INTERSTELLAR H₂O MASERS FROM J SHOCKS

The populations per sub-level n_i are conveniently expressed in terms of $y_i = n_i/n({\rm H_2O})$, so that the overall population of level i is $g_i y_i n({\rm H_2O})$, where g_i is the statistical weight. For each level i included in the pumping scheme, the steady state rate equation is

$$\begin{equation} g_iy_i\sum _j R_{ij} = \sum _j g_jy_j R_{ji}, \end{equation} \tag{ 1 }$$

where R_ij is the overall rate coefficient (cm³ s⁻¹) for transitions from i to j. Introduce the hydrogen nuclei density, n = n(H)  +  2n(H₂). (We ignore the small contribution of collisions with He.) Then the overall collision rate for the i → j transition is $nq_{ij} = n({\rm H})q_{ij}^{{\rm H}}\,{+}\, n({\rm H_2})q_{ij}^{{\rm H_2}}$, determined from the rate coefficients for collisions with H and H₂. Equation (1) is then converted into a relation between rates per unit volume by multiplying both sides by $nn({\rm H_2O})$. While ignoring maser radiation, we include the trapping of all other lines via a slab escape probability formalism. External radiation, such as infrared continuum emission from dust, is assumed negligible (recall that the dust is relatively cold in shocks; see also Section 3) and in this case

$$\begin{equation} R_{ij} = {\beta _{ij}A_{ij} \over n} + q_{ij}. \end{equation} \tag{ 2 }$$

Here β_ij and A_ij are, respectively, the escape probability and A-coefficient for the i → j transition; the notation is such that A_ij = 0 if i is a level below j (i < j). The escape probability β_ij is a function of the transition optical depth at line center

$$\begin{equation} \tau _{ij} = {g_i A_{ij}\lambda ^3 \over 8\pi ^{3/2}}\, {n({\rm H_2O})d \over \Delta v_D}\,(y_j - y_i), \end{equation} \tag{ 3 }$$

where λ is the transition wavelength, d is the source dimension that controls photon escape (thickness in the slab geometry) and Δv_D is the width of the local (thermal and microturbulent) velocity field, which is assumed to be of the form $\exp [-(v - v_0)^2/\Delta v_D^2]$; when the velocity field is dominated by ordered motions across the slab, $\Delta v_D/d$ is replaced by the velocity gradient dv_z/dz. Therefore, for a given mix of H and H₂ the distribution of populations y_i depends only on the following three parameters: n, $n({\rm H_2O})d/\Delta v_D$ (which determines τ_ij), and the temperature T (which determines q_ij). When a transition becomes optically thick, β_ij∝1/τ_ij and the corresponding radiative term in Equation (2) does not depend separately on n and τ_ij but only on their product $n\tau _{ij} \propto n\,n({\rm H_2O})d/\Delta v_D$ (EHM); that is, R_ij depends only on temperature T and the maser emission measure ξ, defined as

$$\begin{equation} \xi \equiv {x_{-4}({\rm H_2O})n_9^2d_{13}\over \Delta v_{D5}}. \end{equation} \tag{ 4 }$$

Here n₉ = n/(10⁹ cm⁻³), $x_{-4}({\rm H_2O})=x({\rm H_2O})/10^{-4}$ where $x({\rm H_2O})$ is the abundance of H₂O molecules relative to hydrogen nuclei, d₁₃ = d/(10¹³ cm), and $\Delta v_{D5} = \Delta v_D/(1\,{\rm km s^{-1}}$). When the transition is thermalized, as a result of further increase of either optical depth or density, the ξ-dependence disappears too, and only the temperature dependence remains.

The populations of the 45 lowest rotational levels of ortho-H₂O were solved for steady state from the set of rate equations in Equation (1). Collision cross sections are not well known. Recent calculations of H₂O rotational excitations explored the low temperature regime in cloud interiors (see Dubernet et al. 2006, and references therein), but the latest available tabulation at maser temperatures (T ≳ 200 K) is from Green et al. (1993). We employ here these cross sections, assuming for simplicity that hydrogen is purely molecular since separate atomic and molecular hydrogen coefficients are not available.

From the solution for the normalized sublevel populations, y_i, we determine the input properties of the standard maser pump model (see Appendix A). Denote by m (=1, 2) the two maser levels. All other levels constitute the maser "reservoir," and interactions with the reservoir levels populate each maser level at a pump rate per unit volume and sub-level, p_m, and deplete it at the loss rate, Γ_m. These pump terms can be read directly off Equation (1),

$$\begin{equation} \Gamma _m = n \sum _{j \ne m} R_{mj}, \quad p_m = n\,n({\rm H_2O}) \sum _{j \ne m} (g_j/g_m)\,y_jR_{jm}; \end{equation} \tag{ 5 }$$

note that the sums do not include transitions between the two maser levels, which are handled separately (see Equations (A1) and (A2), Appendix A). Here we replace the loss rates of the two maser levels, which our numerical results show to be slightly different, with the common rate Γ = (g₂Γ₂ + g₁Γ₁)/(g₂ + g₁). The three quantities p₁, p₂ and Γ fully describe the maser behavior under all circumstances; in particular, the steady-state population of each maser sub-level in the unsaturated regime is simply p_m/Γ. It is convenient to replace the individual pump rates p₂ and p₁ with the rate coefficient for their mean, q (not to be confused with the collisional rate coefficients q_ij in Equation (2)), and the inversion efficiency of the pumping scheme, η, defined from

$$\begin{equation} {1 \over 2}(p_2 + p_1) \equiv n^2 x({\rm H_2O}) q, \quad \eta \equiv {p_2 - p_1\over p_2 + p_1}. \end{equation} \tag{ 6 }$$

Note that this definition of q, which measures the pump rate per sublevel, differs from that in EHM, which was per level; that is, q_EHM∝(1/2)(g₂p₂ + g₁p₁). The functions q, η and Γ/n (= ∑_jR_mj) are expected to display the scaling property first noted in EHM: they depend only on ξ (and T) when the relevant rotational transitions become optically thick at large H₂O columns, with the density dependence totally incorporated into ξ.

Figure 1 shows our results for the three pump parameters as functions of ξ for a range of n and T relevant to observable masers. The scaling behavior of Γ/n, q and η is evident from the left column panels, which show their variation with ξ at a fixed temperature. Even though the plots span five orders of magnitude in density, all three quantities are largely independent of n whenever ξ ≳ 0.1; scaling breaks down only for η when n ≳ 10¹⁰ cm⁻³. Moreover, both Γ/n and q are further independent of ξ when ξ ≳ 0.1, indicating that all level populations are close to thermal equilibrium. The temperature variation of these pump parameters, shown in the right-column panels, is well described by the simple analytic approximations

$$\begin{equation} \Gamma _{-1} \simeq 2.6n_9\, e^{-400/T}, \quad q_{-13} \simeq 3.2\, e^{-460/T}, \end{equation} \tag{ 7 }$$

where Γ₋₁ = Γ/(10⁻¹ s⁻¹) and q₋₁₃ = q/(10⁻¹³ cm³ s⁻¹). The accuracy of both expressions is within a few percent at all T ≳ 300 K and ξ ≳ 0.1; at T = 200 K, the deviations reach only ∼25%.

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As first noted by de Jong (1973), rotation levels on the "backbone" ladder (the lowest level for each J) carry the bulk of the H₂O population and establish a thermal equilibrium among themselves. Levels off the backbone, including the J_{K − K +} = 6₁₆ and 5₂₃ maser levels, are populated predominantly by decays from higher backbone levels. This pattern leads to a number of inverted transitions, with the 22 GHz having the longest wavelength (1.35 cm) among them. Located 644 K above ground, the off-backbone 22 GHz maser system contains such a tiny fraction of the H₂O molecules (∼1% even at the highest temperatures considered here) that it can be inverted with little impact on the overall thermal distribution of level populations. The inversion occurs because small contributions from radiative decays provide sufficient competition with the collisions to maintain p₂ > p₁ over a wide range of parameters; we find that inversion is produced up to a density n = 2 × 10¹² cm⁻³, although significant suppression of the maser line occurs for n ≳ 10¹⁰ cm⁻³. The bottom panels of Figure 1 show the inversion efficiency η. The right panel shows that η is largely temperature independent for T > 200 K, while the left panel displays the scaling property first noted in EHM: when expressed as a function of ξ, η is independent of density as long as n ≲ 10¹⁰ cm⁻³. Indeed, η is well described over the entire displayed range by the analytical approximation

$$\begin{equation} \eta _{-2} \simeq {4.5\over \xi ^{0.5}}\, c_\eta \end{equation} \tag{ 8 }$$

where η₋₂ = η/10⁻² and where the correction factor

$$\begin{equation} c_\eta = \frac{1}{1 + 0.01\, n_9^{1.15}\xi ^{-0.5}}\hbox{ $\times $ }\frac{1}{1 + 0.015\,n_9^{0.2}\xi ^{1.5}} \end{equation} \tag{ 9 }$$

displays explicitly the deviations from scaling and the thermalizing, inversion-quenching effects of high densities and large optical depths. This approximation reproduces the numerical results to within ∼20% over the entire phase space volume displayed in Figure 1, except for its very edge at large ξ.

The quantities η, q and Γ fully determine the pumping scheme, enabling a complete solution of any maser model once its geometry is specified. The planar geometry of the slab is the key to strong maser action. It allows easy escape for the thermal photons through the slab thickness d, enabling inversion everywhere. Simultaneously, maser amplification in the plane can proceed along distance ad, where in principle the aspect ratio a is arbitrarily large but in practice is limited either by the curvature of the shock or by the path length in the shock plane where velocity gradients shift the component in the plane by the thermal width (see Section 5). The resulting radiation is strongly beamed in the plane of the slab, and the strongest masers will be seen from edge-on orientations. Indeed, Marvel et al. (2008) find that the outflows of the water maser associated with IRAS 4A/B in the star-forming region NGC 1333 are nearly in the plane of the sky, with inclination of only 2° for IRAS 4A and about 13° for IRAS 4B.

