Core shift in parabolic accelerating jets

ABSTRACT

The core-shift method is a powerful method to estimate the physical parameters in relativistic jets from active galactic nuclei. The classical approach assumes a conical geometry of jets and a constant plasma speed. However, recent observations have shown that neither may hold close to the central engine, where the plasma in the jet is effectively accelerating and the jet geometry is quasi-parabolic. We modify the classical core-shift method to account for these jet properties. We show that the core-shift index may assume values in the range 0.8−1.2 or 0.53−0.8 depending on the jet geometry and viewing angle, and indices close to both values are indeed observed. We obtain expressions to estimate the jet magnetic field and the total magnetic flux in a jet. We show that the obtained magnetic field value can be easily recalculated down to the gravitational radius scale. For M87 and NGC 315, these values are in good agreement with those obtained by different methods.

1 INTRODUCTION

It is now generally accepted as a model that the ordered magnetic field in the vicinity of supermassive black holes (BHs) residing in the centres of active galactic nuclei plays the key role in launching relativistic jets (Blandford, Meier & Readhead 2019). It is the presence of the magnetic field that either threads the BH ergosphere and establishes the Blandford–Znajek process (Blandford & Znajek 1977), responsible for the extraction of BH rotational energy to supply the jet power (Tchekhovskoy, Narayan & McKinney 2011), or launches the jet from the disc via the Blandford–Payne mechanism (Blandford & Payne 1982). The Poynting flux at the jet base defines the maximum Lorentz factor that can be achieved if all the energy of the electromagnetic field is transformed into plasma bulk motion kinetic energy (e.g. Beskin 2010).

Many efforts have been made to understand the magnitude and geometry of the jet magnetic field at different scales. Polarization measurements in the radio band yield information about the possible B-field geometry. Multiple studies at parsec scales have revealed the helical magnetic field expected from theoretical modelling (Gabuzda et al. 2017; Gabuzda, Nagle & Roche 2018). The unprecedented radio images of M87 confirmed that the helical magnetic field B is maintained at scales from 300 pc up to 1 kpc (Pasetto et al. 2021). The phenomenon of the observed Faraday rotation measure (RM) changing sign can be explained by the presence of a toroidal magnetic field around the jet in 3C 273 up to distances of the order of 500 pc de-projected (Lisakov et al. 2021). The detection of a large-scale toroidal magnetic field at kiloparsec scales (Knuettel, Gabuzda & O’Sullivan 2017) indicates the strong electric current flowing in relativistic jets far from the expected acceleration and collimation domain (Blandford, Meier & Readhead 2019; Kovalev et al. 2020), showing the importance of the Poynting flux even at such distances from the central engine. Direct polarimetry of the event horizon scales in the M87 jet conducted by Event Horizon Telescope Collaboration (2021) gives the B-field magnitude to be of the order of a few to tens of gauss.

One of the most robust methods to estimate the magnetic field amplitude on an observational basis is to use core-shift measurements. The core-shift effect is observed in jets owing to the changing physical properties (flow velocity, magnetic field, and emitting-plasma number density) along an outflow, so that the surface with optical depth equal to unity appears at different positions for different observational frequencies (Lobanov 1998; Hirotani 2005). Together with the model assumptions, core-shift measurements provide typical values for the magnetic field of the order of 1 G at 1-pc distance (Lobanov 1998). The results obtained using this method are in general agreement with other expected jet parameters, such as initial magnetization and jet composition (Hirotani 2005; O’Sullivan & Gabuzda 2009; Pushkarev et al. 2012; Nokhrina et al. 2015; Zdziarski et al. 2015).

The core-shift method as its important ingredients makes the following assumptions: equipartition between the proper-frame emitting particles and the magnetic field energy density, certain scalings with the distance of the magnetic field B and plasma number density n (Blandford & Königl 1979), and a conical jet geometry. The observed jet boundary shape – the dependence of the jet width d on the distance along a jet r – can be approximated by the power law

An extensive study of jet shapes (see e.g. Pushkarev et al. 2017, and references within) reveals an approximately constant power k ≈ 1 in many sources, corresponding to a conical or quasi-conical geometry. However, recent studies of a dozen nearby jets demonstrate that their shapes are consistent with two power laws, smoothly transitioning from one to another on a more or less short scale of the overall length range (Asada & Nakamura 2012; Tseng et al. 2016; Akiyama et al. 2018; Hada et al. 2018; Nakahara et al. 2018, 2019; Kovalev et al. 2020; Nakahara et al. 2020; Boccardi et al. 2021; Park et al. 2021; Okino et al. 2022). For these sources, typical values of k closer to the jet launching point are in the range k ≈ 0.39−0.59. We refer to this as quasi-parabolic, or parabolic for short. Further downstream, the shape becomes closer to conical (quasi-conical), with k ≈ 0.73−1.4 (e.g. Asada & Nakamura 2012; Kovalev et al. 2020; Boccardi et al. 2021). The implicit study of the geometry of more distant sources, which uses the plasma velocity instead of the direct measurements of a jet shape, shows that cores at 15 and 8 GHz belong to the quasi-parabolic part of a jet (Nokhrina, Pashchenko & Kutkin 2022). This means that one may expect the radio cores observed at high frequencies 230−15 GHz, and maybe down to 8 GHz, to be in the parabolic domain and with the plasma still being accelerated. This may impact the results obtained by the classical core-shift method, which uses typical observational frequencies of 8–15 GHz (Pushkarev et al. 2012).

