H-alpha and Free-Free Emission from the WIM
aa r X i v : . [ a s t r o - ph . GA ] N ov H α and Free-Free Emission from the WIM Ruobing Dong and B. T. Draine
Department of Astrophysical Sciences, Princeton University, Princeton, NJ, 08544
ABSTRACT
Recent observations have found the ratio of H α to free-free radio continuum tobe surprisingly high in the diffuse ionized ISM (the so-called WIM), corresponding toan electron temperature of only ∼ α , and collisionally-excited lines such as[N II ]6583. To reproduce the low observed value of free-free to H α , the PAH abundancein the photoionized regions must be lowered by a factor ∼
3, and ∼
20% of the diffuseH α must be reflected from dust grains, as suggested by Wood & Reynolds (1999). Subject headings: atomic processes, ISM
1. INTRODUCTION
Low density ionized regions, often referred to as the warm ionized medium (WIM), account for ∼
90% or more of the ionized interstellar hydrogen in the Galaxy. The WIM occupies a substan-tial volume fraction and is a major component of the interstellar medium (ISM) at ∼ α , H β , He I λ II ] λ , , II ] λ , I ] λ II ] λ III ] λ II ]12.81 µ mand [Ne III ]15.55 µ m) have been made in the last two decades, aiming to understand the physicalconditions in the WIM (Greenawalt et al. 1997; Rand 1997, 2000; Haffner et al. 1999; Otte et al.2002; Miller & Veilleux 2003; Madsen et al. 2006). Although the observed line ratios vary fromregion to region, optical line ratios such as [N II ] λ α appear to indicate that the WIMmaterial has temperature T ≈ ± n e ≈ . − , with a volume filling fraction of ∼ III ] λ α indicate that compared with H II regions powered by early O-typestars, the WIM seems to have fewer ions present that require ionization energies greater than 23 eV,with n (He + )/ n He ranging from 0.3 to 0.6 (Reynolds 1985; Rand 1997; Haffner et al. 1999; Reynolds2004; Madsen et al. 2006; Haffner et al. 2009).Recently, Dobler et al. (2009, hereafter DDF09), using the WMAP five-year data, deduceda surprisingly low temperature ( ∼ α emission. Similar results had previouslybeen reported based on the free-free to H α ratio in the WMAP one-year data (Davies et al. 2006)and three-year data (Dobler & Finkbeiner 2008). The low value came as a surprise, because thediffuse ionized gas had been expected to have T ≈ not in equilibrium, andthat it is possible to understand the observed line ratios and weak free-free emission if part of theWIM is assumed to be gas in the process of cooling and recombining after removal of a photoionizingsource, as a consequence of either stellar evolution or changes in the opacity along the sightlinebetween the parcel of gas and the O star providing Lyman continuum photons. We propose thatobservations of free-free and line emission from the diffuse ISM at high latitudes typically sumover multiple components: (1) gas currently being photoionized, (2) gas that was photoionizedin the past but which is currently cooling and recombining, and (3) neutral gas (the cold neutralmedium and warm neutral medium). Dust grains mixed with the gas also contribute scattered lightthat includes emission lines from H II regions in the disk. For plausible choices of parameters, wefind that this model can account for the low ratio of free-free to H α found by DDF09, while alsoreproducing observed optical line ratios, such as [N II]6583/H α .The structure of this paper is as follows. In Section 2 we describe the model used to simulatecooling and recombination in the WIM. The simulation results are presented in Section 3, wherewe explore various factors which affect this process. In section 4 we propose a model using threeISM components to explain the observed line ratios and free-free emission. We discuss the mainresult and our models in Section 5, followed by a brief summary in Section 6.
2. Model Description
We model the evolution of temperature, ionization, and emission from cooling, recombining gas.As initial conditions we take the ionization and temperature to be consistent with photoionizationby a distant OB association, but at t = 0 we assume that the radiation with hν > . hν < . − , and includes all important coolants. The evolution is assumed to take place at constant volume, with H nucleon density n H = const .The abundance x A,r ≡ n ( A + r ) /n H of ion A + r evolves according to dx A,r dt = ζ A,r − x A,r − + h n e (cid:16) α ( rr ) A,r +1 + α ( dr ) A,r +1 (cid:17) + n H α ( gr ) A,r +1 i x A,r +1 − h ζ A,r + n e (cid:16) α ( rr ) A,r + α ( dr ) A,r (cid:17) + n H α ( gr ) A,r i x A,r for r ≥ , (1) dx A, dt = h n e (cid:16) α ( rr ) A, + α ( de ) A, (cid:17) + n H α ( gr ) A, i x A, − ζ A, x A, , (2)where ζ A,r is the probability per unit time of ionization A + r → A + r +1 + e − due to either photoion-ization, cosmic rays, or secondary electrons; α ( rr ) A,r is the rate coefficient for radiative recombination; α ( dr ) A,r is the rate coefficient for dielectronic recombination; and α ( gr ) A,r is the effective rate coefficientfor recombination on dust grains. The present model includes only ions with r ≤
