The Energetics of Launching the Most Powerful Jets in Quasars: A Study of 3C 82

The BELs in Figure 1 are of particular interest because they provide clues to the nature of the gas near the quasar and the luminosity of the quasar. We provide a standard three-Gaussian component decomposition of the BELs, the broad component BC, (also called the intermediate broad line or IBL; Brotherton et al. 1994), the redshifted VBC (very broad component, following Sulentic et al. 2000), and a blueshifted excess (Brotherton 1996; Marziani et al. 1996; Sulentic et al. 2000). In summary, there are three broad components defined by their velocity relative to the quasar rest frame. The redshifted BC is the broadest of the three components. It will be abbreviated in the following with a prefix "red" as "the red VBC." The blueshifted broad excess is designated BLUE. The component that is close to the quasar rest frame, the IBL/BC, will have no prefix to the BC and just be called "BC" in the following. These designations are used in Table 1 to describe the three component decompositions of the broad lines.

The decompositions are shown in Figure 2, after continuum subtraction. The BC are the black Gaussian profiles in Figure 2. The red VBC is shown in red and BLUE is shown in blue. Only the sum of the three components is shown for both N v λ1240 (contaminating the red wing of Lyα) and He ii λ1640 (merging with red wing of C iv and creating the appearance of a flat-topped profile). In the ${\chi }_{\nu }^{2}$ minimization fit, the decomposition was carried out in a consistent way, with similar initial guesses of the line shift- and -width values for the three components in C iv, He ii, Lyα, and N v, derived from the decomposition of the C iv profile. Their relative intensity was allowed to vary freely (i.e., the relative intensity of the three components is not constrained by the C iv decomposition). With this approach it was possible to obtain a minimum ${\chi }_{\nu }^{2}$ fit that leaves no significant residuals in the decomposition of the main blends (see Figure 2) for both C iv + He ii and Lyα + N v fit. The Lyα + N v blend is especially problematic, as the blue side of the line is contaminated by the Lyα forest. The absorptions may have eaten away most of the flux of the BLUE, whose intensity is therefore especially uncertain. In quasars with broader emission lines (such as the Population B sources described below), the C iv + He ii blend flat-top appearance (e.g., Fine et al. 2010) can be explained by the blending of the C iv red wing and He ii blue wing without the assumption of any additional emission (Marziani et al. 2010). The flat top implies that the He ii BC is weak with respect to the He ii BLUE and red VBC: it is not possible to use a scaled C iv profile to model He ii. A similar condition is apparently occurring also for N v.

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The C iii] complex is affected by telluric absorption. We used the standard star, HD 84937, which was observed on the same night, to correct for this. We could not fully correct for the A-band absorption resulting in the dip on the red side of the complex (indicated in the bottom-right-hand panel of Figure 2). Regions affected by absorption lines are avoided in the fitting. The fit in Figure 2 was obtained by restricting the fitting range to the intervals 1795–1845, 1849–1950, 1965–1980, and 1990–2010 Å. Table 1 labels Al iii with insufficient S/N for an accurate decomposition. We report the total luminosity but note that the Al iii doublet is potentially affected by telluric residuals. The S/N is adequate to expose the complexity of the ∼1900 Å blend, and to reveal C iii] and Si iii] profiles typical of quasars with very broad low-ionization emission lines (see the discussion on Population B sources, below). There is a prominent BC and red VBC, and no BLUE detected in these lines.

These line decompositions (shown in Figure 2) are described quantitatively in Table 1. The table is organized as follows. The line designation is defined in the first two columns. The next three columns define the properties of the Gaussian fit to the red VBC, the shift of the Gaussian peak relative to the vacuum wavelength in km s⁻¹, followed by the FWHM and line luminosity. Columns (6)–(8) are the same for the BLUE. The BC FWHM and luminosity are columns (9) and (10). The last column is the total luminosity of the BEL.

The broad-line decomposition has a physical context. In order to explore this, we note that there are two different, useful, ways of segregating the quasar population. One is to split the population into radio-quiet and radio-loud quasars and the other is to split the quasars into Population A or B (Sulentic et al. 2000): Population A (Hβ FWHM < 4000 km s⁻¹) and Population B (defined by Hβ FWHM > 4000 km s⁻¹). The C iv BLUE has been found to be dominant relative to the C iv red VBC in radio-quiet quasars. It tends to be less prominent in radio-loud quasars and can be completely absent (Richards et al. 2002; Punsly 2010). The trend was shown to be deeper than just the radio-loud—radio-quiet dichotomy, but related more to the Eddington accretion rate onto the central supermassive black hole. The Population B quasars include most of the radio-loud quasars and typically have very large black hole masses and low Eddington ratios: ${R}_{\mathrm{Edd}}\equiv {L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}\sim 0.01\mbox{--}0.1$, where L_bol is the thermal bolometric luminosity of the accretion flow and ${L}_{\mathrm{Edd}}\,=1.26({M}_{\mathrm{bh}}/{M}_{\odot })\times {10}^{38}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is the Eddington luminosity expressed in terms of the central supermassive black hole mass, M_bh (Sulentic et al. 2007). The Population A quasars are generally radio quiet and have typically higher Eddington ratios than Population B quasars (Sulentic et al. 2007). Population B C iv profiles tend to have less blue excess at their bases than Population A quasars, but this distinction is less pronounced for high-luminosity Population B quasars (Richards et al. 2002; Sulentic et al. 2017). These patterns are a strong indication that the BLUE is related to wind driven by radiative luminosity.

The physical insight provided by the Population A/B discussion, above, sheds light on the nature of the central engine of 3C 82. In radio-loud quasars, even though BLUE of C iv is often detectable, it is usually significantly weaker than the red VBC (Punsly 2010). Yet, the BLUE of the C iv broad line in 3C 82 is 2.3 times as luminous as the red VBC. This is very extreme for a radio-loud quasar. The implication is that 3C 82 has a high L_bol for a radio-loud quasar, and this is related to the strong BLUE. In Section 8, we interpret this in terms of an outflowing high-ionization wind that is typically found in high-Eddington-rate quasars.