The parabolic accelerating jet model alters the k_r-index in the frequency dependence of the core position |$r_\mathrm{core}\propto \nu ^{-1/k_\mathrm{r}}$| (Ricci et al. 2022). Multifrequency observations for M87 by Hada et al. (2011) and for 20 sources by Sokolovsky et al. (2011) demonstrate that k_r is indeed close to unity. This may be explained either by the conical jet model with a constant Lorentz factor Γ, which may be the case for the sample by Sokolovsky et al. (2011), or by the quasi-parabolic accelerating jet model observed at the maximum Doppler factor (Ricci et al. 2022) for the M87 jet with a clear quasi-parabolic shape. On the other hand, a dedicated search in NGC 315 by Ricci et al. (2022) yielded a change in the k_r- index from 0.57 to 0.93 at approximately the jet-shape transition point. Using different methods, Kutkin et al. (2014) found the exponent k_r in the range 0.6–0.8 for quasar 3C 454.3. The typical k_r-index for the MOJAVE (Monitoring Of Jets in Active galactic nuclei with Very Long Baseline Array Experiments) sample was estimated to be approximately 0.83 by Kravchenko et al. (in preparation).

Accounting for parabolic accelerating jets alters the scalings of the magnetic field B and plasma number density n as functions of distance from the central source r, introduces the non-constant plasma bulk flow motion Lorentz factor Γ, and changes the way of extrapolating the magnetic field amplitude down to the gravitational radius scale (Ricci et al. 2022). In this paper, we modify the core-shift method for a parabolic jet boundary geometry and estimate the magnetic filed amplitude at the gravitational radius for 10 sources.

The paper is organized as follows. In Section 2 we present the model for the magnetic field and emitting-particle number density to obtain the possible core-shift indices k_r for parabolic accelerating jets. In Section 3 we discuss the relationship between the magnetic field and plasma number density and obtain expressions to estimate both values. In Section 4 we review the data to estimate B. We discuss our results and methods in Section 5. Finally, we summarize our work in Section 6.

2 FREQUENCY-DEPENDENT APPARENT CORE SHIFT IN A PARABOLIC ACCELERATING FLOW

The emission and opacity of a jet results from the emitting-plasma population with an assumed power-law energy distribution in the plasma proper frame, designated by the asterisk:

where α is the power of the spectral flux in the optically thin regime: S_ν ∝ ν^−α. Here, K_e* is the amplitude of the emitting-plasma energy distribution:

The relationship between the observational frequency ν and the local properties of a jet at the surface of the peak spectral flux (which approximately coincides with the surface with the optical depth equal to unity) is as follows (e.g. equation 21 in Hirotani 2005):

Here m, e, c, r₀ are the electron mass and charge, the speed of light, and the classical electron radius, respectively; ν₀ = c/r₀; and C₁(0.5) = 2.85 for α = 0.5 (Gould 1979). The local jet width is d, the emitting-electron number density amplitude is K_e*, and the local magnetic field in the plasma proper frame is B_*. The jet viewing angle is θ, and the Doppler factor for a bulk Lorentz factor Γ = (1 − β²)^−0.5 is

In order to obtain the dependence of the core position on the observational frequency, we need to set the dependence of K_e*, B_*, Γ, d, and δ on the distance along the jet r.

The typical frequencies for the core-shift measurements are 1.4−43 GHz (Kovalev et al. 2008; Hada et al. 2011; Sokolovsky et al. 2011; Boccardi et al. 2021) and may be as high as 86 and 230 GHz for M87 (Doeleman et al. 2012; Hada et al. 2016). This means that either all the observational frequencies or the highest ones may probe the jet in its parabolic accelerating domain. Thus, we assume here that the jet shape is given by (1), with the value for power k = 0.5 for a strictly parabolic outflow, and k ≈ 0.4−0.6 for a quasi-parabolic one.

