## 3. The system of ordinary differential equationsThe solution vector to be found from integration of the system of ordinary differential equations consists of variables and is where is the specific translational kinetic energy of heavy particles (neutral atoms and ions), is the specific translational kinetic energy of free electrons,
is the number of free electrons per unit mass,
is the number of hydrogen atoms in the 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 where is the partial pressure of heavy particles, is the electron pressure, and are the rates of energy gain by electrons in elastic and inelastic collisions, respectively, and 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 whereas the time derivative of the specific volume is determined from Eq. (2): Expressing the time derivative of the specific volume in terms of
the gas pressure according to Eq. (19),
according to Eq. (18) in terms of integrated quantities
and 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 , is isothermal sound speed, 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) where and are ionization and excitation energies per unit mass, is the divergence of radiative flux, is the mean intensity of radiation, and 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, where and 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) where is the time of equipartition given by and The rate of energy gain by electrons in elastic collisions with neutral hydrogen atoms per unit mass is where and are the mass of electron and the mass of hydrogen atom, respectively, and the elastic scattering cross section is (Narita 1973) Here is the Bohr radius. Rate equations (23) and (24) imply that the number density of free electrons and atomic level populations change due to bound-free transitions, that is, due to ionizations and recombinations. The total ionization rate is where the rate of collisional ionizations is given by is energy of ionization from the The rate of photoionizations is where is an absorption cross-section at
frequency in bound-free transition from the
The total recombination rate is where the collisional recombination rate is given by The radiative recombination rate is Substituting Eqs. (42) and (43) into Eq. (41) we obtain that the total recombination rate is 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). © European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |