DETERMINATION OF CENTRAL ENGINE POSITION AND ACCRETION DISK STRUCTURE IN NGC 4261 BY CORE SHIFT MEASUREMENTS

Multifrequency observations of NGC 4261 were conducted using the VLBA (Napier 1994) at four frequency bands (1.4, 2.3, 5.0, and 8.4 GHz) on 2003 July 3, and at three frequency bands (15, 22, and 43 GHz) on 2003 June 26. The two sets of observations were separated by only 7 days. Each observation period was approximately 12 hr. At 1.4 GHz, dual-circular polarizations at two intermediate frequencies (IFs) were received, covering a total bandwidth of 16 MHz. At both 2.3 and 8.4 GHz, two IFs covered a total bandwidth of 16 MHz in right-hand circular polarization. Other frequency bands collected four IFs, covering a total bandwidth of 32 MHz in left-hand circular polarization. Digital sampling of 2 bits was applied at all observational frequency bands. Correlations were processed using the VLBA correlator in Socorro, New Mexico.

The radio core positions of NGC 4261 were measured by the nodding-style phase-referencing technique, relative to a nearby reference source, J1222+0413, separated by less than 2° from NGC 4261:·-T-C1-T-C1-T-·, where T and C1 represent the target and primary calibrator, respectively. Switching between the target and calibrator was repeated after scanning for 20 s. Observations were conducted for approximately 20 minutes at each frequency in turn. Under this measurement condition, the uv-coverage among frequencies was almost identical, thereby minimizing the residual systematic errors in astrometry. The total on-source time on the target was approximately 1 hr at each frequency.

The antenna parallactic angles and Earth orientation parameters were corrected. Furthermore, ionospheric dispersive delays were corrected using the ionospheric model from GPS data provided by the Jet Propulsion Laboratory (Mannucci et al. 1999). Instrumental delays/phases and bandpass were calibrated using 3C 273 data. Assuming a point-source model, we performed fringe-fitting on J1222+0413 as a phase calibrator C1. Subsequently, we calibrated the visibility amplitude and phase using a source structure model available in the Difmap software (Shepherd et al. 1994) procedures. This model is based on iterative deconvolution and self-calibration algorithms. Applying these solutions to the visibilities of NGC 4261, we obtained phase-referenced images of NGC 4261. On the other hand, we also made self-calibrated images for NGC 4261 using the Difmap (Figure 1).

Zoom In Zoom Out Reset image size

The source position in the phase-referenced images was measured using JMFIT in AIPS (Greisen 2003), adopting a single-ellipse Gaussian profile model. However, we consider that, because the core and jets were blended, the core position potentially drifted toward the jet side from its true position. The jet contribution was evaluated by comparing results by two methods of measuring the core position in final (self-calibrated) images of NGC 4261 and J1222+0413: one-component JMFIT and fitting of visibility data to multiple circular Gaussian components using modelfit in Difmap (Figure 2). The final core position was defined as the positions between the core in the phase-referenced image of NGC 4261 with the correction of jet contribution on its final image and the core of J1222+0413 in the final images with the correction of jet contribution.

Zoom In Zoom Out Reset image size

Following the method of Hada et al. (2011), we estimated the error in our astrometric measurements. The error incorporates ionospheric residuals, tropospheric residuals, antenna positional error, Earth orientation error, a priori source coordinate error, core identification error, and thermal noise. The error contributions are listed in Table 1. Based on GPS data, we assume a total electron content of $3\times {10}^{17}\;{{\rm{m}}}^{-2}$, which is reducible to 1/4 equivalent following ionospheric residual correction by TECOR in AIPS (Mannucci et al. 1999). The zenith excess path error in tropospheric residuals is assumed to be 2 cm (Beasley & Conway 1995). We adopted proper geometric errors such as antenna location error (3 mm), Earth orientation parameter error (VLBA memo 69), and a priori source coordinate error that is determined from source declination (Pradel et al. 2006). The core identification error was estimated from the difference in core positions yielded by the above-described two fitting methods. The thermal noise error was defined by the beam size over the dynamic range of phase-referenced maps.

Figure 1 shows the (self-calibrated) CLEANed images from uniformly weighted visibilities at each frequency; the corrections of relative positions in R.A., which were determined by phase-referencing (Section 2.3), have been applied in Figure 3. The beam sizes and image qualities are listed in Table 2. Approaching jets and counterjets, extending west and east from the brightest region (core), respectively, appear in the images at $\leqslant$15 GHz clearly. The counterjet is not so prominent in the image recorded at 22 GHz, while no obvious jet-like structures are visible in the 43 GHz image. However, both approaching jets and counterjets were clearly detected as model components of modelfit in Difmap. The lower the frequency, the more extended and symmetric the jets.

