EMISSION LINES FROM THE GAS DISK AROUND TW HYDRA AND THE ORIGIN OF THE INNER HOLE

Our theoretical disk models start from a prescribed disk surface density distribution and stellar spectrum and proceed to self-consistently calculate the disk structure from the gas temperature and density distribution. The gas disk models simultaneously solve for the thermal and chemical properties of the gas and vertical hydrostatic equilibrium. Gas and dust temperatures are computed separately and can deviate significantly in the surface layers where most of the line emission originates. The gas radiative transfer follows an escape probability formalism. Dust radiative transfer is done using a modified two-layer approach (e.g., Chiang & Goldreich 1997; Dullemond et al. 2001; Rafikov & De Colle 2006). Our chemistry consists of ∼85 species and ∼600 reactions, includes photodissociation and photoionization by EUV, FUV, and X-rays, and is focused toward accuracy in the disk surface layers. Our standard models do not include freeze-out of species on grain surfaces, but we test for the effects of freeze-out where relevant below. We also assume that H₂ forms on grain surfaces in our disk models for all grain temperatures below ∼400 K (Cazaux & Tielens 2004). For further details, see GH04, GH08, and HG09.

Stellar parameters and the incident radiation field are key inputs to our theoretical models and are reasonably well determined for TW Hya. The source is at a distance of 51 pc (Mamajek 2005) and the disk is almost face-on with a very low inclination angle (7° for the outer disk, Qi et al. 2004; 4° for the inner disk, Pontoppidan et al. 2008). We adopt M = 0.7 M_☉, R = 1 R_☉, and T = 4200 K for the stellar mass, radius, and effective temperature (Webb et al. 1999). We construct a composite spectrum for the star by adding observed X-ray and FUV spectra. We use the Far-Ultraviolet Spectroscopic Explorer/International Ultraviolet Explorer spectrum from the NASA HEASARC archive (Valenti et al. 2003; Herczeg et al. 2002). The FUV region is split into several energy bins for accurately computing gas opacity as described in GH04. Herczeg et al. (2002) derive a reconstructed Lyα flux ∼30% higher than the observed flux and we use this value for the Lyα band of our FUV spectrum. The total FUV (6–13.6 eV) flux is approximately 3 × 10³¹ erg s⁻¹. The X-ray spectrum is from the XMM-Newton archive (Robrade & Schmitt 2006) and extends from 0.3 to 5keV, and with a total X-ray luminosity of 1.3 × 10³⁰ erg s⁻¹. We extrapolate the observed spectrum to lower and higher energies (0.1–10 keV) and this results in a total X-ray luminosity of 1.6 × 10³⁰ erg s⁻¹. Although the star might also have an EUV component (13.6–100 eV), perhaps indicated by the observed photoevaporative flow (Alexander 2008; Pascucci & Sterzik 2009), its strength can only be crudely estimated by indirect arguments (Alexander et al. 2005) and we do not include this uncertain contribution to the high energy flux. We discuss below, where relevant, the effect on line emission fluxes if an EUV flux were to be added. Figure 1 shows the composite spectrum used as model input and the vertical dashed lines in the figure indicate the neglect of an EUV radiation field.

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Two different dust disk models have been proposed for TW Hya, by Calvet et al. (2002) and Ratzka et al. (2007). C02 developed detailed models of the TW Hya dust SED to constrain many aspects of the disk structure, and these models have been validated by subsequent observational studies (e.g., Qi et al. 2004; Eisner et al. 2006; Hughes et al. 2007). The flux deficit at ≲ 20 μm and a sharp increase or excess at ∼20–60 μm led them to propose the presence of an inner hole in the disk at r ≲ 4 AU. The deficit is ascribed to a lack of dust in the hole, and the excess arises from direct illumination of the rim of the optically thick dust disk beyond 4 AU. There is also a small mass of dust, 0.5 lunar masses, in the inner cavity which gives rise to the observed spectral feature at 10 μm. Ratzka et al. (2007) situate the inner edge of the outer optically thick dust closer to the star at ∼0.5–0.8 AU. They model both the SED and the 10 μm visibilities in detail by including sedimentation in the disk, where the dust is reduced in the upper atmosphere due to settling processes. Our dust models at present do not include settling. We defer gas modeling of the disk using the Ratzka et al. (2007) dust distribution to future work, and keep our dust model relatively uncomplicated at present.

We avail ourselves of the dust models of C02. For simplicity, we adopt many features of this dust model, specifically the dust size and spatial distribution in our standard dust model. We consider their favored model with a maximum grain size a_max = 1 cm and disk mass 0.06 M_☉, which corresponds to a total dust mass of 7.8 × 10⁻⁴ M_☉ for their assumed gas/dust ratio. As we ignore dust particles with sizes larger than 1 mm which are not very relevant for gas physics, our model disk "dust" (a_max = 1 mm) mass is reduced accordingly to 2.4 × 10⁻⁴ M_☉ (for a grain size distribution n(a)∝a^−3.5, where a is the grain size). Our chemical composition of dust for the outer disk is simpler than the detailed models of C02. We use a mixture of silicates and graphite for the optically thick disk and verify the assumption by the SED fit from the resulting model dust continuum emission. We note that the disk structure (especially in the regions where gas emission lines originate) is eventually determined by the gas temperature structure, although the dust distribution is sometimes important for the heating and cooling of gas. We use pyroxene grains for the inner dust hole as suggested by C02 to better fit the continuum SED and the 10 μm feature. As we show later, the amount of dust present in this inner region is in any case too low to significantly affect the gas thermal balance or chemical calculations. The details of the adopted dust model are listed in Table 2.

Polycyclic aromatic hydrocarbons (PAHs) can efficiently heat gas via the grain photoelectric heating mechanism (Bakes & Tielens 1994) and gas heating depends on the assumed PAH abundance. Disks around low mass stars are believed to be deficient in PAHs by a factor of ∼10–100 (Geers et al. 2006). We typically scale the PAH abundance with the decrease in cross-sectional area of the dust per H atom compared with the interstellar value. We adopt a PAH abundance of ≃ 10⁻⁷ per H in the interstellar medium (e.g., Habart et al. 2003). Therefore, the abundance of PAHs in the optically thick disk (r ≳ 4 AU) around TW Hya is ≃ 10⁻⁹. Such a depleted abundance means that X-ray heating usually dominates FUV-induced PAH heating of gas in the outer disk. The abundance of small grains (a ≲ 0.1 μm) in the inner disk (r ≲ 4 AU) is either negligible (C02) or very small (Eisner et al. 2006). It is therefore likely that the abundance of very small dust like the PAHs is low in the inner disk. Such low PAH abundances do not influence the heating in the inner disk and the PAH abundance in the inner disk is therefore set to zero.

We do not consider turbulence in the disk beyond 4 AU and all cooling line widths are thermal in the optically thick outer disk. This assumption is supported at large radii by submillimeter observations and modeling of molecular emission from TW Hya by Qi et al. (2006) and Hughes et al. (2011). Hughes et al. (2011) infer a small turbulent linewidth that is approximately a tenth of the thermal linewidth in the outer disk. The situation may be different in the inner disk, r ≲ 1–2 AU. Salyk et al. (2007) assume higher turbulent linewidths (v_turb ∼ 0.05v_K ∼ 4 km s⁻¹, where the Keplerian velocity v_K ∼ 80 km s⁻¹ at 0.1 AU) in their analysis of the rovibrational CO lines from the disk. CO_rovib emission arises from hot gas, and the thermal speed of CO for ∼2000 K gas is ∼1 km s⁻¹. The expected thermal linewidth is therefore ∼2 km s⁻¹. Observed line widths are typically ∼15 km s⁻¹ (Salyk et al. 2007), with presumably a large Keplerian contribution for gas located at small radii (∼0.1 AU) in a face-on disk with a small inclination (4°; Pontoppidan et al. 2008). We follow Salyk et al. (2007) in assuming v_turb = 0.05v_K in the inner disk (r < 4 AU) regions.

We also add viscous heating as a mechanism to heat gas in the inner disk. The dust-depleted inner cavity might still contain dense gas and viscous heating may be an important contributor to heating the dust-poor gas, especially in the shielded interior near the midplane. We follow the prescription given in Glassgold et al. (2004, Equation (12) of that paper) for the viscous heating term. The viscosity parameter α is unknown, but is set in our models by assuming steady state accretion onto the star at the observed rate of ∼10⁻⁹ M_☉ yr⁻¹ and from the solutions obtained for the density and temperature of the gas.

