4 Discussion

4.1 The structure of the envelope

The first result of our modeling is its capacity to reproduce the observed water emission. In the following we show that the derived parameter values are in agreement with previous estimates obtained through different studies. Before discussing in detail this comparison we describe here the derived density and temperature profiles, as well as the gas heating and cooling mechanisms. The reconstructed density and temperature profiles are shown in Fig. 6. In the outer region the density follows a r^-2 law (static region), while in the inner region the density is proportional to r^-3/2 (free-fall region), with a transition region starting at $\sim$1500 AU, that covers a substantial range of radii and is flatter than r^-3/2. The absolute density is relatively large, $4 \times 10^6$ cm^-3 at 1500 AU.

Figure 6: Density and temperature profiles of the envelope as computed by the best fit model. In the lower panel the dashed line refers to the dust temperature, while the solid line refers to the gas temperature.

The gas temperature closely follows the dust temperature in the outer and intermediate regions of the envelope. At $r \sim 80$ AU, dust and gas decouple. This is caused by the evaporation of the icy mantles when the dust temperature reaches 100 K, which injects large amounts of water in the gas phase (about a factor ten more) increasing the cooling efficiency of the gas. The gas temperature drops by about 20 K and remains lower than the dust temperature until at $r \sim 30$ AU the dust FIR pumping of the water molecules counterbalances the water line cooling and couples again the gas and dust temperatures. The heating and cooling of the gas in the envelope follow the general properties discussed in CHT96 (see also Ceccarelli et al. 2000a). The heating is dominated by the NIR water pumping, gas-grains collisions and the compression of the gas in the innermost regions, while gas-dust collisions are the main heating factor in the outer parts of the envelope and when the dust and gas decouple at $r \sim 80$ AU. The gas cooling is dominated by the line emission of water, oxygen and CO. In the outer parts of the envelope the cooling is dominated by the CO line emission, while in the intermediate region the cooling is dominated by the oxygen and water lines, as the CO lines become rapidly thick. At $r\leq 80$ AU, the cooling by the water lines takes over, due to the icy mantle evaporation, and dominates the cooling by orders of magnitude with respect to O and CO. Note that before the ices evaporation (i.e. at $r \ge 80$ AU), the heating is dominated by the gas compression and the cooling by the H₂O line emission. The increase of the gaseous water abundance by a factor ten causes an increase of about the same amount in the gas cooling rate, whereas the compression heating rate just increases by 20%. The gas heating becomes hence dominated by the gas-grains collisions, which tend to couple dust and gas. In the specific case of IRAS 4, the water cooling rate $\Lambda_{{\rm H_{2}O}}$ and gas-grains collisions heating rates $\Gamma_{{\rm dg}}$ can be approximated by the following expressions, at the radius just before the evaporation (for the details see CHT96):

$$\Lambda_{\rm {H_{2}O}} = 3.4 \times 10^{-15} ~\frac{x({\rm H_... ...rm gas}}{90 ~{\rm K}} \right)^{3/2} {\rm erg~s^{-1} cm^{-3}}$$ (6)

$$\Gamma_{\rm {dg}} = 3.6 \times 10^{-16} ~({T_{\rm dust} - T_{... ...rm gas}}{90 ~{\rm K}} \right)^{1/2} {\rm erg~s^{-1} cm^{-3}.}$$ (7)

The increase of a factor ten in the water cooling rate after the ice evaporation is only in partly counter-balanced by the increased heating rate due to the gas-grains collisions, and the difference between the two temperatures is $\sim$20 K.

The CO and atomic oxygen abundances are constant across the envelope, within the studied range, i.e. 30 to 3000 AU. We will discuss in the next paragraph the effect of varying the CO abundance across the envelope to take into account the CO depletion when the dust temperature is below the CO-rich ice evaporation temperature. As widely discussed previously, the water abundance undergoes a jump of about a factor ten at $r\leq 80$ AU, when H₂O-rich ices evaporate (dust temperature larger than 100 K). One interesting prediction of this study is the existence of a hot core like region in the innermost parts of the envelope, where the dust temperature reaches the sublimation temperature of the grain mantles.

4.2 CO depletion

Here we want to address in some detail the issue of the CO depletion claimed in IRAS 4 by comparing low-lying, millimeter CO transitions and dust continuum emission (Blake et al. 1995). Blake et al. used the CO J = 3-2 transitions of the ¹²C, ¹⁸O and ¹⁷O isotopes and found that the emission is accounted for a CO abundance of $\sim$ $2.5\times 10^{-5}$, i.e. a factor five lower than the "canonical'' value. Their explanation is that CO molecules freeze out on the grain mantles and the gas phase CO results therefore depleted across the envelope. This result has been recently confirmed by JSD02, who quote CO/H $_2 \sim 10^{-5}$ in the outer envelope of IRAS 4. The question arises whether the CO is depleted across the entire envelope or not, as also remarked by JSD02. We checked with our model if the millimeter observations would be sensitive to higher abundances of CO in regions where the dust temperature exceeds the CO-rich ice evaporation temperature, i.e. $\sim$25-30 K. We found that if the C¹⁸O J = 3-2 line is optically thin, the bulk of the J = 3-2 line is emitted at $\sim$3000 AU. This is because of the excitation of the CO J = 3-2 line itself. In the optically thin and thermalized line approximation it is possible to derive the following approximate analytical expression (strictly speaking, valid only for the CO millimeter lines):

$$I_J \propto \int \frac{{\rm exp}[-E_J/k{T_{\rm gas}}]}{T_{\rm gas}} {\rm d}r.$$ (8)