We wish to estimate L_bol in a manner that does not include reprocessed radiation in the infrared from molecular clouds that are far from the active nucleus. This would be double counting the thermal accretion emission that is reprocessed at midlatitudes (Davis & Laor 2011). The most direct method is to use the UV continuum as a surrogate for L_bol. From the spectrum in Figure 1 and the formula expressed in terms of quasar cosmological rest-frame wavelength, λ_e, and spectral luminosity, L_λe, from Punsly et al. (2016),

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{bol}} & \approx & (4.0\pm 0.7){\lambda }_{e}{L}_{{\lambda }_{e}}({\lambda }_{e}=1350\,\mathring{\rm A} )\\ & \approx & 4.9\pm 0.9\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 1 }$$

The bolometric correction was estimated from a comparison to Hubble Space Telescope (HST) composite spectra of quasars with L_bol ≈ × 10⁴⁶ erg s⁻¹ (Laor et al. 1997; Zheng et al. 1997; Telfer et al. 2002). The estimate is 20% lower if one uses the lower-S/N spectra from 1988 and 2002. One can also use the luminosity of the C iv BEL in Table 1 as a more indirect means of estimation (Punsly et al. 2016),

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}=(107\pm 22)L({\rm{C}}\,{\rm{\small{IV}}})\approx 5.4\pm 1.1\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 2 }$$

The uncertainty in Equations (1) and (2) arises from the uncertainty in the HST composite continuum level and the uncertainty in L(C iv) and L_λe(λ_e = 1350 Å), respectively, added in quadrature (Telfer et al. 2002; Punsly et al. 2018).

One must differentiate between quantities measured in the plasmoid frame of reference and those measured in the observer's frame of reference. The physics is evaluated in the plasma rest frame. Then, the results are transformed to the observer's frame for comparison with observation. The underlying power law for the flux density is defined as ${S}_{\nu }(\nu ={\nu }_{o})=S{\nu }_{o}^{-\alpha }$, where S is a constant. Observed quantities will be designated with a subscript, "o," in the following expressions. The observed frequency is related to the emitted frequency, ν_e, by ν_o = δν_e/(1 + z). The bulk flow Doppler factor of the plasmoid, δ,

$$\begin{eqnarray}&&\delta =\displaystyle \frac{{\gamma }^{-1}}{1-\beta \cos \theta },{\gamma }^{-2}=1-{\beta }^{2},\end{eqnarray} \tag{ 3 }$$

where β is the normalized three-velocity of the bulk motion and θ is the angle of the motion to the LOS to the observer. The SSA attenuation coefficient is computed in the plasma rest frame (Ginzburg & Syrovatskii 1969),

$$\begin{eqnarray}\begin{array}{rcl}\mu ({\nu }_{e}) & = & \overline{g(n)}\displaystyle \frac{{e}^{3}}{2\pi {m}_{e}}{N}_{{\rm{\Gamma }}}{\left({m}_{e}{c}^{2}\right)}^{2\alpha }\\ & & \times \,{\left(\displaystyle \frac{3e}{2\pi {m}_{e}^{3}{c}^{5}}\right)}^{\tfrac{1+2\alpha }{2}}{\left(B\right)}^{(1.5+\alpha )}{\left({\nu }_{e}\right)}^{-(2.5+\alpha )},\end{array}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\overline{g(n)}=\displaystyle \frac{\sqrt{3\pi }}{8}\displaystyle \frac{\overline{{\rm{\Gamma }}}[(3n+22)/12]\overline{{\rm{\Gamma }}}[(3n+2)/12]\overline{{\rm{\Gamma }}}[(n+6)/4]}{\overline{{\rm{\Gamma }}}[(n+8)/4]},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&N={\int }_{{{\rm{\Gamma }}}_{\min }}^{{{\rm{\Gamma }}}_{\max }}{N}_{{\rm{\Gamma }}}{{\rm{\Gamma }}}^{-n}\,d{\rm{\Gamma }},n=2\alpha +1,\end{eqnarray} \tag{ 6 }$$

where Γ is the ratio of lepton energy to rest mass energy m_ec², $\overline{g(n)}$ is the Gaunt factor averaged over the angle, and $\overline{{\rm{\Gamma }}}$ is the gamma function. B is the magnitude of the total magnetic field. The low-energy cutoff, ${E}_{\min }={{\rm{\Gamma }}}_{\min }{m}_{e}{c}^{2}$, is constrained loosely by the data in Figure 4. The fact that 3C 82 is very luminous at ν = 200 MHz means that the lepton energy distribution is not cut off near this frequency. This is not a very stringent bound. Note that the SSA opacity in the observer's frame, μ(ν_o), is obtained by substituting ${\nu }_{e}=(1+z){\nu }_{o}/\delta$ into Equation (2). In the following analysis, δ ≈ 1, so ${\nu }_{e}\approx \nu =(1+z){\nu }_{o}$. Thus, we will drop the subscript "e" in order to streamline the notation i.e., the plasmoid frame of reference and the quasar cosmological frame of reference are the same up to negligible relativistic corrections.

A simple solution to the radiative transfer equation occurs in the homogeneous approximation (Ginzburg & Syrovatskii 1965; van der Laan 1966)

$$\begin{eqnarray}\begin{array}{rcl}{S}_{{\nu }_{\unicode{x000F8}}} & = & \displaystyle \frac{{S}_{o}{\nu }_{o}^{-\alpha }}{\tau ({\nu }_{o})}\times \left(1-{e}^{-\tau ({\nu }_{o})}\right),\\ \tau ({\nu }_{o}) & \equiv & \mu ({\nu }_{o})L,\ \tau ({\nu }_{o})=\overline{\tau }{\nu }_{o}^{(-2.5+\alpha )},\end{array}\end{eqnarray} \tag{ 7 }$$

where τ(ν) is the SSA opacity, L is the path length in the rest frame of the plasma, S_o is a normalization factor, and $\overline{\tau }$ is a constant. There are three unknowns in Equation (7), $\overline{\tau }$, α and S_o. These are effectively three constraints on the following theoretical model that can be estimated from the observational data. These three constraints are used to establish the uniqueness and the existence of physical solutions in Sections 5 and 6.

To connect the parametric spectrum given by Equation (7) to a physical model, an expression for the synchrotron emissivity (Tucker 1975) is required:

$$\begin{eqnarray}&&{j}_{\nu }=1.7\times {10}^{-21}[4\pi {N}_{{\rm{\Gamma }}}]a(n){B}^{(1+\alpha )}{\left(\displaystyle \frac{4\times {10}^{6}}{\nu }\right)}^{\alpha },\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&a(n)=\displaystyle \frac{\left({2}^{\tfrac{n-1}{2}}\sqrt{3}\right)\overline{{\rm{\Gamma }}}\left(\tfrac{3n-1}{12}\right)\overline{{\rm{\Gamma }}}\left(\tfrac{3n+19}{12}\right)\overline{{\rm{\Gamma }}}\left(\tfrac{n+5}{4}\right)}{8\sqrt{\pi }(n+1)\overline{{\rm{\Gamma }}}\left(\tfrac{n+7}{4}\right)},\end{eqnarray} \tag{ 9 }$$

where the coefficient a(n) separates the pure dependence on n (Ginzburg & Syrovatskii 1965). One can transform this to the observed flux density, S(ν_o), in the optically thin region of the spectrum (i.e., our VLA and JVLA measurements in the case of 3C 82) using the relativistic transformation relations from Lind & Blandford (1985),

$$\begin{eqnarray}&&S({\nu }_{o})=\displaystyle \frac{{\delta }^{(3+\alpha )}}{4\pi {D}_{L}^{2}}\int {j}_{\nu }^{{\prime} }{dV}^{\prime} ,\end{eqnarray} \tag{ 10 }$$

where D_L is the luminosity distance and in this expression, the primed frame is the rest frame of the plasma. Equations (4)–(10) are used in Section 5 to fit the data in Figure 4.

