SPITZER SPECTRAL LINE MAPPING OF PROTOSTELLAR OUTFLOWS. II. H₂ EMISSION IN L1157

We have used the H₂ line maps to obtain the two-dimensional distribution of basic H₂ physical parameters, through the analysis of the rotational diagrams in each individual pixel. As described in N09, who analyzed the global H₂ excitation conditions in L1157, the distribution of upper-level column densities of the S(0)–S(7) lines as a function of their excitation energy, does not follow a straight line, indicating that a temperature stratification in the observed medium exists. The exact form of this temperature stratification depends on the type of shock the H₂ lines are tracing, as they probe the post-shock regions where the gas cools from ∼1000 K to ∼200 K.

The simplest way to parameterize the post-shock temperature stratification is to assume a power-law distribution: this approach was applied by Neufeld & Yuan (2008), who also show that this type of distribution is expected in gas ahead of unresolved bow shocks. On this basis, and also following N09, we fit the observations assuming a slab of gas where the H₂ column density in each layer at a given T varies as dN ∝ T^−βdT.

This law is integrated, to find the total column density, between a minimum (T_min) and a maximum (T_max) temperature. For our calculations we have kept T_max fixed at 4000 K, since gas at temperatures larger than this value is not expected to contribute to the emission of the observed pure rotational lines. T_min was instead assumed to be equal, in each position, to the minimum temperature probed by the observed lines. This T_min is taken as the excitation temperature giving rise to the observed ratio of the S(0) and S(1) column densities, assuming a Boltzman distribution. The T_min value ranges between ∼150 and 400 K.

We found that the approach of a variable T_min always produces fits with a better χ² than assuming a fixed low value in all positions. We also assume the gas is in LTE conditions. Critical densities of rotational lines from S(0) to S(7) range between 4.9 cm⁻³ (S(0)) and 4.4 × 10⁵ cm⁻³(S(7)) at T = 1000 K assuming only H₂ collisions (Le Bourlot et al. 1999): critical densities decrease if collisions with H and He are not negligible. Deviations from LTE can therefore be expected only for the high-J S(6) and S(7) transitions: the signal-to-noise ratio (S/N) of these transitions in the individual pixels is however not high enough to disentangle, in the rotational diagrams, NLTE effects from the effects caused by the variations of the other considered parameters. In particular, as also discussed in N09, there is a certain degree of degeneracy in the density and the β parameter of the temperature power law in an NLTE treatment that we are not able to remove in the analysis of the individual pixels. This issue will be further discussed in Section 3.2. An additional parameter of our fit is the OPR value. It is indeed recognized that the 0–0 H₂ lines are often far from being in OPR equilibrium, an effect that in a rotational diagram is evidenced by a characteristic zigzag behavior in which column densities of lines with even-J lie systematically above those of odd-J lines. In order not to introduce too many parameters, we assume a single OPR value as a free parameter for the fit. In reality, the OPR value is temperature dependent (e.g., N09 and Section 3.2), and therefore the high-J lines might present an OPR value closer to equilibrium then the low-J transitions. Our fit therefore gives only a value averaged over the temperature range probed by the lines considered (i.e., ∼200–1500 K).

In summary, we have varied only three parameters, namely the total H₂ column density N(H₂), the OPR, and the temperature power-law index β, in order to obtain the best model fit through a χ² minimization procedure and assuming a 20% flux uncertainty for all the lines. The fit was performed only in those pixels where at least four lines with an S/N larger than 3 have been detected. Before performing the fit, the line column densities were corrected for extinction assuming Av = 2 (Caratti o Garatti et al. 2006) and adopting the Rieke & Lebofsky (1985) extinction law. At the considered wavelengths, variations of Av of the order of 1–2 mag do not affect any of the derived results.

