DIAGNOSTIC LINE EMISSION FROM EXTREME ULTRAVIOLET AND X-RAY-ILLUMINATED DISKS AND SHOCKS AROUND LOW-MASS STARS

Consider an axisymmetric disk described by cylindrical coordinates r, z. If f_EUV is the fraction of ionized photons from the star absorbed by the disk, then the Strömgren condition is

$$\begin{equation} f_{{\rm EUV}}\Phi _{\rm EUV} = 2 \alpha _{r,H} \int _{z_{IF}}^{\infty } dz \int _{r_i}^{r_o} 2 \pi r n_{\rm e}^2 dr, \end{equation} \tag{ 5 }$$

where the electron density n_e is a function of r and z but is negligible below z_IF, the ionization front, and where we have ignored dust attenuation in the H ii surface region above z_IF. We will justify the neglect of dust post facto below, as well as show that f_EUV ∼ 0.7. In Equation (5) r_i and r_o are the inner and outer radii of the disk, and α_r,H is the case B recombination coefficient for electrons with protons (α_r,H = 2.53  ×   10⁻¹³ cm³ s⁻¹ at T = 10⁴ K; Storey & Hummer 1995).

For simplicity, we treat the specific example of a simple two-level fine structure system such as [Ne ii] 12.8 μm. Then, for a transition t the escaping line luminosity L_t from the disk is given as

$$\begin{equation} L_t \simeq \gamma _t \Delta E_t \int _{z_{IF}}^{\infty } dz \int _{r_{\rm i}}^{r_{\rm o}} 2 \pi r \left({{n_{\rm e} n(i)} \over {1+{n_{\rm e} \over n_{{\rm ecr},t}}}}\right) dr, \end{equation} \tag{ 6 }$$

where γ_t is the collisional excitation rate coefficient of the transition, ΔE_t is the photon energy, n_ecr,t is the critical electron density for the transition, and n(i) is the density of the ionized species i that produces the transition. Note that here we have included the fact that half the emitted IR photons are directed toward the disk midplane, where they are absorbed by the (assumed) optically thick disk and half escape. If we set n(i) = x_sf(i)n_e, where x_s is the abundance of species s (in all ionization states) in the EUV layer, and f(i) is the fraction of that species in ionization state i, then we can write using Equation (5)

$$\begin{equation} L_t= \gamma _t \Delta E_t \left({{f_{\rm EUV} \Phi _{\rm EUV} x_s f(i)}\over {2 \alpha _{r,H}}}\right) \left({1 \over {1+ {n_{\rm e} \over n_{{\rm ecr},t}}}}\right). \end{equation} \tag{ 7 }$$

We note that in taking x_sf(i) out of the integral in Equation (6) we implicitly assume that in Equation (7) x_sf(i) is the density-weighted average of this product in the EUV layer. Thus, we find that if n_e < n_ecr,t and if f(i) does not depend on Φ_EUV (e.g., for the dominant ionization state where f(i) ≃ 1), then the line luminosity L_t is directly proportional to the ionizing photon luminosity Φ_EUV of the central star. If EUV is the sole excitation source, the measurement of L_t directly measures the uncertain parameter Φ_EUV if we have knowledge of x_s and f(i). The main unknown is the fraction f(i) in a particular ionization state i, since the gas phase abundance of an element can often be estimated from observations of H ii regions. For example, neon is often found as Ne⁺ and Ne⁺⁺. We therefore rewrite Equation (7) as

$$\begin{equation} L_t= C_t f(i) f_{\rm EUV} \Phi _{\rm EUV} \left({1 \over {1+ {n_{\rm e} \over n_{{\rm ecr},t}}}}\right), \end{equation} \tag{ 8 }$$

where C_t ≡ γ_tΔE_tx_s/(2α_r,H). Note that C_t and n_ecr,t are known quantities and therefore constants in the equation. If the transition is from the dominant ionization state of the species, then f(i) ∼ 1. Our modeling of disks suggests f_EUV ∼ 0.7. C_t is therefore the main constant of proportionality that links the observed IR line luminosity to the EUV photon luminosity; for n_e < n_ecr,t and taking the dominant ionization state, C_t is the energy per absorbed EUV photon that emerges from the disk in the fine structure transition. Table 1 lists the value of C_t for the various transitions considered in this paper, along with n_ecr,t and the assumed x_s that is used in C_t.

The above equations show that it is important to estimate the electron density in the ionized surface of the disk at the radius where most of the emission in a given line is produced. We follow the results of Hollenbach et al. (1994). Because the H ii region surface is isothermal, the electron density at a given r decreases from z_IF upward as

$$\begin{equation} n_{\rm e}(r,z)=n_{\rm e}(r, z_{{\rm IF}}) e^{-\big({{z-z_{\rm IF}}\over {2H}}\big)^2}, \end{equation} \tag{ 9 }$$

where H is the isothermal scale height of the 10⁴ K gas given by

$$\begin{equation} H(r) = r_{\rm g}\left({r \over r_{\rm g}}\right)^{3/2} \end{equation} \tag{ 10 }$$

for r < r_g, and where r_g is given by

$$\begin{equation} r_{\rm g} \simeq 7 \left({M_* \over {1\ {M}_\odot }}\right) \ {\rm AU}. \end{equation} \tag{ 11 }$$

Note that in Equation (9), both z_IF and H are functions of r. The gravitational radius r_g is where the sound or hydrogen thermal speed is equal to the escape speed from the gravitational potential of the star. For r>r_g, where the gas is freely evaporating, the effective height of the disk is H ∼ r, since the initially vertically flowing gas turns over on radial streamlines by the time z = r. Hollenbach et al. also showed that the base electron density at z_IF falls off radially as a power law:

$$\begin{eqnarray} n_{\rm e}(r, z_{\rm IF}) &\simeq& 5 \times 10^4 \left({{\Phi _{\rm EUV}}\over {10^{41} \ {\rm s}^{-1}}}\right) ^{1/2}\left({r_{\rm g} \over r}\right)^{3/2}\nonumber\\ &&\times \left({{ 1 \ {M}_\odot }\over {M_*}}\right)^{3/2} \ {\rm cm}^{-3} \ \ {\rm for} \ r < r_{\rm g} \end{eqnarray} \tag{ 12 }$$

and

$$\begin{equation} n_{\rm e}(r, z_{\rm IF}) \simeq n_{\rm e}(r_{\rm g}, z_{\rm IF}) \left({r_{\rm g}\over r}\right)^{5/2} \ \ {\rm for} \ r > r_{\rm g}. \end{equation} \tag{ 13 }$$

Note that if n_e(r, z_IF) < n_ecr everywhere, then the amount of luminosity L(r) from a logarithmic interval of r is proportional to the volume emission measure n²_eV at that r, or to n²_er²H. For r < r_g we then obtain L(r) ∝ r^1/2 whereas for r>r_g we obtain L(r) ∝ r⁻². In other words, the line luminosity originates mostly from r ∼ r_g. If the electron densities in the very inner regions exceed n_ecr but are less than n_ecr at r_g, the luminosity is relatively unaffected, and our conclusion does not change. However, if n_e(r_g, z_IF)>n_ecr, then the line luminosity will drop. The line emissivity will be suppressed by a factor of approximately n_e(r_g, z_IF)/n_ecr at r_g. However, now L(r) ∝ n_eV as long as n_e>n_ecr, so that L(r) ∝ r^1/2 beyond r_g until the density drops below the critical density, when it reverts to its former r⁻² dependence. The luminosity of a low critical density species will therefore originate from the radius

$$\begin{equation} r_{\rm max}= \left({{n_{\rm e}(r_{\rm g}, z_{\rm IF})}\over {n_{{\rm ecr}}}}\right)^{2/5} r_{\rm g}, \end{equation} \tag{ 14 }$$

where n_e(r_max, z_IF) = n_ecr. As a result,

$$\begin{equation} L \propto n_{\rm e}(r_{\rm max},z_{\rm IF}) r_{\rm max}^3 \propto \Phi _{\rm EUV}^{3/5} \end{equation} \tag{ 15 }$$

as long as n_e(r_g, z_IF)>n_ecr and f(i) is constant.