The energy content is separated into two parts. The first is the kinetic energy of the protons, $\mathrm{KE}(\mathrm{proton})$,

$$\begin{eqnarray}&&\mathrm{KE}(\mathrm{protonic})=(\gamma -1){{Mc}}^{2},\end{eqnarray} \tag{ 11 }$$

where M is the mass of the plasmoid. The other piece is the lepto-magnetic energy, E(lm), and is composed of the volume integral of the leptonic internal energy density, U_e, and the magnetic field energy density, U_B. The lepto-magnetic energy in a uniform spherical volume is

$$\begin{eqnarray}\begin{array}{rcl}E(\mathrm{lm}) & = & \displaystyle \int ({U}_{B}+{U}_{e})\,{dV}\\ & = & \displaystyle \frac{4}{3}\pi {R}^{3}\left[\displaystyle \frac{{B}^{2}}{8\pi }+{\displaystyle \int }_{{{\rm{\Gamma }}}_{\min }}^{{{\rm{\Gamma }}}_{\max }}({m}_{e}{c}^{2})({N}_{{\rm{\Gamma }}}{E}^{-n+1})d\,E\right].\end{array}\end{eqnarray} \tag{ 12 }$$

It has been argued that the lepto-magnetic energy is often nearly minimized in the hot spots and enveloping lobes of large FR II radio sources (Hardcastle et al. 2004; Croston et al. 2005; Kataoka & Stawarz 2005). However, this condition must be evaluated on a case-by-case basis. The leptons also have a kinetic energy analogous to Equation (11),

$$\begin{eqnarray}&&\mathrm{KE}(\mathrm{leptonic})=(\gamma -1){{ \mathcal N }}_{e}{m}_{e}{c}^{2},\end{eqnarray} \tag{ 13 }$$

where ${{ \mathcal N }}_{e}$ is the total number of leptons in the plasmoid.

In terms of estimating the relevant Doppler factor for the lobes, we have no detection of lobe plasma motion in any radio lobe 10 kpc from the central engine in any active galactic nucleus. We must rely on theoretical arguments such as those based on synchrotron cooling and the spatial evolution of spectral breaks across the radio lobe (Liu et al. 1992; Alexnader & Pooley 1996). The velocity of the diffuse lobe plasma, v_diffuse, is a superposition of hot spot advance speed v_hs, the back flow speed v_bf, and the lateral expansion speed v_exp. It has been deduced that v_hs and v_bf are the largest contributors and v_hs ≈ v_bf, almost cancel (Liu et al. 1992; Alexnader & Pooley 1996). The instantaneous v_hs is a different quantity from the lobe advance speed averaged over the entire jet history, v_adv, used in statistical studies (Scheuer 1995). Estimating v_hs at a given time in a given radio source's history is difficult and necessarily speculative. In order to make an estimate of v_hs requires high-resolution images at multiple frequencies in order to deduce the synchrotron cooling as plasma flows away from the hot spot. It also requires high-resolution X-ray observations that can be used to estimate the enveloping density and temperature. These elements are available for Cygnus A for which v_hs ≈ 0.005c has been estimated (Alexnader & Pooley 1996). This best-understood example can help to constrain 3C 82, which has none of the relevant information. v_hs decelerates as the jet propagates (Alexnader 2006). Cygnus A is an order of magnitude larger than 3C 82, so we expect v_hs to be larger in 3C 82. It is expected that v_hs is at least factor of a few larger than 0.005c and significantly less than the v_adv estimated in Section 7. So, we crudely guess 0.05c < v_hs < 0.1c, with the condition v_diffuse ≪ v_hs (Alexnader & Pooley 1996). Because the rest-frame UV is only mildly variable and the radio core is very weak, we assume a typical nonblazar LOS of 30° (Barthel 1989). Using the expression for δ in Equation (1), the ratio of the Doppler enhancement of the approaching hot spot to that of the receding hot spot from Equation (10) is $1.4\lt {[\delta (\beta ={v}_{\mathrm{hs}}/c)/\delta (\beta =-{v}_{\mathrm{hs}}/c)]}^{4}\lt 2$. By contrast, v_diffuse is a superposition of canceling subrelativistic velocities and ${[\delta (\beta ={v}_{\mathrm{diffuse}}/c)/\delta (\beta =-{v}_{\mathrm{diffuse}}/c)]}^{4}\approx 1$. If we could segregate the hot spot flux density with very high-resolution, high-sensitivity images, the Doppler enhancement would lower the energy in our model of the approaching lobe (because the intrinsic luminosity is less than observed) and vice versa for the receding lobe (this will be shown explicitly in Section 6.5 for our models). But we do not have these images, and we have no basis to make this decomposition. So, initially we choose δ = 1. Then in Section 6.5, we consider two cases in which the entire approaching (receding) lobe has a single velocity, v_lobe = 0.05c (v_lobe = −0.05c) and v_lobe = 0.1c (v_lobe = −0.1c) in order to assess the effects. The total energy of the system is unchanged under this range of properties.

Setting δ = 1 and treating E_max as basically infinite effectively adds two more known quantities, making the situation five unknowns with three constraints. The system is still underdetermined, and we would like to improve this situation. In order to restrict the size of the solution space, we need to constrain E_min. We choose E_min ≈ m_ec². This assumption needs to be checked on a case-by-case basis for internal consistency. We will show in Section 6.7 that this is the preferred value of E_min for various physical reasons in our models of 3C 82.

For the assumed values, δ = 1 and E_min, and recalling that the solutions are insensitive to E_max, one has four unknowns in each lobe, N_Γ, B, R, and α. Yet each fitted SSA power-law model has three fitted constraints, $\overline{\tau }$, α, S_o. Thus, for each lobe there is an infinite one-dimensional set of solutions for each preassigned δ and E_min that results in the same spectral output. First, a power-law fit to the high-frequency optically thin synchrotron tail determined by our Gaussian component fits fixes the power-law parameters in each lobe, S_o and α in Equation (7). An arbitrary B is chosen in each spheroid. Then, N_Γ and the spheroid radius, R, are iteratively varied to produce this fitted S_o and the fitted value of $\overline{\tau }$ in each lobe that provides the fit to the "double-humped" SSA region between 250 and 800 MHz (in the quasar rest frame). Another value of B is chosen, and the process is repeated in order to generate two new values of N_Γ and R that reproduce the spectral fit. The process is repeated until the solution space of B, N_Γ, and R is spanned for each lobe.