Figure 3 shows the derived map of the H₂ column density, while in Figures 4 and 5 maps of the OPR and β are displayed together with temperature maps relative to the "cold" and "warm" components (T_cold and T_warm), i.e., the temperature derived from linearly fitting the S(0)–S(1)–S(2) and S(5)–S(6)–S(7) lines, once corrected for the derived OPR value. In Figure 6, we also show the individual excitation diagrams for selected positions along the flow, obtained from intensities measured in a 20'' FWHM Gaussian aperture centered toward emission peaks (Table 1). Values of the fitted parameters in these positions are reported in Table 2. In addition to T_cold and T_warm, we also give in this table the values of the average temperature in each knot, derived through a linear fit through all the H₂ lines (T_med).

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The maps show significant variations in the inferred parameters along the outflow. The H₂ column density ranges between 5 × 10¹⁹ and 3 × 10²⁰ cm⁻³. The region at the highest column density is located toward the B1 molecular bullet (see Figure 2).⁵ This is consistent with the higher column density of CO found in B1 with respect to other positions in the blueshifted lobe (Bachiller & Perez Gutierrez 1997), and might suggest that this is a zone where the outflowing gas is compressed due to the impact with a region of higher density (Nisini et al. 2007). Toward the NW, redshifted outflow, the column density has a more uniform distribution, with a plateau at ∼10²⁰ cm⁻² that follows the H₂ intensity distribution. The N(H₂) decreases at the apex of the redshifted outflow, with a value slightly below 10²⁰ cm⁻² at the position of the D near-IR knot.

T_cold ranges between ∼250 and 550 K. The highest values are found at the tip of the northern outflow lobe, while local maxima correspond to the positions of line intensity peaks. T_warm ranges between ∼1000 and 1500 K. In this case the highest values are in the southern lobe, at the position of the A NIR knot. As a general trend, the T_warm value decreases going from the southern to the northern peaks of emission, with the minimum value at the position of the D NIR knot.

β values range between ∼4 and 4.5 in the blueshifted lobe while they are larger in the redshifted lobe, with maximum values of ∼5.5 at the tip of the flow. Due to the degeneracy between β and density discussed in the previous section, these values can be considered as upper limits because of our assumption of LTE conditions. Neufeld & Yuan (2008) have discussed the β index expectations in bow shock excitation. A β index of ∼3.8 is expected in paraboloid bow shocks having a velocity at the bow apex high enough to dissociate H₂, in which case the temperature distribution extends to the maximum allowed temperature. Slower shocks that are not able to attain the maximum temperature produce steeper temperature distributions, i.e., with values of β greater than 3.8. This is consistent with our findings: low values of β (of the order of 4) are found in the blueshifted lobe: here, evidence of H₂ dissociation is given by the detection of atomic lines (i.e., [Si ii], [Fe ii], and [S i]) in the IRS spectrum. In the redshifted lobe, where values of β larger than 4 are derived in the LTE assumption, no atomic emission is detected and the T_warm values are lower than those measured in the blueshifted lobe, indicating a maximum temperature lower than in the blueshifted flow.

The OPR varies significantly along the outflow, spanning from ∼0.6 to 2.8. Hence it is always below the equilibrium value of 3. Although a one-to-one correlation between temperature and OPR cannot be discerned, some trends can be inferred from inspection of Figure 5. In general, the OPR minima are observed in plateau regions between two consecutive intensity peaks, where the cold temperature also has its lowest values. At variance with this trend, the emission filaments delineating the outflow cavity within ± 20'' from the millimeter source, where the cold temperature reaches a minimum value of ∼250 K, show rather high values of the OPR, ∼2.4–2.8. This might suggest that this region has experienced an older shock event that has raised the OPR, though not to the equilibrium value, and where the gas had time to cool at a temperature close to the pre-shock gas temperature. On the other hand, at the apex of both the blueshifted and redshifted lobes, where the cold temperatures are relatively high (i.e., 500–550 K), the OPR is rather low, 1.5–2.0. Evidence of regions of low OPR and high temperatures at the outflow tips has been already given in other flows (Neufeld et al. 2006; Maret et al. 2009). It has been suggested that these represent zones subject to recent shocks where the OPR has not had time yet to reach the equilibrium value.