Zoom In Zoom Out Reset image size

The determined core positions at 1.4–43 GHz are plotted on the 5 GHz self-calibrated image of NGC 4261 in Figure 4. The core appears to shift northwestward as the frequency decreases. The jet of the reference source J1222+0413 is oriented nearly perpendicular to that of NGC 4261 (toward the south, whereas the NCG 4261 jet tends to the west; see Figure 5). For this reason, the observed core shifts compose the sum of core shifts of both sources. Therefore, we decomposed the observed core shift into a westward shift for NGC 4261 and a northward shift for J1222+0413.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The east–west structure of NGC 4261's jet appears slightly fluctuating in the north–south direction. On the approaching side, the jet gradually inclines toward south beyond ∼4 mas from the 43 GHz core. We analyzed the ridge line in the 5 GHz image and measured its deviation to be −0.24 mas southward at −5 mas, where the core at 1 GHz is found. This southward deviation is used to correct the core position at 1 GHz for the final image to be used for making a spectral index map (Section 3.3). On the other hand, in the innermost region, the jet shows an S-shaped structure at 15–43 GHz images. The measured position angle of the jet is $-70^\circ$ in the 43 GHz image, while it is $-85^\circ$ up to 1.5 mas from the core in the 15 GHz image. The amplitude of the S-shape is only ∼50 μas, which is much smaller than the scale of core shifts in J1222+0413 and NGC 4261. We do not take into account this north–south deviation for the final results of the core shift and the spectral index maps.

Figure 4 reveals that the core position of NGC 4261 shifts at all other observational frequencies relative to the 43 GHz core. By least-squares fitting, the core position is found to shift as a power-law function of frequency, that is,

$$\begin{eqnarray}&&r\left(\nu \right)={\rm{\Omega }}\cdot {\nu }^{-k}+b,\end{eqnarray} \tag{ 1 }$$

where $r(\nu )$, Ω, k, and b are projected (east–west) distance from the 43 GHz core, free-fitting parameters representing a scale factor, a power-law index, and an asymptotic limit at infinite frequency, respectively (solid curves in Figure 4). At the approaching-jet side, these values are determined as ${\rm{\Omega }}=-8.42\pm 0.86$, $k=1.22\pm 0.06$, and $b=0.082\pm 0.016$, respectively, with residuals ${\chi }^{2}$/n.d.f = 0.16 (see Table 4). This reduced ${\chi }^{2}$ is very small because we estimated the core positional errors conservatively. The core positional errors could be actually smaller than our estimations. As a result, the free-fitting parameters, Ω, k, and b, are determined with an uncertainty of 10%, 5%, and 20%, respectively. The value of k suggests a possible excess over the theoretical value of unity predicted for SSA under the equipartition condition and as frequently observed (Lobanov 1998; O'Sullivan & Gabuzda 2009; Hada et al. 2011; Sokolovsky et al. 2011). The asymptotic parameter b was determined within a tolerance of 32 μas (≈120 R_s) and separated by 82 μas (≈310 R_s) from the 43 GHz core position.

We also measured and fitted the core shift at the counterjet side of NGC 4261. In this study, the core of the counterjet is defined as the closest east-side component to the core of the approaching jet, the region of maximum intensity yielded by modelfit. As shown in Figure 6, the distance between the cores in the approaching jets and counterjets reduces as the frequency increases. When the shifting core of the counterjet was fitted to a three-parameter model, we obtained ${\rm{\Omega }}=-8.34\pm 6.40$, $k=0.16\pm 0.23$, and $b=-4.34\pm 7.38$, respectively. However, these errors have little contribution for confining the position of the central engine. This suggests that a simple power law is insufficient to represent the profile of the counterjet side. Hence, we applied two alternative model fitting procedures imposing further assumptions, which will be discussed in Section 4.3.

Zoom In Zoom Out Reset image size

Spectral index (α, $S\propto {\nu }^{\alpha }$) maps were generated from final (self-calibrated and CLEANed) images between two adjacent frequencies using the AIPS task COMB. To ensure proper construction of spectral index maps, the images between the adjacent frequencies must spatially align, and their angular resolutions must match. The map coordinates at each frequency were aligned using the core shift relative to the 43 GHz core in right ascension and the jet axis of NGC 4261 in declination. The images at adjacent frequencies were convolved by rendering both beams equivalent to the beam at the lower frequency. The intensity below 3σ was clipped out before calculating the spectral index.