As examples, consider [Ne ii] 12.8 μm and [S iii] 19 μm. The critical density for [Ne ii] is given as n_{ecr,[Ne ii]} ≡ A₂₁/γ₂₁ ≃ 6 × 10⁵ cm⁻³, where we have taken A₂₁ = 0.00859 s⁻¹ and a collisional de-excitation rate coefficient at 10⁴ K of γ₂₁ = 1.355 × 10⁻⁸ cm³ s⁻¹ (Griffin et al. 2001). Thus, comparing this critical density with the electron density at r_g (Equation (12)), we see that [Ne ii] is subthermal at r_g typically and that therefore it mostly originates from r_g and tracks Φ_EUV linearly as long as f(Ne⁺) is constant. However, [S iii] has a critical density of approximately 5 × 10³ cm⁻³ which is about 10 times less than n_e(r_g, z_IF) for a solar mass star with Φ_EUV ∼ 10⁴¹ s⁻¹ (see Equation (12)). Therefore, r_max ≃ 10^2/5r_g ≃ 2.5r_g. The [S iii] line luminosity is down by a factor of 10^4/5 ∼ 6.3 from the value it would have had if it had been subthermal at r_g, rather than the factor of 10 drop at r_g, because most of the emission comes from further out where there is more volume. The luminosity in the line will not linearly track Φ_EUV because of the significant (and variable with Φ_EUV) collisional de-excitation of the transition. In fact, as long as n_e(r_g, z_IF)>n_ecr and f(S⁺⁺)is not dependent onΦ_EUV, the luminosity in the line scales as Φ^3/5_EUV as shown in Equation (15). However, we find in our numerical analysis that f(S⁺⁺) does depend significantly on Φ_EUV, and the Φ^3/5_EUV dependence is not seen (as we show below in our numerical models).

We have shown above that the fine structure emission from the H ii surface region arises from ∼r_g as long as n_e(r_g, z_IF) < n_ecr. However, this conclusion and the important analytic derivation of the line luminosities (Equation (8)) both require that the surface H ii region be ionization-bounded (i.e., there be a neutral layer underneath the completely ionized surface). The minimum mass of gas required to fully absorb the incident EUV photons and create an ionized surface region with a neutral midplane region for r < r_g is given as

$$\begin{equation} M_{{\rm H}\,\hbox{\scriptsize {\sc ii}}}(\mbox{min}) = \int _{z_{\rm IF}}^\infty 2dz \int _0^{r_{\rm g}} m_{\rm H}n_{\rm e}(r,z) 2\pi rdr, \end{equation} \tag{ 16 }$$

where m_H is the mass of the ionized gas per electron (∼2 × 10⁻²⁴ g). The z integral can be approximated as $2H(r)\equiv 2r_{\rm g}\big({r \over r_{\rm g}}\big)^{3/2}$ with the electron density fixed at the density at the ionization front n_e(r, z_IF). The density n_e(r, z_IF) falls as r^−3/2 for r < r_g. Therefore, for r < r_g the mass is mostly at r_g. Performing the integral, we obtain

$$\begin{eqnarray} M_{{\rm H}\,\hbox{\scriptsize {\sc ii}}}(\mbox{min})&\simeq& 2 \pi r_{\rm g}^3 m_{\rm H} n_{\rm e}(r_{\rm g}, z_{\rm IF}) \simeq 10^{-10}\left({\Phi _{\rm EUV} \over {10^{41} \ {\rm s^{-1}}}}\right)^{1/2}\nonumber\\ &&\times \left({{M_*} \over {1\ {M}_\odot }}\right)^{3/2} \ {M}_\odot. \end{eqnarray} \tag{ 17 }$$

Thus, assuming n_e(r_g) < n_ecr,t, an extremely small gas mass, of order 10⁻⁷ Jupiter masses, inside of r_g will provide the luminosities given by Equation (8), using the values of C_t given in Table 1. These lines, then, are very sensitive diagnostics of the presence of trace amounts of gas at radii of order 1–10 AU in disks. If there is less mass than M_H ii(min) at r_g, then the resulting luminosities will be reduced by a factor M_H ii/M_H ii(min).

We now address whether dust extinction is important in the H ii surface region. Since H(r) ∝ r^3/2, the 10⁴ K surface of the disk is flared. Most of the emission comes from r_g or beyond as shown above. Using Equations (11) and (12), the attenuating column at r_g is roughly

$$\begin{eqnarray} N_{\rm att} &\sim& n_{\rm e}(r_{\rm g}, z_{\rm IF}) \times r_{\rm g} \sim 5 \times 10^{18} \left({{\Phi _{\rm EUV}} \over {10^{41} \ {\rm s}^{-1}}}\right) ^{1/2}\nonumber\\ &&\times \left({{ 1 \ {M}_\odot }\over {M_*}}\right)^{1/2} \ {\rm cm}^{-2}. \end{eqnarray} \tag{ 18 }$$

The dust at the surface of the disk is expected to have less opacity than interstellar dust, so that the column for EUV optical depth unity is greater than 10²¹ cm⁻². The above equation shows that dust will not be important until Φ_EUV>10⁴⁶ s⁻¹ for a 1 M_☉ star. Low-mass stars do not produce such high EUV luminosities. Therefore, our neglect of dust opacity in the H ii surface is justified in disks around low-mass stars.

Finally, we estimate the fraction f_EUV of the ionizing photons emitted by the star that are absorbed by the disk. Most of the absorption (and consequent emission, as shown above) comes from r ∼ r_g or beyond, and here H ∼ r. As we will show below in our numerical disk models, the underlying neutral disk also has considerable height. At r = 10 AU, z_IF ≃ 7.5 AU (see Section 4.2). Thus, the disk is opaque to EUV photons from the midplane to an angle from midplane of about 40°. In addition, the recombining hydrogen in the ionized gas above 40° also absorbs EUV photons. Therefore, we estimate that the fraction f_EUV of EUV photons absorbed by the disk is about 0.7. A detailed hydrodynamical study is needed to more accurately determine this fraction.

As our prime examples for this analytic analysis, consider specifically the case of the [Ne ii] 12.8 μm, [Ne iii] 15 μm, and [Ar ii] 7 μm luminosity emerging from a young disk. We choose these lines because [Ne ii] and [Ne iii] have been observed, and [Ar ii] is predicted to be the brightest of the unobserved lines (see Table 1). In addition, they all have high critical densities so that n_e(r_g, z_IF) < n_ecr for these lines as long as Φ_EUV < 10⁴²–10⁴³ s⁻¹. Assuming that this condition is met and substituting f_EUV = 0.7 and the atomic constants into Equation (8), we obtain

$$\begin{equation} L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} \simeq 3.8 \times 10^{-6} f({\rm Ne}^{+})\left({{\Phi _{\rm EUV}}\over {10^{41} \ {\rm s} ^{-1}}}\right) \ {L}_\odot. \end{equation} \tag{ 19 }$$

$$\begin{equation} L_{[{\rm Ne}\,\hbox{\scriptsize {\sc iii}}]} \simeq 6.4 \times 10^{-6} f({\rm Ne}^{++})\left({{\Phi _{\rm EUV}}\over {10^{41} \ {\rm s} ^{-1}}}\right) \ {L}_\odot. \end{equation} \tag{ 20 }$$

$$\begin{equation} L_{[{\rm Ar}\,\hbox{\scriptsize {\sc ii}}]} \simeq 3.2 \times 10^{-6} f({\rm Ar}^+)\left({{\Phi _{\rm EUV}}\over {10^{41} \ {\rm s} ^{-1}}}\right) \ {L}_\odot. \end{equation} \tag{ 21 }$$

Recall that f(Ne⁺) is the fraction of neon in the singly ionized state in the region near r_g which produces most of the emission. Luminosities greater than about 10⁻⁷ L_☉ are detectable from nearby (<100 pc) sources by the Spitzer Space Telescope, as long as the line-to-continuum ratio is sufficiently large to enable detection of the line above the bright continuum.

The effect on IR luminosity caused by holes in disks. The above analytic results apply for a disk that extends inward to r ≲ r_g∼ 7 AU from the star. However, disks have been observed with inner holes, devoid of dust, that extend to r_i>r_g (e.g., Najita et al. 2007; Salyk et al. 2009). Regardless of the cause of these holes, if gas is absent inside of r_i and r_i>r_g, the disk vigorously photoevaporates at r_i, a process which evaporates the disk from inside out (Alexander et al. 2006a, 2006b). Alexander et al. showed that the flux of EUV photons striking the inner wall of the disk creates a thin (thickness ≃H, the scale height of the neutral disk at r_i) ionized layer. The Strömgren condition gives the electron density in the layer:

$$\begin{equation} F_{\rm EUV} \equiv {\Phi _{\rm EUV} \over {4 \pi r_{\rm i}^2}} = \alpha _{r,H} n_{\rm e}^2 H \end{equation} \tag{ 22 }$$

Assuming H ≃ 0.1r_i (Alexander et al. 2006a, 2006b), we obtain

$$\begin{equation} n_{\rm e}(r_{\rm i}) \simeq 2.5 \times 10^5 \left({\Phi _{\rm EUV} \over {10^{41} \ {\rm s^{-1}}}}\right)^{1/2} \left({10\ {\rm AU} \over r_{\rm i}}\right)^{3/2} \ {\rm cm}^{-3}. \end{equation} \tag{ 23 }$$

Note that the electron density decreases as the inner hole size increases. If r_i ≫ r_g, this leads to an increase in the luminosity of low critical density lines with respect to high critical density lines (see Equation (8)). However, for lines whose critical densities are larger than the electron density at r_g, the presence of a hole of size r_i ≫ r_g does not appreciably affect the IR line luminosity. Essentially, this arises because the IR line luminosity is proportional to the number of EUV photons absorbed, f_EUVΦ_EUV, and this remains constant, with f_EUV ∼ 0.7, regardless of r_i. Therefore, the IR line luminosity tracks Φ_EUV as presented in Equation (8).