Additional constraints on the physical conditions responsible for the H₂ excitation are provided by combining the emission of the mid-IR H₂ pure-rotational lines from the ground vibrational level with the emission from near-IR ro-vibrational lines. It can be seen from Figure 2 that the 2.12 μm emission follows quite closely the emission of the 0–0 lines at higher J. In addition to the 2.12 μm data presented in Figure 2, we have also considered the NIR long-slit spectra obtained on the A and C NIR knots by Caratti o Garatti et al. (2006). These knots, at the spatial resolution of the NIR observations (i.e., ∼08) are separated in several different sub-structures that have been individually investigated with the long-slit spectroscopic observations. For our analysis we have considered the data obtained on the brightest of the sub-structures, that coincide, in position, with peaks of the 1–0 S(1) line.

In order to inter-calibrate in flux the Spitzer data with these NIR long-slit data, obtained with a slit width of 0.5 arcsec, we have proceeded as follows: we first convolved the 2.12 μm image at the resolution of the Spitzer images and then performed photometry on the A and C peaks positions with a 20'' diameter Gaussian aperture, i.e., with the same aperture adopted for the brightness given in Table 1. We have then scaled the fluxes of the individual lines given in Caratti o Garatti et al. (2006) in order to match the 2.12 μm flux gathered in the slit with that measured by the image photometry. In doing this, we assumed that the average excitation conditions within the 20'' aperture are not very different from those of the A–C peaks. This assumption is observationally supported by the fact that the ratios of different H₂ NIR lines do not change significantly (i.e., less than 20%) in the A–C knot substructures separately investigated in Caratti o Garatti et al. (2006). We have considered only those lines detected with S/N larger than 5; in practice this means considering lines from the first four and three vibrational levels for knots A and C, respectively. The excitation diagrams obtained by combining the Spitzer and NIR data for these two knots are displayed in Figure 8.

In order to model together the 0–0 lines and the near-IR ro-vibrational lines, we have implemented two modifications to the approach adopted previously. First of all, the NIR lines probe gas at temperatures higher than the pure rotational lines, of few thousands of K, at which values it is expected that the OPR has already reached equilibrium. Thus, the ortho-para conversion time as a function of temperature needs to be included in the fitting procedure, since lines excited at different temperatures have different OPRs. We have here adopted the approach of N09 and used an analytical expression for the OPR as a function of the temperature, considering a gas that had an initial value of the OPR, OPR₀, and has been heated to a temperature T for a time τ. Assuming that the para-to-ortho conversion occurs through reactive collisions with atomic hydrogen, we have

$$\begin{eqnarray} \frac{\rm OPR(\tau)}{1+\rm OPR(\tau)} &=& \frac{\rm OPR_0}{1+\rm OPR_0}\,e^{-n({\rm H})k\tau }\nonumber\\ &&+ {\frac{\rm OPR_{\rm LTE}}{1+\rm OPR_{\rm LTE}}}\,(1 - e^{-n({\rm H})k\tau }). \end{eqnarray} \tag{ 1 }$$

In this expression, n(H) is the number density of atomic hydrogen and OPR_LTE is the OPR equlibrium value. The parameter k is given by the sum of the rates coefficients for para-to-ortho conversion (k_po), estimated as 8 × 10⁻¹¹exp(−3900/T) cm³ s⁻¹, and for OPR conversion, k_op ∼ k_po/3 (Schofield 1967). Thus, the dependence of the OPR on the temperature is implicitly given by the dependence on T of the k coefficient. The inclusion of a function of OPR on T introduces one additional parameter to our fit: while we have previously considered only the average OPR of the 0–0 lines, we will fit here the initial OPR₀ value and the coefficient K = n(H)τ.

