## 2. Simple theory of X-rays from stellar windsOwocki & Cohen (1999) present a scaling analysis for the X-ray emission from hot star winds. They considered an exospheric approximation, where the observed X-ray emission arising from "hot" gas emerges only from radii exterior to the optical depth unity surface of radius , with X-rays at smaller radii being completely attenuated. The radius is determined primarily by K-shell photoelectric absorption in the "cool" wind. The extent of is energy dependent, with with the opacity for the ratio the relative
abundance by number for atomic species Owocki & Cohen (1999) showed that for a constant expansion
wind, the exospheric approximation overestimates
by a factor of 2 only, as compared to
an exact integration. Since over a
broad range of X-ray energies for the WR stars, a constant
expansion wind is an excellent approximation. We therefore assume a
spherical wind with density and
constant filling factor . We take the
filling factor to be the same as Kudritzki et al. (1996), such
that the emitted energy from volume The parameter is the cooling
function and the assumed constant
temperature of the hot gas. The electron and ion number densities
and
appearing in Eq. (3) are for the
cool wind. For a constant , the
equality holds for any infinitesimal
volume element The emergent X-ray luminosity arises from a spherical volume integral over the observable wind, with where the parenthetical accounts for occultation by the optically thick surface of radius and a factor of corrects for the overestimation made in the exospheric approximation. Applying the assumption of constant expansion, the integration can be evaluated analtyically, yielding where is the mean molecular weight per free electron of the cool wind. There are several crucial factors that determine the total emergent X-ray luminosity from the wind: the opacity, the cooling function, and the filling factor. The opacity can be taken as , with in the range 2-3 and a constant that depends on abundances as with , , and the abundances of H, He, and metals relative to all nucleons. The K-shell absorption for a given atom depends on the fourth power of the proton number, hence the coefficient of 16 is just for , and assuming CNO are the dominant metals, is a mean for . Thus is most appropriate at energies above the CNO edges. For temperatures in which
is dominated by line emission (in
contrast to thermal Bremsstrahlung at
K), the cooling function is
roughly given by , where
is a factor relating to the emitted
power in the line where is the mean molecular
weight per proton, the same for both the cool and hot gas. In the case
is not constant for every We assume the filling factor is constant throughout the wind, but it's value can vary between stars. First, it can vary with abundance as . The filling factor also varies with the ratio . For example, Kudritzki et al. (1996) has analyzed ROSAT observations for 42 O stars and empirically determined . They attribute this result to the expectation that larger ratios of result is more efficient cooling, shorter cool zones, and consequently smaller filling factors (see also Hillier et al. 1993). The end result is that the volume filling factor scales as Combining Eqs. (5)-(8) and integrating over energy yields the overall dependence of X-ray luminosity on composition and wind parameters, viz where and
are assumed. In this expression the
dependence of on
has cancelled out (although an
implicit dependence may exist through
). We have satisfied the minimum
requirement of our theory by reproducing the observed independence of
on
with our scaling result of Eq. (9). This independence is a consequence
of the fact that the emissivity scales with density squared, hence
, but
and
are each proportional to
, thus leaving no net dependence on
the wind flow parameters. We next consider how well our scaling
reproduces the observed © European Southern Observatory (ESO) 1999 Online publication: July 26, 1999 |