2, and gas tem-peratures T ≤ K. We assume case B recombination for H and He. The recombination rate forH is taken to be (Draine 2011) α B = 2 . × − T − . − . T cm s − , (3)where T ≡ T / K, and for recombination of He II → He I we take α B (He) = 2 . × − T − . cm s − . (4)For other elements, radiative recombination rates are evaluated by subroutine rrfit.f fromVerner (1999), which uses rates from Pequignot et al. (1991) for ions of C, N, O, and Ne (refittedwith the formula of Verner & Ferland (1996)), and from Shull & van Steenberg (1982) for ions ofMg, Si, S, Ar, and Fe. Rate coefficients α ( dr ) for Mg and Si were from Nussbaumer & Storey (1986),for S were from Shull & van Steenberg (1982), and for Fe were from Arnaud & Raymond (1992).Rate coefficients α ( gr ) A,r for recombination on grains are taken from Weingartner & Draine (2001a).The electron density n e ≡ n H P A P r rx A,r , and the free particle density n = n e + n H P A P r x A,r .For isochoric evolution, the temperature T evolves according to dTdt = Γ − Λ(3 / nk − T n dndt , (5)where (Γ − Λ) is the net rate of change of thermal kinetic energy per unit volume due to heatingand cooling processes (including ionization and recombination), and dn/dt is the net rate of changeof the free particle density due to ionization and recombination processes. The heating rate per 4 –volume Γ includes heat deposition by cosmic ray ionization and by photoelectrons emitted fromatoms, ions, and grains. The cooling rate per volume Λ includes kinetic energy removed from thegas by inelastic collisions and by recombination of ions and electrons. The mean kinetic energy perrecombining electron is taken to be (Draine 2011, eq. 27.23) h E rr i ≈ [0 . − . T ] kT . (6)Radiative cooling processes consist of free-free emission, and line emission following collisionalexcitation of various atoms and ions. Collisional deexcitation is included, although it is unimportantat the densities considered here.Grain-related processes are included in our simulation. From their study of the “spinningdust” emission from the WIM, DDF09 concluded that the PAH abundance in the WIM is a factor ∼ I . Because the PAHs account for a large fraction of the photoelectric heat-ing (Bakes & Tielens 1994; Weingartner & Draine 2001b), and are also thought to dominate thegrain-assisted recombination (Weingartner & Draine 2001a), depletion of the PAHs will affect theheating and recombination. To explore this, we multiply the rates for dust photoelectric heatingand grain-assisted recombination by a factor g , where g = 1 gives the rates estimated for normalgrain abundances in the diffuse ISM for photoelectric heating and grain-assisted recombination(Weingartner & Draine 2001a). Our standard model for the WIM assumes g = 1 /
3, which pre-sumably results primarily from reduced abundances of the small grains that account for most ofthe grain surface area. We will explore the sensitivity to the reduction factor g by also performingsimulations with g = 1 (no reduction in PAH abundance) and g = 0 .
1, a factor of 10 suppression ofgrain photoelectric heating and grain-assisted recombination. Grain photoelectric heating is calcu-lated following Weingartner & Draine (2001b), and the g = 1 rates for grain-assisted recombinationare taken from Weingartner & Draine (2001a).We calculate collisional excitation and resulting radiative cooling by ions of C, N, O, Ne, Si,S, Ar, and Fe. We include a total of 159 lines, but the 26 lines of listed in Table 1 account for morethan 95% of the radiative cooling at each point in the thermal evolution of our models. Sources ofthe collisional rate coefficients for species used in our calculations are listed in Table 2.Table 1: Principal Cooling Lines (wavelengths in vacuo )[C II ]157.7 µ m [O I ]145.5 µ m [S II ]6733˚A [Fe II ]25.99 µ m[C II ]2328˚A [O I ]63.19 µ m [S II ]6718˚A [Fe II ]5.340 µ m[C II ]2326˚A [O I ]6302˚A [S II ]4070˚A [Fe II ]1.644 µ m[N II ]205.3 µ m [O II ]3730˚A [S III ]33.48 µ m [Fe II ]1.321 µ m[N II ]121.8 µ m [O II ]3727˚A [S III ]18.71 µ m [Fe II ]1.257 µ m[N II ]6585˚A [Ne II ]12.81 µ m [S III ]9533˚A[N II ]6550˚A [Si II ]34.81 µ m [S III ]9071˚A 5 –Table 2: Sources for Collisional Rate CoefficientsIon Transition e − H C II P o1 / − P o3 / Tayal (2008) Barinovs et al. (2005)N II P J − P J ′ Hudson & Bell (2005) —N II P J − D Hudson & Bell (2005) —O I P J − P J ′ Pequignot (1996) Abrahamsson et al. (2007)O I P J − D Pequignot (1996) —O II S o3 / − D o J Tayal (2007) —Ne II P o3 / − P o1 / Griffin et al. (2001) —Si II P o1 / − P o3 / Bautista et al. (2009) —S II S o3 / − D o J Tayal & Zatsarinny (2010) Barinovs et al. (2005)S
III P J − P J ′ Tayal & Gupta (1999) —S
III P J − D Tayal & Gupta (1999) —Fe II D J − D J ′ Ramsbottom et al. (2007) —Fe II D J − F J ′ Ramsbottom et al. (2007) —Fe II D J − D J ′ Ramsbottom et al. (2007) —For the diffuse interstellar radiation field we use the estimate of Mathis et al. (1983). Pho-toionization rates for this radiation field were taken from Draine (2011, Table 13.1), calculatedusing photoionization cross sections from Verner & Yakovlev (1995) and Verner et al. (1996).We include secondary ionization and heating by cosmic rays following Dalgarno & McCray(1972). Charge exchange between H and O is included in our model, with rate coefficients fromStancil et al. (1999). α We calculate the free-free emission at 41 GHz, I ν (41 GHz), to compare with observations(DDF09): j ν = 5 . × − g ff T − . e − hν/kT n e erg cm s − Hz − sr − . (7)where n e is the electron number density, and the Gaunt factor (Hummer 1988) is accurately ap-proximated by (Draine 2011, eq. 10.9) g ff ≈ ln " exp . − √ π ln( ν T − . ) ! + e . (8)where ν ≡ ν/ GHz. The H α emission rate is: j H α = 2 . × − T − . − .