We choose the western lobe to be associated with the lowest-frequency spectral peak as deduced above. The volume of the eastern lobe is therefore less. We note that as the spectral index of the synchrotron spectrum steepens, there are more electrons at lower energy, by Equation (6). This creates an increase in energy density that can compensate for the smaller volume in the eastern lobe and help meet constraint 1, the long-term bilateral symmetry in the ejected energy. Similarly, one can choose α as small as possible for the western lobe. This is the strategy for Fit A. In addition, there is little flexibility in the error bars at ν_o = 151 MHz and ν_o = 74 MHz in order to make the SSA opacity as small as possible for the eastern lobe (slightly larger size) and as large as possible for the western lobe (slightly smaller size). Thus, constraint 1, on long-term bilateral symmetry, tends to drive the SSA opacity to the maximum (minimum) possible value in the western (eastern) lobe that is consistent with the ν_o = 74 MHz (ν_o = 151 MHz) lower (upper) error bar in our fits. The result of this strategy is shown in Figure 4 and the upper panel of Figure 5.

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The relevant details are shown as entry I in Table 3 for direct comparison with other fitting strategies. Table 3 describes the overall fit to the power-law region as well as the details of the component fits. Column (1) indicates the name of the particular fit. The details of the combined east lobe plus west lobe solution, the total solution, is described in columns (6)–(10). Column (6) is ${\sigma }_{\mathrm{rms}}^{2}$ from Equation (17) for the fit to the power law from ν_o = 432 MHz to ν_o = 8.44 GHz in the observer's frame. Because ${\sigma }_{\mathrm{rms}}^{2}$ is driven largely by the outlier ν_o = 686 MHz GBT data, we remove this point from the ${\sigma }_{\mathrm{rms}}^{2}$ calculation in Column (7) in order to test whether this one point is changing our fit-to-fit comparison. Column (8) tabulates the ratio of the lepto-magnetic energy (E(lm) from Equation (12)) of the east lobe to the E(lm) of the west lobe assuming a minimum energy solution in both lobes as the source of the resultant spectrum. The next column is the total E(lm) and the last column identifies the largest lobe based on the physical model. Indented below these data are the details of the east and west lobe fits, the peak frequency, the luminosity at the spectral peak, and the power-law spectral index. The only entries for the individual lobes are columns (3)–(5); the other columns are not applicable, N/A.

Figure 6 describes the physical parameters of the model. The top panels of Figure 6 show the dependence of E(lm) in Equation (12) on R (the lobe radius) for the leptonic lobes that produce the spectra in Figure 4. The middle row shows the dependence of E(lm) on the total number of particles in the lobe, ${{ \mathcal N }}_{e}$. The bottom row shows the dependence on the turbulent magnetic field strength.

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Alternatively, Fit B adjusts α to be as flat as possible in the eastern lobe and α to be as steep as possible in the western lobe. The fit is shown in the top frame of Figure 5. The corresponding minimum energy model is entry II in Table 3. Note that this choice produces less total flux in the ν_o = 432 MHz to ν_o = 686 MHz range compared to Fit A in Figure 4. This results in an increase of ${\sigma }_{\mathrm{rms}}^{2}$ of the fit relative to Fit A in Columns (6) and (7). The choice also increases the E(lm) of the western lobe and decreases E(lm) in the eastern lobe for the minimum energy solution. Column (8) shows that the spectral index difference between the lobes is no longer large enough to compensate for the volume difference in the lobes, and the solution deviates significantly from bilateral symmetry in the ejected energy. Thus, we consider this solution less plausible than Fit A, but it is not formally excluded by any of the data.

Finally, we consider the possibility that our lobe assignments with the SSA peaks is backwards in Fits A and B. Fit C reverses these assignments and is shown in the bottom frame of Figure 5. The corresponding minimum energy model is entry III in Table 3. This fit has the lowest flux in the ν_o = 432 MHz to ν_o = 686 MHz range and the largest ${\sigma }_{\mathrm{rms}}^{2}$. It is also the farthest from bilateral energy ejection symmetry in Table 3. Furthermore, the fit is below the lower bound VLBA data at ν_o = 327 MHz. For these reasons, it is the least plausible fit.

In this section, we consider a different model of the lobes in order to understand the effects of abandoning the minimum energy and the ${E}_{\min }\approx {m}_{e}{c}^{2}$ assumptions. This solution radiates the same spectrum as Fit A in Figure 4. The eastern lobe solution is the same as the minimum energy solution described in Figure 6. The western lobe solution is altered from that in Figure 6 and the previous subsections. We now choose ${E}_{\min }\approx 5{m}_{e}{c}^{2}$. The solutions are still governed by constraints (1) and (2) listed at the beginning of Section 6. However, the western lobe violates minimum energy, constraint (3).

The characteristics of the solution are plotted in Figure 7. Note that the minimum energy in the two lobes is drastically different in Figure 7. There is no minimum energy solution that fulfills constraint (1). We choose a solution with U_B > U_e in the western lobe. The horizontal dashed line in the top panel of Figure 7 connects the minimum energy solution in the eastern lobe to the magnetically dominated condition in the western lobe, under the time-averaged, bilateral symmetry condition that is embodied in constraint (1): $E(\mathrm{lm})(\mathrm{western}\ \mathrm{lobe})\approx E(\mathrm{lm})(\mathrm{eastern}\ \mathrm{lobe})$. The blue dot designates the location of this solution in the three frames of Figure 7. The condition that ≈90% of the lobe energy is in the magnetic component might seem extreme and not plausible. However, a detailed analysis of the ejection of large leptonic plasmoids (using the same modeling as here) in the Galactic black hole GRS 1915+105 showed that the plasmoids evolve from being magnetically dominated toward equipartition as they propagate. CSS sources are young for an FR II radio source, so it is not unreasonable that it has not relaxed to a minimum energy configuration. The fact that the eastern lobe has reached a minimum energy configuration and the western lobe has not yet reached a minimum energy configuration might be a consequence of a different life history during their propagation from the central engine. The details of the solution are described as entry IV in Table 3.

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This solution suggests studying a model in which ${E}_{\min }\approx 5{m}_{e}{c}^{2}$ in both lobes. If we assume minimum energy in both lobes, this model produces a minimum energy 2.5 times larger in the west lobe than in the east lobe. This is a consequence of the east lobe spectrum being much steeper; the low-energy cutoff removes more leptons from components with steeper spectral indices. It is clear that due to the different values of α, assigning the same ${E}_{\min }\ne {m}_{e}{c}^{2}$ in each lobe does not produce solutions with the same minimum energy. Only when both lobes have ${E}_{\min }\approx {m}_{e}{c}^{2}$ can equal minimum energy be achieved in our models for 3C 82.

