**Intro/References**: I will discuss the five main diagnostics from line ratios and continuum fluxes for H II regions that can be applied to other photoionized emission regions including active galactic nuclei, planetary nebulae, and any other emission line sources such as Wolf-Rayet stars, etc. For a more detailed description of photoionization modeling, statistical mechanics / thermodynamics, and quantum mechanics, please see Chapter 18 of Draine, this link (from which the summary and figures below are based), and the all-time best reference AGN^{2 }: Astrophysics of Gaseous Nebulae and Active Galactic Nuclei 2nd Ed. by Osterbrock and Ferland.

**Motivation**: Line ratios and relative continuum fluxes of photoionized emission regions can give much physical insight regarding the properties of the nebula and its vicinity including the amount and type of dust extinction along the line of sight, the number density, temperature, physical radius / volume, mass, and metallicity of the gas, and even the temperature of the ionizing source. These indicators may also give further clues as to the surrounding environment, including and certainly not limited to the star formation rate and history, metallicity and mass of the host stellar population / galaxy / AGN, and depletion rates and abundances of the dust and gas.

**(1) Dust Extinction and Reddening from Balmer Decrement**

Photoionization of hydrogen is fairly well understood as it relies on already quantified atomic properties such as the photoionization cross-section, the recombination rate, the cascade matrix toward lower energies after recombination, oscillator strengths / Einstein A coefficients of the various transitions, etc. Consider the following table of theoretical emission line strengths j_{n’n }from various Balmer transitions (relative to Hβ):

where Case A and Case B represent the limits of two idealized situations so that actual H II regions must lie between these two extremes. Namely, Case A assumes the optically thin limit where hydrogen emission from all energy levels (Lyman, Balmer, etc.) can escape the H II region unabsorbed while Case B assumes all transitions more energetic than Lyα are absorbed and re-radiatved via Lyα and longer wavelengths. This is why the Hα emission line strength is greater in Case B. Nevertheless, the line ratios between Hβ and Hγ , for example, only vary by ~10% despite the temperature changing by a factor of two and not knowing in which optical depth regime the H II region exists. Thus, this Balmer decrement of decreasing relative line strengths toward more energetic transitions is fairly insensitive to temperature and number densities across the parameter space of H II regions.

If the actual line strengths of an H II region are measured and since we know the theoretical Balmer decrement, we can infer the amount of dust reddening toward the H II region. If only two Balmer lines are measured, the amount of dust reddening along with a theoretical dust extinction curve can give us a dereddened spectrum corrected for dust extinction. Additional Balmer lines can be used to constrain the type and functional dependence on wavelength of the actual dust extinction. This information has its own merit for inferring the dust properties along the line of sight and possibly dust properties in the vicinity of the H II region itself, but perhaps more important it gives us corrected emission line strength ratios which can be used for other diagnostics (see below).

**(2) Electron Number Density and Temperature from Forbidden Lines**

The forbidden transitions of [OIII] and [S II] can serve as excellent diagnostics of both temperature and density of the gas. Theoretically, observations of [OII] and [NII] can also constrain these properties but in reality the transitions of [OII] are too closely spaced to be resolved while the [NII] lines are typically contaminated by Hα. Consider the energy diagrams of [OIII] and [S II] below, where you can see all the important listed transitions are forbidden due to violation of one or more of the selection rules of quantum mechanics: ΔL = ±2 (such as transitioning from a D state to S state or vice versa), ΔJ = ±2 (which is intrinsically tied to ΔL), and/or no change in parity π_{f }= π_{i }(both initial and final states in these cases have even parity). Obviously, these transitions are only forbidden to electric dipole radiation, and do occur in nature when densities are sufficiently low such that timescales of collisional deexcitation are longer than the lifetime before spontaneous decay via these forbidden transitions.

