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Astron. Astrophys. 332, 367-373 (1998)

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4. Ionization of minor species

It is of some interest to consider the photoionization of minor singly ionized species in the presence of the hydrogen-proton background whose layer structure we have just analyzed. Since the minor species is ionized by its own photon flux [FORMULA] and also by the photon flux, [FORMULA] say, which ionizes hydrogen, the decay of [FORMULA] and the production of ions can be described by the equations

[EQUATION]

[EQUATION]

[EQUATION]

in which [FORMULA] is the cross section for photoionization of species [FORMULA], and where [FORMULA], [FORMULA] are respectively the density and speed of the neutrals (and their ionized counterparts). The hydrogen-proton background is described by the previous set of equations in which [FORMULA] assumes the role of J. In a similar way to the hydrogen-proton layer we assume that the incoming subsonic neutrals are very subsonic so the pressure gradients of the neutrals and their ion counterparts are balanced by frictional forces with the hydrogen-proton background enabling us to write,

[EQUATION]

[EQUATION]

where [FORMULA]
Again we may specify the temperatures and proceed to solve these equations for the structure of a layer of minor species and their ion counterparts superimposed on the hydrogen-proton layer which is unaffected by the minor constituents. However it is clear from Eqs. (19) that the requirement of homogeneity on either side of the layer (i.e. no gradients at [FORMULA]) implies that the neutrals enter the layer at the hydrogen speed [FORMULA] and exit the layer at the proton speed [FORMULA] (given by Eq. (14b)). Therefore from continuity (Eq. 18c) the ratio of the density of ions at the top to the density of their neutral counterparts at the bottom is simply

[EQUATION]

which exactly mimics the hydrogen-proton layer. Hence no fractionation can occur contrary to the argument made by Marsch et al. (1995) and Peter (1996). Their results appear to be an artefact of the application of an unsuitable boundary condition applied at some arbitrarly chosen lower boundary and an inappropriate description of the depletion of the photon flux. It is clear that in a one dimensional steady model, such as is described by the above equations, no enhancements or depletions of one type of ion over another can occur because continuity demands what goes in must come out. Therefore the density ratios are given by (20) for all species, because the partial pressures are constant on either side of the layer.

[FIGURE] Fig. 3. Distributions of J, [FORMULA], N and n as functions of z as given by the asymptotic solution for the case [FORMULA] = 2 (corresponds to [FORMULA]).

To solve the above system of equations one could proceed as before by making use of the [FORMULA], [FORMULA], [FORMULA] phase space. For the present purposes it is sufficient to employ the asymptotic solution given in Appendix B and assume that collisions are sufficiently frequent to maintain equal speeds for all particles. Thus with [FORMULA], where V is given by Eq. (B2), the decay and continuity Eqs. (18) tell us that

[EQUATION]

[EQUATION]

Substituting (21b) for Nm into (18a) yields the decay equation for [FORMULA] in the form

[EQUATION]

in which [FORMULA] is given by the asymptotic solution provided by Eqs. (B4). The results of numerical integration of (22) and (21) are shown in Figs. (4 and 5) for [FORMULA], [FORMULA] and [FORMULA] as functions of z within the layer, for various species.

[FIGURE] Fig. 4. The normalized photon fluxes [FORMULA] as functions of z using the asymptotic solution for the photon flux [FORMULA] which ionizes hydrogen.

[FIGURE] Fig. 5. The neutral ([FORMULA]) and ion ([FORMULA]) distributions with z for various species. Note that those ions with larger photoionization cross sections stand to the left of hydrogen whereas those with lower cross sections stand to the right. The exception is He since although its cross section is slightly greater than that of hydrogen its ionizing energy is about twice that of hydrogen with the result that it decays according to its own photoionization length which is much greater than of hydrogen.

The height z has been normalized to the hydrogen photoionization length and the curves show that species with larger photoionization cross sections than hydrogen (Ar, Al, Si, C) stand to left of hydrogen (H), whereas those with lower cross sections stand to the right (Fe, O, Mg).

Helium is an exception since the the photon flux (with [FORMULA]) which ionizes helium also ionizes hydrogen because its first ionization potential (FIP) is about double that of hydrogen. Therefore the roles of [FORMULA] (for helium) and [FORMULA] are reversed in Eqs. (18). However since we assume He is still a minor constituent and therefore its effect on the hydrogen equations is neglible this implies that the depletion of [FORMULA] for He takes place on its own photoionization length scale [FORMULA] which is much greater than that of hydrogen. This effect is shown in Figs. 4 and 5 in which the He curves are shown standing well to the right of those of hydrogen. A maximum in the ion density [FORMULA] (for those ions with the larger photoionization cross sections) can arise because [FORMULA] attains its asymptotic value before [FORMULA] does, after which the acceleration of the flow takes over to reduce [FORMULA] to its final value ([FORMULA] in the isothermal case [FORMULA]). This calculation should be regarded as merely illustrative because the situation is artificial in that it requires that we have just the right amount of photon flux [FORMULA] for all species. In reality there exists an excess of [FORMULA] in the form of [FORMULA] so the layer would not be stationary and in fact would move down (to the left), so that the metals would come in ionized from deep down. Furthermore since the ionizing of the metals is done by an enormous [FORMULA] flux it means we should regard the metals as coming into the layer fully ionized with the same speed as hydrogen and hence no fractionation can take place.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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