031 ln( T )4 n e n (H + ) erg cm s − sr − . (9) 6 – −2 −1 T (K) ψ = j ν ( G H z ) / ( j H α / h ν ) ( kJy s r − R − ) X H :110 −3 −4 F =0 F =0.8
Fig. 1.—
The ratio of j ν (41GHz) / (j H α / h ν ) as a function of gas temperature T for H ionization fraction x H = 1 (solidcurve), 10 − (dash curves), and 10 − (dot curves). For each x H , upper curves employ gas phase elemental abundances for F ⋆ = 0 to simulate WIM, lower curves employ gas phase elemental abundances for F ⋆ = 0 . α ratio ∼ .
85 kJy sr − R − from WMAP (DDF09) is indicated by the horizontal dash-dot line,corresponding to gas temperature ∼ corresponding to an effective rate coefficient α H α ≡ πj H α hν = 1 . × − T − . − .
031 ln( T )4 cm s − . (10)The ratio of free-free to H α depends on both the temperature and the fraction of the electronscontributed by H + : ψ ( T ) ≡ j ν (41 GHz) j H α /hν ≈ . n e n (H + ) T . .
031 ln T ln " exp .
913 + 3 √ π ln T ! + e kJy sr − R . (11)Free-free emission comes from all ions, therefore in low-ionization gas we have free-free emissionfrom ions such as C + + e − , whereas H α comes only from H + + e − . Therefore, if H becomesalmost neutral, ψ is sensitive to the gas-phase abundance of elements that can be ionized by hν < . ψ ( T ) as a function of gas temperature from 50 − K,for three different H ionization fractions x H ≡ n (H + ) / n H : x H = 1 (solid curve), 10 − (dash curves)and 10 − (dot curves), and for two gas phase elemental abundances: F ⋆ = 0, representing WIM,and F ⋆ = 0 .
8, representing CNM. The horizontal dash-dot line indicates the observed free-free/H α ratio ∼ .
085 kJy sr − R − determined from the WMAP ∼ α and free-free emission fromH I clouds. The H I gas in the interstellar medium is found at a wide range of temperatures. The 7 –majority is in the CNM phase, at T ≈ K, with n e /n (H + ) ≈ ζ CR /n (H) ∼ × − cm s − ,then ∼
50% of the free electrons are from C + , S + , and other metal ions.) From this gas we expect ψ ≈ . n e /n (H + )] kJy sr − R − ≈ .
034 kJy sr − R − . In addition, a substantial fraction of theH I is in the WNM phase, with T ≈ K; this gas will have ψ ≈ .
047 kJy sr − R − . Overall, weestimate that the H α and free-free emission from the H I phase will have ψ H I ≈ . Gas-phase elemental abundances are based on the recent study by Jenkins (2009). For ourstandard model, we use abundances from the model of Jenkins (2009) with F ⋆ = 0, representinga relatively low level of depletion (see Section 5 for a discussion of the choice of F ⋆ ). For Neand Ar (not covered in Jenkins’ study), we assume solar abundances from Asplund et al. (2009).He/H=0.1 is also assumed. In addition, we construct a model with F ⋆ = 0 .
25 (reduced gas phaseabundance for elements that deplete) to explore the sensitivity to coolant abundances. Carbonrequires special attention, since recent work (Sofia & Parvathi 2010) indicates that the oscillatorstrength of C II ]2325˚A is larger than previously estimated, implying that gas phase C abundancesestimated from measurements of C II ]2325˚A could be lowered by a factor ∼
2. For this reason, inaddition to the standard model and the F ⋆ = 0 .
25 model, we consider the “Reduced C” model inwhich C has an abundance two thirds of its F ⋆ = 0 value and other elements all have their F ⋆ = 0values.For our standard model we take n H = 0 . − and initial temperature T i = 8000 K. Underthese conditions the thermal pressure of the photoionized state, p/k ≈ . n H T ≈ × cm − K,is comparable to, although somewhat higher than, current estimates for pressures in the diffuseISM. In our standard model, the initial ionization fractions (IIF) of all the elements (Table 3)are adopted from Sembach et al. (2000), where we use the value in their standard model withparameter χ edge = 0 . ζ CR , is controversial. Direct mea-surement of the cosmic ray flux (Wang et al. 2002) at E . ζ CR ∼ . × − s − . However,studies of the ionization conditions inside molecular clouds (Black & van Dishoeck 1991; Lepp 1992;McCall et al. 2003; Indriolo et al. 2007) indicate primary ionization rates ζ CR ∼ . − × − s − ,and we will consider values of ζ CR within this range. We adopt ζ CR = 1 × − s − for the standardmodel, and will then explore varying ζ CR from 2 × − s − to 5 × − s − . 8 – n H t (10 cm −3 yr) T ( K ) n H =0.5 cm −3 n H =2.5 cm −3 X H =0.1 X H =0.03X H =0.01X H =0.01X H =0.03X H =0.1X H =0.9 X H =0.3 50.21350.21 ζ CR (10 −16 s −1 )0.001 0.01 0.1 1 100.070.090.110.130.150.17 n H t (10 cm −3 yr) Ψ ( kJy s r − R − ) X H =0.9 X H =0.3 X H =0.03X H =0.01X H =0.1n H =0.5 cm −3 ζ CR (s −1 )5 × −16 × −16 × −16 × −17 n H =2.5 cm −3 , ζ CR =1 × −16 s −1 Fig. 2.—
Effect of different density and cosmic ray ionization rate. All cases have g = 1 / F ⋆ = 0. Toppanel: The temperature evolution for n H = 0 . − (solid curves) and n H = 2 . − (dashed curves), and differentcosmic ray ionization rate ζ CR . Bottom panel: The ratio of cumulative free-free emission at 41 GHz to cumulative H α emission for models with different ζ CR for n H = 0 . − . Values of x H = n (H + ) / n H are given at several points alongthe curves by different tick marks. Note that when ζ CR /n H > × − cm s − , the final temperature T f > K,and Φ( n H t = 10 cm − yr) > .