Now consider the relative population of ions in the various electronic configurations and the measured transitions that arise from these upper energy levels. For example in [OIII], the ratio of the λ4363 to either or the sum of the λ4959, λ5007 intensities can allow you to infer the relative populations of the ^{1}S_{0 }to ^{1}D_{2 }states. Considering the energies between the two levels are widely spaced, the relative population will be almost completely dictated by the temperature of the gas, so the ratio of these [OIII] line intensities can be used to infer the temperature of the gas with only a slight dependence on number density. Conversely, the upper energy levels that give rise to the λ6716 and λ6731 transitions in [S II] are closely spaced in terms of energy so that the relative population will be controlled by collisional excitation / deexcitation between the two levels which depends significantly on the electron density. Thus, the relative intensities of these two transitions can be used to infer the number density of electrons. Combining observations of both pairs of transitions can further refine the precision of electron density and temperature via detailed photoionization modeling that relies on thermodynamic and quantum mechanical properties such as Boltzmann, Saha, Einstein A coefficients, collisional strengths and cross sections, and incident radiation.

**(3) Mass and Radius from Hβ Flux
**

Once the temperature and density of the gas can be ascertained, other physical properties can be modeled from atomic physics such as the recombination rate α_{B}, the subsequent probability of then cascading via Hβ emission P_{Hβ }so that the total volumetric rate of Hβ emission is α_{Hβ }= α_{B }P_{Hβ}, and thus finally the emissivity of Hβ emission ε_{Hβ }= α_{Hβ }hν_{Hβ}. (Historically, Hβ is used instead of Hα because it is in the green part of the optical spectrum where the sensitivity of photographic film peaks and since Hα is typically contaminated on either side by transitions of [N II] in real HII regions.) The functional fit of Hβ emissivity is given by:

which then gives the following relationship between Hβ luminosity, flux, and emissivity:

Since n_{e }is known from the forbidden transitions, ε_{Hβ }is calculated based on the gas temperature also inferred from the forbidden transitions, the total F_{Hβ }can be measured and corrected for dust extinction according to the Balmer decrement, and the ratio of radius to distance R/d can be measured by the angle the HII region subtends on the sky, then the absolute values of d and R can be calculated. Once the radius of and distance to the HII region are known, the mass of ionized hydrogen within the HII region is simply given by:

where as stated we already know the specified parameters on the right hand side.

**(4) Zanstra Temperature of Ionizing Source from Continuum Flux**

Using a simple Stromgren sphere analysis, the temperature of the central ionizing star can be inferred from the ratio of continuum flux (e.g. in the visual V filter) to line flux (e.g. Hβ) in a manner that is independent of the distance to the ionizing source and HII region. The actual relation is the following inequality:

because we must make the assumption that we are in the optically thick regime, i.e. Case B, where all ionizing photons are absorbed within the HII region and reradiated at longer wavelengths. In the equation, f is the fraction of the flux of photons that are capable of ionizing hydrogen, and can be simply modeled while simultaneously fitting the temperature of the ionizing source T_{*}. So to infer the temperature of the star, you must measure the broadband continuum V magnitude and know the response of the V filter with respect to frequency. You must also estimate the emissivity which you have already measured from the ratios of forbidden lines. Nevertheless, in all these input parameters you never had to assume the distance to the HII region, so the Zanstra temperature of the ionizing source is distance independent.

**(5) Metallicity / Abundances from Line Ratios**

By comparing line ratios from ions of different elements, the metallicity / chemical abundances of the gas can be estimated. Granted, there is an implicit correction for excitation and ionization, but empirical observations have shown that there is a measured correspondence between certain line ratios and metallicities. For example, the R23 ratio defined by:

has been found empirically to have the following dependence on the oxygen to hydrogen abundance:

Note that the relationship is double valued because of ionization and excitation effects. Nevertheless, observations of line ratios of other various ions can help constrain the abundances. Moreover, detailed photoionization modeling relying on the observed ratios of line intensities of the same ion can be used to derive the actual corrections for excitation and ionization.