The Physical Properties of Low-redshift FeLoBAL Quasars. III. The Location and Geometry of the Outflows

The SimBAL model fits for the sample are given in Choi et al. (2022), and the details can be found in that paper (Tables 2 and 3). We extracted the following parameters from those results: the ionization parameter logU, the gas density $\mathrm{log}n\,[{\mathrm{cm}}^{-2}]$, the broad-absorption-line velocity V_off (km s⁻¹), the broad-absorption-line velocity width V_width (km s⁻¹) ⁹ , the total column density integrated over the BAL feature $\mathrm{log}{N}_{H}\,[{\mathrm{cm}}^{-2}]$, the covering-fraction parameter loga, and the radius of the outflow logR [pc].

Several additional and derived parameters were also produced that are not reported in tables in Choi et al. (2022). These include the largest and smallest outflow velocities V_max , V_min , the force multiplier (FM; the ratio of the total opacity to the electron scattering opacity), the thickness of the outflow ΔR [pc], the filling factor $\mathrm{log}{\rm{\Delta }}R/R$, the net outflow rate $\mathrm{log}\dot{M}\,[{M}_{\odot }\,{\mathrm{yr}}^{-1}]$ (per component), the net kinetic luminosity $\mathrm{log}{L}_{\mathrm{KE}}\,[\mathrm{erg}\,{{\rm{s}}}^{-1}]$, and the ratio of the net kinetic luminosity to the bolometric luminosity. The thickness of the outflow ΔR is the ratio of the total column density and the density. Making the simple assumption that the azimuthal size of the outflowing gas is comparable to the thickness of the gas (i.e., the gas is distributed into individual clouds and the clouds are approximately spherical), and using the radius of the accretion disk at 2800 Å described in Leighly et al. (2022), we computed the log of the number of clouds required to cover the continuum-emission region (see Section 4.3.2 for more details).

Finally, we used the parameters from the SimBAL best-fitting solution to compute the luminosity of the predicted [O iii] emission line assuming an emission-line global covering fraction of 0.1 (discussed in Section 4.4). We also computed a parameter called the covering-fraction correction, described in that section, which parameterizes the comparison of the predicted [O iii] luminosity from the wind with the observed [O iii] luminosity.

The following optical emission-line parameters were taken from Table 1 of Leighly et al. (2022). We used the Hβ FWHM and equivalent width to parameterize the Hβ line. Leighly et al. (2022) also defined a parameter called the Hβ deviation that measures the systematically broader Hβ FWHM observed among the FeLoBALQs compared with unabsorbed quasars. Fe ii was parameterized using R_FeII defined as the ratio of the Fe ii equivalent width in the range 4434–4684 Å to the broad Hβ equivalent width (e.g., Shen & Ho 2014). The [O iii] emission line was parameterized using the equivalent width and luminosity, along with profile parameters v₅₀ and w₈₀ defined according to the prescription of Zakamska & Greene (2014). Briefly, from the normalized cumulative function of the broad [O iii] model profile, the velocities at 0.1, 0.5, and 0.9 were identified. The velocity at 0.5 is assigned to v₅₀, and w₈₀ is the difference between the velocities at 0.1 and 0.9. Leighly et al. (2022) also fit the FeLoBAL objects and the unabsorbed objects with eigenvectors created from the continuum-subtracted spectra of the unabsorbed objects (Section 2.3, Leighly et al. 2022). The fit coefficients for the first four eigenvectors (SPCA1–4) serve as a parameterization of the spectra. The first eigenvector displays the relationship between the strength of Fe ii and [O iii] that is commonly found (e.g., Grupe 2004; Ludwig et al. 2009; Wolf et al. 2020), and none of the other ones display any particular anomalies. We estimated the bolometric luminosity using the rest-frame flux density at 3 μm and a bolometric correction of 8.59 (Gallagher et al. 2007). BAL quasars tend to be reddened (e.g., Krawczyk et al. 2015), and evidence for reddening is present in this sample (0 ≲ E(B − V) ≲ 0.5; Figure 20 in Choi et al. 2022; K. M. Leighly et al. in preparation). Therefore, we used the 3 μm luminosity density as representative, rather than the luminosity in the optical or UV. The black hole mass and Eddington ratio were computed using standard methods and as described in Leighly et al. (2022; Section 2.1), and the calculation of the location of the 2800 Å emission from the accretion disk follows the method used in Leighly et al. (2019b).

Finally, we continue to use the E1 parameter, which was defined in Section 3.1 of Leighly et al. (2022). This parameter is a function of the measured values of R_FeII and the [O iii] equivalent width. As described in Section 3.1 of Leighly et al. (2022), we normalized and scaled R_FeII and the [O iii] equivalent width of the 132 object comparison sample and then derived the bisector line. We performed a coordinate rotation so that E1 is a parameter that lies along the bisector. Our E1 parameter is therefore related to the Boroson & Green (1992) Eigenvector 1. Boroson & Green (1992) performed a principal component analysis of the emission-line properties in the vicinity of Hβ and found that the variance is dominated by an anticorrelation between R_FeII and the [O iii] equivalent width. Moreover, Eigenvector 1 has been shown to dominate the variance in quasar emission-line properties (e.g., Francis et al. 1992; Boroson & Green 1992; Brotherton et al. 1994; Corbin & Boroson 1996; Wills et al. 1999; Sulentic et al. 2000; Grupe 2004; Yip et al. 2004; Wang et al. 2006; Ludwig et al. 2009; Shen 2016). E1 is very strongly correlated with SPCA1 (p = 6.5 × 10⁻¹⁵ and p = 10⁻⁵⁰ for the FeLoBALQs and the unabsorbed comparison sample that was analyzed in parallel, respectively). It is also correlated with the Eddington ratio (p = 1.2 × 10⁻⁵ and p = 9.3 × 10⁻⁷ for the FeLoBALQs and the unabsorbed comparison sample, respectively). Here, p is a measure of the statistical significance of the correlation. More specifically, it is a probability that the observed correlation could have been produced from draws from two uncorrelated samples (e.g., Bevington 1969). A low (negative) value of the E1 parameter corresponds to a low accretion rate, while a high (positive) value of the E1 parameter corresponds to a high accretion rate. The 90% range of $\mathrm{log}{L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}$ among the sample is −1.4 to 0.84 (Figure 9 Leighly et al. 2022). As discussed in Section 3.3 of Leighly et al. (2022), the apparent bimodal distribution of FeLoBALQs in E1 shows that there are two types of FeLoBALQs: objects with E1 < 0 are characterized by a low Eddington ratio, and objects with E1 > 0 are characterized by a high Eddington ratio. We divided the FeLoBALQs into two groups based on this parameter, where the dividing line E1 = 0 corresponds to $\mathrm{log}{L}_{\mathrm{Bol}}/{L}_{\mathrm{Edd}}=-0.5$, i.e., an Eddington ratio of about 0.3. Throughout this paper, we use a consistent coloring scheme to denote E1: red (blue) corresponds to E1 < 0 and a low Eddington ratio (E1 > 0 and a high Eddington ratio).

Leighly et al. (2022) presented comparisons of the distributions of the emission-line and derived properties of the FeLoBALQs with those of the comparison sample of unabsorbed quasars using cumulative distribution plots. We applied the two-sample Kolmogorov–Smirnov (K-S) test, and the two-sample Anderson–Darling (A-D) test. The K-S test reliably tests the difference between two distributions when the difference is large at the median values, while the A-D test is more reliable if the differences lie toward the maximum or minimum values (i.e., the median can be the same, and the distributions different at larger and smaller values). ¹⁰ We also compared the samples divided by the sign of the E1 parameter. Note that the values of the E1 parameter and plots of the spectral model fits are given in Leighly et al. (2022). In this paper, we present the comparisons of the SimBAL parameters for the FeLoBALQs segregated by the E1 parameter (Table 1).