09 kJy sr − R − .
3. Results
The temperature as a function of time for our standard model is shown in the top panel ofFigure 2. Initially the gas cools rapidly, reaches a minimum temperature ∼
100 K within 0 . n H t & × cm − yr occurs because the [C II]158 µ m cooling declines as free electronsrecombine. Similar late time reheating has been seen previously (e.g., Draine 1978). Define thecumulative emission ratio as: Ψ( t ) ≡ R t j ν (41 GHz) dt ′ R t ( j H α /hν ) dt ′ ; (12)Ψ starts from ∼ − R − , corresponding to the initial temperature T i = 8000 K, thendrops as the gas cools, to ∼ cm − yr . n H t . cm − yr as shown in the bottom panelof Figure 2. This low value of Ψ is consistent with the observed value of ∼ .
085 kJy sr − R − ζ CR , n H , abundance of metal elements and grain depletion,and minor factors: the initial temperature and IIF. Figure 2 shows the sensitivity to ζ CR and n H . For fixed n H , the final asymptotic temperature( T f ) increases with increasing cosmic ray ionization rate, while for fixed ζ CR , T f decreases withincreasing n H . At a higher density the final temperature and free-free to H α emission ratio becomeless sensitive to ζ CR . For ζ CR ranging from 2 × − s − to 5 × − s − , T f increases from ∼
240 K to ∼ n H = 0 . − , but only from 120 K to 190 K when n H = 2 . − (asshown in top panel of Figure 2). As ζ CR is varied from 2 × − to 5 × − s − , the emissionratio Ψ( n H t = 10 cm − yr) changes about 20% for n H = 0 . − (as shown in bottom panel ofFigure 2), but only about 1% for n H = 2 . − . For each density, there is a critical value of ζ CR ,above which the gas remains warm, with asymptotic temperature T > > . T f , and consequently will raise Ψ. As we discussed in Section 2.1, evidencefrom analyzing spinning dust emission in the WIM (DDF09) suggests that the small polycyclicaromatic hydrocarbons (PAHs) may be underabundant in the WIM relative to the general ISM.We explore the effect of different grain reduction factors, with g = 1 / g .When g is reduced, the photoelectric heating rate drops; since the grain-assisted recombination ratealso drops, more electrons and ions will be present in the gas phase, leading to increased coolingthrough collisionally-excited lines. The combined result of the two factors is that the ionized gasrecombines more slowly, cools faster, and reaches lower values of Ψ when g is small. The initial temperature and ionization fractions also affect the cooling process. Quantities like T and x H for models with different initial T and ionization will evolve differently at the initialstage of the simulations, but will all converge to the same asymptotic steady-state values. On theother hand, the cumulative ratio of free-free to H α , Ψ, will differ because it involves integrationover time. We study the effect of different T i by running a model with T i = 10 K (instead of8000 K), and we study the effect of different IIF by running models with IIF2 and IIF3. IIF3assumes Orion Nebula values (Baldwin et al. 1991), formed by a much harder ionization field than 10 – n H t (10 cm −3 yr) T ( K ) ModelReduced CStandardg=0.1f =0.25g=10.001 0.01 0.1 1 100.070.090.110.130.150.17 n H t (10 cm −3 yr) Ψ ( kJy s r − R − ) Modelg=0.1StandardReduced Cf =0.25g=1
Fig. 3.—
Effect of varying gas-phase abundances ( F ⋆ = 0 .
25 vs. F ⋆ = 0), PAH abundance ( g = 1 vs. g = 1 / F ⋆ = 0). Top panel shows the temperature as a function of time for differentmodels (indicated by model name), and bottom panel show the ratio of cumulative free-free emission at 41 GHz tocumulative H α emission as a function of time. the one that forms the WIM, as we discussed above in Section 1. IIF2 is a set of values based onthe literature (Madsen et al. 2006; Haffner et al. 1999, 2009; Reynolds 2004), falling between IIF3and our standard model. These three models cover a large range of IIF. As shown in Figure 4, theresults are insensitive to changes in the different initial ionization conditions.