The general solution of planar masers is presented in EHM92. Its essentials are reproduced in Appendix B, together with a glossary of key dimensions in Table 1. The aspect ratio a is the most important geometrical property of such masers; for a circular disk with radius ℓ and thickness d (Figure 2) it is

$$\begin{equation} a = {2\ell \over d}. \end{equation} \tag{ 10 }$$

The shape of the masing material in the plane is largely irrelevant once the maser saturates. For a circular H₂O maser disk, the aspect ratio required to bring about maser saturation is

$$\begin{equation} a_{\rm sat}\simeq 3.6{n_9\over \xi ^{1/2}c_\eta }\,e^{60/T} \left[1 - {21\over T} + 0.12\ln \,{n_9\over \xi ^{1/2}}\right], \end{equation} \tag{ 11 }$$

obtained by inserting into the general expression for this geometry, reproduced in Equation (B3), the results of the H₂O pumping scheme (Equations (A8) and (A12)). For comparison, the analogous expression for a cylindrical maser with diameter d is listed in Equation (B4) in Appendix B. The two saturation aspect ratios are nearly identical in H₂O masers, the differences mostly involving small logarithmic corrections. Since the saturation condition is the same for the two extremes of planar geometrical shape, this ensures that maser saturation is controlled solely by the length of the velocity coherent region.

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Table 1. Glossary of Maser Dimensions

d	⋅⋅⋅	Thickness of the masing region; determined by the shock properties.
dthin	⋅⋅⋅	Diameter of a spherical maser that has just reached saturation; determined by the pump properties (Equation (B7)). Planar masers with $d \ < \ d_{\rm thin}$  are "thin" and can be described by the EHM92 disk maser solution when they also obey the filamentary condition a ≫ max [1, κ0d/8]. "Thick"  disk masers have $d > d_{\rm thin}$ and can be described by the solution for a saturated sphere with the same diameter as the disk.
$d_\Vert$	⋅⋅⋅	Maser observed size in the direction parallel to the shock propagation; equal to d for thin disk masers.
d⊥	⋅⋅⋅	Maser observed size in the direction perpendicular to the shock propagation; given in Equation (B6) for thin disk masers.
dℓ	⋅⋅⋅	Diameter of the observed circular shape of a thick disk maser. Given by Equation (B11) when the core is unsaturated, and Equation (B13) when it  is saturated.

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With κ₀ the unsaturated absorption coefficient, the quantity $a_{\rm sat}\,\kappa_0d$ is the maser optical depth along the disk diameter at saturation (cf. Equation (A12)); it is a measure of the amplification required along the maser longest path in order to bring saturation. As is evident from Equation (B3), this quantity has similar values for all pumping conditions, varying only logarithmically with the pumping parameters; some general arguments show that the intrinsic properties of the H₂O molecule imply $a_{\rm sat}\,\kappa_0d \sim$ 15 (E92; see also Equation (B3)). Strong masers can be expected when saturation is reached at realistic elongations, i.e., moderate aspect ratios a_sat (≲10). As we show below (see Section 4, in particular Figure 11), J shocks produce a_sat ≲ 5 over a large volume of parameter space, ensuring strong maser action for a wide range of conditions. The near constancy of a_sat over such a large parameter region implies that κ₀d (roughly proportional to 1/a_sat; see Equation (B3)) too has only moderate variation there.

As noted in EHM92, saturated masers can be distinguished by two types of beaming. For amplification-bounded masers, whose prototype is the spherical maser, the beaming angle depends on the amplification. These masers are characterized by observed sizes significantly smaller than their projected physical size. Furthermore, the observed size increases with frequency shift from line center (Elitzur 1990). Such increases have been reported in a recent study of H₂O masers around evolved stars (Richards et al. 2011). For matter-bounded masers, whose prototype is the filamentary maser, the beaming angle depends only on the geometry of the maser. They are characterized by observed sizes that are equal to their projected physical size and constant across the line profile. In principle, saturated planar masers produced by shocks can display both types of behavior. For a shock moving across the LOS (shock velocity vector v_s in the plane of the sky), denote by || the direction parallel to v_s and by ⊥ the direction orthogonal to both v_s and the LOS. The dimension of the masing medium along the ||-direction is the slab thickness, d, and the dimension along the LOS is 2ℓ = ad. Whereas d is determined by the structure of the shock, the dimensions of the masing medium in the two directions in the slab plane are controlled by other factors, such as velocity coherence and shock curvature. We term planar masers that are matter bounded in the ||-direction "thin," and those that are amplification bounded in that direction "thick."

Appendix B presents a detailed description of both thin and thick disk masers, and Table 1 provides a glossary of maser dimensions relevant for the two cases. As we shall see below, most interstellar shocks are "thin," with the maser structure as depicted in Figure 2. Let d_|| denote the observed size of the maser parallel to the shock velocity and d_⊥ the observed size in the plane of the sky normal to the shock velocity. A thin maser is matter bounded in the ||-direction and amplification bounded in the ⊥-direction, therefore d_|| = d but d_⊥ is smaller than ad, the physical size in the ⊥-direction. Inserting the results of the H₂O pumping scheme (Equation (A12)) into Equation (B6) and utilizing Equation (11), the ratio d/d_⊥ is given by

$$\begin{equation} {d\over d_\perp} \simeq {3.3\over a_{\rm sat}}\hbox{ $\times $ }\frac{\displaystyle 1 - {21\over T} + 0.12\ln \,{n_9\over \xi ^{1/2}}}{\displaystyle \left[1 - {14\over T} + 0.26\ln \,{n_9\over (\xi c_\eta)^{1/2}} - 0.13\ln a\right]^{1/2}}. \end{equation} \tag{ 12 }$$

The maser will appear elongated either in the plane of the shock or along the shock propagation, depending on the value of a_sat that the pumping scheme generates.

We now discuss the predictions of the H₂O pumping scheme for observable radiative quantities. These results are applicable only for the emission from resolved individual maser spots. The brightness temperature at line center is given by Equations (B14) and (B20), respectively, for the unsaturated and saturated regimes. While the pump properties can be specified in terms of density-independent scaling quantities, the onset of saturation does involve the density (Equation (11)). Figure 3 shows the variation of brightness temperature with ξ for a wide range of densities for disk masers with a = 10, chosen for illustration; the behavior for other aspect ratios can be deduced from the explicit expressions for T_b shown below. Each curve shows a steep exponential rise at the low-ξ end, corresponding to unsaturated maser growth. The break in the slope marks the onset of saturation, and the behavior of T_b at higher values of ξ is controlled by the variation of the maser pump properties. Saturation is reached for all the displayed densities except for n = 4 × 10⁹ cm⁻³, which falls just short of saturation—in that case a_sat = 13 around the peak of the T_b curve. Therefore, n ≃ 3 × 10⁹ cm⁻³ is the highest density that produces saturated disk masers with a = 10 at T = 400 K. The curves for n = 10⁷ and 10⁸ cm⁻³ show an additional break at ξ = 0.01 and 0.3, respectively. This break marks the transition to a thick disk regime, a transition that occurs only at lower densities (see Equation (B8)). Masers with n ⩾ 6 × 10⁸ cm⁻³ are in the thin-disk domain for all values of ξ.

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On each curve in the figure, an "X" marks the value of ξ generated by a J shock with v_A⊥ = 1 km s⁻¹ and Δv_D = 1 km s⁻¹ that produces a maser density corresponding to that curve, as described below (see Section 3). This shows that J shocks with a = 10 produce saturated thin-disk masers in the entire range 10⁷ cm⁻³ ⩽ n ⩽ 3 × 10⁹ cm⁻³. For thin-disk masers with a > a_sat, the brightness temperature at line center can be written as

$$\begin{equation} T_b \simeq 4.7\hbox{ $\times $ }\hbox{10^{11}}\,\xi ^{1/2}c_\eta (d/d_\perp) e^{-460/T}a_1^3\, {\rm K}, \end{equation} \tag{ 13 }$$

where a₁ = a/10 (see Equation (B22)). This result amplifies our earlier findings (EHM; EHM92): apart from the gas temperature, the brightness temperature of J-shock produced masers is determined almost exclusively by ξ and the maser aspect ratio a. Further discussion of this effect appears in Section 4.1. The relevant quantity for maser detectability from a distance D_kpc in kpc is its observed monochromatic flux. From Equation (B19), the observed flux at line center is

$$\begin{equation} F_{\nu _0} \simeq 74\,\xi ^{1/2}c_\eta e^{-460/T} {d_{13}^2\over D_{\rm kpc}^2}a_1^3 \ \rm Jy. \end{equation} \tag{ 14 }$$

Measured flux is frequently expressed in terms of the equivalent isotropic luminosity L_iso = 4πD²F, where F is the flux integrated over the spectral range of the maser feature. Then

$$\begin{eqnarray} L_{\rm iso} &=& 4\pi D^2F_{\nu 0}\cdot \frac{\pi ^{1/2}\Delta v_D}{\lambda } \nonumber\\ & \simeq & 3\hbox{ $\times $ }\hbox{10^{-6}}\xi ^{1/2}c_\eta e^{-460/T}\Delta v_{D5}d_{13}^2 a_1^3\, L_\odot. \quad \end{eqnarray} \tag{ 15 }$$

Because of the beaming, the actual luminosity of a planar maser is only L_m = L_iso/2a (EHM92).

