Astron. Astrophys. 324, 203-210 (1997)

5. Discussion

5.1. Derived parameters from NH₃ observations

The rotational temperature () is defined by applying the Boltzmann distribution to the populations of different (J,K) rotational levels and has been calculated from the (1,1) and (2,2) data following the procedure outlined by Hüttemeister et al. (1993 ) and Lemme (1995 ). Given the rotational temperatures, it is possible to estimate the kinetic temperature (). Since the derived values (see Table 5) are all around 10 K, the differences between kinetic and rotational temperatures, as shown by Walmsley & Ungerechts (1983 ) and Danby et al. (1988 ), are within the errors of our measurements. We thus can assume = .

Table 5. Derived parameters from NH₃ observations

The excitation temperature (), derived assuming a beam-filling factor of one, is defined in an analogous way to , but it is related to the population across an inversion doublet (J,K). Using and , it is possible to derive the total ammonia column density ( ; cm^-2) following the equations given by Ungerechts et al. (1986 ). For this, we have neglected non-metastable (J K) inversion doublets and have assumed a common rotational temperature for all the metastable (J = K) states. These are reasonable approximations for rotational temperatures around 10 K (e.g. Harju et al. 1993 ). The derived temperatures and column density values (see Table 5) are consistent with NH₃ observations towards other molecular cores in the TMC (e.g. Myers & Benson 1983 ; Benson & Myers 1989 ) and more isolated Bok globules (Lemme et al. 1996 ; Scappini & Codella 1996 ). The measured (1,1) excitation temperatures are only slightly smaller than the rotational temperatures (see Table 5) suggesting that the (1,1) transition is close to being thermalised. also requires that the beam-filling factor is near unity. This implies that, for the sources of the present sample, most of the ammonia flux is arising from an extended structure.

By balancing collisions and stimulated emission against spontaneous emission, the hydrogen density ( ; cm^-3) can be estimated (Ho & Townes 1983 ). The derived values, also accounting for photon trapping, range from 0.6 10⁴ to 19.9 10⁴ cm^-3 (see Table 6) and confirm that ammonia traces regions with densities 10⁴ cm^-3. Following Ho & Townes (1983 ), the errors on are 17% and 64% for L1521D and L1507A, respectively. It is worth noting that the errors associated with become large when is close to . Thus, the hydrogen densities derived for L1521F and L1524, i.e. for those sources with , may be lower limits only.

Table 6. Derived parameters related to the core stability of the NH₃ condensations

It is well known that intrinsic motions in molecular clouds and, in particular, in molecular cores are complex. The spectral lines of thermally excited transitions also contain a non-thermal contribution to the linewidth (e.g. Myers et al. 1991 ), which can provide information about the dynamics of the molecular gas. In order to derive , the actual intrinsic linewidth of the (1,1) line, we have corrected the observed intrinsic linewidth for the spectral resolution of the autocorrelator used during the observations, , by means of , assuming the filter (and the line) being gaussian shaped. The actual intrinsic linewidth is composed of a thermal component () and a non-thermal one (): (e.g. Myers et al. 1991 ). It is possible to see (Tables 2 and 5) that: (i) the actual intrinsic linewidths are between 0.17 and 0.27 km s^-1 ( 0.28 km s^-1) and (ii) the non-thermal velocity component is small, with a ratio between and the thermal linewidth component / 1.3. The results of the NH₃ observations, performed by Benson & Myers (1989 ) using the larger beamwidths of the Haystack and Green Bank antennas, indicate that, on average, ammonia cores with embedded IRAS source(s) have larger linewidths (0.45 km s^-1, not corrected for the spectral resolution) than cores without IRAS point source (0.27 km s^-1). Moreover, Myers et al. (1991 ) studied the linewidths of 61 dense condensations associated with star forming regions. They showed (see their Fig. 1) that the non-thermal component of the NH₃ core motions increases with IRAS luminosity more rapidly than does the thermal component and that in cores which can form stars more massive than 2 , non-thermal motions dominate thermal motions ( / 4). Thus, the linewidths derived suggest that our NH₃ clumps are either not associated with YSOs or that they are sites of low-mass star formation and are presumably associated with IRAS point sources of low luminosity ( 10 ; Myers et al. 1991 ), corresponding to a stellar mass of less than 2 .