The surface density distribution of gas in the disk is the primary input to our disk models. We begin with the dust distribution of C02 which has a rapid increase in surface density over ∼3.5–4 AU. We then vary the gas surface density distribution, Σ(r), so that the resulting gas emission lines provide a good match to observational data. The resulting profile for the SML disk is shown in Figure 2. The SML disk has a gas mass ∼10⁻⁵ M_☉ within 3.5 AU and 0.06 M_☉ for 3.5 AU < r < 200 AU. Although the dust surface density drops by ∼10³ inward of r ∼ 4 AU, the gas surface density drops only by ∼100 for the SML disk. The alternate model SMH has a gas mass ∼10⁻⁴ M_☉ within 3.5 AU, and in this case the gas surface density only drops by ∼10. The outer disk surface density distribution is the same for both SML and SMH cases. We discuss later in Section 3.3 how deviations from the models SML and SMH worsen the agreement between model results and observational data.

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For descriptive purposes, we divide the disk into three distinct regions, (1) the inner disk (r ≲ 3.5 AU), (2) the mid-disk region (3.5 AU ≲ r ≲ 20 AU), and the (3) outer disk (r ≳ 20 AU). Our modeling procedure first constrains the gas in the inner disk region, as this gas provides opacity to stellar radiation, shields gas in the partially exposed rim region at 3.5–4 AU, and affects the structure of the outer disk. The inner region of the SML disk is not only depleted in dust but also in gas, by ≳2 orders of magnitude compared to that expected if the outer optically thick disk was smoothly extrapolated radially inward. A power law is assumed for the inner disk surface density distribution, Σ(r)∝r⁻¹; 0.06 AU  ⩽  r < 3.5 AU. We place the inner disk edge at 0.06 AU, at the location of the magnetospheric truncation radius (Eisner et al. 2006). As there is continued accretion onto the star, presumably by funnels along magnetic field lines, this is a logical choice for this parameter. Between 3.5 and 4 AU, Σ(r) rises sharply and these radii constitute the "rim" region. The functional form for Σ(r) is modeled to fit the appearance of the rim of a photoevaporating disk (Gorti et al. 2009) and is given by log Σ(r) = 18 − exp (11/r_AU) g cm⁻²; 3.5 < r_AU < 4. Here, r_AU = r/1 AU. The dust disk turns optically thick over this region. Beyond 4 AU, the surface density is assumed to follow a self-similar density profile, as expected from a photoevaporating and viscously evolving disk (e.g., Hughes et al. 2007; Gorti et al. 2009). For r > 4 AU, Σ(r) = 500r^−0.7_AUexp (− r_AU^1.3/100) g cm⁻².

We briefly discuss the disk density and temperature structure, the dominant heating and cooling mechanisms at three representative disk radial positions, and then describe the line emission calculated from the disk models.

The density and temperature distribution as a function of spatial location in the SML disk is shown in Figure 3. Gas temperatures in the surface regions where emission lines originate are typically ∼2000–200 K in the inner and mid-disk regions, r ≲ 30 AU. The near absence of dust in the inner disk and the sudden rise in dust surface density at the rim causes an increase in gas heating and results in a small vertically extended region at this radius ∼4 AU. However, gas in the inner disk provides opacity that shields the midplane regions at ∼4 AU, and there is only a slight "puffing-up" of the inner rim. (We find that for the upper layers of the outer disk where the gas temperature decouples from the dust temperature, the column density of gas to the star at a given angle (height of the rim) changes by less than 10% as the mass of the inner disk changes from 10⁻⁵ M_☉ to 10⁻⁴ M_☉.) As seen in Figure 3 (see A_V = 1 surface), the disk structure is overall quite flat, with only modest flaring of the outer disk. Gas at depths where A_V ∼ 0.1–1 to the star, and at radii r ≳ 50 AU, only attains temperatures of ∼50–200 K and the density in these warm layers is low, n ∼ 10⁵–10⁷ cm⁻³.

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Disk gas temperature and density is shown as a function of height at typical positions in the three different disk regions, inner disk at r = 0.1 AU, mid-disk at r = 6 AU, and the outer disk at r = 30 AU in Figures 4–6.

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In the inner disk (Figure 4), heating is mainly by X-rays all through the disk, and in the midplane FUV is significant only via chemical heating (primarily due to photodissociation of water and ionization of carbon). However, to attain the high surface temperatures achieved in this region, FUV photons are essential to photodissociate molecules capable of efficiently cooling the surface gas. The gas temperature increases when H₂ photodissociates and molecular (H₂O and OH, formed via H₂) cooling decreases, and also when CO photodissociates and can no longer cool the gas efficiently. In the inner disk, the transition from C to CO occurs higher up than the transition from H to H₂, as is typical in low dust regions (e.g., Glassgold et al. 1997). The inversion in height can be seen in Figure 4 where the H₂ and CO photodissociation fronts are marked. Rovibrational emission of CO (CO_rovib) dominates the cooling with some additional cooling by H₂O and OH, until CO photodissociates. At higher z, O i 6300 Å and Lyα are the main coolants.

Figure 5 shows the disk vertical structure in the mid-disk at r = 6 AU. In these regions where dust is abundant, dust collisions keep gas at the ambient dust temperature in the optically thick midplane region. Near the optically thin surface layer and near the H₂ photodissociation front, heat released from the formation of H₂ is significant. Chemical heating due to photoreactions (FUV photodissociation of water) and FUV-induced grain photoelectric heating by PAHs is of significance over a limited column (A_V ∼ 0.1–1). At the surface, X-rays dominate the heating. Cooling is mainly due to CO and H₂ in the molecular layer, and O i 6300 Å, [O i] 63 μm, [Ne ii], and [Ar ii] fine structure lines cool the surface gas.

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The outer disk vertical structure at r = 30 AU is shown in Figure 6. Dust is thermally coupled to the gas by collisions in the midplane where gas and dust temperatures are equal. Near the A_V = 1 layer, heating by PAHs, photodissociation reactions and formation and dissociation of H₂ are important. X-rays dominate the heating at vertical column densities of ≲ 10²² cm⁻². FUV photoionization of carbon contributes to the heating (∼15%) in the region above the CO photodissociation layer. Main coolants in the outer disk are CO in the molecular region and [O i] 63 μm fine structure emission at the surface.

We adopt the following procedure to determine the surface density distribution that is described in Section 3.1. In order to obtain a gas disk surface density model that best matches observed gas line emission, we first constrain the surface density distribution in the inner disk, i.e., within 4 AU. This is because gas in the inner disk can shield the outer disk regions, affecting the density and temperature structure of the disk beyond 4 AU, where the dust disk is optically thick. There clearly is some gas in the inner hole, based on the observed accretion onto the central star, and on observations of CO rovibrational lines and H₂ ultraviolet lines. We discuss these emission line constraints and the undetected water lines below, and their implications for the gas mass in the inner hole.

Gas opacity. Although the dust is optically thin to stellar radiation, the column density of gas may be sufficient to provide significant optical depth in the inner disk. In fact, the gas may be optically thick to its own infrared emission. We use our disk chemical models to obtain chemical abundances for gas in the inner disk and follow Tsuji (1966) in calculating the monochromatic opacity due to each gas species. At the likely pressures and temperatures of our inner disks, H⁻ and H⁻₂ bound-free and free–free, CO and H₂O all can provide significant opacity. H⁻ and H⁻₂ bound-free and free–free continuum opacity can be significant at wavelengths shorter than about 4 μm, and produce near IR radiation from the inner edge of the disk at ∼0.06 AU. The ground state CO vibrational band occurs near 4.7 μm and the blend of CO rovibrational lines provide opacity near 5 μm. Beyond about 6 μm, the blend of pure rotational H₂O lines dominate the opacity.

