicity but high-z systems have high HD / H atmetallicity 0.04 . Z . ff ects. The abruptH transition to H / H ≈ .
16 cm − . Interior H fractions are substantially increased by dust extinction below .
32 cm − . Atsmaller n(H), ζ H , small increases in H triggered by dust extinction can trigger abrupt increases in N(HD). Subject headings: astrochemistry . ISM: molecules . ISM: clouds. Galaxy
1. Introduction
Like molecular hydrogen H , the much rarer deuter-ated isotopologue HD has been studied and observedacross cosmic time. A survey of HD / H ratios along41 galactic sightlines has recently been publishedby Snow et al. (2008), revising and greatly extend-ing the work of Lacour et al. (2005) and yet-earlierresults summarized by Liszt (2003). Oliveira et al.(2014) recently detected HD in a low-redshift, low-metallicity Damped Lyman Alpha (DLA) systemwith column densities N(HD) and N(H ) very muchlike those seen in the Milky Way. HD and H have also been detected at eight redshifts towardsix DLA and sub-DLA systems at z > / N(H ) ratios well above those seen in theMilky Way (Noterdaeme et al. 2008; Balashev et al. 2010; Ivanchik et al. 2010; Noterdaeme et al. 2010;Tumlinson et al. 2010).The chemistry of deuterium and HD plays a spe-cial role in the formation of structure and the firststars in the early Universe (Gay et al. 2011). Althoughit is generally understood now that observations ofHD can not provide a direct determination of the el-emental [D / H] ratio (Le Petit et al. 2002; Snow et al.2008; Ivanchik et al. 2010), [D / H] is well-determinedby other means, with [D / H] = . × − in primor-dial gas (Pettini & Cooke 2012; Cooke et al. 2014) and[D / H] = . × − locally (Linsky et al. 2006).Given that the intrinsic [D / H] ratio is reflected onlyindirectly in the HD / H ratio, the study of HD is nowof interest owing to its value as a probe of the micro-physics of the di ff use atomic gas and the more general1roblem of H -formation in relatively low-density dif-fuse neutral atomic gas. The formation rate of HD de-pends primarily on the local proton (or deuteron) den-sity and hence on the strength of penetrating hydrogen-ionizing radiation (usually the cosmic-ray ionizationrate). In turn, the proton density is balanced by theprocesses whereby atomic ions recombine, basicallyvia grain-assisted recombination mediated by the samesmall particles that heat the gas via the photoelectrice ff ect (Draine & Sutin 1987; Bakes & Tielens 1994;Wolfire et al. 1995). The interaction of H + and D + with HD and H to equilibrate the HD / H ratio cou-ples the microphysics and HD chemistry to the generalH formation problem, highlighting the separate rolesof shielding of H and HD by themselves and by dustextinction.Here we discuss these observations of HD in thecontext of models of the coupled heating / cooling- H -HD- formation in di ff use neutral atomic gas. Section2 discusses models of H and HD formation and self-shielding in di ff use clouds. In Section 3 observationsof HD in the Milky Way are discussed and comparedwith the model results, which are explored in some de-tail in Section 4 in order to separate the various phys-ical and chemical processes involved. Section 5 dis-cusses what is known observationally of the H I-H transition in di ff use gas in nearby systems havingsub-Solar metallicity and Section 6 discusses the ob-servations of H and HD in high-redshift Damped Ly-man Alpha Systems (DLA). Section 7 is a summary.
2. Model calculations2.1. H and HD formation and self-shielding This work is an update of our earlier investigationof the formation of HD and H + (Liszt 2003), revisedto study the wealth of new observations of HD notedin the Introduction. As before (Liszt 2003, 2007)we model the formation of H self-consistently in aspherical gas cloud of uniform density immersed inthe average ambient, isotropic galactic radiation field,and we compute the local kinetic temperature T K fol-lowing the methods of Wolfire and his collaborators(Wolfire et al. 1995, 2003). The equations of chemicaland thermal balance are solved iteratively over a modelwith 128 or more equi-spaced radial shells, computingthe radiation field in each shell averaged over the sur- In this work we refer to neutral atomic hydrogen as H I followingthe usage of Savage et al. (1977) f H n(H)
816 1632 3264 64128 n ( HD ) / n ( H ) Fig. 2.— Radial variation of the H -fraction (lowerpanel) and HD / H ratio (upper panel) for models withN(H) = × cm − and number density n(H) = − . Dotted (red) lines are resultswithout dust extinction of dissociating photons.2
014 1015 1016 1017 1018 1019 1020 1021 102210910101011101210131014101510161017 N(H2) [cm-2] N ( HD ) [ c m - ] [D]/[H]=2.54x10-5 MILKY WAY ζ H DLA,sub-DLAB0120-282E-192E-182E-172E-16
Fig. 1.— Observed and model HD and H column densities for Milky Way sightlines from Snow et al. (2008) andDamped and sub-Damped Lyman Alpha systems (DLA) at z = > =
16 cm − andprimary cosmic ray ionization rates per H-atom 2 × − s − ≤ ζ H ≤ × − s − . Numerical annotations show the H fraction along the curves. The companion dotted lines show the results when the explicit shielding by dust extinctionis neglected. The orange dashed-dotted line shows results for n(H) =