Table 1.  SimBAL E1 < 0 versus E1 > 0 Parameter Comparisons

Parameter Name	K-S a	A-D b
	Statistic/Probability	Statistic/Probability
logU	0.21/0.28	0.64/0.18
$\mathrm{log}n$ [cm−3]	0.24/0.57	−0.07/> 0.25
Voffset (km s−1)	0.50/0.016	4.4/5.9 × 10−3
Vmax (km s−1)	0.50/0.014	4.2/7.0 × 10−3
Vmin (km s−1)	0.55/4.7×10−3	5.0/3.6 × 10−3
Vwidth (km s−1)	0.30/0.33	0.67 / 0.17
loga	0.36/0.14	1.1/0.11
Net log NH [cm−2]	0.33/0.21	0.094/> 0.25
log Force Multiplier	0.35/0.17	0.32/> 0.25
logR [pc]	0.26/0.45	−0.10/> 0.25
ΔR [pc]	0.46/0.032	3.4/0.013
log Volume Filling Factor	0.51/9.8 × 10−3	5.7/2.0 × 10−3
log Number of Clouds	0.46/0.032	4.7/4.6 × 10−3
Net M˙ (M⊙ yr−1)	0.31/0.28	0.084/> 0.25
LKE [erg s−1]	0.42/0.10	1.4/0.09
LKE/LBol	0.30/0.41	0.37/0.24
Predicted [O iii] Luminosity	0.26/0.49	0.56/0.19
Covering-fraction Correction	0.58/2.1 × 10−3	9.3/< 0.001

Notes.

^a Kolmogorov–Smirnov two-sample test. Each entry has two numbers: the first is the value of the statistic, and the second is the probability p that the two samples arise from the same parent sample. Bold type indicates entries that yield p < 0.05. ^b Anderson–Darling two-sample test. Each entry has two numbers: the first is the value of the statistic, and the second is the probability p that the two samples arise from the same parent sample. Note that the implementation used does not compute a probability larger than 0.25 or smaller than 0.001. Bold type indicates entries that yield p < 0.05.

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The results for the SimBAL parameters are shown in Figure 1, and the statistics are given in Table 1. The parameters that exhibit statistically significant differences between the E1 > 0 and E1 < 0 groups are all three velocity outflow parameters (V_off, V_max, and V_min), the thickness of the outflow ΔR, the log filling factor, the log of the number of clouds covering the continuum-emitting source, and the covering-fraction correction factor. The E1 > 0 objects show systematically larger opacity-weighted outflow velocities than the E1 < 0 objects, with median values of −1520 km s⁻¹ and −500 km s⁻¹, respectively. For the E1 < 0 objects, the outflows are thicker, have larger filling fractions, and fewer ¹¹ are required to cover the continuum source (median $\mathrm{log}{\rm{\Delta }}R=-1.8$ pc, log volume filling fraction = −3.8, and log number of clouds = −0.54). The E1 > 0 objects have thinner outflows, smaller volume filling fractions, and more clouds are necessary to cover the continuum emission (ΔR = − 2.4 [pc], log volume filling fraction = −5.6, and log number of clouds = 0.34). These properties are discussed further in Section 4.3.2. The covering-fraction correction factor is lower for the E1 < 0 than for the E1 > 0 objects (median values of 0.6 and 1.5, respectively); this parameter was defined in Section 2.1 and is discussed in detail in Section 4.4.

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The ionization parameter, loga, and the force multiplier show suggestions of differences. Specifically, while the highest ionization parameter represented by E1 > 0 objects is $\mathrm{log}U=-0.5$, 25% of E1 < 0 objects have ionization parameters larger than that value (Figure 1). Likewise, lower values of the force multiplier are dominated by E1 < 0 objects (Figure 1). In addition, the kinetic luminosity L_KE and the ratio of the kinetic to bolometric luminosity are consistently lower for the E1 < 0 objects (Figure 1).

In Leighly et al. (2022), we examined the relationships among the optical and derived parameters such as the bolometric luminosity. In this paper, we compare those parameters with the SimBAL parameters, both for the full sample, and for the sample segregated by the E1 parameter. We used the Spearman rank correlation for our comparisons.

We first correlated the SimBAL parameters with one another. A correlation analysis of the SimBAL results for the full 50 object sample is given in Section 6.1 of Choi et al. (2022). Here we considered only the low-redshift subsample. The results are shown in Figure 2. The plots represent the log of the p value for the correlation, where the sign of the value gives the sense of the correlation. That is, a large negative value implies a highly significant anticorrelation.

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Parameter uncertainties were propagated through the correlations using a Monte Carlo scheme. We made 10,000 normally distributed draws of each parameter, where the distribution was stretched to the size of the error bar. Asymmetrical errors were accounted for by using a split-normal distribution (i.e., stretching the positive draws according to the positive error, and the negative draws according to the negative error). We chose p < 0.05 as our threshold for significance. In most cases, taking the errors into account did not dramatically change the significance of a correlation, if present.

There are several extremely strong correlations among the SimBAL parameters (Figure 2) that were also observed for the full sample Choi et al. (2022). As we discuss only FeLoBALQs in this paper, most gas columns extend beyond the hydrogen ionization front ¹² in order to include sufficient Fe⁺ ions to create an observable absorption line. The thickness of the H ii region increases with the ionization parameter, a fact that explains the strong correlation between the column density and the ionization parameter. The net mass outflow rate is a function of the velocity, explaining the strong correlation between those two parameters. The force multiplier depends inversely on the ionization parameter.

We also correlated the SimBAL parameters for the E1-divided samples (Figure 2). As noted above, E1 is related to the Boroson & Green (1992) Eigenvector 1, which dominates the variance in quasar properties. The motivation for looking for correlations among the E1-divided samples is that by removing that dominant dependence, more subtle parameter dependencies may be revealed.

Finally, Figure 3 shows the correlations between the 17 optical and global parameters and the 18 SimBAL parameters. A special technique was used to handle these data because there are 30 objects and 36 outflow components. Five of the objects showed multiple outflow components ¹³ ; Choi et al. (2022, Section 6.2) discussed the need for multiple outflow components in these objects. We assumed that physically one of the components is more representative than the other one or two. For example, [O iii] may be produced by one component in the outflow but not another. Likewise, the outflow location may be correlated with the outflow velocity for components that share some fundamental property (i.e., perhaps they are the main outflow in the system), but other subsidiary outflows do not obey this trend. Therefore, we computed the correlations among all combinations and present the statistically most significant ones.

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An interesting set of patterns observed among the SimBAL parameters is shown in Figure 4. We plot the velocity of the outflow as a function of the log of the radius divided by the dust sublimation radius (${R}_{d}=0.16{L}_{45}^{1/2}\,\mathrm{pc}$; Elitzur & Netzer 2016); note that R_d is close to 1 pc in these samples. It is immediately apparent that among the compact outflows $\mathrm{log}R/{R}_{d}\lt \sim 2$, the E1 > 0 objects have systematically larger velocity outflows than the E1 < 0 objects.