All of the spectral index maps exhibit common features, in particular, the spectral index peaks in the regions between the cores of the approaching jets and counterjets. As the jet extends further from the core, the spectral index becomes smaller. Figure 7 displays the spectral index maps of the 1–2 GHz and 5–8 GHz adjacent frequency pairs. Some regions show $\alpha \gt 2.5,$ which is the theoretical limit of SSA. This result is consistent with other observations in which the circumnuclear structure favored FFA over SSA (Jones et al. 2001). The spectral index gradients are clearly parallel to the jet axis for all frequency pairs. The restored beams in each map are too large to decompose the absorption structures across the jet. However, a transverse gradient marginally appears in 1–2 GHz and 2–5 GHz spectral index maps. We cannot rule out that the combination of the north–south fluctuation in the jet structure (Section 3.2) and remaining position errors in the east–west direction causes the apparent transverse gradient.

Zoom In Zoom Out Reset image size

Figure 8 shows slice profiles of spectral index along the jet axis (from the east to the west). We estimated error components in the spectral index. We tentatively displaced one image eastward and westward with respect to the other image by our conservative positional error, in order to evaluate a spectral index error due to the positional error in overlaying two images. We also used the spectral error map, simultaneously produced during COMB processing, and assumed a conservative flux calibration error of 5% at $\nu \leqslant 15$ GHz and 10% at $\nu \geqslant 22$ GHz. The total error represented by an envelope is the square root of the summed squares of these error components.

Zoom In Zoom Out Reset image size

At 8–15 GHz and lower-frequency pairs, a region of $\alpha \gt 2.5$ appears in the slice profiles, suggesting an FFA mechanism, although the error is too high for a conclusive assessment. Another feature is that the curve at ν $\geqslant$ 5 GHz peaks at the counterjet side. The peaks are located at the same positions, regardless of the error-induced displacement of the core position. At ν $\leqslant$ 5 GHz, the peaks are located around the origin.

First, we consider the relationships between the black hole positions and limiting core positions. Hada et al. (2011) showed that the central engine in M87 is located $23\;{R}_{{\rm{s}}}$ distant from the observed core at 43 GHz in their analyses of core shift measurements. This study assumed that the asymptotic line, representing the origin of a jet, spatially coincides with the central engine. However, the core location in BL Lac, indicated as the standing shock point (Marscher et al. 2008), is actually 10⁴R_S–${10}^{6}{R}_{{\rm{s}}}$ from the black hole, as determined from γ-ray monitoring observations (Albert et al. 2007). On the other hand, a core shift was discovered downstream of the BL Lac jet. Core shift measurements of BL Lac and other blazar sources revealed that the base of the jet is located ${10}^{4}{R}_{{\rm{s}}}$ from the 43 GHz core position (O'Sullivan & Gabuzda 2009). Recently, core shift data have accumulated at an accelerated pace for blazars (Sokolovsky et al. 2011; Pushkarev et al. 2012). However, the above-cited examples highlight the difficulty of proving assumptions from one-sided jets.

To overcome these limitations, we located the central engine in NGC 4261 by investigating a double-sided system and measuring the core positions of the approaching jet and the counterjet. As shown in Figure 9, the core (most upstream component) on the counterjet side resides at 0.17 ± 0.03 mas ($\approx 640\pm 110\;{R}_{{\rm{s}}}$) from the approaching-jet core at 43 GHz. The central engine and black hole must be situated at most 750 R_s distant from this core to ensure its location between the cores of both jets. Moreover, because the origin of the counterjet coincides with that of the approaching jet (as discussed later), the assumption that the origin of a jet spatially coincides with the central engine within the error tolerance is reasonable for NGC 4261. The center of activities was discussed using cross-identification of the Gaussian component models in multifrequency maps in other two-sided jet systems (e.g., Cygnus A, Krichbaum et al. 1998; NGC 1052, Kadler et al. 2004). For Cygnus A, a relatively distant source, the core shift in the counterjet has not been measured. Kadler et al. (2004) determined the position of the central engine in NGC 1052 with an uncertainty of ∼0.03 pc, corresponding to $\sim 2000\ {R}_{{\rm{s}}}$ assuming $\mathrm{log}{M}_{\mathrm{BH}}/{M}_{\odot }={10}^{8.19}$ (Woo & Urry 2002), from both sides of the jets. Our result for NGC 4261 is of higher accuracy and includes coordinates relative to 1222+0413 rather than relative to optically thin components in the source itself.