We consider the model of Section 6.1 for Fit A with the modification that we do not assume δ = 1. Both lobes are at minimum energy with ${E}_{\min }\approx {m}_{e}{c}^{2}$. Based on the discussion in Section 5, we consider a single lobe speed that is a crude attempt to average the hot spot speed, $0.05c\lt {v}_{\mathrm{hs}}\lt 0.1c$, and the diffuse lobe plasma velocity, v_diffuse ≪ v_hs (neither of which is observed or well known). We choose the entire western (eastern) lobe to advance (recede) at a speed of 0.05c. Assuming that the western lobe is the advancing lobe is consistent with it being brighter and being adjacent to the putative jet. The E_lm of the eastern lobe increases by ∼10% from 6.76 × 10⁵⁹ to 7.62 × 10⁵⁹ erg. By contrast E_lm of the western lobe decreases ∼10% from 8.70 × 10⁵⁹ to 7.84 × 10⁵⁹ erg. The size of the approaching (receding) lobe gets slightly smaller (larger); most of the energy change is accomplished by a density change. The details of the model are found in entry V in Table 3. Notice that the system is much closer to bilateral symmetry in the total energy ejected into each jet arm than entry I of Table 3. This would appear to be an improvement to the accuracy of the model. We also added the case where the lobes advance at 0.1c as entry VI in Table 3. Based on entries V and VI in Table 3, the Doppler enhancement has no effect on the total energy in the system for these modest velocities.

Fit A has the lowest ${\sigma }_{\mathrm{rms}}^{2}$ for the fit to the power law, columns (6) and (7) of Table 3. It is the best fit of any two-component SSA model. It is the closest to having bilateral symmetry in the ejected energy (constraint 1, above), column (8), and it has the west lobe larger than the east lobe (constraint 2, above) per column (10). For these reasons, we consider this the most plausible choice for the best fit within the context of the minimum energy assumption. We note why ${\sigma }_{\mathrm{rms}}^{2}$ for the fit to the power law is lower in this case. The ν_o = 432 MHz, ν_o = 602 MHz, and ν_o = 686 MHz flux densities require a significant contribution from the smaller lobe in order for them to be attained by the sum of the two lobe flux densities. This is best achieved by moving the spectral peak to lower frequency and increasing the flux at low frequency by making the spectral index as steep as possible.

Table 3 indicates that the process of adjusting the individual lobe spectral indices so as to minimize the excess variance of the power-law fit to the radio data naturally drives the solution to one in which there is bilateral symmetry in the ejected energy in the minimum energy limit. The excess variance is very sensitive to the fitting of the high-frequency edge of the SSA local maximum. This is the most complicated and constraining region of the fit—both lobes contribute significantly, and the fitting of this spectral hump is constrained by the nearby ν_o = 432 MHz, ν_o = 602 MHz, and ν_o = 686 data. This is the region that produces most of the excess variance in Fits B and C. But Table 3 verifies that the identification of Fit A as the best fit is robust; it still holds even if the apparent outlier data at ν_o = 686 MHz is removed.

The solutions depend critically on the ν_o = 151 MHz and ν_o = 74 MHz data that define the SSA region. We corroborated the ν_o = 151 MHz 6C data with the GMRT ν_o = 150 MHz data as discussed above. We also downloaded the VLA B-array ν_o = 74 MHz image,¹⁹ and there are no confusing sources in the field of view, nor is there any evidence of strong side lobes resulting from u–v coverage issues (Lane et al. 2014). We can find no reason to doubt the accuracy of these data.

Considering the six solutions in Table 3, it is clear that there is only one solution with all three of the following properties:

• 1.  
  Both lobes are near the minimum energy state (constraint 3).
• 2.  
  The magneto-leptonic energy, E_lm, is approximately the same in both lobes (constraint 1): ${E}_{\mathrm{lm}}(\mathrm{West}\ \mathrm{Lobe})\,\approx {E}_{\mathrm{lm}}(\mathrm{East}\ \mathrm{Lobe}).$
• 3.  
  The low-energy cutoff to the lepton power law is the same in both lobes: ${E}_{\min }(\mathrm{West}\ \mathrm{Lobe})={E}_{\min }(\mathrm{East}\ \mathrm{Lobe})$.

That solution has ${E}_{\min }(\mathrm{West}\ \mathrm{Lobe})={E}_{\min }(\mathrm{East}\ \mathrm{Lobe})={m}_{e}{c}^{2}$ with Fit A. This solution is unique and has the properties that we posited as relevant at the start of this section. It also has the esthetic that there are no unexplained spectral breaks or low-energy cutoffs in the lepton spectra. Of the models in Table 3 that have these properties, the model with a lobe advance speed of 0.05c (entry V from Table 3) in Section 6.5 might be preferred because it has almost exact bilateral symmetry in the energy output.

The main goal of this section will be to find an estimate of the mean lobe advance speed, v_adv, that is applicable to 3C 82. There is no evidence of motion of the lobes in the images, so one must use indirect means. In particular, we will use the statistics of estimates of v_adv for "similar" objects. The main method that is implemented to this end is jet arm-length asymmetry. This method requires identifying the approaching jet and knowing the core position. Even so, the method cannot be reliably applied to single objects because the results are heavily skewed by intrinsic asymmetry in the ambient environment. If one does have perfect bilateral symmetry in emission and in the enveloping medium, one can formulate the arm-length ratio of the approaching lobe, L_app, to receding lobe, L_rec, as (Ginzburg & Syrovatskii 1969; Scheuer 1995)

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{asym}} & = & \displaystyle \frac{{L}_{\mathrm{app}}}{{L}_{\mathrm{rec}}}=\displaystyle \frac{1+({v}_{\mathrm{adv}}/c)\cos \theta }{1-({v}_{\mathrm{adv}}/c)\cos \theta },\mathrm{where}\\ {L}_{\mathrm{app}} & = & \displaystyle \frac{({v}_{\mathrm{adv}}T)\sin \theta }{1-({v}_{\mathrm{adv}}/c)\cos \theta }:\ {L}_{\mathrm{rec}}=\displaystyle \frac{({v}_{\mathrm{adv}}T)\sin \theta }{1+({v}_{\mathrm{adv}}/c)\cos \theta },\end{array}\end{eqnarray} \tag{ 16 }$$

where T is the time measured in the quasar rest frame and projected length on the sky plane of an observer on Earth (corrected for cosmological effects) is ${ \mathcal L }\equiv {L}_{\mathrm{app}}+{L}_{\mathrm{rec}}$. Because this formula is not reliably applicable to single sources, the preferred method is to define samples of "similar" objects and look at the statistical distribution of v_adv (Scheuer 1995). But, how does one define the notion "similar" in the case of 3C 82? The jet propagation has been studied in terms of simple self-similar models which assume an ambient density that scales like ${n}_{\mathrm{ambient}}={n}_{o}{r}^{-\psi }$, where r is the distance from the quasar (Kaiser & Alexander 1997; Willott et al. 1999). The following relevant scalings have been shown in Equations (10) and (11) of Willott et al. (1999), respectively,