The LSR velocity information of the four mapped cores has been analysed in order to search for velocity gradients: does not vary greatly within L1524 and L1507A, since the maximum differences between the values of the map positions are 0.16 km s^-1 and 0.14 km s^-1, respectively. These values are comparable to the channel spacing of 0.154 km s^-1. The map of L1521F shows a significant variation (0.46 km s^-1), but there is no clear indication for a systematic velocity gradient along a preferred axis. In contrast, L1521D displays a large variation of across the map revealing a velocity gradient roughly along the north-south axis with a change of about 0.43 km s^-1 over 120 (cf. Fig. 1; weak line emission from outside the outer 1.4 K contour supports this trend). This corresponds to 5 km s^-1 pc^-1, which is a value definitely larger than those reported by Goodman et al. (1993 ) for dense cores ( 4 km s^-1 pc^-1). It is also worth noting that the direction of the gradient is not related to the major-axis of the elongated NH₃ distribution of L1521D (Fig. 1). This is consistent with the work of Goodman et al. (1993 ), who studied the occurrence of velocity gradients in a sample of 43 ammonia cores, suggesting that the motion connected with the gradient, either rotation or shear, is not the major factor in the core dynamics.

5.2. IRAS counterparts

It is well known that the IRAS PSC offers a precious opportunity to identify star forming regions in optically obscured objects, such as molecular cores. Using the IRAS PSC, Kenyon et al (1990 ) have recently performed a survey of star forming regions in the TMC, identifying about 100 YSOs. They have shown that YSOs are scattered in a large area of 15 pc 20 pc and that star formation in the TMC is restricted to the dense molecular cores. With the aim to collect more information about the nature of the four molecular cores of the present list, we have cross-correlated their positions with the IRAS PSC. Clark (1987 ) has analysed the spatial distribution of 396 IRAS PS located up to 2 pc from 60 molecular cores. He found that the distribution is strongly peaked on the ammonia cores, with the central peak reduced by an order of magnitude at 0.18 pc. At distances larger than this value, a long tail is present and the population of IRAS PS can be significantly affected by background sources. Considering that the typical ammonia core diameter is 0.1 pc, 0.18 pc corresponds to 3.6 radii. Taking into account this result, we have looked for a coincidence within a radius of 1.8 times the size of the four molecular cores of the present sample. Moreover, IRAS sources known to be associated with extragalactic sources have been excluded (IRAS 1985 ). The result of the cross-correlation shows that L1521F and L1507A have no IRAS counterparts within the selected radius. We find two IRAS PS around L1521D, IRAS 04181+2655 (at a distance of 108 ) and IRAS 04181+2654 (138 ), and one near L1524, IRAS 04263+2426 (120 ). It is worth noting that all these IRAS PS satisfy the typical IRAS colour distribution for IRAS counterparts of molecular cores, since they have [25-12] ([i-j] log [ / ], where are the IRAS flux densities in bands i and j, in m) in the range between 0.42 and 0.47 and [60-12] between 0.60 and 1.23 (for comparison, see figure 3 of Codella & Palla 1995 ).

We thus conclude that L1521D and L1524 are associated with at least one IRAS counterpart (and therefore presumably with YSOs), while L1521F and L1507A appear not to be connected with any IRAS PS. Following Henning et al. (1990 ), the IRAS luminosities of the associated IRAS PS has been computed. The values of are 0.3 for IRAS 04181+2655 and IRAS 04181+2654 (L1521D) and 6.1 for IRAS 04263+2426 (L1524). This confirms that the ammonia molecular cores are sites of low-mass star formation, in accord with our results from the ammonia linewidths (Sect. 5.1).

5.3. NH₃ core stability

In order to obtain estimates of the energy terms of the NH₃ cores of our sample, the mean NH₃ total column density has been obtained as the average of all observed positions inside the FWHP contour of the ammonia cores (Figs. 1-4). This allows to calculate, using the ammonia abundance and assuming spherical cloud geometry, estimates of the mean hydrogen density and of the mass M () of a molecular core (Table 6). It is worth noting that, since M has been derived from the NH₃ total column density without virial equilibrium, we have made no assumption about the stability of the NH₃ cores. Ammonia abundances in interstellar molecular clouds cover a wide range: the [NH₃ ]/[H₂ ] ratio is believed to vary from a few 10^-8 in small dark clouds up to 10^-5 in the dense cores like Orion-KL (e.g. Ho & Townes 1983 and references therein). In particular, Benson & Myers (1983 ) studied the NH₃ abundance in a sample of quiescent dense molecular clouds and found that [NH₃ ]/[H₂ ] ranges between 3 10^-8 and 2 10^-7.

