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

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3. Solution of the structure equations

Using the concept of a phase plane it is useful to analyze the structure in terms of how v, V, n and N vary with J throughout the layer. This is accomplished by dividing the stress balance Eqs. (6) and (7) by the photon flux decay Eq. (1) to obtain the following two differential equations for V and v, in which the spatial coordinate z is eliminated in favour of J,

[EQUATION]

[EQUATION]

in which we have used Eqs. (4) and (5) to eliminate n and N in terms of J, and V. The constants [FORMULA] and [FORMULA] are given by,

[EQUATION]

[EQUATION]

In fact it is only necessary to solve one of the above differential equations, say Eq. (10) for the hydrogen speed V as a function of the ionizing photon flux J, since constancy of total pressure, Eq. (8), along with flux conservation Eqs. (4) and (5), yield v in terms of V and J, namely

[EQUATION]

In this approach the structure of the layer is determined by Eq. (10) with the photon flux decay Eq. (1) merely serving as a reference which can be consulted for J as a measure of z. (For an explicit expression for [FORMULA] see the asymptotic solution given in Appendix (B)). In order to avoid a singularity at the upper boundary, where [FORMULA], the numerator of the expression on the right hand side of Eq. (10), must vanish, so that, although there are no hydrogen atoms left, they must exit asymptotically at a speed given by

[EQUATION]

The protons exit the layer at the speed [FORMULA] which follows from (13) as

[EQUATION]

Similarly at the lower boundary, where [FORMULA], Eq. (11) tells us that the protons must enter the layer at the speed,

[EQUATION]

The dimensionless parameter which characterizes this problem is,

[EQUATION]

The ratio of the cross sections is large since,

[EQUATION]

Therefore even if hydrogen enters the layer very subsonically, say [FORMULA] (corresponding to [FORMULA] = 1 km/sec and [FORMULA] 10 km/sec), the parameter given by (16) is indeed very much greater than unity. In these circumstances the asymptotic solution, given in Appendix B, provides a simple, analytic description of the structure of the photoionization layer.

The results of the numerical integration of the solution of Eq. (10) for the illustrative case in which [FORMULA] = 4 are shown in Fig. (1).


[FIGURE] Fig. 1. The critical solutions for the hydrogen (V) and proton (v) speeds as a function of the photon flux (J) in the layer for the illustrative case of [FORMULA] = 4 . The upper dotted curve ([FORMULA]) corresponds to the locus on which the phase trajectories [FORMULA] and the lower broken curve is the asymptotic solution (Appendix A) for large [FORMULA].

Hydrogen atoms enter at speed [FORMULA] and density [FORMULA] and are depleted on exit. Protons enter the layer at speed [FORMULA] (given by (15)) and exit at twice [FORMULA] with half the density of the entering hydrogen atoms. The asymptotic solution, which is shown by the broken curve for comparison, is indeed an excellent approximation even for the moderate value of 4 for the characteristic dimensionless parameter of the problem, Eq. (16). The curves of V and v as functions of J represent the critical solution of Eq. (10) which uniquely links the bottom of the layer to the top (but see also the discussion given in Appendix A, in which the structure is analyzed in the [FORMULA] plane). The corresponding distributions of hydrogen and protons are shown in Fig. (2).


[FIGURE] Fig. 2. The full curves show the hydrogen (N) and proton (n) densities in the layer as a function of J for the illustrative case [FORMULA] = 4. The broken curves correspond to the asymptotic solution ([FORMULA] 1).

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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