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

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Appendix A: the structure equation in the (N, J) plane

It is instructive to analyze the structure of the photoionization layer in the N (hydrogen), J (photon flux) plane. This is accomplished by dividing Eq. (7) by Eq. (1) and using (4), (5) and (8) to eliminate n, v, and V in terms of J and N so as to obtain.

[EQUATION]

where [FORMULA] is given by (12b) and [FORMULA] is the speed ratio (14b). The point [FORMULA] = 1, J = 0 is a critical point of the differential equation and the phase trajectories in this neighbourhood can be approximated by the bundle

[EQUATION]

where c, the constant of integration, generates the family of curves emanating from ([FORMULA] = 1, J = 0). The phase trajectories can exhibit a minimum ([FORMULA] = 0) on the locus corresponding to the zero of the numerator of (A1). Phase trajectories lying above this locus have [FORMULA] whereas below it [FORMULA]. The point N = 0, [FORMULA] is also a critical point of the differential equation for which there is one critical solution which permits N to smoothly go to zero according to

[EQUATION]

[EQUATION]

The phase trajectories are shown in Fig. (6). The structure of the layer is described by the critical solution which is the only phase trajectory which links the bottom of the layer ([FORMULA], J = 0) to the top of the layer (N = 0, [FORMULA]). In Appendix (B) we show that the locus [FORMULA] = 0 (given by the broken curve on Fig. (6)) provides a useful asymptotic form for the critical solution.

Appendix B: asymptotic solution for layer structure for large cH

If [FORMULA], or more precisely its normalized version given by (16), is very much greater than unity the critical solution in the ([FORMULA]) plane is well approximated (to O([FORMULA])) by the zero of the numerator of (A1), i.e.

[EQUATION]

Physically this corresponds to the situation in which there are many hydrogen-proton collisions in a characteristic photoionization length so that the speeds are almost equal, i.e. [FORMULA], and given by

[EQUATION]

Hence the proton density is given by

[EQUATION]

Using (B1) to eliminate N in favour of J in the photon flux decay Eq. (1) facilitates its exact integration

[EQUATION]

where [FORMULA]. Eqs. (B1) to (B4) provide a neat analytic form for the structure of the ionization layer. Eq. (B4) shows that at the lower boundary ([FORMULA]) the photon flux decays exponentially to zero with the photonionization length scale L, whereas it approaches its asymptotic value [FORMULA] at the upper boundary ([FORMULA]) with that length scale augmented by the speed ratio [FORMULA]. This asymptotic solution is shown in Figs. (3).

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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