4. A Three Component Model for the Diffuse Emission
In this section we try to compose a model to simultaneously explain the observed low free-freeto H α emission ratio (corresponding to T ∼ II ] λ α and[N II ] λ II ] λ T & α emission ratio (DDF09) comes from measurements which integrateover a large fraction of the high-latitude sky, while the measurement of various other diagnostic 11 – n H t (10 cm −3 yr) T ( K ) Standard ModelIIF2IIF3T i =10 K0.001 0.01 0.1 1 100.070.090.110.130.150.17 n H t (10 cm −3 yr) Ψ ( kJy s r − R − ) Standard ModelIIF2IIF3T i =10 K Fig. 4.—
Effect of different initial conditions (initial temperature and ionization fractions) on the evolution oftemperature T and the cumulative ratio Ψ of 41 GHz free-free to H α . lines from the WIM, like [N II ] λ II ] λ T ≈ K, nearly fully ionized, producing free-free emission,recombination radiation, and collisionally-excited lines such as [N II ] λ II ] λ α and free-free emission as well as a small amount of collisionally-excitedlines, such as [N II ] λ T ≈ K, and partiallyin the “warm neutral medium” (WNM), with T ≈ K; together these emit a small amountof H α and free-free emission due to cosmic ray ionization, but negligible metal line emissionin the optical. 12 –In addition to the actual emission from the H I gas, there will also be reflected light – both H α and metal lines such as [N II ] λ II regions, and then reflected by dust grains present in the H I gas. Let f (refl)H α be the fraction of theobserved H α that is scattered light. Wood & Reynolds (1999) estimated that f (refl)H α ≈ . − . II ] λ II ] λ II ] λ α , because the wavelengths are close to H α .The observed intensities are weighted averages over the three components. Here we ask whatweighting factors are needed to reproduce the observed low ratio of free-free/H α , as well as otherline ratios.The relative contributions of the three components can be determined from the observed emis-sion ratios. Among all the diagnostic line ratios in the WIM, [N II ] λ α and [S II ] λ α are the two best-studied cases, while other lines have been studied only in a few select directions(Haffner et al. 2009). These two line ratios depend on temperature and ionization fraction of N andS in the gas. The high second ionization potential of N (29.60 eV) protects it from being doublyionized in the WIM, which has a generally soft ionization field (Madsen et al. 2006). This factoralong with the similar first ionization potentials of N and H (14.53 and 13.60 eV) make N + /N closeto unity in the H II gas (Madsen et al. 2006; Haffner et al. 1999). On the other hand, some of theS in H II regions will be doubly ionized due to its low second ionization potential (23.33 eV), whichleads to uncertainty in the predicted [S II ] λ α , reducing the utility of [S II ] λ α as aconstraint. Nevertheless, we have calculated the integrated value of [S II ] λ α for the coolinggas (see Table 4).Let f (hot)H α and f (H I)H α be, respectively, the fraction of the observed diffuse H α emitted from thephotoionized gas and H I clouds, and let f (refl)H α be the fraction of the observed H α that is actuallyreflected from dust in H I . Then f (cooling)H α = 1 − f (hot)H α − f (H I)H α − f (refl)H α is the fractional contribution ofthe cooling material. The emission from this three component model depends on four parameters: f (hot)H α , f (H I)H α , f (refl)H α , and the temperature T hot of the photoionized component. For an adoptedvalue of f (refl)H α , a physical solution must have 0 ≤ f (hot)H α ≤ − f (refl)H α , 0 ≤ f (cooling)H α ≤ − f (refl)H α , and0 ≤ f (H I)H α ≤ − f (refl)H α . We take the observed free-free/H α and [N II ] λ α emission ratios astwo constraints to solve for the relative fraction of the three components.The instantaneous [N II ] λ α line ratio is: φ ( T ) ≡ j [NII] λ j H α ≈ . T . .
040 ln T e − . /T n (N + ) /n (H + )7 . × − . (13)where we use the collision strength Ω( P , D ) = 0 . T . .
009 ln T from Hudson & Bell(2005). The cumulative ratio of [NII]6583 to H α for cooling gas isΦ( t ) ≡ R t j [NII]6583 dt ′ R t j H α dt ′ = R t φ ( t ′ ) j H α dt ′ R j H α dt ′ . (14) 13 – hot (10 K) f H α Hot componentHI component (including reflection)Allowed range for f H α >0 Cooling component20% reflection + 1.4% emission from the HI Fig. 5.— H α fractions for the three gas components based on two constraints ( I ν (41 GHz) / H α = 0 .
85 kJy sr − R − and [N II] λ α =0.4), as a function of hot (photoionized) gas temperature T hot . The equations are solved underthese assumptions: f (refl)H α = 0 . f (H I)H α = 0 . n (H + ) /n e = 0 . ζ CR ∼ × − s − inCNM), and He + /He=0.3 in the hot ionized gas, as in IIF1. The fact that f (refl)H α + f (H I)H α ≈ f (hot)H α is coincidental. Cooling gas in our standard model gives Φ ≈ .
06 (see Table 4). The WNM and CNM contributenegligible [N II ] λ II ]6583/H α = 0 .
4, based on observations (Reynolds 2004;Reynolds et al. 2001; Madsen et al. 2006). Then we must have0 . f (hot)H α + f (refl)H α ) φ ( T hot ) + (1 − f (hot)H α − f (H I)H α − f (refl)H α )Φ . (15)Note that in this equation, we assume the reflected component has the same [N II ] λ α ratioas the hot ionized component, as discussed previously.The ratio of all-sky free-free to H α emission is I ν (41 GHz) /I (H α ) ∼ .
085 kJy sr − R − (DDF09). Thus we must have0 .
085 kJy sr − R − = f (hot)H α ψ ( T hot ) + (1 − f (hot)H α − f (H I)H α − f (refl)H α )Ψ + f (H I)H α ψ H I . (16)For trial values of T hot and f (refl)H α ≥
0, we can use the observed I ([N II]6583) /I (H α ) and I ν (41GHz) /I (H α )to determine f (hot)H α and f (H I)H α , by solving the two linear equations (15) and (16). The coefficients ψ ( T hot ) and φ ( T hot ) are obtained from eq. (11) and (13), while Ψ ≈ .
08 and Φ ≈ .
06 are ob-tained from the appropriate simulations as their asymptotic values (see Table 4 as well as Figure 2).Based on the discussion in § ψ H I ≈ .
04. Physical solutions must have 0 ≤ f (hot)H α ≤ ≤ f (H I)H α ≤
1, and 0 ≤ (1 − f (hot)H α − f (H I)H α − f (refl)H α ). Figure 5 shows one example, where the reflectedH α fraction is set to 20%, n (H + ) /n e = 0 . I phase and n (He + ) /n (He) = 0 . (refl)H α T H o t ( K ) (n H + /n e ) cold =1(n H + /n e ) cold =0.5(n H + /n e ) cold =1/3All assume f (HI)H α =0.014 a 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.70.80.9 Reflection Fraction f (refl)H α f H α (n(H + )/n e ) cold =0.5 bCooling component HI component(including reflection)Hot component Fig. 6.— (a) Temperature T hot and (b) fractional H α contributions from different ISM components, as func-tions of the reflected H α fraction f (refl)H α , based on two constraints — I ν (41 GHz) / H α = 0 .
