One of the more common techniques used in the literature is to estimate the gas mass based on the luminosity of a single emission line at constant density. A recombination line such as Hα or Hβ is typically chosen as these are more stable than the forbidden lines across a wide range of physical conditions. When accessible, Hβ is the preferred option as it is less sensitive to collisional effects than Hα and is generally not blended with other strong emission lines. This is important, as this technique is essentially “photon counting” to determine the mass, which assumes that the emission is dominated by pure radiative recombination and neglects other emission processes. This simplifies the analysis and allows for a simple multiplicative factor known as the recombination coefficient to relate the number of photons to the number of hydrogen atoms and thus the total ionized gas mass, i.e. M ∝L/ne.
The main differences between our modeling approach and this single emission line technique are illuminated further by deriving the exact expression that relates the Hβ luminosity to the gas mass. A simplified form of this derivation for pure hydrogen can be found in Peterson (1997), with additional expressions and physical insight gathered from Osterbrock & Ferland (2006). The physical setup is as follows. First, we consider an unresolved region containing discrete gas clouds that each contribute line emission to the observed spectrum. The total mass M in these clouds is given by
M = 4π
3 l
3 n
e mp Nc (4.1)
where l is the radius of a cloud, ne is the electron density,mp is the mass of a proton, and
Nc is the total number of clouds. This framework establishes the mass by defining the cloud
volume (4πl3/3) multiplied by the density to get the total number of particles. This result is
multiplied by the mass per particle to get the mass of a single cloud, which is then summed over the total number of clouds.
Next, the emission released by these gas clouds is derived from the gas emissivity (jHβ), which is the luminosity per unit volume per solid angle, defined as
jHβ = 1
4π nenp α
ef f
Hβ hνHβ (erg s
−1 cm−3 ster−1) (4.2)
where ne and np are the electron and proton number densities and αHβef f is the effective
recombination coefficient that describes all transitions from levels n≥4 that will eventually transition ton= 2 and release an Hβphoton. The recombination coefficient is a weak function of temperature due to collisional ionization effects1, approximately following αef fHβ ∝T−0.9. Exact values for various temperatures can be found in Tables 4.2 and 4.4 of Osterbrock &
1The effective recombination coefficient is also a very weak function of density owing to collisional effects,
which is increasingly negligible for higher temperatures. Over the density range of nH = 102−106 cm−3 the change is∼4.1% atT = 5,000 K,∼1.7% atT = 10,000 K, and∼0.6% atT = 20,000 K. See Table 4.4 in Osterbrock & Ferland (2006).
Ferland (2006). Assuming optically thick (Case B) recombination for several temperatures these values are:
αef f Hβ = 5.37×10 −14 (cm3 s−1, T = 5,000 K) (4.3) αef f Hβ = 3.03×10 −14 (cm3 s−1, T = 10,000 K) (4.4) αef f Hβ = 2.10×10 −14 (cm3 s−1, T = 15,000 K) (4.5) αef f Hβ = 1.62×10 −14 (cm3 s−1, T = 20,000 K) (4.6)
The emitted luminosity is then the emissivity integrated over all angles (dΩ) and volume (dV). L(Hβ) = Z Z jHβ dΩ dV = 1 4π nenp α ef f Hβ hνHβ ×4π× 4π 3 Nc l 3 , (4.7) which reduces to L(Hβ) = 4π 3 Nc l 3 n enp αef fHβ hνHβ. (4.8)
Using the original expression for the total gas mass (Equation 4.1, M = 4π 3 l
3n
empNc) we can
identify the first portion of this expression as M/mp and write the luminosity as
L(Hβ) = 4π
3mp
M np αef fHβ hνHβ. (4.9)
Solving for the mass and introducing mp = mef fp andnp =nef fp to account for elements other
than hydrogen, M = L(Hβ) αef f Hβ hνHβ ! mef f p nef fp ! . (4.10)
We found that the gaseous abundances of the NLR for our AGN areZN LR≈1.3Zand taking
into account elements heavier than hydrogen yields mef f
p =µmp ≈1.4mp andnef fp ≈1.1ne.
The remaining physical quantity in this expression is the energy of an Hβ photon, which is
Incorporating these constants into Equation 4.10 we arrive at the final expression relating the Hβ luminosity to the ionized gas mass
M = 5.21×10−13 L(Hβ) αef f Hβne ! (g), (4.12) or equivalently, M = 2.62×10−46 L(Hβ) αef f Hβ ne ! (M). (4.13)
We may assign uncertainties to these expressions by adopting a range of effective re- combination coefficients that are appropriate for the range of temperatures observed in the NLRs of nearby AGN. Various studies have derived the electron temperature for statistically significant samples of Seyfert galaxies using the standard [O III] λλ4363/5007 emission line ratio that is sensitive to the gas electron temperature as shown in Figure 4.1. These grids were originally presented in Chapter 2 and are shown here for clarity.
Reported NLR temperatures span an order of magnitude with T ≈ 5,000−50,000 K (Dors et al. 2015). However, most studies find a mean [O III] emission line temperature of T ≈15,000 K with a standard deviation∼2,500−5,000 K (Bennert et al. 2006; Vaona et al. 2012; Zhang et al. 2013). Owing to the large range of reported temperatures we conservatively adopt the upper limit of this range as the formal uncertainty, yielding a final mean NLR temperature of T ≈15,000±5,000 K.
As discussed in Chapter 2, it is critical to note that these temperatures probe the [O III] emission line gas that may be hotter and more highly ionized than the [S II] emission line gas that we use to constrain the gas density in one of the following simplified methods. Our photoionization modeling results showed that the [S II] gas is on average ∼ 60% cooler than the [O III] gas, which is in excellent agreement with the observational results of Vaona et al. (2012). If we adopt this temperature for the gas then the effective recombination coefficients
Figure 4.1 The theoretical relationship between the [O III]λλ4363/5007 emission line ratio and the corresponding electron temperature for several typical NLR densities. These relationships were calculated using a grid of Cloudy photoionization models.
Inserting the effective recombination coefficients that correspond toT ≈15,000±5,000 K we obtain the useful expressions
M = (24.81 +7.35−7.62) L(Hβ) ne (g), (4.14) and, M = (12.48+3.70−3.83×10−33) L(Hβ) ne (M). (4.15)
This result is conceptually elegant as it only requires a measurement of the emission line luminosity and the electron density in order to determine the gas mass within any particular region. However, there are still several observational considerations that add an underlying level of complexity in practice. Specifically, deriving the intrinsic Hβ luminosity from the observed flux requires an accurate correction for extinction from dust and geometric dilution. In general, at least two hydrogen or helium recombination lines are required to derive the
reddening (or color excess, i.e. the differential extinction between the photometric B and V bands due to dust), while correcting for geometric dilution requires an accurate estimate of the distance to the galaxy. In addition, deriving a single electron density for all of the emitting material may not be realistic based on the results of our multicomponent models that required multiple density and ionization states at each spatial location. The uncertainties in the above equation are entirely due to the dependence of αef f
Hβ on a broad range of temperatures
(T ≈10,000−20,000 K) if the latter is not well constrained by emission line diagnostics or photoionization models.