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Astron. Astrophys. 345, 949-964 (1999)

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4. Discussion and interpretation

CFTWO concluded that IRAS 20126+4104 consists of a disk-outflow system originating from a young early type massive (proto)star. Is this scenario confirmed by the present observations? And, if so, do they improve the understanding of the jet/outflow and disk structure? In the following we shall give an answer to these questions. In particular, in Sect. 4.1 we show how the new SiO measurements are fundamental to understand the geometry and kinematics of the outflow/jet, while in Sect. 4.2 we make use of the CH3CN(12-11) sub-arcsecond resolution observations to improve our knowledge of the disk dynamics.

4.1. The outflow/jet system

As already noted in Sect. 3.1.1 and discussed in CFTWO, the morphology of the outflow in IRAS 20126+4104 is very intriguing: the blue-shifted lobe of the flow corresponding to the inner blue wing lies to the SE, whereas the same lobe lies to the NW in the maps of the outer wings; a similar statement holds for the red-shifted lobe (see Fig. 3, left). The tentative explanation proposed by CFTWO and supported by models such as that by Cabrit & Bertout (1986) is that the axis of the outflow lies very close to the plane of the sky. As we shall see, such an interpretation is confirmed by the SiO observations.

In order to better illustrate the kinematics and morphology of the outflow/jet system, we show in Fig. 12 the position-velocity plots along the flow axis for the HCO+(1-0) and SiO(2-1) lines. The difference between the two plots is striking: although in both transitions the lowest contour describes an S-shaped pattern, the peaks in the two maps lie on opposite sides with respect to the bulk velocity. This can be seen for example looking at the peak to the NW, which is blue-shifted in SiO, but red-shifted in HCO+. Another difference consists of the fact that most of the HCO+ emission arises inside about [FORMULA]8" from the core position, whereas for SiO it is strong only outside that range. Finally, a similar difference holds for the velocity range, because the SiO emission extends up to [FORMULA]25 km s-1 with respect to the bulk velocity, whereas only very faint HCO+ emission is detected beyond [FORMULA]5 km s-1.

[FIGURE] Fig. 12. Position-velocity plot along the axis of the flow for the HCO+(1-0) (top panel ) and SiO(2-1) (bottom ) lines. Contour levels correspond to -0.05, 0.05 to 4.45 by 0.4 Jy/beam for HCO+, and to -0.002, 0.002 to 0.05 by 0.006 Jy/beam for SiO. The crosses in the bottom left of each panel indicate the angular and spectral resolution. The horizontal and vertical dotted lines correspond respectively to the bulk velocity and to the peak position of the 3 mm continuum

A possible explanation for these characteristics is that the relatively wide outflow seen in HCO+ is fed by the narrow, well collimated jet traced by the SiO and H2 line emission. In this scenario the SiO molecules would be more abundant in the shocked layer of gas compressed and accelerated by the jet, whereas HCO+ would dominate in the molecular gas further away from the shock. In other words, the molecular outflow would roughly consist of two components: one (the jet) ejected with high velocity in a narrow angle around the axis of the flow; the other expanding at lower velocity and much less beamed. The SiO abundance would be strongly enhanced in the shocked region between the two components and would hence trace the high velocity gas. On the other hand, the bulk of the HCO+ line would trace the low velocity component, although also some faint high velocity HCO+ emission is clearly detected: this probably traces the gas closer to the SiO emitting region. A similar example is given by the flow structure seen in the CO(2-1) line towards HH 211 (Guilloteau et al. 1997).

In order to confirm the previous scenario, one should elaborate a detailed physical model describing the line emission in an expanding flow with given density and excitation temperature gradients. Such a model goes beyond the purpose of this work; however, it is possible to fit the S-shaped pattern of the SiO position-velocity plot using a simple-minded approach. Our assumption is that the jet traced by SiO is conical in shape and that the gas in it is uniformly accelerated up to a maximum distance from the centre, where its velocity drops abruptly to zero. The latter hypothesis is justified by the clear existence of a bow shock at the end of the SiO jet (see Figs. 3 and 4). This simple model allows to easily reproduce the maximum extent of the emission pattern in Fig. 12 bottom, namely the shape of the faintest contour level. Note that the assumption of velocity proportional to the distance from the centre ("Hubble" law) is consistent with what observed in other outflows such as that in HH 211 (see Guilloteau et al. 1997).