85 kJy sr − R − and[N II] λ α = 0 . f (refl)H α ranging from 5% − f (H I)H α ≈ .
014 of the H α isemitted by the CNM and WNM components (see text). Panel (a) also shows the dependence of the solution ( T hot )on the value of n (H + ) /n e in the H I gas. Our standard model assumes ζ CR = 1 × − s − , which corresponds to n (H + ) /n e ≈ . In addition to the requirement that the fractions of all the three components have to be positive,which, for f (refl)H α ≈ .
2, requires 8300 ≤ T hot ≤ α emission from the H I phase is determined bythe cosmic ray ionization rate. Consider a uniform layer of cold neutral gas with a column density N (HI), with cosmic ray ionization of H balanced by case B recombination as well as grain assistedrecombination; the H α intensity from this component at latitude b is I H α hν = 14 π × cm − | sin b | ζ CR (1 + φ CR ) α H α /α B α gr /x e α B , (17)where x e ≡ n e /n H , φ CR ∼ .
67 is the number of secondary ionization per primary ionization inneutral gas (Dalgarno & McCray 1972), and N (HI) ≈ × | sin b | − cm − (Radhakrishnan et al.1972; Dickey et al. 1978). Thus I H α hν ≈ . | sin b | (cid:18) ζ CR − s − (cid:19) R , (18)where the H I is taken to be a 1:1 mixture of CNM and WNM material. The distribution of H α for | b | ≥ ◦ has recently been measured by Hill et al. (2008), whofound the full WIM to be fitted on average by I H α /hν ≈ (0 . ± . | sin b | − R. Comparing thesetwo results, the fraction of H α emitted by the H I should only be f (H I)H α ≈ . ζ CR / − s − ). We have taken φ CR ≈ . α H α /α B ≈ . α gr /x e α B ≈
10 for CNM with T ≈
100 K, n e ≈ . n H ≈
30 cm − (Draine 2011); φ CR ≈ . α H α /α B ≈ . α gr /x e α B ≈ . T ∼ N CNM ≈ N WNM ≈ N HI .
15 – (refl)H α [ N II] λ /[ N II] λ Pure hot gas phasePure cooling gas phaseWeighted over hot and cooling components(n(H + )/n e ) cold =0.5 Observedrange Fig. 7.— [N II] λ λ λ λ In Figure 6 we take f (H I)H α = 0 .
014 and consider different values of f (refl)H α . For each f (refl)H α , theH α /free-free and [N II ]/H α constraints serve to determine f (hot)H α and T hot , as shown in Figure 5.For f (refl)H α ranging from 5% to 30%, panel (a) in Figure 6 gives T hot ranging from 8000–15000 K.For f (refl)H α = 0 .
2, we find T hot ≈ n (H + ) / n e in the H I , which affects ψ H I (equations 11). Assuming ζ CR ∼ − s − , T CNM ∼ K and n CNM ∼ − , about half ofthe electrons come from metal elements, so the free-free/H α ratio for the H I will be Ψ HI ≈ .
04, asdiscussed in Section 2.2. We explore the effect of this uncertainty on T hot in the range 0 . − . n (H + ) /n e ) cold = 1 and , as shown in panel (a) of Figure 6. We find thisuncertainty does not significantly affect the derived solutions.In addition to [N II ] λ α , the [N II ] λ II ] λ II ] λ α , [N II ] λ II ] λ II ] λ II ] λ II ] λ II ] λ II ] λ f (refl)H α ranging from 5%–30%, under the assumption that the [N II ] λ II ] λ weighted average value over all the components in the WIM – measure-ments along individual sightlines will vary. The observed [N II ] λ II ] λ α and [N II ] λ α ratio. The dashed curve and the dotted curve in Figure 7 show [N II ] λ II ] λ II ] λ II ] λ II ] λ II ] λ II ] λ II ] λ α ratio;2. reproduces the observed [N II ] λ α ratio3. has [N II ] λ II ] λ ∼ α emitted from H I gas;5. includes a reflected component accounting for ∼
20% of the total H α We emphasize again that the three-component model presented in this section is aimed toexplain averaged observational results; individual sightlines will generally differ from the averagebecause (1) the relative weights of its components differ from the average values, the temperature T hot and the ionization conditions in the H II component may vary, and (3) the cooling componentalong an individual sightline may not be an average over the entire cooling and recombination.There is evidence that the physical conditions in the WIM do vary from sightline to sightline,and even among different velocity components within one sightline (Madsen et al. 2006). Differentregions in the WIM will have different densities, cosmic ray ionization rates, elemental abundances,and histories. The three component model presented here is highly idealized, but it appears toprovide a physical framework that is consistent with the observations.
