*Astron. Astrophys. 332, 367-373 (1998)*
## 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.
where is given by (12b) and
is the speed ratio (14b). The point
= 1, *J* = 0 is a critical point of the
differential equation and the phase trajectories in this neighbourhood
can be approximated by the bundle
where *c*, the constant of integration, generates the family
of curves emanating from ( = 1, *J* = 0).
The phase trajectories can exhibit a minimum (
= 0) on the locus corresponding to the zero of the numerator of (A1).
Phase trajectories lying above this locus have
whereas below it . The point *N* = 0,
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
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
(, *J* = 0) to the top of the layer
(*N* = 0, ). In Appendix (B) we show that
the locus = 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 *c*_{H}
If , or more precisely its normalized version
given by (16), is very much greater than unity the critical solution
in the () plane is well approximated (to
O()) by the zero of the numerator of (A1), i.e.
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.
, and given by
Hence the proton density is given by
Using (B1) to eliminate *N* in favour of *J* in the
photon flux decay Eq. (1) facilitates its exact integration
where . Eqs. (B1) to (B4) provide a
neat analytic form for the structure of the ionization layer.
Eq. (B4) shows that at the lower boundary
() the photon flux decays exponentially to zero
with the photonionization length scale *L*, whereas it approaches
its asymptotic value at the upper boundary
() with that length scale augmented by the speed
ratio . This asymptotic solution is shown in
Figs. (3).
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998
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