The IR hydrogen recombination lines can be analytically determined by noting that the luminosity in a given line produced by the transition n_u to n_l is given as

$$\begin{equation} L_{ul} = \alpha _{ul} \Delta E _{ul} \int ^{\infty } _{z_{\rm IF}} dz \int ^{r_{\rm o}} _{r_{\rm i}} 2 \pi r n_{\rm e}^2 dr, \end{equation} \tag{ 24 }$$

where α_ul is the rate coefficient for recombinations through the n_u–n_l transition and ΔE_ul is the energy of the photon produced in this transition. Clearly, the hydrogen recombination lines also count EUV photons (see Equation (5)) and could be used to measure Φ_EUV. However, hydrogen recombination produces weak IR lines compared to the fine structure lines such as [Ne ii] if the electron density n_e is less than the critical density n_ecr of the fine structure transition, as can be seen by taking the ratio of the predicted line luminosities:

$$\begin{equation} {{L_{ul}}\over {L({\rm Ne}\,\hbox{\sc ii})}} = {{ \alpha _{ul} \Delta E _{ul}} \over { \gamma _{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} \Delta E_{{\rm Ne}\,\hbox{\scriptsize {\sc ii}}} x_{\rm Ne} f({\rm Ne}^+)}}. \end{equation} \tag{ 25 }$$

The hydrogen recombination lines we are most interested in are the 7–6 (Humphreys α) and 6–5 (Pfund α) at wavelengths of 12.37 and 7.46 μm, respectively. These two lines have been reported observed toward stars with disks (Pascucci et al. 2007; Ratzka et al. 2007). Substituting the atomic constants for these transitions, we obtain predicted ratios for the EUV-induced surface H ii layer:

$$\begin{equation} {L_{7\hbox{--}6}\over L({\rm Ne}\,\hbox{\sc ii})} = 0.008 \end{equation} \tag{ 26 }$$

and

$$\begin{equation} {L_{6\hbox{--}5}\over L({\rm Ne}\,\hbox{\sc ii})} = 0.02. \end{equation} \tag{ 27 }$$

The observed ratios are close to unity! The predicted ratios are small because of the low ratio of the radiative recombination rate α_ul of hydrogen to the electronic collisional excitation rate of [Ne ii] γ_[Ne ii]. Thus, we predict that these IR hydrogen recombination lines must originate from another source if [Ne ii] originates from the H ii region surface of the disk. One place which would provide copious recombination line emission without producing even more [Ne ii] emission would be very high electron density regions, $n_{\rm e} \gg n_{{\rm ecr}, [{\rm Ne}\,\hbox{\scriptsize {{\sc ii}}}]}$. In these regions, [Ne ii] is suppressed relative to the hydrogen recombination lines due to the collisional de-excitation of the upper level of the [Ne ii] transition. Therefore, these observed recombination lines are possibly produced in very dense plasma very close to the star, in the stellar chromosphere, the accretion shock, or an internal wind shock if it is both high speed (v_s ≳ 100 km s⁻¹ so that it produces ionized hydrogen) and occurs so close to the wind origin (≲1 AU) that the postshock density is high enough to suppress [Ne ii] relative to the recombination lines. In any of these cases, the prediction is that the H recombination lines will be much broader (≳100 km s⁻¹) than the [Ne ii] lines (∼20 km s⁻¹) in face-on disks.

Glassgold et al. (2007) and Meijerink et al. (2008) have presented models of the [Ne ii] 12.8 μm emission and emission from other lines, such as [Ne iii], [O i], [S iii], and [S iv], produced in the X-ray-heated layer that lies just below the ionization front created by EUV photons. This layer is predominately neutral, x_e ∼ 0.001–0.1, depending on r and z, but with a higher ratio of Ne⁺/Ne. Typically, the [Ne ii] emitting layer has T ∼ 1000–4000 K. In Section 4, we also present numerical results from our models of the X-ray-induced fine structure emission, and in Section 5 we discuss differences between our models and those of these authors. Here, we present a simple analytic estimate of the strengths of the fine structure transitions in X-ray-illuminated regions. These estimates are more approximate than those presented above for the EUV-dominated regions because of the uncertainties in estimating the gas temperature in this mostly neutral gas illuminated by a spectrum of X-ray photons. Nevertheless, they provide insight into the X-ray process and into the relative strengths of X-ray-induced fine structure emission in the X-ray layer as opposed to that produced by EUV photons in the surface EUV layer.

The simplest derivation arises if we assume our "hard" X-ray spectrum, dominated by 1–2 keV photons. These photons are sufficiently energetic to ionize the K shell of Ne, and the ionization of Ne is dominated by direct X-ray photoabsorption, and not by collisions with secondary electrons. If we make the assumption in the X-ray layer that the atomic Ne absorbs a fraction f^X_Ne of all ∼1 keV X-rays and that Ne⁺ radiatively recombines with electrons, and we assume that n_e < n_{ecr,[Ne ii]}, we obtain in a manner completely analogous to the EUV layer's Equations (5)–(7)

$$\begin{equation} L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^X = {{\gamma _{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^X \Delta E_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} f_{\rm Ne}^X f_X \Phi _X} \over {2\alpha _{r,{\rm Ne}}}}. \end{equation} \tag{ 28 }$$

Here, γ^X_[Ne ii] is the collisional rate coefficient for [Ne ii] by electrons in the X-ray layer (i.e., it is only different from γ_[Ne ii] in Equation (7) because the X-ray layer is cooler than the EUV layer), Φ_X is the ∼1 keV X-ray photon luminosity of the central source, f_X is the fraction of X-rays absorbed by the disk in the X-ray layer, and α_r,Ne is the recombination rate coefficient of Ne⁺ with electrons in the X-ray layer. The fraction of ∼1 keV photons absorbed by neon, f^X_Ne, is approximately the neon cross section at 1 keV divided by the total X-ray absorption cross section at 1 keV; using Wilms et al. (2000), we obtain f^X_Ne ≃ 0.21. We see that L^X_[Ne ii] scales linearly with Φ_X, just as L_[Ne ii] scales linearly with Φ_EUV in the EUV layer. The ratio of the [Ne ii] luminosity from the EUV layer to that in the X-ray layer is then given as

$$\begin{equation} {L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} \over L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^X} = \left({\gamma _{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}\over \gamma _{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^X}\right) \left({f_{\rm EUV} \over {f_{\rm Ne}^X f_X}}\right) \left({\alpha _{r,Ne} \over \alpha _{r,H}}\right) \left({\Phi _{\rm EUV} \over \Phi _X}\right) x_{\rm Ne}f({\rm Ne}^+), \end{equation} \tag{ 29 }$$

where f(Ne⁺) is the fraction of neon that is singly ionized in the EUV layer. We take T ∼ 10⁴ K for the EUV layer and T_X ∼ 2000 K for the X-ray layer to estimate the recombination coefficients, and assuming that f_X is approximately the height of the layer which becomes optically thin to 1 keV X-rays from the star (roughly N ∼ 10²¹ cm⁻², or a column of about 10²² to the star) divided by r or f_X ∼ 0.25 (see Figure 5). The EUV layer is more flared; hence f_EUV ∼ 0.7. In the EUV layer, f(Ne⁺) ≃ 1. Substituting into Equation (29), we obtain

$$\begin{equation} {L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} \over L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^X} \simeq 2 \times 10^{-3} e^{{1100\ {\rm K} \over T_X}} \left({\Phi _{\rm EUV} \over \Phi _X}\right). \end{equation} \tag{ 30 }$$