The second important change that we have introduced with respect to the previous fitting procedure is to include an NLTE treatment of the H₂ level column densities. In fact, the critical densities of the NIR lines are much higher than those of the pure rotational lines (see Le Bourlot et al. 1999). For example, n_crit of the 1–0 S(1) 2.12 μm line is 10⁷ cm⁻³ assuming only collisions with H₂ and T = 2000 K. Therefore, the previously adopted LTE approximation might not be valid when combining lines from different vibrational levels. This is illustrated in Figure 7, where we plot the results obtained by varying the H₂ density between 10³ and 10⁷ cm⁻³ , while keeping the other model parameters fixed. The observed column densities in the A position are displayed for comparison. For the NLTE statistical equilibrium computation we have adopted the H₂ collisional rate coefficients given by Le Bourlot et al. (1999).⁶ This figure demonstrates the sensitivity of the relative ratios between 0–0 and 1–0 transitions to density variations. For example, in this particular case, the ratio N(H₂)_0−0S(7)/N(H₂)_1−0S(1) is 64.6 at n(H₂) = 10⁴ cm⁻³ and 1.9 at n(H₂) ≳ 10⁷ cm⁻³. The figure also shows that the observational points display only a small misalignment in column densities between the 0–0 lines and the 1–0 lines, already indicating that the ro-vibrational lines are close to LTE conditions at high density.

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In Figure 8, we show the final best-fit models for the combined mid- and near-IR column densities in the A and C positions. As anticipated, the derived n(H₂) densities are large, of the order of 10⁷ and 6 × 10⁶ cm⁻³for the A and C positions, respectively. The two positions indeed show very similar excitation conditions: only the column density is a factor of 3 smaller in knot C. Hence, we conclude that the lack of detection of rotational lines with v > 3 in knot C, in contrast to knot A (Caratti o Garatti et al. 2006), is purely due to a smaller number of emitting molecules along the line of sight and not to different excitation conditions.

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The derived H₂ densities are much higher than previous estimates based on other tracers. Nisini et al. (2007) derive a density of 4 × 10⁵ cm⁻³ at the position of knot A from multi-line SiO observations, thus more than an order of magnitude smaller than those inferred from our analysis. The high-J CO lines observed along the blueshifted lobe of L1157 by Hirano & Taniguchi (2001), indicate a density even smaller, of the order of 4 × 10⁴ cm⁻³. SiO is synthesized and excited in a post-shock cooling zone where the maximum compression is reached; therefore, it should trace post-shock regions at densities higher than H₂ (e.g., Gusdorf et al. 2008). One possibility at the origin of the discrepancy is our assumption of collisions with only H₂ molecules, and thus of a negligible abundance of H. This can be considered roughly true in the case of non-dissociative C-shocks, where H atoms are produced primarily in the chemical reactions that form H₂O from O and H₂, with an abundance n(H)/n(H₂+H) ∼ 10⁻³ (e.g., Kaufmann & Neufeld 1996). However, if the shock is partially dissociative, the abundance of H can increase considerably and collisions with atomic hydrogen cannot be neglected, in view of its large efficiency in the H₂ excitation. This situation cannot be excluded at least for the knot A, where atomic emission from [Fe ii] and [S i] has been detected in our Spitzer observations. Since we cannot introduce n(H) as an additional independent parameter of our fit, we have fixed n(H₂) at the value derived from SiO observations (4 × 10⁵ cm⁻³; Nisini et al. 2007) and varied the H/H₂ abundance ratio (see Figure 9). The best fit is in this case obtained with a ratio H/H₂ = 0.3; this indicates that our observational data are consistent with previous H₂ density determinations only if a large fraction of the gas is in atomic form.

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Turning back to the inferred OPR variations with temperature, our fit implies that the OPR in the cold gas component at T = 300 K is significantly below the equilibrium value, while the value of OPR = 3 is reached in the hot gas at T = 2000 K traced by the NIR lines. The parameter K = n(H)τ is constrained to be ∼10⁶ and 10⁷ yr cm⁻³ for knots A and C, respectively. We can also estimate the time needed for the gas to reach this distribution of OPR, from the limits on the atomic hydrogen abundance previously discussed. Our data implies a high value of the n(H) density: a minimum value of n(H) ∼ 0.6–1 × 10⁴ cm⁻³ (for knots C and A, respectively) is given if we assume H/H₂ ∼ 10⁻³ (and thus n(H₂) ∼ 10⁷ cm⁻³ given by our fit), while a maximum value of ∼10⁵ cm⁻³ is derived from the fit where n(H₂) is kept equal to 4 × 10⁵ cm⁻³. The high abundance of atomic H ensures that conversion of para- to ortho-H₂ proceeds very rapidly; the fitted values of the K parameter indeed imply that the observed range of OPR as a function of temperature have been attained in a timescale between 100 and 1000 yr for both the knots.