A number of authors, including Hollenbach & McKee (1979, 1989, hereafter HM79, HM89), Neufeld & Dalgarno (1989), Neufeld & Hollenbach (1994), Smith & Rosen (2003), Guillet et al. (2009), and Flower & Pineau Des Forêts (2010), have discussed J-shock structure in dense molecular gas. EHM discussed the particular structure found in the very dense J shocks that may give rise to 22 GHz water masers. In fast J shocks, the molecules are first completely dissociated by the extremely high postshock temperatures (∼10⁵ K) immediately behind the shock front. In this very hot region, dust may be partially or totally destroyed by thermal sublimation, sputtering, and grain-grain collisions. Further downstream, where the material cools down, H₂ molecules reform on the surviving dust grains and are ejected to the gas phase with sizable internal energies, which provides a source of heating for the gas if the postshock densities are sufficiently high (≳ 10⁶ cm⁻³) to convert this internal energy into heat. In other words, the rovibrationally excited H₂ molecule needs to be collisionally de-excited, rather than suffering radiative decay, for the energy to be converted to heat. This heating produces an "H₂ re-formation plateau," a nearly isothermal column of gas at a temperature T_p ∼ 300–400 K. The plateau gas is warm enough to drive all oxygen not locked in CO to form H₂O and to collisionally populate the 22 GHz maser levels, which lie 644 K above ground. The dust temperature in the masing region is typically 50–100 K. The hydrogen column density, N_p, of the heated plateau region can be as large as ∼10^22–23 cm⁻², and the H₂O column as high as 3 × 10¹⁹ cm⁻².⁴ The H₂ re-formation plateau is an ideal site for relatively low-lying H₂O masers like the 22 GHz masers; the temperature T_p may be too low to significantly excite higher excitation H₂O maser levels, and C shocks have been proposed as sites of those masers (Melnick et al. 1993; Kaufman & Neufeld 1996).

In both C and J shocks, the component of the magnetic field normal to the shock velocity, B_0⊥, serves to limit the compression of the postshock gas. This component of the magnetic field is related to the corresponding preshock Alfvén speed v_A⊥ by

$$\begin{equation} B_{0\perp } = 1.7 v_{A\perp,5}n_{0,7}^{1/2}\ {\rm mG}, \end{equation} \tag{ 16 }$$

where v_{A⊥, 5} = v_A⊥/(1 km s⁻¹) and where n_{0, 7} = n₀/(10⁷ cm⁻³) is the density of hydrogen nuclei in the preshock gas [i.e., n₀ = n₀(H)+2n₀(H₂)]. If the shock velocity and the orientation of the magnetic field are uncorrelated, the median value of B_0⊥ equals (√3/2)B₀, so the distinction between B₀ and B_0⊥ is not numerically important. Nonetheless, we shall retain this distinction here since future observations might be able to determine the relative orientations of the shock velocity and the upstream magnetic field. Typical preshock magnetic fields in molecular clouds of widely varying density are characterized by preshock Alfvén speeds v_A ∼ 1–2 km s⁻¹ (Heiles et al. 1993). Fields at high densities have recently been measured by Falgarone et al. (2008) who observed the Zeeman effect in CN. For the 8 measurements with positive detections, the median value of v_A5 is 1. Including the 6 measurements with no detections, but using the quoted error as the value, we find a median of 0.6. The dispersion is large, however: a factor six. Correcting for inclination, we estimate v_A5 ≃ 1 ± 0.8 dex. This is quite crude, however, since our treatment of the upper limits is very approximate and since the Zeeman technique averages over fluctuations in the LOS field. As noted above, typically v_A⊥ ≃ v_A if the orientations of the shock velocity and the magnetic field are uncorrelated, so we shall adopt v_{A⊥, 5} = 1 as a fiducial value.

The density n_p in the masing ("plateau") region of a J shock⁵ is usually limited by the value of B_0⊥ (HM79), and can be written as n_p = √2n₀v_s/v_A⊥. In terms of the flux of H nuclei through the shock, j ≡ n₀v_s, we have

$$\begin{equation} n_{p9}\equiv \frac{n_p}{10^9\ {\rm cm^{-3}}} = 1.4 \left(\frac{j_{14}}{v_{A\perp,5}}\right), \end{equation} \tag{ 17 }$$

where

$$\begin{equation} j_{14}\equiv n_{0,7}v_{s7}, \end{equation} \tag{ 18 }$$

v_s7 ≡ v_s/(100 km s⁻¹) is the shock speed in units of 10⁷ cm s⁻¹, and j₁₄ = j/(10¹⁴ cm⁻² s⁻¹). We shall find that many of masing parameters mainly depend on j. The magnetic field in the masing region of a J shock balances the ram pressure of the shock and is therefore independent of the preshock field,

$$\begin{equation} B_p\simeq 0.24n_{0,7}^{1/2}v_{s7}\ \mbox{G} = 0.24j_{14}^{1/2}v_{s7}^{1/2}\ \mbox{G} = 0.24j_{14}n_{0,7}^{-1/2} \ \mbox{G}. \end{equation} \tag{ 19 }$$

The analytic formulae for the density n_p and the magnetic field B_p in the maser region apply when the magnetic pressure dominates there, or when $v_{A\perp,5}\gtrsim 2\times 10^{-2} v_{s7}^{-1}$, assuming the plateau temperature is 300–400 K (HM79; EHM). Since v_{A⊥, 5} is usually ≳ 0.2, this condition is readily met.

Table 2 summarizes analytic solutions and approximations previously obtained in HM79, HM89, and EHM, or, in the case of T_b and L_iso, taken from Section 2. We define T_{b, 11} ≡ T_b/10¹¹ K and L_{iso, -6} ≡ L_iso/(10⁻⁶ L_☉). The factor γ = 10⁻¹⁷γ₋₁₇ cm³ s⁻¹ is the average rate coefficient for H₂ formation on grains in the temperature plateau region. In our numerical shock computations we use the formulation for γ from HM79. Note that because of the gas and dust temperature sensitivity of γ and because partial destruction of dust near the shock front reduces the area of grains and therefore γ, there is a "hidden" additional dependence on n₀ and v_s, or on d and v_A⊥, when γ appears in Table 2. The same holds true for x₋₄(H₂O). However, Table 2 is general to any formulation for γ and for any postshock H₂O abundance, in contrast to the tables presented later in Section 4.2 that are specific to our particular formulation of γ and to our shock chemistry, which derives x₋₄(H₂O) at each point in the postshock gas. The column density in the masing region (or the H₂ re-formation plateau region), N_p, is analytically determined by finding the timescale $t_{\rm H_2}$ for H₂ formation in the plateau and then taking $N_p=n_0v_s t_{\rm H_2}$ (EHM). The timescale $t_{\rm H_2}$ is given by

$$\begin{equation} t_{\rm H_2}={1\over {n\gamma }} \end{equation} \tag{ 20 }$$

leading then to

$$\begin{equation} N_p= 7\times 10^{21} \left({v_{A\perp,5} \over \gamma _{-17}}\right) \ {\rm cm^{-2}}. \end{equation} \tag{ 21 }$$

The thickness d of the postshock masing region (the spot size parallel to the shock velocity) is given by

$$\begin{equation} d=\frac{N_p}{n_p}=5.0\times 10^{12}\;\frac{v_{A\perp,5}^2}{\gamma _{-17} j_{14}}. \end{equation} \tag{ 22 }$$

The formulae for N_p and d are quite accurate when applied to the numerical results, but require a knowledge of γ₋₁₇. The value of γ₋₁₇ is of order 0.1–3.0 (HM79) for the gas and dust temperatures typical of the masing plateau. Note that the column density in the plateau is independent of the preshock density and shock velocity or of j, for fixed γ. The analytic formulae for ξ, T_b, and L_iso in the masing region come from Equations (4), (13), and (15), respectively. The expressions for n_p, T_p, d, ξ, T_b and L_iso in Table 2 depend only on j and not separately on n₀ and v_s. The parameter B_p is the sole parameter dependent on the additional parameter v_s, but only as the square root.

Table 2. Equations from Analytic Shock and Saturated Maser Slab Model

Parameter	Preshock Variables j = n0vs and vA⊥	Observable Variables d, d/d⊥ and vA⊥
np	$1.4\times 10^9\left({\displaystyle j_{14} \over \displaystyle v_{A\perp,5}}\right)$ cm−3	$7\times 10^8\left({\displaystyle v_{A\perp,5} \over \displaystyle \gamma _{-17}d_{13}}\right)$ cm−3
Bpa	$0.24j_{14}^{1/2}v_{s7}^{1/2}$ G	$0.17\left({\displaystyle v_{A\perp,5}v_{s7}^{1/2} \over \displaystyle \gamma _{-17}^{1/2}d_{13}^{1/2}} \right)$ G
Np	$7\times 10^{21}\left({\displaystyle v_{A\perp,5} \over \displaystyle \gamma _{-17}}\right)$ cm−2	$7\times 10^{21}\left({\displaystyle v_{A\perp,5} \over \displaystyle \gamma _{-17}}\right)$ cm−2
d	$5\times 10^{12}\left({\displaystyle v_{A\perp,5}^2 \over \displaystyle \gamma _{-17}j_{14}}\right)$ cm	d
j14	j14	$0.5 \left({\displaystyle v_{A\perp,5}^2 \over \displaystyle \gamma _{-17}d_{13}}\right)$
ξ	$1.0\left[{\displaystyle x_{-4}({\rm H_2O})j_{14} \over \displaystyle \gamma _{-17}\Delta v_{D5}} \right]$	$0.5\left[{\displaystyle x_{-4}({\rm H_2O})v_{A\perp,5}^2 \over \displaystyle \gamma _{-17}^2d_{13}\Delta v_{D5} }\right]$
$T_{b,11}^+$	$4.7\left[{\displaystyle x_{-4}({\rm H_2O})j_{14} \over \displaystyle \gamma _{-17}\Delta v_{D5}}\right]^{1/2}\left({d\over d_\perp }\right) c_\eta e^{-460 {\rm K}/T_p} a_1^3$	$3.3 \left[{\displaystyle x_{-4}({\rm H_2O})v_{A\perp,5}^2 \over \displaystyle \gamma _{-17}^2 d_{13}\Delta v_{D5}}\right]^{1/2}\left({d\over d_\perp }\right) c_\eta e^{-460 {\rm K}/T_p} a_1^3$
$L_{\rm iso,-6}^+$	$0.75\left[{\displaystyle x_{-4}({\rm H_2O})^{0.5} \Delta v_{D5}^{0.5}v_{A\perp,5}^4 \over \displaystyle \gamma _{-17}^{2.5}(j_{14})^{1.5}}\right] c_\eta e^{-460 {\rm K}/T_p} a_1^3$	$2.1\left[{\displaystyle x_{-4}({\rm H_2O})\Delta v_{D5}v_{A\perp,5}^2 d_{13}^3 \over \displaystyle \gamma _{-17}^2}\right]^{1/2} c_\eta e^{-460 {\rm K}/T_p} a_1^3$