$$\begin{eqnarray}&&{v}_{\mathrm{adv}}\propto {T}^{(\psi -2)/(5-\psi )}{\overline{Q}}^{1/(5-\psi )}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{v}_{\mathrm{adv}}\propto {D}^{(\psi -2)/3}{\overline{Q}}^{[\psi -1]/[3(5-\psi )]},\end{eqnarray} \tag{ 18 }$$

where D is the size of the source projected on the sky plane. One problem is that we do not know ψ, but this has been estimated to be 1.5 (Willott et al. 1999). This value is consistent with azimuthally averaged density profiles estimated for elliptical galaxies (Mathews & Brighenti 2003). However, for a jet traversing a given direction through the galaxy, a simple power law is a crude approximation. In any event, Equations (17) and (18) indicate that for these simple models, v_adv increases with jet power and decreases (decelerates) with distance from the quasar. 3C 82 is at the high end of the $\overline{Q}$ distribution and is smaller than most of the FR II quasars in the 3CRR catalog (Laing et al. 1983). In terms of the first requirement, the most-luminous quasar sample considered in Scheuer (1995) was the 3CRR sample, for which his Monte Carlo simulations found a median ${v}_{\mathrm{adv}}=0.115c$. However, these objects have a median length of ≳110 kpc versus the 11 kpc found for 3C 82. For ψ = 1.5 in Equation (17), ${v}_{\mathrm{adv}}\sim {D}^{-0.167}{\overline{Q}}^{0.047}$. We will find that the $\overline{Q}$ of 3C 82 is ∼100 times that of the median 3CRR source. Using this relation and the 3CRR analysis of Scheuer (1995), we estimate a most likely value for 3C 82 based on its size and luminosity of

$$\begin{eqnarray}&&{v}_{\mathrm{adv}}\approx 0.115c{\left(110/11\right)}^{0.167}{\left(100\right)}^{0.047}\approx 0.21c.\end{eqnarray} \tag{ 19 }$$

In order to corroborate the estimate in Equation (19), we assembled a sample of CSS quasars of similar size ($2\,\mathrm{kpc}\lt D\,\lt 25\,\mathrm{kpc}$). We require very straight jets that would be indicative of negligible interaction with the enveloping media. We need a tight mathematical constraint on straightness. To this end, we require: if the vertex of a cone with a half angle of 15° is placed on one of the lobes, then the other lobe, the jet, and the core in all high dynamic range radio images can be contained within the cone. We also require a very high lobe luminosity; thus, we restricted the sample to 3C objects. We eliminated objects previously identified as CSS that turned out to be much larger based on higher dynamic range imaging and objects that turned out to be powerful blazars with strong lobe emission that only appeared small due to the polar LOS. We then computed R_asym from Equation (16) using the highest sensitivity and dynamic range images in the public domain. We estimated the arm lengths based on the methods of Scheuer (1995). The results are tabulated in Table 4.

Table 4.  Arm-length Asymmetry in Straight 3C CSS Quasars

Source	Linear Size (kpc)	Rasym	vadv/ca	References
3C 138	4.9	1.66	0.30 ± 0.03	Akujor & Garrington (1995)
3C 186	18.1	1.21	${0.11}_{-0.01}^{+0.02}$	Akujor & Garrington (1995)
3C 277.1	17.2	1.83	${0.34}_{-0.03}^{+0.04}$	Reid et al. (1995), Pearson et al. (1985)
3C 298	9.4	1.87	${0.35}_{-0.03}^{+0.04}$	Akujor & Garrington (1995)

Note.

^aAssumes $20^\circ \lt \mathrm{LOS}\lt 40^\circ$.

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The advantage of restricting the CSS sources to quasars is that it eliminates the very oblique LOS of radio galaxies. The LOS to the jet in quasars is believed to be <45° with an average of 30° (Barthel 1989). By eliminating blazar LOSs, <10°, we have a narrow range of LOSs that will provide only a few percent variation of the v_adv in column (4). The values in Table 4 are slightly larger than what was expected from Equation (19). Based on Table 4, the 3CRR analysis of Scheuer (1995), and Equation (19), we conclude that

$$\begin{eqnarray}&&{v}_{\mathrm{adv}}\approx 0.2c\pm 0.1c\end{eqnarray} \tag{ 20 }$$

should cover the lobe advance speeds that might be applicable to 3C 82. Assuming bilateral symmetry and an LOS ≈ 30°, Equations (16) and (20) yield a loose bound on T,

$$\begin{eqnarray}\begin{array}{rcl}T & = & \displaystyle \frac{{ \mathcal L }\left[1-{\left(\tfrac{{v}_{\mathrm{advcos}\theta }}{c}\right)}^{2}\right]}{2{v}_{\mathrm{adv}}\sin \theta }\approx 11.3\,\mathrm{kpc}{/v}_{\mathrm{adv}},\mathrm{or}\\ & & 3.86\times {10}^{12}\,{\rm{s}}\lt T\lt 1.16\times {10}^{13}\,{\rm{s}}.\end{array}\end{eqnarray} \tag{ 21 }$$

The uncertainty in T in Equation (21) is driven by the large uncertainty in v_adv in Equation (20) that is estimated from the spread in values in Table 4.

From the minimum energy solution corresponding to Fit A in Table 3 and Figure 6, the total lepto-magnetic energy stored in both lobes is approximately

$$\begin{eqnarray}&&E(\mathrm{lm})\approx 1.55\times {10}^{60}\,\mathrm{erg}.\end{eqnarray} \tag{ 22 }$$

Equations (15) and (21) combined with Equation (22) imply a very large lower bound on the jet power of

$$\begin{eqnarray}&&\overline{Q}\gt 1.33\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 23 }$$

This is a conservative lower bound because it does not include work done by the expansion into the ambient medium, which can be of comparable magnitude (Willott et al. 1999). There is no observational data that can reliably constrain this for 3C 82. Ignoring the contribution from the work on the external environment, Equations (15) and (21) imply

$$\begin{eqnarray}&&\overline{Q}\approx 2.66\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\pm 1.33\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 24 }$$

This value is similar to that obtained using the same methods applied to the nonminimum energy solution of Section 6.4, $\overline{Q}\approx 2.32\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\pm 1.16\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Similarly, from Table 3, for the minimum energy solution for Fit B, $\overline{Q}\approx 2.32\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\pm 1.16\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Thus, for a wide range of assumptions the estimated jet power is very similar.