Zoom In Zoom Out Reset image size

The central engine of NGC 4261 is located at 82 ± 16 μ as, corresponding to $(310\pm 60)\;{R}_{{\rm{s}}}$ from the 43 GHz core. The core–central engine separation differs by orders of magnitude among NGC 4261, M87, and BL Lac, being larger and smaller in M87 and BL Lac, respectively, than in NCG 4261. The separation caused by a core shift is correlated with the strength of SSA and depends on jet characteristics such as luminosity, velocity, structure, radial profiles of density and magnetic field, the geometric relationship to our line of sight, velocity gradients, and/or their time variation. According to Lobanov (1998), if a jet is continuous, is uniform, is stationary, and travels with constant velocity, and under equipartition condition, the (projected) core shift from the jet origin is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{r}_{\mathrm{mas}}\approx \frac{{C}_{{\rm{r}}}\left(1+z\right)}{{D}_{L}{\gamma }^{2}{\phi }_{o}{\nu }_{1}}{\left[\frac{{L}_{\mathrm{syn}}\mathrm{sin}\theta }{\beta \left(1-\beta \mathrm{cos}\theta \right){\rm{\Theta }}}\right]}^{2/3},\end{eqnarray} \tag{ 2 }$$

where ${C}_{{\rm{r}}}=4.56\times {10}^{-12}$ and $\gamma \equiv 1/\sqrt{1-{\beta }^{2}}$, $\beta \equiv v/c$, ${\phi }_{o}$, and θ denote the Lorentz factor, intrinsic jet velocity, jet opening angle from an observer (intrinsic opening angle $\phi ={\phi }_{o}/\mathrm{sin}\theta$), and viewing angle, respectively. D_L and L_syn are the luminosity distance to the source (pc) and synchrotron luminosity (erg s⁻¹). In this case, Θ depends on the jet structure, whether conical or parabolic, and does not usually exceed ln 3 for parsec-scale jets. The frequency ν is given in Hz. In the present study, the separation between the central engine and core position is determined at 43 GHz. Using Equation (2), we now compare the separation in NGC 4261 with those estimated in M87 and BL Lac. In place of L_syn, we use the intrinsic luminosity at millimeter wavelengths, L_m. In order to remove the beaming effect from the observed luminosity ${L}_{{\rm{m}}}^{\dagger }$, we calculate the beaming factor ${D}^{3-\alpha }$, where $D={\left[\gamma (1-\beta \mathrm{cos}\theta )\right]}^{-1}$ is the Doppler factor, using the viewing angle and intrinsic velocity for each source. Using $\theta =63^\circ$, $\beta =0.46$ for NGC 4261 (Piner et al. 2001) and ${L}_{{\rm{m}}}^{\dagger }=3.9\times {10}^{40}$ erg s⁻¹ (Geldzahler & Witzel 1981), we estimated a beaming factor of 1.5 and ${L}_{{\rm{m}}}=2.4\times {10}^{40}$ erg s⁻¹, assuming $\alpha =-0.7$. The parameters of NGC 4261, M87, and BL Lac are provided in Table 5.

Table 5.  Parameters for Comparisons

Object Name	${\rm{\Delta }}{r}_{43}$	MBH	β	θ	D	${L}_{{\rm{m}}}^{\dagger }$	Lm
	(Rs)	(108M⊙)				(1040 erg s−1)	(1040 erg s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
NGC 4261	310 ± 60	4.9a	0.46d	63°d	1.1	3.9h	2.5
M 87	23 ± 6	60b	0.38–0.62e	15°–25°g	1.4–1.7	5.2i	1.4
BL Lac	104–106	1.7c	≧0.99f	6°–10°f	5.6–9.1	3.6 × 103j	1.0

Notes. Columns are as follows: (1) object name; (2) distance of core at 43 GHz from putative central engine; (3) black hole mass; (4) jet velocity in the unit of speed of light; (5) estimated viewing angle; (6) Doppler factor, $D\equiv {\left[\gamma (1-\beta \mathrm{cos}\theta )\right]}^{-1}$; (7) observed luminosity at millimeter wave range; (8) estimated intrinsic luminosity.

^aFerrarese et al. (1996). ^bGebhardt & Thomas (2009). ^cWoo & Urry (2002). ^dPiner et al. (2001). ^eCalculating from ${\beta }_{\mathrm{app}}=0.25$–0.40 (Ly et al. 2007). ^fJorstad et al. (2005). ^gAcciari et al. (2009). ^hGeldzahler & Witzel (1981). ⁱDavies et al. (2009). ^jAgudo et al. (2010).