Notes. ^aNote that B_p is the only one of these parameters to depend directly on the shock velocity v_s in addition to the indirect dependence through j. ⁺ These expressions for T_b and L_iso are only valid if a > a_sat (see Equation (31)). Recall that a₁ = a/10.

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Table 2 tabulates quantities in terms of the preshock variables j and v_A⊥ in column 1 and in terms of the observable quantities d, d/d_⊥ (the "shape" of the maser spot) and v_A⊥ in column 2. Essentially, we have eliminated j = n₀v_s from column 1 in favor of d in column 2. Column 2 is added to aid observers in estimating the shock parameters. However, it must be noted that the average values of γ₋₁₇, x₋₄(H₂O), and Δv_D5 in the masing plateau appear in these equations. As we will see in the subsections below, these all have values near unity for most cases, enabling an estimate to be made of the shock parameters. The numerical results presented in this paper can be understood, interpolated and extrapolated by applying these formulae.

HM79, HM89, and Neufeld & Hollenbach (1994) describe in detail the physical processes included in the one-dimensional steady state shock code we have used in this paper. The fundamental input parameters to this code are the preshock hydrogen nucleus density n₀, the shock velocity v_s, the Alfvén speed v_A⊥ in the preshock gas, the velocity dispersion Δv_D in the line-emitting gas, and the gas phase abundances of the elements. In our standard runs we take v_A⊥ = 1 km s⁻¹, Δv_D = 1 km s⁻¹, and gas phase abundances listed in HM89 (the main number abundances relative to hydrogen nuclei that are relevant here are those for carbon, 2.3 × 10⁻⁴, and oxygen, 5.4 × 10⁻⁴).

The code uses the Rankine–Hugoniot jump conditions to set the physical parameters immediately behind the shock front, and the various continuity equations to numerically solve for the temperature, density and chemical structure in the cooling postshock gas. The chemistry includes 35 species and about 300 reactions. For the fast (v_s ≳ 20 km s⁻¹), dense (n₀ ≳ 10⁵ cm⁻³) shocks considered in this paper, important chemical processes include collisional dissociation and ionization, photodissociation and photoionization by the UV photons produced in the (upstream) hot postshock gas, neutral-neutral reactions with activation barriers, and the re-formation of H₂ molecules on warm (T_gr ≃ 50–150 K) dust grains. All but the last are either well determined experimentally or well understood theoretically.

We discuss here the formation rate coefficient of H₂ on warm dust grains in some detail, as this process is critical to forming the high temperature plateau where the H₂O maser is produced. We use the theoretical model of HM79 for the formation of H₂ on warm grains. In this formulation the formation rate coefficient γ is a function of both the gas temperature T and the dust temperature T_gr. At relatively low T (≲ 100 K) and T_gr (≲ 30 K), the rate coefficient has been inferred observationally in diffuse clouds and molecular cloud surfaces to be ∼3 × 10⁻¹⁷ cm³ s⁻¹. At the somewhat higher gas temperatures T ∼ 400 K in the plateau, the coefficient drops by about a factor of ∼2 due to the decreased sticking probability of incoming H atoms (e.g., HM79, Cuppen et al. 2010). However, a more important effect in the plateau is caused by the increased T_gr ∼ 100 K, which causes γ to drop even more because of the evaporation of H atoms from grain surfaces prior to H₂ formation. The exact amount of this drop cannot be well determined for realistic interstellar dust. However, HM79, Cuppen et al. (2010), and Cazaux et al. (2011) have used theoretical modeling to try to estimate the effect. All three of these studies are in quite good agreement, given the inherent uncertainties. Including both the sticking probability and the probability of H₂ formation on the grain surface, and normalizing to obtain the above standard rate at low T and T_gr, Cuppen et al. get a H₂ rate coefficient for 400 K gas and 100 K dust of 2.2 × 10⁻¹⁸ cm³ s⁻¹, whereas HM79 find 3.8 × 10⁻¹⁸ cm³ s⁻¹. In addition, Cazaux et al. find a rate coefficient for 400 K gas and 125 K dust of 1.5 × 10⁻¹⁸ cm³ s⁻¹, whereas HM79 find 1.4 × 10⁻¹⁸ cm³ s⁻¹. Therefore, the HM79 formation rate coefficient agrees well with the more recently obtained values in the region of parameter space (T ∼ 400 K, T_gr ∼ 100 K) where the H₂ re-forms in the postshock gas, and where the H₂O maser is produced. This agreement is far better than the uncertainties in these models, and therefore there could be fairly large differences between these values and the values for real shocked grains at high dust temperatures. Because of these uncertainties, we consider the sensitivity of our results to the H₂ formation rate coefficient in the next subsection.

Our model also includes the partial destruction of dust grains in the shock, which reduces the grain surface area per H nucleus, and which therefore also reduces the rate coefficient for H₂ formation on grains. Our J-shock maser model relies on at least some grains surviving the shock, since the H₂ re-formation plateau is caused by H₂ re-formation on grain surfaces. However, shocks with v_s ≳ 200 km s⁻¹ will completely destroy dust grains by sputtering and grain-grain collisions (Jones et al. 1996). We therefore only consider shocks with v_s < 200 km s⁻¹. In addition, shocks with $n_{0,7} v_{s7}^3 \gtrsim 100$ will sublimate all grains with sublimation temperatures T_sub ≲ 1500 K, which is the maximum sublimation temperature of a likely interstellar grain material. Therefore, no dust exists above this constraint as well. For n₀ and v_s values that are low enough to provide at least some dust survival, we adopt the grain composition mixture of Pollack et al. (1994), and allow for the sublimation of the less refractory material at lower values of $n_{0,7} v_{s7}^3$ as each sublimation temperature is exceeded.

In summary, the conditions $n_{0,7} v_{s7}^3 \lesssim 100$ and v_s7 ≲ 2 provide upper limits for J-shock masers produced by the H₂ re-formation plateau. We find below, however, somewhat more stringent conditions on preshock density occur due to the quenching of the H₂O maser by high postshock densities and H₂O line optical depths. An example of this is shown in Figure 3, which shows the quenching that occurs in the case of slabs with a = 10 and v_{A⊥, 5} = Δv_D5 = 1; the maser is quenched for n_p ⩾ 4 × 10⁹ cm⁻³ since then $a_{\rm sat}>a$. The upper limit on the preshock density for effective maser emission in this particular case is therefore n₀ ≲ 3 × 10⁷(v_{A⊥, 5}/v_s7)cm⁻³ from Equation (17).

The cooling of the postshock gas is treated with the escape probability formalism (including the effects of dust absorption), since a number of important cooling transitions become optically thick in the lines. For this study, we focus mainly on the postshock temperature region bounded by 3000 K >T > 50 K, where molecular formation occurs and H₂O masers may be produced. In the masing region, the gas cooling is dominated by optically thick rotational transitions of H₂O and by gas collisions with the cooler dust grains. The gas heating in the masing region is dominated by the re-formation of H₂. There are two main contributions to this heating process. The newly formed molecules can be ejected from the grain surfaces with kinetic energies greater than kT, thereby heating the gas. In addition, a newly formed and ejected molecule may carry with it rovibrational energy that can be transferred to heat by collisional de-excitation in the gas. These processes are not well determined. We adopt the theoretical formulation of HM79, in which the newly formed molecule is ejected with 0.2 eV of kinetic energy and 4.2 eV of rovibrational energy and use the de-excitation rate coefficients for H and H₂ collisions quoted in HM79. However, Tielens & Allamandola (1987) speculate that the H₂ molecule may lose a significant portion of its formation energy (the rovibrational energy) to the grain, before leaving the grain surface. Since the formation heating is proportional to the H₂ formation rate times the energy delivered per H₂ to the gas, we test the sensitivity of our results to the energy partition when we test the sensitivity of the plateau temperature to the uncertain formation rate coefficient.