128 cm − and ζ H = × − s − including dustextinction. The primordial ratio [D / H] = . × − (Pettini & Cooke 2012; Cooke et al. 2014) is shown as a greendashed line. 3ounding 4 π solid angle.Following the prescription of Spitzer (1978) (seealso Sternberg et al. (2014) the rate constant forH -formation on grain surfaces is taken as R G = × − cm s − √ T K but the thermal balance andtemperature-dependent rate constant are not of cru-cial importance to the H -fraction. The same re-sults are obtained using a fixed rate constant R G = . × − cm s − that is the average of the valuesobtained toward three stars by Gry et al. (2002) oftencited in other work, as discussed in Section 5.2 (thosesightlines are called out in Figure 5). The thermal bal-ance is however very important to several endothermicreactions driving the oxygen and deuterium chemistryand to the overall ionization balance in the models.The models described here di ff er from our previ-ously published results in that they employ the H photodissociation scheme of Draine & Bertoldi (1996)which explicitly treats dust attenuation of the radia-tion field at the wavelengths of the Lyman and Wernerbands of H (90 - 110 nm). The optical depth fordust absorption is τ d = . × − N(H) (Draine 2003)as in Sternberg et al. (2014). Regarding our previ-ous models based on the shielding factors of Lee et al.(1996), we note that incorporation of dust extinctionis implicit and somewhat ambiguous in the formula-tion of Lee et al. (1996) where only an overall H self-shielding function is employed, more similar to theearlier work of Federman et al. (1979).The accuracy of the Draine & Bertoldi (1996) for-mulation was recently been verified in great detail bySternberg et al. (2014) using an exact calculation inthe context of the Meudon PDR code. Separation ofdust extinction-related phenonema is important for un-derstanding the HD formation problem, and perhapseven more important for understanding the general H -formation problem in media having low number den-sity and / or low metallicity. The separate e ff ects ofdust extinction, and the shielding factors and dust ex-tinctions for our models are explicitly shown and dis-cussed here.Direct HD formation on grains is modelled follow-ing the precepts of Le Petit et al. (2002) with a rateconstant 40% larger than for H . This is manifest in themodels shown in Figure 1 at very small N(H ) but thedirect formation of HD on grains is of almost no rel-evance to the observations of HD, as noted in Section3. Self-shielding of HD is important in some cases,and may have contributed to the observed HD; the self-shielding of HD is the same function of N(HD) as the self-shielding of H employing N(H ) .For reference, note that the cosmic-ray ioniza-tion rate of atomic hydrogen has been taken as ζ H = × − s − as seems appropriate for the dif-fuse molecular ISM (McCall et al. 2002; Liszt 2003;Hollenbach et al. 2012; Indriolo et al. 2012). As astandard value we take Γ H = . × − s − as thefree-space photodissociation rate of H , as in the workof Draine & Bertoldi (1996) from which our H self-shielding scheme was drawn.Other values in the recent literature are Γ H = . × − s − (over 2 π steradians) in the work of Lee et al.(1996) and Γ H = . × − s − in Sternberg et al.(2014). In this work, number and column densitiesimplicitly refer to hydrogen nuclei when stated in thetext, unless otherwise noted. HD does not enjoy the high degree of self-shieldingwhich is the sine qua non of high H -fractions.Hence the HD / H ratio might be expected to bevery small in di ff use gas, well below the inher-ent [D / H] ratio. That this is not so reflects thefractionation and charge exchange processes deter-mined by the ambient proton and deuteron density asnoted by Black & Dalgarno (1973), Watson (1973),Jura (1974), O’Donnell & Watson (1974) and Spitzer(1978).The basic chemistry of H -HD interconversion hasbeen sketched out by those authors and by Stancil et al.(1998) in the context of early-Universe chemistry;the rates used in this work were taken from Table1 of Stancil et al. (1998). Although the fractiona-tion / deuteration chemistry can be quite complex incold, fully-molecularized dark clouds, it is fairly sim-ple in warmer, lower-density di ff use regions whereprotons are abundant and only H and HD need be con-sidered, with exchange of protons or deuterons as themeans of interconversion between the two molecularhydrogen isotopologues.In a purely atomic gas ionized by cosmic rays theionization and recombination rates of H and D atomswould be very nearly the same (the grain neutralizationof D + is slower by a factor √ + + D +∆ E → D + + H (rate constant k = − cm s − , ∆ E / k =
41 K) tends to force n(D + ) / n(DI) ≈ n(H + ) / n(H I) exp(-41 K / T). In the presence of H + + H → HD + H + forms HD with rate constant k = . × − cm s − .If only charge transfer and H fractionation neutral-ize D + (a highly reductive assumption), a relativelycompact expression gives the proton density n(p) re-quired to reproduce a given HD / H ratio in terms ofobserved quantities and physical constants, with no ex-plicit dependence on either the density or recombina-tion rates: we have n ( p ) = n(HD) / n(H )[D / H] Γ HD k [1 + ( k k −
2) n(H )n(H) ] exp ( 41 T )(1)where n(H) = n(H I) + ) and the photodis-sociation rate of HD in free space is Γ HD = Γ H = . × − s − (Draine & Bertoldi 1996; Le Petit et al.2002). The required proton density n(p) derived fromEq. 1 is nearly independent of the molecular fraction inthe gas for k / k = . fraction is fixed. Of coursethe actual proton density may have quite strong depen-dence on n(H) and the assumptions used to derive Eq.1 are rather archaic. Below we discuss the actual pro-ton density in the models but the chief means by whichan adequate proton density is achieved is the high de-fault cosmic ray ionization rate that we have adopted,see Liszt (2003). / H ratio
The models whose results are shown here use thecosmic ratio [D / H] = . × − (Pettini & Cooke2012; Cooke et al. 2014) which is near the Milky Wayvalue [D / H] = . ± . × − determined byLinsky et al. (2006). The actual gas phase [D / H] maybe slightly smaller than the overall [D / H] value in theMilky Way but this is a small di ff erence comparedto the e ff ects of the chemistry. Moreover the modelresults are intended to be generally relevant, for in-stance in Figure 1 where the local and high-z resultsare shown together.