Download table as:  ASCIITypeset image

As a rough discussion, assuming that the jet opening angle is nearly constant⁶ under Equation (2), the ratio of the core–central engine (deprojected) separation to that of NGC 4261 in R_s units is given by

$$\begin{eqnarray}{\rm{\Delta }}{r}_{\mathrm{ratio}} & \approx & 1.92\times {\left(1+z\right)}^{-1}{\left(\frac{{\left(1-{\beta }^{2}\right)}^{3/2}\mathrm{sin}\theta }{\beta \left(1-\beta \mathrm{cos}\theta \right)}\right)}^{2/3}\\ & & \times {\left(\frac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{-1}{\left(\frac{{L}_{{\rm{m}}}}{{10}^{40}\mathrm{erg}/{\rm{s}}}\right)}^{2/3}.\end{eqnarray} \tag{ 3 }$$

For the M87 case, the viewing angle of the jet is likely 15°–25° (Acciari et al. 2009), and apparent jet velocities of ${\beta }_{\mathrm{app}}=0.25$–0.40 have been reported for the innermost region at 43 GHz (Ly et al. 2007). Under these situations, the intrinsic velocity would be $\beta =0.38$–0.62. Assuming $\theta =25^\circ$, $\beta =0.38,$ and ${L}_{{\rm{m}}}^{\dagger }=5.2\times {10}^{40}$ erg s⁻¹ (Davies et al. 2009), we obtain ${L}_{{\rm{m}}}\approx 1.5\times {10}^{40}$ erg s⁻¹. Adopting a black hole mass of $6\times {10}^{9}{M}_{\odot }$ (Gebhardt & Thomas 2009), we obtain ${\rm{\Delta }}{r}_{\mathrm{ratio}}=0.05$. For $\theta =15^\circ$ and $\beta =0.62$, we obtain ${L}_{{\rm{m}}}\approx 0.43\times {10}^{40}$ erg s⁻¹ and ${\rm{\Delta }}{r}_{\mathrm{ratio}}=0.01$. On the other hand, based on the core shift measurements, the 43 GHz core–central engine distance ratio of $23{R}_{{\rm{s}}}$ (M87) to $310{R}_{{\rm{s}}}$ (NGC 4261) is 0.07. The difference between the two results is not so large. The discrepancy may originate in uncertainties of the black hole mass or the intrinsic luminosity. Another possibility is the difference of opening angles between different sources. From Equation (2), the separation is inversely proportional to the opening angle. If the opening angle of M87 is widely larger (see Junor et al. 1999) than that of NGC 4261, it can explain the difference of the separation between them.

By contrast, the core shift in BL Lac substantially differs from that of NGC 4261. The estimated jet velocity and inclination are $\beta =0.99$ and $\theta =6^\circ$–10°, respectively (Jorstad et al. 2005). Such conditions should enhance the relativistic beaming effect and indicate a beaming factor of $3.6\times {10}^{3}$. Since the observed luminosity was reported to be $3.6\times {10}^{43}$ erg s⁻¹, we estimate an intrinsic luminosity of ${L}_{{\rm{m}}}\approx 1\times {10}^{40}$ erg s⁻¹. The mass of the black hole associated with BL Lac is estimated to be $1.7\times {10}^{8}{M}_{\odot }$ (Woo & Urry 2002). Substituting these values into Equation (3), we obtain ${\rm{\Delta }}{r}_{\mathrm{ratio}}\approx 0.08$ as an expected core shift, showing a large discrepancy from the result from γ-ray monitoring observations (∼30–3000; Albert et al. 2007; Marscher et al. 2008). To explain the large discrepancy, a much larger intrinsic luminosity of ${L}_{{\rm{m}}}\geqslant 7\times {10}^{43}$ erg s⁻¹ or a much smaller opening angle ${\phi }_{o}\leqslant 0\buildrel{\circ}\over{.} 08$ is necessary.

However, such a wide difference in intrinsic luminosity between NGC 4261 and BL Lac is quite unlikely if both FR-I radio galaxies and BL Lac objects are regarded as a common AGN population intrinsically (Urry et al. 1991). More likely, the appearance of FR-I radio galaxies and that of BL Lac objects differ because of the beaming effect imposed by jet inclination. The jet of M87 has a limb-brightened structure, which is probably associated with a fast spinal flow surrounded by a slow layer (sheath) (Ghisellini et al. 2005). If the jet axis inclines significantly from the line of sight, such as for M87 and NGC 4261 as FR-Is, sheath emission dominates because spinal emission is deboosted by the relativistic effect. On the other hand, if the jet is viewed from nearly pole-on, such as for BL Lac, spine emission from a small opening angle dominates. This phenomenon can explain the varying distance between the central engine and the 43 GHz core between FR-I radio galaxies and BL Lac objects.

The physical causes of the core shift are considered as SSA and/or FFA. The core shift measurement for a two-sided system allows us to utilize a symmetry assumption for understanding the jet structure and surrounding circumstances and offers a critical advantage over studies for one-sided systems. Only the counterjet itself could not well describe its core shift profile (Section 3.2). In the following subsections we establish the model that best explains the core shift of NGC 4261, focusing on the results of fitting parameters in Equation (1).