Figure 4 presents the shock profile for our standard model: n₀ = 10⁷ cm⁻³, v_s = 100 km s⁻¹, v_A⊥ = 1 km s⁻¹, and Δv_D = 1 km s⁻¹. The column density of hydrogen nuclei, N, and the position, z, are measured from the shock front. The ultraviolet radiation from the shock has processed the preshock gas before it enters the shock front. As a result, the molecular preshock gas is photodissociated and partially photoionized prior to being shocked. Using the results of HM89, we take the initial abundances at the shock front for the standard case to be x(H⁺) = 0.47, x(H) = 0.36, x(H₂) = 0.087; the trace species are largely atomic and singly ionized as well. For slower shocks, the precursor field is less important and the shock front abundances are initially largely molecular. The gas then collisionally dissociates and partially ionizes in the hot postshock gas just downstream of the shock front. Figure 4 shows that the 100 km s⁻¹ shock heats the plasma to about 2 × 10⁵ K, and the gas cools by collisional ionization and by UV and optical emission to 10⁴ K in a column N ≃ 4 × 10¹⁷ cm⁻². The Lyman continuum photons from the ∼10⁵ K gas maintain a Strömgren region at T ∼ 10,000 K to a column N ≃ 10^19.5 cm⁻². Once the Lyman continuum photons are absorbed, the electrons and protons recombine and the gas cools until the heating due to H₂ re-formation maintains the temperature at T_p ≃ 300–400 K. This is the "H₂ re-formation plateau." Note that the size scale of this plateau, shown at the top of the figure, is d ∼ 10¹³ cm for v_A⊥ = 1 km⁻¹ and for our assumed formulation for γ₋₁₇. After the molecular hydrogen has nearly completely reformed, at a column of about 2 × 10²² cm⁻², the heating rate drops and the gas temperature drops to ≲ 100 K.

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The H₂O number abundance relative to hydrogen nuclei, x(H₂O), is also plotted in Figure 4. The abundance is negligible for N ≲ 10²¹ cm⁻², but the H₂O abundance rapidly climbs once the H₂ abundance rises in the re-formation plateau. CO re-forms even more rapidly; typically all the gas phase carbon is incorporated into CO once x(H₂) ≳ 10⁻³. Therefore, the abundance of H₂O is limited to the abundance of oxygen that remains once an oxygen atom has combined with every gas phase carbon atom. We have taken for elemental gas phase abundances x_O = 5.4 × 10⁻⁴ and x_C = 2.3 × 10⁻⁴; thus, x_max(H₂O) ≃ 3 × 10⁻⁴. Typically, x(H₂O) ≃ x_max(H₂O) once x(H₂) ≳ 0.25. However, the reactions that lead to H₂O have large activation energies (ΔE/k ∼ 4000 K) and proceed slowly in the plateau region. In many cases the timescales spent in the plateau are insufficient to reach chemical equilibrium; as a result, the H₂O abundance varies somewhat with T_p and N_p and consequently with n₀ and v_s, as shall be demonstrated below.

We have also plotted the grain temperature, T_gr, in Figure 4 to emphasize the fact that the grain temperature is significantly below the gas temperature in the postshock gas if $n_{0,7} v_{s7}^3 \lesssim 100$ (HM79). The grains are only weakly coupled to the gas through gas collisions and through the line radiation from the gas. At the same time, radiative grain cooling is very efficient; the result is that the grains are considerably cooler than the gas. Because the dust is optically thin in the H₂O rotational transitions and the lines themselves have finite opacity, the effective temperature of the radiation field is cooler than the gas kinetic temperature. Consequently, collisions with H atoms and H₂ molecules excite the H₂O and the escaping IR photons from H₂O rotational transitions create non-LTE populations and the population inversion of the maser levels. Absorption by dust competes with escape of the IR photons when the dust optical depth reaches unity for the IR photons; for ${\rm H_2O}$ IR photons with typical wavelengths of 50 μm, this occurs at a column density N_p ≃ 3 × 10²³ cm⁻² if we account for some reduction in dust abundance in the shock (HM79). Using the expression for N_p in Table 2, we estimate that dust absorption is not important for

$$\begin{equation} {\gamma _{-17} \over v_{A\perp,5}} \gtrsim 0.02, \end{equation} \tag{ 23 }$$

which is generally the case. If dust absorption is important, and the dust is cooler than the gas, the presence of dust enhances the effective escape probability of the H₂O IR photons (Collison & Watson 1995).

Figure 5 plots contours of T_p, the temperature of the H₂ re-formation plateau, as a function of the shock parameters n₀ and v_s. The gas temperature declines slightly with N in the H₂ re-formation plateau; we have defined T_p as the temperature of the gas when x(H₂) = 0.375 (i.e., when 75% of the hydrogen is molecular). Figure 5 and the subsequent three figures are the results of a grid of shock models (n₀ = 10⁵, 10⁶, 10⁷, 10⁸, and 10⁹ cm⁻³; v_s = 20, 40, 80, 100, and 160 km s⁻¹; v_A⊥ = 1 km s⁻¹; Δv_D = 1 km s⁻¹). This grid is sufficiently coarse that the contours are somewhat approximate. As noted above, we have taken v_s = 160 km s⁻¹ as our upper limit because shocks with v_s ≳ 200 km s⁻¹ destroy essentially all of the dust grains. Once there are no grain surfaces upon which to form H₂, molecular re-formation in the postshock gas effectively ceases, no warm H₂O is produced, and postshock H₂O masing action is destroyed. In addition, we have carried out calculations up to n₀ = 10⁹ cm⁻³ since dust grains sublimate at higher preshock densities, but in fact maser emission is generally quenched at considerably lower pre-shock densities as discussed above.

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The main results from Figure 5 are that T_p ≃ 300–400 K and that T_p is very insensitive to n₀ and v_s as long as n₀ ≳ 10⁵ cm⁻³ and v_s ≳ 30 km s⁻¹. EHM discussed this insensitivity as due to the balance between the gas heating by H₂ formation being balanced by H₂O and grain cooling of the gas. An analytic fit to the numerical results gives:

$$\begin{equation} T_p \simeq 350n_{0,7}^{0.12}v_{s7}^{-0.12}\Delta v_{D5}^{-0.22}\ {\rm K}, \end{equation} \tag{ 24 }$$

accurate to a factor of 1.3 for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹. For preshock densities n₀ ≲ 10⁵ cm⁻³, the H₂ formation heating rapidly drops because the newly formed, vibrationally excited H₂ molecules radiate away their vibrational energy before collisions can transform this excitation energy into heat. However, for n₀ ≳ 10⁵ cm⁻³, the H₂ re-formation plateau provides a temperature environment where the chemical production of H₂O is efficient and where collisional excitation of the 22 GHz H₂O maser, which lies 644 K above ground, is possible.

Figure 6 provides contour plots of the H₂O abundance at the point where x(H₂) = 0.375 and x(H) = 0.25, the same position in the re-formation plateau where we measure T_p. The main result is that an appreciable fraction of the available oxygen is converted to water for n₀ ≳ 10⁶ cm⁻³. At n₀ = 10⁵ cm⁻³, the water abundance is only ∼10⁻⁵, caused by a combination of lower plateau temperature (see Figure 4) and lower plateau column density N_p. The former suppresses the rate of H₂O formation because of the activation barriers present in this process. The latter reduces the protective shielding by the dust of the dissociating UV photons, and reduces the time available for H₂O to form. However, for n₀ ≳ 10⁶ cm⁻³, a simple analytic fit to the numerical results gives:

$$\begin{equation} x_{-4}({\rm H_2O}) \simeq 1.6n_{0,7}^{0.2}v_{s7}^{-0.3}, \end{equation} \tag{ 25 }$$

accurate to a factor of 1.2 for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹.

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The postshock density, n_p, as a function of n₀ and v_s in the masing plateau is accurately given by the equation in Table 2, and we therefore do not present a contour plot for it. However, d depends on γ₋₁₇ and ξ depends on γ₋₁₇ and x(H₂O), and therefore these parameters are accurately determined only by numerical solutions of shock structure. Figure 7 plots the thickness d₁₃ = d/(10¹³ cm) of the masing plateau as a function of n₀ and v_s. Comparing the numerical results with the equation in Table 2, we see that γ₋₁₇ declines as n₀ and v_s increase, because denser and faster shocks have higher grain temperatures, which reduces the rate of H₂ formation on grain surfaces. In addition, there is reduction in grain area at high values of $n_{0,7} v_{s7}^3$ due to partial sublimation of grains. A simple fit to the numerical results gives

$$\begin{equation} d_{13}\simeq 1.3n_{0,7}^{-0.7}v_{s7}^{-0.2}v_{A\perp,5}^2, \end{equation} \tag{ 26 }$$

$$\begin{equation} \gamma _{-17}\simeq 0.38n_{0,7}^{-0.3}v_{s7}^{-0.8}, \end{equation} \tag{ 27 }$$

accurate to a factor of 1.5 for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹. We note (see also Table 2) that $d \propto v_{A\perp }^2$, so that the maser spot size may be the observable parameter that is most sensitive to the strength of the preshock magnetic field. Typical preshock densities of n_{0, 7} ∼ 0.1–1 and observed maser spot sizes of ∼10¹³–10¹⁴ cm imply that v_A⊥ ∼ 1 km s⁻¹, in line with the observations discussed above in Section 3.1.

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Figure 8 presents the contours of the maser emission measure ξ (measured from the shock front to the point in the postshock plateau where x(H₂) = 0.375) as a function of n₀ and v_s for v_{A⊥, 5} = 1 and Δv_D5 = 1. The main result is that ξ increases monotonically with increasing n₀, as would be expected from its dependence on the relevant parameters seen in Table 2. From this one might predict that denser masers will be brighter; however, the pump efficiency η decreases with increasing n₀ and ultimately the maser quenches as collisions and line trapping create LTE conditions (see Section 2). A simple fit to the numerical results gives

$$\begin{equation} \xi \simeq 4.0(n_{0,7}v_{s7})^{1.5}\Delta v_{D5}^{-1}=4.0j_{14}^{1.5} \Delta v_{D5}^{-1}, \end{equation} \tag{ 28 }$$

accurate to a factor of 1.4 for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹.