Later on in this subsection, we discuss CO and H₂O lines in some detail. Here, we make more general arguments based on the expected continuum emission from gas in the disk. We use the opacities from Tsuji (1966) to show that if the surface density of the outer disk is extrapolated inward to produce a ∼10⁻³ M_☉ inner disk (we call this the "full disk" or the "undepleted disk"), observable emission would result in the wavelength region 5–15 μm due to the CO and H₂O emission. We find, by a similar analysis, that the H⁻ and H⁻₂ bound-free and free–free produce excess emission at shorter wavelengths (∼2–3μm) where the stellar emission is significant. We use the specific opacity (per molecule) for each opacity contributor from Tsuji (1966) and calculate the opacity at each radial annulus by multiplying this with the vertical column density as obtained from our model disk chemical calculations. For illustration, we calculate the flux at one wavelength and compare that with the observed emission. We choose a wavelength of 8 μm where the stellar contribution is negligible, there is no contamination from the silicate feature and where there is a significant dip in the continuum emission (see C02). At each radial annulus, the photosphere (at 8 μm) is defined to lie at the height (z) above the midplane where the total optical depth at 8 μm becomes of order unity. Near the inner edge of the disk for 0.06 AU < r ≲ 0.08 AU, H⁻ and H⁻₂ free–free are the main opacity contributors. This opacity drops rapidly with decreasing electron density further out in the disk. At 0.08 AU ≲ r ≲ 0.2 AU, H₂O lines dominate the 8 μm opacity. Having defined the photosphere of the disk from the gas opacities, we estimate the 8 μm flux from this surface, where the gas temperature ranges from ∼300 to 2000 K. The disk becomes optically thin to the midplane (λ ∼ 8 μm) at ∼0.2–0.3 AU. We obtain an 8 μm flux of ∼1 Jy, a factor of ∼4 higher than observed. A full gas disk is therefore incompatible with the SED. The H₂O lines at 8 μm therefore yield an estimate of the upper limit on gas mass as 2.5 × 10⁻⁴ M_☉. We discuss later our model results for H₂O line emission and compare them to Spitzer IRS upper limits to the line fluxes to obtain a better limit on the inner disk mass.

A similar opacity analysis at a wavelength of 5 μm suggests that CO rovibrational emission in excess of what is observed will be produced by a full gas disk. We use our model calculations and compare them with observed rovibrational CO line strengths near 5 μm to estimate the actual mass of the inner disk to be ∼10⁻⁴ to 10⁻⁵ M_☉.

In what follows, rather than using an opacity argument, we use our disk model results on emergent line fluxes from the disk surface to infer the disk mass. We describe the emitting regions for each gas emission line diagnostic and also discuss the constraints set in each case, where applicable. The outer disk gas is attenuated by the gas in the inner disk as described earlier. We calculate the outer disk structure using the surface density distribution of the inner disk in the SML model and use the same results for the outer disk component of the SMH model as well, as the attenuation columns of the gas provided at 4 AU are similar in both of these cases (within ∼10%). As our disk models are computationally intensive, we do not calculate disk models for the outer disk in the SMH case separately. In the rest of the paper, the emission in the mid and outer disk regions refer to that calculated using gas optical depths and absorption columns for the 10⁻⁵ M_☉ inner disk mass case (SML). Tables 3 and 4 summarize the contribution of the inner and outer disk to the emission lines for the SML and SMH model disks. Figure 7 shows the radial extent of each of the gas emission lines discussed below. The cumulative line luminosity, L(r), integrated up to the radius r is shown for each emission line and is normalized to the total calculated value (from inner, mid, and outer disk regions). Also marked are the 10% and 90% luminosity levels. We quote the corresponding radii below for each emission line and this defines the radial extent of the emission.

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CO rovibrational emission.CO_rovib emission typically arises from a height in the inner disk where the CO is just beginning to photodissociate. Gas temperatures in the disk generally increase with height and gas temperatures in the emitting layers are high (T_gas ≳ 500 K). In dust-poor gas, the CO and H₂ photodissociation layers are inverted, as first reported by Glassgold et al. (2004), that is, the carbon becomes CO at larger z than the H becomes H₂. We similarly find that the H₂ is photodissociated to deeper layers and that CO is coexistent with large columns of atomic hydrogen, creating conditions favorable for exciting CO_rovib emission (Glassgold et al. 2004). The primary formation route to CO is the oxygenation of atomic carbon (C+OH). In the layers shielded from FUV photodissociation CO is destroyed by collisions with helium ions.

CO vibrational lines are an important constraint for setting the gas mass in the inner disk. A high disk mass results in high total CO vibrational emission if the lines are optically thick, as the luminosity in these lines increases with temperature which increases (albeit weakly) with increasing gas mass. The total luminosity of the observed lines (select P and R branch transitions) is constrained by observations to be ≲ 10⁻⁵ L_☉ (Salyk et al. 2007). We find that inner disk masses of 10⁻⁴, 10⁻⁵, and 10⁻⁶ M_☉ all give total CO luminosities within a factor of ∼2 of the observations and are all hence viable choices. The CO emission is found to be mainly a surface effect dominated by hot gas at the surface of the disk, resulting in approximately the same total CO luminosity for all three disk masses. To distinguish between these cases, the line luminosity of each of the CO rovibrational lines needs to be compared with data. We estimate the CO rovibrational line luminosities of the v = 1–0 P(J) series as follows. We adopt the analysis of Scoville et al. (1980) and perform an approximate non-LTE calculation for the populations of the CO rovibrational levels. We only consider the v = 0 and v = 1 states as higher vibrational states are unlikely to be significantly occupied at typical disk temperatures. We assume that collisions result in a change in the vibrational state, and vibrational level populations are determined from detailed balance assuming an "effective" Einstein A-value from v = 0–1 as described in Najita et al. (1996). We then assume that the rotational levels within each v state are thermally populated (Scoville et al. 1980). Collisional rates with e⁻, H, and H₂ are taken from Najita et al. (1996). Oscillator strengths and transition frequencies are from the compilations of Kirby-Docken & Liu (1978). The optical depth for each transition is calculated as detailed in Hollenbach & McKee (1979) from the column density of CO molecules in a given rovibrational state, and an escape probability formalism is used for line transfer.

Salyk et al. (2007) detect 15 transitions covering rovibrational energies E_u/k from 3000 K to 6000 K, where E_u is the energy of the excited upper level. We also compare our results with earlier CO rovibrational detections by Rettig et al. (2004) in complementary transitions. There was a typographical error in the Rettig et al. (2004) table of line fluxes, and the actual fluxes are a factor of 10 below what has been reported. We also do not include the P(21–25) transitions of that paper, as these data are prone to errors due to photospheric and telluric corrections (S. Brittain 2011, private communication). We compare our model results with the data in Figure 8. We find that a gas depletion from the full disk of 0.1–0.01 corresponding to a gas mass of 10⁻⁴ to 10⁻⁵ M_☉ matches the observed line emission fairly well, with the lower mass inner disk model (SML) being a better fit to the CO data. A factor of 10 lower in mass results in CO emission from warmer gas that produces excess emission at high J levels. All of the CO gas emission comes from r ≲ 0.5 AU (due to rapidly decreasing gas temperatures beyond), in agreement with both the analysis of Salyk et al. (2007) and the spectroastrometric imaging study by Pontoppidan et al. (2008). Our gas mass estimate for the SML disk is consistent with the surface density estimate of Salyk et al. (2007) of 1 g cm⁻² at r ∼ 1 AU where they used a slab model, although in our models the emission is mainly from hot gas in the disk surface layers. Salyk et al. (2007) also assume an H₂/CO conversion ratio of 5000 as observed toward a massive protostar, and caution that their derived mass estimate is likely be a lower limit. In our disk models, the hydrogen is predominantly atomic in the regions where CO_rovib emission originates (also see Glassgold et al. 2004). We explicitly calculate disk chemical abundances and find that the H/CO ratio in the hot inner disk regions where CO_vib emission arises is ∼7000. The gas/dust mass ratio in the inner disk is ∼500–5000 for the dust mass of the Calvet et al. (2002) disk, indicating a relative depletion of the dust with respect to the gas at r ≲ 4 AU.

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H₂O rotational emission. Our gas opacity analysis above suggests that a full undepleted disk would produce excess H₂O rotational emission in lines whose wavelengths range from 6 to 15 μm compared with observations. As we specifically calculate water abundances and the disk density and temperature structure, we can compare the model disk H₂O rotational line fluxes to the corresponding upper limits from Spitzer IRS in the 10–19 μm wavelength region (Najita et al. 2010). No water emission has been detected and line luminosities are therefore expected to be lower than ∼10⁻⁷ L_☉. We find that for the full undepleted inner disk with a mass of 10⁻³ M_☉, many rotational lines in the 10–19 μm region are above the detection limit. Some of the strongest transitions are 7₆₁–6₁₆, 7₆₂–6₁₅, 8₅₄–7₀₇, 8₅₃–7₂₆, and 8₄₄–7₁₇, all with luminosities in the range of (1–3) × 10⁻⁶ L_☉, compared with the upper limits set by Spitzer observations of ∼10⁻⁷ L_☉. From the water rotational line emission constraint alone, we can set a limit on the inner disk mass as ∼10⁻⁴ M_☉, at which the water emission is barely detectable. This limit is consistent with the upper limit set by CO emission in the preceding discussion, where we fix the inner disk mass to be in the range 10⁻⁴ to 10⁻⁵ M_☉ based on CO rovibrational emission.