When J = 3 the integrand has a peak around ${T_{\rm gas}} \sim 30$ K, does not vary by more than 20% with temperatures between 20 K and 40 K, and decreases for temperatures larger than $\sim$40 K. As a result, the line intensity in the outer envelope is linearly proportional to the radius of the emitting gas, and therefore increases going outward, i.e. the emission is dominated by the outer envelope where CO is depleted. We run a case in which there is no CO depletion across the envelope, (i.e. the CO abundance is constant and equal to 10^-4), a case where CO is depleted on the entire envelope (i.e. the CO abundance is constant and equal to $2.5\times 10^{-5}$), and a third case where CO is depleted only in the outer envelope (i.e. the CO abundance is $1 \times 10^{-5}$ when ${T_{\rm dust}} \leq 30$ K and 10^-4 in the rest of the envelope). The last two cases give the same intensity (within 15%) on both C¹⁸O J = 3-2 and J = 2-1 lines, while the first one gives an intensity larger by about a factor 3. Therefore, the last two cases, CO depletion across the entire envelope and depletion in the outer envelope only, are not distinguishable by the C¹⁸O J = 3-2 and J = 2-1 observations. In this sense, we think that the $J_{\rm up} \leq 3$ observations cannot probe the innermost regions and that it is possible that the measured CO depletion is only relative to the external regions of the envelope.

This situation would be similar to what has been claimed to occur in IRAS 16293-2422 (Ceccarelli et al. 2001), based on the indirect evidence provided by the D₂CO emission. The D₂CO molecule is considered a grain mantle product, as gas phase reactions seem unable to form an appreciable amount of this molecule. Ceccarelli et al. showed that in IRAS 16293-2422 the D₂CO emission originates in the region of the envelope where ${T_{\rm dust}} \geq 30$ K. The proposed interpretation is that D₂CO is trapped in CO-rich ices that evaporate when the dust temperature exceeds 30 K. Hence, in IRAS 16293-2422 there is an outer region of the envelope where CO is frozen onto the grain mantles ( ${T_{\rm dust}} \leq 30$ K), and a regions with ${T_{\rm dust}} \geq 30$ K where CO is released into the gas phase and has the standard $\sim$10^-4 abundance. A similar scenario has been also suggested by JSD02 for other Class 0 sources that show CO depletion.

4.3 Comparison with previous studies of IRAS 4

In this paragraph we compare our results with previous studies dealing in a way or in another with some of the issues addressed in the present study. We start with the recent study by JSD02, who derived the density profile of the IRAS 4 envelopes by modeling the continuum emission (spectral energy distribution and 450/850 $\mu$m maps simultaneously) and assuming a single power law index. They found a power law index equal to 1.8 and 1.3 for 4A and 4B respectively, consistent with the Shu inside-out solution adopted in our model. At 1000 AU they estimate a density equal to 6 (4A) and 2 (4B) $\times 10^{6}$ cm^-3, which are quite comparable with our estimate: $6 \times 10^{6}$ cm^-3. We emphasize that the two methods, their and our, are totally independent, and use different data, continuum and line observations respectively. The fact that both predict approximately the same density structure in a certain way validate both methods, or at least increases the probability that both models describe reasonably well IRAS 4.

Apart from the density and temperature profiles, our model also constrains the water abundance profile. It is reasonable to ask whether our predicted water abundance in the innermost and outer regions of the envelope are realistic and if they have any support from different observations. The situation here is somewhat complicated by the fact that there aren't many other independent ways to measure the water abundance. From a theoretical point of view the abundance in the outer envelope, $5 \times 10^{-7}$, can be very well compared with chemistry model predictions (e.g. Lee et al. 1996). In this respect, the value that we derive is certainly not extraordinary and rather plausible. From an observational point of view Bergin et al. (2002) succeeded to detect the 557 GHz water line in the NGC1333 molecular cloud. They estimate the water abundance in the region to be $\sim$10^-7, with unfortunately a relatively large error ($\sim$10) due to the many uncertainties in the excitation of the line. Moneti et al. (2001) derived a water abundance of $3 \times 10^{-7}$ in the clouds in the line of sight of the galactic center. These authors claim that this is very likely the abundance of standard molecular clouds. In summary, the water abundance that we find for the cold region of the IRAS 4 envelope is consistent with other studies. Regarding the abundance in the inner region, $5 \times 10^{-6}$, the value that we obtain seem to be lower than what expected if all the water ice is injected in the gas phase and a large fraction of the oxygen is locked in this ice. A typical water ice abundance is estimated around 10^-4 (Tielens et al. 1991). However, SWAS observations of IRAS 4 and other low mass protostars suggest that the water abundance in their outflows is around 10^-6 (Neufeld et al. 2001; Bergin et al. 2002), i.e. similar to the value that we find. Those estimates are very rough and could easily be off by a factor ten (Neufeld et al. 2000), as they are based on one transition only, but nevertheless have the advantage that the observed emission is certainly dominated by the outflow (the spectral resolution of these observations is $\sim$1.2 km s^-1) so there are no doubts on its origin. Since the water abundance in the outflow would be probably dominated by the grain mantles released in the gas phase, these observations probably measure the water content in the mantles, very much as our observations measure (indirectly) the water content mantles in the inner hot like region. The two measurements seem to be consistent in giving a rather low value. Whether this validates both measures is less certain than the density profile case: it certainly does not discredit the two measures. Finally, even the comparison of our estimate of the accretion rate and central mass are in good agreement with the previous estimates, based on a different method (line profile and molecule H₂CO), by Di Francesco et al. (2001). We derived $\dot{M}$ = $5 \times 10^{-5}$  ${M}_{\odot }$ $~{\rm yr}^{-1}$ against the $11{-}4 \times 10^{-5}$  ${M}_{\odot }$ $~{\rm yr}^{-1}$ quoted by Di Francesco et al. (2001), and M_* = 0.5  ${M}_{\odot }$ against the 0.23-0.71  ${M}_{\odot }$.