In general, we assume the power laws |$K_{\rm e*}\propto r^{-k_\mathrm{n}}$| and |$B_*\propto r^{-k_\mathrm{b}}$|⁠. The magnetic field in a jet is decomposed into poloidal B_p and toroidal B_φ components. The toroidal component dominates the major part of the jet, r > R_L, as there is a universal relationship B_φ = B_pr/R_L for a relativistic flow (see e.g. Lyubarsky 2009). However, as the velocity of the bulk plasma motion is dominated by the poloidal component, the corresponding relationship in the plasma proper frame between B_p* ≈ B_p and |$B_{\varphi *}\approx \sqrt{B_{\varphi }^2-E^2}$| can be different. We focus on the domain where the jet plasma is still effectively accelerating. In this case, either the co-moving magnetic field is dominated by the poloidal component |$B_\mathrm{p}^2\gg B_{\varphi }^2-E^2$| or both components in the plasma proper frame are comparable, namely |$B_\mathrm{p}^2\sim B_{\varphi }^2-E^2$| (Vlahakis 2004; Komissarov et al. 2009). In this case, we adopt

To keep the final expression as simple as possible we will not account here for the jet transversal structure. Thus, the conservation of the total magnetic flux defines the dependence of B_* on the distance r:

where r_ζ is the distance at which we will estimate the magnetic field and the emitting-plasma number density. It is convenient to use r_ζ = 1 pc, except for sources with the jet shape break detected at smaller distances, as in NGC 315 (Boccardi et al. 2021; Ricci et al. 2022).

The plasma number density obeys the continuity equation and can be written as

for the steady state in an accelerating jet. Here, subscripts 1 and 2 indicate two different crosscuts perpendicular to the jet axis. We expect that the jets are still effectively accelerating in the parabolic domain (Nokhrina et al. 2019; Kovalev et al. 2020; Nokhrina, Kovalev & Pushkarev 2020; Nokhrina, Pashchenko & Kutkin 2022), and the plasma bulk motion Lorentz factor

increases linearly with the jet radius d/2 and depends on the light cylinder radius R_L (Beskin & Nokhrina 2006; Komissarov et al. 2007; Lyubarsky 2009). The ideal linear acceleration, Γ ∝ d, is valid while the flow is still highly magnetized. In order to account for the slowing down of this Lorentz factor behaviour as the flow approaches the saturation state with the magnetization equal to unity, we introduce the coefficient ρ. Using relationship (A13) from Nokhrina, Pashchenko & Kutkin (2022) and table 1 from Nokhrina et al. (2019), we conclude that ρ ≈ 0.3–0.55. Using (8) and (9), we obtain the following relationship for the emitting-plasma number density amplitude:

Substituting (7) and (10) into (4), we obtain the relationship for the frequency-dependent core position:

For the different relationships between the viewing angle θ and local Lorentz factor Γ in Ricci et al. (2022), we obtain three different values of the core-shift exponent k_r. For θ ≈ Γ⁻¹, the Doppler factor is approximately constant, and

For k_n = 3k and k_b = 2k, the core-shift exponent does not depend on the index of the emitting-particle distribution α: k_r = 2k. If θ ≪ Γ⁻¹, the Doppler factor is proportional to the Lorentz factor δ ≈ 2Γ, and

Substituting exponents from (7) and (10), we obtain

This expression depends weakly on the emitting-particle distribution for a reasonable range of α (see e.g. Gould 1979). For α = 0.5, this exponent k_r = 4k/3. If θ ≫ Γ⁻¹ and δ ∝ Γ⁻¹, one obtains

For k_n = 3k and k_b = 2k,

which is equal to k_r = 8k/3 for α = 0.5 and, again, in general depends on α very weakly. All the above results were obtained by Ricci et al. (2022), but assuming the dominance of the toroidal magnetic field in the collimation and acceleration region.

Obtained above value k_r = 2k for θ ≈ Γ⁻¹ for a strictly parabolic power k = 0.5 provides k_r = 1, which coincides with the result for a conical outflow with a constant speed. This value of k_r index is supported by multifrequency measurements for a dozen sources (Hada et al. 2011; Sokolovsky et al. 2011). However, for different quasi-parabolic jet collimation profiles and different viewing angles, k_r may vary. The core-shift exponent measurements for NGC 315 favour θ ≪ Γ⁻¹ and k_r = 4k/3 (Ricci et al. 2022).

It is convenient to use the core-shift offset (Lobanov 1998)

where Δr_mas is the measured core shift in milliarcseconds for observational frequencies ν₁ and ν₂ for a source at the luminosity distance D_L and corresponding cosmological redshift z. The core-position offset measure depends on K_e*ζ and B_*ζ. If the relationship between K_e*ζ and B_*ζ is set, the measured core shift allows estimation of the magnetic field and emitting-particle number density (Lobanov 1998; Hirotani 2005; O’Sullivan & Gabuzda 2009; Nokhrina et al. 2015).