5. Discussion
When the hν > . n H = 0 . − , T falls below 10 K within 17 –0.3 Myr. If the photoionized gas was initially overpressured (relative to its surrounding), it wouldinitially be expanding, resulting in adiabatic cooling after the photoionizing source turns off at t = 0. Conversely, if the photoionized gas was in pressure equilibrium with a confining medium at t = 0, it would begin to undergo compression as it cools and recombines at t >
0. However, if the n H = 0 . − , photoionized region is ∼ hν > . T ≤ α and free-free emission, with a ratio of free-free emission to H α appropriateto T . α below the value calculated for the cooling gas model, allowingthe observed low free-free/H α ratio to be explained by a somewhat smaller value of f (cooling)H α , thefraction of the H α contributed by cooling and recombining gas. However, we expect this “escape”of ionizing photons to be minimal: the mean free path of ∼
14 eV photons in partially recombinedH is short, and we expect the case B on-the-spot treatment to be a valid approximation.In our standard model, we choose F ⋆ = 0 for the elemental abundances in the WIM. InJenkins’ model, the nonzero initial depletion ( F ⋆ = 0) is identified with the gas-phase abundancesin a warm, low-density medium (Spitzer 1985; Savage & Sembach 1996), and should therefore applyto the WIM. Also, as mentioned above, there is evidence (DDF09) showing that the PAHs in theWIM are underabundant by a factor of ∼
3. Our calculation shows that both factors are crucialfor cooling the gas quickly enough and to a low enough temperature to be able to reproduce theobserved low free-free/H α ratio. As shown in Figure 3, if there is no grain depletion or there isappreciable depletion of coolants (such as for F ⋆ = 0 . α ratio.Cosmic rays heat the gas directly, and they also raise the electron density n e . If n e is low,photoelectron emission causes dust grains and PAHs to become positively charged, reducing thephotoelectric heating rate. Thus increased cosmic ray ionization, by lowering the charge state of thedust and PAHs, has the effect of increasing the dust photoelectric heating rate. For our cooling gasmodel to be able to reproduce the low free-free/H α ratios that are observed, the ratio of cosmic rayprimary ionization rate to gas density ζ CR /n H should not exceed ∼ × − cm s − . For ζ CR /n H & × − cm s − (e.g, n H = 0 . − and ζ CR = 5 × − s − , with ζ CR /n H = 1 × − cm s − ),the cosmic ray ionization maintains n (H + ) /n H ≥ .
1, and grain photoelectric heating can sustainthe gas at T ≥ K. Observations of H +3 give estimates of ζ CR ≈ (0 . − × − cm s − , withan average of ∼ × − cm s − (Indriolo et al. 2007). If ζ CR ≈ × − cm s − , then the gasdensity cannot be much smaller than n H = 0 . − if the gas is to cool with Ψ ≤ . α . This high initial value of n e = 0 . − seems 18 –at first sight to be at odds with the study by Hill et al. (2008), which concluded that the mostprobable value of n e is only n e ≈ .
03 cm − in turbulent models of the WIM. Note, however, thatin our standard model, the cooling gas ends up with x H < .
03 and n e ≈ . − .Because the observed line emission and free-free emission are weighted sums over 3 compo-nents, one of which (the cooling phase) itself has a range of temperature, there is no way to measureemission ratios associated with different phases unless the components could be separated. Tem-peratures derived from different line ratios need not agree. Moreover, even for a single line ratio,the contributions from different components may vary significantly from sightline to sightline, orwith velocity on a single sightline. This may explain the large scatter in the physical conditionsdeduced from different line ratios.The present model envisages intermittent photoionization events in the diffuse high-latitudegas, followed by cooling and recombination. Suppose that the probability per unit time of a pho-toionization event is τ pi , with the ionizing radiation lasting a time τ H II before the photoionizationswitches off and the gas begins to cool and recombine. During the recombination phase, the numberof H α photons per recombining H + is ∼ α emission from the photoionizedgas to the emission from the cooling gas is f (hot)H α f (cooling)H α ≈ τ − × n e α H α τ H II τ − × . τ H II ≈ . n e α H α f (hot)H α f (cooling)H α (20)If f (refl)H α ≈ .
2, then f (hot)H α /f (cooling)H α ≈ . / . ≈ .
39, and the duration of the photoionizationphase is only τ H II ≈ × (0 . − /n e ) yr.Given that O star lifetimes are ∼ × yr, such a short value of τ H II seems surprising. Longervalues of τ H II can be obtained if f (refl)H α is larger: if f (refl)H α = 0 .
3, then f (hot)H α /f (cooling)H α ≈ . / . ≈ .
1, raising τ H II to ∼ × (0 . − /n e ) yr, but this is still short compared to O star lifetimes.Longer values of τ H II can be obtained if n e is lowered, but if n e is much smaller than 0 . − ,grain photoelectric heating and cosmic ray heating prevent the gas from cooling to low enoughtemperature to be able to account for the observed low free-free/H α ratio, if, as assumed, thecooling takes place at ∼ constant density.O star lifetimes are not the only time scale on which photoionization can vary. The ionizingradiation for the WIM may be provided in large part by runaway O and B stars, with spacevelocities &
100 km s − . Modulation of the photoionization rate at a given point may result frommotion of the ionizing sources through a medium of varying opacity: a star moving at 100 km s − travels 10 pc in 10 yr.Although O stars have been generally favored as the source of ionization for the WIM, itshould be kept in mind that the actual source for WIM ionization has not been securely established. 19 –According to the present analysis, the ionization phase, if it heats the plasma to ∼ K, needs tobe of relatively short duration in order to be able to explain the low observed ratio of free-free/H α .Perhaps other transient processes (such as shock waves or soft X-rays due to supernova blast waves,infalling gas, etc.) contribute to heating and ionization of the WIM.A prediction of the current model is that the H α emission should have a component with thesame radial velocity profile as the 21 cm emission, resulting from cosmic ray ionization of the H I .However, because cosmic ray ionization is thought to account for ≤
2% of the H α , it may not beeasy to recognize this component. The reflected component of H α , [N II ], etc, should also correlatewith N (HI), but the apparent space velocities will differ from 21 cm radial velocities due to motionof the emitting H II relative to the reflecting dust grains.
