Revisited mass-loss rates for the nuclei of the planetary nebulae NGC 6210, NGC 6826 and NGC 6543 - The first order moment W1 of subordinate line profiles

The authors present the results of numerical calculations for the first order moment W[SUB]1[/SUB] of subordinate P Cygni line profiles as a function of the parameter W[SUB]1[/SUB][SUP]0[/SUP]alphaM>n[SUB]g[/SUB]<[SUB]s[/SUB], where Mrepresents the mass-loss rate of the central star and >n[SUB]g[/SUB]<[SUB]s[/SUB] the average fractional abundance of the relevant ion across the region of the envelope where the subordinate line profile is formed. The calculations clearly show that for unsaturated P Cygni line profiles the relation W[SUB]1[/SUB]alphaM>n[SUB]g[/SUB]<[SUB]s[/SUB] holds irrespective of the type of the opacity and/or velocity distributions. For currently observed P Cygni line profiles, the misuse of this relation can lead to an underestimate of the mass-loss rates by more than one order of magnitude. This is illustrated for the case of 3 selected central stars of planetary nebulae (NGC 6210, NGC 6826 and NGC 6543) for which mass-loss rates have been revisited.