4 Discussion

4.1 The atomic line spectrum: CNO

4.1.1 [O III] 53 $\mu$m, [O III] 88 $\mu$m and [N II] 122 $\mu$m

As is evident from Table 2, only upper limits were obtained for the high ionisation lines [O  III] 53 $\mu$m, [O  III] 88 $\mu$m and [N  II] 122 $\mu$m. This is consistent with the luminosities derived by Larsson et al. (2000), indicating the presence of stellar sources generating at best only gentle UV-fields.

We can also exclude the presence of extended strongly shocked regions. For instance, if associated with the fast moving objects of Rodríguez et al. (1989), our data imply that the physical scales of these shocks would be small, $\ll$1 $^{\prime \prime }$ (e.g. for $v_{\rm s}$ $\sim$ 200 km s^-1 and n₀ $\ga$ 10⁵ cm^-3; see: Cameron & Liseau 1990; Liseau et al. 1996a).

We can conclude that the degree of ionisation of the atomic gas is generally low. Lines of low-ionisation species will be discussed in the next sections.

4.1.2 Extended emission: [C II] 157 $\mu$m and [O I] 63 $\mu$m

The spatial distribution of the [C  II] 157 $\mu$m emission is shown in Fig. 3 from which it is evident that the emission varies within a factor of about two. The S 68 nebulosity is pronounced in the [C  II] 157 $\mu$m line, making it likely that its origin is from a photondominated region (PDR), close to the cloud surface.

This idea can be tested quantitatively by invoking also the observed [O  I] 63 $\mu$m emission. The LWS subtends a solid angle $\Omega_{\rm LWS}=9.0\times 10^{-8}$ sr. Disregarding for a moment the peak emission (see below), we find for unit beam filling the line intensities $I_{157}= 2 \times 10^{-5}$ erg cm^-2 s^-1 sr^-1 and $I_{63}= 1 \times 10^{-5}$ erg cm^-2 s^-1 sr^-1, respectively. Hence, the line ratio [O  I] 63 $\mu$m/[C  II] 157 $\mu$m is about 0.5 (Fig. 5). These data are consistent with PDR-emission, where an interstellar radiation field about 10 times as intense as that of the solar neighbourhood, i.e. $G_0 \sim 10$, is impinging on the outer layers of a cloud with densities in the range $n({\rm H})$ = (0.1-1) $\times 10^5$ cm^-3 (cf. Figs. 4 and 5 of Liseau et al. 1999). This estimate of the strength of the UV field is in reasonable agreement with the FIR-background measured by IRAS and ISO (Larsson et al. 2000), which would imply G₀ = 5-25. In the advocated PDR model, the [O  I] 145 $\mu$m line is fainter by two orders of magnitude than the [O  I] 63 $\mu$m line. The observed line ratio, [O  I] 63 $\mu$m/[O  I] 145 $\mu$m > 12, is clearly consistent with this prediction. Finally, no detectable emission from higher ionisation stages would be expected. The PDR model would offer therefore a satisfactory explanation for the observed fine structure line distribution over the map.

If this PDR emission is treated as a large scale background and subtracted from the maps, the resulting line ratio toward the peak (SMM 1) increases dramatically, viz. to [O  I] 63 $\mu$m/[C  II] 157 $\mu$m $=15 \pm 5$. Such large ratios are generally not predicted by PDR models but are a common feature of J-shocks (Hollenbach & McKee 1989). The residual [O  I] 63 $\mu$m flux corresponds to an observed intensity $I_{63}= 1 \times 10^{-4}$ erg cm^-2 s^-1 sr^-1, more than two orders of magnitude below that of J-shock models (see Sect. 4.1.3). Interpreted as a beam filling effect, this would imply the size of the shocked regions to be about 4 $^{\prime \prime }$ to 5 $^{\prime \prime }$.

Figure 5: Emission line intensity ratio map for [O  I] 63 $\mu$m/[C  II] 157 $\mu$m. Observed positions are identified by crosses. The observed flux distributions of the individual lines are shown in Fig. 3.

4.1.3 The [O I] 63 $\mu$m emission toward HH 460

Toward the interstellar shock, HH 460, the [O  I] 63 $\mu$m flux is not conspicuously larger than that of [C  II] 157 $\mu$m, as one might naively expect for shock excitation, and we cannot exclude the possibility that the spatial coincidence with the [O  I] 63 $\mu$m emission spot is merely accidental. However, pursuing the shock idea we find that, for the previously inferred cloud densities, $\ga$10⁵ cm^-3, the [O  I] 63 $\mu$m intensity is roughly constant with the shock speed (a few times 10^-2 erg cm^-2 s^-1 sr^-1; Hollenbach & McKee 1989). These J-shock models do also predict that the accompanying [O  I] 145 $\mu$m emission would not be detectable in our observations and that any [C  II] 157 $\mu$m contribution would be totally insignificant.