The 1 keV X-ray photon luminosity Φ_X from a typical source is of order 10³⁹ photons s⁻¹. The EUV luminosity Φ_EUV is generally of order 10⁴¹ photons s⁻¹. Therefore, assuming that the 1 keV X-rays are absorbed in regions with T_X>1000 K, the [Ne ii] luminosity is expected to be marginally dominated by emission from the X-ray layer as opposed to the EUV layer. We show below in our numerical work that L_[Ne ii]/L^X_[Ne ii] ∼ 0.6 when Φ_EUV = 10⁴¹ s⁻¹ and Φ_X ≃ 10³⁹ s⁻¹ and when the EUV spectrum is such to produce more Ne⁺ than Ne⁺⁺ in the EUV layer; this result agrees with Equation (30). Note that Φ_EUV = 10⁴¹ s⁻¹ and Φ_X ≃ 10³⁹ s⁻¹ corresponds to L_EUV ≃ L_X. In other words, if the central star emits the same EUV and X-ray luminosity, and the EUV has a soft spectrum which produces more [Ne ii] than [Ne iii], there will be roughly 2 times more [Ne ii] luminosity arising from the X-ray layer than from the EUV layer. As we will show in Section 4, where we present results from our detailed numerical models, this conclusion that X-rays are more efficient at producing [Ne ii] emission does not depend strongly on the X-ray spectrum for reasonable choices of the spectrum. If we adopt a softer spectrum, the ionization of Ne is dominated by secondary electrons because most of the X-rays are absorbed by He, C, or O. The gas is also hotter because there is more heating per unit volume due to the higher cross sections for softer X-rays. The net effect is that the [Ne ii] luminosity does not change much for fixed X-ray luminosity even as we vary the X-ray spectrum. In the case of the "hard" X-ray spectrum, the reason X-rays are somewhat more dominant than the EUV is because for high temperature (T>1000 K) gas such as exist in both the H ii region and the X-ray-heated region, the luminosity in the line depends mainly on the number of Ne⁺ ions times the electron density. In the H ii region, the vast number of EUV photons are used ionizing H, an extremely small fraction of the photons are used ionizing Ne, and therefore the number of Ne⁺ ions times the electron density is a small (the x_Nef(Ne⁺) factor in Equation (29); however, we explicitly include the relatively large fraction, f^X_Ne = 0.21 of 1 keV X-ray photons that directly ionize Ne and lead to a large product of electron density times Ne⁺ ions in Equation (29). Therefore, no x_Nef(Ne⁺) term appears in the denominator. Another way of understanding this result is that although the EUV layer is completely ionized with f(Ne⁺) ∼ 1 and x_e ∼ 1, the EUV layer has much less column of Ne⁺ because H and He rapidly absorb the EUV photons; the penetrating X-rays partially ionize a much larger column.

Lahuis et al. (2007) and van Boekel et al. (2009) discussed the possible origin of [Ne ii] emission from shocks generated by protostellar outflows. In order for a shock to produce significant [Ne ii] emission, the shock must ionize most of the preshock gas in order to produce high quantities of Ne⁺ and electrons. The fraction of preshock gas that is ionized by the shock is a very sensitive function of the shock speed v_s (e.g., Hollenbach & McKee 1989, hereafter HM89). HM89 showed that v_s ≳ 100 km s⁻¹ is required to ionize most of the H and Ne and that the [Ne ii] emission rises very sharply with v_s and then plateaus above v_s ≳ 100 km s⁻¹. This suggests that any possible shock must originate from internal shocks in the protostellar wind (which has terminal speeds of ∼200 km s⁻¹) or from the protostellar wind overtaking much slower moving outflow material. It is unlikely to originate from the shock produced by the wind striking the disk, since this shock is so oblique that the (normal) shock speeds are typically ≲20 km s⁻¹ (Matsuyama et al. 2009). The total emission per unit area F_t from the postshock region of a radiative shock is given as

$$\begin{equation} F_t = {1\over 2} m_{\rm H} n_0 v_s^3, \end{equation} \tag{ 31 }$$

where m_H is the mass per hydrogen nucleus and n₀ is the hydrogen nucleus number density of the preshock gas. The numerical results of HM89 can be approximated for the emission per unit area of [Ne ii] for shocks with v_s ≳ 100 km s⁻¹:

$$\begin{equation} F_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]} \simeq \left({{5 \times 10^{-3}}\over {1 + {n_0 \over {10^4 \ {\rm cm}^{-3}}}}}\right) F_t, \end{equation} \tag{ 32 }$$

where the dependence on density at high density arises because of collisional de-excitation of the upper state of [Ne ii] in the postshock gas. Assume that a fraction f_sh of the protostellar wind shocks at speeds v_s ∼ 100 km s⁻¹ with preshock density n₀. It follows that the [Ne ii] shock luminosity is

$$\begin{eqnarray} L_{[{\rm Ne}\,\hbox{\scriptsize {\sc ii}}]}^{sh}&=& {1\over 2} f_{\rm sh} \dot{M}_{\rm w} v_s^2 \left({5\times 10^{-3} \over {1 + {n_0 \over {10^4 \ {\rm cm}^{-3}}}}}\right)\nonumber\\ &\simeq& \left({4\times 10^{-5} \over {1 + {n_0 \over {10^4 \ {\rm cm}^{-3}}}}}\right) f_{\rm sh} \dot{M}_{-8} \ {L}_\odot, \end{eqnarray} \tag{ 33 }$$

where $\dot{M}_{-8} = \dot{M}_{\rm w}/10^{-8}$ M_☉ yr⁻¹ and v_s ≳ 100 km s⁻¹. Therefore, if the protostellar wind mass loss rate $\dot{M}_{\rm w} \gtrsim 10^{-9}$ M_☉ yr⁻¹, v_s ≳ 100 km s⁻¹, f_sh ∼ 1, and n₀ ≲ 10⁴ cm⁻³, then the [Ne ii] luminosity produced in internal wind shocks may be comparable to or greater than the luminosity produced in the EUV or X-ray layer of the disk.

Van Boekel et al. (2009) argued this may be the case in T Tau S. Here, the observed [Ne ii] luminosity is L_[Ne ii] ∼ 10⁻³ L_☉. From the above, this would require, for example, f_sh ∼ 1, $\dot{M}_{\rm w} \sim 2.5 \times 10^{-7}$ M_☉ yr⁻¹, and n₀ ≲ 10⁴ cm⁻³. The preshock density can be estimated with knowledge of the distance of the shock from the star, r_sh, and $\dot{M}_{\rm w}$. The preshock density (the density in the wind at r_sh) is given as

$$\begin{equation} n_0 = 2.5 \times 10^3 \dot{M}_{-8}\ r_{15}^{-2} f_\Omega ^{-1} \ \ {\rm cm}^{-3}, \end{equation} \tag{ 34 }$$

where r₁₅ = r_sh/10¹⁵ cm and f_Ω is the fraction of 4π steradians into which the protostellar wind is collimated. Van Boekel et al. (2009) measured an extent of the emission from T Tau S of approximately 160 AU. If we assume r_sh = 160 AU and f_Ω = 1, presumably upper limits, we obtain a lower limit to n₀ ≳ 1.5 × 10⁴ cm⁻³. Note that for $\dot{M}_{\rm w} > 2.5 \times 10^{-7}$ M_☉ yr⁻¹ and for our specific assumptions on r_sh and f_Ω, the preshock density is so high that the [Ne ii] luminosity is independent of $\dot{M}_{\rm w}$; the luminosity from the shock saturates once the emitting [Ne ii] in the postshock gas is in local thermodynamic equilibrium (LTE). Therefore, although it pushes parameters a bit uncomfortably, if T Tau S has protostellar mass loss rates ≳2.5 × 10⁻⁷ M_☉ yr⁻¹, it is possible that internal wind shocks produce the observed [Ne ii] luminosity. Note that the "dynamical time," v_s/r_sh ∼ 10 yr, is marginally consistent with the observations of no significant time dependence since 1998 (van Boekel et al. 2009).

More recently, Güdel et al. (2009) have assembled [Ne ii] data from a large number of sources and have plotted L_[Ne ii] versus $\dot{M} _{\rm acc}$. For low values of $\dot{M}_{\rm acc} \lesssim 10^{-8}$ M_☉ yr⁻¹ the [Ne ii] luminosity is nearly independent of $\dot{M}_{\rm acc}$, and is typically of order 3 × 10⁻⁶ L_☉. However, for higher mass accretion rates and, in particular, for all the sources with known outflows or jets, L_[Ne ii] increases with increasing $\dot{M} _{\rm acc}$ (arguably linearly, but with a lot of scatter). We present these observational results from Güdel et al. and compare them to Equation (33) in Section 4.4. The correlation of L_[Ne ii] with $\dot{M}_{\rm acc}$ suggests that either the higher luminosity (L_[Ne ii] ∼ 10⁻⁵–10⁻³ L_☉) sources may originate in protostellar shocks or from EUV or soft X-rays produced by the accretion of disk material onto the star. However, in the latter case, these photons must penetrate the disk wind, which seems unlikely.

[O i] 6300 Å emission is often observed in young low-mass stars with disks and outflows (e.g., Hartigan et al. 1995). Two velocity components are seen: a high velocity component "HVC" and a low velocity component "LVC." Hartigan et al. argued that the HVC comes from shocks in the protostellar wind, similar to our above discussion of internal shocks. The typical velocity of this component is ∼100–200 km s⁻¹ and the [O i] 6300 Å luminosity is ∼10⁻⁶–10⁻² L_☉. HM89 showed that for v_s ∼ 100 km s⁻¹ the [O i] 6300 Å emission from the shock is about 10 times more luminous than the [Ne ii] 12.8 μm emission for n₀ ≲ 10⁴ cm⁻³, with even higher ratios at n₀>10⁴ cm⁻³ because [O i] 6300 Å does not collisionally de-excite as readily as [Ne ii] 12.8 μm. Therefore, in agreement with Hartigan et al, we find that mass outflow rates of ≳10⁻⁷M_☉ yr⁻¹ can produce the most luminous [O i] 6300 Å HVC sources (see Equation (33)).