H₂ fluorescent emission. We do not model H₂ UV fluorescence but draw on the results of Herczeg et al. (2004), who conducted a very detailed analysis and modeled the observed emission. Our SML disk corresponds to their thick disk model, and most of the emission in this case arises from the illuminated edge of the disk at r = 0.06 AU. Herczeg et al. (2004) estimate a hot (T_gas ∼ 2500 K), H₂ mass of ∼10¹⁹ g from their analysis. We calculate the mass of H₂ gas at these temperatures in our inner disk to be ∼3 × 10¹⁹ g, in reasonable agreement with the Herczeg et al. (2004) results. All of this gas lies very close to the inner edge at ∼0.06 AU. At this radius the Keplerian speed of the gas from the almost face-on disk (i = 4°) is about ∼7 km s⁻¹, the thermal speed is ∼4 km s⁻¹, bringing the expected FWHM to ∼16 km s⁻¹ very close to the observed FWHM of 18 km s⁻¹ (Herczeg et al. 2002). We note that the inner disk structure near the magnetospheric truncation radius is likely to be quite complicated compared to our simple power-law assumption of the surface density, and our comparison with the Herczeg et al. model is only a consistency check.

OH lines. Sixteen rotational transitions of OH, mainly in the ground vibrational state, were reported by Najita et al. (2010) from their Spitzer observations. The data indicate high excitation states (J_upper ∼ 15–30), with equivalent energies E_u/k up to 23,000 K above ground. OH mid-infrared thermal emission from our model is mainly from the hot, dense (n ∼ 10¹²–10¹³ cm⁻³, T_gas ∼ 700–1000 K) gas very close to the central star at r ≲ 0.1 AU. These lines are, however, weak in the 10–19 μm region and below the sensitivity of the Spitzer IRS instrument. OH thermal emission is also produced from gas in the mid-disk and outer disk regions, but this gas is typically at ∼300 K and incapable of reproducing the observed high excitation lines. As suggested by Najita et al. (2010) for TW Hya, UV photodissociation of water is a possible source of the OH emission (see also Tappe et al. 2008 who suggest this mechanism for HH214).

From our disk chemical network solution, we can estimate the total luminosity in the OH lines produced by the photodissociation of water in the disk. We calculate the total OH line luminosity as follows. For simplicity, in this discussion we assume that Lyα photons (E = 10.2 eV) are mainly responsible for the photodissociation of water molecules—a reasonable assumption because Lyα contributes to a significant fraction of the total FUV luminosity (e.g., Herczeg et al. 2002; Bergin et al. 2003). Harich et al. (2000) studied the photodissociation of water by Lyα photons and the photodissociation energy is determined to be 5.1 eV. The excess energy for a Lyα photodissociation is therefore 5.1 eV. About 66% of the photodissociations of water result in OH and H, while the remaining 34% result in O and H atoms. Harich et al. (2000) further find that most of the OH products are extremely rotationally excited with a peak at a rotational level J = 45 (E_u/k ∼ 45,000 K) and that almost 75% of the available energy goes into pure rotational excitation. This translates to an energy of ∼4 eV per photodissociation (of H₂O to OH) that is then eventually radiated away in a rotational cascade from the high J states (also see Mordaunt et al. 1994; van Harrevelt & van Hemert 2000). We calculate the energy in the rotational cascade in our model disk by counting the total number of water photodissociations by photons with energy equal to or greater than Lyα that lead to OH and multiplying this by the energy available for excitation (4 eV). If $\dot{N}_{{\rm PD}}({\rm H_2O})$ is the rate of the total number of photodissociations of water in the disk by Lyα (and more energetic FUV) photons, then the luminosity in the OH cascade L_{OH, PD} is the product $0.66 \times \dot{N}_{{\rm PD}}({\rm H_2O}) \times 4$ eV. Equating this to the observed OH luminosity then requires that there are ∼10⁴⁰ photodissociations per second or that only ∼1% of the stellar Lyα luminosity of 10⁴² photons s⁻¹ be intercepted by water in the disk (also see Bethell & Bergin 2009). If the solid angle subtended by the disk is ∼0.1, the desired Lyα flux absorption by water is feasible if water abundances are high enough (at the 10% level) to compete with the other dominant sources of opacity such as dust, photoionization ionization of Mg, Fe, and Si, and photodissociation of O₂ and OH. Therefore, photodissociation of water can plausibly account for the observed line luminosities.

In the above approximate analytic calculation, we simply used the Lyα flux. However, in the model we calculate the OH luminosity due to water photodissociation using photodissociation rates obtained at each spatial grid cell. We use the full FUV spectrum, including Lyα, but only count those photodissociations where the FUV photons are energetic enough to lead to OH in highly excited rotational states as described above. The OH luminosity in the SML disk is 4 × 10⁻⁶ L_☉, lower than the total luminosity in the observed lines (upper J levels ranging from ∼30 to ∼15), 6 × 10⁻⁶ L_☉. We note that the cascade must also produce lines that are not observed in the Spitzer IRS band. The energy of a rotational state ∝J(J + 1) and the cascade originates at J ∼ 45. The unobserved transitions from J ∼ 45 to J ∼ 30 and J ∼ 15 to J = 0 therefore have a total energy approximately 1.5 times that in the transitions that are observed. The unseen luminosity must account for another ∼9 × 10⁻⁶ L_☉, bringing the "observed" value to 1.5 × 10⁻⁵ L_☉. Our SML disk results for OH produced by H₂O photodissociation are discrepant by a factor of ∼4. Of this emission, 30% arises from photodissociation of water in the inner disk and the rest comes from the extended middle and outer disk regions to radii r ≲ 40 AU (Figure 7). In the inner disk, the OH is from the edge of the disk near the star where the number densities are ∼10¹²–10¹³ cm⁻³ and gas temperatures are ∼500–1000 K. The main formation route to water is H₂ + OH and it is destroyed mainly by FUV photodissociation. The water abundance in these regions is, however low, with typical vertical column densities ≲ 10¹⁵–10¹⁶ cm⁻² in the photodissociation layer. In the mid-disk and outer disk regions, OH emission is dominated by the rim at 4 AU and beyond the rim is primarily produced from regions where the column density to the star ∼(1–5) × 10²² cm⁻², densities are 10⁷–10¹⁰ cm⁻³, and T_gas ∼ 200–600 K. In the mid-disk regions OH+H₂ is the dominant formation route to water, whereas in the cooler outer disk regions (r ≳ 15 AU), H₃O⁺ recombination leads to the formation of water molecules. Water abundances range from X(H₂O) ∼ 10⁻⁶ to 10⁻⁹ in these photodissociation layers, and typical water columns to the star are ∼10¹⁶ cm⁻². The latter is consistent with our earlier analysis, which suggested that if the water opacity were ∼10% of the total opacity at FUV wavelengths, we would obtain OH luminosities close to that observed. A column density of ∼10¹⁷ cm⁻² of water provides optical depth of unity at the FUV wavelengths needed to photodissociate water.

If the OH is mainly prompt emission following photodissociation of water, the SML disk appears to produce less water by a factor of ∼4 than required by the observations. We consider a few mechanisms by which the water production rate may be enhanced to match observations. Raising the gas temperature and hence increasing the water formation rate by the endothermic H₂+OH reaction is one possibility and this could be achieved by increasing the unknown PAH abundance and FUV grain photoelectric heating in the disk. However, as H₂ pure rotational emission originates at the same spatial location where there is warm molecular gas, any increase in temperature overproduces the H₂ S(1) line flux. We discuss the H₂ S(1) and S(2) line fluxes below.

Another possibility is that the SMH disk is the better solution for the inner disk mass. This disk is a reasonable match to the CO lines (Figure 8) and does not violate other observational constraints such as the lack of mid-IR water emission. Higher densities in the inner disk result in higher water abundances and the prompt OH emission increases by a factor of ∼4 in the inner disk, bringing the total OH luminosity to ∼7.8 × 10⁻⁶ L_☉, and only a factor of two below the observed value (see Table 3). This model slightly overestimates the O i 6300 Å and H₂ S(2) line luminosities, but may nevertheless be a feasible model for the TWHya disk. The SMH disk also has the advantage that it better fits the [O i] 6300 Å linewidth and provides a more typical value for the turbulence parameter α in the inner disk (see Table 3).