The observed line intensity is $1 \times 10^{-5}$ erg cm^-2 s^-1 sr^-1 which, if due to shock excitation, would indicate that the source fills merely a tiny fraction of the LWS-beam (beam dilution of $2.5\times 10^{-4}$). A size of about 1 $^{\prime \prime }$ for the [O  I] 63 $\mu$m emitting regions of the HH object would thus be implied, which is comparable to the dimension of the dominating, point-like, optical knot HH 460 A. From the observed line flux, a current mass loss rate from the HH-exciting source of M $_{\rm loss} = 3\times 10^{-7}$  $M_{\odot} ~ {\rm yr}^{-1}$ would be indicated (Hollenbach 1985; Liseau et al. 1997), which is at the 3% level of the mass accretion rate in the ${\rm Serpens ~ cloud ~ core}$ (Sect. 4.4).

Based on the L₆₃- $L_{\rm bol}$ calibration by Liseau et al. (1997), one would predict that the luminosity of the central source is slightly less than 0.5 $L_{\odot}$. No detailed information about the exciting source of HH 460 is available, though. Based entirely on morphological arguments, Ziener & Eislöffel (1999) associate HH 460 with SMM 1, and Davis et al. (1999) either with SMM 1 or with SMM 9/S 68N. The inferred luminosities of these objects, 71 $L_{\odot}$ and 16 $L_{\odot}$, respectively (Larsson et al. 2000), are however much larger than that inferred for the putative source driving HH 460. Evidently, the present status regarding the identification of the driving source of HH 460 is inconclusive. Proper motion and radial velocity data would be helpful in this context.

4.2 H$\mathsf{_{2}}$, CO, H $\mathsf{_{2}}$O and OH toward SMM 1

We can directly dismiss the PDR of Sect. 4.1.2 as responsible for the molecular line emission observed with the LWS, since gas densities and kinetic temperatures are far too low for any significant excitation of these transitions. Shock excitation would be an obvious option. In the following, we will examine the line spectra of H₂, CO, H₂O and OH.

 

 

Table 5: OH line measurements in the spectrum of SMM 1.

$^2\Pi$ Transition		Wavelength				Flux $\times$ 1019	Err $\times$ 1019	Single/	LWS
		($\mu$m)				(W cm-2)	(W cm-2)	Multi	Detector
$\Omega - \Omega^{\prime}$	$J - J^{\prime}$	$\lambda$	$\lambda_{\rm obs}$	$\Delta \lambda$	$\sigma_{\lambda}$	F	$\Delta F$	fit
$^1\!/_2 - ^1\!\!/_2$	$^3\!/_2 - ^1\!\!/_2$	163.26	163.26 Fix	+0.00	0.60 Fix	1.39 $\pm$ 0.90	0.93	S	LW 5
			163.26 Fix	+0.00	0.60 Fix	1.00 $\pm$ 0.85	0.91	S	LW 4
$^3\!/_2 - ^3\!\!/_2$	$^5\!/_2 - ^3\!\!/_2$	119.34	119.38 $\pm$ 0.20	+0.04	1.31 $\pm$ 0.11	2.07 $\pm$ 0.49	0.56	S	LW 2
			119.42 $\pm$ 0.07	+0.08	0.60 Fix	1.28 $\pm$ 0.22	0.36	S	LW 2
$^3\!/_2 - ^3\!\!/_2$	$^7\!/_2 - ^5\!\!/_2$	84.51	84.43 $\pm$ 0.03	-0.08	0.39 $\pm$ 0.03	2.94 $\pm$ 0.64	0.79	S	LW 1
			84.44 $\pm$ 0.08	-0.07	0.60 Fix	3.49 $\pm$ 0.73	0.87	S	LW 1
			84.51 $\pm$ 0.02	+0.00	0.41 $\pm$ 0.02	4.02 $\pm$ 0.50	0.58	S	SW 5
			84.50 $\pm$ 0.02	-0.01	0.29 Fix	3.26 $\pm$ 0.31	0.42	S	SW 5
$^1\!/_2 - ^3\!\!/_2$	$^1\!/_2 - ^3\!\!/_2$	79.15	79.14 $\pm$ 0.02	-0.01	0.29 $\pm$ 0.02	2.84 $\pm$ 0.38	0.54	S	SW 5
			79.14 $\pm$ 0.02	-0.01	0.29 Fix	2.86 $\pm$ 0.21	0.43	S	SW 5
			79.17 $\pm$ 0.07	+0.02	0.56 $\pm$ 0.07	2.22 $\pm$ 0.80	0.89	S	SW 4
			79.14 $\pm$ 0.04	-0.01	0.29 Fix	1.47 $\pm$ 0.34	0.51	S	SW 4
$^3\!/_2 - ^3\!\!/_2$	$^9\!/_2 - ^7\!\!/_2$	65.21	65.18 $\pm$ 0.03	-0.03	0.24 $\pm$ 0.03	1.51 $\pm$ 0.45	0.47	S	SW 3
			65.18 $\pm$ 0.03	-0.03	0.29 Fix	1.64 $\pm$ 0.26	0.30	S	SW 3
$^3\!/_2 - ^3\!\!/_2$	$^{11}\!/_2 - ^9\!\!/_2$	53.30					0.73	S	SW 2

Note to the table: $\lambda$ is an average wavelength for the various fine structure lines.

