6 Data analysis: $\bf ^{12}$CO($\bf J = 1 \rightarrow 0$)

6.1 Positions

We have used interferometric maps (not combined data) to determine star positions. Mostly VLA positions were used for the primary phase calibrators. We have ignored phase errors due to inaccuracies in the assumed baselines, as the distance $\Delta s$ between the stars and their calibrator(s) was generally below $15\,$degrees. Actually, baseline errors $\Delta B$ at the frequency $\nu$ of the ¹²CO($J=1\rightarrow 0$) transition lead to a maximum phase error $2\pi \nu \Delta B \Delta s/c$ of a few degrees, somewhat smaller than the atmospheric phase fluctuations recorded on the shortest baselines, typically $10-30\,$degrees. On average, this corresponds to absolute positional errors of about 1/10 of the synthesized beam, or equivalently 0.5$^{\prime\prime}$. The star positions listed in Table 4 were obtained by averaging the positions of the fitted emission centroids in the blue ($V \le -V_{\rm exp}/3$) and red ($V \ge V_{\rm exp}/3$) velocity maps. When two-component elliptical gaussians were used to fit the CO brightness distribution (see Sect. 6.4) we used the positions of the gaussian ascribed to the inner, more compact shell surrounding the central star.

 

Table 3:

$\parbox{1cm}{\mbox{}}$ $\parbox{16cm}{\vskip 2mm $^a$\space No (1--0) data.\\ $^b$\space No ... ...,$kpc, $R_{\rm CO} = 2.1\,10^{17}\,$cm and $\dot{M}= 14\,10^{-6}\,M_\odot$/yr.}$

  

Table 4: See text

Except for 12 stars, most of the positions listed in Table 4 are in excellent agreement with those given in Loup et al. (1993). The stars for which the positional difference is larger than 3'' are noted in the caption to Table 4. Single-dish ¹²CO($J=2\rightarrow 1$) maps were used just in the case of CL Mon and HD 187885 for which no ¹²CO($J=1\rightarrow 0$) interferometer data were available.

6.2 Fluxes and main beam temperatures

To properly bootstrap the fluxes of the stronger calibrators (mostly 3C 84 and 3C 273), we have made measurements of these, in conjunction with planets, with the antennas operating in the autocorrelation mode. We have then referred the weaker calibrators to the first ones in the more sensitive interferometric mode. Since more than half of the calibrators were used to calibrate more than one source (e.g. 2005+403 was used alternatively for 10 stars), remaining errors in the flux density scale of those were largely removed by cross-checking the interferometric visibility profiles on a sample of stars across the whole set of array configurations. After readjusting discontinuities in the visibility profiles attributed to errors in the amplitude calibration, we expect the fluxes of the main calibrators to be accurate within 10%. Table 1 lists the flux of all the calibrators used in the survey. Despite an accurate relative flux calibration, the absolute calibration scale is probably not better than 15%.

The spatially integrated ¹²CO($J=1\rightarrow 0$) fluxes have been estimated from the combination of both, single-dish and interferometric visibility profiles. We have fitted either one or two elliptical gaussian components to the real part of the global visibility profile of each star, As a result, we have derived the integrated flux from the fitted value at zero spacing for each velocity channel. When either the uv coverage was inadequate or the line strength insufficient to account for possible departures from circular symmetry, we have fitted only circular gaussian components. Just in the case of U Cam, we have fitted a circular ring to the visibility profiles corresponding to the central velocities.

The spatially integrated fluxes listed in Table 4 are peak values. Each value was determined by averaging the flux in a small range of velocities centered on $V_{\rm lsr}$. This range of velocities is delimited by an horizontal bar in the integrated ¹²CO($J=1\rightarrow 0$) flux vs velocity plots.

Particular care was given to fields with significant emission far from the tracking center (e.g. 04307+6210, CIT6, $\chi$Cyg, IRC+40540) as here the quality of the single-dish contribution appears to be far more important than the interferometric one for the determination of the integrated fluxes.

The main beam temperatures listed in Table 4 were determined directly from the peak integrated ¹²CO($J=1\rightarrow 0$) flux densities. The main beam temperature, $T_{\rm MB}$, is related to the integrated flux by $S = \eta_{\rm C}\times\gamma\times\mbox{$T_{\rm MB}$}$, where $\eta_{\rm C}$ accounts for residual calibration errors and where $\gamma = 4.8$JyK^-1 is the nominal point source sensitivity of the 30 m telescope at the frequency of the ¹²CO($J=1\rightarrow 0$) transition. However, $T_{\rm MB}$is not exactly the main beam temperature as measured by the 30 m telescope in extended sources, it must be much larger in very extended sources.

6.3 Velocities and line shapes

If available, only interferometric data were used to determine the expansion velocity in the ¹²CO($J=1\rightarrow 0$) line. As for the ¹²CO($J=2\rightarrow 1$) lines, the velocity was not determined by profile fitting but from edge channel maps still showing emission features. Terminal $V_{\rm exp}$ and systemic $V_{\rm lsr}$ velocities of the circumstellar envelopes were determined from channel maps prior resampling to the frequency resolution of the 30 m telescope.

Centroid and expansion velocities measured from the ¹²CO($J=1\rightarrow 0$) line are listed in Table 4. The same comments pointed out in Sect. 4.3 on the meaning of our estimation of the expansion velocity and on the presence of profile anomalies also hold here.

  

Table 5: See text

6.4 Sizes and asymmetries

We have obtained envelope sizes by fitting either one or two elliptical gaussian components to the real part of the combined visibility profiles in the range $-V_{\rm exp}/3 \le V \le V_{\rm exp}/3$. The envelope size $\theta_{\rm CO}$ of a circular gaussian visibility profile was derived according to $\theta_{\rm CO} = 2\log 2/(\pi D/\lambda)$ where $D/\lambda$ is the HWHP projected baseline of the profile in units of the observing wavelength. In case of elliptical gaussians a size is fitted along the major and minor axes. The gaussian curves corresponding to the fitted envelope sizes in the direction of maximum and minimum extension are shown for each star on the individual ¹²CO($J=1\rightarrow 0$) pages (central panel, second row) of the atlas.

As for the ¹²CO($J=2\rightarrow 1$) data we have determined the asymmetry of the envelope from the major and minor axes (see Table 5 - consult the atlas for the velocity dependence of the asymmetry). Double elliptical gaussian were used for 17 envelopes, three of which (IRC+10011, HD 235858, RAFGL3068) show a clear asymmetry in the compact, central component, two (RAFGL 2155 and $\chi\,$Cyg) in the extended, outer component, and one (IRC+40540) in both components. IK Tau has a visibility profile which is apparently too complex to be fitted by a double component profile.

Except for U Cam, where we have fitted a circular ring, almost all visibility profiles are well-approximated by gaussian profiles.