Water formation on dust grains is another mechanism that could increase water in the disk. Our disk models do not include freeze-out and desorption processes to properly consider grain-surface reactions. We estimate an upper limit to the effect of water formation on dust in a simple manner as follows. We assume that oxygen atoms stick on grains are instantly hydrogenated and released into the gas phase as water, at a rate equal to the collision rate of O with grains. We applied dust formation of gas phase water only in regions where the dust is cold enough to prevent thermal evaporation of O or OH before H₂O is formed on the grain surface, T_gr > 100 K. We use this relatively high grain temperature for thermal desorption so that we obtain an upper limit on the production of water on grains. We also require that A_V to the star is less than ∼3 so that photodesorption can clear the water ice from the grain surface (Hollenbach et al. 2009). Glassgold et al. (2009) considered a similar mechanism to increase water production in disks but found that it was not significant in increasing column densities of warm, observable water. However, the gas temperature is not relevant for our calculation as the OH lines are a result of photodissociation and not thermal emission. Water formation of grains in the SML disk (with 10⁻⁵ M_☉ within 4 AU) leads to a higher OH luminosity, 7 × 10⁻⁶ L_☉, but still a factor of ∼2 below what is observed. We emphasize that this 7 × 10⁻⁶ L_☉ is an upper limit, given the optimistic assumptions on water formation on grains.

Radial transport of solids or water-bearing ices in the disk, a process not considered by us, has been suggested as a possible agent causing local enhancements and depletion of water abundances in the disk (Ciesla & Cuzzi 2006). Water ice from the midplane may also be transported to greater heights where it is subject to thermal desorption or photodesorption, leading to higher water abundances in the disk surface where it can be photodissociated. These effects, while likely to operate in disks, require time-dependent models that consider radial and vertical transport and are beyond the scope of this paper. However, one simple calculation is illuminating. If water is transported as water ice radially inward to inner regions where it is photodissociated once, the mass influx needed is 10⁴⁰ H₂O molecules per second. Assuming an H/H₂O ratio of 2000 (if every O atom is in H₂O ice) this translates to 2 × 10⁴³ H atoms per second or 5 × 10⁻⁷ M_☉ yr⁻¹. Since TW Hya has an age of 10 Myr, and with a constant influx, this requires an improbable initial mass reservoir of 5 M_☉ in the outer disk and also implies that the current disk will only last 10⁵ years. This is unlikely. On the other hand, if turbulent mixing were to act vertically, then this has the advantage that the water can be reformed at the midplane and brought to the surface to be photodissociated repeatedly. For our adopted gas mass of 0.06 M_☉, we therefore require that this mass be brought to the surface of the disk where it is photodissociated every 10⁵ years. At a typical radius, r = 10 AU, the distance from midplane to the A_V = 1 surface layer where water is photodissociated is ∼1 AU. Thus, the average vertical transport speed is only 1 AU/10⁵ yr = 5 cm s⁻¹. This suggests that vertical mixing may supply the needed water.

We have discovered a new mechanism for the rotational excitation of OH that may be the most promising, but difficult to quantify precisely at this time. Here, we do not need a greater production of gas phase water. The process is initiated by the photodissociation of OH, which leads to atomic oxygen in an excited electronic state (O¹D) approximately 50% of the time (van Dishoeck & Dalgarno 1984b). The ¹D state of oxygen, due to its empty 2p orbital, is more electrophilic than the triplet ground state ³P and is highly reactive to readily undergo bond-forming addition reactions. O(¹D) + H₂ re-forms OH very efficiently (with a rate coefficient γ_O = 3.0 × 10⁻¹⁰ cm³ s⁻¹; NIST Chemical Kinetics database) leading to an effective reduction in the overall destruction of OH. This route has no thermal barrier as in the reaction of the ground state O(³P) atoms with H₂ and is especially important at high gas densities when the rate of formation of OH by this route is faster than the radiative decay of the O¹D atom to the ground state (with A ∼ 8 × 10⁻³ s⁻¹). The fraction f_O of O¹D that reacts with H₂ before radiatively decaying to the ground state is given by f_O = 1/(1 + (n_crit/n(H₂))), where the critical density n_crit ∼ γ_O/A = 3 × 10⁷ cm⁻³. An important consequence of the O(¹D) + H₂ reaction may be that this reaction produces OH in a rotationally excited state (with an efficiency _OH ∼ 0.2 at J ∼ 20–30 states; Lin & Guo 2008) which could then cascade to lower J to produce the observed MIR emission by Spitzer. In regions with n(H₂) > n_crit, photodissociation of OH itself (rather than H₂O as considered earlier) may result in the re-formation of rotationally excited OH. We estimate the contribution of this mechanism to the OH line emission as follows.

$\dot{N}_{{\rm OH}}$ photodissociations per second of OH will produce ${\sim} 0.5 \epsilon _{{\rm OH}} f_O \dot{N}_{{\rm OH}}$ OH molecules in the J ∼ 20–30 state per second. For our SML disk, we estimate that this mechanism could result in ∼5 × 10⁻⁶ L_☉ (nearly all from the inner r ≲ 1 AU region), which recovers most of the observed OH emission which the model was deficient in when we considered the photodissociation of water alone. We will explore this mechanism for producing OH emission in greater detail in a future study where we will treat O¹D as a separate species for better accuracy and treat the efficiencies of _OH(v, J) individually.

We conclude with a footnote to the above discussion. Mandell et al. (2008) recently reported the discovery of OH from two HAeBe stars and concluded that the emission was a result of OH fluorescence. The rotational energy diagram of the transitions observed (in this case from the v = 1 state) indicates a single rotational temperature which leads the authors to conclude that it is unlikely to be the result of "prompt" emission, i.e., OH produced by the photodissociation of water. Their models of fluorescent emission from OH match the observed data from HAeBe stars quite well. In their models, OH is excited to the v = 1 state through rovibrational transitions pumped by near-infrared photons and electronic transitions pumped by UV photons. However, the Spitzer data for TW Hya indicate mainly a downward rotational cascade from the high J states of v = 0 state (Najita et al. 2010). Therefore, in TW Hya the observed OH is more likely to be prompt emission.

H₂ rotational lines. Pure rotational S(2) and S(1) line emission from H₂ was reported by Najita et al. (2010) at 4.8 × 10⁻⁷ and 10⁻⁶ L_☉, respectively. Most of the H₂ emission in our models comes from within radii of 20–30 AU, with ∼35% of the H₂ S(2) emission and 10% of the H₂ S(1) emission from the inner SML disk (see Figure 7). In the mid-disk region, H₂ emission originates in the superheated dust later at A_V ∼ 0.3–0.1 and where the gas temperatures are 100–500 K. H₂ formation in the mid and outer disk (r > 4 AU) is mainly on grains and it is destroyed by FUV photodissociation. We obtain H₂ S(2) and S(1) line luminosities to be 3.4 × 10⁻⁷ and 1.2 × 10⁻⁶ L_☉, respectively. These are in reasonable agreement with the SpitzerIRS measurements. Bitner et al. (2008), in their ground-based survey for H₂ using TEXES, did not detect H₂ S(1) emission and obtained an upper limit of 5.0 × 10⁻⁷ L_☉. This suggests that half of the S(1) emission observed by Spitzer IRS must originate from a more extended region outside the narrower slitwidth of TEXES, i.e., at r ≳ 20 AU. Although the H₂ S(1) emitting region is quite extended (Figure 7), half the H₂ S(1) emission in the SML disk in fact originates at r ≲ 14 AU. The S(1) flux within 20 AU is 8 × 10⁻⁷ L_☉ and the radial extent of emission is in moderate agreement with the TEXES and Spitzer data. The line luminosity in the H₂ S(0) transition is predicted from the SML disk to be 3 × 10⁻⁷ L_☉.

[O i] 6300 Å and 5577 Å emission. We show below that the [O i] 6300 Å line from TW Hya originates from the disk, mainly from the photodissociation of OH (also see Storzer & Hollenbach 1998; Acke et al. 2005) with a small contribution from thermal emission. The line luminosities for the [O i] 6300 Å and [O i] 5577 Å lines from TW Hya are 10⁻⁵ L_☉ and 1.4 × 10⁻⁶ L_☉, respectively (Alencar & Batalha 2002; Pascucci et al. 2011, S. Edwards 2010, private communication). We first show that the observed [O i] 6300 Å/[O i] 5577 Å line ratio of ∼7 makes it unlikely that the origin is thermal.