4.2.1 Rotation diagram: H $\mathsf{_{2}}$ and CO

The analytical technique known as "rotation diagram'' analysis is relatively simple and easy to apply to wavelength integrated molecular rotational line data. The advantages and the shortcomings of this analysis tool have been thoroughly discussed by Goldsmith & Langer (1999).

Assuming the lines to be optically thin and to be formed in Local Thermodynamic Equilibrium (LTE), one can derive the equation of a straight line for the molecular column density as a function of the upper level energy in temperature units. The slope of this line is the reciprocal excitation temperature of the levels (which in LTE is the same for all levels and equals the kinetic gas temperature), viz.

$$% \ln{ \left ( \frac { 4~\pi }{ h~\nu_0~g_{\rm u}~A_{\rm ul} ... ...{Q(T_{\rm ex})} \right )} - \frac{ E_{\rm u} }{ k~T_{\rm ex} }$$ (1)

where the symbols have their usual meaning. The left hand side of Eq. (1) entails the column density of the molecules in the upper levels. For H₂, the upper level energies, $E_{\rm u}$, were obtained from Abgrall & Roueff (1989) and the Einstein transition probabilities, $A_{\rm ul}$, were adopted from Wolniewicz et al. (1998). For CO, these data were taken from Chandra et al. (1996). The statistical weights of the upper levels are given by $g_{\rm u} = (2~I + 1)(2~J_{\rm u} + 1)$, where I is the quantum number of the nuclear spin. Further, for the evaluation of the approximate partition function

$$% Q(T_{\rm ex}) \approx \frac { k~T_{\rm ex}}{ h~c~B_0}$$ (2)

we used the rotational constant, $B_0 = 59.33451~{\rm cm}^{-1}$, for H₂ from Bragg et al. (1982). For CO, $B_0 = 1.9225~{\rm cm}^{-1}$, was obtained from the data by Lovas et al. (1979). Graphs of Eq. (1) are shown in Figs. 6 and 7, fitted to the CAM-CVF and LWS data, respectively.

To obtain a consistent result, the H₂ data need to be corrected for the foreground extinction. Using the data of Ossenkopf & Henning (1994; model for thin ice coating, n =10⁵ cm^-3, t=10⁵ yr), an extinction correction of $A_{\rm V}$ = 4.5 mag resulted in a total column density of warm H₂ gas of $N({\rm H}_2) = (3.7 \pm 0.8) \times 10^{18}$ cm^-2. The rotation temperature is $T_{\rm ex}=(1130 \pm 60)$ K and an ortho-to-para ratio (nuclear spin state population) of o/p = 3 is consistent with these data.

Figure 6: Rotation diagram for the ortho-H2 (filled symbols) and para-H2 (open symbols and upper limit) lines observed toward the flow from SMM 1. A linear regression fit to extinction corrected data is shown by the full drawn line. The physical parameters with their formal errors are given in the figure (see also the text).

Figure 7: Rotation diagram for the CO lines observed with the LWS (diamonds, this work) and for ground based data (filled circles), taken from Davis et al. (1999), Hogerheijde et al. (1999) and White et al. (1995). More than one LWS-value for the same upper energy $E_{\rm u}/k$ refer to measurements with different detectors. The data have been (ad hoc) fit by two linear segments, the solutions of which are given in the figure (see also the text).

In Fig. 7, ground-based CO data from the literature were added for lower lying transitions. Evidently, the high-Jdistribution appears markedly different from that of the low-J lines. If these latter lines were truly optically thin, they could originate in extended cloud gas, where $T_{{\rm low-}J} = (62 \pm 6)$ K, of column density $N_{{\rm low-}J} = (2.0 \pm 0.2)\times 10^{17}$ cm^-2. Seemingly in contrast, the LWS data identify gas at a characteristic temperature of $T_{{\rm high-}J} = (320 \pm 20)$ K, with an LTE-column density of $N_{{\rm high-}J} = (1.9 \pm 0.2)\times 10^{15}$ cm^-2.

