STRUCTURAL TRANSITION IN THE NGC 6251 JET: AN INTERPLAY WITH THE SUPERMASSIVE BLACK HOLE AND ITS HOST GALAXY

We conducted EVN observations of NGC 6251 on 2013 March 10 at 1.6 GHz with the stations at Badary, Svetloe, Zelenchukskaya (Russia), Effelsberg (Germany), Jodrell Bank (UK), Medicina, Noto (Italy), Onsala (Sweden), Shanghai, Urumqi (China), Torun (Poland), and Westerbork (Netherlands). Data with both left and right circular polarization were recorded at each station using eight sub-bands of 8 MHz bandwidth and 2-bit sampling. The data were correlated at the JIVE (Joint Institute for VLBI in Europe) correlator.

Data were calibrated following the standard procedures in AIPS. A priori amplitude calibration for each station was derived from the system temperatures measured during each run and the antenna gain curves. Fringe fitting was performed to remove the residual delays and rates by assuming a point source model. After applying the solutions, the data were exported to Difmap and averaged over 12 s in each sub-band. Then the gain phase and amplitude were self-calibrated by the iterative process of CLEAN to obtain the final image. More detailed observational parameters are described in Table 1.

Table 1.  Image Parameters

Facility	Code	Epoch	Frequency	Beam FWHM	rms noise	Dynamic	Jet PA	Φba
		(yyyy-mm-dd)	(GHz)	(mas × mas, deg)	(μJy/beam)	range	(deg)	(mas)
EVN-ib	ET028	2013 Mar 10	1.658	3.06 × 2.96, 651	127	1874	295.9	3.02
EVN-iib	ET028	2013 Mar 10	1.658	4.81 × 4.01, 356	51	5392	295.9	4.78
EVN-iiib	ET028	2013 Mar 10	1.658	16.7 × 15.8, 337	54	7740	294.6	16.78
VLBA	V105	1998 Apr 30	4.816	0.997 × 0.962, −241	63	3460	297.9	0.975
MOJAVE data
VLBA	BT036	1998 Jun 02	15.365	0.450 × 0.390, −258	204	1701	298.4	0.54c
VLBA	BJ033B	2000 May 30	15.363	0.468 × 0.453, 256	454	844	298.4	0.54
VLBA	BS089	2001 Feb 28	15.365	0.346 × 0.326, 341	370	630	298.4	0.54
VLBA	BL137R	2007 Jan 06	15.365	0.564 × 0.468, −428	967	487	298.4	0.54
VLBA	BL149AA	2007 Jun 03	15.365	0.528 × 0.500, 337	863	532	298.4	0.54
VLBA	BL149AF	2007 Aug 16	15.365	0.557 × 0.481, 897	675	717	298.4	0.54
VLBA	BL149AK	2008 Jul 17	15.365	0.506 × 0.494, −429	547	863	298.4	0.54
VLBA	BL149BF	2008 Nov 26	15.357	0.543 × 0.483, −183	657	686	298.4	0.54
VLBA	BL149BM	2009 Jun 03	15.357	0.524 × 0.496, 633	585	785	298.4	0.54
VLBA	BL149CL	2010 Jul 12	15.357	0.496 × 0.482, −414	390	1167	298.4	0.54
VLBA	BL178AO	2012 Sep 02	15.357	0.552 × 0.486, −618	495	715	298.4	0.54
VLBA	BL178BH	2013 Aug 12	15.357	0.527 × 0.461, −252	470	802	298.4	0.54

Notes.

^aΦ_b: Beam FWHM perpendicular to the local jet axis. ^bWeighting schemes of EVN-i, -ii, and -iii images are uniform, natural, and natural with tapering, respectively. ^cMOJAVE images are convolved with the same circular beam of 0.54 mas in the analysis.

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Archival Very Long Baseline Array (VLBA) data at 5 GHz are used and calibrated in the same manner as the EVN data. Also, 12 epochs of the VLBA data at 15 GHz are obtained from the MOJAVE database (Lister et al. 2009). Observations were conducted during 1998–2013. We reproduce all images in agreement with the MOJAVE database. In order to perform a joint analysis, images of the 12 epochs are convolved with the same circular beam with FWHM of 0.54 mas (see more details in Table 1).