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Utilizing Equations (8), (9), (17), and (28) we find good fits⁶ to the results of the numerical shock runs for c_η and η:

$$\begin{equation} c_\eta \simeq \left({1+0.045{j_{14}^{2.8} \over {v_{A\perp,5}^{0.2} \Delta v_{D5}^{1.5}}}}\right)^{-1}, \end{equation} \tag{ 29 }$$

$$\begin{equation} \eta \simeq 0.032{{\Delta v_{D5}^{0.5} c_\eta } \over {j_{14}^{0.75}}}, \end{equation} \tag{ 30 }$$

which are accurate to within a factor of 1.2 for c_η and 1.4 for η in the parameter range 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹. The fit to c_η will be useful in subsequent analytic fits to a_sat, d/d_⊥, T_b and L_iso.

Figures 4–8 are valid for v_{A⊥, 5} = 1, which corresponds with measured values (to within a factor ∼6) for a wide range of cloud densities in the Galaxy. However, as discussed earlier, there may be environments (such as very dense gas or the nuclei of galaxies) where v_{A⊥, 5} deviates substantially from unity. Therefore, in Figure 9 we have plotted the variation of d, T_p, and ξ as functions of v_A⊥ for the standard case n₀ = 10⁷ cm⁻³, v_s = 100 km s⁻¹ and Δv_D5 = 1. For convenience, we have plotted the ratios of these parameters to their values d_std, T_p, std, and ξ_std at the standard v_{A⊥, 5} = 1. The results follow the predictions from Table 2: d varies as $v_{A\perp }^2$ whereas T_p and ξ are relatively insensitive to v_A⊥.

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Figures 4–9 assume the HM79 model for the formation rate γ of H₂ molecules on grains and for the kinetic and rovibrational energy delivered to the gas per H₂ formation. Because the formation process is uncertain, we test the sensitivity of the results to variations in these parameters. Since the heating rate is proportional to the formation rate, we vary only γ in this test. In the HM79 model γ is a complicated function of gas and grain temperature. Figure 10 presents the results of models with constant γ and shows the sensitivity of T_p and ξ to variations in γ. In this figure we test only the standard case (n_{0, 7} = 1, v_s7 = 1, v_{A⊥, 5} = 1, and Δv_D5 = 1). The plateau temperature varies slowly with γ, changing from 180 K to 550 K as γ increases from 3 × 10⁻¹⁹ cm³ s⁻¹ to 3 × 10⁻¹⁷ cm³ s⁻¹, where the latter corresponds to the maximum H₂ formation efficiency on grains. The emission measure ξ drops rapidly for decreasing γ ≲ 3 × 10⁻¹⁸ cm³ s⁻¹, because of the inefficient production of H₂O when the plateau temperature drops below ∼250 K. The water abundance drops from ∼3 × 10⁻⁴ for γ ∼ 3 × 10⁻¹⁸ cm³ s⁻¹ to ∼3 × 10⁻⁷ for γ ∼ 3 × 10⁻¹⁹ cm³ s⁻¹. Therefore, referring to the equation for ξ in Table 2, ξ∝x(H₂O)/γ decreases by a factor of ∼100 over this range.

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Perhaps the most important parameter for the application to J-shock models is the aspect ratio of the maser (the ratio of the length along the LOS to the thickness) required for saturation, a_sat, which is shown in Figure 11. Shock-produced astrophysical H₂O masers are weak and unobservable as long as they remain unsaturated, so that we shall require $a > a_{\rm sat}$. However, the coherence length in the shock plane is finite and a cannot exceed ∼30–100 because of the curvature of the shock and because of velocity gradients in the plane (e.g., see Section 5). Therefore, $a_{\rm sat}\lesssim 30\hbox{--}100$ is a constraint on observable H₂O masers produced in J shocks. Figure 11 shows the dependence of a_sat on n₀ and v_s for v_{A⊥, 5} = 1. When n₀ ≲ 10⁶ cm⁻³, a_sat rapidly increases to ≳ 100 because of the low value of ξ in the shock due to low densities and low values of x₋₄(H₂O). Since such large aspect ratios are extremely unlikely in astrophysical shocks, we conclude that low-density shocks cannot produce saturated masers beamed in the shock plane and that, therefore, such shocks will produce weak, unsaturated masers that are difficult to detect. They are "starved" for sufficient collisions to the highly excited states that feed the maser. At the other extreme, when n₀ ≳ 10⁸ cm⁻³, a_sat again rapidly increases to ≳ 100 because the maser quenches (levels approach LTE and the inversion is weak) and large coherence paths are needed to reach saturation. At such high preshock densities the plateau density n_p ≳ 4 × 10⁹ cm⁻³ and ξ ≳ 10 (see Table 2 and Figure 8), and therefore quenching is significant (e.g., see η in Figure 1 or the quenching of T_b at high ξ and n in Figure 3). When 10⁶ cm⁻³ ≲ n₀ ≲ 3 × 10⁷(v_{A⊥, 5}/v_s7) cm⁻³ and 0.3 ≲ v_s7 ≲ 1.6, a_sat ≃ 1–10. An analytic fit to a_sat can be obtained using Equations (11), (17), and (28) as guides:

$$\begin{equation} a_{\rm sat} \simeq 2.5 \;{j_{14}^{0.25} \Delta v_{D5}^{0.5}\over {v_{A\perp,5} c_\eta }}, \end{equation} \tag{ 31 }$$

where the expression for c_η given in Equation (29) completes the analytic fit. This expression is good to a factor of 1.3 over the main maser parameter space 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹.

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The ratio of the maser spot diameter in the parallel direction to that in the ⊥ direction, d/d_⊥, behind J shocks is another observational diagnostic of the shock conditions. The shape d/d_⊥ of the maser is given in Equations (11) and (12) in terms of the general parameters n, T, ξ, and a. Figure 12 plots d/d_⊥ for our numerical shock results over the shock parameter space n₀ and v_s, assuming v_{A⊥, 5} = 1, Δv_D5 = 1, and a = 10. The dashed lines demarcate the zone where a_sat < 10; above the top dashed line and below the lower dashed line a_sat > 10. Our expressions for d/d_⊥ are no longer valid if the maser is not saturated and therefore we do not plot d/d_⊥ outside the dashed lines since a = 10 < a_sat there. We see that d_⊥ ∼ d in the strongly masing region of parameter space. Maser spots will be approximately circular and shocked masers can be approximated by equivalent cylinders of diameter d and length ad. However, we predict some variation in the shape of the maser spot. An analytic fit can be obtained using Equations (12) and (31),

$$\begin{equation} \frac{d}{d_\perp } \simeq {3.3 \over a_{\rm sat}} \simeq 1.3\; {{v_{A\perp,5} c_\eta } \over j_{14}^{0.25}\Delta v_{D5}^{0.5}}, \end{equation} \tag{ 32 }$$

good to a factor of 1.5 over the main maser parameter space 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹ as long as a > a_sat. Note that if a > 10, then the dashed lines move to accompany the slightly more allowed n₀, v_s parameter space (see Figure 11). The equation shows that the elongation of the maser in the direction of the shock velocity is directly proportional to the Alfvén speed in the ambient medium.

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Figure 13 shows the dependence of T_{b, 11} = T_b/(10¹¹ K) on n₀ and v_s for v_{A⊥, 5} = 1, Δv_D5 = 1, and a = 10. The dashed lines are the same as in Figure 12. Since this figure applies to a = 10, regions outside the dashed lines with a_sat > 10 will have exponentially reduced T_b (see Equation (B13)) since they will be unsaturated. The effect of entering the unsaturated regime is seen in Figure 3, where very small changes in ξ lead to extremely large changes in T_b. For n₀ ≲ 10⁶ cm⁻³ the maser is unsaturated and "starved" for exciting collisions as discussed above. For n₀ ≳ 3 × 10⁷(v_{A⊥, 5}/v_s7) cm⁻³ in the case of a = 10, the maser is rapidly quenched and T_b precipitously drops. For intermediate densities, where the maser is saturated (a > a_sat), an approximate analytic fit (using Equations (13), (28), and (32)) to these numerical results is

$$\begin{equation} T_{b,11} \simeq 2.5 j_{14}^{0.5} \left({v_{A\perp,5}\over \Delta v_{D5}}\right) c_\eta ^2 a_1^3 \ {\rm K}, \end{equation} \tag{ 33 }$$

which is accurate to a factor of 1.4 for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹. Note that as long as c_η is of order unity, that is, as long as ξ and n are not so large that the maser quenches, T_b increases with increasing j and/or increasing v_A⊥. However, if j becomes too large, the maser quenches, c_η plummets, and T_b drops. Although raising v_A⊥ for fixed a raises T_b, increasing v_A⊥ also has the effect of increasing d; observed maser spot sizes limit the size of d, thereby limiting the possible v_A⊥ and therefore T_b in the region. In addition, since a = 2ℓ/d, increasing d can lower a; this then can lead to lower T_b even as v_A⊥ and d increase.

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Ever since the detailed study of W51 by Genzel et al. (1981), maser spot sizes have been shown to be uncorrelated with brightness temperature, a finding reaffirmed by the thorough investigation of W49 by Gwinn et al. (1992) and Gwinn (1994b). Figure 13 indicates this lack of correlation for fixed v_A. The brightness temperature does not vary much in the strong masing region, even though (see Figure 7) d varies by a factor of roughly 20 from ∼5 × 10¹² cm at the upper boundary to ∼10¹⁴ cm at the lower boundary. In addition, note that Figure 13 is for fixed a = 10. T_b varies with a³ and therefore the T_b spread in a given source arises mostly from variations in a. In summary, at fixed v_A the brightness is practically independent of the observed dimensions and dependent almost entirely on a, in agreement with observations. One can increase T_b and d by holding j, a and Δv_D5 fixed, but increasing v_A. In this case, T_b∝v_A∝d^1/2. Here, there is a weak dependence of T_b on d, but the very strong dependence on the aspect ratio (T_b∝a³) likely washes this out.