These results are based on ad hoc assumptions, i.e. that of unit beam filling and of low optical depth in the lines, potentially underestimating the column densities, and that the level populations are distributed according to their LTE values. LTE may be a reasonably good assumption for the low-J lines. Regarding CO, it is however questionable to what extent these are optically thin. On the other hand, low opacity may come close to the truth for the high-J lines, but LTE is not at all guaranteed a priori for these transitions. Obviously, one needs to check how well these assumptions are justified. In the next sections, this will be addressed by employing first a method based on the Sobolev approximation and then a full Monte Carlo calculation, including gradients for both density and temperature. The latter method takes any (previously neglected) beam dilution effects directly into account.

4.2.2 Large Velocity Gradient models: CO

In the Large Velocity Gradient model (LVG) opacity effects in the lines are explicitly taken into account by introducing the photon escape probability formalism. Crudely speaking, the critical density of the transition, $n_{\rm crit} = A_{\rm ul}/C_{\rm ul}(T)$, can be lowered by means of an effective Einstein-probability, $A_{\rm ul}~\beta_{\rm esc}$, where $\beta_{\rm esc}$is in the range 0 to 1 for infinite and zero optical depth, respectively. This can effectively "delay'' line saturation. For illustrating purposes, $\beta_{\rm esc} \sim 1/\tau_{\rm line}$but, in general, $\beta_{\rm esc}$ is geometry dependent (Castor 1970).

Figure 8: The fluxes of rotational lines of CO, observed with the LWS, are compared to LVG model computations. Filled circles with error bars refer to LWS data, where more than one value for a given J correspond to different detectors. The upper limit is $3\sigma$. In addition, open symbols represent ground based data, where the different sizes refer to different telecope beams (see Fig. 7). The physical parameters of the LVG model are indicated in the figure.

For under-resolved sources, an ambiguity can arise from the fact that hot and tenuous models may be indistinguishable from cool and dense ones. However, assuming that the rotation diagram analysis can provide an estimate of the kinetic gas temperature, LVG can be used to determine the average density of the emitting region. This is shown in Fig. 8, for the resulting $\log{n({\rm H}_2)} = 6.2 \pm 0.2$ cm^-3, which is in good agreement with the results by McMullin et al. (2000).

In these models, the presence of a diffuse radiation field is introduced by the dust temperature $T_{\rm dust} = 40$ K, the wavelength of unit optical depth $\lambda_{\tau = 1} =200$ $\mu$m, the frequency dependence of the dust emissivity $\beta = -1$ and a geometrical covering factor of 0.5 (cf. Larsson et al. 2000). The (clearly detected) high-J lines are all only mildly sub-thermally excited (justifying a posteriori our initial assumption), but have substantial opacity, e.g. $\tau_{0(J=14-13)} = 1.7$. First at $J_{\rm u}=22$ start the lines to become optically thin again ( $\tau_{0(J=22-21)} = 0.14$).

The principle parameter of the LVG model is related to the ratio of the column density to the line width, $N/\Delta v$. For a given density of the collision partners, $n({\rm H}_2)$, this ratio is given by

$$% \frac {N_{\rm CO}}{\Delta v} \propto X_{\rm CO}/\frac{\partial~v}{\partial~r}$$ (3)

where the right hand side is the "LVG-parameter''. There, $\partial~v/\partial~r$ is the (Doppler) velocity gradient in the gas and $X_{\rm CO}$ is the molecular abundance relative to H₂. From Eq. (3), it is clear that LVG models are, in general, not unique, since an increase of the column density could have the same effect as a decrease of the line width.

From the model fit, $N({\rm CO}) = 1.5 \times 10^{18}$ cm^-2 for the adopted $\Delta v = 7.5$ km s^-1 (FWHM of a Gaussian line shape; see Sect. 4.3). A circular source would have a diameter of about 5 $^{\prime \prime }$ (1500 AU), a thickness of about 600 AU and an H₂ mass of about 0.01 $M_{\odot}$ (for $X_{\rm CO}$ = 10^-4). Finally, the total CO cooling rate amounts to $3.6\times 10^{-1}$ $L_{\odot}$.

The hot regions emitting in the H₂ lines (Sect. 4.2.1) are not expected to contribute significantly to the CO emission observed with the LWS. We predict the strongest CO lines from this gas to be the (J=4-3) and the (J=5-4) transitions, with "LWS''-fluxes from a 10 $^{\prime \prime }$ source of about $1 \times 10^{-14}$ erg cm^-2 s^-1.

Figure 9: The fit (smooth line) of the LVG model to the observed spectrum (histogram). All molecules, i.e. CO, 13CO, ortho-H2O, para-H2O and OH, are assumed to share the same density and temperature, viz. $n({\rm H}_2) = 1.6\times 10^6$ cm-3, $T_{\rm kin} = 320$ K; for further model details, see the text. Line identifications are as those given in Fig. 4.