We also use a published VLA image of NGC 6251 at 1.4 GHz to compare with the VLBI measurements in Section 4.1. The image, as well as the calibration processes, is shown in Sambruna et al. (2004). Observations were conducted on 1995 August 15 using the full VLA in its A-configuration. The beam is restored to be circular with an FWHM of 2''.

Figure 1 shows images of NGC 6251 with high dynamic range taken from our EVN observations at 1.6 GHz, and the archival VLBA data at 5 and 15 GHz; Table 1 summarizes the detailed image parameters (e.g., beam size, dynamic range). The EVN images are obtained with different weighting schemes (denoted as EVN-i, -ii, and -iii) to enhance the sensitivity of the emission on various angular scales. The brightest component at the eastern edge of the object is presumably the core, and a continuous jet emission extends towards the northwest. The jet direction and the knotty features are consistent with each other in all images. This is the first time that the jet emission associated with NGC 6251 has been continuously detected at a distance of 50–150 mas from the core.

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To investigate the collimation profile of the jet, we derive the radius (half width) of the jet emission along the jet axis in each image. We mainly adopt the methodology of AN12 as follows; in addition, the change in the jet direction on different length scales (i.e., jet bending) is taken into consideration:

• 1.  
  The position angle (PA) of the jet direction is determined from the averaged PAs of all CLEAN components excluding the core region, weighted by their flux density.
• 2.  
  The cross sections of the jet are fitted with a single Gaussian function of FWHM Φ₀ at every 1/5 of the beam size (as independent measurements, the resolution limit was studied by Lobanov 2005) along the jet axis to evaluate the transverse structure.
• 3.  
  The jet radius r (i.e., half of the jet width) is thus defined as half the FWHM of the resultant fitted Gaussian, which is deconvolved from the beam FWHM Φ_b taking into account the beam orientation, i.e., $2r=\sqrt{{{\rm{\Phi }}}_{0}^{2}-{{\rm{\Phi }}}_{b}^{2}}$ (see also Perley et al. 1984, Section IV-d).
• 4.  
  The uncertainty is estimated from the Gaussian fitting error (in step 2), and the imaging error that is evaluated as:

  • 1.  
    The dispersion (standard deviation) of the 12-epoch measurements for the VLBA data at 15 GHz.
  • 2.  
    Φ_b/5 for the other frequencies due to the lack of sufficient independent observations. Particularly for the EVN data, the jet radii are further averaged over both polarizations (i.e., RR and LL).

The measurements are presented in a machine-readable format in Table 2. In our analysis, the jet PA decreases mildly as the length scale increases, ranging from 294°–298° (see also Table 1); a similarly decrease is detected by Jones & Wehrle (2002). All jet cross sections are well represented with a single Gaussian component.

Table 2.  Measurements of the Jet Radii and Estimated Uncertainties

z	Φ0	${e}_{{{\rm{\Phi }}}_{0},\mathrm{fit}}$	r	${e}_{r,\mathrm{fit}}$	er
(mas)	(mas)	(mas)	(mas)	(mas)	(mas)
EVN-i, 1.6 GHz, RR pol.
16.2	3.73	0.05	1.09	0.04	0.61
16.8	3.74	0.05	1.11	0.04	0.61
17.4	3.62	0.05	1.00	0.04	0.61
18.0	3.47	0.04	0.85	0.04	0.61
18.6	3.34	0.04	0.71	0.05	0.61

