A generalized method for deriving mass-loss rates - The first order moment of unsaturated P Cygni line profiles

For the case of optically thin lines, we show that the relation existing between the first order moment W1 ~ j (E(2)/E~ - 1)(2 - 212) d2 of a P Cygni profile and the quantity ii~in(1eve1), where M is the mass-loss rate of the central star and n(level) the fractional abundance of the ion under study, is in fact independent of any Sobolev-type approximations used for the transfer of line radiation, contrary to what has been assumed in some previous works (Castor eta!., 1981; Surdej, 1982). Consequently, all results established in the context of'very rapidly' expanding atmospheres and mainly referring to the non-dependence of W1 vs various physical (underlying photospheric absorption line, limb darkening, etc.) and geometrical (velocity field v(r), etc.) effects remain unchanged for arbitrary (e.g. non-Sobolev type) outward-accelerating velocity laws. Whenever applied with caution, Equation (50) thus provides a very powerful means of deriving mass-loss rates - with a total uncertainty less than 60 percent - from the measurement W1 of unsaturated P Cygni profiles observed in the spectrum of early - as well as late - type stars, quasars, etc