3 MAGNETIC FIELD ESTIMATES BASED ON THE CORE-SHIFT EFFECT

The classical equipartition between the emitting plasma and magnetic field |$K_{e*}\propto B_{*}^2$| can no longer hold everywhere because of relationships (7) and (10) being incompatible with the conical approach (see Appendix  A). To relate K_e*ζ and B_*ζ we introduce the local magnetization – the ratio of the Poynting flux to the plasma bulk motion kinetic energy flux (see e.g. Nokhrina et al. 2015):

Setting σ = 1, which corresponds to Γ = Γ_max/2, and using the universal relationship for relativistic flows between the toroidal and poloidal magnetic field components (e.g. Vlahakis 2004; Komissarov et al. 2009; Lyubarsky 2009), namely

we obtain

In order to relate the plasma number density and magnetic field at a given distance ζ and the point σ = 1, we use (7) and (10). This provides the following relationship for n_*ζ and B_*ζ:

Assuming that just a fraction |$\eta \in (0;\, 1]$| of all the plasma emits, we set

With the above relationship, we are able to write the final expression for estimating the jet magnetic field using the core-shift effect in the parabolic accelerating jet with δ ≈ const (k_r = 2k):

In the expression for the magnetic field B_*ζ, all the distances (r_ζ, d_ζ, d_br and R_L) are in parsecs. It can be compared with the classical expression for a magnetic field in the plasma proper frame at a distance 1 pc from the jet base (e.g. Lobanov 1998; Hirotani 2005):

where φ is the intrinsic half-opening angle for a conical jet.

The expression for the magnetic field in the case of δ ≈ 2Γ (k_r = 4k/3) can be written as:

Although the case of k_r = 8k/3 (δ ∝ Γ⁻¹) is unlikely, based on current observations, we still present the expression for B:

For the domain of effectively accelerating outflow, the co-moving magnetic field is dominated by the poloidal component (Vlahakis 2004; Komissarov et al. 2009). This makes it straightforward to estimate the total magnetic flux contained in a jet and to extrapolate the magnetic field down to the gravitational radius scale.

The total magnetic flux is estimated as

Using magnetic flux conservation, we obtain the magnetic field amplitude at the gravitational radius scale as

4 MAGNETIC FIELD ESTIMATES

Below we apply the above relationships to sources with measured jet shape and core shift. Additionally, we use the black hole mass estimate to assess the magnetic field amplitude at the gravitational radius. The required parameters are summarized in Tables 1–3.

To calculate the parameter Ω_rν using expression (17) we adopt the core-shift measurements by Pushkarev et al. (2012) between the observational frequencies of 15.4 and 8.1 GHz. We also set k_r = 2k, where the power k in the parabolic regime was measured in Kovalev et al. (2020) (see Tables 2 and 4). For the M87 jet we use the core-shift measurements reported by Hada et al. (2011). The core position dependence is

For homogeneity of the results, we calculate the core shift between the pair of frequencies 8.1 and 15.4 GHz (see Table 3) using this expression. The k_r-index obtained by Hada et al. (2011) assumes the value |$1.06^{+0.12}_{-0.09}$| and is in good agreement with the relationship k_r = 2k, corresponding to a constant Doppler factor result (12). We set the measured k_r = 1.06 for M87 (Hada et al. 2011). For NGC 315 we use the recent results by Ricci et al. (2022). In particular, multifrequency measurements are consistent with k_r = 0.57 ± 0.17 at 22−8 GHz, and with k_r = 0.92 ± 0.01 at 8−5 GHz. This result is in agreement with formula (14), k_r = 4k/3, with 4k/3 = 0.6 for k = 0.45 (Boccardi et al. 2021). We adopt the core-shift measurement by Boccardi et al. (2021) between the frequencies of 15.4 and 8.4 GHz. For 3C273, the core shift was measured by Okino et al. (2022). We chose the pair 23.7 and 15.4 GHz as the closest to the frequency pair in other sources. We also adopt k_r = 2k, because the authors fitted the core shift with assumed k_r = 1. The opacity parameters are listed in Table 3. Note that Ω_rν here is calculated with k_r, which differs from unity. In order to check that the corresponding cores are indeed lying in the quasi-parabolic domain, we calculated the positions of cores from a jet base at the high and low frequencies: |$r_{\nu _2}$| and |$r_{\nu _1}$| , as listed in columns (7) and (8) of Table 3. We see that these cores lie closer to the jet base than the geometry break-point position r_break (see column 5 in Table 2).

The jet geometry data – the jet opening parameter a₁, the power k in the parabolic domain, the jet width at the break d_br, and the position of a break r_break – are presented in Table 2.