Note. This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. z: deprojected axial distance. Φ₀: fitted Gaussian FWHM of the jet cross section. ${e}_{{{\rm{\Phi }}}_{0},\mathrm{fit}}$: fitting uncertainties of Φ₀. r: jet radius. ${e}_{r,\mathrm{fit}}$: fitting uncertainties of r. e_r: uncertainties of r taking the imaging error into account, which is described in Section 3.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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To properly evaluate the uncertainty, we utilized the multi-epoch VLBA data at 15 GHz. The jet radii derived in 12 epochs are in good agreement with each other so that we obtain the standard deviation (${\sigma }_{r,15}$) of the jet radii as the imaging error. We estimated the distribution of ${\sigma }_{r,15}$ as a function of signal-to-noise ratio (S/N), and found it to be well described by a power-law function, ${\sigma }_{r,15}/{{\rm{\Phi }}}_{b}=(0.66\pm 0.09)\times {\rm{S}}/{{\rm{N}}}^{-0.68\pm 0.05}$. ${\sigma }_{r,15}$ is about 0.01 mas in the innermost jet region where ${\rm{S}}/{\rm{N}}\simeq 200$, and 0.1 mas downstream of the jet where ${\rm{S}}/{\rm{N}}\simeq 5$, corresponding to 1/50 and 1/5 of the beam size, respectively. Therefore, we consider the upper limit of the imaging error to be ${{\rm{\Phi }}}_{b}/5$ because the jet radii are measured where the jet is detected when ${\rm{S}}/{\rm{N}}\geqslant 5$ (see also Homan et al. 2002, Appendix A for similar analysis). Note that we assume the time variability of the jet radius is negligible.

Figure 2 shows the distribution of jet radius (r) as a function of the deprojected distance to the core (z), which we refer to as the radius profile of the jet. The jet viewing angle (θ) for deprojection is uncertain within an upper limit of 47° (Jones & Wehrle 2002). We adopt θ = 19° derived from a detection of a sub-parsec scale counter-jet (Sudou et al. 2000). An independent model of the spectral energy distribution (SED) suggests a consistent result of θ ≃ 18° (Chiaberge et al. 2003). We use the deprojected length scale ($\mathrm{27,000}\,{r}_{s}$ mas⁻¹) along the jet throughout the rest of the paper. Note that the jet radius, as defined perpendicular to the jet axis, is not affected by projection effects.

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Collimation profiles are generally described by power-law functions with some local features, as a manifestation of the global evolution of jet structure. We perform regression analyses with single and double power-law models throughout the whole VLBI data set and show the results and their relative residuals in Figure 2. Note that the power-law indices, a, are defined as $r\propto {z}^{1/a}$ throughout the paper.

First, we fit the data with a single power-law model, r_SP,

$$\begin{eqnarray}&&{r}_{\mathrm{SP}}(z)={A}_{0}\,{z}^{1/a},\end{eqnarray} \tag{ 1 }$$

which gives the best fit using a power-law index of a = 1.25 ± 0.03, with ${\chi }^{2}/\mathrm{dof}\simeq 0.98$ (see also Table 3). The radius profile can be described well with this power law in the range z₀ = (10⁵–10⁶) r_s, but it deviates systematically at each end.

Table 3.  Best Fit Parameters of the Power-law Models

a or au	A0	ad	z0	r0	n	${\chi }^{2}/\mathrm{dof}$
	(${r}_{s}^{1-1/a}$)		(${10}^{5}\,{r}_{s}$)	(${10}^{3}\,{r}_{s}$)
Single power law, Equation (1)
1.25 ± 0.03	0.18 ± 0.05	⋯	⋯	⋯	⋯	0.98
Double/broken power law, Equation (2)
6.6 ± 6.0	⋯	0.83 ± 0.06	1.3 ± 0.7	1.0 ± 0.4	1	0.58
3.0 ± 0.7	⋯	0.88 ± 0.05	1.7 ± 0.5	1.6 ± 0.4	2	0.57
2.5 ± 0.4	⋯	0.90 ± 0.04	1.8 ± 0.5	1.9 ± 0.4	3	0.57
2.3 ± 0.3	⋯	0.91 ± 0.04	1.9 ± 0.4	2.1 ± 0.3	4	0.57
2.2 ± 0.2	⋯	0.92 ± 0.04	2.0 ± 0.4	2.2 ± 0.3	5	0.57
2.0 ± 0.1	⋯	0.94 ± 0.04	2.1 ± 0.3	2.5 ± 0.2	100	0.57

Note. a: power-law index of the single power-law model, A₀: normalization parameter, a_u: upstream power-law index (pre-break) of the double power-law model, a_d: downstream power-law index (post-break), z₀: deprojected axial distance of the break position, r₀: jet radius at the break position, n: controlling parameter for the sharpness of the break, and ${\chi }^{2}/\mathrm{dof}$: reduced chi-square of the fitting.