The oxygen atom has a triplet ground state (³P) and singlet D and S states which are at E_u/k ∼ 22, 850 K and 48, 660 K, respectively. The O i 6300 Å transition from the ¹D to the ground state and the O i 5577 Å transition from the ¹S state to the ¹D state therefore require high gas temperatures for collisional excitation to these high energy levels and subsequent radiative de-excitation. If the gas is collision dominated (in LTE), then the ratio of the two lines can be easily obtained from the ratio of n_uAΔE, where n_u is the upper level population, A is the Einstein A-value, and ΔE is the energy of the photon emitted. In LTE, n(¹S) ∼ (1/5)n(¹D)e^ΔE/kT, where (1/5) is the ratio of the statistical weights and ΔE/k ∼ 26,000 K is the energy difference between the two levels. Therefore

$$\begin{equation} { {L_{6300}}\over {L_{5577}}} = { {n(^1D) A_{6300} \Delta E_{6300}}\over { n(^1S) A_{5577} \Delta E_{5577} }} \sim 5.4 \times 10^{-3} e^{26000/T}, \end{equation} \tag{ 1 }$$

where A₆₃₀₀ ∼ 6 × 10⁻³ s⁻¹ and A₅₅₇₇ ∼ 1.3 s⁻¹. To obtain the observed line ratio of 7, gas temperatures therefore need to be ∼3600 K, which is possible with X-ray heating of the disk surface. However, the gas densities have to be high enough to produce LTE conditions. Collision partners can either be electrons or neutral hydrogen atoms and typical collision rate coefficients are γ_e ∼ 5 × 10⁻⁹(T/10⁴)^0.6 cm³ s⁻¹ for electrons for both the transitions. The collisional rate for O(¹D) with H atoms is γ_H ∼ 10⁻¹² cm³ s⁻¹ (Krems et al. 2006). γ_H for O(¹S) is not known. However, the rates for the relaxation of the O(¹S) state are expected to be smaller than those for the relaxation of the ¹D state, because the spin–orbit couplings in electronically excited atoms are generally weaker (R. Krems 2011, private communication). We conservatively assume γ_H ∼ 10⁻¹² cm³ s⁻¹ for O(¹S) as well. Therefore, electrons dominate collisions if x_e > 4 × 10⁻⁴. For levels to achieve LTE, electron densities n_e > A/γ_e ∼ 10⁸ cm⁻³ are needed for the 5577 Å line (n_e > 10⁶ for the 6300 Å line). Collisions with H atoms require neutral densities about 2500 times these values. Even for gas with high electron fractions x_e ∼ 0.1, gas densities therefore need to be high, n_H ≳ 10⁹ cm⁻³. These conditions are likely to be attained only in the inner disk, where gas at the surface may be dense, be partly ionized with sufficient neutral oxygen, and at T ≳ 3600 K.

For gas at lower densities a balance between radiative de-excitation and collisional excitation and de-excitation of the different levels determines the level populations. We calculate the line ratios as a function of electron density and gas temperature and Figure 9 shows the contours for the observed value of 7 and other values about this number. From the contour plot, one can see that even for relatively high gas temperatures of ∼8000 K the required electron densities are ∼10⁷ cm⁻³ implying gas densities of ∼10⁹ cm⁻³ for an electron fraction of x_e ∼ 0.01, which is typical. Such conditions are unlikely, except perhaps in the innermost regions, r ≲ 0.1 AU. We also note that soft X-rays that can heat the gas to such high temperatures typically have very small absorption columns ∼10¹⁹ cm⁻³, and very little mass is expected in gas at these high temperatures and densities. Such low masses may not be able to reproduce the observed high optical O i line luminosities.

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For typical densities in the hot disk surface, n ∼ 10⁵–10⁷ cm⁻³, and electron fractions x_e ∼ 0.01–10⁻³ the electron densities are n_e ≲ 10⁵ cm⁻³, and the observed line ratio requires T ≳ 30,000 K. We consider this extremely unlikely as it is very difficult to heat disk gas to these high temperatures and moreover retain a significant amount of neutral oxygen. The recent calculations of Ercolano & Owen (2010) also show that the 5577 Å line is appreciable relative to the 6300 Å line only when gas densities are high, as in their primordial disk models where the O i emission is from the inner r ≲ 1 AU region. In their models, the O i emission is thermal in origin and they quote L₆₃₀₀/L₅₅₇₇ ∼ 13–50 for the primordial disk models and ∼300–700 for the transition disk models, supported by the above analysis. Only one of their many models, a primordial disk with L_X ∼ 2 × 10³⁰ erg s⁻¹ has a low line ratio ∼13, where the emission presumably originates from gas with high electron density.

In the SML disk, thermal emission contributes to about 20% of the observed [O i] 6300 Å line and 10% of the 5577 Å line. In the inner disk, the luminosities are 1.2 × 10⁻⁶ L_☉ and 10⁻⁷ L_☉ for the thermal component of the 6300 Å and 5577 Å lines, respectively, and the emission comes from regions where n_e ∼ 10⁵–10⁸ cm⁻³ and T ∼ 4000–8500 K at the disk surface. In the SMH disk, the calculated thermal [O i] 6300 Å line luminosity is higher, 2 × 10⁻⁶ L_☉ (Table 3). There is also thermal [O i] 6300 Å emission from the mid and outer disk surface that is heated by X-rays to T ≳ 5000 K, the luminosity here is ∼1.1 × 10⁻⁶ L_☉. There is negligible thermal [O i] 5577 Å emission from the mid and outer disk regions.

We propose here that most of the O i forbidden line emission from TW Hya is the result of photodissociation of OH in the disk. The OH photodissociation layer in the SML disk is almost spatially coexistent with the H₂O photodissociation layer, and this generally is located near the H/H₂ transition in the disk. OH photodissociates about 50% of the time to produce an oxygen atom in the ¹D state and occasionally (5% of the time) in the higher ¹S state (e.g., Festou & Feldman 1981; van Dishoeck & Dalgarno 1984a). Subsequent electronic de-excitation to the ground state produces the oxygen doublet at 6300 Å and 6363 Å (¹D−³P) and the ¹S−¹D transition produces the 5577 Å emission. The atoms in the O¹D state produce the 6300 Å+6363 Å doublet in the branching ratio of 3 (Storey & Zeippen 2000). The ratio of these line strengths is the same as the branching ratio because both lines originate from the same upper state. These calculations are supported by comet observations of the photodissociation products due to solar Lyα (Morgenthaler et al. 2007). We again count the number of photodissociations of OH in the model disk as we did for water, and multiply this by the energy of the transition and the theoretically calculated branching ratio.

Our inclusion of the O(¹D) + H₂ → OH reaction leads to a lower effective photodissociation rate of OH as discussed earlier. If this reaction were not considered, then prompt emission produces a 6300 Å/5577 Å line ratio ∼7. However, in regions where the densities are high, such as in the inner disk, the line ratio is expected to be lower as the reactive nature of the O¹D atom will result in a suppression of 6300 Å emission. Although the ¹S state of oxygen is also more electrophilic than the triplet ground state, the unpaired electrons likely make the O(¹S)+H₂ reaction less favorable. Further, the radiative decay of O¹S is faster (A₅₅₇₇ > 200A₆₃₀₀) requiring even higher densities for the reaction with H₂ to be competitive. The 5577 Å line is therefore not expected to be lowered significantly. In dense regions of disks, the ratio of 6300 Å/5577 Å is therefore expected to be lower than ∼7. $\dot{N}_{{\rm OH}}$ photodissociations per second of OH will produce ${\sim} 0.5 (1-f_{\rm O}) \dot{N}_{{\rm OH}}$ 6300 Å photons, where (1 − f_O) is the reduction factor due to O(¹D) + H₂ → OH which suppresses the radiative transition leading to the 6300 Å line. We therefore calculate that the [O i] 6300 Å and [O i] 5577 Å line luminosities due to OH photodissociation are 1.1 × 10⁻⁶ L_☉ and 9.3 × 10⁻⁷ L_☉ in the inner disk, respectively, and these lines originate in the same region as the OH cascade, at r < 0.1 AU, where n ∼ 10¹²–10¹³ cm⁻³ and T ∼ 500–1000 K. OH photodissociation in the mid-disk regions produces line luminosities 5 × 10⁻⁶ L_☉ (6300 Å) and 7 × 10⁻⁷ L_☉ (5577 Å). This emission is produced from gas with n ∼ 10⁷–10¹⁰ cm⁻³ and T ∼ 200–600 K out to radii r ∼ 40 AU. Therefore, most of the O i luminosity from TW Hya comes from photodissociation of OH. Ercolano & Owen (2010) recently propose that the O i forbidden line emission seen in disks may arise from an X-ray driven wind. Our picture differs from theirs in two ways—in our model the O i emission arises through a non-thermal process and the emission arises from the disk atmosphere rather than in a wind. This perspective is consistent with the O i 6300/5577 Å line ratio and the lack of a blueshift for the emission.