Hartigan et al. (1995) attributed the LVC to [O i] emission emanating from the disk surface, probably in a relatively slow outflow since the emission is observed to be slightly blueshifted (∼−5 km s⁻¹ with great dispersion). However, there are red and blue wings extending to ±60 km s⁻¹ in the LVC, presumably due to a combination of Keplerian rotation and outflow. The [O i] 6300 Å luminosity in the LVC ranges from ∼10⁻⁶ to ∼10⁻³ L_☉, with "typical" values of ∼10⁻⁴L_☉ (Hartigan et al. 1995). The exact origin of the LVC [O i] emission, and its heating source, remains a mystery. In Section 4, we present our numerical model results for [O i] emission from the EUV layer and the X-ray layer. In agreement with previous work by Font et al. (2004), we find that the EUV layer can only provide [O i] 6300 Å luminosities ≲10⁻⁶ L_☉. Meijerink et al. (2008) were able to produce [O i] luminosities as high as 5 × 10⁻⁵ L_☉ in their models of the X-ray layer. Therefore, they found it very difficult to explain the most luminous [O i] LVC sources, but were able to produce luminosities in accordance with many of the observations. However, our more detailed models with an X-ray spectrum similar to that assumed by Meijerink et al. produce lower [O i] luminosities, primarily because we obtain lower gas temperatures in the X-ray-heated layer (see below). However, if we use a softer X-ray spectrum such as the one proposed by Ercolano et al. (2009), we do obtain luminosities of order 10⁻⁴ L_☉. In summary, it appears that emission from the EUV and X-ray layers can only explain the lower and typical luminosity LVCs, but not the highest luminosity LVCs.

It is instructive to estimate what physical conditions are required to produce [O i] luminosities in the LVC as high as 10⁻⁴–10⁻³ L_☉. Consider a layer on the disk surface of thickness ℓ, temperature T, and extending to radius r_o from which most of the [O i] emanates. This top and bottom layer of the disk has hydrogen nucleus density n and vertical column N. Because the [O i] 6300 Å transition is ΔE/k = 23, 000 K above the ground state, we require T≳ several thousand degrees K for significant emission. The emerging [O i] 6300 Å luminosity produced by the surface layers is then given as

$$\begin{equation} L_{[{\rm O}\,\hbox{\scriptsize {\sc i}}]}= \pi r_{\rm o}^2n_{\rm e} n(O)\gamma _{[{\rm O}\,\hbox{\scriptsize {\sc i}}]} \ell, \end{equation} \tag{ 35 }$$

where γ_[O i] is the collisional excitation rate for electrons on atomic oxygen, n(O) is the density of atomic oxygen, and n_e is the electron density. We account here for both the top and bottom of the disk, but recall that half of the luminosity (that directed to the midplane) is absorbed by the optically thick disk. Equation (35) assumes n_e to be less than the critical density n_ecr. (HM89 gave n_ecr ∼ 10⁶ cm⁻³, so n_e < n_ecr is generally satisfied.) HM89 gave $\gamma _{[{\rm O}\,\hbox{\scriptsize {\sc i}}]} =8.5 \times 10^{-9} T_4^{0.57} e^{-2.3/T_4}$ cm³ s⁻¹, with T₄ = T/10⁴ K. Oxygen rapidly charge exchanges with hydrogen and therefore at high temperatures (T ≫ 200 K) O⁺/O = H⁺/H. Therefore, n_e = x_en and n(O) = x(H)n_O, where n_O ≃ 3 × 10⁻⁴n is the gas phase density of oxygen in both O and O⁺ and x(H) is the abundance of atomic hydrogen. It follows that

$$\begin{equation} L_{[{\rm O}\,\hbox{\scriptsize {\sc i}}]} \simeq 6.5 \times 10^{-4} x_e x(H) n_5 r_{14}^2 N_{20} T_4^{0.57} e^{-2.3/T_4} \ {L}_\odot, \end{equation} \tag{ 36 }$$

where r₁₄ = r_o/10¹⁴ cm, n₅ = n/10⁵ cm⁻³, and N₂₀ = N/10²⁰ cm⁻². This analytic exercise shows that to produce L_[O i] ∼ 10⁻⁴–10³ L_☉ in the LVC requires, for example, surface layers with n ∼ 10⁵cm⁻³, N ∼ 10²⁰ cm⁻², r_o ∼ 60 AU, T ∼ 10⁴ K, and x_e ∼ 0.5. In the EUV layer x_e ∼ 1, T₄ ∼ 1, and r²₁₄ ∼ 1 (recall that most of emission arises from r ∼ r_g). Therefore, the [O i] luminosity from the EUV layer is approximately

$$\begin{equation} L_{[{\rm O}\,\hbox{\scriptsize {\sc i}}]}^{\mbox{,}} \simeq 6.5 \times 10^{-5} x(H) r_{14}^2 n_5 N_{20} \ {L}_\odot. \end{equation} \tag{ 37 }$$

This equation shows the difficulty in producing the observed [O i] emission from the EUV layer: the fraction of neutral gas x(H) is very low in the EUV layer or, equivalently, the fraction of atomic oxygen is very low. In addition, the emission mostly arises from r ∼ r_g ∼ 10¹⁴ cm, which is not large, and n₅N₂₀ rarely exceeds unity.

On the other hand, in the X-ray layer the temperature is of order T₄ ∼ 0.1–0.4 and x(H) ∼ 1 so that

$$\begin{equation} L_{[{\rm O}\,\hbox{\scriptsize {\sc i}}]}^X \simeq 6.5 \times 10^{-4} x_e n_5 r_{14}^2 N_{20} T_4^{0.57} e^{-2.3/T_4} \ {L}_\odot. \end{equation} \tag{ 38 }$$

The X-ray layer has x_e ≲ 0.1, r²₁₄ ≲ 10, n₅ ≲ 100, and N₂₀ ≲ 10. Even inserting these upper limits, we find that the [O i] luminosity is at most 10⁻³ L_☉ if T = 4000 K and 3 × 10⁻⁶L_☉ if T = 2000 K. Therefore, the [O i] luminosity is extremely sensitive to the gas temperature (and its variation in r and z) in the X-ray layer. The Meijerink et al. (2008) model has high temperatures, ∼3000–4000 K, in the X-ray layer out to 10–20 AU in their case with a relatively high X-ray luminosity of 2 × 10³¹ erg s⁻¹. For this case, they therefore find L^X_[O i] ∼ 5 × 10⁻⁵ L_☉. Our model for the same case, however, gets temperatures in the X-ray layer in the range 1500–2500 K, and therefore we get an [O i] luminosity of only ∼5 × 10⁻⁷ L_☉ (see Section 4.4). However, keeping the X-ray luminosity the same but assuming a much softer power-law spectrum (L(E) ∝ E⁻¹ from 0.1 to 2 keV), we do find that the X-ray-heated gas becomes hotter and the [O i] luminosities approach 10⁻⁴ L_☉. We discuss in Section 4.3 the reasons for the differences in temperature in our model relative to that of Meijerink et al. We conclude that it is unlikely that the X-ray layer can provide the highest [O i] luminosities observed in some LVC sources, but that soft X-ray sources can produce the typical luminosities.

The "transition layer" (x_e ∼ 0.5) between the fully ionized EUV layer and the partially ionized X-ray layer is also unlikely to produce either the highest observed [O i] luminosities or even the typical luminosities. The density in this layer is similar to the density at the base of the EUV layer, or n ∼ 10⁵Φ^1/2₄₁ cm⁻³ at r = r_g = 7 AU (see Equation (12)). However, beyond r_g = 7 AU the density falls as r^−5/2 so that most of the emission arises from r ∼ 7 AU. In addition, the column N of this transition layer is roughly the column for optical depth unity in EUV photons, or N ∼ 10¹⁸ cm⁻². Therefore, the small r₀ and low N conspire to produce only L_[O i] ∼ 10⁻⁶ L_☉.

We plan to investigate the possibility that the source of the LVC arises from the shear layer produced when the protostellar wind strikes the surface of the disk obliquely and sets up an outward moving layer of shocked wind and entrained disk surface gas (a "shear" layer; see Matsuyama et al. 2009, hereafter MJH09). MJH09 showed that this layer can have columns N ∼ 10¹⁹–10²⁰ cm⁻² out to 100 AU. As noted earlier, the oblique wind shock (v_s ≲ 20 km s⁻¹) is insufficient to ionize hydrogen or helium to provide the electrons needed for [O i] excitation. Therefore, we require the EUV and soft X-rays to ionize this layer. This shear layer is likely turbulent so that there may be rapid mixing of the bottom (cooler and more neutral layers) with the top shear layers, perhaps allowing x_e ∼ 0.5 and x(H) ∼ 0.5 in the entire layer (the most efficient for producing [O i]) and maintaining a relatively isothermal layer. The heating would be a combination of shock/turbulent heating plus the heating due to photoionization by EUV and soft X-rays. The density n in the shear layer (MJH09) is approximately $n \sim 3000\, \dot{M}_{-8} r_{15}^{-2}$ cm⁻³. Note that we cannot allow $\dot{M}_{-8}$ to exceed unity or the EUV and soft X-rays will not be able to penetrate the base of the wind to heat and ionize the shear layer. Therefore, although the shear layer may provide sufficient column N, electron fraction x_e, and temperature T, it appears unlikely to produce sufficient nr² to give the observed [O i] luminosities in the more luminous sources. Further work is needed to confirm this rough argument.

