2. Simple theory of X-rays from stellar winds
Owocki & 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 j, and the mean molecular weight per nucleon of the cool wind. Note that for early-type winds that have essentially no neutral gas, , the mean molecular weight per ion, and is the same for both the cool and hot gas components. Typical values of at 1 keV can range from the stellar radius for B and some O star winds (entirely optically thin), to around for other O star winds, to hundreds or thousands of in the WR case (extremely optically thick). Thus, the wind attenuation can have a major influence on the emergent X-ray spectrum of stellar winds.
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 dV is , where the emissivity for X-ray emission is
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 dV.
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 k, and is the ratio of the number density population for the line k to the total ion number density of the hot gas. For solar abundances, the Raymond & Smith (1977; hereafter RS) cooling function is used, with , where is the hydrogen proton density of the hot gas. Assuming the 's vary weakly with density and temperature, and further that the ratio is constant for every line k, a scaling correction to the known RS cooling function for non-solar abundances is
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 k, is an overall average enhancement (or reduction) to the RS cooling function.
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 ratio of X-ray luminosities between WN and WC stars.
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999