The total luminosities from the SML disk (thermal +photodissociation) are therefore 8.4 × 10⁻⁶ L_☉ for the [O i] 6300 Å line and 1.6 × 10⁻⁶ L_☉ for the [O i] 5577 Å line. Our 6300/5577 Å line ratio is ∼6. The SMH disk which was a better match for the OH emission gives slightly higher [O i] emission, 1.5 × 10⁻⁵ L_☉ for the 6300 Å line and 1.92 × 10⁻⁶ L_☉ for the 5577 Å line for a ratio of 7.8 (also see Table 3).

The intrinsic FWHM linewidth of the observed O i lines are of order 10 km s⁻¹ (Pascucci et al. 2011). This corresponds to a velocity dispersion of about 5 km s⁻¹. In the SML disk, the O i lines arise mostly in the mid-disk regions where the projected Keplerian speeds are low, <2 km s⁻¹. In addition, the gas is relatively cool and turbulent speeds are expected to be low (Hughes et al. 2011). Hence, the contribution to velocity dispersion from thermal and turbulent velocity fields is expected to also be low, <1 km s⁻¹. However, the photodissociation of OH leads to a recoil of the resultant O(¹D) or O(¹S) atoms at speeds of order 2 km s⁻¹, and the [O i] photons are emitted prior to collisional relaxation (van Dishoeck & Dalgarno 1984a). We estimate the linewidth of the O i 6300 Å line by adding the various components and the Keplerian, thermal and recoil contributions as appropriate. We obtain a linewidth of ∼6 km s⁻¹, narrower that the observed value. In the SMH disk, the calculated linewidth is more in agreement with observations, ∼11 km s⁻¹, because of the increased prompt emission close to the star, although the total O i 6300 Å line luminosity is overproduced by a factor of ∼1.5 (Table 3).

We end with a caveat that although the above production mechanism via OH photodissociation explains the O i 6300 Å and O i 5577 Å line strengths, ratio, and lack of an observed blueshift in TW Hya, it may not be a more general explanation valid for all disks in which optical forbidden line emission has been detected, especially if these lines are blueshifted and originate in a flow. For disks with different surface density distributions and FUV and X-ray luminosities, there may be a more dominant thermal contribution to [O i] emission. Further, our disk models are static and do not include any emission that might arise in an extended wind (e.g., Ercolano & Owen 2010). Although we expect that our disk density and temperature structure is reasonably accurate within our grid, we do not account for any emission that might arise from heights above our vertical extent of z = r. Densities in these regions are typically low, and usually not a significant source of emission. For the optical lines (O i and S ii, discussed below) however, the upper states are at ∼20,000–40,000 K and the emission is very sensitive to temperature, and the contribution from the wind in this case might be significant.

S ii optical lines. Pascucci et al. (2011) re-analyze the optical data of Alencar & Batalha (2002) and report the detection of the [S ii] 4069 Å line (∼3 × 10⁻⁶ L_☉) from TW Hya. They do not detect other lines of S ii and place an upper limit of 1.7 × 10⁻⁷ L_☉ on the 6731 Å line. We find that in our SML disk, most of the S ii emission arises from the inner disk in hot, dense gas that also results in the thermal O i emission calculated above. Our calculated S ii line luminosities are 3.9 × 10⁻⁶ L_☉ for the 4069 Å lines, and 8.6 × 10⁻⁷, 5.6 × 10⁻⁸, and 1.1 × 10⁻⁷ L_☉ for the undetected lines at 4076 Å, 6716 Å, and 6731 Å, respectively. Our [S ii] 4069 Å line is slightly stronger than observed, however, sulfur may be somewhat depleted compared to our assumed abundance (2.8 × 10⁻⁵).

Ne ii and Ne iii. [Ne ii] emission from TW Hya is strong and has been detected by Ratzka et al. (2007), Herczeg et al. (2007), Pascucci & Sterzik (2009), and Najita et al. (2010). There is some discrepancy between the measured values, with more flux observed by the Spitzer study. Najita et al. (2010) conclude that [Ne ii] emission arising from a more extended region (r ≳ 10 AU) explains their higher flux at all three observational epochs, which the narrow slits of the ground-based observations exclude (Pascucci & Sterzik 2009). A revised calibration (Pascucci et al. 2011) recovers the Spitzer flux and suggests that most of the [Ne ii] emission comes from within ∼10 AU. Earlier work by Herczeg et al. (2007) measured a line luminosity of 4.4 × 10⁻⁶ L_☉ with their ground-based study, similar to the Spitzer data. There is therefore some indication that the [Ne ii] line flux (or the underlying continuum) might be variable (Pascucci et al. 2011). Here we use the Spitzer IRS measurements of Najita et al. (2010) for the [Ne ii] and [Ne iii] emission, (3.9–4.7) × 10⁻⁶ L_☉ and 2 × 10⁻⁷ L_☉.

[Ne ii] emission from the disk models is found to be quite insensitive to the surface density distribution. This is because in X-ray heated regions, the [Ne ii] emission mainly depends only on the intercepted X-ray photon flux and the temperature of the emitting gas (HG09). A simple estimate of the expected [Ne ii] line luminosity can be made using the analytical expression derived in HG09,

$$\begin{equation} L_{[{\rm Ne\,\mathsc{ii}}]}^X \sim 10^{-6} \;\;T^{0.26} e^{-1120/T} \left({f}\over {0.7}\right) \left({\phi _X}\over {10^{39}\,{\rm s}^{-1}}\right) {L}_{\odot }, \end{equation} \tag{ 2 }$$

where ϕ_X ∼ 10³⁹ s⁻¹ is the 1 keV X-ray photon luminosity. In the above, we also assume that the disk intercepts a fraction f ∼ 0.7 of the stellar X-ray flux (our grid in the numerical work typically extends to z_max = r). A typical gas temperature in the [Ne ii] emitting layer is T ∼ 2000 K. The expected [Ne ii] luminosity for the TW Hya disk is therefore 4 × 10⁻⁶ L_☉.

From our full, detailed numerical calculations, we obtain total [Ne ii] and [Ne iii] line luminosities for the SML disk as 3 × 10⁻⁶ L_☉ and 3 × 10⁻⁷ L_☉, respectively. Approximately 25% of the [Ne ii] comes from gas in the inner disk (see Figure 7). The mid-disk region (3.5 AU < r < 20 AU) contributes the remaining 75% to the [Ne ii] luminosity in the SML disk. The [Ne ii] arises from the X-ray heated and partially ionized surface of the disk where gas temperatures range from 1000 to 4000 K. [Ne ii] emission from the SML disk is restricted radially to r ≲ 10 AU (Figure 7) and peaks strongly at the vertically extended 4 AU rim. The X-ray-produced [Ne ii] is somewhat lower than the observed line flux, which could be explained either by the presence of an EUV component and/or by variability of the stellar high-energy photon flux. We note that the accretion rate of TW Hya is observed to be variable on timescales similar to that for the [Ne ii] line flux (Eisner et al. 2010), ∼a few years. As the X-ray and any [Ne ii] producing EUV radiation fields may be a result of accretion-induced stellar activity (Robrade & Schmitt 2006; HG09), this may naturally explain the variability of the [Ne ii] line.

Pascucci & Sterzik (2009) resolve [Ne ii] in their spectra, and the observed linewidths and blueshifted emission are consistent with a photoevaporative flow from the star. The line profile is very well reproduced by EUV (hν > 13.6 eV) photon-driven photoevaporation models (Alexander 2008). However, we do not include EUV in our source spectrum for TW Hya and can recover most of the observed flux from X-ray heated gas alone. The remaining ∼25% of the observed Ne ii flux may be due to the EUV layer. We can thus estimate the EUV photon luminosity as being ≲ 3 × 10⁴⁰ s⁻¹. The [Ne ii]/[Ne iii] line ratio may provide some measure of the origin of the emitting gas (the completely ionized EUV-heated layer or the partially ionized X-ray-heated layer below the EUV-heated layer; 10,000 K in the EUV case and ∼1000–3000 K for X-rays), but is not definitive because of the uncertain nature of the EUV spectrum (HG09). However, the most natural explanation is an origin in the X-ray heated gas, where rapid charge exchange of Ne⁺⁺ with H atoms quenches the [Ne iii] line and produces the observed [Ne ii]/[Ne iii] ratios. (For a more detailed discussion, see HG09.) The observed blueshift may also arise for an X-ray induced photoevaporative flow (Gorti et al. 2009; Owen et al. 2010), and the lack of need for significant EUV to reproduce the [Ne ii] suggests that any photoevaporative flows may originate in primarily neutral, X-ray heated gas.