We conclude that the origin of the very luminous LVC [O i] emission is not from the EUV layer, the X-ray layer, or the transition layer. The typical LVC [O i] emission, however, may be produced by soft X-rays. The LVCs are also unlikely to originate from the shear layer set up by the impact of the protostellar wind. Perhaps a model that invokes ambipolar diffusion as a heating source, such as those that Safier (1993a, 1993b) has constructed for the HVC, might be applicable for the most luminous LVCs. However, the EUV layer is capable of producing the lowest luminosity sources, and the X-ray layer may produce the typical luminosity, so we proceed with detailed numerical studies of the [O i] luminosity from these layers in Section 4.

Gorti & Hollenbach (2008) described the numerical code that we use to calculate, self-consistently, the gas temperature, gas density, and chemical structure of the predominantly neutral gas in the disk. To summarize, the code includes ∼600 reactions among 84 species, gas heating by a number of mechanisms including FUV grain/polycyclic aromatic hydrocarbon (PAH) photoelectric heating and the heating caused by X-ray ionization of the gas, and cooling not only from collisional excitation of the species followed by radiative decay, but also from gas–grain collisions when the dust is colder than the gas. In some instances, for example, deep in the disk below the surface layers, the dust is warmer than the gas in which case gas–grain collisions can be an important heating source for the gas. The vertical structure of the disk is calculated self-consistently by using the computed gas temperature and density to calculate the thermal pressure and then balancing thermal pressure gradients with the vertical (downward) gravitational force from the central star.

For this paper we consider the "chemistry" of the fully ionized (H ii) surface region, photoionized by the EUV radiation field from the star. By "chemistry," we mean the computation of the different ionic states of a given element by balancing photoionization with electronic recombination and charge exchange reactions. Photoionization rates are computed using cross sections from Verner et al. (1996). Recombination rates are taken from Aldrovandi & Pequignot (1973), Shull & van Steenberg (1982), and Arnaud & Rothenflug (1985). Charge exchange rates are from Kingdon & Ferland (1996). We assume a gas temperature of 10⁴ K and do not perform thermal balance calculations in the H ii region. In short, our code for the surface of the disk is an H ii region code where we assume an EUV spectrum from the central star and then compute the abundances of, for example, Ne⁺, Ne⁺⁺, and Ne⁺⁺⁺ at each point r, z in the surface H ii region. At each point r, z, we compute both the direct EUV flux from the star and the diffuse EUV field caused by recombinations to the ground state of atomic hydrogen in the surface H ii region. We use the method described by Hollenbach et al. (1994) and utilized by Font et al. (2004) to do both these computations. The code finds the electron density n_e(r, z_IF) at the base of the surface H ii region (in other words, at the ionization front separating the ionized surface from the predominantly neutral disk below). The thermal pressure at the IF is then P_IF ≃ 2n_e(r, z_IF)kT_II, where T_II = 10⁴ K and the factor of 2 includes the pressure from protons, He⁺, and electrons. This pressure then determines the height z_IF where the thermal pressure in the predominantly neutral gas has dropped from its midplane value to P_IF. The parameter z_IF is the height of the base of the H ii region: above z_IF the emission is mostly EUV-induced and we call this region the EUV surface layer; below z_IF the emission is mostly X-ray-induced and we call this region the X-ray layer.³

Implicit in our model is the assumption that the EUV luminosity and the X-ray luminosity are generated close to the stellar surface. This assumption then allows us to determine the column density of wind that the EUV and X-ray fluxes must traverse (see Section 2) as well as the angle of incidence of the EUV and X-ray flux on the flared disk surface. Since we assume that the protostellar disk wind originates near r ∼ 10¹² cm, our estimate of the attenuation column density at the wind base is valid as long as the EUV and X-ray source of luminosity originates roughly within this distance from the stellar surface. Models of X-ray flares indicate that the X-rays probably arise from flares whose size ranges from 0.1 to 10 times the stellar radius (e.g., Favata et al. 2001, 2005; Grosso et al. 2004; Franciosini et al. 2007; Stelzer et al. 2007; Getman et al. 2008a, 2008b). Therefore, our estimate of the attenuation column is likely valid. Similarly, the line emission that we model usually arises from r∼ 1–10 AU in the disk, and so the placement of the EUV or X-ray source within 10¹² cm of the star does not affect our results. However, in DG Tau, a soft X-ray source has been imaged and seen to arise about 20 AU from the star, probably from shocks in a jet (Güdel et al. 2008; Schneider & Schmitt 2008). Such a geometry would certainly lower the column of attenuating wind, because of the spherical divergence of the wind. In addition, the X-ray flux would strike the disk from above, nearly normal to the surface. This latter effect, however, will likely not significantly affect the luminosities in the lines, since as we have shown in Section 3, the line luminosities are really an emission measure effect, and mainly depend on the fraction of energetic photons that the disk absorbs. If the source is 20 AU from the star, roughly half of the energetic photons are absorbed. We show below that in the case of a flared disk with a central source of energetic photons, nearly 0.7 of the photons are absorbed. Therefore, the fraction of photons absorbed is nearly the same.

Also implicit in our model is that the X-ray luminosity is the mean value of the time variable X-ray luminosity. Getman et al. (2008a) showed that typical decay times of flares is of the order of hours to days. In the EUV and X-ray layers where the modeled lines originate, recombination and cooling timescales are of the order of 1–10 yr. Thus, the gas generally does not have time to respond to the flares, but settles to a state given by the mean value of the X-ray luminosity.

Figures 1 and 2 show the results of models as we vary the EUV luminosity. We assume two forms for the EUV spectrum. The first form (Figure 1) is a relatively hard spectrum; we assume a power-law spectrum νL_ν= constant from 13.6 eV to the X-ray regime (∼0.1 keV). This spectrum is motivated by the fact that νL_ν in the FUV band is observed to be similar to νL_ν in the X-ray band, and each band typically has νL_ν ∼ 10⁻³ L_*, where L_* is the stellar bolometric luminosity. On the other hand, the EUV spectrum is very uncertain. The Ribas et al. (2005) observations of older, but very nearby, solar mass stars show EUV spectra, which can drop rapidly from 13.6 eV to 40 eV, even though the overall trend from the FUV to the X-ray tends to roughly an L_ν ∝ ν⁻¹ spectrum. To simulate a softer spectrum than the first form, we take a blackbody spectrum with an effective temperature of 30,000 K (Figure 2). We are further motivated to adopt a softer spectrum by the observations in one source of the ratio [Ne iii] 15 μm/[Ne ii] 12.8 μm ≲ 0.06 (Lahuis et al. 2007) and because [Ne ii] has been detected in more than 25 sources and none of them show [Ne iii]. Our first form of the EUV spectrum produces a ratio >1! This either indicates very little production of [Ne ii] by the EUV layer or the fact that the EUV spectrum is much softer. We therefore have chosen a blackbody EUV spectrum that provides ratios in accord with measured values or upper limits on the ratio.

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Figures 1 and 2 show the nearly linear rise in L_[Ne ii] or L_[Ne iii] with L_EUV as predicted in Equations (19) and (20). The absolute values are also in good agreement with these equations. At very high L_EUV this linear relationship breaks down for [Ne iii], and L_Ne iii begin to saturate because the electron densities in the dominant emitting regions begin to exceed n_{ecr,[Ne iii]}. For our power-law spectrum in Figure 1, we find that by fitting our analytic results to the model, f(Ne⁺⁺) ∼ 0.75 and f(Ne⁺) ∼ 0.25, that is, 75% of the emitting neon is in Ne⁺⁺ and only 25% in Ne⁺. However, in Figure 2 we see that a softer EUV spectrum (blackbody with T_eff = 30, 000 K) will reverse the situation so that [Ne ii] dominates. Another mechanism, not treated here, that would quench [Ne iii] and raise [Ne ii] would be turbulent mixing of neutral gas into the H ii region. The charge exchange reaction Ne⁺⁺ + H → Ne⁺ + H⁺ is very rapid (e.g., Butler & Dalgarno 1980), and even a neutral fraction x(H) ∼ 10⁻² would lead to [Ne iii]/[Ne ii] <1.