3. Observations and models of HD in the MilkyWay3.1. Observations of HD
Shown in Figure 1 are the observational resultsfor N(HD) and N(H ) along the 41 Milky Waysightlines in the recent omnibus FUSE survey of Snow et al. (2008). Results for the DLA sightlinesat z > = and HD, N(HD) / N(H ) ≈ / H] ≈ × − . Inthe opposite limit when only insignificant amounts ofH and D are molecular, N(HD) / N(H ) ≈ . / H](Le Petit et al. 2002). By contrast the galactic HD col-umn densities lie about a factor 10 below the cosmic[D / H] ratio in Fig. 1, falling nearly parallel to a lineof constant N(HD) / N(H ) = × − . The regressionanalysis of Snow et al. (2008) found a power-law slope1 . ± .
03. The N(HD) / N(H ) values of Snow et al.(2008) are about three times larger than those consid-ered in our simillar analysis of the same phenomena(Liszt 2003). The slightly super-linear empirical slope deter-mined by Snow et al. (2008) means that N(HD) / N(H )increases with increasing molecular fraction f H = ) / N(H) with N(H) = N(H I) + ) (note theannotations in Fig. 1 showing f H along the curves).Snow et al. (2008) pointed out that the models ofLe Petit et al. (2002) seemed to predict the oppositebehaviour except at f H & H . H & -fraction in di ff use clouds because of its dependenceon the presence of a relatively high proton density.Shown in Figure 1 are our equilibrium model re-sults for sightlines through the centers of the uniform-density gas spheres discussed in Section 2. Resultsfor a family of models with n(H) =
16 cm − andvarying primary cosmic-ray ionization rate per H-atom2 × − s − ≤ ζ H ≤ × − s − are shown with andwithout the attenuation of Lyman and Werner Bandphotodissociating radiation by dust. Also shown aremodel results for n(H) =
128 cm − .At the lowest cosmic-ray ionization rate consid-ered, ζ H = × − s − , the chemistry is essentiallyswitched o ff and the predicted HD abundance is somethree orders of magnitude below observed values. Thisdemonstrates that when HD is seen in the Milky Wayit overwhelmingly originates in situ in the gas phase5y deuteration of H . Although the actual [D / H] ra-tio in the gas phase is important, and n(D) / n(H) in thegas phase is a ff ected by the n(HD) / n(H ) ratio (Liszt2006), considerations of HD formation on grains areirrelevant to HD-formation in the observed amounts.HD molecules formed on grains comprise a negligiblefraction of the observed HD.At the highest cosmic-ray ionization rate consid-ered, ζ H = × − s − , the observations are boundedby the models with n(H) =
16 cm − and 128 cm − .If the cosmic ray ionization rate is ubiquitous, the re-gion occupied by the observations in Figure 1 is under-stood in terms of the relatively slow variations of theN(HD) / N(H ) ratio with number density, and cloudswith n(H) >>
128 cm − apparently were not sampledin the observations. Alternatively, if the cosmic rayionization rate is assumed to vary, it should generallybe in the sense of increasing above ζ H = × − s − ,implying somewhat higher number density, becauseneutral atomic gas with number density much below16 cm − is not generally understood to be thermallystable in multi-phase models of the ISM , see e.g. Fig-ure 7 of Wolfire et al. (2003).Our models with ζ H = × − s − and 16 cm − . n(H) ≤
128 cm − reproduce the observations, in-cluding the slightly super-linear slope derived bySnow et al. (2008), but the slope of the observed varia-tion of N(HD) with N(H ) must also reflect the under-lying distribution of physical environments that weresampled. The HD column density results from a va-riety of influences including the gas-phase chemistry,the individual self-shielding of HD and H , their cou-pling via the chemistry, and the extinction of photodis-sociating UV radiation by dust. The consequential dif-ference between our models and those of Le Petit et al.(2002) lies most nearly in the treatment of the overallionization balance in the presence of grain-assisted re-combination, which both allows and requires a highercosmic ray ionization rate. H and n(HD) / n( H ) Although our models have uniform total numberdensity n(H), the molecular fraction, proton densityand n(HD) / n(H ) ratio vary considerably within them.Figure 2 shows the n(HD) / n(H ) ratio and molecularfraction as functions of radius for models with N(H) = × cm − having column densities N(HD) andN(H ) that are typical of the Milky Way data shown inFigure 1. The models have 128 radial shells but only barelyresolve some sharp variations near the outer cloudedges. Typical variations in n(HD) / n(H ) are a factortwo or three with higher values at the edge and cen-ter. At the highest density, n(H) =
128 cm − , the H -fraction varies less and generally in the opposite sensefrom n(HD) / n(H ) so that HD is distributed rather uni-formly throughout the model. At the lowest density,n(HD) / n(H ) varies much less than the molecular frac-tion and the bulk of the HD resides near the center ofthe model.Two e ff ects directly attributable to dust extinctionare shown in Figure 2. First, the n(HD) / n(H ) ratiois slightly smaller in the outer regions of the mod-els when dust attenuation is negelected, presumablybecause the attenuation by dust has a larger e ff ecton HD than on H , given that HD is so much moreweakly self-shielded. Second and most noticeably, then(HD) / n(H ) ratio does not rise near the center of mod-els in which the extinction of dissociating photons bydust is neglected (but note that the scale in the up-per panel of Figure 2 is much expanded compared tothat at bottom). An analogous situation appears inFigure 1 for smaller values of ζ H when N(HD) onlyrises sharply at N(H ) & cm − in the presenceof dust extinction. Although the H -fraction alwaysreaches higher values toward the center of modelswhen the explicit attenuation of dissociating photonsby dust is included, sharp increases in n(HD) / n(H )and N(HD) / N(H ) result primarily from the onset ofstrong HD self-shielding, triggered by the extinctiondue to dust. Whether the onset of H self-shieldingand high H -fractions is triggered by dust extinction isthe subject of Section 4.