To conclude, these studies show that the values we derive of the four parameters of our model are plausible and nothing of particularly surprising, with the possible exception of the water abundance in the innermost regions. In other words, if we had to choose a priori those values we would have chosen exactly what we found. The conclusion is that it is very probable that at least most of the observed water emission in IRAS 4 originates in the envelopes. If any, just a small fraction should therefore be associated with the outflow. Our final comment is therefore that care should be taken when interpreting the observed water emission towards low mass, Class 0 protostars as due to shocks (e.g. Ceccarelli et al. 1998; Nisini et al. 1999, GNL01), as we showed in two out two cases that the massive envelopes surrounding these sources dominate the water emission, just because of the large total column density. As a matter of fact, Class I sources, which are characterized by less massive envelopes, do not show up strong water emission (Ceccarelli et al. 2000a).

4.4 Comparison with IRAS 16293-2422

The mass and accretion rate we derived for IRAS 4A and B are of the same order of magnitude of those found in IRAS 16293-2422 (Ceccarelli et al. 2000a). IRAS 16293-2422 seems more massive (0.8  ${M}_{\odot }$) than IRAS 4A (0.5  ${M}_{\odot }$), and accreting at a slightly lower accretion rate (3 against 5 $\times 10^{-5}$  ${M}_{\odot }$ $~{\rm yr}^{-1}$). Assuming a constant accretion rate, those values give an age of 10 000 years and 27 000 for IRAS 4 and IRAS 16293-2422 respectively. Hence IRAS 16293-2422 seems more evolved than IRAS 4. Moreover, IRAS 4 possesses an hot core like region about two times smaller than IRAS 16293-2422 (80 AU against 150 AU). Ground-based H₂CO and CH₃OH observations (Blake et al. 1995; Maret et al. in preparation) confirm that IRAS 4 is in fact colder, and therefore less bright in these molecular transitions than IRAS 16293-2422, and that indeed the IRAS 4 hot core like region is very small. This fact coupled with the larger distance of IRAS 4 from the Sun may explain the apparent difference in the molecular emission of these two sources, which is much richer in IRAS 16293-2422. This conclusion is also in agreement with the relatively higher millimeter continuum observed in IRAS 4, which implies a larger amount of cold dust surrounding this source than IRAS 16293-2422. In addition, the region where the dust temperature is higher than 30 K is smaller in IRAS 4 ($\sim$1500 AU) than in IRAS 16293-2422 ($\sim$4000 AU), i.e. the CO depleted part of the envelope is relatively larger in IRAS 4 than in IRAS 16293-2422. This may explain why the CO depletion has been observed towards IRAS 4 and not in IRAS 16293-2422 (van Dishoeck et al. 1995; Ceccarelli et al. 2000b).

Finally, despite this difference in the age, the water abundance in the envelope is remarkably similar in the two sources, both in the outer part of the envelope and in the inner ones, where ice mantles are predicted to evaporate. This is an important piece of information, suggesting that the ice mantle formation in the two sources underwent a similar process, despite the macroscopic difference between the two molecular clouds which the two sources belong to. In the case of IRAS 16293-2422, the cloud seems very quiescent, shielded from strong UV and/or X-ray radiation (e.g. Castets et al. 2001) and with even evidence of large CO depletion (Caux et al. 1999b). In the other case, IRAS 4, the cloud presents cavities excavated by the several young stars of the region (e.g. Lefloch et al. 1998), and it is probably permeated by the X-rays emitted by them. A forthcoming study will allow to measure the H₂CO and CH₃OH abundances in the inner hot core like region of IRAS 4 (Maret et al. in preparation) and make comparisons with that found in IRAS 16283-2422 (Ceccarelli et al. 2000b). This study will hence help to understand in more detail how apparently different conditions in the parental clouds affect the grain mantle composition.