The details of the model are given in Appendix A; however, it is easy to understand that the emitting region in the position-velocity plot is included between two straight lines (corresponding to the expansion velocity along the surface of the cone) and an ellipse (corresponding to the maximum velocity reached by the expanding gas). The inclination of the straight lines and the size and eccentricity of the ellipse depend on the free parameters of the model, namely the semi-aperture angle of the cone, its inclination with respect to the plane of the sky, the maximum velocity of the gas, and the angular distance from the centre at which this velocity is reached. In Appendix A we demonstrate that these parameters can be reasonably determined from the observational data and we present the best fit obtained in Fig. 13.

[FIGURE] Fig. 13. Same as Fig. 12 for the SiO(2-1) line only. Contour levels correspond to -2.5, 2.5 to 44.5 by 6 mJy/beam. The dashed lines and ellipse identify the region of the plot where line emission is expected in the best fit model described in Appendix A

The most important result of the model is the accuracy of the determination of crucial quantities such as the inclination angle of the jet axis. This is due to the sensitive dependence of the fit on the input parameters. Clearly the inclination angle is very small, thus confirming the hypothesis of CFTWO that the flow axis lies close to the plane of the sky.

Incidentally, we note that the faint emission seen in Fig. 13 along the line of sight through the centre at [FORMULA][FORMULA]-70, 50, and 90 km s-1 is very unlikely to be blue- and red-shifted SiO. We believe that this is to be attributed to lines of other species such as C2H5CN and C2H5OH arising from the core.

Finally, we note that the previous treatment cannot be applied to HCO+, which presents a much more complicated morphology. For this, a more complex model than the one presented here is needed.

4.1.1. Time scale and physical parameters of the jet

As done by CFTWO for the HCO+ outflow, we can use the SiO emission to estimate the relevant parameters of the jet. These are given in Table 6, where the mass was obtained by integrating the SiO line emission over the whole jet, in the outer and inner wings, i.e. from -28 to -5 km s-1 and from 2 to 21 km s-1. Optically thin emission was assumed, with an excitation temperature of 30 K as in CFTWO. The SiO abundance relative to H2 was chosen equal to [FORMULA], similar to that found by Acord et al. (1997) for the flow in the high-mass YSO G5.89-0.39. The momentum and kinetic energy of the jet were estimated by summing the contribution of each velocity channel. The time scale, t, of the jet was calculated assuming uniform acceleration of the ejected gas, as suggested by the model previously discussed: in this case [FORMULA], where [FORMULA]=0.099 pc is the maximum distance from the centre reached by the ejected gas and [FORMULA]=100 km s-1 is the velocity of the gas at that distance (see Appendix A for an estimate of these values).


Table 6. Physical parameters of jet[FORMULA].
a) assuming [FORMULA]=30 K and [SiO/H2]=[FORMULA]
b) values in parentheses are corrected for an inclination angle [FORMULA] of the jet axis with respect to the plane of the sky (see Appendix A)
c) obtained assuming a true jet expansion velocity [FORMULA]=100 km s-1 (see Appendix A)

We note that all the quantities in Table 6 computed from the observed velocity (i.e. energy, momentum, and luminosity) must be corrected for the inclination of the flow axis: this can be obtained by dividing the momentum by [FORMULA] and the energy and luminosity by [FORMULA], where [FORMULA] is the angle between the flow axis and the plane of the sky. In Appendix A we show that a reasonable guess is [FORMULA] and hence the correction factor amounts to [FORMULA]6.4 for the momentum and momentum rate, and [FORMULA]41 for the energy and mechanical luminosity.

4.2. The rotating disk

As shown in Sect. 3.1.2, the CH3CN(12-11) observations confirm that the hot core in IRAS 20126+4104 is elongated perpendicularly to the jet/outflow direction and presents a velocity gradient along its symmetry axis. Also, we note that the peak of the millimeter continuum emission lies to a good approximation at the centre of such an elongated structure (see Fig. 5b). The CFTWO interpretation is that the methyl cyanide core is indeed a rotating disk around a high-mass (proto)stellar object. A similar conclusion is reached by Zhang et al. (1998b), who detect a rotating disk on a much larger scale ([FORMULA]4", i.e. [FORMULA]0.03 pc) in the ammonia (1,1) and (2,2) inversion transitions. If we are really observing such a well defined entity and not just the resulting effect of two or more distinct cores unresolved within the PdBI beam, then one should expect to see systematic trends in other parameters than velocity. In other words, if one deals with a well defined symmetric disk with a (proto)star at its centre, then it makes sense to ask a couple of more questions, namely: is there any temperature gradient in the disk, due to the embedded central stellar object? and, is it possible to obtain some better information on the velocity field in the disk? In this section we address an answer to these questions.