4.2.3 LVG models: $\mathsf{^{13}}$CO, H $\mathsf{_{2}}$O and OH

The CO-model of the previous section can be used (by keeping $\partial~v/\partial~r$ constant) to investigate whether it is applicable also to other molecular species. Such a "one-size-fits-all'' model would have the advantage of permitting the straightforward estimation of the relative abundance of these species (see Liseau et al. 1996b for an outline of this method). The reasonably satisfactory result of such computations for ¹³CO, ortho-H₂O, para-H₂O and OH is presented in Fig. 9.

The ¹³CO spectrum has been computed under the assumption that ¹²CO/¹³CO is as low as 40 (Leung & Liszt 1976). The data are clearly consistent with this value, but the S/N is insufficient to conclusively provide a better defined value. Since the ¹³CO lines are all optically thin, the cooling in these lines ( $1.3\times 10^{-2}$ $L_{\odot}$) is relatively more efficient than that in CO (by almost a factor of two).

The H₂O model is based on considering 45 levels for both ortho- and para-H₂O, including 164 transitions each. The radiative rates are from Chandra et al. (1984) and the scaled collision rates from Green et al. (1993). The model fit of the observed spectrum requires an o/p = 3for H₂O and the derived H₂O-abundance is X(H₂O) = $1\times$10^-5. As expected, the excitation is sub-thermal and the lines are very optically thick (e.g., $\tau_{0({\rm o}~179.5~\mu {\rm m})} = 433$, $\tau_{0({\rm p}~101~\mu {\rm m})} = 218$). Both the 380 GHz ortho-transition (4₁₄-3₂₁) and the 183 GHz para-transition (3₁₃-2₂₀) are predicted to be masing ( $\tau_0 =-1$). The total cooling rate due to water vapour is $L({\rm H_2O}) = 2.1\times 10^{-1}$ $L_{\odot}$, i.e. at the 60% level compared to the CO cooling rate.

For OH, the Einstein A values were computed from the data provided by D. Schwenke, who also gives the energy levels. The collision rate coefficients for 50 transitions were obtained from Offer et al. (1994). As before, the excitation is sub-thermal and the lines are optically thick (e.g., $\tau_{0(119~\mu {\rm m})} \sim 170$ in each line of the doublet). This refers to the derived, relatively high, value of the OH-abundance of $X({\rm OH}) = 2\times 10^{-6}$(OH/H₂O = 0.2). The OH lines cool the gas as efficiently as H₂O, viz. $L({\rm OH}) = 2.0\times 10^{-1}$ $L_{\odot}$. The model is overpredicting the 119 $\mu$m line flux (whereas the 113 $\mu$m H₂O line is underpredicted), perhaps indicating a distribution of temperatures (and densities). However, these lines fall in one of the least well performing LWS detectors (LW 2) and instrumental effects cannot be excluded.

So far, we have considered only models of a homogeneous source at a single kinetic temperature in a plane-parallel geometry. The relaxation of these, likely unrealistic, assumptions is the topic of the next sections.

4.2.4 2D radiative transfer model of the dusty torus

In our previous paper, we presented a self consistent radiative transfer model for the SED of SMM 1 (Larsson et al. 2000). For a simplified analysis and for a direct comparison with previous spherical models of the object, we adopted spherical geometry of the dusty envelope. The model provided a good fit to the observations longward of about 60 $\mu$m, but resulted in too low fluxes in the mid-IR. As already noted in that paper, the spherical symmetry may not be a very good assumption for SMM 1, the source driving the bipolar outflow. In this paper, we performed detailed modelling of the dusty object using our 2D radiative transfer code (Men'shchikov & Henning 1997), which enabled us to quantitatively interpret existing dust continuum observations and to derive accurate physical parameters of SMM 1. In the next section, the density and temperature structure of the model will be used in a Monte-Carlo calculation of the CO line radiation transfer in the envelope. Our approach and the model geometry are very similar to those for two other embedded protostars: HL Tau (Men'shchikov et al. 1999) and L1551 IRS 5 (White et al. 2000); we refer to the papers for more details on the general assumptions, computational aspects, and uncertainties of the modelling.

Figure 10: Model geometry of the dusty torus of SMM 1 (see Sects. 4.2.4 and 4.2.5). Different shades of gray show schematically the density falling off outwards. The radius of the compact dense torus is $\sim$100 AU, whereas the outer radius of the envelope is based on our maps.

Figure 11: Comparison of the observed SED of SMM 1 and the model of the dusty torus. The individual fluxes (see Larsson et al. 2000 for details) are labelled by different symbols, to distinguish between beams of different sizes. The model assumes that we observe the torus at an angle of 31$^{\circ }$, relative to its midplane. The effect of beam sizes is shown by the vertical lines and by the difference between the dotted and solid lines in the model SED at mid-IR wavelengths. Whereas only the lower points of the vertical lines are relevant, we have connected them to the adjacent continuum by straight lines, to better visualise the effect. The SED for the equivalent spherical envelope is also shown, to illustrate the influence of the bipolar outflow cavities.