Finally, given the column density and particle density discussed above, we can estimate the H₂ cooling length (L∼ N(H₂)/n(H₂)). If we consider the case of n(H₂) ∼ 10⁷ cm⁻³ and negligible n(H), we have L∼ 10¹³ cm while a length of ∼10¹⁵ cm is inferred in the case of n(H₂) ∼ 4 × 10⁵ cm⁻³. All the parameters derived from the above analysis are summarized in Table 3 and they will be discussed in the following section in the framework of different shock models.

The copious H₂ emission at low excitation observed along the L1157 outflow indicates that the interaction of the flow with the ambient medium occurs prevalently through non-dissociative shocks. Both the Spitzer-IRS maps of N09 and the NIR narrow band images of Caratti o Garatti et al. (2006) show that significant gas dissociation in L1157 occurs only at the A peak, where both mid-IR lines from [Fe ii], [S ii], and [S i], and weak [Fe ii] at 1.64 μm have been detected. Weaker [S i] 25 μm and [Fe ii] 26 μm emission have also been detected on the C spot, but overall the atomic transitions give a negligible contribution to the total gas cooling, as pointed out in N09. These considerations suggest that most of the shocks along the outflow occur at speeds below ∼40 km s⁻¹, as this is the velocity limit above which H₂ is expected to be dissociated. The knot A is the only one showing a clear bow shock structure. Here, the velocity at the bow apex is probably high, causing H₂ dissociation and atomic line excitation, consistent with the fitted temperature power law β of ∼4, as discussed in Section 3.1, while the bulk of the H₂ emission comes from shocks at lower velocities originating in the bow wings.

Constraints on the shock velocity that gives rise to the molecular emission in L1157 have been already given in previous works. The submillimeter SiO emission and abundances, measured in different outflow spots, suggest shock velocities of the order of 20–30 km s⁻¹ (Nisini et al. 2007). The comparison of SiO and H₂ emission against detailed shock models performed by Gusdorf et al. (2008) confirm a similar range of velocities in the NIR-A knot, although the authors could not find a unique shock model that well represents both the emissions.

Cabrit et al. (1999) found that the column density of the mid-IR H₂ emission lines, from S(2) to S(8), observed by ISO-ISOCAM was consistent either with C-shocks having velocities of ∼25 km s⁻¹ or with J-shocks at a lower velocity of the order of ∼10 km s⁻¹. Gusdorf et al. (2008), however, conclude that stationary shock models, either of C- or J-type, are not able to reproduce the observed rotational diagram on the NIR-A position, constructed combining ISOCAM data and NIR vibrational lines emission. A better fit was obtained by these authors considering non-stationary shock models, which have developed a magnetic precursor but which retain a J-type discontinuity (the so-called CJ shocks, Flower et al. 2003). Similar conclusions, but on a different outflow, have been reached by Giannini et al. (2006) who studied the H₂ mid- and near-IR emission in HH54: in general, steady-state C- and J-type shocks fail to reproduce simultaneously the column densities of both the ro-vibrational and v = 0, pure-rotational H₂ levels.

A different way to look at the issue of the prevailing shock conditions in the observed regions is to compare the set of physical parameters that we have inferred from our analysis to those expected from different shocks. With this aim, we summarize in Table 3 the physical properties derived on the A and C H₂ knots. In addition to the parameters derived from the NLTE analysis reported in Section 2, namely H₂ post-shock density, H/H₂ fraction, initial OPR, cooling length, and time, the table reports also the average values of OPR and rotational temperature, as they are measured from a simple linear fit of the rotational diagrams presented in Figure 6.