4.3.1. SSA in Jets

First, we assume a pure SSA model. The physical conditions in the jet reflect the index of the core shift profile in Equation (1). In this model, k_s (representing k for the SSA case) is given by

$$\begin{eqnarray}&&{k}_{{\rm{s}}}=\frac{5-2\alpha }{\left(3-2\alpha \right)m+2n-2},\end{eqnarray} \tag{ 4 }$$

where α, m, and n denote the spectral index at the optically thin regime, the profiles of the magnetic field B (where $B\propto {r}^{-m}$), and the electron density N (where $N\propto {r}^{-n}$), respectively (Blandford & Königl 1979). Assuming m = 1 and n = 2 as the profile indices, corresponding to the stabilization of jet energy in equipartition, k_s invariably becomes unity regardless of the spectral index value. However, the fitting parameter on the approaching-jet side is ${k}_{{\rm{s}}}=1.22\pm 0.06$, which possibly exceeds that from previous studies for other sources (${k}_{{\rm{s}}}\approx 1$; i.e., M87, Hada et al. 2011; BL Lac, O'Sullivan & Gabuzda 2009; and 3C 345, Lobanov 1998), indicating that m and n in Equation (4) can take different values in NGC 4261.

We next predict the core shift profile on the counterjet side assuming the SSA model. If the jets are intrinsically symmetric, k_s and b of the counterjet and approaching-jet sides should be identical. On the other hand, comparing Equations (1) and (2), we find that Ω depends on geometrical and intrinsic parameters of a jet. The parameter Ω differs between the approaching jet and counterjet only through the inclination, θ. Therefore, Ω of the counterjet and approaching-jet sides, denoted ${{\rm{\Omega }}}_{\mathrm{CJ}}$ and ${{\rm{\Omega }}}_{\mathrm{AJ}}$, respectively, must satisfy ${{\rm{\Omega }}}_{\mathrm{AJ}}\gt$ ${{\rm{\Omega }}}_{\mathrm{CJ}}$. In other words, the position of the core at the same frequency is farther from the black hole in the approaching jet than in the counterjet. The parameter ${{\rm{\Omega }}}_{\mathrm{CJ}}$ should be connected with ${{\rm{\Omega }}}_{\mathrm{AJ}}$ by a difference of 180° in terms of viewing angle. The curve predicted by the pure SSA model (dashed line in Figure 6) can explain the core position at both low frequency ($\leqslant$2 GHz) and high frequency ($\geqslant$22 GHz). On the other hand, the observed positions of the core at 5, 8, and 15 GHz are significantly larger than those predicted by this model, implying that the SSA model alone lacks sufficient absorption to explain the core shift in these regions.

4.3.2. Spherical FFA Plasma

Another alternative is the FFA model. The spectral index maps indicate that FFA plasma exists at least in regions near the putative black hole. For simplicity, and from the parameters of the core shift measurement, we assume a spherical FFA plasma (see Appendix B). In this case, k depends on the structure of the spherical FFA plasma. When the FFA opacity ${\tau }_{f}$ becomes unity, k_f (representing k in Equation (1) for the FFA model) is given by

$$\begin{eqnarray}&&{k}_{f}=\frac{2.1}{2s-\frac{3}{2}t-1},\end{eqnarray} \tag{ 5 }$$

where s and t represent the profiles of the electron density n_e and temperature T_e (${n}_{{\rm{e}}}\propto {r}^{-s}$, ${T}_{{\rm{e}}}\propto {r}^{-t}$) in the FFA absorber, respectively. These parameters can be determined from the opacity profile; the denominator on the right-hand side of Equation (5) represents the index value of the opacity profile (${\tau }_{f}=2\;s-\frac{3}{2}t-1$). We obtain the radial profile of FFA opacity on the approaching-jet side by fitting to a power-law function: ${\tau }_{f}\propto {r}^{-1.6}$ (Appendix B). Substituting this value into Equation (5), we obtain ${k}_{f}=1.3$. It is consistent with the result of the core shift measurements (${k}_{f}=1.22\pm 0.06$), in spite of independent measurements of the core shift and the opacity. Theoretical speculations have indicated that k_f exceeds unity (Lobanov 1998; Kadler et al. 2004). Thus, the core shift on the approaching-jet side can also be sufficiently interpreted by the spherical FFA model.

However, the spherical FFA is inconsistent with the radial profile of FFA opacity on the counterjet side. The parameter k_f should be identical in both the approaching-jet and counterjet sides if the FFA absorber is symmetric. Ω reflects the path length of the absorber, that is, the geometrical structure of the absorber (for details, see Appendix B). If the absorber covers the counterjet side, Ω must satisfy ${{\rm{\Omega }}}_{\mathrm{AJ}}\lt {{\rm{\Omega }}}_{\mathrm{CJ}}$ because the path length of the counterjet side is longer than that of the approaching one. The core-shift curve predicted by the spherical FFA plasma model (dot–dashed line in Figure 6) meets some of the core positions at $\geqslant$ 5 GHz on the counterjet side. However, the core shifts at 1 and 2 GHz are inconsistent because their shifts are significantly smaller than those predicted by this model.