Figure 14 plots the contours of L_{iso, -6} ≡ L_iso/(10⁻⁶L_☉) as a function of n₀ and v_s for our standard values of v_{A⊥, 5} = 1, Δv_D5 = 1, and a = 10. The dashed lines are the same as in Figures 12 and 13 (i.e., they demarcate a_sat = 10). For a fixed aspect ratio, the luminosity peaks at somewhat lower n₀ compared with T_b because L_iso/T_b is proportional to d², and d increases as n₀ decreases (see Table 2 or Figure 7). Using Equation (15) and the analytic fits (28) for ξ and (32) for d/d_⊥, we find a fit for a > a_sat:

$$\begin{equation} L_{\rm iso,-6} \simeq 2.2\left({{v_{s7} v_{A\perp,5}^4 \Delta v_{D5}^{0.5}c_\eta }\over {j_{14}^{0.65} }}\right) a_1^3, \end{equation} \tag{ 34 }$$

which is accurate to a factor of two for 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹. Recalling that j = n₀v_s, we see that L_iso is proportional to $n_0^{-0.65}v_{s7}^{0.35}a_1^3$ as long as c_η is of order unity. Therefore, for fixed aspect ratio a, the luminosity increases with decreasing preshock density due to the increase in d, as discussed above. In addition, regions with high preshock magnetic fields (i.e., high v_A⊥) will produce much more luminous maser spots, because the maser spot size $d \propto v_{A\perp }^2$. In both cases the masers will not be much brighter, but bigger and more luminous. However, there is a very important caveat to this discussion. Recall that the aspect ratio a = 2ℓ/d, the coherence path length divided by the shock thickness. Therefore, as d gets larger, it is likely that a gets smaller. In both Figures 13 for T_b and Figure 14 for L_iso, a is held constant, even as d increases from roughly 3 × 10¹² cm at n₀ ∼ 10⁸ cm⁻³ to 10¹⁴ cm at n₀ ∼ 10⁶ cm⁻³. This implies that the coherence length is assumed to increase by a factor of roughly 30 as n₀ decreases over this range. This is likely not physically plausible; ℓ will not exactly scale as d, and, in fact, a will likely decrease as d increases. Both T_b and L_iso scale with a³. Therefore, the regions of higher preshock density may be more likely to give larger T_b and L_iso.

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All the equations derived in this subsection assume that the shocked slab is geometrically thin, that is, $d<d_{\rm thin}$ (see Section 2.3, Table 1, and Equations (B7) and (B8)). In this case, the maser spot size observed in the parallel direction is d, that is, it is limited by the thickness of the masing region—it is "matter" bounded. In Appendix C we justify this assumption, using the approximate analytic formulae we have derived. The equations in this section also assume the H₂ formation rate coefficient, γ, determined by HM79. As discussed at the end of Section 3.3, the value of γ has a significant effect on the shock structure. For example, if γ drops from 3 × 10⁻¹⁸ cm³ s⁻¹ to 3 × 10⁻¹⁹ cm³ s⁻¹, Table 2 shows that this decrease in γ leads to a decrease in the brightness temperature by a factor of ∼30 when we include the T_p dependence. On the other hand, the maser spot size d increases by a factor of 10; as a result, for a fixed aspect ratio a, the isotropic luminosity L_iso, which is proportional to d²T_b, actually increases by a factor of ∼3 for this decrease in γ. This increase in the maser luminosity is primarily due to the increased area of the maser spot and the increased length of the coherence path 2ℓ associated with the assumption of a constant aspect ratio.

We summarize the approximate analytic fits to the numerical results of Sections 2, 3 and 4 in two tables. Table 3 presents the fits to the parameters d, γ, ξ, T_p, x₋₄(H₂O), c_η, η, a_sat, d/d_⊥, T_b and L_iso as functions of j = n₀v_s, v_s, v_A⊥, Δv_D and a. The fits are good to better than a factor of two (see Section 4.1 for the individual error estimates) over the range 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³ and 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹, which is the range that produces strong J-shock H₂O masers.

Table 3. Approximations Derived from Numerical Shock Results

Parameter	Approximation (Preshock Variables j, vs, vA⊥)a
d	$1.3\times 10^{13}j_{14}^{-0.7} v_{s7}^{0.5}v_{A\perp,5}^2\ \ {\rm cm}$
γ	$3.8\times 10^{-18}j_{14}^{-0.3}v_{s7}^{-0.5}$ cm3 s−1
ξ	$4.0j_{14}^{1.5}\Delta v_{D5}^{-1}$
Tp	$350j_{14}^{0.12}v_{s7}^{-0.24}\Delta v_{D5}^{-0.22}\ \ {\rm K}$
x−4(H2O)	$1.6j_{14}^{0.2}v_{s7}^{-0.5}$
cη	$\left(1+0.045 {j_{14}^{2.8}\over {v_{A\perp,5}^{0.2} \Delta v_{D5}^{1.5}}}\right)^{-1}$
η	$0.03 j_{14}^{-0.75}\Delta v_{D5}^{0.5} c_\eta$
asat	$2.5j_{14}^{0.25} \Delta v_{D5}^{0.5}v_{A\perp,5}^{-1} c_\eta ^{-1}$
d/d⊥	$1.3 j_{14}^{-0.25} \Delta v_{D5}^{-0.5}v_{A\perp,5} c_{\eta }$
Tb	$2.5 \times 10^{11} j_{14}^{0.5} \Delta v_{D5}^{-1} v_{A\perp,5} c_\eta ^2 a_1^3 \ \ {\rm K}$
Liso	$2.2 \times 10^{-6} j_{14}^{-0.65} v_{s7} \Delta v_{D5}^{0.5} v_{A\perp,5}^4 c_\eta a_1^3 \ \ {L_\odot }$

Notes. ^aRecall that j = n₀v_s so that, equivalently, these expression show the preshock density dependence. These expressions assume that the HM79 prescription for the rate coefficient γ of H₂ formation is correct. These approximations are good to better than a factor of two (see text for individual error estimates) in the range 10⁶ cm⁻³ ≲ n₀ ≲ 10⁸ cm⁻³, 30 km s⁻¹ ≲ v_s ≲ 160 km s⁻¹, 0.5 ≲ v_{A⊥, 5} ≲ 5, and 0.5 ≲ Δv_D5 ≲ 3. The approximations for d/d_⊥, T_b and L_iso assume a > a_sat (see Equation (31)).

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Table 4 inverts the equations in Table 3 so that the shock parameters n₀, v_s, j, B_0⊥ or v_A⊥ and a are derived in terms of the potential observables v_A⊥ or B_0⊥, d, d/d_⊥, B_p, and either T_b or L_iso.⁷ We also give an expression for c_η since it is needed in the expression for a. It may be the case that v_A⊥ is not directly observable, but that a rough estimate of n₀ can be obtained. In this case we can use

$$\begin{equation} v_{A\perp,5}\simeq n_{0,7}^{0.45} d_{13}^{0.5} B_p^{0.1}, \end{equation} \tag{ 35 }$$

which is obtained by inverting the expression for n_{0, 7} given in the top line of Table 4. Recall B_p is measured in Gauss. Note the weak dependence on n₀ and especially B_p, which enables an estimate of v_A⊥ even when n₀ is only roughly estimated and B_p is even more uncertain.

Table 4. Approximations Derived from Numerical Saturated Maser Slab and Shock Models

Parameter	Approximationa
	(Observable Variables d, vA⊥ or B0⊥, Tb, Liso, Bp)
n0, 7	$1.0 d_{13}^{-1.11} v_{A\perp,5}^{2.22}B_p^{-0.22}$
vs7	$4.2d_{13}^{0.56} v_{A\perp,5}^{-1.11}B_p^{1.11}$
j14	$4.2 d_{13}^{-0.56} v_{A\perp,5}^{1.11} B_p^{0.89}$
B0⊥	$1.7 \times 10^{-3} d_{13}^{-0.56} v_{A\perp,5}^{2.11}B_p^{-0.11} \ {\rm Gauss}$
vA⊥, 5	$20.5 d_{13}^{0.27} B_{0\perp }^{0.47} B_p^{0.052}$
cη	$\left(1 + 2.5 d_{13}^{-1.57} v_{A\perp,5}^{2.91} \Delta v_{D5}^{-1.5} B_p^{2.49} \right)^{-1}$
a	$5.8 T_{b,11}^{0.33} d_{13}^{0.094} v_{A\perp,5}^{-0.52} \Delta v_{D5}^{0.33} B_p^{-0.15} c_\eta ^{-0.67}$
a	$6.5 L_{\rm iso,-6}^{0.33} d_{13}^{-0.31} v_{A\perp,5}^{-0.72} \Delta v_{D5}^{-0.17}B_p^{-0.18} c_\eta ^{-0.33}$

Notes. ^aAs in Table 3, these expressions assume that the HM79 formulation for the rate coefficient of H₂ formation on grain surfaces is correct. These expressions are good to better than a factor of two in the same range quoted in Table 3. In column 2, B_p and B_0⊥ are measured in Gauss.

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Figure 15 graphically plots the J-shock parameter space that produces strong 22 GHz H₂O masers, and indicates the physical mechanisms that intercede to reduce maser activity in J shocks. Above n₀ ≳ 10⁸ cm⁻³, the maser inversion is quenched in J shocks by the high densities and high optical depths in the H₂O infrared transitions, which drive the H₂O rotational levels to LTE and reduce the inversion in the maser levels. Below about n₀ ∼ 10⁶ cm⁻³, masers with a ≲ 10–100 are weak and unsaturated ("starved").