The variability Doppler factor estimates are summarized in column (7) of Table 1 (Hovatta et al. 2009). To estimate the Doppler factor for the M87 jet we employ the maximum speed in the acceleration region measured by Mertens et al. (2016) and Γ ∼ 3 with the viewing angle θ = 18° from Nakamura et al. (2018). For sources with an unknown Doppler factor, we set it as δ = 1 based on the relatively large viewing angles of these jets.

In order to compare the magnetic field estimates based on the model of parabolic accelerating jets (23) with those based on conical geometry (24), we use the apparent opening angle measurements by Pushkarev et al. (2017). These angles reflect the jet geometry at the 15-GHz core. For the M87 jet we use the apparent opening angle φ_app ≈ 5° reported by Mertens et al. (2016) on parsec scales. To obtain the intrinsic half-opening angles, we use the relationship

Viewing angles are collected in Pushkarev et al. (2017). However, for M87 we use the value θ = 18° from Nakamura et al. (2018), and for NGC 315, that from Ricci et al. (2022). We set the parameters η = 0.01, based on the modelling of synchrotron emission of non-uniform jets (Frolova, Nokhrina & Pashchenko 2023), and ρ = 0.33 (Nokhrina, Pashchenko & Kutkin 2022).

The results are presented in Table 4. Using equation (23) and the measured parameters, we estimate the magnetic field |$B_{*\zeta }^{\mathrm{par}}$| at the distance ζ (see column 2) from the jet base for parabolic accelerating jets. The conical estimate |$B_{*\zeta }^{\mathrm{cone}}$| is presented in column (6) to enable comparison of the results. We observe that for all the sources except for 3C 273 the magnetic field at the given distance ζ is several times smaller that the estimates based on the conical jet geometry.

Special attention must be given to NGC 315. As the core-shift index k_r at high frequencies clearly points to the case θ ≪ Γ⁻¹, we should apply formula (25). We adopt the light cylinder radius R_L = 4.7 × 10⁻⁴ pc, corresponding to Γ_max = 20, and calculate the magnetic field at 0.2 pc using relationship (25). Equation (25) yeilds the values of B_*ζ and B_g an order of magnitude lower than the relationship (23).

5 RESULTS AND DISCUSSION

First of all, we observe that the magnetic field estimates within the model of a parabolic accelerating jet are 1.4 to 32 times smaller than those for a classical conical model (Lobanov 1998). The only exception is 3C 273, with the largest jet width at the break d_br in the sample.

These differences in the magnetic field calculated within two different models – conical with a constant velocity and a quasi-parabolic accelerating flow – result from two effects. The first is the difference in the dependence of K_e* and B_* on the distance r. For the ideal parabolic power k = 0.5, the magnetic field B_* ∝ r⁻¹ as in the Blandford–Königl model. On the other hand, the emitting-plasma number density amplitude K_e* ∝ r^−1.5 decreases more slowly than in the flow with a constant Lorentz factor. In the majority of our sources, only the first effect plays a role: the core-shift parameter Ω_rν and the core positions |$r_{\nu _2}$| and |$r_{\nu _1}$| for k_r ≈ 1 are close to their values for the non-accelerating conical outflow. Thus, if we observe the core of an accelerating parabolic jet at the same frequency and distance, as in the conical case, than the expression |$K_{\rm e*}B_*^{1.5+\alpha }$|⁠, which defines the local opacity (see equation 4), must also be the same as in the conical non-accelerating jet. But, as the plasma number density decreases more slowly with distance, then the magnetic field amplitude must be lower to balance the larger plasma number density. The second effect is the possible difference between the parabolic and conical core positions for large enough departures of k_r from unity. In this case, both effects play a role. If we set k_r = 1 in NGC 315 (005+300), the core-shift offset would be equal to |$\Omega _{r\nu }=2.56\,\,\mathrm{pc\, GHz}$|⁠, and the core positions are |$r_{\nu _1}=0.27$| and |$r_{\nu _2}=0.50$| pc. These values are larger than those calculated with k_r = 0.57, and such a difference in their positions works in the same direction as the first effect, making the magnetic field estimate in a parabolic jet even smaller than that in a conical one. In 3C 273 (1226+023), for k_r = 1,|$\Omega _{r\nu }=4.27\,\,\mathrm{pc\, GHz}$|⁠, |$r_{\nu _1}=3.13$|⁠, |$r_{\nu _2}=4.82$| pc. The cores are situated closer to the jet base than those in the model with k_r = 1.32. In this case, there is an interplay between the effects of a slower decrease of K_e* with distance and the required higher amplitudes in both K_e* and B_* for the surface with optical depth equal to unity to be situated farther from the jet origin.

