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

Abstract

For the case of optically thin lines, we show that the relation existing between the first order momentW ₁α∫ (E(λ)/E _c −1)(λ−λ₁₂)dλ of a P Cygni profile and the quantityMn(level), whereM is the mass-loss rate of the central star andn(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 (Castoret al., 1981; Surdej, 1982). Consequently, all results established in the context of ‘very rapidly’ expanding atmospheres and mainly referring to the non-dependence ofW ₁ vs various physical (underlying photospheric absorption line, limb darkening, etc.) and geometrical (velocity fieldv(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 measurementW ₁ of unsaturated P Cygni profiles observed in the spectrum of early—as well as late—type stars, quasars, etc.