Annu. Rev. Astron. Astrophys. 1999. 37:
487-531
Copyright © 1999 by . All rights reserved |

**2.4. Basics of Abundance Analysis**

**2.4.1. Collisionally Excited Lines**

Collisionally excited lines form by the internal excitation of an ion after electron impact. Their emissivities, that is, the energy released per unit volume and time, follow from the statistical equilibrium of the energy levels. For example, the equilibrium (detailed balance) equation for a 2-level atom is,

(1) |

where *n _{e}* is the electron density,
is
the probability for line photons escaping the local region (0
1),

(2) |

where _{o} is the line
frequency. This emissivity has a strong temperature dependence because
*q*_{ul}
*T*^{-1/2} and
(*q*_{lu} / *q*_{ul}) = (*g*_{u}
/ *g*_{l}) exp(-*h*
_{o} / *kT*),
where *g*_{u} and *g*_{l} are the statistical
weights. In the high-density limit we have,

(3) |

and the levels are said to be thermalized. Line thermalization, where
_{coll}
no longer depends on the transition strength, additionally requires
>> 1.
(*A*_{ul} and
are both proportional to
the oscillator strength, which therefore drops out of the
factor
*A*_{ul}
*A*_{ul}/
in Equation 3 if >>
1.) At low densities we have,

(4) |

Note that
_{coll}
scales here like the density
squared, compared with the linear dependence in Equation 3. The critical
density, *n*_{crit}, between these two limits is
the density at which the two terms in the denominator of Equation 2 are
equal,

(5) |

where the approximate relation holds only if
>> 1. Physically,
*n*_{crit} is the
density at which the upper level is as likely to be
de-excited by collisions as by radiative decay. Note that significant
optical depths have the effect of lowering *n*_{crit}. Also
note that transitions with very different oscillator strengths
(but similar collision strengths) will have similar
*n*_{crit} in the limit
>> 1 (because
*A*_{ul} / is
independent of oscillator strength).

The most prominent recombination lines belong to HI, HeI, and HeII, with HI Ly being typically strongest. These lines form by the capture of free electrons into excited states, followed by radiative decay to lower states. In the simplest case, in which every photon escapes freely and competing processes are unimportant, the emissivity is,

(6) |

where _{rad} is
the radiative
recombination coefficient into the upper energy state and
*n*_{i} is the number density of parent ions. The temperature
dependence is approximate and derives from
_{rad} (see
Osterbrock 1989).

**2.4.3. Deriving Abundance Ratios**

These two types of lines can be combined to form three types of
ratios for abundance analysis. The general idea is that, for any
element *a* in ion
stage *i*, the observed line intensity
*I*(*a*_{i}) is proportional to the density in that
ion, *n*(*a*_{i}), times a
function of the overall gas density and temperature
*F*(*a*_{i}, *T, n*), such
that *I*(*a*_{i}) =
*n*(*a*_{i})*F*(*a*_{i}, *T,
n*). The ionic abundance ratios are then given by,

(7) |

Abundance studies require line pairs for which the ratio of the two
functions *F* is nearly constant or has a limiting
behavior that still allow for abundance constraints. The last step is to
convert the ionic abundances into elemental abundances, which we express
logarithmically relative to solar ratios as
^{(2)},

(8) |

where *f*(*a*_{i}) is
the fraction of element *a* in ion stage *i*, etc. The
middle term on the right-hand side is the ionization correction
(*IC*), which can be deduced from numerical simulations
or set to zero (in the log) based on the similarity of the species
(Peimbert 1967).
Another strategy is to
compare summed combinations of lines from different ion stages so
that *IC* tends to zero on average
(Davidson 1977).

Ratios of pure recombination lines are simplest because they are least
sensitive to the temperature and density. In principle, we could derive
the He/H abundance from these ratios. However, in practice, all of the
strong HI and HeI recombination lines in QSOs, most notably
Ly, are affected
by collisions and thermalization effects. Moreover, because
H^{0}, He^{+} and He^{+2} have
different ionization energies, they need not be cospatial in the BELR, and
their levels of ionization depend on the different numbers of photons
available to produce each ion
(Williams 1971). As a
result, the H and He recombination spectra are most useful as
indicators of the shape of the ionizing continuum (e.g.
Korista et al. 1997a).
We do not expect substantial deviations from solar He/H
abundances anyway, based on normal galactic chemical evolution, and
the BEL data are grossly consistent with that expectation.

The second possibility involves the ratio of collisional to recombination lines. These ratios have strong temperature dependences (compare Equations 3 and 4 to Equation 6). Nonetheless, they can still be used for abundance work if the temperature sensitivities are quantified by explicit calculations. For example, there is an upper limit on the line ratio NV 1240/HeII 1640 related to the maximum temperature attained in photoionized BELRs. That upper limit sets a firm lower limit on the N/He abundance (Section 2.6.3 below).

The last ratio and the one most often used involves two
collisionally excited lines. Roughly a dozen collisionally excited
BELs are routinely measured in the UV spectra of quasars, so there is
a variety of possibilities. The ideal collisionally excited line pair
would have similar excitation energies, so their ratio has a
small *h*
_{o} / *kT* and
thus a small temperature dependence
(Equations 3 and 4). Similar values of *n*_{crit} and
similar ionization energies further minimize the sensitivities to
density and BELR structure. Well-chosen ratios that meet these criteria
can sometimes provide abundance estimates without recourse to detailed
simulations (e.g.
Shields 1976;
see Section 2.6.1 below).

^{2} Our notation here
is based on the usual definition of logarithmic abundances
normalized to solar ratios, [*a*/*b*]
log(*a* /
*b*) - log(*a* /
*b*)_{}.
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