4. What triggers the onset of high H and HDfractions?4.1. The outermost self-shielding layer In this work, dust extinction of dissociating pho-tons is expressed using a single absorption cross-section σ d = . × − N ( H ) as in the work ofDraine & Bertoldi (1996) and Sternberg et al. (2014)and scattering is ignored. As shown in the Figure 2,dust extinction raises n(HD) / n(H ) and f H near thecenters of the models but does not play a large role intriggering the onset of higher n(H ) in the outer shield-ing layers except perhaps at the very lowest density:the curves showing the radial variation of f H with andwithout dust extinction di ff er little in the outer radial6
018 1019 1020 10211018 1019 1020 1021101310141015101610171018101910201021 10-710-610-510-410-30.010.11N(H) [cm-2] N ( H ) [ c m - ] T r a n s m i ss i on a t ce n t e r ζ H=2x10-16 s-18128n(H) dust8 832128 32 total
Fig. 3.— Calculated H column densities and interiorattenuations. The solid black curves represent the H column density toward the geometric centers of mod-els with n(H) =
8, 32 and 128 cm − over a wide rangeof cloud column density N(H) from front to back ofthe model, with ζ H = × − s − . As in Figure1 each solid curve has a dotted red companion rep-resenting the same model neglecting dust attenuation.Reading the scale at right, the blue dashed curves sep-arately show the fraction of ambient H -dissociatingphotons that penetrates to the centers of the models af-ter H self-shielding and dust extinction (labelled “to-tal”) and the much smaller amount which the dust ex-tinction contributed to that total. The grey horizontalline represents the attenuation at which the cosmic rayflux indicated becomes the dominant cause of H de-struction at the rate indicated. regions where f H abruptly increases by factors of 100or more. The reason for this can be better understoodin the context of Figure 3 showing the attenuation ofdissociating photons in models with di ff ering numberand H column density, along with the particular con-tribution from dust extinction of dissociating photons.In Figure 3 the onset of higher H fractions isscarcely a ff ected by the presence of dust attenuation,rather, that is determined by the onset of strong H self-shielding which already occurs at such small A V that there is very little dust opacity even in the UV. Ifphysical conditions foster small H -fractions at highN(H), the dust extinction and its e ff ect on f H may beappreciable for H and this is the situation for HDas well at very high N(H) in Figure 1. The detailsdepend on the slopes of the variation of the attenua-tion with column density: if the self-shielding of H varies slowly with N(H ), especially when f H is larger,dust attenuation and small increases in N(H) may bee ffi cient at increasing the molecular fraction locally.At the lowest number density in Figure 3, N(H ) to-ward the center of the models about doubles for N(H) & cm − and the strong increases in HD were notedearlier. Whether clouds with such high N(H ) exist atsuch low density is another matter.Also shown in Figure 3 is the transmission at whichthe attenuation of the radiation field is so great that thedestruction rate of H by cosmic rays equals that dueto photodissociation in the Lyman and Werner bandswhen ζ H = × − s − . In fact the destruction of H near the centers of our models is dominated by cosmicrays even for clouds having total A V = H inside the outer self-shielding layer. The overall contribution of dust extinction of Ly-man and Werner band photons is summarized in Fig-ure 4 where for models with n(H) =
8, 16, ..., 128cm − we show the total molecular fraction integratedover the model and the fraction of that total that isdirectly attributable to dust extinction, calculated sym-bolically as ((Mass with dust extinction)-(Mass with-out)) / Mass(with). For conditions approximating aSpitzer Standard H I cloud with n(H) =
32 cm − andN(H) = × cm − some 20% of the hydrogen is inH and 30% of that is attributable to the presence of7ust extinction. H formation at Solar and sub-Solar metallicity5.1. Observations In Figure 5 we show N(H ) and N(H) as determinedin UV absorption in the Milky Way and the MagellanicClouds. The galactic data include observations towardbright stars as studied by Copernicus (Savage et al.1977) and toward bright stars (Rachford et al. 2002,2009) and AGN (Gillmon & Shull 2006) using FUSE,extending to much higher N(H) than were available toCopernicus. The Southern Hemisphere data is fromTumlinson et al. (2002). The Milky Way data in thisplot shows the familiar jump in N(H ) at N(H) ≈ − × cm − whereby the fraction of hydrogenin H abruptly increases to ≥ . ) against N(H) is unlike the situationshown in Figure 1, where sightlines have been or-dered according to their H column densities. Viewingthe onset of H formation along sightlines harboringa mixture of conditions involves several separate as-pects of the ISM; the cold neutral medium of the ISMis clumped into di ff use “clouds” (usually called H Iclouds by radio astronomers); H forms in apprecia-ble quantities in H I clouds having su ffi ciently highN(H); and lines of sight with N(H) & × cm − cross at least one of these clouds. The column densi-ties at which the jump occurs has been increased, andsomewhat spread out by the contribution from unre-lated, less-molecular material along the line of sight,see Spitzer (1985).The sightlines used by Gry et al. (2002) to de-termine R G are marked in Figure 5; they all havevery high H -fractions at their respective values ofN(H): nonetheless the value of the grain surface H -formation rate constant derived in that work repro-duces our temperature-dependent results extremelywell as noted in Section 5.2 and as shown in Figure6. For the Large Magellanic Cloud (LMC), the metal-licity is some 2.5 times smaller than that of the MilkyWay (Z = occurs around N(H) = cm − , roughlythree times that of the Milky Way. According toWeingartner & Draine (2012) such N(H) correspondsto A V ≈ .
12 mag, as against A V = = occurs at even higher N(H) ≈ × cm − , but correspond-ing to A V = H self-shielding In Section 5.1 we showed that the surface area oflarge grains (as represented by A V in the small range0.12 mag . A V . .
18 mag) is very nearly constantat the onset of strong self-shielding in H in three sys-tems of di ff erent metallicity, in principle leading to thequestion whether it is the extinction by these grainsor their aggregate H -catalytic grain surface area thatis responsible for the increase in the H -fraction withN(H).The behaviour seen in Figure 3 suggests that onlythe grain catalytic area is important, because the dustextinction per se is small at the onset of H self-shielding. This is further illustrated in Figure 6 show-ing the results of varying several parameters in mod-els with n(H) =
32 cm − . This is the density of aSpitzer H I cloud and for typical ISM pressures p / k = − − K (Jenkins & Tripp 2011) it is con-sistent with the kinetic temperatures that have been in-ferred from the J = = , ie 77 K forthe original Copernicus survey (Savage et al. 1977),86 ±
20K and 124 ±
8K for FUSE sightlines towarddistant AGN in the galactic disk and halo, respectively(Gillmon & Shull 2006) and 67 ±
15K for the high-est column density translucent FUSE sightlines towardbright stars (Rachford et al. 2002, 2009). The kinetictemperatures in our models are in the range 50 - 160K, varying inversely with n(H) and having somewhathigher pressure at higher density, as is generally thecase for phase diagrams in multi-phase models of thedi ff use ISM. This is all consistent with the heating-cooling model that we adopted and is discussed ingreater detail by Wolfire et al. (2003).The curves shown in Figure 6 cluster in two groupsaccording to whether the H-H transition is shiftedappreciably. Some parameters have little e ff ect. Re-placing our temperature-dependent H formation rateby the value derived by Gry et al. (2002) has littlee ff ect. Increasing the cosmic-ray ionization rate to ζ H = − s − increases H formation and hastensthe onset of H self-shielding at small N(H) becausethe models are somewhat warmer but H formation issuppressed at the very highest N(H) because cosmicray ionization so greatly dominates the H destruction.The column density at the onset of H self-shielding8able 1: HD in high-z DLA and sub-DLA systemsSource Z / Z ⊙ z N(H I) N(H ) N(HD) 2N(H ) / N(H I) N(HD) / N(H ) HD Ref.log cm − log cm − log cm − / [D / H]Q1232 0.04 2.34 20.90(0.08) 19.68(0.08) 15.43(0.15) 0.121 2.21 1Q1331a 0.04 1.78 21.20(0.04) b . a a Prochaska et al. (2003); Balashev et al. (2010) b Prochaska & Wolfe (1999)References (1) Ivanchik et al. (2010); (2) Balashev et al. (2010); (3) Tumlinson et al. (2010); (4) Noterdaeme et al.(2008); (5) Noterdaeme et al. (2010) f r ac t i on n(H)
16 16 32 32 64 64 128 128
Fig. 4.— Molecular mass fractions and fractions ofthe molecular mass attributable to dust extinction. Thesolid curves show the fraction of the cloud model massthat is molecular at the indicated number and columndensity. The dash-dotted red curves nearer the top ofthe plot show the fraction of the molecular mass thatis attributable to dust extinction of Lyman and WernerBand photons as discussed in Section 4. scales inversely with the radiation field as expectedfrom the discussion in Appendix B but dependenceon the radiation field is extremely complex becausethe thermal and ionization balance is strongly a ff ected.Stronger radiation heats the models, increasing the H formation rate, somewhat compensating the increasedphotodissociation.Also shown in Figure 6 are curves correspondingto decreasing the grain surface H -formation rate con-stant R G by a factor of 10, and separately, a calculationdepleting the quantity of large, H -forming grains bythe same amount and perforce also lessening the ex-tinction by dust of dissociating photons by the sameamount. Figure 6 shows that the onset of H self-shielding in the outer shielding layer depends linearlyon the grain surface area, but only to the extent that thisarea is available for catalytic H formation; reducingthe grain formation rate and the surface area of largegrains both result in self-shielding at very nearly thesame N(H), confirming that the extinction provided bythe grain surface area is a small e ff ect on the initialonset of strong H self-shielding and the H I - H tran-sition.The e ff ects directly attributable to the grain surfacearea di ff er noticeably inside the self-shielding layer.When large grains are removed entirely the H -fractiondoes not exceed 10% except at very large N(H), whichis the case in Figure 5 for the sightlines with sub-Solarmetallicity. Inside the self-shielding layer, dust ex-tinction drives the H -fraction higher, and dust extinc-tion will have a strong e ff ect whenever the H frac-tion is substantially below unity well inside the outer9elf-shielding layer; this would be the case at lown(H) and large N(H) if those conditions actually oc-cur in the ISM. This has interesting consequences forthe time-evolution of the H fraction, because regionswith lower molecular fraction equilibrate earlier, and,even though dust extinction can increase the equilib-rium molecular fraction inside the self-shielding layer,it does not hasten the approach to equilibrium. Whendust extinction matters to the H fraction the times toreach H equilibrium will be long.
