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

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2. Model equations for a photoionization layer

We adopt a simple one dimensional model of a photoionization layer in the solar chromosphere in which a flux of EUV ([FORMULA]) photons incident from above ionizes neutral hydrogen flowing up with subsonic speeds from below. The photon flux J decays into the layer according to

[EQUATION]

in which z is directed upwards and J decays downwards, [FORMULA] is the cross section ([FORMULA] cm2, see Allen (1976)) for photoionization by [FORMULA] and N is the neutral hydrogen density in the layer. The decay of the photon flux is accompanied by the production of protons with density n and speed v according to

[EQUATION]

The total particle flux of hydrogen and protons is conserved throughout the layer, that is

[EQUATION]

where V is the neutral hydrogen speed within the layer. At the bottom of the layer ([FORMULA]) there are no protons (n = 0) whereas at the top ([FORMULA]) there are no neutral hydrogen atoms (N = 0). The constant in (3) is then just the hydrogen flux at the bottom, [FORMULA], say. Eq. (1) and (2) yield the simple integral

[EQUATION]

since the ionizing photon flux is assumed to be completely extingiushed at the bottom of the layer. Therefore if [FORMULA] denotes the total incident ionizing photon flux, particle conservation can be written

[EQUATION]

because [FORMULA].

Since the ionizing length scale ([FORMULA] 40 km) is much smaller than the gravitational scale height (300 km) and also because we assume the hydrogen is flowing upwards at low subsonic speeds, the momentum equations for neutral hydrogen and protons can be reasonably approximated by a balance between their pressure gradients and friction arising from collisions between protons and hydrogen atoms. Thus we can write

[EQUATION]

[EQUATION]

Because total momentum is conserved, which in the subsonic approximation reduces to the constancy of total pressure. i.e.,

[EQUATION]

the collision frequencies must obey the relation,

[EQUATION]

[EQUATION]

Thus the collision times are equal and can be expressed as

[EQUATION]

where [FORMULA] is thermal speed of hydrogen ([FORMULA]) and [FORMULA] is the cross section ([FORMULA] cm2, see Halsted (1972) for charge exchange collisions between hydrogen atoms and protons).

For simplicity we assume the temperatures [FORMULA], [FORMULA] and [FORMULA] are given throughout the layer (for example isothermal). Thus we have five Eqs. (1), (4), (5), (6) and (7) for the five physical variables n, N, V, and J, which determine the structure of the layer.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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