The dominance of the poloidal magnetic field component in the accelerating parabolic jet domain provides a convenient and straightforward way to recalculate the magnetic field on scales of the order of the gravitational radius. The results are presented in column (10) of Table 4. For NGC 315, both the obtained values |$B_\mathrm{g}^\mathrm{par}=21$| and 480 G can be compared with the independent estimate based on an assumption of a magnetically arrested disc launching a jet (Ricci et al. 2022). The estimate for k_r = 2k is in excellent agreement with B_MAD = 125−480 G at the lower-limit event-horizon radius. On the other hand, the magnetic field, obtained assuming k_r = 4k/3, when recalculated to the upper limit magnetosphere radius |$10.6\, r_\mathrm{g}$| has the value 2.5 G, which corresponds very well to the independent magnetic estimate of 5−18 G at this scale. The |$B_\mathrm{g}^\mathrm{par}=80$|-G estimate for M87 is corroborated by the Event Horizon Telescope (EHT) results B ≈ 1−30 G in the vicinity of a central BH (Event Horizon Telescope Collaboration 2021). The rest of the sources in the sample have amplitudes of the magnetic field at the gravitational radius of the order of 10³−10⁴ G, with only 3C 273 reaching a few times 10⁵ G owing to the relatively large core shift measured by Okino et al. (2022). The estimated total magnetic flux in jets in our sampleassumes the expected values of 10³²−10³⁵ G cm².

The results presented above depend on the modelling parameters, such as the assumed fraction η of emitting plasma, the coefficient ρ describing the deviation from a strictly linear acceleration, the maximum possible Lorentz factor Γ_max of the bulk plasma motion, and the Doppler factor δ, which is unknown for several sources in the sample. The fraction η = n_syn/n is the most weakly constrained parameter. It is estimated to be from the order of 1 per cent, based on particle-in-cell simulations (Sironi & Spitkovsky 2011; Sironi, Spitkovsky & Arons 2013; Sironi & Spitkovsky 2014) and the modelling of synchrotron intensity maps from stratified jets (Frolova, Nokhrina & Pashchenko 2023), to the order of 100 per cent, based on numerical modelling by Kramer & MacDonald (2021). The parameters ρ and Γ_max are usually better constrained. We can estimate Γ_max with a precision of up to a factor of a few (see discussion in Nokhrina et al. 2019). Both numerical (Chatterjee et al. 2019) and analytical (Nokhrina, Pashchenko & Kutkin 2022) modelling provide about the same accuracy for ρ ≈ 0.2−1. Finally, the Doppler factor is unknown for four of the sources in our sample and assumed to be unity.

The dependence of estimates (23) and (25) on the unknown ratio of the emitting to full plasma number density and Γ_max is very weak: as the 1/4 power. This means that even if all the plasma emits (η = 1 instead of 0.01), magnetic field estimates would increase by a factor of about 3. This would make estimates based on the parabolic and conical geometry closer to each other. The same holds for the total magnetic flux Ψ and the magnetic field at the gravitational radius |$B_\mathrm{g}^\mathrm{par}$|⁠. The magnetic field depends on the particular values of ρ and δ as a power of 1/2. Thus, the larger-than-assumed δ = 1 Doppler factors will lower the magnetic field estimate as |$1/\sqrt{\delta }\approx 3$| for, for example, δ = 10 instead of 1. Thus, even the extreme values of these parameters will affect the result for B within a factor of a few.

Errors in estimates for the magnetic field and emitting-plasma number density arise from observational errors in the measured values Δr, θ, δ, d_break and from model-assumed values η, ρ, Γ_max. Typical errors in observational data are of the order of tens per cent (Pushkarev et al. 2012, 2017; Kovalev et al. 2020), with the major source being the error in Δr. The largest expected uncertainty of the model parameters is in the fraction of the emitting-plasma number density η. As discussed above, this may lead to a drop in the estimate of |$B_{*\zeta }^\mathrm{par}$| up to 3.2 times. Thus, the results presented in column (5) must be treated as estimates with accuracy up to, but no more than, a factor of a few.

The estimate |$B_{*\zeta }^\mathrm{cone}$| based on the conical non-accelerating jet model (24) depends on the apparent intrinsic opening angle φ_app, which reflects the local jet width at r₁ = 1 pc. It changes in the acceleration and collimation zone. We checked that accounting for the true jet width |$a_1 r_1^{k}$| provides a smaller jet width and at the distance r₁, than using the intrinsic opening angle and conical geometry. This leads to an effectively smaller φ and, thus, an even larger estimate for |$B_{*\zeta }^\mathrm{cone}$|⁠. However, owing to the weak dependence of B_*1 on φ in (24), its value changes at most by a factor of 2.