6. HD formation at high redhshift and / or lowmetallicity The observations of HD in DLA and sub-DLA sys-tems at high redshift are summarized in Table 1 andshown in Figures 1 and 7. Although DLA and sub-DLA systems typically have low metallicity, the sys-tems in which HD has been detected cover a widerange of metallicity from Z / Z ⊙ = / N(H ) seen athigh z in terms of higher density when the metallicityis small and rather puzzling that the high redshift sys-tems are so similar in having such high HD / H ratioswhile the metallicity varies over such a wide range.Figure 6 shows that a given N(H ) will be seen atsmaller molecular fractions in systems of lower metal-licity because the required N(H) is larger; indeed,the DLA and sub-DLA systems at high redshift havecomparable N(HD) to those seen in the Milky Waysightlines, but generally at smaller molecular fractionsoverall. In Table 1 systems with high and low metal-licity have comparable values of N(H ) but with muchsmaller f H at the low-metallicity end.Unlike the case at Solar metallicity illustrated inFigure 1 where N(HD) / N(H ) decreases with density,larger proton densities and higher HD / H ratios are ex-pected at higher number density when the metallicityis small. Higher proton densities are always more read-ily available in principle at higher number densities butradiative recombination with electrons from ionizedcarbon and neutralizations by small grains suppress theproton fraction at higher density when the metallicityis near-Solar. When the metallicity is small, the e ff ectsof both electrons and small grains are sharply curtailedand the proton density better tracks the number densityoverall.The situation is illustrated in Figure 7 where wevaried the metallicity in our calculations by linearlyscaling the abundance of all metals or metal-bearing N ( H ) [ c m - ] StandardG0=1.4 ζ H ζ H=10-15 s-1GryRH2/10Large Grains/1010%
Fig. 6.— Other influences on N(H ) for models at n(H) =
32 cm − . The group of curves at the left includes a“standard” model (solid black), a model with the ra-diation field increased by 40% (dotted grey), a modelusing the temperature-independent H -formation rateof Gry et al. (2002) (dash-dot green) and a model with ζ H = − s − (dashed, blue). At right are mod-els with a diminished grain formation rate coe ffi cient(dashed red) and a model in which 90% of the largegrains are entirely removed from the calculation (dash-dot orange). The green dashed line indicates the lo-cus where 10% of the hydrogen is in molecular form.These models were computed along sightlines dis-placed 5 / transition has beendisplaced toward higher N(H) compared with modelsobserved through the central sightline.10
019 1020 1021 1022101310141015101610171018101910201021 N(H) [cm-2] N ( H ) [ c m - ]
10% The Milky WayGry et al.LMC SMC
Fig. 5.— H column densities for the Milky Way and Magellanic Clouds. The Milky Way data include results fromSavage et al. (1977), Rachford et al. (2002) and Gillmon & Shull (2006) and are shown as open green rectangles; thethree lines of sight used by Gry et al. (2002) to determine the H -formation rate coe ffi cient are separately marked.Results for the SMC and LMC are taken from Tumlinson et al. (2002).11pecies (ie the large and small dust grains). At left it isshown that the N(HD) / N(H ) ratio increases at least asfast as 1 / Z at n(H) =
128 cm − in the range of N(H )where HD is observed at high redshift. At right inFigure 7 the increase of N(HD) / N(H ) with density isshown for Z / Z ⊙ = /
16. The functional dependence ondensity is weaker than linear but the senseis quite con-trary to the variation with density shown in Figure 1 atSolar metallicity. + Balashev et al. (2010) derived T K = ± = = towardJ0812 + / N = -1.93 in the J = = the J = = . × − s − (Flower et al. 2000). Balashev et al.(2010) pointed out that the HD level populations canbe used to derive the ambient density and they quoten ≈
50 cm − using a two-level atom approach underthe assumption that radiative pumping of the J = in their unnumbered expression for thenumber density (their Section 3.2) when the smallerupward rate is actually called for.The collisional rate constants for HD excitation byH, He, and H are given by Flower et al. (2000). Theyallow a number of important simplifications in theanalysis below 100 K, because the rate constants arethe same for H and He, and the same for ortho- andpara-H . As well the rate for H excitation is just twicethat for atomic hydrogen so that the analysis is inde-pendent of the molecular fraction. Denoting the up-ward rate constant for excitation by atomic hydrogenas C and the spontaneous emission coe ffi cient as A ,the total hydrogen number density may be written as n ( H ) = N N A C × + [He / H] C = g g e − E / kT K C = e − / T K C C = . × − cm s − T / K in the limit of weak excitation n(H) C << A .With T K =
48K and [He] / [H] = =
240 cm − . Such a density is consistent with observed, relatively high HD / H at low metallic-ity (Figure 7) and indeed Balashev et al. (2010) con-sidered that log Z / Z ⊙ = -1. However Prochaska et al.(2003) actually derived [O] / [H] ≈ / [H], so that J0812 is not an espe-cially low-metallicity system.