Figure 3 shows the results for a number of other fine structure transitions listed in Table 1. One sees that [Ar ii] 7 μm is the strongest predicted line not yet observed. Again, the analytic estimates (Equation (8) and Table 1) are very good.

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Figure 4 shows the radial origin of the EUV-induced emission in the [Ne ii], [Ne iii], [Ar ii], and [S iii] lines. We use here as a standard case Φ_EUV = 10⁴¹ s⁻¹ (L_EUV = 2 × 10³⁰ erg s⁻¹) and the blackbody spectrum, and L_X = 2 × 10³⁰ erg s⁻¹ (with our standard X-ray spectrum; see Section 4.3) although the radial origin is quite insensitive to these parameters. We see that most of the emission arises from r ∼ r_g ∼ 10 AU, as predicted in Section 3. We plot 4πr² times the emergent flux from one side of the disk. This roughly gives the luminosity arising from both sides of the disk and from a region extending from 0.5r to 1.5r. For dominant ions such as Ne⁺, the luminosity scales as n²_eH(r)r² ∝ r^1/2 for r < r_g and n_e < n_ecr and as r² for n_e>n_ecr. The luminosity scales as n²_er³ for r>r_g and n_e < n_ecr, so that here it scales as r⁻² (see Equation (13)), as seen in [Ar ii] and [Ne ii]. For nondominant ions such as Ne⁺⁺, the scalings change because the fraction of neon in Ne⁺⁺ changes with radius.

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Hollenbach et al. (1994) and Gorti & Hollenbach (2009) showed that significant photoevaporation flows proceed in the 10⁴ K gas in the EUV layer at ≳1 AU. Thus, although our models here are static, the emitting gas is actually rising (and rapidly turning radial; see Font et al. 2004) at speeds of the order of the sound speed (or 10 km s⁻¹) from the surface of the disk. As discussed in Hollenbach et al. (1994), the electron density at the base is not much affected by this flow. Photoionization and recombination timescales are sufficiently short that the steady state still applies to the computation of the ionization state of each element in the gas. Therefore, we expect our model results on the emitted luminosities in each fine structure line to be well approximated by the static model solution. However, the observed line profiles will be affected by this flow. For a disk viewed edge-on, the lines will be broadened not only by the Keplerian rotation but also by the radial outward flow. For a disk viewed face-on, the lines will be broadened mostly by the radial outward flow, and since the far side of the disk is obscured, one would expect a blue shift. Alexander (2008b) has recently modeled [Ne ii] line profiles from photoevaporating disks. He predicted broad (30–40 km s⁻¹), double-peaked profiles from edge-on disks due to rotation and a narrower (∼10 km s⁻¹) profile with a significant blue shift (5–10 km s⁻¹) from face-on disks. He argued that the observed line widths in TW Hya (Herczeg et al. 2007) are consistent with a photoevaporative wind (see also Pascucci & Sterzik 2009). Resolved [Ne ii] observations can thus provide a test of EUV photoevaporation models.

Figure 5 shows the vertical origin of the EUV-induced and X-ray-induced emission at r = 10 AU for the standard case. We have plotted gas temperature T, the dust temperature T_dust, and the hydrogen nucleus density n as a function of the hydrogen nucleus column N measured from z = r (the putative "surface" of the disk) downward. On the top of the figure, we give the values of z that correspond to those of N. The completely ionized EUV layer extends to N ∼ 3 × 10¹⁸ cm⁻² and has electron densities n_e ∼ n ∼ 3 × 10⁴ cm⁻³ (see Equation (12)). The X-ray-heated (T ∼ 1000 K) and ionized layer extends from N ≃ 3 × 10¹⁸ cm⁻² to N ≃ 3 × 10²⁰ cm⁻² with hydrogen atom densities ∼3 × 10⁶ cm⁻³. FUV photons also contribute to the heating of this layer.

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We model the X-ray layer for both a "soft" X-ray spectrum and a "hard" X-ray spectrum. Our standard ("hard") X-ray spectrum (Gorti & Hollenbach 2004, 2008) is based on observed X-ray spectra from young stars, with an attempt to correct for extinction at the softer energies (Feigelson & Montmerle 1999). Our fit to this spectrum is that of a power law L_ν ∝ ν from 0.1 keV <hν < 2 keV, fitting to another power law L_ν ∝ ν⁻² for hν>2 keV. This spectrum is similar to that adopted by Glassgold et al. (2007) and Meijerink et al. (2008). We also model disks illuminated by a softer X-ray spectrum: L_ν ∝ ν⁻¹ for 0.1 keV <hν < 2 keV and L_ν ∝ ν^−1.75 for hν>2 keV.⁴ Ercolano et al. (2009) recently provided evidence that such a spectrum might be expected from young, low-mass stars. We note that our soft X-ray spectrum has equal energy flux in equal logarithmic intervals of photon energy between 0.1 keV and 2 keV, that is, there is as much energy flux between 0.1 and 0.2 keV as there is between 1 and 2 keV. We do not consider here a harder spectrum than our "hard" case, although recently there have been observations of "superhot" flares (Getman et al. 2008a) that indicate significant emission in the 3–8 keV region of the spectrum. We do extend our "soft" and "hard" spectra to 10 keV, but there is insignificant energy flux beyond a few keV. If there were, then for the same X-ray luminosity as our two cases, we would obtain less emission in the lines we model in this paper. The higher energy photons penetrate more column of gas, depositing less energy per unit volume, and therefore lead to cooler gas than in our current X-ray layer. In addition, because of the increased penetration, the heat is deposited in molecular regions, where the cooling is enhanced by the molecular transitions. Therefore, the emitting gas is substantially cooler and most of the X-ray heating energy presumably emerges in molecular rotation lines of, for example, CO, OH, and H₂O or possibly, if grains are abundant, as IR continuum emission from grains heated by collisions with the warmer gas. However, we emphasize that the heating and cooling timescales are long, of order 1–10 yr, so that superhot flares that are much more short-lived than this timescale will not produce a significant effect.

Glassgold et al. (2007) first modeled and discussed the physics of the X-ray layer, and our results are in basic accord with theirs, except that, as discussed below, we obtain somewhat cooler temperatures in the X-ray layer. Figure 6 shows the vertical structure of the same case plotted in Figure 5 at the same radius, r = 10 AU; only the N range is shrunk to emphasize the X-ray layer. In Figure 6, we plot the electron abundance x_e relative to H nuclei and the fraction of neon in Ne⁺, f_Ne(Ne⁺). The [Ne ii] 12.8 μm line luminosity is proportional to x_ef_Ne(Ne⁺) and very sensitive to T in the X-ray layer (see Figure 5 and Equation (28), where the T dependence comes in the collisional rate coefficient, which is proportional to e^−1100/T). Below the EUV layer to a depth N ∼ 3 × 10²⁰ cm⁻², X-rays maintain a relatively high fraction of Ne⁺, f_Ne(Ne⁺) ≳ 10⁻², and X-ray ionization of H and He as well as FUV ionization of C maintains a relatively high ionization fraction, x_e ≳ 2 × 10⁻⁴ (see Figure 6). Note that the column attenuating the X-rays in this layer is the column through the disk to the star, which is typically ∼10N, where N is the vertical column to the disk surface. In the X-ray layer, 10¹⁹ cm⁻² < N < 10²¹ cm⁻², the gas is heated by a combination of FUV grain photoelectric heating and X-ray photoionization heating. It is cooled mainly by [O i] 63 μm, [O i] 6300 Å, [Ne ii] 12.8 μm, [Ar ii] 7 μm, and gas–grain collisions (see Gorti & Hollenbach 2008). The resultant temperature is ∼1000–2000 K, dropping with increasing r and N as dilution and attenuation of the X-rays and FUV lower the heating rates. Because T, f_Ne(Ne⁺), and x_e drop with increasing N (see Figure 6) and r, most of the [Ne ii] 12.8 μm emission arises from r < 20 AU and N ∼ 10²⁰–10²¹ cm⁻², where T ∼ 1000 K, f(Ne⁺) ∼ 10⁻¹, and x_e ∼ 10⁻³.

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In addition to showing the radial origin of the [Ne ii] emission in the EUV layer, Figure 4 also shows the [Ne ii] "luminosity" plotted as a function of r for the X-ray layer (hard spectrum X-rays). Note that there is greater contribution from inner (∼1 AU) regions of the disk compared to the EUV layer. In addition, there is more luminosity emerging from the X-ray layer than the EUV layer. Figure 4 shows that the X-ray-induced emission arises mostly from r ∼ 1–10 AU, also in agreement with Glassgold et al. Beyond this radius, the X-ray and FUV heating is insufficient to maintain significant quantities of T ≳ 1000 K gas.