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While there is no correlation between the location of the outflow $\mathrm{log}R/{R}_{{\rm{d}}}$ and the outflow velocity for the sample as a whole, we found a tentative or marginal correlation for the E1 > 0 subsample and an anticorrelation for the E1 < 0 subsample (p values are reported later in the text). E1 < 0 objects with outflows close to the central engine generally have no net outflow velocity, while those located far from the central engine show modest outflows. In contrast, some of the largest velocities in the sample are found in E1 > 0 object outflows located close to the central engine, while at larger distances, the velocities tend to be lower.

The correlation between the outflow velocity and $\mathrm{log}R/{R}_{{\rm{d}}}$ observed among the E1 > 0 objects might be expected from a high-accretion-rate quasar. The terminal velocity for a radiatively driven outflow is larger at smaller radii because of the higher available photon momentum, which means that larger velocity at small radii would be needed for a sustained (i.e., not failed) wind.

Choi et al. (2022) identified a new class of FeLoBALQs called "loitering" outflows as objects that have $\mathrm{log}R\lt 1$ [pc] and a velocity offset of the excited-state Fe ii λ2757 of ∣v_off∣_{FeII excited} < 2000 km s⁻¹ (Figure 18 of Choi et al. 2022). They found that the loitering outflow objects had distinct photoionization properties too: they have larger ionization parameters and densities compared with the full sample. Leighly et al. (2022) found that, in the low-redshift subsample, almost all loitering outflow objects had E1 < 0 and were therefore categorized as low-accretion-rate objects.

We next try to understand the origin of the low velocities among the loitering outflows. The high ionization parameter ($-2\lt \mathrm{log}U\lt 0.5$; Figure 6 of Choi et al. 2022) and accompanying large column density ($\mathrm{log}{N}_{H}\gt 21.5$ (cm⁻²); Figure 7 of Choi et al. 2022) yield a low force multiplier; basically, the slab has very large column of gas where the illuminated face and a significant fraction of the total column density are too ionized to contribute much to resonance scattering (e.g., Arav et al. 1994). The E1 < 0 objects are characterized by a low Eddington ratio ($\mathrm{log}{L}_{\mathrm{Bol}}/{L}_{\mathrm{Edd}}\,\lt -0.5$), which means that the radiative flux is small relative to the gravitational binding of the black hole and therefore it is less able to accelerate the outflow gas. Thus, the combination of the large outflow column and low radiative flux compared with gravity may explain the low velocities (V_off < − 2000 km s⁻¹; Figure 18 of Choi et al. 2022).

However, not all E1 < 0 objects have loitering outflows; at larger radii, near $\mathrm{log}R/{R}_{{\rm{d}}}=3$, the E1 < 0 objects merge with the E1 > 0 objects, and outflows of both E1 groups have velocities near − 1000 km s⁻¹. The combination of near-zero velocity for small $\mathrm{log}R/{R}_{{\rm{d}}}\lt 2$ and outflows for larger $\mathrm{log}R/{R}_{{\rm{d}}}\gt 2$ results in the anticorrelation between the outflow velocity and $\mathrm{log}R/{R}_{{\rm{d}}}$ among the E1 < 0 objects shown in Figure 4.

As seen in Figure 1, it is clear that the E1 > 0 objects have systematically larger velocity outflows than the E1 < 0 objects; the probability that they are drawn from the same population is less than 1.6% (Table 1). However, correlations between the outflow velocity and the location of the outflow are barely or arguably not significant, and might be construed to fall in the realm of p hacking. For example, if we examine logR as a function of V_off, then the E1 < 0 correlation is significant (p = 0.025) but the E1 > 0 is not (p = 0.076), while for $\mathrm{log}R/{R}_{d}$, the E1 < 0 correlation is not significant (p = 0.052) while E1 > 0 is significant (p = 0.050). On the other hand, these correlations are principally driven by the differences in the velocity distributions at $\mathrm{log}R\lt 2$ [pc], which are very clear from Figure 4. Regardless, these differences in the behavior of the outflow velocity among the E1 < 0 and E1 > 0 objects may point to a difference in formation and acceleration of outflows among those two classes of objects. In particular, if the E1 parameter is considered to be a proxy for L_Bol/L_Edd, these differing behaviors may point to a difference in the formation and acceleration of outflows as a function of that parameter. The relationship between potential acceleration mechanisms and the origin and location of FeLoBAL winds is discussed extensively in Choi et al. (2022) Section 7.2.

Using the black hole masses estimated in Leighly et al. (2022) we computed the Keplerian velocities at the location of the outflow. We then computed the ratio of the Keplerian velocity to the outflow velocity V_off, as well as the ratio to V_min and V_max, and plot the results in Figure 5. This plot shows several interesting features. Large values of V_Kep/∣V_off∣ indicate absorption features that have line-of-sight velocities much lower than the local Keplerian velocity (e.g., SDSS J1125+0029, SDSS J1321+5617). A number of E1 < 0 objects fall into this category. Because the continuum-emission region is so small compared with the outflow location (Section 4.3), this phenomenon could be achieved if the outflow velocity vector lies strictly along the line of sight to the nucleus, and the local Keplerian motion is principally tangential.

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On the other hand, very low values of V_Kep/∣V_off∣ indicate objects with outflow velocities very much larger than the local Keplerian velocity (e.g., SDSS J1039+3954, SDSS J1044+3656). Small values suggest that the outflow is kinematically decoupled from the gravitational potential of the black hole. It is possible that these small ratios signal a distinct acceleration mechanism for their outflows, such as the "cloud-crushing" mechanism suggested by Faucher-Giguère et al. (2012), rather than radiative line driving or dust acceleration, where the magnitude of those mechanisms scales with the location of the outflow.

Finally, a number of the outflows show V_Kep/∣V_off∣ commensurate with unity. Most of the E1 > 0 objects fall into this category. Of particular interest is SDSS J1448+4043. Choi et al. (2022) found that the SimBAL solution required three separate outflow components in this object (Section 6.2 of that paper). The lowest velocity component could be seen to be kinematically distinct, as it is narrow and shows a prominent ground-state Mg i λ2853 line. The other two features are kinematically blended in this overlapping trough object and were inferred to be distinct based on their photoionization properties (Figure 11 of Choi et al. 2022). They found that the outflows lie at dramatically different distances ($\mathrm{log}R\,=0.84$, 2.05, and >3.1 [pc]); yet their outflow velocities are commensurate with Keplerian velocity at those locations. These outflows could be considered kinematically coupled in some way to the gravitational potential of the black hole.

The plot illustrating the results of the correlation analysis (Figure 3) reveals the most frequently observed relationship among BAL quasar outflows: the correlation of the luminosity and Eddington ratio with the outflow speed (p = 5 × 10⁻⁴ and p = 1.2 × 10⁻³, respectively). The relationship between the BAL outflow velocity and Eddington ratio is shown in Figure 6. These relationships have been previously reported for HiBALQs (Laor & Brandt 2002; Ganguly et al. 2007; Gibson et al. 2009) and are generally found among objects with outflows (e.g., Fiore et al. 2017). Ganguly et al. (2007) noted that the terminal velocity of an outflow should scale with the Eddington ratio as ${v}_{\mathrm{terminal}}\propto {({L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}})}^{1/2}$ (Hamann 1998; Misawa et al. 2007); this dependence arises from the solution of the conservation of momentum equation (e.g., Leighly et al. 2009) assuming that the outflow is accelerated by radiation. Thus the outflow velocity is predicted to be correlated with the luminosity relative to the Eddington value. However, a linear relationship between these two quantities is not what is observed; rather, there is usually an upper-limit envelope of velocity as a function of the luminosity or L_bol/L_Edd. That is, at any luminosity, there is a range of outflow velocities up to some upper-limit value, and those upper-limit values are correlated with the luminosity. For example, see Figure 6 in Laor & Brandt (2002) and Figures 6 and 7 in Ganguly et al. (2007). This upper-limit relationship is seen in our data too (Figure 6, top).