7. Summary7.1. HD
In Section 3 (see Figure 1) we showed that the ra-tios N(HD) / N(H ) ≈ × − in the Milky Way areprimarily diagnostic of the cosmic-ray ionization rateand secondarily of the number density in the sense of ζ H / n(H) tending to be constant, with 16 cm − . n(H) .
128 cm − for ζ H = × − s − . The gas sampled ingalactic HD measurements has moderate density andmolecular fraction f H ≈ . − .
5, because the protondensities necessary to ensure adequate protonation aremore di ffi cult to maintain at the higher densities thatare more favorable to H formation overall (see Fig-ures 2 and 8).In Section bf 6 we discussed N(HD) and N(H )observed in high-redshift DLA and sub-DLA sys-tems. DLA and sub-DLA systems observed at z > / N(H ) ratios compared tothe Milky Way, ie with comparable N(HD) but orderof magnitude smaller N(H ) and somewhat smallerN(H ) / N(H I). In Section 5 (Figure 5) we discussedhow smaller H fractions are observed in nearby sys-tems with sub-Solar metallicity: H becomes self-shielding at progressively larger N(H I) and smallerH fractions, but with nearly-fixed A V = . ± . self-shielding at progres-sively higher N(H I) and fixed A V is a consequenceof the smaller dust / gas ratio at smaller metallicity, andthe consequent smaller grain surface area per unit hy-drogen that is available for H formation.In Section 6 (see Figure 7) we showed that higherN(HD) / N(H ) and smaller N(H ) / N(H) can be ex-plained at higher density in systems of smaller metal-licity, because the proton density increases with in-creasing number density, opposite to the Milky Waycase at Solar metallicity. Moreover, in one case wherethe J = ≈
240 cm − is relatively high. However this system is not obvi-ously of very low metallicity (Z / Z Solar . .
36) and12
017 1018 1019 1020 10211012101310141015 n(H)=128 cm-31016 N(H2) [cm-2] N ( HD ) [ c m - ] DLA z > 1.7B0120-28Z=1.03 .07 .14 .21 .29 .38.47.56.64.72.78.83.86.89.90.01 .04 .08 .13 .20 .28 .36.46.55.64.71.76.08 .13 .19 .27 .36.46.551/16.01 .02 .03 .05 .08 .13
DLA z > 1.7
Z/ZSolar=1/16n(H)
B0120-2832 64128256
Fig. 7.— Influence of density and metallicity on N(HD) / N(H ). Left: at n(H) =
128 cm − the metallicity is variedin steps of two from Solar to 1 /
16 Solar. Values of f H are indicated in gray along the curves. Right:n(H) is varied insteps of two over the range 32 cm − ≤ n(H) ≤
256 cm − at 1 /
16 Solar metallicity.high N(HD) / N(H ) ratios ranging from 80-200% ofthe cosmic [D / H] are observed in high redshift systemsover a wide range of metallicity ranging from 0.04 to 2times Solar (Table 1) for the 6 out of 8 high z systemswith N(H ) > × cm − and smaller quoted col-umn density errors. Conversely, a low-redshift DLAsystem at z = ) values typ-ical of Milky Way gas and a high H fraction, but atmetallicity only 7% Solar.It seems odd that the high-redshift systems shouldshare such high N(HD) / N(H ) values while having solittle else in common in terms of the fractionationchemistry expected at their quoted metallicities. Forthe high-z systems it is tempting to abandon the frac-tionation scenario, with its implication of a high H-ionization rate, in favor of an ad hoc scenario in whichthe molecules are in some tight knot where the con-version of both hydrogen and deuterium to molecularform is nearly complete. But this begs the question ofwhy such similar and unusual conditions would existover such a wide range of metallicity. H and the e ff ect of dust extinction Motivated by e ff ects like the sharp central upturnsin HD / H observed in the upper panel of Figure 2, oras seen in Figure 1 at smaller ζ H and high N(H ), webroke out the explicit extinction of H -dissociating Ly- man and Werner Band photons by dust. As shown inFigures 2 and 3 and discussed in Sections 3 and 4, theouter self-shielding layer in H is just that, triggered bynon-linear e ff ects inherent in the cross-section for H photo-absorption (Figure 3) with little dependence onthe existence of extinction by dust. Only at very smalln(H) in a regime that is probably not thermally stablein two-phase ISM is the location of the onset of H -selfshielding appreciably shifted by dust extinction.By contrast, there is a somewhat wider range ofnumber density where dust extinction has an apprecia-ble e ff ect on the H fraction inside the outermost H self-shielding layer. As shown in Figure 4, about one-third of the H in a cloud at n(H) =
32 cm − is directlyattributable to dust extinction.The e ff ects of dust extinction on HD are somewhatgreater owing to its lesser degree of self-shielding. Asshown in Figure 2, HD / H ratios are increased slightlyeven in the outer portions of our models, with muchstronger e ff ects toward the center and even moreso atlower density. Increases in either the amount of H orthe attenuation of HD-dissociating photons can tip theHD over into a non-linear regime where its local abun-dance is strongly dependent on its own self-shieldingand somewhat less on the local fractionation chem-istry.In discussing the dependence of H formation on13etallicity, Figure 5 shows how the location of the on-set of strong H self-shielding shifts monotonically tohigher N(H) with decreasing metallicity, in such a wayas to keep constant the inferred A V representing the to-tal column of grain surface area, see Sections 5 and 6.This constancy of A V could be interpreted as implyingthat the onset of H self-shielding is caused by the ex-tinction due to dust, but that is not the case. As shownin Figure 6, it is the lessening of the grain surfacearea available for H formation that causes the shift,while the dust extinction per se increases the H frac-tion at yet-higher N(H), inside the H self-shieldinglayer. As shown in Figure 6, producing H fractionsabove 10% requires even much higher N(H) in low-metallicity systems owing to the diminished dust ex-tinction cross section per H-atom.The National Radio Astronomy Observatory is cur-rently operated by Associated Universities, Inc. un-der a cooperative agreement with the National ScienceFoundation. The hospitality of the ITU-R and Ho-tel Bel Esperance in Geneva and the UKATC and theApex City Hotel in Edinburgh are appreciated duringthe writing of this manuscript. 14 . Intra-cloud variation of the proton and electron densities Evaluating Eq 1 we find n(p) ≈ .