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Above v_s ≳ 200 km s⁻¹, the J shocks destroy most of the dust grains, leaving no grain surface upon which H₂ can reform. As a result, insufficient columns of warm H₂O are produced in the postshock gas, and no observable H₂O masers are excited.

For v_s ≲ 40 km s⁻¹, C shocks rather than J shocks may form in dense molecular gas (cf. Draine & McKee 1993, and references therein). We have marked the boundary of C shocks with J shocks with a dashed vertical line in Figure 15 at v_s = 40 km s⁻¹. This is appropriate if the gas is weakly ionized (ionization fractions of ≲ 10⁻⁷), so that charged grains mediate the C shock. However, if the dense preshock gas is more highly ionized, perhaps by the UV photons from nearby faster J shocks, this boundary between C and J shocks moves to lower values of v_s, and the J-shock maser parameter space is extended to lower v_s. For example, for ionization fractions of about 10⁻⁵ the boundary between J and C shocks occurs near v_s ∼ 10 km s⁻¹ (Smith & Brand 1990). To the left of the solid line (v_s ≲ 15 km s⁻¹, marked "too cold"), no H₂ re-formation plateau is produced in J shocks because too few preshock H₂ molecules are dissociated in J shocks.

Figure 15 also roughly indicates the region of parameter space where C shocks may produce water masers. Kaufman & Neufeld (1996) model H₂O maser emission from such C shocks. Here, the H₂ is not dissociated, but is kept warm over a large column by the ambipolar heating of the neutrals as they drift through the ions. In non-dissociating C shocks, the low density boundary (marked by solid horizontal line at n₀ = 10⁷ cm⁻³) is at a higher density than in J shocks because of less compression of the gas in the warm region. Similarly, the upper boundary marked by quenching is raised in C shocks to n₀ ∼ 10⁹ cm⁻³.

Several constraints bound the velocity range of C shock masers. For C shocks with low ionization fraction, the postshock peak temperatures are too cold to excite the maser for v_s ≲ 15 km s⁻¹. However, for higher ionization fractions, C shocks are warm enough to excite maser emission at shock velocities as low as v_s ∼ 5 km s⁻¹. Another factor affecting the low-velocity boundary of C-shock masers is the velocity required to sputter water-ice mantles off the grains. The C-shock masers occur in regions of high density, n₀ ≳ 10⁷ cm⁻³, and the freeze-out times for gas phase molecules is very short, ≲ 100 yr. The grains are likely warm enough to thermally desorb CO, but may not be warm enough (T_gr ≲ 100 K) to prevent the formation of water-ice mantles. In addition, the FUV radiation fields may sufficiently attenuated to prevent photodesorption of the ice. Therefore, for C shocks to produce strong water maser emission, they must sputter the ice mantles off the grains in these regions, and Draine (1995) estimates that only 10% of the water ice is sputtered off by C shocks with v_s = 20 km s⁻¹. Hence, unless the radiation field in the C-shock maser region is high enough to warm the grains to T_gr ≳ 100 K, C shocks must have v_s ≳ 20 km s⁻¹ to produce strong water-maser emission. As noted above in our discussion of J shocks, the high-velocity boundary for C shocks also depends on the ionization fraction in the gas, and is likely of order v_s ≲ 30–50 km s⁻¹.

Are most water masers produced in J shocks or in C shocks? This is a difficult question to answer with certainty. One measure might be the ram pressure ${\propto} n_0v_s^2$ needed to drive masing shocks. C shocks require lower shock velocities but higher preshock densities than J shocks. As seen in Figure 15, these two effects roughly cancel each other, suggesting J shocks and C shocks require roughly the same driving pressure and, thus, from this point of view, could be equally likely. However, there may be more gas in the density range that can produce J-shock masers than in the higher density range required for C-shock masers, which would favor J shocks. Water masers require densities of ∼10⁸–10⁹ cm⁻³. Regions of this density are rare, especially at distances ≳ 100 AU from a central protostar along the jet axis, where many masers are observed. The maser emission from C shocks must come from gas that is close to this density, whereas the emission from J shocks comes from gas that has been compressed from a density (only) ∼10⁶–10⁷ cm⁻³. Another factor to consider is the relative values of the key maser parameter ξ, which is proportional to the product of warm postshock column N_p times postshock density n_p. J shocks not only have the advantage in producing higher postshock densities for a given preshock density, as discussed above, but J shocks also produce larger columns of warm gas. In J shocks, the column is determined by the time to reform H₂ in the postshock gas, and N_p ∼ 10²² cm⁻², as we have shown. In C shocks the warm column is determined by the column needed for ions to collide with neutrals and drag the neutrals up to the shock speed. In dense regions, the ionization fraction is low and small charged grains mediate the C shock. The warm coupling column here is only N_p ∼ 10²¹ cm⁻² (e.g., Kaufman & Neufeld 1996). If gas ions dominate rather than charged grains, the column is even smaller. The smaller value of N_p results in a smaller value of ξ and therefore of T_b and L_iso, both of which vary as ξ^1/2 (Equations (13) and (15)).

Another method of distinguishing between the two types of shocks would be to infer the shock velocity, with the idea that slower masers might be C-shock masers. However, in shock masers the maser is beamed perpendicular to the shock velocity (in the plane of the shock). Therefore, even if the shock velocity is high, the Doppler velocity observed will be low. Proper motion studies are needed to try to estimate the shock velocity. Unfortunately, these studies determine the velocity of the shocked gas, not of the shock itself. For example, if high-velocity gas containing dust grains impacts a stationary dense clump or a protoplanetary disk and produces a J shock, the postshock gas would be decelerated to a speed similar to that of the dense gas, resulting in a small proper-motion velocity. The contrary could also occur: if the observed masing gas had a high velocity, one cannot be sure that the maser was induced by J shocks since the emission could originate in fast moving clumps with slow C shocks moving through them. In short, it is difficult to distinguish J-shock masers from C-shock masers by velocity information alone.

Liljeström & Gwinn (2000) observed 146 maser spots in W49N in the 22 GHz water maser line. Although no attempt was made to compare their observations with C-shock models, they found good agreement with our J-shock models, inferring shock velocities of order 30–100 km s⁻¹ and aspect ratios of 30–50. In addition, their inferred values of T_p and d matched the predictions of J-shock maser models. Many of their masers features had Doppler velocities in excess of 30 km s⁻¹ and up to ±200 km s⁻¹, again suggesting, but not proving, that the masers were produced by J shocks.

J-shock masers might be distinguished by their atomic and ionic infrared line emission. The shocks producing the maser spots likely have typical sizes of order the masing region, or ≳ 100 AU. This is a lower limit; in massive star-forming regions like W49 the size is likely of order 10¹⁷ cm. The shock area A_shk therefore at least ≳ 10³⁰ cm⁻² in low mass star-forming regions and could be as high as 10³⁴ cm⁻² in high mass star-forming regions. The shock is very embedded, so that the emergent cooling lines must lie in the mid to far infrared, so that they can penetrate the high dust extinction. J shocks differ from C shocks in that they create singly ionized and atomic species, which are strong coolants. C shocks are molecular and mainly cool via molecular rotational lines. One observational test of a J-shock origin is therefore to look for strong infrared cooling transitions from atomic or singly ionized species. For example, in our standard model of a J shock with n₀ = 10⁷ cm⁻³ and v_s = 100 km s⁻¹, we find that the luminosities in the [Ne ii] 12.8 μm, [Fe ii] 26 μm, and [O i] 63 μm lines are 3.1 × 10⁻⁵(A_shk/10³⁰ cm⁻²) L_☉, 2.1 × 10⁻⁴(A_shk/10³⁰ cm⁻²) L_☉, and 1.7 × 10⁻³(A_shk/10³⁰ cm⁻²) L_☉, respectively. Even with an area of A_shk = 10³⁰ cm⁻², these lines can be detected by the Stratospheric Observatory for Infrared Astronomy (SOFIA) in nearby (≲ 1 kpc) maser regions. The angular resolution of SOFIA for these lines may not be sufficient to spatially resolve these lines, but SOFIA has the sensitivity to detect the fluxes from these lines. We note that although the maser lines originate from portions of the shock nearly edge on, the full shock will likely have considerable portions directed along the LOS, and therefore the shock IR emission lines could be distinguished from background photodissociation regions (PDRs) or H ii regions by their width or velocity shifts, which will be of order 30–200 km s⁻¹. The [Ne ii] 12.8 μm line can be also observed with high spectral resolution by 8 meter class telescopes from the ground, at least in the nearest (≲ 1 kpc) masing regions. Here, the spatial resolution is roughly three times better than the obtained on SOFIA, and this will also help to disentangle the J shock masing region from background PDRs or H ii regions. We note, however, that the [Ne ii] line is very sensitive to the J shock velocity, and is strong only for v_s ≳ 100 km s⁻¹.

Finally, one might appeal to ratios of different masing lines to determine the temperature of the masing gas. As discussed in this paper, J-shock masers likely cannot heat the masing gas to temperatures greater than about T_p ∼ 400 K. However, C-shock masers can heat the masing regions to T_p ≳ 1000 K. As discussed in the Introduction, maser regions as hot as 1000 K will excite not only the 22 GHz maser, but also a number of submillimeter masers (Kaufman & Neufeld 1996). These authors have applied their results to observations of submillimeter masers, which almost certainly are produced by C shocks. However, there are many more regions where only 22 GHz masers are seen (see Introduction), and these masers are likely produced in cooler T_p ∼ 400 K gas. Although this is suggestive of J-shock masers, again the proof is not definitive since C shocks can also produce dense molecular gas with maximum temperatures of about 400 K.