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Theoretically, the speed of a radiatively driven outflow should also depend on two factors: 1) the launch radius (because the flux density of the radiation drops as R⁻²) and 2) the force multiplier (because the ability of an outflow to use the photon momentum depends on which ions that can scatter the photons are available). Using the momentum conservation equation it can be shown that

$$\begin{eqnarray*}&&v\sim {{FM}}^{1/2}{(L/{L}_{\mathrm{Edd}})}^{1/2}{({R}_{\mathrm{launch}}/{R}_{{\rm{S}}})}^{-1/2},\end{eqnarray*}$$

where R_S is the Schwarzschild radius. The launch radius is difficult to determine, but we can estimate it by assuming that the measured velocity is some representative fraction of the terminal velocity. For a β velocity law ¹⁴ , the launch radius is a constant fraction of the radius at which the outflow is observed, i.e., R_launch/R_S ≈ R/R_S. While this approximation would be inadequate to solve an equation of motion for a single object (e.g., Choi et al. 2020), it may be sufficient to compare a sample of objects. The force multiplier, defined to be the ratio of the total opacity to the electron scattering opacity, is an output of Cloudy; it includes both line and continuum opacity.

Armed with these relationships and approximation, we can investigate whether the scatter in the outflow velocity for a given L_bol/L_Edd could be a consequence of intrinsic scatter, whether there are trends in either the launch radius or the force multiplier, or whether both cases may apply. To put it another way, if the velocity upper-limit envelope describes the behavior of an optimal outflow, perhaps the objects below that optimal level may be deficient in either their force multiplier or their launch radius.

We note that there is another factor that could be important: the angle of the line of sight to the velocity vector. We do not discuss that factor here, which means that there should be additional intrinsic scatter associated with that parameter. In addition, dust scattering may be important in accelerating quasar outflows (e.g., Thompson et al. 2015; Ishibashi et al. 2017), and our use of the force multiplier means that we are only considered continuum and resonance scattering for the acceleration mechanism. Additional discussion of the potential role of dust scattering in acceleration in FeLoBALQ outflows can be found in Choi et al. (2022) Section 7.2.

We investigated these questions by first defining a parameter V_ledd=1 that describes how far below the optimal outflow velocity an object lies on the Eddington ratio versus the velocity as shown in Figure 6. That is, we solved ${V}_{\mathrm{off}}\,={V}_{\mathrm{ledd}=1}{({L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}})}^{1/2}$ for V_ledd=1, where V_off is plotted as the y axis in the top panel of Figure 6. Then, for any particular object, V_ledd=1 is the velocity that it would have if L_bol/L_Edd = 1. In essence, we have derived a parameter that can be used to compare objects as though they have the same L_bol/L_Edd (similar to the concept of the absolute magnitude). Traces for representative values of V_ledd=1 are shown in Figure 6. We then plotted this parameter against the force multiplier and R/R_S (Figure 6).

We first considered the force multiplier. We found that the force multiplier is not correlated with V_ledd=1 for the whole sample, but it is marginally anticorrelated for the E1 < 0 objects (r_s = − 0.46, p = 0.047), and correlated for the E1 > 0 objects (r_s = 0.62, p = 6.7 × 10⁻³). Keeping in mind that V_ledd=1 is negative for outflows, this means that E1 < 0 objects with larger FM have higher outflow velocities. We interpret this behavior to mean that unfavorable FM (too small) is responsible for nonoptimal (i.e., below the envelope) outflows in the E1 < 0 objects. In contrast, the correlation for the E1 > 0 objects means that the E1 > 0 objects with low FM have higher outflow velocities, opposite of the physical expectation. We interpret this to mean that E1 > 0 objects produce their outflows despite nonoptimal FM values, and therefore another factor is causing the below-envelope scatter in the E1 > 0 objects.

Turning to R/R_S , we found that, while there is no correlation between this parameter and V_ledd=1 for the sample as a whole, there is an anticorrelation for the E1 < 0 objects (r_s = − 0.52, p = 0.023) and a correlation for the E1 > 0 objects (r_s = 0.51, p = 0.035). Again keeping in mind that V_ledd=1 is negative for outflows, the correlation between R/R_S and V_ledd=1 means that objects with smaller R/R_S have larger outflow speeds, as predicted by the solution to the momentum conservation equation above. This correlation suggests that the below-envelope scatter among the E1 > 0 objects is caused by unfavorable (too large) R/R_S (Figure 6, bottom panel). The anticorrelation seen among the E1 < 0 objects can be interpreted as implying that R/R_S is less important in determining their outflow velocity.

Finally, we can compare the V_ledd=1 values for the E1 < 0 and the E1 > 0 objects. It turns out that the V_ledd=1 distributions for the two classes are statistically indistinguishable. That is, both types of objects lie along the same set of V_ledd=1 traces in Figure 6. The E1 < 0 objects have both systematically lower outflow velocities (Figure 1) and systematically lower L_bol/L_Edd (Figure 7, Leighly et al. 2022). This result implies that E1 > 0 objects reach a larger velocity because of their larger L_bol/L_Edd, i.e., L_bol/L_Edd is primary for both classes of objects.

We have shown that in the low-redshift subsample of FeLoBALQs the outflow velocity depends on L_bol/L_Edd. This is not a new result; it has been seen before in other samples (e.g., Ganguly et al. 2007). Such a result is relatively simple to extract from any set of BAL quasars, depending as it does only on estimation of the outflow velocity (e.g., from the C iv trough) and an estimate of the Eddington luminosity. That requires an estimate of the black hole mass, which is arguably most reliably extracted from Hβ but can also be estimated using Mg ii and C iv, in principal, although difficult in FeLoBALQs due to the heavy absorption. The difference in our analysis is that, because SimBAL yields the physical conditions of the outflow (including logU, $\mathrm{log}n$, $\mathrm{log}{N}_{H}$), we can investigate this relationship in more detail and parse the dependence on the subsidiary parameters: the force multiplier and the estimated launch radius. We discovered a difference in the velocity dependence of these two parameters among the two accretion classes. So while L_bol/L_Edd is principally responsible for determining the outflow velocity, the force multiplier (launch radius) is responsible for producing the scatter in the velocity at a particular value of L_bol/L_Edd in the E1 < 0 (E1 > 0) objects. This result provides additional evidence for differences in the acceleration mechanism.