004 cm − for T =
75 K, using the photodissociation rate of HD Γ HD = Γ H = . × − s − in free space (Draine & Bertoldi 1996) and N(HD) / N(H ) ≈ × − as shown in Figure 1. This canbe compared with the results shown in Figure 8, where we have plotted the proton and electron densities just insidethe outer edge and at the center of the models whose results were depicted in Figure 1.The electron and proton densities behave very di ff erently with respect to changes in density, largely because carbonremains ionized under all conditions, putting a floor on n(e) that does not exist for n(p). At the outer edge, n(e)increases with density while n(p) remains nearly fixed owing to a balance between the volume ionization n(H) ζ H and the neutralization by small grains, whose density and neutralization rate change in fixed proportion to n(H). Theelectron density decreases toward the centers of all the models because the ionization fraction due to ionized hydrogen(ie, n(p)) decreases there. The proton density decreases by nearly the same factor at all density while the decrease inthe electron density at the center is small at high density when ionization of hydrogen is weak.The decline of n(p) toward the cloud center means that the chemistry has an innate tendency to segregate protonsand H , shown in Figure 2, complicating the HD formation problem. In Section 3.3 we noted that the distribution ofHD is more uniform in radius at higher density, and the models have higher f H at higher n(H) but overall it is theproton density that is decisive, and Figure 1 shows that lower density clearly wins the HD battle at Solar metallicity. B. Scaling parameters for the H -formation problem Simply writing down the equation for the growth of H dn (H ) / dt = − n (H ) Γ H + R G n ( H ) n ( HI ) p T K (1)and setting dn(H ) / dt = = n(H I) + ) and Γ H is function-ally dependent on N(H )) n (H ) / n ( HI ) = R G p T K n ( H ) / Γ H shows that Γ H / (R G n ( H ) √ T K ), the n(H ) / n(H I) ratio in free-space, is a dimensionless scaling parameter for thisproblem. It serves as the basis of the discussion of Federman et al. (1979) who in terms of our quantities defined 1 /ǫ = n(H I) / ) = Γ H / (2R G n ( H ) √ T K ) and showed that the data of Savage et al. (1977) (see Figure 6 here) couldbe reproduced with ǫ = × − corresponding to n(H) ≈
33 cm − for R G √ T K = . × − cm − (Gry et al.2002) and Γ H = . × − s − . The formulation by Federman et al. (1979) does not consider shielding due to dustand indeed, our models showed that dust shielding contributes modestly at that density at Solar metallicity (Section4.2). The formulation of Federman et al. (1979) is numerically intensive but it provides a description of the widthand location of the self-shielding layer without additional assumptions because the equilibrium conditions are solvedexactly, although not analytically.The e ff ects of dust extinction, shown in many instances in our models but most important at small n(H), wereincorporated in the analytic formulation of Sternberg and his collaborators (Sternberg 1988; Sternberg et al. 2014)whose most recent description defines parameters α = /ǫ and G that is the mean H self-shielding factor includingthe dust-extinction associated with N(H ) (ie, not N(H)). The total dust column is considered to have two parts, calledthe H I and H dust, whose columns are proportional to N(H I) and N(H ). In this formulation the structure of the HI - H transition is determined by the product α G. The α G << α G >> α G ≈ -dominated regimes asindeed seen from the basic definition of α or ǫ as H I / H ratios; one replaces the small free-space H / H I ratio by thatin the strongly shielded transition zone. In e ff ect, our models lie in the transition between two regimes, as understoodin the context of the underlying physics of the heating, cooling and ionization balance.Krumholz et al. (2008) explicitly introduced the total attenuation due to dust into the equation of detailed balance(akin to eqn 1), resulting in the creation of a hybrid (chimerical?) parameter χ that is in e ff ect a redefinition of 1 /ǫ .15 np [ c m - ] n e [ c m - ] Fig. 8.— Proton (upper) and electron densities for some of the models employed in Figure 1. The solid lines are forlocations just inside the outer edge of the models; the dashed lines are at the center. In all cases the primary cosmicray ionization rate per H-atom is ζ H = × − s − . The electron and proton densities move oppositely in response tochanges in density at solar metallicity, see Section 7. 16n χ , the H photoabsorption cross section that determines Γ H is replaced by the cross-section for dust extinction.Because the dust cross section is so much smaller, one deals with χ -values of order unity in the H I - transition regionas with Sternberg’s formulation. Sternberg et al. (2014) show that χ = α G in the limit of low metallicity.17
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