As mentioned in Section 3.2, the high fraction of atomic hydrogen inferred by our analysis rules out excitation in a pure C-shock. In fact, dissociation in C-shocks is always too low to have an H/H₂ ratio higher than 5 × 10⁻³, irrespective of the shock velocity and magnetic field strength (Kaufman & Neufeld 1996; Wilgenbus et al. 2000). C-shocks are not consistent with the derived parameters even if we consider the model fit with the high H₂ post-shock density of the order of 10⁷ cm⁻³ and negligible atomic hydrogen: in this case, we derive an emission length of 10¹³ cm, which is much lower than the cooling length expected in C-shocks, which, although decreasing with the pre-shock density, is never less than 10¹⁵ cm (Neufeld et al. 2006).

Stationary J-shock models better reproduce some of our derived parameters. For example, in J-type shocks the fraction of hydrogen in the post-shocked gas can reach the values of 0.1–0.3 we have inferred, provided that the shock velocity is larger than ∼20 km s⁻¹. In general, a reasonable agreement with the inferred post-shock density and H/H₂ ratio is achieved with models having v_s = 20–25 km s⁻¹and pre-shock densities of 10³ cm⁻³ (Wilgenbus et al. 2000). Such models predict a shock flow time of the order of 100 yr or less, which is also in agreement with the value estimated in our analysis at least in knot A. In such models, however, the cooling length is an order of magnitude smaller than the inferred value of ∼10¹⁵ cm. In addition, the gas temperature remains high for most of the post-shocked region: the average rotational temperature of the v = 0 vibrational level is predicted to be, according to the Wilgenbus et al. (2000) grid of models, always about 1600 K or larger, as compared with the value of about 800–900 K inferred from observations. The consequence of the above inconsistencies is that J-type shocks tend to underestimate the column densities of the lowest H₂ rotational levels in L1157, an effect already pointed out by Gusdorf et al. (2008).

As mentioned before, Gusdorf et al. (2008) conclude that the H₂ pure rotational emission in L1157 is better fitted with a non-stationary C+J shock model with either v_s between 20 and 25 km s⁻¹ and pre-shock densities n_H = 10⁴ cm⁻³, or with v_s ∼ 15 km s⁻¹ and higher pre-shock densities of n_H = 10⁵ cm⁻³. Such models, however, still underestimate the column densities of the near-IR transitions: the post-shocked H₂ gas density remains lower than the NIR transitions critical density and the atomic hydrogen produced from H₂ dissociation is not high enough to populate the vibrational levels to equilibrium conditions.

The difficulty of finding a suitable single model that reproduce the derived physical conditions is likely related to possible geometrical effects and to the fact that multiple shocks with different velocities might be present along the line of sight. It would be indeed interesting to explore whether bow shock models might be able to predict the averaged physical characteristics along the line of sight that we infer from our analysis.

H₂ emission represents one of the main contributors to the energy radiated away in shocks along outflows from very young stars. Kaufman & Neufeld (1996) predicted that between 40% and 70% of the total shock luminosity is emitted in H₂ lines for shocks with pre-shock density lower than 10⁵ cm⁻³ and shock velocities larger than 20 km s⁻¹, the other main contributions being in CO and H₂O rotational emission. This has also been observationally tested by Giannini et al. (2001) who measured the relative contribution of the different species to the outflow cooling in a sample of class 0 objects observed with ISO–LWS (Long Wavelength Spectrometer).

We will discuss here the role of the H₂ cooling in the global radiated energy of the L1157 outflow. From the best-fit model obtained for the knots A and C, we have derived the total, extinction corrected, H₂ luminosity by integrating over all the ro-vibrational transitions considered by our model. $L_{{\rm H}_2}$ is found to be 8.4 × 10⁻² and 3.7 × 10⁻² L_☉ for the A and C knots, respectively. Out of this total luminosity, the contribution of the rotational lines is only 5.6 × 10⁻²(A) and 2.7 × 10⁻²(C) L_☉, which means that in both cases they represent about 70% of the total H₂ luminosity.

N09 have found that the total luminosity of the H₂ rotational lines from S(0) to S(7), integrated over the entire L1157 outflow, amount to 0.15 L_☉. If we take into account an additional 30% of contribution from the v > 0 vibrational levels, we estimate a total H₂ luminosity of 0.21 L_☉. This is 30% larger than the total H₂ luminosity estimated by Caratti o Garatti et al. (2006) in this outflow, assuming a single component gas at temperature between 2000 and 3000 K that fit the NIR H₂ lines.

If we separately compute the H₂ luminosity in the two outflow lobes, we derive $L_{\rm H_2}$ = 8.5 × 10⁻² L_☉ in the blueshifted lobe and 1.3 × 10⁻¹ L_☉ in the redshifted lobe. Comparing these numbers with those derived in the individual A and C knots, we note that the A knot alone contributes to most of the H₂ luminosity in the blueshifted lobe. In contrast, the H₂ luminosity of the redshifted lobe is distributed among several peaks of similar values. This might suggest that most of the energy carried out by the blueshifted jet is released when the leading bow shock encounters a density enhancement at the position of the A knot. On the other hand, the redshifted gas flows more freely without large density discontinuities, and the corresponding shocks are internal bow shocks, all with similar luminosities.

The integrated luminosity radiated by CO, H₂O, and O i in L1157 has been estimated, through ISO and recent Herschel observations, as ∼0.2 L_☉ (Giannini et al. 2001; Nisini et al. 2010), which means that H₂ alone contributes about 50% of the total luminosity radiated by the outflow. Including all contributions, the total shock cooling along the L1157 outflow amounts to about 0.4 L_☉, i.e., L_cool/L_bol ∼ 5 × 10⁻², assuming L_bol = 8.4 L_☉ for L1157 mm (Froebrich 2005). This ratio is consistent with the range of values derived from other class 0 sources from ISO observations (Nisini et al. 2002).

The total kinetic energy of the L1157 molecular outflow estimated by Bachiller et al. (2001) amounts to 0.2 L_☉ without any correction for the outflow inclination angle, or to 1.2 L_☉ if an inclination angle of 80° is assumed. Considering that the derivation of the L_kin value has normally an uncertainty of a factor of 5 (Downes & Cabrit 2007), we conclude that the mechanical energy flux into the shock, estimated as L_cool, is comparable to the kinetic energy of the swept-out outflow and thus that the shocks giving rise to the H₂ emission have enough power to accelerate the molecular outflow.

The total shock cooling derived above can also be used to infer the momentum flux through the shock, i.e., $\dot{P}$ = 2L_cool/v_s, where V_s is the shock velocity that we can assume, on the basis of the discussion in the previous section, to be of the order of 20 km s⁻¹. Computing the momentum flux separately for the blueshifted and redshifted outflow lobe, we derive $\dot{P}_{\rm red} \sim$ 1.7 × 10⁻⁴ and $\dot{P}_{\rm blue} \sim$ 1.1 × 10⁻⁴ M_☉ yr⁻¹ km s⁻¹. In this calculation, we have assumed that the contribution from cooling species different from H₂, as estimated by ISO and Herschel, is distributed among the two lobes in proportion to the H₂ luminosity. If we assume that the molecular outflow is accelerated at the shock front through momentum conservation, then the above-derived momentum flux should show results comparable to the thrust of the outflow, derived from the mass, velocity, and age measured through CO observations. The momentum flux measured in this way by Bachiller et al. (2001) is 1.1 × 10⁻⁴ and 2 × 10⁻⁴ M_☉ yr⁻¹ km s⁻¹in the blueshifted and redshifted lobes, respectively, i.e., comparable to our derived values. It is interesting to note that the $\dot{P}$ determination from the shock luminosity confirms the asymmetry between the momentum fluxes derived in two lobes. As shown by Bachiller et al. (2001), the L1157 redshifted lobe has a 30% smaller mass with respect to the blueshifted lobe, but a higher momentum flux due to the larger flow velocity. The northern redshifted lobe is in fact more extended than the southern lobe; however, given the higher velocity of the redshifted gas, the mean kinematical ages of the two lobes are very similar.