After substitution of the relationship (22) between K_e*ζ and B_*ζ into equations (23) and (25), we obtain the dependence of K_e*ζ on observed values such as δ, and on the implied (modelling) parameters such as η and ρ. In the case of k_r = 2k, the emitting-plasma number density depends on the parameters η, Γ_max, d_br, ρ, and δ as

For k_r = 4k/3, the expression for the emitting-plasma number density amplitude is

We see that the K_e*ζ estimate would increase by a factor of 10 if all the plasma emitted (η = 100 instead of 1). The dependence of K_e*ζ on the Doppler factor is even stronger. If, for example, we take for four sources δ = 10 instead of 1, the estimate in the emitting-plasma number density will decrease by the same factor 10. It is worth noting that the sources with assumed δ = 1 have the largest estimates for the emitting-plasma number density. Using a fiducial δ = 10 would decrease these values to much closer to those for the sources with observational estimates of the Doppler factor.

The new relationship between K_e* and B_* presented in this work |$K_{\rm e*}\propto B_*^2d$| is very important because it differs from the usually accepted equipartition,

Here, Λ = ln (γ_max/γ_min) for α = 0.5, and we used Λ = 10 in our calculations. If we assume this classical equipartition regime, then the core-shift offset Ω_rν should depend on the distance along the jet, or, correspondingly, on the particular frequency pair. For details, see Appendix  A.

The relationship (22) means that while a jet is still accelerating, the ratio K_e*/B increases in proportion to the jet width until the acceleration ceases. This is due to the dependence of the particle number density in the plasma proper frame on the local Lorentz factor. As soon as the flow reaches an almost constant Γ, the additional dependence on the jet width, connected with Γ, vanishes, and the relationship between K_e* and |$B_*^2$| naturally assumes the classical form (33). In order to get a feeling for the difference from the classical equipartition, we calculate the values

(presented in column 9 of Table 4), which should be equal to unity for the classical equipartition regime. Equations (23), (25), and (26) are similar to the equations (13), (14), and (15) by Ricci et al. (2022), with one important distinction. While the assumed dependences of both B_* and K_e* on r are the same in both works, the difference comes from the equipartition assumptions, and it affects the dependence of B_* on d_ζ. Ricci et al. (2022) assumed |$K_{\rm e*\zeta }\propto B_{*\zeta }^2$| everywhere, and here we show that it is essential to set the relation |$K_{\rm e*\zeta }\propto B_{*\zeta }^2d_\zeta$| for the consistency of the results (see Appendix A). This means that equations (13)–(15) by Ricci et al. (2022) can be applied for comparable jet widths at the cores |$r_{\nu _i}$| and at the point where we estimate the magnetic field ζ. Only in this case are the obtained results self-consistent. This condition holds for NGC 315 in Ricci et al. (2022), because d_break = 0.072, |$d_{\nu _1}=0.057$|⁠, and |$d_{\nu _1}=0.035$| pc. Extrapolating our magnetic field estimate from the distance ζ = 0.2 pc to ζ = 0.58 pc, we get B_* = 0.2 G, which is very close to the result of 0.18 G by Ricci et al. (2022).

There is a zone along a jet where the k-index changes more or less smoothly from a quasi-parabolic value to a quasi-conical one. Observational data provide a change in k on short length-scales (see e.g. Kovalev et al. 2020; Boccardi et al. 2021), while analytical modelling provides a wider range for such a change (Kovalev et al. 2020). Here we assume constant k in a quasi-parabolic domain, because the radio cores lie much closer to the jet origin than the geometry break position (see Tables 2 and 3). However, if the core at the lower frequency lies in the transition domain with the k-index being between the parabolic and conical values, it will boost the magnetic field estimate closer to the results from the conical non-accelerating model.

6 SUMMARY

This work is dedicated to the modification of the core-shift method developed by Lobanov (1998) for parabolic accelerating jets in the region of observed radio cores. We apply our results to a sample of sources with the detected jet shape break. Our results are summarized as follows.

1. We use the poloidal component of a magnetic field as the dominant component B_* in the plasma proper frame. This is true for a still-accelerating jet (Vlahakis 2004; Komissarov et al. 2009). We account for the changing particle number density in the plasma proper frame K_e* as the flow Lorentz factor increases. As a result, the relationship between K_e* and |$B_*^2$| deviates from the classical equipartition. The values of the core-shift offset Ω_rν for different pairs of frequencies do not show any significant trend with r. This supports our use of relationship (22) between K_e* and |$B_*^2$|⁠.