The far-infrared and submillimeter emission lines discussed hereon originate only in the mid and outer-disk regions and are the same for the two disk models, SML and SMH. The line luminosities are listed in Table 3.

[O i ] and [C ii] fine structure lines. We obtain a [O i] 63 μm line emission of 6 × 10⁻⁶ L_☉, a factor of ∼2 higher than the Herschel PACS measurement of Thi et al. (2010). Our calculated [O i] 145 μm line luminosity is 3 × 10⁻⁷ L_☉, and consistent with the upper limit of 4 × 10⁻⁷ L_☉ obtained by Thi et al. (2010). Emission is extended and produced by ∼50–100 K gas at r ∼ 30–120 AU.

Our model overproduces [O i] 63 μm emission, and we suggest that inclusion of water freezing on grains might alleviate this problem. Freezing of water ice on grains colder than 100 K removes gas phase elemental oxygen from the outer disk where dust grains are cold. Photodesorption would release some of this ice and at A_V < 3, all water ice is photodesorbed from grains (Hollenbach et al. 2009). Including these processes reduces the [O i] 63 μm emission from the disk compared to a disk with no ice formation, to 4 × 10⁻⁶ L_☉, in better agreement with the PACS observations. The [O i] 145 μm line is reduced from 3 × 10⁻⁶ L_☉ to 2 × 10⁻⁶ L_☉. A more detailed treatment of freeze-out of molecules on dust grains and desorption processes will be a subject of future work. Another possibility is that the PAH abundance in the outer disk is even lower than we assume, with a depletion factor of nearly ∼1000. This lowers the temperature in the outer disk sufficiently and decreases the [O i] 63 μm line luminosity to 4 × 10⁻⁶ L_☉. If freezing were also included, this would bring the O i emission even closer to the observed value. However, we note that decreasing the PAH abundance everywhere in the disk alters the vertical structure of the mid-disk region (r < 20 AU) sufficiently to deteriorate fits to observation of other species such as H₂, OH, and O i forbidden lines. There may, in fact, be a radial gradient in the PAH abundance in the disk, with a higher degree of depletion in the outer disk, if PAHs were to enter the disk frozen out on dust grains as suggested by Geers et al. (2009). Another possibility is that the PAH abundance is low everywhere in the disk, but that other sources of heating such as mechanical viscous dissipation are effective at the surface in the disk(r ≲ 30 AU, e.g., Glassgold et al. 2004).

Thi et al. (2010) modeled [O i] emission from TW Hya and concluded that the observed flux together with their ¹³CO observations indicate a low gas disk mass by factors of 10–100 compared with that expected from the disk dust mass and assuming a gas/dust ratio of 100. However, we find that lowering the gas disk mass by 10 and retaining the dust mass only marginally improves model fits to the observed [O i] emission, and worsens the other line emission fits considerably. Most of these discrepancies can be attributed to the increased grain absorption cross section per H in a disk with a low gas/dust ratio. H₂ emission increases (FUV heating by PAHs becomes significant) by a factor of ∼2–3, OH prompt emission is reduced due to the increased (relative to gas) absorption of Lyα photons by dust, and there is excess emission from CO pure rotational lines. The HCO⁺4–3 line requires high densities for emission, and this line luminosity is almost a factor of 10 lower for the low mass disk than the observed value. Increasing the PAH abundance by 10 as in the preferred model of Thi et al. would raise gas temperatures even higher and further increase H₂ and CO emission. Although we do not concur with the Thi et al. (2010) results, we note that our calculated model line fluxes are in good agreement with results from other models they consider. Our model is most similar to their low PAH abundance models (their series 2 and 3) and if they were to base their conclusions on these alternate models, a disk mass close to ∼0.06 M_☉ is inferred from their results. For these series, they also seem to find that the O i line luminosity is quite insensitive to disk mass in the range we consider. Their conclusion of a low gas disk mass appears to be largely based on the ratio of the ¹²CO/¹³CO line fluxes. In order to accurately interpret this measurement, a more accurate disk chemical model which includes ¹³CO as a species is necessary. Our model does not include ¹³CO. We also note that Thi et al. assume a constant ¹³CO to ¹²CO ratio to calculate the ¹³CO flux and do not include a proper chemical model of ¹³CO as well. Moreover, the ¹³CO/¹²CO line ratio will be significantly affected by processes such as selective photodissociation (Lyons & Young 2005) and freeze-out processes, which are not included in our disk models or those of Thi et al. (2010). It remains to be seen whether a low gas disk mass is needed to reproduce the observed CO isotope flux ratio.

The calculated [C ii] 158 μm line luminosity is 3 × 10⁻⁷ L_☉ and is consistent with the upper limit of 4.5 × 10⁻⁷ L_☉ set by the Herschel observations.

CO rotational lines. The observed line luminosities of the CO 6–5, 3–2, and 2–1 lines are 5.0 × 10⁻⁸, 2.5 × 10⁻⁸, and 7.9 × 10⁻⁹ L_☉, respectively, whereas we obtain 8 × 10⁻⁸, 2.4 × 10⁻⁸, and 7.9 × 10⁻⁹ L_☉ for these lines. Purely thermal line broadening in the outer disk provides better correspondence between data and model results. This is in fair agreement with the results of Qi et al. (2006) who assume only a small amount of turbulence (ΔV ∼ 0.3 km s⁻¹) in their analysis and with the more recent results of Hughes et al. (2011) who find that the turbulent linewidths are low. A power-law surface density profile also matches the data, but has to be arbitrarily truncated at ∼70 AU (GH08). We consider it more realistic to assume that the surface density declines more smoothly with radius, as in a disk evolving viscously and subject to photoevaporation (also see Hughes et al. 2008; Isella et al. 2009).

CO rotational line emission constrains the distribution of gas in the outer disk. We do not model disk masses higher than 0.06M_☉, because we expect that appreciably higher disk masses will make the disk gravitationally unstable. Lower disk masses by a factor of 10 (M_disk = 0.006 M_☉) do not fit the CO rotational line emission well. We find that although it is possible to match the observed CO 2–1 and 3–2 line emission by altering the surface density profile in the outer disk for the lower mass disk, the CO 6–5 line emission is higher by a factor of ∼4 than the SML value, and hence discrepant with data. We expect freezing will not affect the emission line luminosities calculated. A simple estimate made by discounting contributions from spatial locations where the dust temperature is ≲ 20 K only slightly lowers the calculated line luminosities by ∼15%. Photodesorption may also keep the CO molecules from freezing on dust grains in the surface regions of the outer disk. We note that we do not need severe radial depletion of CO to explain the observed line ratios (e.g., Qi et al. 2006), although some degree of depletion (less than a factor of three) in the mid-disk region may help lower the CO 6–5 line luminosity from the model disk.

HCO⁺ 4–3 line. van Zadelhoff et al. (2001) measured the line intensity of the HCO⁺ 4–3 transition from TW Hya using the James Clerk Maxwell Telescope (JCMT), and the line luminosity is 7.7 × 10⁻⁹ L_☉. Our calculation yields 4 × 10⁻⁹ L_☉, nearly a factor of two lower than observed. We could obtain a better match by changing the surface density distribution in the outer disk to increase the strength of the HCO⁺4–3 line, but this then also changes the CO rotational line intensities and makes that agreement poorer. Further, increasing the outer disk surface density also results in a high overall disk mass, bringing the disk closer to being gravitationally unstable. The HCO⁺ line mainly originates in relatively warm (T ∼ 20–100 K) and high density gas (n ≳ 10^7–8 cm⁻³) at r ≲ 50 AU. The abundances of HCO⁺ calculated by the model in the emitting layers of the disk are a factor of 20–70 lower than the typical dark cloud abundance of 5 × 10⁻⁹ adopted by van Zadelhoff et al. (2001) in their analysis, and thus consistent with their derived "depletion" factors of 10–100.

We conclude this section with the caveat that our models may not be unique solutions to the gas surface density distribution in the disk. There are theoretical uncertainties that are inherent in chemical rates and the microphysics, as well as several important processes (such as freezing of ices) have been either simplified or ignored. The complicated and computationally intensive modeling procedure also makes a full exploration of parameter space impossible at this time. Nevertheless, we find it promising that theoretical models with reasonable parameters can reproduce observed emission ranging from the optical to the submillimeter wavelengths with an accuracy to within a factor ≲2.