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Then, we test a double power-law model, in the form of the broken power-law function, r_DP,

$$\begin{eqnarray}&&{r}_{\mathrm{DP}}(z)={r}_{0}{\left[{\left(\displaystyle \frac{z}{{z}_{0}}\right)}^{n/{a}_{u}}+{\left(\displaystyle \frac{z}{{z}_{0}}\right)}^{n/{a}_{d}}\right]}^{1/n},\end{eqnarray} \tag{ 2 }$$

so that we can simultaneously solve for a_u, a_d, z₀, and r₀, which are the upstream and downstream power-law indices (pre- and post-break), the break position, and the jet radius at the break position, respectively, and n is a controlling parameter for the sharpness of the break, i.e., the sharper the break, the larger n. The radius profile is best fitted with the model parameters listed in Table 3, with χ²/dof ≃ 0.57–0.58 by adopting different $n\geqslant 1$.

Note that the different values of sharpness (i.e., choices between n) result in changes of the best fitted parameters (see also Table 3). In particular, a small value of n ≳ 1 implies a smooth transition with a significant amount of data located amid the two pure power-law branches, causing an increase in the uncertainty for both a_u and z₀. However, the resultant fittings are all fairly consistent (in 1σ, and in 2σ for the cases n = 1 and n = 2) with power-law indices ${a}_{u}\geqslant 2$ and a_d = 0.9–1, and the distance of the transition (break position) z₀ = (1–2) × 10⁵ r_s regardless of the sharpness. In addition, their relative residuals are uniformly distributed on all distance scales. The oscillatory pattern in the radius profile, presumably corresponding to a local enhancement of the emissivity, potentially affects the quantification of the power-law index upstream. In order to have a better constraint, it is important to extend structural studies of the jet towards the inner regions. Future (sub)millimeter VLBI observations, with unprecedented angular resolution, are expected to provide a unique chance to unveil the initial point of collimation and even the genesis of AGN jets.

Next, we perform a statistical test (an f-test) to check whether the double power-law model is necessary. Given the f-ratio of 1.71 (i.e., the ratio of ${\chi }^{2}/\mathrm{dof}$) with the degrees of freedom of the single and double power-law fitting, dof = 189 and 187, respectively, the null hypothesis that the two models fit the population of the data equally well corresponds to a small probability p ∼ 1.4 × 10⁻⁴. As a result, the double power-law model is significantly better than the single power law at describing the radius profile of the NGC 6251 jet. Also, this is further supported by the measurements in both the outer and inner regions taken with the VLA and VLBI core data (see Section 4.1).

In summary, we conclude that the structure of the NGC 6251 jet is described by a parabolic shape ($a\simeq 2$) upstream, a conical expansion ($a\simeq 1$) downstream, and the structural transition takes place at a characteristic distance scale of (1–2) × 10⁵ r_s. The NGC 6251 jet is the second piece of observational evidence of the structural transition from parabolic to conical shape occurring at a scale of ${10}^{5}\,{r}_{s}$ in AGN jet systems, following the result of M87 (AN12).

We take a step further to investigate the jet structure beyond the two ends of the distance scale, which are shown as the gray data points in Figure 2. The radius profile on kiloparsec scales is derived from an archival VLA image (as introduced in Section 2.3), which was similarly measured by Perley et al. (1984, Figure 10). Figure 2 shows their innermost portion (with distances $z\leqslant 20$ kpc) to guide the comparison; this portion is smoothly connected to the radius profile predicted by the double power-law model, while largely deviating from the single power-law model. The VLA measurements additionally support the idea that the VLBI jet downstream is better described by a conical shape (a_d ∼ 1) that propagates three orders of magnitude in distance, if the jet structure is indeed linked together on intermediate scales (z = 10⁶–10⁸ r_s). Note that the VLA data are not included in the previous regression analysis. To study the missing link of the jet structure requires facilities with an adequate angular resolution (e.g., VLA, MERLIN), which will be investigated in future work.