4.3.1. The Full Sample

The volume filling factor (ΔR/R, ΔR = N_H /n_H ) gives us information about the physical size scales of the outflowing gas. It is most directly interpreted as the fractional volume of space occupied by the outflow. Typically, using the values for the column density, density, and radius derived using excited-state absorption lines, small log volume filling factors, mostly ranging between −6 and −4 are found (e.g., Korista et al. 2008; Moe et al. 2009; Dunn et al. 2010). The volume occupied by the absorbing clouds ranges from 0.01% to 1%. The volume filling factor tells us how thin or extended in the radial direction the BAL cloud structure is and provides us with information about the BAL physical conditions. A small volume filling factor ($\mathrm{log}{\rm{\Delta }}R/R\sim -5$) for BALs may imply a pancake- or shell-like geometry that is very thin in the radial direction (e.g., Gabel et al. 2006; Hamann et al. 2011, 2013). These BAL absorbers with $\mathrm{log}{\rm{\Delta }}R/R\lesssim -3$ may be composed of smaller gas clouds (e.g., Waters & Proga 2019) that are potentially supported by magnetic confinement in order to avoid dissipation (e.g., de Kool & Begelman 1995). In contrast, Murray & Chiang (1997) proposed that a continuous flow from the accretion disk is the origin of broad emission lines and BAL features; such a flow would have a volume filling factor of 1. We emphasize that our results are not consistent with a direct observation of a disk wind because the size scales that we measure are too large. The minimum distance of the outflow from the central engine found in our sample is R ∼ 1 pc in SDSS J1125+0029, whereas reasonable size scales for disk wind outflows should be comparable to the size of the accretion disk (R ≪ 0.01 pc). That does not imply that disk winds do not exist but rather that we do not find them to have rest-UV BAL outflow signatures. This result is consistent with the literature; among the FeLoBALQs previously subjected to detailed analysis, typical outflow distances lie between 0.4 and 700 pc (e.g., de Kool et al. 2001, 2002a, 2002b; Moe et al. 2009; Dunn et al. 2010; Aoki et al. 2011; Lucy et al. 2014; Shi et al. 2016; Hamann et al. 2019; Choi et al. 2020), i.e., no closer than the broad-line region.

We calculated the volume filling factors for the full sample and examined the dependence on BAL properties (Figure 7). The strong correlation seen between $\mathrm{log}{\rm{\Delta }}R/R$ and logU can be explained by the mathematical relationship between the parameters as follows. First, the BAL physical thickness (ΔR) is proportional to the hydrogen column density (N_H ), which is also proportional to the ionization parameter U as $\mathrm{log}{N}_{H}\,-\mathrm{log}U$ is nearly constant in the FeLoBALQ sample, and the distance of the outflow from the central SMBH (R) is inversely proportional to U^1/2, both for a fixed density. Dividing the BAL thickness by its distance from the center, we obtain the volume filling factor ΔR/R ∝ U^1.5, and we find a slope of ∼1.5 in the left panel of Figure 7. We observe a scatter around that line because of the range of $\mathrm{log}{N}_{H}-\mathrm{log}U$ and $\mathrm{log}n$ for the FeLoBALs in our sample (Figure 4, Choi et al. 2022). There is also a range in photoionizing flux Q, which we assume to be proportional to L_Bol. This parameter enters through U =Q/4π R² nc. Therefore, larger values of $\mathrm{log}{N}_{H}-\mathrm{log}U$ (thicker outflows), smaller density, or smaller $\mathrm{log}{L}_{\mathrm{bol}}$ correspond to a larger value of $\mathrm{log}{\rm{\Delta }}R/R$.

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The distribution of $\mathrm{log}{\rm{\Delta }}R/R$ is not uniform across logR. At large radii, corresponding to $\mathrm{log}U\lesssim -1$, the volume filling factors mostly range between −6 to −4. These values are similar to those reported in the literature for samples of high-ionization BAL quasars (e.g., Gabel et al. 2006; Hamann et al. 2011, 2013). In contrast, the outflows that are found at $\mathrm{log}R\lesssim 1$ have a very wide range of $\mathrm{log}{\rm{\Delta }}R/R$, ranging from −6 to nearly almost zero. These are mostly the special types of BALs that were identified in Choi et al. (2022), including the overlapping trough and loitering BALs.

The analysis for the full sample shows significant differences in $\mathrm{log}{\rm{\Delta }}R/R$ as a function of radius. This result suggests that BAL winds may favor different models at different radii (Choi et al. 2022; K. M. Leighly et al. 2022, in preparation). The compact winds at $\mathrm{log}R\lesssim 1$ [pc] showed a wide range of $\mathrm{log}{\rm{\Delta }}R/R$ that agrees with the predictions of the various BAL physical models that explain either thin shell-like outflows (small volume filling factor) or stream-like outflows (large volume filling factor). On the other hand, the properties of distant BAL winds only favor the physical model with a thin pancake-like BAL geometry. Specifically, Faucher-Giguère et al. (2012) proposed that FeLoBALs with large $\mathrm{log}R\gtrsim 3$ [pc] and small $\mathrm{log}{\rm{\Delta }}R/R\sim -5$ are formed by "cloud crushing" where the ambient ISM is shocked by the supersonic energy-conserving quasar outflow and FeLoBALs are formed in situ at kiloparsec scales rather than formed near the accretion disk. In addition, in order for distant BALs to have large filling factors, the BAL clouds would need to have large physical radial thicknesses proportional to their distances from the central engine (ΔR ≳ 10 pc). Maintaining such a large structure is physically challenging due to cloud destructive processes (e.g., Proga & Waters 2015).

4.3.2. The z < 1 FeLoBALQs and the E1 Dependence

In this section, we discuss the relationship between the BAL outflow parameters involving the geometry of the outflow and the optical-band emission-line properties. We have emission-line properties only for the 30 object z < 1 subsample, so this discussion only involves that subsample. In particular, we investigated how the parameters that describe the geometrical properties of the outflow depend on the E1 parameter and, by extension, the accretion rate.

We first investigated the physical thickness of the gas ΔR. We showed in Section 3.1 that the ΔR parameter is significantly different for the E1 < 0 and E1 > 0 subsamples (Figure 1 and Table 1); the median log thickness is about 1 dex larger for the E1 < 0 objects. This result arises because although there is no statistical difference in logU between the E1 < 0 and E1 > 0, there is a tendency for E1 < 0 objects to have larger logU and therefore thicker outflows. The thickness is also anticorrelated with both R_FeII and the E1 parameter (Figure 3) for the same reason.

We next considered the volume filling factor for the low-redshift subsample. We found that E1 < 0 objects have significantly larger volume filling factors than E1 > 0 objects because of the significant difference in thickness $\mathrm{log}{\rm{\Delta }}R$ but also that E1 < 0 objects tend to have larger logU and therefore smaller logR, i.e., to be located closer to the central engine.

Because we have black hole mass and accretion rate estimates (Leighly et al. 2022), we can estimate the number of spherical clouds required to completely cover the continuum-emission region. This parameter is useful to visualize the BAL absorption region in the quasar. The first ingredient in this computation is the size of the emission region R₂₈₀₀. We calculated the size of the 2800 Å continuum-emission region using the procedure described in Section 6.1 of Leighly et al. (2019b). To summarize, we used a simple sum-of-blackbodies accretion disk model (Frank et al. 2002) and assigned the 2800 Å radius to be the location where the radially weighted brightness dropped by a factor of e from the peak value. Among the objects in this sample, this parameter spans a rather small range of values with 90% of the objects having $-2.9\lt \mathrm{log}{R}_{2800}\lt -2.3$ [pc] (Figure 8, Leighly et al. 2022). This is likely a consequence of the T⁴ dependence of the accretion disk.

The second consideration is the wide range of angular diameters that the continuum-emission region will subtend at the location of the outflows. For example, the continuum-emission region will subtend an angular diameter that is 1000 times larger to a wind located at 1 pc than to one located at 1000 pc. Folding in the small difference in size of the continuum-emission region we found that the largest angular diameter is presented to the higher-velocity component of SDSS J1125+0029 at 18 arcmin ¹⁵ , and the smallest is the higher-velocity component of SDSS J1044+3656 at 4.5 ×10⁻⁴ arcmin.