4.2.1. Temperature gradient

If a temperature gradient does exist, this should affect the distribution of lines with different excitation energies. In order to investigate this effect, we plot in Fig. 14 three quantities measured for each K line of the CH3CN(5-4) (from CFTWO) and (12-11) transitions, as a function of the corresponding excitation energy. These quantities are the peak velocity, the full width at half maximum ([FORMULA]), and the angular diameter ([FORMULA]) of the map obtained by integrating the emission under the line. [FORMULA] is computed from the measured FWHP with a simple gaussian deconvolution; also, the position of the emission peak is the same for the different K lines, within the uncertainty. The values of [FORMULA] and [FORMULA] have been obtained with a gaussian fit: note that the formal errors of the fit are much less than the spectral resolution and must hence be taken with some care. The errors on [FORMULA], instead, have been computed from the FWHP obtained after increasing and decreasing the value of the peak emission by an amount equal to the 1[FORMULA] RMS noise of the corresponding map.

[FIGURE] Fig. 14. Plot of the peak velocity (top panel ), FWHM (middle ), and angular diameter (bottom ) measured in each K component of the CH3CN(5-4) (from CFTWO) and (12-11) lines, versus the energy of the corresponding lower level of the transition. In the bottom panel we plot also the points for the K=2 and 4 lines of C[FORMULA]CN(12-11). Note that the number of points in the bottom and middle panels differs from that in the bottom panel because for a few K lines the estimate of [FORMULA] was affected by too large an error to be reliable, although it was possible to estimate [FORMULA] and [FORMULA] from the spectrum. The dashed and dotted horizontal lines in the bottom panel indicate the size and the associated error of the 1.3 mm continuum emission

Looking at Fig. 14, one sees that [FORMULA] is clearly decreasing with increasing excitation energy of the transition. This demonstrates two facts: that higher energy lines arise from smaller regions; and that the excitation temperature increases towards the centre of the core. We stress that the trend shown in Fig. 14 bottom, cannot be due to optical depth effects: in fact, in this case the size measured in the high energy lines should be greater or equal to that measured in an optically thin tracer such as C[FORMULA]CN(12-11), whereas Fig. 14 shows that [FORMULA] is marginally greater in the latter (filled circles above [FORMULA]400 K) than in the former (empty circles) lines. Also, the ratio between [FORMULA] in the highest and that in the lowest energy lines is [FORMULA]0.4, whereas it is possible to demonstrate that in a spherical homogeneous core the ratio between [FORMULA] in the optically thick and thin limit cannot be less than [FORMULA].

4.2.2. Velocity field: rotation

The fact that higher excitation lines trace inner regions of the core (and hence of the disk) can be used to study the velocity field in it. The trend of [FORMULA] and [FORMULA] in Fig. 14 reflects a variation of these quantities as a function of distance from the centre. In particular, if we are dealing with a rotating disk whose mass is not much larger than that of the embedded star, the rotation velocity should increase towards the centre. Such an effect cannot be seen in Fig. 5c because the core is only barely resolved; however, just for this reason the whole disk falls into the beam and hence one should see an increase of the line width in the higher energy transitions, which arise closer to the centre of rotation. This is exactly what the middle panel of Fig. 14 shows. In order to clarify this issue, in the top panel of Fig. 15 we plot [FORMULA] versus [FORMULA] on a logarithmic scale. A linear fit to the data gives the following relation:


The dependence on [FORMULA] is very close to that expected for Keplerian rotation: in this case the line width is equal to twice the velocity at an angular distance [FORMULA]/2 from the centre and is given by the expression


where d is the distance from the observer, [FORMULA] the mass of the star, and [FORMULA] the angle between the disk axis and the plane of the sky. In order to reproduce the fit of Eq. (1), one must pose [FORMULA] in Eq. (2). In Appendix A we demonstrate that [FORMULA]-[FORMULA]: we thus conclude that the mass of the star is [FORMULA]24 [FORMULA]. Such a value is in very good agreement with the 20 [FORMULA] derived by Zhang et al. (1998b).