Figure 7 plots the [Ne ii] 12.8 μm and [Ne iii] 15 μm emissions from the X-ray layer. In agreement with Glassgold et al. (2007) and Meijerink et al. (2008) we find that [Ne iii]/[Ne ii] ≲0.1, mainly caused by the rapid charge exchange of atomic H with Ne⁺⁺. We also plot outlines of the observed 54 sources tabulated by Güdel et al. (2009). The vertical dotted lines shade the region where the sources have known outflows and jets. The horizontal solid lines shade the regions with no outflows or jets detected. The L_X tabulated by Güdel et al. is a two-component fit with an attempt to correct for absorption of the softer X-rays by material on the line of sight from star to observer. However, many of the observations do not extend to hν < 0.3 keV and extinction is severe at the lower energies, so that a luminous soft X-ray source that is weak at 0.5–1 keV could exist undetected. The effect of such a "soft" X-ray component would be to move the data points to the right on Figure 7 and comparison should be made to our "soft" X-ray spectrum results (dashed line). We also find that L_Ne ii and L_Ne iii scale with L_X, as predicted in Section 3 and also as found by Meijerink et al. (2008). Comparison with Figure 2 shows that if the X-ray luminosity is about the same as the EUV luminosity from the central star, and if the EUV spectrum is soft enough that [Ne ii] dominates [Ne iii] in the EUV layer, the [Ne ii] luminosity is roughly 2 times stronger from the X-ray layer as from the EUV layer, as we estimated analytically. The main conclusion from comparing the data to the model results is that although the X-ray layer may explain the origin of the [Ne ii] emission in some (perhaps most if a strong soft X-ray excess is common) sources, there are a significant number of sources, especially those with observed outflows and jets, where the X-ray luminosity seems insufficient to explain the [Ne ii] luminosity. In Section 4.4, we compare the observational data with our analytic results on the [Ne ii] luminosity expected from internal shocks in the jets or winds, and find that this is a plausible origin for these sources.

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As noted above, our results on the IR fine structure emission from the X-ray layer do not differ appreciably from previous results (Glassgold et al. 2007; Meijerink et al. 2008). Overall, we tend to find somewhat (factor of ∼2) lower IR line luminosities. This agreement is a bit fortuitous, arising from a cancellation of several effects and the insensitivity of the fine structure lines with variations in T if T ≳ 1000 K. Our models self-consistently calculate the vertical density structure of the gas by using the computed gas temperatures (which differ from the dust temperature) to calculate the gas density structure rather than assuming that the gas density structure is fixed by the calculation of vertical pressure balance when one assumes the gas temperature to equal the dust temperature, as done in the previous work. Our self-consistent model produces significantly different results at columns N ≲ 10²¹ cm⁻², where the gas temperature rises above the dust temperature (Gorti & Hollenbach 2008). The net effect is that our gas disk is more flared, intercepting a larger fraction of the X-ray luminosity. This tends to raise the emission from our models. In addition, we include FUV grain photoelectric heating which also raises the emission. However, counteracting these effects is the inclusion of more gas coolants in our model, especially [Ne ii] 12.8 μm and [Ar ii] 7 μm. In addition, our treatment of the gas heating by X-rays follows Maloney et al. (1996), which is somewhat different than the approach used by Glassgold et al. and Meijerink et al, and our X-ray heating rates are lower than these authors by a factor of 3–10. We believe that this may arise because we include the loss of "heat" due to escape of Lyman α and other photons created by recombining hydrogen or to the absorption of these photons by dust. Overall, our X-ray layer tends to be a factor of about 2 cooler than the previous models (roughly 1000–2000 K versus 2000–4000 K in the previous models), thereby lowering the fine structure emission from this layer. This lower temperature has a relatively small effect on the fine structure lines, because their upper states lie only ΔE/k ∼ 1000 K above the ground state. However, it has an enormous effect on our predictions of the [O i] 6300 Å emission, whose upper state lies ΔE/k ∼ 23, 000 K above the ground state, as we will discuss in Section 4.5.

Figure 8 plots the [Ne ii] luminosity versus the mass accretion rates assembled by Güdel et al. (2009). As in Figure 7, the vertical dotted lines shade the region that includes sources with known jets or outflows, whereas the solid horizontal lines denote sources with no detected jets/outflows. We plot here our predicted [Ne ii] luminosities from internal shocks in the winds/jets, using our analytic expression (Equation (33)). The solid line represents the expected [Ne ii] luminosity when $\dot{M}_{\rm w} = 0.1\dot{M}_{\rm acc}$, the entire wind or jet passes through a shock or f_sh = 1, the shock velocity is in excess of about 100 km s⁻¹, and the preshock density is less than 10⁴ cm⁻³. The upper dashed line makes the same assumptions except that $\dot{M}_{\rm w} = \dot{M}_{\rm acc}$ and the lower dashed line assumes $\dot{M}_{\rm w} = 0.01\dot{M}_{\rm acc}$. Note that the [Ne ii] luminosity is proportional to the product of f_sh and $\dot{M}_{\rm w}$, so that, for example, the lower dashed line also corresponds to f_sh = 0.1 and $\dot{M}_{\rm w} = 0.1\dot{M}_{\rm acc}$. The main conclusion is that internal wind or jet shocks very likely explain the origin of [Ne ii] from the outflow and jet sources. In fact, the figure might suggest that these shocks could explain [Ne ii] observed in nearly all of the sources, if it were not for the fact that in some cases (e.g., Herczeg et al. 2007; Najita et al. 2009; Pascucci & Sterzik 2009) where the lines have been spectrally resolved, they are narrower than what a shock origin would predict. We do note that in some of these cases, the integrated flux seen with the high spectral and spatial resolution ground-based instruments is significantly less than the flux seen by the low resolution Spitzer Space Telescope. Najita et al. speculated that perhaps there are two components comprising the total flux: a strong but broad and extended shock component and a weaker, but narrow and spatially unresolved disk component arising from the X-ray layer. On the other hand, it is quite possible that X-rays or EUV produce most of the [Ne ii] luminosity in the sources with no observed winds or jets. Note that these sources in Figure 8 are distributed in a nearly horizontal line with no apparent dependence on $\dot{M}_{\rm acc}$ over a two orders of magnitude increase in this parameter.

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Figure 9 plots the [O i] 6300 Å luminosity from the EUV layer versus Φ_EUV for both our harder L_EUV(ν) ∝ ν⁻¹ spectrum and our softer L_EUV(ν) blackbody spectrum. A harder spectrum gives more [O i] luminosity in the EUV layer because although the gas is almost entirely ionized, there is a greater fraction of neutral H and O in the gas due to the smaller photoionization cross section of these atoms with higher photon frequency. However, even the harder EUV spectrum results in [O i] 6300 Å luminosities ≲10⁻⁶ L_☉, which can only explain the weakest LVC sources. Recall that L_[O i] ranges from 10⁻⁶to10⁻³ L_☉ in LVCs.

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Figure 10 plots the [O i] 6300 Å luminosity in the X-ray layer versus L_X for both our harder and our softer X-ray spectra. The [O i] luminosity does increase with L_X, due to the higher temperatures and higher ionization fractions in the X-ray layer. Recall that the gas is primarily neutral, so the higher ionization fraction increases the luminosity by increasing the density of the electrons that excite [O i]. However, with our standard (harder) X-ray spectrum, which is quite similar to that adopted by Meijerink et al. (2008), we obtain [O i] luminosities that are a factor of nearly 100 times lower than those of Meijerink et al. (2008) from the X-ray layer. This is primarily because of the extreme sensitivity of the [O i] luminosity to the temperature of the X-ray layer (see Equation 38). As noted above, our temperatures are roughly a factor of 2 lower than those of Meijerink et al.

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The main point, however, is that in our standard models neither the EUV layer nor the X-ray layer can produce [O i] luminosities as high as 10⁻⁵–10⁻³ L_☉ as observed in many LVC sources. This confirms the analytic estimates made in Section 3.5. However, as discussed above, the [O i] luminosity is very sensitive to the temperature of the X-ray layer. One way to increase the temperature is to assume a softer X-ray spectrum. Softer X-rays have much higher absorption cross sections and therefore deposit much more heat per unit volume in the upper layers. In addition, they create higher electron abundances, and these lead to increased efficiency in converting the absorbed X-ray energy into heat. Therefore, we also plot in Figure 10 the results for cases with similar X-ray luminosities, but with our "soft" X-ray spectrum where L_ν ∝ ν⁻¹ from 0.1 keV to 2 keV. This spectrum has many more 0.1–0.3 keV X-rays than our standard case, and we find that we do indeed get higher temperatures and electron abundances in the upper parts of the X-ray layer and consequently much higher [O i] luminosities. L_[O i] can be as high as ∼10⁻⁴L_☉ if L_X ∼ 10³² erg s⁻¹ (∼2 × 10⁻² L_☉), a likely upper limit to the soft X-ray luminosity. Therefore, soft X-rays may be able to explain "typical," L_[O i] ∼ 10⁻⁴ L_☉, LVC sources, but not the most luminous sources. We note that if the soft X-rays caused photoevaporation, then high spectral resolution observations of the [O i] line might diagnose the flow parameters.