6. Summary
The principal points of this paper are as follows:1. We simulated the cooling and recombination of initially photoionized gas in the WIM followingremoval of hν > . α , and emission lineratios such as [N II ] λ α , were calculated, and various factors which influence the coolingwere explored. The result strongly depends on the abundances of elements, gas density,cosmic ray ionization rate, and the abundance of very small grains and PAHs, but dependsonly weakly on the initial temperature and ionization fractions, as shown in Figure 2, 3 and4.2. Based on these calculations, we propose a three component model — emission from hotionized gas, cooling gas and neutral H I, plus reflected light – to explain multiple observationsin the WIM. With plausible weighting factors for these components, the model simultaneously yields the low free-free to H α ratio, which indicates a low temperature ( ∼ II ] λ α and [N II ] λ II ] λ ∼ K, Madsen et al. (2006)). The reflected component is crucial —there must be a fraction ( ∼ α coming from the reflected light, consistentwith the estimate by Wood & Reynolds (1999), to explain the observation.3. For our model to successfully reproduce the low free-free/H α ratio, some restrictions of thephysical conditions in the WIM are required:(a) The ratio of cosmic ray primary ionization rate to gas density ζ CR /n H should not exceed ∼ × − cm s − (Figure 2), consistent with current observational estimates of ζ CR if n H ≈ . − .(b) The gas phase element depletion parameter F ⋆ ≤ .
25 (Figure 3) so that there aresufficient gas phase coolants to cool the gas in the presence of heating by cosmic raysand photoelectrons from grains, consistent with Jenkins (2009). 20 –(c) The abundance of ultrasmall grains (including PAHs) should be suppressed by a factor ∼ I (Figure 3), consistent with DDF09.4. Sightline-to-sightline variations of emission ratios are expected in our model. First, the pa-rameters which affect the cooling model, such as the gas density, elemental abundances,cosmic ray ionization rate and the depletion factor of small grains, may vary spatially withinthe WIM. Second, different sightlines will have different proportions of the three componentsin our model. Third, the cooling component is time-dependent, and individual sightlines maynot present a full average over the entire cooling history. These effects will lead to the vari-ation of physical quantities deduced from observations on different sightlines, or for differentvelocity components on an individual sightline. Acknowledgments
We thank R. Benjamin for helpful discussions, and the anonymous referee for comments thatled to improvements in the manuscript. This research was supported in part by NSF grants AST-0406883 and AST-1008570.
REFERENCES ∼ verner/fortran.htmlVerner, D. A., & Ferland, G. J. 1996, ApJS, 103, 467Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJ, 465, 487Verner, D. A., & Yakovlev, D. G. 1995, A&AS, 109, 125Wang, J. Z., et al. 2002, ApJ, 564, 244Webber, W. R., & Yushak, S. M. 1983, ApJ, 275, 391Weingartner, J. C., & Draine, B. T. 2001a, ApJ, 563, 842—. 2001b, ApJS, 134, 263Wood, K., & Reynolds, R. J. 1999, ApJ, 525, 799 This preprint was prepared with the AAS L A TEX macros v5.2.
24 –Table 3. Initial Ionization Fractions
IIF = 1 a IIF = 2 b IIF = 3 c A n A /n Hd I II III I II III I II III IVH 1 0.05 0.95 · · · · · · · · · · · ·
He 0.1 0.68 0.32 0 0.5 0.5 0 0.04 0.96 0 · · ·
C 10 − . − . − . − . − . − . − . − . − . a Standard Model in Sembach et al. (2000) with χ edge = 0 . b Estimate based on Haffner et al. (1999); Reynolds (2004); Madsen et al. (2006); Haffner et al.(2009); between IIF1 and IIF3 c Orion nebula values (Baldwin et al. 1991) d Elemental abundance for standard Model ( F ⋆ = 0)
25 –Table 4. Models for Recombining Gas
Name a n Hb ζ CRc F ⋆ d g e IIF f T i g Ψ h Φ i [NII] λ λ j [SII] λ α k ( cm − ) (10 − s − ) ( K )standard 0.5 1.0 0 0.33 1 8000 0.081 0.060 0.0039 0.15 n H = 2 . n H = 2 . ζ CR = 0 . n H = 2 . ζ CR = 2 . n H = 2 . ζ CR = 3 . n H = 2 . ζ CR = 5 . ζ CR = 0 . ζ CR = 2 . ζ CR = 3 . ζ CR = 5 . F ⋆ = 0 .
25 0.5 1.0 0.25 0.33 1 8000 0.099 0.085 0.0040 0.14Reduced C 0.5 1.0 0+C l g = 1 0.5 1.0 0 1.0 1 8000 0.109 0.083 0.0038 0.27 g = 0 . T i = 10 K 0.5 1.0 0 0.33 1 10000 0.086 0.099 0.0069 0.23 a Model name b H nucleon density c Cosmic ray primary ionization rate. d Depletion parameter from (Jenkins 2009) e The grain reduction factor. The grain assisted recombination rate and photoelectric heating rate reduce to this fraction of theirfull values. f Initial ionization fractions (see Table 3) g Initial gas temperature. h Integrated ratio of free-free at 41 GHz to H α , as in Equation 11, for n H t = 10 cm − yr. i Integrated ratio of [NII] λ α , as in Equation 14, for n H t = 10 cm − yr. j Integrated ratio of [NII] λ λ n H t = 10 cm − yr. k Integrated ratio of [SII] λ α , for n H t = 10 cm − yr. l Element abundance identical to the standard model ( F ⋆ = 0) except for C, for which we take the F ⋆ = 0 value × (2 / n C /n H = 1 . × −−