[FIGURE] Fig. 15. Top panel: line FWHM of the CH3CN(5-4) (from CFTWO) and (12-11) K components as a function of the corresponding angular diameter of the emission integrated under the line. The straight line represents a least square fit to the data. Bottom panel: same as top panel with the addition of the points corresponding to the H13CO+(1-0) transition (this paper) and to the NH3(1,1) line (Zhang et al. 1998b). The straight line is the same as in the top panel

We caution that the result obtained above needs further confirmation because the spectral resolution of our observations is much larger than the formal errors reported in Fig. 15. However, a couple of facts support this interpretation. First of all, Fig. 5c shows a maximum dispersion along the velocity axis towards the centre of the core (indicated by the dashed vertical line): this is just what one expects for Keplerian rotation. Secondly, the trend shown by the CH3CN data (Fig. 15 top) is consistent with the points obtained from the NH3(1,1) inversion transition (Zhang et al. 1998b) and the H13CO+(1-0) line, as shown in Fig. 15 bottom. For the H13CO+ emission, the diameter is the FWHP of the map represented by the grey scale in Fig. 6 bottom, whereas [FORMULA] is the FWHM of the line obtained by integrating the H13CO+ emission over the FWHP of such a map. In conclusion, we favour the hypothesis that the structure seen in the NH3(1,1) (Zhang et al. 1998b) and H13CO+ (Fig. 6 top) lines represents the outer layers of a rotating Keplerian disk, the inner layers of which are seen in the CH3CN lines.

If we are indeed observing a Keplerian disk, then its mass must be less than that of the star. An estimate of the disk mass on the scale traced by CH3CN can be obtained from the millimeter continuum flux measured with the PdBI. We use the expression of Wilking et al. (1989) to compute the mass from the measured millimeter flux, assuming a kinetic temperature of 200 K (see CFTWO) and a dust absorption coefficient proportional to [FORMULA], as suggested by the slope of the continuum spectrum between 1.3 and 3 mm ([FORMULA]): we thus obtain [FORMULA], much less than the mass estimated above for the star. However, this refers only to the inner 0.007 pc whereas Keplerian rotation seems to hold up to the size seen in the H13CO+ line ([FORMULA]0.055 pc), as suggested by Fig. 15 bottom. It is thus interesting to estimate the mass traced by the H13CO+ disk. Assuming that the H13CO+(1-0) line is optically thin with an excitation temperature equal to 30 K, (i.e. the same assumed by CFTWO for the HCO+(1-0) line), and an abundance [H13CO+/H2]=[FORMULA], one obtains a mass of [FORMULA]30 [FORMULA] for the H13CO+ disk. Although such a value is very uncertain, it indicates that even the disk mass inside a radius of 0.028 pc could be not much greater than the mass of the star. We thus conclude that the hypothesis of a Keplerian disk as large as 0.055 pc cannot be excluded a priori.

4.2.3. Velocity field: collapse

So far, we have presented an explanation for the trends in the bottom and middle panels of Fig 14, namely for the dependence of [FORMULA] and [FORMULA] on the excitation energy of the transition. However, the rotating disk model cannot explain the trend of [FORMULA], shown in the top panel of the same figure. As already discussed for [FORMULA], also for [FORMULA] an increase with excitation energy implies an increase with decreasing distance from the centre: such a red-shift of the line is not expected if the disk is purely rotating, but might be explained if the inner regions of it are collapsing towards the centre. In this scenario, the line profile arising from the outer regions is quite symmetric because the collapse is less enhanced than close to the centre and because both the front (blue-shifted) and the rear (red-shifted) parts of the disk contribute to it; on the contrary, a line tracing the part of the disk closer to the centre must be fainter in the blue, if the rear emission is partly suppressed by the continuum absorption due to the dust in the core.

In order to check this effect one needs high spectral resolution in a high excitation line: the K=8 transition of CH3CN(12-11) can be used to this purpose. In Fig. 16 we show the K=8 line profile observed towards the centre. There is little doubt that the peak of the profile is shifted towards the red with respect to the systemic velocity, although some faint emission down to -15 km s-1 is detected. This seems to confirm that the blue-shifted emission is strongly affected by absorption. Note that the spectral feature at [FORMULA][FORMULA]13 km s-1 is unlikely to be CH3CN emission, because it should be seen at the same velocity also in the other K components of the CH3CN(12-11) transition: a comparison with the K=7 line (dashed histogram in Fig. 16) shows that this is not the case.

