Astron. Astrophys. 350, 529-540 (1999)

3. Analysis of molecular emission

To understand the origin of the observed strong molecular emission, we first want to derive the physical parameters, averaged over the rather large LWS beam ( 75"), which characterises the different gas components. For this purpose, the line emission from CO, H₂O and OH have been analysed by solving the radiative transfer in the line simultaneously with the level populations, under the Large Velocity Gradient (LVG) approximation in a plane-parallel geometry. Given the average nature of the observed emission and the uncertainty on the exact geometry involved, we did not attempt to consider a more realistic geometrical model. In principle, the emission from warm dust could be efficient in the radiative excitation of H₂O and OH; however, the influence of the local radiation field has not been taken into account in the radiation transfer calculations considering that the continuum source has a temperature of about 30 K and a dust opacity of only 0.07 at 200µm (Barsony et al. 1998), while gas temperatures in excess of 500 K are implied by our observed lines (see Fig. 6).

Fig. 6. Energy level diagram of H2O with the transitions observed by LWS in L1448-mm and their wavelengths in microns indicated.

In these approximations, the expected flux of a transition at frequency is given by:

where is the spontaneous radiative rate, is the fractional population of the upper level of the transition, is the number density of the considered molecular species, is the depth of the emitting region subtending a solid angle , and is the angle-averaged escape probability which depends on the line optical depth through the relationship (valid for a plane-parallel geometry, Scoville & Solomon 1974):

The optical depth can be expressed as:

where is the velocity gradient in the region and and are the populations and the statistical weights of the upper and lower levels respectively.

If we assume an homogeneous slab where the total velocity dispersion is this last expression can also be written:

making explicit the dependence on the column density () of the particular molecule.

The fractional population is given by the equations of the statistical equilibrium which, for a collisionally excited molecule, depend on the local temperature and density as well as on .

From the above expressions we can see that the line intensity depends on several free parameters (temperature, density, size and depth of the emitting region, velocity dispersion) some of which are related together, and thus difficult to be simultaneously constrained even with the large number of observed lines. However, the high-J CO lines are likely to be optically thin (), in which case their relative ratios depend only on the excitation temperatures and densities. In addition, the H₂ lines, being optically thin and thermalised, can further constrain the gas temperature. The temperature and density derived in this way can then be used as fixed parameters in modelling the water and OH lines, which are very sensitive to opacity effects because they have their energy levels connected by strong radiative transitions.

3.1. CO and H₂

The downward collisional rates for CO levels with 60 and 100 K were calculated using the coefficients taken from McKee et al. (1982), while the upward rates were computed using the principle of detailed balance. Radiative decay rates were taken from Chackerian & Tipping (1983). If we look at fluxes of the observed CO lines plotted as a function of the upper level rotational quantum number (Fig. 7), we note that the transitions with J=24, 25 and 26 have flux levels which are too high, compared to the lower transitions, to be excited in the same gas component. We have therefore attempted only to fit the transitions with 21. From their ratios we derive a temperature ranging from 700 to 1400 K and a density in the range 3 10⁴ to 5 10⁵ cm^-3. These limits were obtained after considering all the model fits which were compatible with the errors associated with the lines. In Fig. 7 we also report the J=21 measurement taken from Bachiller et al. (1990): we see that both the intensity integrated over all velocities (open triangle) and the contribution just of the EHV gas (open circle) lie inside our considered parameter range. In particular, temperatures greater than 1400 K would not agree with the total flux in the J=21 line while the 700 K fit already miss the J=21 completely; temperatures lower than this value would also fail to reproduce both the lower and higher J lines observed by LWS, giving rise to curves narrower than the observed flux distribution.

Fig. 7. Model fits through the observed CO lines. The range of temperatures and densities compatible with the observations are indicated. The total flux in the J=21 line from Bachiller et al. (1990) is also reported (open triangle) together with the J=21 emission relative only to the EHV component (open circle). The higher observed J lines (J=24, 25 and 26) have fluxes too high to be fitted by the same parameters as the other lines, suggesting the presence of a second component.

Given the high uncertainty on the flux determination of the J=24 and 26 lines due to their blending with the much stronger H₂O lines (see previous section), we have not attempted to fit the second component. However, the fact that it should peak at J24, indicates that its temperature cannot be lower than about 2000 K.

The analysis of the H₂ (0-0) lines gives a further way to constrain the gas temperature. Indeed, these lines originate from quadrupole transitions and are therefore optically thin. In addition, they have critical densities lower than 10⁴ cm^-3 1, implying that they are very quickly thermalised; their ratios can therefore be used to directly estimate the gas temperature. This can be better visualised in a rotational diagram (Fig. 8), where the natural logarithm of , the column density for a given upper level divided by its statistical weight, is plotted against the excitation energy of the upper level. To construct this diagram, the spontaneous radiative rates were taken from Turner et al. (1977), and an ortho- to para- H₂ equal to 3 was adopted. For a Boltzmann distribution, the slope of this plot is inversely proportional to the kinetic temperature of the gas. We note that the three detected lines were all observed with the same beam size, and therefore no beam filling correction is needed.

Fig. 8. H2 excitation diagram. The open circles are the observed line fluxes to which a dereddening of =5 mag has been applied, while open squares are dereddened with =11 mag

The S(3) line at 9.7 µm is particularly sensitive to extinction corrections, lying in the middle of the silicate absorption feature. An average visual extinction of about 5 mag was estimated towards the region covering the entire outflow (Bachiller et al. 1990), but of course the reddening could be much larger than this value on smaller scales, particularly in the surroundings of the mm source. Given the fact that the S(4) and S(5) lines are much less affected by reddening than the S(3) line, we can make a rough estimate of the extinction correction by imposing that the S(3) transition should lie on the same line as the other two transitions in the excitation diagram. In this way we estimate 11 mag. Fig. 8 shows the derived excitation temperatures both assuming =5 mag and =11 mag. Given also the uncertainty associated with the measurements, a range of temperatures from 1300 to 1700 K can be deduced. Therefore, in the assumption that the H₂ lines are emitted in the same region as the CO lines, a temperature between 1300 and 1400 K would be favourite as the best estimate to simultaneously reproduce the H₂ and the CO observations. The corresponding density values from the CO fits would in this case range between 3 and 6 10⁴ cm^-3. However, given the different beam sizes of the LWS and SWS instruments, the possibility that they are actually tracing different gas components cannot be ruled out. We will therefore conservatively assume that CO temperatures as low as 700 K cannot be excluded.

3.2. H₂O and OH

Assuming that the water lines originate in the same region as the CO lines, we can take the temperature and density as derived from the CO analysis to model the observed H₂O fluxes. We will in particular consider the two extreme conditions derived from CO, 1400 K and 10⁴ cm^-3, and 700 K and 10⁵ cm^-3, referring to them as the "high" and "low" temperature cases respectively. The water spectrum has been computed considering 45 levels for both the ortho and para species (i.e. considering the levels up to excitation temperatures of 2000 K) and adopting radiative rates taken from Chandra et al. (1984) and H₂O-H₂ collisional rates from Green et al. (1993). We assume an ortho/para abundance ratio of 3, equal to the ratio of the statistical weights of their nuclear spins.

Having fixed the temperature and density, we see from Eqs. (1), (2) and (4) that the other parameters affecting the absolute flux intensity of the water lines are the velocity dispersion, the H₂O column density and the projected area of the emitting region. The optical depth in the lines is proportional to N(H₂O)/ and since the ratios of different lines depend on their relative optical depths through the escape probability (Eqs. [2] and [4]), we can use them to constrain a value of N(H₂O)/. The dependence of the line ratios on this parameter is shown in Fig. 9 for two different sets of lines. From the figure it appears clear that a unique value of N(H₂O)/ can be effectively derived from a single ratio providing that the opacity of the considered lines is neither too high () so that their absolute intensities approach the blackbody value, nor too low () that their relative ratios become independent from the column density. The best fit consistent with all the observed ratios gives N(H₂O)/ = 2 10¹⁶ cm^-2 km^{- 1} s for either the "high" and "low" temperature cases. A comparison of the observed line fluxes with those predicted by these models is shown in Fig. 10. We see that both cases reproduce quite well the observations, reflecting the difficulty in determining the physical parameters of the emission using the water lines alone; however, the "high" temperature case seems to give a slightly better agreement with the data, especially because it is able to reproduce the observed 179.5 µm line, which is the strongest line observed in the spectrum.

Fig. 9. Ratio of intensities of the 303-212 174.6 µm to the 221-110 108.1 µm (upper panel) and of the 221-212 179.5 µm to the 505-414 99.5 µm (lower panel) H2O lines, as a function of the N(H2O)/ parameter for a gas temperature T= 1400 K and a volume density =3 104 cm-3. The straight line indicates the observed value while the dashed region delimits the range of uncertainty.

Fig. 10. Comparison of the modelled H2O line fluxes (triangles and squares indicate the lines in the ortho and para form respectively) with the observations (open circles) in the two extreme conditions derived from the CO fit. We note that the theoretical fluxes for the lines not observed in the LWS spectrum are all below the detectability threshold.

The absolute line intensity depends on both N(H₂O) and , which can be independently derived once a linewidth is assumed. To estimate we assume that the observed emission is related to the outflow activity taking place in the close environment of the millimeter source. This has been traced in many different molecular lines (e.g. H₂, CO, SiO), having widths ranging from 20 to 50 km s^-1. Taking this range of values we derive a water column density of (0.4-1)10¹⁸ cm^-2; the projected area of the emitting region in the "high" temperature regime is then (50-120) arcsec² (i.e. 7-12" diameter assuming a single emitting region), implying that about 2% of the LWS beam is filled with emitting gas. Adopting this source size, the CO column density can be now estimated from the observed CO line fluxes, giving N(CO)= (0.6-3)10¹⁷ cm^-3 (a range which also takes into account the absolute flux calibration errors). We find therefore that the H₂O/CO abundance ratio is 5. Hence, for the standard CO abundance of 10^-4, the water abundance is 5 10^-4, implying an enhancement with respect to the ambient gas of a factor 10³ (Bergin et al. 1995).