 

 

Table 6: Main input parameters of the dusty torus model.

Parameter	Value
Distance	310 pc
Central source luminosity	140 $L_{\odot}$
Stellar effective temperature	5000 K
Torus opening angle	100$^{\circ }$
Viewing angle	31$^{\circ }$
Torus dust melting radius	2 AU
Torus outer boundary	1.4 $\times$ 104 AU
Torus total mass (gas+dust)	33 $M_{\odot}$
Density at melting radius	2.5 $\times$ 10-13 g cm-3
Density at outer boundary	1.4 $\times$ 10-18 g cm-3
Outflow visual $\tau_{\rm v}$	71
Midplane $\tau_{\rm v}$	2200

The model assumes that SMM 1 consists of an axially-symmetric (quasi-toroidal), dense inner core surrounded by a similarly-shaped "envelope'' (Fig. 10). A biconical region of much lower density with a full opening angle of 100$^{\circ }$ is presumed to be excavated in the otherwise spherical envelope by the outflow from SMM 1. The structure, for brevity called "torus'', is viewed at an inclination of 30$^{\circ }$ with respect to the equatorial plane of the torus. Main input model parameters are summarised in Table 6.

As very little is known about dust properties in SMM 1, we adopted a dust model very similar to that applied by Men'shchikov & Henning (1999) for HL Tau and by White et al. (2000) for L1551 IRS 5. The dust population consists of 4 components: (1) large dust particles of unspecified composition, with radii 100-6000 $\mu$m, (2) core-mantle grains made of silicate cores, covered by dirty ice mantles, (3) amorphous carbon grains, and (4) magnesium-iron oxide grains. The latter 3 components of dust grains have the same radii of 0.08-1 $\mu$m. The dust-to-gas mass ratios of the components are 0.01, 0.0005, 0.0068, and 0.0005, respectively. The first component of very large grains is present only in the compact dense torus ($r~\le~$120 AU), where all smaller grains are assumed to have grown into the large particles. Note that although unknown properties of dust generally introduce a major uncertainty in the derived model parameters, extremely high optical depths in SMM 1 make the model results not very sensitive to the specific choice of the grain properties, except for the presence of very large grains in the dense central core.

Figure 12: Comparison of the model visibilities at 0.8 mm, 1.4 mm, 2.7 mm, and 3.3 mm with available measurements of Brown et al. (2000) and Hogerheijde et al. (1999). The upper and lower curves in each panel show the visibilities for two directions in the plane of sky, parallel and orthogonal to the projected axis of the model.

Figure 13: Density and temperature structure of the model torus of SMM 1. Left panel: the total densities in the midplane and in polar outflow regions, and dust densities of different dust grain components (for only their smallest sizes). Right panel: the temperature profiles correspond to the midplane of the torus; they were obtained self-consistently from the equation of radiative equilibrium.

In the modelling of the dusty torus, we fitted all available photometry of SMM 1, paying special attention to the effect of different beam sizes. Important constraints for the density structure were provided by the available submm and mm interferometry of the object. The model SED, compared to the observations in Fig. 11, fits almost every single individual flux in the entire range from the mid-IR to mm wavelengths. Note that it would be wrong to fit the observed data with the total model fluxes, because the angular size of SMM 1 is generally much larger than the photometric apertures. In fact, the model demonstrates that the effect of beam sizes on the fluxes may be as large as an order of magnitude.

Comparison of the model visibilities to the interferometry data shown in Fig. 12 demonstrates that the model is also consistent with the observed spatial distribution of intensity. The visibilities indicate that there is a dense core inside of a lower density envelope. The radial density and temperature profiles of the model, are shown in Fig. 13. The innermost dense core has a $\rho~\propto~r^{-1}$ density gradient in the model, whereas the outer parts of the lower-density envelope have a steeper density distribution ( $\rho~\propto~r^{-1.5}$). The temperature distribution was obtained in iterations as a solution of the energy balance equation.

4.2.5 2D radiative transfer of the molecular emission

We have used the density and temperature distributions of this dusty torus model in combination with a Monte Carlo scheme to compute the radiative transfer of the CO lines, and its isotopomers, through the source.