[FIGURE] Fig. 16. Spectrum of the CH3CN(12-11) K=8 line (full histogram) towards the peak position. The spectrum of the K=7 line (dashed histogram) is also shown to demonstrate that the feature at [FORMULA]13 km s-1 is not red-shifted CH3CN emission. The horizontal and vertical dotted lines correspond respectively to the zero intensity level and to the systemic velocity (-3.5 km s-1)

Incidentally, we note that self-absorption can be probably excluded because it should be more evident in the low energy transitions (i.e. low K), which better trace the low temperature diffuse material, than in the high energy ones (such as K=8), which mostly arise from the high temperature gas.

The explanation proposed above requires that a suitable region close to the centre of the core is optically thick in the continuum at 1.3 mm. Is this realistic? Following Wilking et al. (1989), one can express the dust optical depth as [FORMULA], where we have assumed a dust absorption coefficient proportional to [FORMULA]. A mean value of the column density over the core can be computed using the mass estimate of Sect. 4.2.2 (1.7 [FORMULA]) and the diameter as derived from Fig. 5b (0:004 or 0.0033 pc): one obtains [FORMULA]=[FORMULA] cm-2 and hence [FORMULA]=0.64. This result suggests that the optical depth along a line of sight close to the centre of the core could indeed be larger than 1.

4.2.4. The nature of the YSO

In Sect. 4.2.2 and 4.2.3 we have illustrated the evidence in favour of IRAS 20126+4104 being a YSO of 24 [FORMULA] surrounded by a collapsing and rotating Keplerian disk. Now, we use this result to investigate the nature of such a YSO.

Density distribution. If the disk is Keplerian up to a distance of 3:003 (0.028 pc) from the star (as suggested by the H13CO+ emission), then the mass inside this distance cannot be larger than, say, half the stellar mass, i.e. 12 [FORMULA]. On the other hand we have measured 1.7 [FORMULA] inside the core radius of 0.0026 pc. If the surface density of the disk is proportional to [FORMULA] (with R distance from the centre), then it is possible to demonstrate that the previous constraints turn into a lower limit on [FORMULA], namely [FORMULA]. Such a value is consistent with the expected surface density distribution in an accretion disk ([FORMULA]; see Hayashi 1981).

Mass accretion rate and luminosity of the YSO. One of the key questions about IRAS 20126+4104 is whether this is a young embedded zero age main sequence (ZAMS) early type star or a massive protostar. To this purpose, we note that the stellar mass of 24 [FORMULA] estimated above is larger than expected for a ZAMS star with the luminosity of IRAS 20126+4104. In fact, the latter is [FORMULA][FORMULA] which corresponds to a B0.5 ZAMS star (Panagia 1973), the mass of which should not exceed [FORMULA]15 [FORMULA]. Instead, a protostar derives its luminosity from accretion according to the expression (see Molinari et al. 1998)


where [FORMULA]=24 [FORMULA] is the mass of the protostar and [FORMULA] the mass accretion rate on its surface. An estimate of [FORMULA] can be obtained from the equation


with [FORMULA] and v surface density and collapse velocity of the disk at radius R. On the basis of the previous result, we can reasonably assume [FORMULA]: it is possible to demonstrate that, under such an assumption, [FORMULA] is 1/4 of the average surface density within R. The latter can be easily derived from the mass of the disk computed from the continuum emission (1.7 [FORMULA]; see Sect. 4.2.2) and the corresponding radius (550 AU; see Table 4). The value of v is estimated from the difference between the minimum and maximum [FORMULA] of Fig. 14 top ([FORMULA]3 km s-1). One thus obtains [FORMULA] yrs-1 and [FORMULA], which agrees very well with the bolometric luminosity of IRAS 20126+4104.

We conclude that the previous results favour the hypothesis that IRAS 20126+4104 is a massive protostar . However, one cannot exclude the possibility is that we are dealing with a circumbinary disk: the luminosity of a binary (or multiple) system with a total mass of 24 [FORMULA] can be much less than that of a single star with the same mass, thus explaining the discrepancy between the stellar mass and luminosity.

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Online publication: April 28, 1999