The final ingredient is the transverse size of the absorber. This parameter cannot be measured directly from these data as absorption is a line-of-sight measurement. Instead, we make the simplifying order-of-magnitude assumption that the absorbing gas occurs in clouds, and the clouds are approximately spherical. Making this assumption yields a transverse size that is equal to ΔR, the thickness of the absorbing gas.

Employing these three ingredients (the size of the continuum-emission region R₂₈₀₀, the angular diameter that the continuum-emission region will subtend at the location of the outflow, and the transverse size of the absorbing cloud), we can determine the number of clouds required to cover the continuum-emission region. For example, if the angular size of the cloud from the perspective of an observer located at the continuum-emission region is the same as the angular size of the continuum-emission region at the location of the cloud, then only one cloud is required to cover the continuum-emission region. ¹⁶ Conversely, if the angular size of the cloud from the perspective of an observer located at the continuum-emission region is much smaller than the angular size of the continuum-emission region from the perspective of a viewer located at the outflow, then many clouds are required. The number of clouds is the ratio of the area of the continuum-emission region and the transverse area of the cloud. For this order-of-magnitude computation, we assume that the continuum-emission region is viewed face on.

The resulting number of clouds spans a huge range for the sample. Ninety percent of the values fall between 0.016 (which means that the inferred size of the cloud is about 60 times larger than the emission region) to ∼1000, a factor of more than 60,000. Moreover, there is a significant difference between E1 < 0 and E1 > 0 objects (Figure 1; Table 1); the median values of the log of the number of clouds are 0.3 and 2.2, respectively.

The E1 > 0 outflows located close to the central engine (lower right in Figure 8) require a large number of clouds (100–1000 s) to cover the continuum-emission region. Their outflows are characterized by a lower ionization parameter (typically less than −1.5) compared with the E1 < 0 objects ($\mathrm{log}U\sim 0$), and therefore the physical thickness of the outflow is smaller in these objects. Physically, this scenario suggests that the outflow is a fine mist of cloudlets.

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Objects in which the number of clouds is less than 1 are split between those at large distances from the central engine ($\mathrm{log}R/{R}_{{\rm{D}}}\gt 2$) and those at small distances ($\mathrm{log}R/{R}_{{\rm{D}}}\lt 2$). In these objects, the angular size of the BAL cloud structure is comparable or larger than the projected angular size of the continuum emission region. One possibility is that our simplifying estimate that the clouds are approximately spherical is wrong. For example, the number of clouds required to cover the continuum emission region could be much larger if each cloud were long and needle-like, with the long axis pointed toward the central engine. This scenario might be somewhat reasonable physically if the clouds are confined magnetically along field lines that are bent radially by radiation pressure (e.g., de Kool & Begelman 1995) or sheared by radiation pressure.

Another possibility is that the nature of the partial covering is different in some of the objects. The SimBAL model includes a power-law partial covering parameter loga that may parameterize a mist of clouds uniformly covering the continuum emission region (see Leighly et al. 2019b, and references therein for discussion and visualization). The presence of power-law partial covering does not preclude the presence of step-function partial covering as well. The step-function partial covering can be understood as a partial occultation of the continuum emission region. There is some evidence for the presence of step-function partial covering in objects lying in the lower left corner of Figure 8. In several of these objects, the SimBAL models required that a portion of the continuum and/or emission lines be unabsorbed by the outflow (Section 6.4, Figure 17 Choi et al. 2022). Physically, this result might be expected when the continuum emission region has a large angular size from the perspective of the absorber, i.e., an absorber with small $\mathrm{log}R/{R}_{{\rm{d}}}$. It may mean that the outflow is not a mist of clouds uniformly covering the source, but a distribution of nearly continuous gas.

Outflows in quasars are also seen in ionized emission lines. For example, kiloparsec-scale outflows observed in emission lines such as [O iii] are known to be common among luminous active galactic nuclei (e.g., Harrison et al. 2014; Bischetti et al. 2017; Vayner et al. 2021). More compact ionized emission-line outflows have been resolved in nearby objects; for example, ionized gas outflows have been found 0.1–3 kpc from the nucleus (Revalski et al. 2021). It is possible that ionized emission-line outflows and BAL outflows are related. The outflowing broad-absorption-line gas is photoionized, and therefore it must produce line emission. In particular, [O iii]λ5007 line emission is an important coolant in photoionized gas (e.g., Osterbrock & Ferland 2006). It is possible that in some objects the same gas produces absorption lines along the line of sight as well as emission lines from all lines of sight.

There are several fundamental problems that make finding a connection difficult. While absorption is a line-of-sight effect, an emission line is an aggregate of many lines of sight, so that emission from gas not associated with the BAL would be included in any observed line emission. In other words, the covering fraction of the emitting gas needs not be the same as that of the absorbing gas. [O iii] emission is observed to have a very large range of equivalent widths (6–84 Å; Shen et al. 2011) potentially originating in a range of gas covering fractions (Baskin & Laor 2005; Ludwig et al. 2009; Stern & Laor 2012). Moreover, line emissivity depends on density squared below the critical density, i.e., n_cr = 6.8 × 10⁵ cm⁻² for [O iii] λ5007] (Osterbrock & Ferland 2006) and on the density above the critical density. Thus, the line emission might not be seen against the continuum if the density is too low. Finally, if the absorption lines are broad, then the line emission may be distributed over a large range of velocities, and the line may be too broad to be seen against the continuum.

We do not have information about the extent of the line emission for our objects. However, it is interesting to see if there is a correspondence or relationship between the predicted [O iii] emission from the BAL gas and the observed [O iii] emission. Xu et al. (2020) tackled this problem using a sample of seven z ∼ 2 quasars. Those objects were chosen to have r-band magnitudes ≲ 18.8 and deep Si iv λ λ1393.76, 1402.77 troughs. They found evidence for a link between the [O iii] emission and the BAL absorption. However, they presented a conceptual error: they suggested that [O iii] emission is suppressed at high densities. In fact, [O iii] emission always increases with density. Rather than a decrease or suppression of [O iii] at high densities, the property that decreases is the ratio of the [O iii] emission with respect to lines with higher critical densities. This physics is the basis of the Si iii]λ1893/C iii]λ1909 density diagnostic used in the near-UV (e.g., Leighly2004).

We used Cloudy to predict the [O iii] line emission from the outflowing gas in each of the 36 BAL components of the 30 z < 1 objects. Each component is characterized by a single ionization parameter and density, but is generally split into multiple bins with different column densities (Leighly et al. 2018; Choi et al. 2022). The [O iii] emission was computed for each bin and then summed. The local covering fraction loga for the BAL outflows was not taken into account in the computation of the [O iii] flux, as it is not clear how it would manifest in the observations of line emission. The luminosity of the [O iii] emission was then computed for each component assuming a global covering fraction of 10% for the line-emitting gas.

We expect the luminosity of the predicted [O iii] could depend on several parameters. The [O iii] flux density should be larger for higher densities. It should be larger for higher ionization parameters as a consequence of the larger column density required to include the hydrogen ionization front in FeLoBAL outflows. Both of these conditions are met in outflows closer to the central engine. However, the volume included increases as R² for the absorbing material for a fixed emission-line-region covering fraction. As ${R}^{2}\propto \tfrac{1}{{nU}}$, the effects cancel out, leaving the Cloudy L _[OIII] uncorrelated with n, U, or R (Figure 2).