Observations of the ${\rm Serpens ~ cloud ~ core}$ in the J=2-1 transitions of the CO-isotopomers C¹⁸O and C¹⁷O are present in the archive of the James Clerk Maxwell Telescope (JCMT). These potentially optically thin lines could trace the embedded core SMM 1. Our disk model reproduces the observed line intensities of these low-J isotopomers fairly well (Fig. 14). There, the averaged background emission of the surrounding gas has been subtracted, in order to reveal the line profiles of SMM 1 itself. From the figure it is evident that the C¹⁸O and C¹⁷O lines are optically thick out to a point, where the temperature falls below 15 K and where substantial condensation of the CO gas onto dust grains occurs. This CO freeze-out was treated following Sandford & Allamandola (1993 and references therein), where the ice-to-gas ratio $\eta$ is proportional to the dust density n, and to functions of the gas and dust temperatures, $T_{\rm g}$ and $T_{\rm d}$respectively, viz.

$$% \eta \propto n~T_{\rm g}^{1/2}~ e^{1/T_{\rm d}}.$$ (4)

In the inner ("core'') regions, line opacities are very high so that the photons are essentially trapped. Therefore, the excitation temperature of the molecules follows the kinetic temperature of the H₂ gas. Beyond the core, i.e. in the "torus'', $\tau$ drops quickly below unity for the high-J lines, but there, the dust starts to contribute significantly to the (small) opacity (Fig. 15).

Figure 14: Upper panel: for background emission corrected line profiles of low-J CO isotopomers, viz. C18O (2-1) and C17O (2-1), toward SMM 1 are shown as histograms. The observations were retrieved from the JCMT archive. The results from 2D-Monte Carlo radiative transfer calculations for the disk/torus model are shown by smooth lines. The shown integrated line intensity refers to the model, for which adopted abundances are 12CO:C18O:C $^{17}{\rm O} =$ 1:550:2750 with X(12CO ) = 10-4, and freeze-out of the molecules is treated in accordance with the work by Sandford & Allamandola (1993). Lower panel: the total line centre optical depth in C18O (2-1) and C17O (2-1), along the line-of-sight toward the central region of the source, is shown by the solid line. Similarly, the dotted lines display the line opacities and the broken lines the dust opacities.

Figure 15: Same as in Fig. 14, but for three high-J CO transitions, which fall in the ISO-LWS spectral band.

The CO lines falling into the LWS range are all formed in the inner, hotter parts of the source ( $r \ll 10^3$ AU, $T \gg 10^2$ K). This small line forming region is insufficient to produce the observed flux levels, i.e. the model underpredicts observed high-J line fluxes by more than two orders of magnitude. Irrespective of the geometry, we can conclude quite generally that the CO lines observed with the LWS do not originate from the central regions of SMM 1, be it an accretion disk, be it infalling gas (we have also computed "inside-out'' collapse models).

For the excitation of this gas we need to consider alternative mechanisms and, since outflows are known to exist in this region, shock heating of the gas offers a natural option. Our temperature determinations for the molecular gas (Sect. 4.2.1) are also consistent with this idea.

4.3 Shock heating

From the discussion of the preceding sections we can conclude that the heating of the gas is most likely achieved through shocks. These shocks are generated by flows within the LWS beam. Comparing the observed and predicted molecular line emission with the J-shock models by Hollenbach et al. (1989) and Neufeld & Hollenbach (1994), we find that these models are in conflict with our observations.

In Fig. 16, we compare our observations of rotational lines of H₂, CO, H₂O and OH with predictions of the C-shock models by Kaufman & Neufeld (1996). The models for $\log n_0 = 5.5$ (cm^-3) and $v_{\rm s}$ $\sim$ 15-20 km s^-1 are in reasonable agreement with the for extinction corrected ($A_{\rm V}$ = 12 mag) observed values for H₂ and for a flux from $6^{\prime \prime} \times 6^{\prime \prime}$ (1 CVF-pixel). For CO, the model fits the observations for an adopted circular source of diameter 11 $^{\prime \prime }$. To achieve agreement for H₂O, the model fluxes would need to be adjusted downwards by a factor of 2.5, whereas an increase by more than one order of magnitude (a factor of 12) would be required for OH. Evidently, OH is largely underproduced by these models, a fact also pointed out by Wardle (1999). If on the other hand the Wardle model is essentially correct, this would suggest that the ionisation rate in the ${\rm Serpens ~ cloud ~ core}$ is significantly higher (up to $10^{-15}~{\rm s}^{-1}$) than on the average in dark clouds, $\zeta =(10^{-18}$- $10^{-17})~{\rm s}^{-1}$. High X-ray activity is known to be present within the ${\rm Serpens ~ cloud ~ core}$ (Smith et al. 1999 and references therein). It is conceivable that such a high ionisation rate could also have considerable consequences for the cloud chemistry and its evolution. For instance, a relatively higher H⁺₃ abundance could be expected, the effects of which (in addition to the enhanced abundance of OH) may in fact have already been observed (e.g., HCN/HNC $\sim$ 1; McMullin et al. 2000 and references therein).

Figure 16: Comparison of our molecular line observations with the predictions of theoretical models of C-shocks (Kaufman & Neufeld 1996). The pre-shock density is always $\log n_0 = 5.5$ (cm-3). Solid and dashed lines are for models with $v_{\rm s}$ = 15 km s-1 and 20 km s-1, respectively. The H2 models refer to the CVF pixel size, whereas the other panels are for a circular source of diameter 11 $^{\prime \prime }$. The scaling factors, necessary to bring the observed and model data into agreement, are indicated in each frame by " $\times~{\rm factor}$''.