On the other hand, VLBI cores are considered to be the innermost jet emissions, as originally suggested by Blandford & Königl (1979). The frequency-dependent core-shift (${{\rm{\Delta }}}_{z}\propto {\nu }^{-1}$) due to synchrotron self-absorption enables us to estimate the exact location of the innermost components with respect to the black hole, and hence the innermost jet structure and its width. This technique has been previously demonstrated in M87 (e.g., Hada et al. 2011, 2013, NA13). In NGC 6251, the core-shift between 15 and 5 GHz was estimated to be ${{\rm{\Delta }}}_{z}\simeq 0.3\,$ mas by image registration with the optically thin jet (Sudou et al. 2000). Together with the transverse widths (FWHM, deconvolved from the beam) of the cores, the sub-parsec-scaled jet radii can be estimated by the VLBA and EVN cores (see also Figure 2). Once again, they are relatively consistent with the upstream region that is predicted by the double power-law model rather than the single power law. As a result, we expect that the jet maintains a parabolic shape over at least two orders of magnitude in distance. Interestingly, the power-law index of the jet upstream (a_u ∼ 2) is very close to the genuinely parabolic field configuration in which the jet extracts electromagnetic energy from a spinning black hole (Blandford & Znajek 1977). Core-shift measurements of NGC 6251 at higher frequencies (i.e., submillimeter bands) can be very helpful in understanding the jet formation process as well as in extending the structural study.

We consider the gravitational potential of the environment of NGC 6251: the SMBH and the host galaxy. As indicated in Figure 2, it is found that the structural transition is close to the boundary of the sphere of gravitational influence (SGI, ${r}_{\mathrm{SGI}}\simeq 5\times {10}^{5}\,{r}_{s}$) in order of magnitude, where the central SMBH dominates the gravitational potential. The SGI is defined by r_SGI = GM/${\sigma }_{v}^{2}$ under the assumption that the stellar distribution is supported by random motion rather than rotationally, where σ_v is the velocity dispersion of the collective stars, i.e., the elliptical galaxy here (Peebles 1972). It is measured by optical spectroscopy that σ_v ≃ 290 ± 14 km s⁻¹ in NGC 6251, and the ratio of rotational velocity to velocity dispersion is v/σ_v = 0.16 (Ferrarese & Ford 1999; Kormendy & Ho 2013), which gives our estimates of the r_SGI above. The estimated SGI can be an upper limit, since one may expect much larger σ_v towards the nucleus of NGC 6251, for we cannot spatially resolve the region at present. Note that the evaluation of the SGI may be affected by the triaxiality of the bulge, or the warped disky structure in the nucleus possibly due to galaxies merging, as discussed in Ferrarese & Ford (1999). A study of the stellar velocity structure with an angular resolution ∼10 mas will be helpful in narrowing down this issue. However, we believe that both effects are insignificant in order-of-magnitude estimates. Approximately, the cool core is expected to be thermodynamically stable, so that virial equilibrium is considered for the gas as well as for the stars. As a consequence, the Bondi radius (r_B = 2GM/c_s², where c_s is the local sound speed) is expected to be close to the SGI. For instance, in M87, which is the very limited case that we can directly compare those two numbers with thanks to its proximity, the Bondi radius is indeed close to the SGI (${r}_{{\rm{B}}}\simeq 3.8\times {10}^{5}\,{r}_{s}$, AN12 and references therein, and ${r}_{\mathrm{SGI}}\simeq 3.1\times {10}^{5}\,{r}_{s}$ with ${\sigma }_{v}\simeq 380$ km s⁻¹, Murphy et al. 2014; we adopt an SMBH mass of $6.6\times {10}^{9}\,{M}_{\odot }$, Gebhardt et al. 2011).

Therefore, the Bondi radius in NGC 6251 is expected to be a few times ${10}^{5}\,{r}_{s}$ with a virial temperature ${k}_{B}{T}_{\mathrm{vir}}\simeq \mu {m}_{{\rm{p}}}{\sigma }_{v}^{2}\simeq 0.5\,\mathrm{keV}$ derived from the velocity dispersion, where μ (≃0.6) and m_p are the mean molecular weight and proton mass. Independently, we estimate ${c}_{s}=\sqrt{\gamma p/\rho }=\sqrt{\gamma {k}_{B}T/{m}_{{\rm{p}}}}\simeq 500$ km s⁻¹ with an upper limit of temperature ${kT}\simeq 1.6\,\mathrm{keV}$, which is extrapolated from X-ray observation on a scale of 20 kpc (or 10'', Sambruna et al. 2004), where γ (=5/3), p, and ρ are the adiabatic index, pressure, and density of the gas. This estimation gives the lower limit of the Bondi radius as ${r}_{{\rm{B}}}\simeq 4\times {10}^{5}\,{r}_{s}$ and agrees well with the former expectation. Note that the extrapolation is made towards the Bondi radius assuming a flat temperature profile; it was demonstrated in many simulations that the profile is nearly flat in the cool core of elliptical galaxies (e.g., Gaspari et al. 2012, 2013).