The left panel of Figure 9 shows the Cloudy predicted [O iii] luminosity as a function of the observed [O iii] luminosity. All of the predicted line emission from objects with multiple BAL outflow components are plotted with respect to the single observed [O iii] luminosity. There is clearly no relationship between these two luminosities. Moreover, while both the observed and predicted [O iii] luminosities each span about 1.7 dex, the ratio of the two spans almost 2.7 dex. In other words, the difference between the observed and predicted [O iii] emission is not subtle.

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It is possible that the observed and predicted [O iii] luminosities could be reconciled if the global covering fraction of outflowing line-emitting gas were not constant for all outflows or for all emission-line regions. For example, the location of the outflow in this sample spans 3 orders of magnitude, and it is conceivable that the emission-line covering fraction is different in the vicinity of the torus (R ∼ 1 pc) compared with on galaxy scales ($\mathrm{log}R\sim 3$ [pc]).

We parameterized the profound difference between the observed and predicted [O iii] luminosities by defining the covering fraction correction factor. The covering fraction correction factor is the difference between the log of the Cloudy predicted [O iii] luminosity and the log of the observed [O iii] luminosity. The covering fraction correction factor measures how much larger or smaller than the assumed value of 0.1 that the covering fraction needs to be to reconcile the observed and predicted [O iii] luminosities. In Section 3.1 we showed that the covering fraction correction factor is significantly different for E1 < 0 versus E1 > 0 objects. The median values were 0.6 (1.5) for the E1 < 0 (E1 > 0) objects, respectively, implying that, on average, the emission-line region covering fraction needs to be 4 (32) times smaller than 0.1.

We explored the possibility that the covering fraction depends on other parameters using multiple regression analysis. We considered seven independent variables, and our reasons for choosing these parameters follows. We included logR for the reasons outlined above. We also considered the Seyfert type using the E1 parameter, and the log of the estimated bolometric luminosity because there is evidence for a reduction in the [O iii] emission at larger luminosities (Baskin & Laor 2005). Objects with a high Eddington ratio might produce more powerful winds and thereby evacuate a larger fraction of their reservoir of gas, so we also consider $\mathrm{log}{L}_{\mathrm{Bol}}/{L}_{\mathrm{Edd}}$. There is evidence that the BAL partial covering fraction loga is local (Leighly et al. 2019b), but it is possible there are global trends as well. In any particular outflow component, loga is a function of the velocity. For this experiment, we chose the representative loga to be the one in the bin with the deepest ground-state Fe ii absorption. If the velocity of the outflow is very large, then the resulting line may be very broad and blend with the continuum, so we also considered the opacity-weighted v_off. Finally, the observed [O iii] velocity offset might reveal a connection between the outflow and the observed emission.

The result forms a multiple regression problem with the seven independent variables listed above. As in Section 3.2, we accounted for the multiple components in five of the objects by running the regression analysis for all 48 combinations. We used mlinmix_err (Kelly 2007), which accounts for measurement errors in both the dependent variable (the covering fraction) and the independent variables (the design matrix) using the Pearson correlation coefficient. To determine which independent variables can best reproduce the variance in the covering fraction, the multiple regression procedure was iterated, each time removing the independent variable with the largest p value until all of the remaining variables showed a p value no larger than the cutoff, which was chosen to be 0.025.

A statistically significant correlation was found between the E1 parameter and the covering fraction correction factor for all 48 combinations. The next most significant parameter was logR (39), followed by the velocity offset of [O iii] (23). As more than half of the combinations found significant regression with the E1 parameter and logR, we proceeded to extract the regression parameters for these two parameters and all combinations.

The right panel of Figure 9 shows the results of the regression analysis between the covering fraction correction factor and the independent parameters E1 and logR. The Cloudy covering fraction was adjusted using the best-fitting regression parameters. Points are shown for each of the 48 combinations. The relationship between the observed and SimBAL-predicted [O iii] emission is now linear, although considerable scatter remains.

We next examine how large the log covering fraction correction factors are and how they depend on the regression variables. The results from the regression are seen in Figure 10. The points are plotted for all 48 combinations, and the gray shaded region shows the 90% confidence regions from the mlinmix_err procedure. The left panel shows the results in three dimensions, while the right two panels show the two-dimensional projections for each of the regression variables. The assumed emission-line covering fraction was 0.1; so, for example, a log covering fraction correction value of 1 would imply that the covering fraction of 0.01 is needed to reconcile the observed and predicted [O iii] values. Many of the E1 > 0 objects have large covering fraction correction factors (1.5–2), which would seem to imply that the emission-line-gas covering fraction needs to be be very small (0.001–0.003) in order to reconcile the observed and predicted [O iii] emission.

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In contrast, several of the loitering outflow objects (E1 < 0 with a low-velocity and compact outflow; Choi et al. 2022, Section 6.5, Figure 18) show very low covering fraction correction factors near zero, which means that the observed [O iii] emission is consistent with being produced in the outflow. Figure 11 compares the [O iii] emission profile and the absorption opacity profiles (Choi et al. 2022) for the six objects with $\mathrm{log}R\lt 1$ and log covering fraction correction less than 0.5. In these objects, the data are roughly consistent with the line emission and absorption being produced in the same gas. The profiles are not identical, but there are consistent trends: objects with broader [O iii] emission lines show broader absorption profiles. For example, in SDSS J1128+0113, the [O iii] line has a velocity width of w₈₀ = 830 km s⁻¹ (Leighly et al. 2022), and the Mg ii absorption line has a velocity width of 3000 km s⁻¹ (Choi et al. 2022). In contrast, in SDSS J0916+4534, the [O iii] line has a velocity width of w₈₀ =460 km s⁻¹ (Leighly et al. 2022), and the Mg ii absorption line has a velocity width of 500 km s⁻¹ (Choi et al. 2022). SDSS J1321+5617 is particularly interesting: both the [O iii] emission-line and absorption-line optical-depth profiles are narrow with a blue wing.

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However, there is a flaw in this analysis. We showed in Section 4.1 that the BAL outflow velocity in some objects was much smaller than the local Keplerian velocity (Figure 5). The [O iii] emission-line width in the same objects is also much less than the Keplerian velocity, by factors of 5–24. While we could explain the small outflow velocity if the orientation is directly along the line of sight, the same argument does not work for the emission lines, as they are composed of emission from all lines of sight.

The correlation analysis discussed in Section 3.2 indicates significant correlations between the BAL outflow velocity offset and the [O iii] velocity offset and width (Figure 3). Figure 12 (left and middle) explores these relationships. In E1 > 0 objects, the [O iii] emission line is sometimes very small and difficult to discern amid the sometimes strong and broad Fe ii emission; therefore, points with the smallest symbols are less robustly measured. The correlation between the BAL outflow velocity and [O iii] velocity offset (p = 1.9 × 10⁻⁴) and the anticorrelation between the BAL outflow velocity and [O iii] velocity width (p = 2.7 × 10⁻⁴) are apparent.

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Our regression analysis above explored the relationships between the observed and predicted [O iii] luminosities for seven selected measurements. We also looked for relationships with other measured parameters via correlation analysis between the covering fraction correction factor and the SimBAL parameters (Figure 2) and the optical parameters (Figure 3). Interestingly, although the predicted [O iii] emission must increase with density, we found no correlation between the covering fraction correction factor and the BAL $\mathrm{log}n$. This is explained by the fact that lower density BAL gas is found at larger radii, where, for a fixed global covering fraction of the line-emitting gas, the volume of the emitting gas is larger.

