Circumstellar envelopes of semi-regular long-period variables: mass-loss rate estimates and general model fitting of the molecular gas

Aims. We aim to study the main properties of a volume-limited unbiased sample of well-characterized semi-regular variables (SRs) in order to clarify important issues that need to be further explained, such as the formation of axially symmetric planetary nebulae (PNe) from spherical circumstellar envelopes (CSEs), which takes place during the mass-loss process along the asymptotic giant branch (AGB) phase. Methods. We present new high-S/N IRAM 30m observations of the (CO)-C-12 J = 2-1, (CO)-C-12 J = 1-0, and (CO)-C-13 J = 1-0 lines, in a volume-limited sample of SRs for which the HIPPARCOS distances are between 100 and 500 pc and the declinations are above -25 degrees. We analyzed the data by characterizing the main properties of the CSEs. The (CO)-C-12 J = 2-1 data were used to study the profiles, while the (CO)-C-12 J = 1-0 data were used to estimate mass-loss rates for the complete sample. Moreover, the (CO)-C-12 J = 2-1 line has been used to determine the possible structures responsible for such profiles. Results. We have classified the sources into four groups according to the different profiles and final gas expansion velocities. Type 1 and 2 profiles are broad and narrow symmetric lines, respectively. Furthermore, type 1 profiles are more related to previously studied, standard, spherically symmetric envelopes. Type 3 profiles on the contrary are strange profiles with very pronounced asymmetries. Finally, type 4 profiles are those showing two different components: a narrow line profile superimposed on a broad pedestal component. We find that for sources with this latter kind of profile, the variation amplitude is very low, which means that these SRs do not have a well-developed inner envelope differentiated from the outer one. Interestingly, we report a moderate correlation between mass-loss rates and (CO)-C-12 J = 1-0/(CO)-C-12 J = 2-1 line intensity ratios for O-rich SRs, suggesting a different behaviour between C- and O-rich SRs. Using SHAPE+shapemol, we find a unified simple model based on an oblate spheroid placed in different orientations that may explain all the (CO)-C-12 profiles in the sample, indicating that the gas expansion is in general predominantly equatorial. Moreover, in order to explain the type 4 profiles, we define an extra component which may somehow be a biconical structure or similar according to the structures already found in this kind of source. Type 1 and 2 profiles, curiously, may also be explained by standard spherically symmetric envelopes, but often requiring anomalously low velocities. Type 3 and 4 profiles however, need axial symmetry to be explained. We conclude that most circumstellar shells around SRs show axial, strongly nonspherical symmetry. More interferometric observations are needed in order to make firm conclusions about mass-loss processes and possible morphologies of SRs.