We compare the collimation (radius) profile of the jet in NGC 6251 to that in M87, because the structural transition of the M87 jet is also located at around the Bondi radius (AN12). Figure 3 shows their jet radii and deprojected distances normalized respectively by their Schwarzschild radii. Located in the same type of galaxy, the two jets show remarkably similar structures. They both undergo a structural transition at a scale of ${10}^{5}\,{r}_{s}$, suggesting an interplay with the galactic environment, i.e., the SMBH and its host galaxy. The collimation process of AGN jets may be fundamentally characterized by the external galactic medium, for which it has been argued that the stratified ISM is responsible (AN12; NA13). Recently, it has also been shown that the jet can be confined by the wind generated from the accretion flow within the Bondi radius (Yuan et al. 2015; Globus & Levinson 2016).

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The internal pressure of the jet is estimated in order to compare it with the external medium. We consider the NGC 6251 jet to be a relativistic system organized by helical magnetic field, i.e., the toroidal component of the field (denoted by B_ϕ in the observer's frame, and ${B}_{\phi }^{\prime }={B}_{\phi }/{\rm{\Gamma }}$ in the fluid comoving frame, where Γ is the bulk Lorentz factor) is dominant farther from the black hole.

Assuming the jet satisfies the synchrotron minimum energy condition (Pacholczyk 1970), the magnetic field strength B_ϕ is estimated as

$$\begin{eqnarray}{B}_{\phi } & = & [6\pi (1+k){c}_{12}(\alpha ,{\nu }_{\min },{\nu }_{\max })\\ & & \times {{L}_{\mathrm{syn}}(\alpha ,{\nu }_{\min },{\nu }_{\max })/({\rm{\Phi }}V)]}^{2/7},\end{eqnarray} \tag{ 3 }$$

(equivalently, the internal pressure ${p}_{\mathrm{eq}}\simeq {B}_{\phi }^{2}/8\pi$) by using the radio emissivity along the jet. First, we adopt a spectral index $\alpha \simeq 0.6$ (${S}_{\nu }\propto {\nu }^{-\alpha }$) that has been found in the jet at both kiloparsec scale and parsec scale (optically thin) (Perley et al. 1984; Jones & Wehrle 2002), and a nominal frequency integrand of 10–10⁵ MHz to estimate the synchrotron luminosity L_syn, as well as to determine the factor c₁₂; we set the proton-to-electron energy ratio k = 1 as a strict minimum; the emitting volume ΦV is estimated with a filling factor Φ = 1 and a circular jet cross section characterized by the jet radius, which is derived in Section 3. Second, we estimate the bulk Lorentz factor Γ by adopting an empirical correlation ${\rm{\Gamma }}\,{\theta }_{j}\simeq 0.2$ with the half opening angle ${\theta }_{j}={\tan }^{-1}(r/z)$, which has been found in large samples of AGN jets (Jorstad et al. 2005; Pushkarev et al. 2009; Clausen-Brown et al. 2013). This relation is also demonstrated in the MHD models, indicating a causally connected jet in the acceleration zone (e.g., Komissarov et al. 2009; Tchekhovskoy et al. 2009; Pu et al. 2015). As a result, Figure 4 shows the field strength (${B}_{\phi }^{\prime }={B}_{\phi }/{\rm{\Gamma }}$) together with the jet pressure (${p}_{\phi }^{\prime }={{B}_{\phi }^{\prime }}^{2}/8\pi$) in the fluid frame. Note that true minimum-energy field strength can be one order of magnitude higher than the strict lower limit by tuning the parameters mentioned above.

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4.3.1. The Parabolic Jet

4.3.2. The Conical Jet