Therefore, neither of these simple models can explain the core shifts in this two-sided system.

4.3.3. SSA in Jet + FFA by the Disk

A possible solution is to combine the SSA core shift with an additional external absorber, which compensates for the inadequate core shift. Indeed, our results show that the pure SSA model fitted fairly well on both approaching-jet and counterjet sides at all frequencies, except for the counterjet components at 5, 8, and 15 GHz. When the observed core shifts on the counterjet side are fitted without the cores at these frequencies, the best-fit values of $k=1.15\pm 0.06$ and $b=0.08\pm 0.02$ (representing the index and the asymptotic limit in this condition) approximate those of the approaching-jet side, as expected from SSA theory (Table 4). This suggests the presence of an absorber in a limited region. Thus, it appears that the cores at 5, 8, and 15 GHz are additionally affected by FFA. FFA opacity ${\tau }_{f}$ at the positions of 2–22 GHz cores in the counterjet is determined by spectral analysis of multifrequency images (Appendix A) and shown in Figure 10. On the 5–15 GHz core, ${\tau }_{f}$ is well represented by a power-law profile. On the other hand, ${\tau }_{f}$ at other frequencies, where SSA alone can describe the core shift profile, should be ${\tau }_{f}\lt 1$.

Zoom In Zoom Out Reset image size

Such a limited range of FFA regions is consistent with a gap found in the intensity profile on the counterjet side in previous studies (e.g., Jones et al. 2001) and an obscuring disk proposed by them. Thus, the geometrically thin disk predicted is a candidate to explain the core shift of the counterjet side.

We consider the structure of a disk to explain the core shift and the profile of ${\tau }_{f}$ on the counterjet side.

4.4.1. Disk Model

Figure 11 shows the geometric relationships between the jet and disk, assuming that $\theta =63^\circ$ and the rotation axis of the disk is aligned with the axis of the parsec-scale jet. As discussed in the combined model, if the disk plasma causes the core shift at 5–15 GHz, the density and temperature of the disk require the radial gradient (${n}_{{\rm{e}}}={n}_{1}{x}^{-s}$, ${T}_{{\rm{e}}}={T}_{1}{x}^{-t}$, where x is the radius of the disk in R_s units, $r={{xR}}_{{\rm{s}}}$). In addition, the thickness of the disk and the path length through the disk to observers are also described as $H={H}_{1}{x}^{-h}$ and $L=H\mathrm{sec}\theta$. By substituting n_e, T_e, and L into Equation (A2), we obtain ${\tau }_{{\rm{f}}}$ by the disk (hereafter ${\tau }_{\mathrm{disk}}$), that is,

$$\begin{eqnarray}&&{\tau }_{\mathrm{disk}}=0.46{n}_{1}^{2}{T}_{1}^{-3/2}{H}_{1}\mathrm{sec}\theta {x}^{-(2s-3\;t/2-h)}.\end{eqnarray} \tag{ 6 }$$

Zoom In Zoom Out Reset image size

In Figure 10, the least-squares fitting at 5–15 GHz data reveals the profile of disk opacity to the power-law function of the disk radius, that is, ${\tau }_{{\rm{f}}}\propto {r}^{-3};$ then we obtain

$$\begin{eqnarray}&&2s-\frac{3}{2}t-h=3.\end{eqnarray} \tag{ 7 }$$

4.4.2. Core Shifts at $\nu \;\lt$ 2 GHz and $\nu \;\gt$ 22 GHz Do Not Require Disk Absorption

To explain the small opacity at low frequency ($\leqslant$2 GHz), the low density, the short path length, or high temperature is required because the FFA effects depend on frequency ($\tau \propto {\nu }^{-2.1};$ see Appendix A). In these requirements, the condition of the path length and the temperature are physically unrealistic because the low-frequency core positions correspond to the outer radii of the disk. On the other hand, it is realistic that the density is sufficiently small and/or the temperature of the plasma is too low to be ionized at the outer radii.