In order to derive the parameters reported in this paper, we have assumed [NH₃ ]/[H₂ ] = 10^-7. The resulting masses lie between 0.2 and 1.0 , placing the NH₃ cores with the lowest M values (L1507A and L1521D) towards the edge of the mass distribution for ammonia cores in the TMC, which ranges from a fraction to a few tens of solar masses, with a median value of 4 (Benson & Myers 1989 ).

Studying C₃ H₂ and H¹³ CO in TMC molecular cores, Mizuno et al. (1994 ) found that, contrary to the NH₃ results of Benson & Myers (1989 ), the cores without young stars (i.e. IRAS counterparts) are less dense than those with stars. The results of Mizuno et al. (1994 ) and Benson & Myers (1989 ) are consistent (average values) regarding the hydrogen density of the cores not connected with stars (4 10⁴ and 3 10⁴ cm^-3, respectively), while for those which host YSOs Mizuno et al. (1994 ) find 5 10⁵ cm^-3, a value larger than that given by Benson & Myers (1989 ), 1 10⁴ cm^-3. It is worth noting that Mizuno et al. (1994 ) determined the hydrogen density from H¹³ CO data, while the Benson & Myers (1989 ) analysis is based on a large velocity gradient model for the NH₃ cores. Mizuno et al. (1994 ) suggest that the disagreement of their results with those of Benson & Myers (1989 ) could be caused by differences in angular resolutions: 20 vs. 1 .2, respectively. We stress that our results have been obtained with a resolution of 40 , which is still lower that that of the Mizuno et al. (1994 ) observations, but which is definitely higher than that used by Benson & Myers (1989 ). The hydrogen densities derived from our data show no remarkable difference between the values of NH₃ cores with (L1521D and L1524) or without (L1521F and L1507A) IRAS counterparts.

In Table 6 we display the energy terms of the cores of our sample: turbulent ( ; J), thermal ( ; J) and gravitational ( ; J). The last column is for the parameter /(), which is a measure of core stability. The turbulent, thermal and gravitational (assuming a homogeneous sphere) energy terms have been derived using the formulae reported by Harju et al. (1993 ).

Star formation occurs most readily in molecular clumps in which the internal motions are sufficiently reduced to allow the occurrence of gravitational collapse. Table 6 shows that the thermal energy is always larger than the turbulent one. The values of for the four cores lie between 0.2 (L1521D) and 1.2 (L1524); = 1 implies gravitational equilibrium in the absence of a magnetic field. The values obtained here are only slightly different from unity. Taking into account the contribution of a magnetic field to the energetic support against gravitational collapse, considering also uncertainties in the assumed [NH₃ ]/[H₂ ] value and deviations from the adopted (spherical) geometry of the cores, we can conclude that the four ammonia cores are close to equilibrium. These considerations are in agreement with the results of Myers & Goodman (1988 ) and Harju et al. (1993 ) who found that the NH₃ cores in the TMC are approximately virialized.

5.4. Derived parameters from HC₅ N observations

An estimate of the total HC₅ N column density ( ; cm^-2) has been derived through the standard equation (e.g. Olano et al. 1988 ), using the molecular parameters given, e.g., by Alexander et al. (1976 ) and Churchwell et al. (1978 ). Table 4 lists the derived cyanodiacetylene column densities at the ammonia peak positions of the four molecular cores: the values range from 1.6 10¹² to 9.2 10¹² cm^-2, in agreement with the values of other molecular cores located in the TMC (Benson & Myers 1983 ).

In order to estimate the HC₅ N abundances a comparison between HC₅ N and NH₃ column densities has been made. We presently do not know the size and geometry of the HC₅ N distribution in the ammonia cores. However, the HC₅ N and NH₃ maps given by Benson & Myers (1983 ) show that the size and shape of the maps are similar and that the peak positions differ significantly only for the TMC-1 molecular core. Therefore, we have estimated the [HC₅ N]/[H₂ ] ratio using the cyanodiacetylene and ammonia column densities and the NH₃ abundance. The obtained [HC₅ N]/[NH₃ ] values are: 8.3 10^-3 (L1521D), 4.6 10^-3 (L1521F), 2.7 10^-3 (L1524) and 10.7 10^-3 (L1507A). Thus, the derived HC₅ N abundances relative to H₂ are: 8.3 10^-10 (L1521D), 4.6 10^-10 (L1521F), 2.7 10^-10 (L1524) and 1.1 10^-9 (L1507A), in agreement with the suggestion of Benson & Myers (1983 ), that relative cyanodiacetylene abundances in dense cool cores vary from 10^-10 to 10^-9.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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