2.1 Equations to be Solved

These routines make use of the fact that that most of the common ions that dominate the nebular cooling rate have either , , or ground-state electron configurations, which have five low-lying levels. The major physical assumption within this algorithm is that only these five levels are physically relevant for calculating the observed emission line spectrum; higher levels in these ions are not significantly populated through collisions, recombinations, or other mechanisms. Schematics of the energy-level diagrams for the ground electron configurations are shown in Figure 1 .

For such ions, collisional and radiative transitions can occur between any of the levels. For each excitation level for a given ion, the equations of statistical equilibrium may be written:

where is the fraction in level , is the electron density (cm), are the electron (de)excitation rate coefficients (cm s), and are the radiative transition probabilities (s). The first term on the left includes the collisional (de)excitation rate from the (upper) lower levels, and the second term gives the radiative transition rate from an upper level. The third term is the collisional (de)excitation rate from (upper) lower levels, and the last term is the radiative transition rate from the level itself.

The are independent of temperature, and are inversely proportional to the lifetimes of the upper level, but the are temperature-dependent. The de-excitation rate (i.e., ) is given by:

where is the statistical weight of level , and is the mean (dimensionless) collision strength which is temperature-dependent. The collisional excitation rate is related to the de-excitation rate via:

where is the excitation energy difference between levels and ; and is Boltzmann's constant. It is equation , with the additional constraint that the relative level populations sum to unity, that is solved within the nebular library to determine the level populations.

The atomic data that are independent of temperature-, , and (the energy level separations above the ground state)-are tabulated within the nebular source code and are selected at run time for a given ion. The temperature-dependent atomic data-i.e, the collision strengths for each transition-are computed at run time for a specified temperature. While the collision strengths are really continuous functions of temperature, they are often tabulated in the literature at only a few fixed temperatures between 5000 K and 20,000 K. In the nebular routines, the actual collision strengths for a given are derived from low-order polynomial fits of the published as a function of temperature. The allowed range of temperature in the nebular tasks is therefore restricted to 2000 K <T < 36,000 K for most ions (to avoid excessive extrapolations of the polynomials), unless the published cross sections for a particular ion are tabulated over a wider range. The atomic data were generally taken from the compilation by Mendoza (1983), except where noted in Table 1 for several ions. Note, however, that the atomic data (particularly the cross sections) are likely to be updated in the future (see § 5 ).

For a given emission line, the emission rate of line photons resulting from a downward transition from , , is given by:

where is the relative population in the upper level of ion , is Planck's constant, and is the frequency of the photon emitted in the transition. As the density increases, collisional de-excitation becomes important. A benchmark called the ``critical density'' for a level is defined as the density at which the collisional de-excitation rate equals the radiative transition rate. That is:

In the low density limit it can be shown that the emissivity is proportional to , whereas for densities exceeding the critical density, the emissivity goes as . Thus, line emission in a nebula occurs most efficiently near the critical density.