4.4 Summarising discussion

Based on their 0.8 mm JCMT-CSO interferometry, Brown et al. (2000) obtained estimates of the size, mass and average (dust) temperature of the disk of SMM 1. The estimated mass is larger and the size of the disk is smaller by one order of magnitude than what is required to account for the observed level of line emission (Sect. 4.2.2). Unless the disk (extended atmosphere?) is heated to very much higher temperatures (by an as yet to be identified mechanism) than the 60 K determined by Brown et al., we find it unlikely that the molecular line spectrum of SMM 1 is of circumstellar disk origin. Our own calculations (Sect. 4.2.4) confirm this conclusion.

It is intriguing that the luminosity of the spherical model of SMM 1 (71 $L_{\odot}$, Larsson et al. 2000) is close to the "magic number'' of the classical main accretion phase of solar mass stars (Shu et al. 1987). At the elevated cloud temperature of the ${\rm Serpens ~ cloud ~ core}$ ($\sim$ 40 K, White et al. 1995), the isothermal sound speed is 0.4 km s^-1 and, hence, the (time averaged, cf. Winkler & Newman 1980) mass accretion rate corresponds roughly to M $_{\rm acc} = 10^{-5}$  $M_{\odot} ~ {\rm yr}^{-1}$, yielding $~L_{\rm acc} = 70$ $L_{\odot}$, where we have used the mass-radius relationship of Palla & Stahler (1990). In this scenario, the age of SMM 1 would be about 10⁵ yr or less, depending on the details of the acquired mass of the (presumably deuterium burning) central core. Regarding the data presented in this paper, we find it however difficult to reconcile this accretion shock model with our observations. As concluded in Sect. 4.2.4, the excitation of the observed lines requires significantly larger volumes at elevated densities and temperatures.

The H₂ observations are partially resolved and there exists no ambiguity as to where, with respect to SMM 1, the emission arises (cf. Fig. 2). These lines trace a collimated outflow toward the northwest of SMM 1, which is also seen in ro-vibrationally excited H₂ line emission (Eiroa & Casali 1989; Hodapp 1999). In the graphs of Fig. 16, we have assumed that also the LWS lines originate essentially at the location of the H₂ spots (i.e. we have artificially introduced another factor of two for the fluxes). However, the dereddened data of Eiroa & Casali (with the $A_{\rm V}$-value determined in Sect. 4.2.1) could potentially present an additional difficulty for the C-shock model (Kaufman & Neufeld 1996). The estimated 1-0S(1) line intensity would in this case be larger by more than two orders of magnitude than that predicted by the model. We cannot exclude at present, however, the possibility that the 1-0S(1) emission observed by Eiroa & Casali (1989) is essentially unextinguished. Photometrically calibrated data at higher spatial resolution would be required to settle this issue.

The mechanical energy input by the flow is $L_{\rm mech} = 0.5~\mu_{\rm gas}~m_{\rm H}~n_0~v_{\rm s}^3 \times {\rm area}$, which for a pre-shock density of $\log n_0 = 5.5$, a shock velocity $v_{\rm s}$ = (15-20) km s^-1, and a 5 $^{\prime \prime }$ source size yields $L_{\rm mech} = (6.4$- $15.1) \times 10^{-2}$ $L_{\odot}$. From the Kaufman & Neufeld (1996) C-shock model, this gas is cooled by H₂ at a rate of (1.0- $4.7) \times 10^{-2}$ $L_{\odot}$. From the LWS data, we inferred the total cooling rate through the lines of CO, ¹³CO, H₂O  and OH of $78\times 10^{-2}$ $L_{\odot}$ (Sects. 4.2.2 and 4.2.3), corresponding to 0.5% to 1% of the total dust luminosity. This is larger by factors of 5 to 12 and it is thus not excluded that the shocked regions observed in the H₂ lines and those giving rise to the FIR lines are not the same. We reached the same conclusion on the basis of our excitation and radiative transfer calculations.

The observed and background-corrected [O  I] 63 $\mu$m emission toward SMM 1 suggests a contribution also by J-shocks within the LWS beam (Sect. 4.1.2). Intriguingly, the derived dimensions are practically identical to those determined for the LWS-molecular emission, albeit existing J-shock models do not predict the relative intensities correctly. At present, we can merely conclude that shocks, in general, provide a plausible energy input mechanism, although the details of the shock type(s) are less clear. We propose that predominantly slow shock waves in the dense medium surrounding SMM 1 provide the heating of the molecules we have observed with ISO, whereas dynamical collapse is not directly revealed by our data.