3.2. Photoionization Models
Consider a cloud of material, modeled by a plane parallel slab with a certain total column density of Hydrogen, N(H) = N(HI) + N(H II), and with a constant total number density nH = n(H I) + n(H II) along the line of sight. The cloud is also characterized by its metallicity, Z, which is the ratio of Fe/H expressed relative to the solar value, Z, and by an abundance pattern (the abundance ratios of all other elements to Fe). The degree of ionization in the gas depends upon the intensity and shape of the spectrum of ionizing radiation. The intensity is characterized by the ionization parameter, U = n / nH, which is the ratio of the number density of photons at the Lyman edge to the number density of Hydrogen (nH = ne, where ne is the total number density of electrons). The larger the value of U, the more ionized the gas. Collisional ionization can also be an important process for some absorption systems with gas at high temperatures (hundreds of thousands of degrees). Photoionization equilibrium models typically yield temperatures of tens of thousands of degrees.
Once the metallicity, abundance pattern, ionization parameter, and spectral shape are specified the equations of radiative transfer can be solved to find the column densities of all the different ionization states of various chemical elements. Figure 6 illustrates, for N(HI) = 1016 and 1019 cm-2, the dependence of column densities of various transitions on the ionization parameter, U. For optically thin gas [N(HI) < 1017.2 cm-2], the column density ratios of the various metal transitions are not dependent on the overall metallicity, i.e. the curves shift vertically in proportion to Z. For optically thick gas, ionization structure develops, with an outer ionized layer around a neutral core, and there is no simple scaling relation with metallicity.
Figure 6. Photoionization model predictions of the column densities of Mg II, Fe II, and C IV as a function of the ionization parameter (the ratio of ionizing photons to the electron number density in the gas). The spectrum incident on the cloud, represented by a constant density slab, is the ``Haardt-Madau'' spectrum (attenuated spectrum due to integrated effect of quasars and young galaxies). The predicted column densities are presented in two series of models with N(HI) = 1016 cm-2 and with N(HI) = 1019 cm-2, the optically thin and optically thick cases. For both, the metallicity is fixed at 10% of the solar value. For the optically thin case, the column densities scale with metallicity, i.e. the ratios remain constant, but for the optically thick case the situation is more complex.
In practice, if we assume that a cloud has a simple, single phase structure, the ratios of the column densities can be used to infer the ionization parameter, which relates to the density of the gas. However, the abundance pattern can differ from the solar abundance pattern because of differing degrees of depletion onto dust, or because of different processing histories. Most of the so-called particle nuclei (such as Mg and Si) are synthesized primarily by Type II supernovae during the early history of a galaxy when most massive stars form and quickly evolve to reach their end states. On the other hand, the Fe-group elements are primarily produced by Type Ia supernovae, and therefore build up over a longer timescale. In the basic picture of galaxy evolution, the halo stars are formed early, have been enriched only by Type II supernova, and therefore are -element enhanced. Younger disk stars have incorporated also the Type Ia processed material and therefore have relatively larger Fe-group abundances. Ideally, several different ionization states of the same chemical element are observed so that there is no ambiguity between the ionization parameter and the abundance pattern, but this has generally not yet been possible because of limited wavelength coverage at high resolution.
Figure 7. HIRES/Keck Fe II and Mg II absorption profiles for the z = 1.325 system in the spectrum of the quasar Q0117+213. The six clouds in this system show a range of more than an order of magnitude in N(Fe II) / N(Mg II), given below each cloud in the lower panel. These variations could be due to cloud to cloud variations of ionization parameter (density) or of abundance pattern within the system.
Examples of the variation of column density ratios with velocity in two absorption systems are shown in Figures 7 and 8. In Figure 7, N(Fe II) / N(Mg II) varies by an order of magnitude over the four components in the z = 1.325 system toward the quasar Q0117+213. This represents a variation of an order of magnitude in the ionization parameter (10-4 < U < 10-3), or an order of magnitude variation in the abundance pattern. Figure 8 is a very unusual system with two clouds separated by only 20 km s-1 in velocity, one of which has a Silicon to Aluminum ratio similar to the Milky Way ISM, and the other which requires a significant enhancement of Aluminum.
Figure 8. An unusual Aluminum-rich cloud is apparent in the z = 1.93 system toward the quasar Q1222+228, and it is close in velocity space to a normal (relative to Galactic clouds) cloud which has detected Si II. Note the different kinematic structure in the higher ionization transitions. The excess of Al II and Al III in the cloud at v = 9 km s-1 is best explained by an abundance pattern variation, since Si II and Al II are transitions with very similar ionization states.