On the other hand, assuming the "low" temperature regime a more compact region of 3-5" of diameter is derived from the water lines; since however the intrinsic CO line intensity of the considered transitions is higher than in the "high" temperature case, the implied CO column density is similar in the two cases and consequently the H₂O/CO ratio remains almost unchanged. This makes us confident that the derived value of the water abundance does not depend significantly on specific values adopted for the physical conditions and projected area, the only assumption affecting this value being that CO and water are emitted by the same gas component.

Finally, we also remark that our model predicts a population inversion in the 22 GHz - transition, which has been detected in the form of strong masering spots aligned with the stellar outflow (Chernin 1995), but we are unable to reproduce the large line luminosity observed; the 22 GHz maser is however localised in dense ( 10⁶) and very compact regions probably having the optimal geometry for producing the strong masering, while the LWS observations probe emission on larger scales.

Very high abundances of gas-phase H₂O have so far been estimated only for the Orion region (Harwit et al. 1998) but never in low mass protostars similar to L1448, since the typical values in these sources are of the order of (1-5) 10^-5 (Liseau et al. 1996; Saraceno et al. 1998; Ceccarelli et al. 1998). In this respect, L1448-mm seems to have an exceptionally high water abundance, indicating that the processes responsible for the production of gas-phase water are particulary efficient here. We will discuss this point further in the next section.

Finally, adopting the same parameters as derived from the above analysis also for OH, from the observed 119 µm flux we deduce a column density N(OH) of (0.7-1.3) 10¹⁶ cm^-2 and therefore an X(OH)10^-5, i.e. 2 10^-2 times the water abundance. For the molecular parameters and assumptions in the modelling of the OH, we refer to Giannini et al. (1999).

Table 3 summarises the derived quantities for the observed molecular species.

Table 3. Ranges of physical parameters for the molecular emission.
Notes:
a) The given range takes into account the lower and upper limits for the emitting region deduced by the different models (see text).
b) Obtained assuming the same emission area as the other components; see text for a discussion on this point.

3.3. The N(H₂) column density

The absolute intensities of the observed H₂ lines can in principle be used to derive the H₂ column density if the beam filling factor is known. Adopting the hypothesis that the high-J CO and H₂ pure rotational line emission is fully enclosed within the relatively smaller SWS beam, we can use the size of the emitting region derived from the above analysis to estimate the N(H₂) of the warm gas, obtaining a value of about 10¹⁹ cm^-2. When compared with the estimated CO column density, this value would imply a CO/H₂ abundance of 10^-2. This result does not in fact depend on the specific model adopted; a direct measure of the CO/H₂ abundance ratio in the warm gas can be independently given simply by comparing the ratio of two observed lines emitted in the same conditions. Indeed, both the high-J CO and the H₂ lines are optically thin and therefore their ratio is directly proportional to the ratio of their relative column densities. Measuring the ratio for different sets of lines, we always obtain a value of CO/H₂ of the order of 2 10^-2. One possible explanation for this anomalous abundance ratio (the interstellar CO/ H₂ abundance is of the order of 10^-4, while in dense cores some of the CO can be depleted on grains, further reducing this value, see e.g. van Dishoeck et al. 1993) could be that the region strongly emitting the molecular lines observed by the LWS, has been totally or partially missed by the smaller SWS beam. This in turn would imply, giving the low beam-filling derived for the LWS observations, that the region of emission is not located in the center of the two beams, i.e. coincident with the mm source.

Alternatively, the extinction in the region could be much higer than assumed in our analysis. An value of about 250 mag would be necessary to increase the H₂ S(4) and S(5) observed fluxes by a factor of one hundred, which is needed to find a CO/H₂ abundance ratio similar to the commonly assumed interstellar value. Such a high extinction, which is found only if the observed lines originate from very close to the protostar itself, would strongly affect the (0-0) S(3) line intensity, causing it to deviate too much from the straight line in the rotational diagram. Inconsistencies would be also found with the upper limits of the S(1) and S(2) lines, which, by contrast, are not greatly affected by a large amount of extinction.

A different interpretation could be that in the considered region the hydrogen is not fully molecular but part of it is present in atomic form. This circumstance could happen for instance in protostellar winds, where a significant fraction of atomic hydrogen can be present even if the heavy atoms are processed into molecules (Glassgold et al. 1991). Alternatively, also in the presence of dissociative shocks a significant part of the post-shock gas can be composed of hydrogen which is mainly in atomic form (Hollenbach & McKee 1989). That the flow from young stars can be characterised by an appreciable amount of atomic hydrogen has indeed been demonstrated by the detection of the 21 cm hydrogen line associated with high-velocity molecular outflows (e.g. Lizano et al. 1988).

The different possibilities will be discussed in Sect. 5.

© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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