It is of interest to compare this with more traditional estimates of $\overline{Q}$. The spectral luminosity at 151 MHz per steradian, L₁₅₁, provides a surrogate for the luminosity of the radio lobes in a method that assumes a relaxed classical double radio source (Willott et al. 1999). The method assumes a low-frequency cutoff at 10 MHz, the jet axis is 60° to the LOS, there is no protonic contribution, and 100% filling factor. The plasma is near minimum energy and a quantity, f, is introduced to account for deviations of actual radio lobes from these assumptions as well as the energy lost by expanding lobe into the external medium, back flow from the head of the lobe, and kinetic turbulence. $\overline{Q}$ as a function of f and L₁₅₁ is plotted in Figure 7 of Willott et al. (1999),

$$\begin{eqnarray}&&\overline{Q}\approx 3.8\times {10}^{45}{{fL}}_{151}^{6/7}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 25 }$$

Note that the exponent on f is 1 not 3/2 as was previously reported (Punsly et al. 2018). The quantity f was estimated to be in the range of 10 < f < 20 for most FR II radio sources (Blundell & Rawlings 2000). Using ${L}_{151}\approx 5.8\times {10}^{28}\,{\rm{W}}\,{\mathrm{Hz}}^{-1}\,{\mathrm{sr}}^{-1}$ from Figure 4, Equation (31), and 10 < f < 20

$$\begin{eqnarray}&&\overline{Q}=2.07\pm 0.69\times {10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 26 }$$

Note the close agreement between our estimate in Equations (24) and (26).

In order to place this jet power estimate in context, we compare this result with other quasars with extremely large values of L₁₅₁ in Table 5. The first column is the name of the quasar followed by the redshift. Column (3) is the L₁₅₁ of the radio lobes; this is the appropriate luminosity to use in Equation (25) in order to calculate $\overline{Q}$ (Punsly et al. 2018). Columns (4) and (5) are $\overline{Q}$ and the method used for estimation. Column (6) is the reference to where this was done. Columns (7) and (8) are an estimate of L_bol from Equations (1) and (2) with a reference to the spectrum that was used. The last column is an estimate of $\overline{Q}/{L}_{\mathrm{bol}}$ from columns (4) and (7). Table 5 shows that 3C 82 has the most-luminous radio lobes, within observational uncertainty, and one of the largest values of jet dominance, $\overline{Q}/{L}_{\mathrm{bol}}=5.45\pm 2.73$. This expression used the average L_bol from Section 2. If we include the variability over the three epochs and uncertainty in Equation (1), $3.2\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\ \lt \ {L}_{\mathrm{bol}}\lt 5.8\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, with the uncertainty in $\overline{Q}$ from Equation (24) then we have a more rigorous estimate, $\overline{Q}/{L}_{\mathrm{bol}}=5.91\pm 3.41$, where the uncertainties are added in quadrature. The source, 3C 298, is likely less powerful than indicated in the table because it has a strong radio core and strong knots in the jet indicating increased dissipation relative to that of relaxed radio lobes (Ludke et al. 1998). Thus, we could be overestimating the jet power based on Equation (25) (Willott et al. 1999). By contrast, there are no detected strong knots or radio core in 3C 82, just the very steep-spectrum lobes with no highly delineated hot spots. It is conspicuous that all powerful radio sources in Table 5 (the most powerful known radio quasars) have $\overline{Q}/{L}_{\mathrm{bol}}\gtrsim 1$.

Table 5.  Quasars with Time-averaged Jet Powers Exceeding 10⁴⁷ erg s⁻¹

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Quasar	z	L151	$\overline{Q}$	Method	References	Lbol	Reference	$\overline{Q}/{L}_{\mathrm{bol}}$
		${10}^{28}\,{\rm{W}}\,{\mathrm{Hz}}^{-1}\,{\mathrm{sr}}^{-1}$	1047 erg s−1			1046 erg s−1	Spectrum
3C 298	1.44	6.04a	2.10 ± 0.70b	Equation (31)	This Paper	7.20	HST FOS G270H	2.92 ± 0.97
3C 82	2.87	5.94	2.07 ± 0.69	Equation (31)	This Paper	4.88	This Paper	4.24 ± 1.41
3C 82	2.87	5.94	2.66 ± 1.33	SSA Plasmoid	Sections 4–7	4.88c	This Paper	5.45 ± 2.73c
3C 9	2.01	5.01	1.79 ± 0.60	Equation (31)	This Paper	8.97	SDSS	2.00 ± 0.67
4C+11.45	2.18	3.08	1.16 ± 0.39	Equation (31)	This Paper	2.11d	SDSS	5.41 ± 1.80
PKS 0438–436	2.86	3.00d	1.15 ± 0.39e	Equation (31)	Punsly et al. (2018)	4.7	Punsly et al. (2018)	2.45 ± 0.82

Notes.

^aThe lobe luminosity is estimated from higher-resolution, higher-frequency images due to the existence of a very prominent radio core and jet. ^bThe use of Equation (25) for 3C 298 is uncertain due to the small size and the luminous core plus jet. $\overline{Q}$ needs to be verified by independent means, as we did for 3C 82 in this paper. ^cThis estimate does not include all the sources of uncertainty in L_bol that were developed in this paper (see Section 7.2 for such an estimate) in order to provide a consistent comparison to the other quasars. ^dBased on two SDSS spectra. The continuum is 0.45 of the level measured in Barthel et al. (1990), which they said was "roughly calibrated." ^eL₁₅₁ and $\overline{Q}$ are derived from the unbeamed radio lobes. The Doppler-boosted jets and blazar core are not included to improve the accuracy of Equation (25) in Punsly et al. (2018).

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In principle, the positive charges in the ionized lobes can be protonic matter instead of positronic matter. Based on Equations (11) and (19), the kinetic energy of the lobes would be much larger than E(lm). The spectrum in Figure 4 is the same and is created by electrons in the magnetic field. Using these same methods to describe a relativistic plasmoid ejection, it was determined that this was a viable solution for GW 170817 and the associated gamma-ray burst, GRB 170817A (Punsly 2019). However, in this case, consider the mass stored in the lobes based on Figure 6 for the minimum energy, minimum E(lm), solution, M_lobe = 5.44 × 10⁸ M_⊙. We can compare this to the accreted mass, ${M}_{\mathrm{acc}}=T\dot{M}={{TL}}_{\mathrm{bol}}/(\eta {c}^{2})$, where η is the radiative efficiency of the accretion flow which we take to be 10% and $\dot{M}$ is the accretion rate (Novikov & Thorne 1973). From L_bol in Equation (1) and the average value of the jet age in Equation (21), the accreted mass during the jet lifetime is ${M}_{\mathrm{acc}}=1.58\times {10}^{6}\,{M}_{\odot }$. Thus, in the minimum energy solution ${M}_{\mathrm{lobe}}/{M}_{\mathrm{acc}}=341$. For the nonminimum energy solution in Section 6.4, ${M}_{\mathrm{lobe}}/{M}_{\mathrm{acc}}=165$. These solutions require two orders of magnitude more mass to be transported to the radio lobes than is accreted toward the supermassive central black hole. For this reason, the protonic solutions for the lobes in 3C 82 are disfavored.

