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Astron. Astrophys. 333, 687-701 (1998)

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3. The system of ordinary differential equations

The solution vector to be found from integration of the system of ordinary differential equations consists of [FORMULA] variables and is

[EQUATION]

where L is the number of bound atomic states treated in non-LTE, U is the gas material velocity with respect to the discontinuous jump,

[EQUATION]

is the specific translational kinetic energy of heavy particles (neutral atoms and ions),

[EQUATION]

is the specific translational kinetic energy of free electrons, [FORMULA] is the number of free electrons per unit mass, [FORMULA] is the number of hydrogen atoms in the i -th state per unit mass. Hereafter the prime implies that the quantity is expressed per unit mass. The solution vector [FORMULA] does not contain the gas density [FORMULA] because this variable can be easily evaluated from the mass conservation relation (1).

Thus, the system of ordinary differential equations consists of the momentum equation, two energy equations for heavy particles and free electrons, the rate equation for free electrons and L rate equations for non-LTE bound states of the hydrogen atom:

[EQUATION]

where [FORMULA] is the partial pressure of heavy particles, [FORMULA] is the electron pressure, [FORMULA] and [FORMULA] are the rates of energy gain by electrons in elastic and inelastic collisions, respectively, [FORMULA] and [FORMULA] are the total, that is collisional plus radiative, ionization and recombination rates, respectively. It should be noted that in the present study we consider only bound-free transitions, so that Eqs. (16) and (17) contain only ionization and recombination terms.

The system of ordinary differential equations (13) - (17) written in the form of derivatives with respect to time t is not appropriate for calculation of the spatial structure, so that these equations should be rewritten in order their left-hand sides are replaced by derivatives with respect to the spatial coordinate X. Furthermore, the space derivative of the gas pressure [FORMULA] in Eq. (13) and the time derivative of the specific volume V in Eqs. (14) and (15) have to be expressed in terms of integrated variables. To this end we write the gas pressure as a sum of translational kinetic energies:

[EQUATION]

whereas the time derivative of the specific volume is determined from Eq. (2):

[EQUATION]

Expressing the time derivative of the specific volume in terms of the gas pressure according to Eq. (19), [FORMULA] according to Eq. (18) in terms of integrated quantities [FORMULA] and [FORMULA] and substituting Eqs. (18) and (19) into Eqs. (13) - (17), we obtain the following system of ordinary differential equations with right-hand sides depending only on the independent variable X and integrated variables [FORMULA]:

[EQUATION]

where

[EQUATION]

[FORMULA], [FORMULA] is isothermal sound speed, [FORMULA]  eV is the ionization potential of the hydrogen atom.

In obtaining Eq. (27) we expressed the rate of energy gain by electrons in inelastic collisions as (Murty 1971)

[EQUATION]

where [FORMULA] and [FORMULA] are ionization and excitation energies per unit mass,

[EQUATION]

is the divergence of radiative flux, [FORMULA] is the mean intensity of radiation, [FORMULA] and [FORMULA] are the total emission and absorption coefficients.

Free electrons acquire the energy from heavy particles in elastic collisions with hydrogen ions and neutral hydrogen atoms, hence,

[EQUATION]

where [FORMULA] and [FORMULA] are the corresponding rates of energy gain. The rate of energy gain by electrons in elastic collisions with hydrogen ions per unit mass is (Spitzer & Härm 1953)

[EQUATION]

where [FORMULA] is the time of equipartition given by

[EQUATION]

and

[EQUATION]

The rate of energy gain by electrons in elastic collisions with neutral hydrogen atoms per unit mass is

[EQUATION]

where [FORMULA] and [FORMULA] are the mass of electron and the mass of hydrogen atom, respectively, and the elastic scattering cross section is (Narita 1973)

[EQUATION]

Here [FORMULA] is the Bohr radius.

Rate equations (23) and (24) imply that the number density of free electrons [FORMULA] and atomic level populations [FORMULA] change due to bound-free transitions, that is, due to ionizations and recombinations. The total ionization rate is

[EQUATION]

where the rate of collisional ionizations is given by

[EQUATION]

[FORMULA] is energy of ionization from the i -th level, [FORMULA] is a slowly varying function of T evaluated with approximation formulae by Mihalas (1967).

The rate of photoionizations is

[EQUATION]

where [FORMULA] is an absorption cross-section at frequency [FORMULA] in bound-free transition from the i -th state and [FORMULA] is a threshold frequency for ionization from the i -th state.

The total recombination rate is

[EQUATION]

where the collisional recombination rate is given by

[EQUATION]

The radiative recombination rate is

[EQUATION]

where

[EQUATION]

Substituting Eqs. (42) and (43) into Eq. (41) we obtain that the total recombination rate is

[EQUATION]

The system of ordinary differential equations (20) - (24) is stiff because it is characterized by very different time constants due to the rate equations (23) and (24). In order to obtain the stable and enough correct solution of Eqs. (20) - (24) we used the Livermore solver for ordinary differential equations based on the GEAR package (Hindmarsh 1979).

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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