One correlation stands out: the outflow velocity is anticorrelated with the covering fraction correction factor. This anticorrelation is significant (p < 0.025) in 27 of 48 cases using Spearman's rank; it did not appear to be significant in the multiple regression analysis, as mlinmix_err uses Pearson's R. This dependence offers a way to reconcile the large covering-fraction correction factors required by E1 > 0 objects: the [O iii] emitted by the BAL could be so broadened that it blends with the continuum, especially in low signal-to-noise spectra or amidst strong Fe ii emission.

This is the third in a sequence of four papers that discuss the properties of low-redshift FeLoBALQs. Taken together, they build a picture of the properties and physical conditions of the BAL gas and explore links between these properties to the accretion and emission-line properties of the quasars.

This paper combines SimBAL (Choi et al. 2022) and optical emission-line analysis (Leighly et al. 2022). The most significant result is the discovery that the E1 parameter division discovered by Leighly et al. (2022) carries over to the outflow properties (Section 4.1). Among the E1 > 0 high-accretion-rate objects, the outflow velocity decreases with the distance from the central engine (Figure 4). This is consistent with the expectation of radiative line driving acceleration. Among the E1 < 0 low-accretion-rate objects, the outflow velocity increases with the distance from the central engine.

We confirmed the relationship between the outflow velocity and Eddington ratio previously reported by Ganguly et al. (2007); i.e., at a particular Eddington ratio, a range of velocities are observed up to a maximum velocity, which is itself a function of the Eddington ratio (Section 4.2). Using the physical properties of the outflows obtained from the SimBAL analysis, we investigated whether the scatter in the velocity at a particular Eddington ratio could be a consequence of a scatter in the force multiplier or the launch radius. We found that among E1 > 0 objects, the scatter in the velocity could plausibly be attributed to a scatter in the launch radius, while among E1 < 0 objects, the scatter in the velocity could be attributed to scatter in the force multiplier.

We investigated the volume filling factor of the outflows, both for the full sample (Section 4.3.1) and for the z < 1 subsample (Section 4.3.2). The full sample reveals a large range of log volume filling factors, from −6 to −1. At large distances from the central engine, the log volume filling factors were less than −3, similar to those inferred in HiBALQs (e.g., Gabel et al. 2006; Hamann et al. 2011, 2013). Closer to the black hole, for $\mathrm{log}R\lesssim 1$, the full range of volume filling factors was found. We also found that the special BAL classes identified by Choi et al. (2022; the loitering outflows and the other overlapping trough objects) are also divided by their E1 parameter and, by extension, their Eddington ratio. The loitering outflow objects have E1 < 0 and low Eddington ratios, while other overlapping trough objects have E1 > 0 and high Eddington ratios. Moreover, although both special types of FeLoBALQ outflows are compact, with typical locations of ∼10 pc from the central engine, a dramatically different number of assumed spherical clouds would be required to occult the continuum emission region (Figure 8). For the loitering outflows, a single cloud (or continuous outflow) would be sufficient. For the other overlapping trough absorbers, 100–1000 s of clouds would be required. Of course, the outflow may not have the structure of discrete spherical clouds; the point is that the differing physical conditions of these two categories of outflows tell us that the structure of the outflows is dramatically different.

We also investigated the relationship between the observed [O iii] emission line and the [O iii] emission predicted to be produced by the BAL outflow gas (Section 4.4). This analysis first underlines the very large range of equivalent widths observed in this sample. The observed [O iii] luminosity and the estimated bolometric luminosity both span about 1.6 dex, but the ratio of the two luminosities spans 2.1 dex. We found that, in order to reconcile the observed and predicted log [O iii] luminosities, the emission-line-gas global covering fraction may depend on the E1 parameter and the location of the outflow logR (Figure 9). At the same time, the covering fraction correction factor (defined as the difference between the observed and predicted [O iii] luminosity) was observed to be correlated with the outflow velocity, which may imply that the predicted strong [O iii] emission in E1 > 0 objects is broadened and hidden under their typically strong Fe ii emission (Figure 12). Most intriguing were the six E1 < 0 objects with the lowest covering-fraction correction factors, whose [O iii] profiles resembled the BAL absorption profiles (Figure 11).

The final paper in this series of four papers (K. M. Leighly et al. in preparation) includes an analysis of the broadband optical/IR properties and discusses the potential implications for quasar evolution scenarios.

This sample was limited to objects with redshifts less than 1.63 for the full 50 object sample from Choi et al. (2022) and less than 1.0 for the optical emission-line analysis from Leighly et al. (2022). FeLoBALQs can be observed up to z ∼ 3 in ground-based optical-band spectra. Because the SDSS is a flux-limited survey, we generally expect higher-redshift objects to be more luminous; a sample currently being analyzed has bolometric luminosities about 1 order of magnitude larger than the z < 1 sample. Assuming a similar distribution of Eddington ratios, the higher-redshift objects will have larger black hole masses and therefore a softer (more UV-dominant) SED. The softer SED could influence the properties of the BAL outflows in two ways: 1) the ions present in the outflowing gas would be different, e.g., tend toward lower ionization species (e.g., Leighly et al. 2007), and 2) the velocities might be larger, as the SED will produce a relatively larger number of UV photons that can transfer momentum by resonance scattering in the outflowing gas. Preliminary analysis of a higher-redshift sample that is being observed in the near-infrared shows evidence of higher outflow velocities (Voelker et al. 2021). Another interesting feature of the high-redshift sample is that the larger redshifts provide access to the high-ionization lines (e.g., C iv, Si iv) that these objects share with the much more common HiBALQs. Preliminary analysis shows that HiBALQs seem to be very different than FeLoBALQs (Hazlett et al. 2019; Leighly et al. 2019a); in particular, the high ionization lines sometimes have complicated velocity structures and certainly extend to higher velocities. The potential link between the low-ionization line and high-ionization line properties may also prove to be very illuminating.

Our investigation of whether there could be a relationship between the ionized outflows manifest in [O iii] emission and the BAL outflows showed that in most objects such a relationship would be possible only if the emission-line covering fraction is extremely low, or if the [O iii] is broad and blended with the Fe ii emission. An exception was several of the loitering outflow objects, where the emission-line and absorption-line profiles appeared to resemble one another. All of these objects had compact outflows, so the possibility that there is a direct relationship might be able to be tested with spatially resolved observations of the [O iii] emission; unlike many quasars (e.g., Harrison et al. 2014; Bischetti et al. 2017; Vayner et al. 2021), the [O iii] emission in these objects should be unresolved if it does indeed originate in the BAL outflow. Proving a relationship would be very difficult in general. However, we might be able to falsify one. At a redshift of 0.9, the angular scale is about 7.9 kpc per arcsec, so it would be possible to determine whether the [O iii] emission was extended or compact on the scale of a kiloparsec. Many of the SimBAL solutions indicate BAL outflows with size scales much less than 1 kpc. The [O iii] emission should be unresolved in such objects.

Regardless of the details, the analysis presented here has shown that again, key physical properties of the outflows differ as a function of the location in the quasar and as a function of the accretion rate as probed by the E1 parameter. These patterns and differences have broad potential implications. We may finally be able to understand the acceleration mechanisms that operate in quasars. In addition, we may be able to use the accretion properties measured from the emission lines to statistically infer the outflow properties in objects that do not show outflows along the line of sight. These ambitious goals will require much more work and analysis of many more objects, but at least we have identified a promising path forward.