The line-driven continuous wind has the advantage in that it does not require a cloud confinement mechanism. It avoids large optical depths because the wind is accelerating, and gas is constantly being Doppler-blueshifted relative to the accretion disk and the inner gas. The acceleration scale length is ∼2 × 10¹³ cm, basically the ad hoc cloud dimension in other models (Murray et al. 1995). The outflowing wind is more highly ionized than cloud models. This is quantified in terms of the ionization parameter, U,

$$\begin{eqnarray}&&U=\displaystyle \frac{{\int }_{3.3\times {10}^{15}\,\mathrm{Hz}}^{\infty }({L}_{\nu }/h\nu )\,d\nu }{4\pi {r}^{2}{n}_{w}c},\end{eqnarray} \tag{ 27 }$$

where n_w is the hydrogen number density in the wind. In the line-emitting region, U ≈ 1–10. Most of the emission comes from the base of the wind where U is smaller; U increases as the wind accelerates away from the source. Based on Figure 1, we approximate the flux density ${F}_{\lambda }(\lambda =916\mathring{\rm A} )\approx 2\times {10}^{-15}$ erg s⁻¹ cm⁻² Å⁻¹ in the quasar rest frame. The spectral index in the extreme ultraviolet in frequency space is chosen to be very steep, α_ν = 2, as is indicated for powerful radio-loud quasars (Zheng et al. 1997; Telfer et al. 2002; Punsly 2015). Thus, the numerator in Equation (27), the flux of ionizing photons, ${N}_{\mathrm{ion}}=2\times {10}^{56}\,{{\rm{s}}}^{-1}$. From Equation (27), we get the following constraint on the C iv-emitting gas

$$\begin{eqnarray}&&4\pi {r}^{2}{n}_{w}\approx 2.2\times {10}^{45}\,{\mathrm{cm}}^{-1}\,\displaystyle \frac{3}{U}.\end{eqnarray} \tag{ 28 }$$

This equation is useful because the wind power (kinetic energy flux) is

$$\begin{eqnarray}&&{P}_{w}=4\pi {r}^{2}{n}_{w}{f}_{c}{m}_{p}\displaystyle \frac{1}{2}{v}_{w}^{3},\end{eqnarray} \tag{ 29 }$$

where m_p is the mass of a proton, v_w = 2618 km s⁻¹ is from the peak of the C iv BLUE Gaussian component in Table 1, and the wind covering factor is f_c ≈ 0.04 (Murray et al. 1995). This is based on the estimate that quasars are viewed with an LOS within 45° of the accretion disk normal (Barthel 1989). If broad absorption-line quasars are those viewed through the wind and their rate of occurrence is ≈10%, then f_c ≈ 0.04. Thus, Equations (28) and (29) yield

$$\begin{eqnarray}&&{P}_{w}=1.36\times {10}^{45}\displaystyle \frac{3}{U}\displaystyle \frac{{f}_{c}}{0.04}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 30 }$$

In the cloud outflow model, the force that drives the clouds outward is radiation pressure. Thus, high-luminosity accretion flows produce the most-luminous BLUE outflow components. The wind power is given by (Marziani et al. 2017)

$$\begin{eqnarray}\begin{array}{rcl}{P}_{w} & = & 2.44\times {10}^{44}{k}^{2}\displaystyle \frac{L(\mathrm{BLUE})}{{10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\\ & & \times \,{\left[\displaystyle \frac{{v}_{\mathrm{clouds}}}{5000\mathrm{km}{{\rm{s}}}^{-1}}\right]}^{3}\displaystyle \frac{1\,\mathrm{pc}}{{r}_{\mathrm{BLR}}}\displaystyle \frac{{10}^{9}\,{\mathrm{cm}}^{-3}}{{n}_{\mathrm{clouds}}}\displaystyle \frac{5\,{Z}_{\odot }}{Z}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&k\equiv \displaystyle \frac{{v}_{\mathrm{terminal}}}{{v}_{\mathrm{clouds}}}\end{eqnarray} \tag{ 32 }$$

where ${v}_{\mathrm{terminal}}$ is the terminal velocity of the outflow, v_clouds = 2618 km s⁻¹ is the BLUE cloud velocity from Table 1, L(BLUE) = 1.78 × 44 erg s⁻¹ is the luminosity of the BLUE from Table 1, r_BLR is the distance to the BLUE, and the metallicity of the clouds is Z. Reverberation mapping of high-redshift quasars of similar luminosity indicates (Lira et al. 2018)

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{\mathrm{BLR}}}{10\,\mathrm{lt} \mbox{-} \mathrm{day}}=(0.22\pm 0.10){\left[\displaystyle \frac{\lambda {L}_{\lambda }(1345\mathring{\rm A} )}{{10}^{43}\mathrm{erg}{{\rm{s}}}^{-1}}\right]}^{0.48\pm 0.08}.\end{eqnarray} \tag{ 33 }$$

However, it is not clear if this can be applied to a wind-driven BEL region. Based on CLOUDY models, we expect that U ≳ 1 will provide a strong C iv and a very weak C iii] as desired. From modeling of cloud dynamics using the simulations in Netzer & Marziani (2010), we find that a hydrogen column density of N_H ≲ 10²² cm⁻² is required for efficient acceleration by the radiation field of the quasar. A column density of N_H ≈ 10²³ cm⁻² will not be efficiently accelerated. Thus, we have a constraint on the model N_H ≲ 10²² cm⁻² that is used in combination with an assumed cloud size of ∼10¹³ cm, U ≈ 1 and N_ion = 2 × 10⁵⁶ s⁻¹ as was estimated for 3C 82, above. For N_H ≈ 2.7 × 10²² cm⁻², r_BLR ≈ 4.4 × 10¹⁷ cm, consistent with the high end of the range of reverberation estimates in Equation (33). For these parameters, the models of outflow in a system of clouds in pressure equilibrium, under the combined effect of radiation and gravity, yield k ≈ 10. From these we can estimate the wind power in Equation (31) for the case of 3C 82,

$$\begin{eqnarray}&&{P}_{w}\approx 1.12\times {10}^{45}{\left[\displaystyle \frac{k}{10}\right]}^{2}\displaystyle \frac{5\,{Z}_{\odot }}{Z}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 34 }$$

The fundamental nature of high-ionization outflow is not well understood; it could be a continuous line-driven wind or an outflow of clouds. The details of both scenarios have significant uncertainty as discussed in the previous two subsections. However, both models yield estimates of P_w ≳ 10⁴⁵ erg s⁻¹. The wind kinetic power is approximately 10 times the line luminosity. Note that the time-averaged jet power is two orders of magnitude larger than the wind power.