We consider the condition that the temperature at the radius corresponding to the 5 GHz core position and at the 2 GHz position are higher and lower than 10⁴ K, respectively. Assuming that $t=3/4$ predicted by the standard disk model at the outer region (Shakura & Sunyaev 1973) and 10⁴ K at 5 GHz, we obtain

$$\begin{eqnarray}&&{T}_{{\rm{e}}}=1\times {10}^{7}{x}^{-3/4}\ \left({\rm{K}}\right).\end{eqnarray} \tag{ 8 }$$

In addition, if h = 0 (wherever the path length is constant) and ${H}_{1}=0.01$ pc, for simplicity, we obtain s = 2 from Equation (7). It is very close to $s=15/8$ predicted by the standard disk model. Inserting these value and ${\tau }_{\mathrm{disk}}$ into Equation (6), we obtain

$$\begin{eqnarray}&&{n}_{{\rm{e}}}=8\times {10}^{12}{x}^{-2}\;\;\left({\mathrm{cm}}^{-3}\right).\end{eqnarray} \tag{ 9 }$$

The density and temperature profile in this condition are shown in Figure 11.

We note that the assumptions of h and H₁ may be unsupported and thus other values are acceptable. Substituting $h=-9/8$ (predicted by the standard disk model) into Equation (7) and ${H}_{1}=2\times {10}^{-6}$ pc (corresponding to L = 0.01 pc at the 5 GHz core), we obtain s = 2.6 and ${n}_{1}=2\times {10}^{15}$ cm⁻³. Although the index s increases the gap to the value predicted by the model, n_e at $x={10}^{3}$–10⁵ is almost the same, within a factor of a few, as the case of h = 0. In the case of H₁, ${n}_{{\rm{e}}}\propto {H}_{1}^{-0.5}$ is obtained from Equation (6), indicating that the thinner disk geometrically requires the higher density (i.e., ${n}_{1}=8\times {10}^{13}$ cm⁻³ at ${H}_{1}={10}^{-4}$ pc). However, if the disk thickness was much larger (i.e., ${H}_{1}\gt 1$ pc), the disk would obscure the approaching jet as well as the counterjet. Thus, ${H}_{1}=0.01$ pc provides a reasonable upper limit, indicating the lower limit of n_e.

On the other hand, this density profile cannot explain the small opacity at the high frequency ($\geqslant$22 GHz, $x=1.5\times {10}^{3}).$ The interpretation under a simple power-law profile requires a large t such as $t\geqslant 3$, which leads to a sufficiently high temperature of plasma to reduce the FFA effect at the inner region (22 GHz). However, this is impractical because the disk temperature at 3 R_s should be too high (10¹⁴ K). The weak effects of FFA at the 22 GHz core indicate that the disk structure changes dramatically at a characteristic radius.

The RIAF model is suitable for describing this situation of the low density at inner radii of an accretion. In fact, X-ray observations suggest that a low luminosity and flux and spectral variations of the nuclear X-ray emission of NGC 4261 can be caused by an accretion disk with a low accretion rate with a low radiative efficiency such as RIAF (Gliozzi et al. 2003). Therefore, we propose that the accretion flow of NGC 4261 has a two-component structure: one is an inner, hot, optically thin RIAF, and the other is an outer, truncated, cool, optically thick accretion disk. This structure is considered to be not expedient for NGC 4261 but common for LLAGNs based on broad SED and X-ray spectral studies (Nemmen et al. 2006; Nandra et al. 2007).

4.4.3. Comparison with Theoretical Models for Transition Radius in an Accretion Flow/Disk

According to Liu et al. (1999), the transition radius, x_c, in an accretion disk is given by

$$\begin{eqnarray}&&{x}_{c}=18.3{m}_{b}^{0.17/1.17}{\dot{m}}_{b}^{-1/1.17}\end{eqnarray} \tag{ 10 }$$

where m_b and ${\dot{m}}_{b}$ denote the black hole mass in units of solar masses, ${M}_{\mathrm{BH}}={m}_{b}{M}_{\odot },$ and the mass accretion rate in Eddington units, $\dot{M}={\dot{m}}_{b}{\dot{M}}_{\mathrm{Edd}}$, ${\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{Edd}}/0.1{c}^{2}=1.39\times {10}^{18}{m}_{b}$ g s⁻¹. Inserting ${x}_{c}=2\times {10}^{3}$ and ${m}_{b}=4.9\times {10}^{8}$ into Equation (10), we obtain ${\dot{m}}_{b}=0.12$, requiring a significantly higher accretion rate compared to that of the typical LLAGNs (${\dot{m}}_{b}={10}^{-2}$–10⁻³). Recently, the equation of the transition radius is modified by the disk evaporation model taking account of the effects of viscosity and magnetic field in accretion flow (Qiao et al. 2013). In this case, ${x}_{c}=2\times {10}^{3}$ is easily acceptable for ${\dot{m}}_{b}\sim {10}^{-3}$, which is an expected accretion rate for NGC 4261.

Consequently, this combined model of RIAF and truncated disk is a possible solution to explain the core shift of NGC 4261 on both sides simultaneously. Figure 11 shows this schematic illustration.

4.4.4. The Structure of Absorber and the Location of X-Ray Emission