2. We show that for a jet boundary geometry given by a power law, d ∝ r^k, the core-shift index in the parabolic accelerating jet may assume values of k_r = 2k, k_r = 4k/3, and k_r = 8k/3 for θ ≪ Γ⁻¹ (Doppler factor δ ≈ 2Γ), θ ∼ Γ⁻¹ (Doppler factor δ ≈ const) and θ ≫ Γ⁻¹ (Doppler factor δ ∝ Γ⁻¹), respectively. Thus, for quasi-parabolic jets, we should expect k_r ≈ 0.8−1.2 and k_r ≈ 0.53−0.8 for most of the sources. Both values are observed (e.g. Hada et al. 2011; Porth et al. 2011; Sokolovsky et al. 2011; Kutkin et al. 2014), and we explain these values by the different viewing angles with respect to the emission opening angle ∼Γ⁻¹. In fact, this means that multifrequency observations of radio cores coupled with kinematics can provide one more instrument to constrain the jet viewing angle.

3. We obtained formulas to estimate the magnetic field B_* (23) and (25), and the emitting-particle number density amplitude K_e* (22) using the measurements of the core-shift effect and the jet geometry. We show that B_* can be straightforwardly used to calculate the magnetic field at the gravitational radius scale and the total magnetic flux in a jet. This is due to the dominance of the poloidal component in the accelerating jet region.

4. We have estimated the magnetic field in a jet B_* and at the gravitational radius B_g, the particle number density K_e*, and the total magnetic flux Ψ for a sample of 10 sources. The magnetic field at 1-pc distance along a jet is 1.4 to 32 times smaller than the classical estimates. For NGC 315 we used both equations (23) and (25), although the k_r = 0.57 measured by Ricci et al. (2022) strongly points to the case of θ ≪ Γ⁻¹. For M87 and NGC 315, B_g can be compared with independent estimates by different methods, and the correspondence is very good.

ACKNOWLEDGEMENTS

We thank the anonymous referee for suggestions that helped to improve the paper. This study has been supported by the Russian Science Foundation: project 20-72-10078, https://rscf.ru/project/20-72-10078/. This research made use of data from the MOJAVE database,² which is maintained by the MOJAVE team (Lister et al. 2018). This research made use of NASA’s Astrophysics Data System.

DATA AVAILABILITY

There are no new data associated with the results presented in the paper. All the previously published data has the appropriate references. Intereferometric raw data for the project code BK134 are available from the VLBA (Very Long Baseline Array) archive.¹

Footnotes

References

APPENDIX A: POSSIBLE RELATIONSHIPS BETWEEN PLASMA NUMBER DENSITY AND MAGNETIC FIELD

Suppose there is some kind of relationship between the emitting-particle number density amplitude k_e* and the magnetic field B_* in the region of the observed radio cores. The one usually used is the equipartition between the energy density of emitting plasma and the magnetic field |$k_{\rm e*}\propto B_*^2$| (Burbidge 1956; Blandford & Königl 1979; Lobanov 1998). Suppose also that the multifrequency observations are indeed consistent with the power-law dependence (11), which means a constant (at least for the implied distances along a jet) Ω_rν (e.g. Hada et al. 2011; Sokolovsky et al. 2011). Let us discuss the possible relationships between the emitting-plasma number density and the magnetic field in the plasma proper frame that ensure that Ω_rν is approximately constant for different frequency pairs. This, in turn, means a constant value at different distances r_ζ.

Substituting the relationships |$K_{\rm e*}\propto r^{-k_\mathrm{n}}$|⁠, |$B_*\propto r^{-k_\mathrm{b}}$| and (1) into equation (4), we obtain the following:

where the power p and the coefficient C depend on the assumed Doppler factor dependence on the Lorentz factor.

Case 1. Approximately constant Doppler factor δ.

where r₀, ν₀, are e are in CGS units, and a₁ is in pc^{1 − k}.

Case 2. Doppler factor δ ≈ 2Γ.

where R_L is in pc.

Case 3. Doppler factor δ ∝ Γ⁻¹.

where R_L is in pc.

We see that |$\Omega _{r\nu }^p$| is proportional to the expression

which, in turn, we do not expect to depend on the particular distance along a jet ζ.

Suppose that the relationship |$K_{\rm e*}\propto B_*^2$| holds together with |$K_{\rm e*}\propto r^{-k_\mathrm{n}}$| and |$B_*\propto r^{-k_\mathrm{b}}$|⁠. If the expression (A8) does not depend on distance r_ζ, the following equality must be true:

This equality is independent of the particular index of the plasma energy distribution α. It holds, for instance, for the classical approach (Lobanov 1998) with k_n = 2 and k_b = 1. However, it does not hold for assumptions (7) and (10). The self-consistent dependence of Ω_rν on K_e*ζ, B_*ζ, and r_ζ exists only for certain relations between the emitting-plasma number density and magnetic field. Suppose that one can relate the number density amplitude and the magnetic field as

In order that the combination (A8) does not depend on ζ, the following relationship between exponents l and m must hold:

In expression (21), l = 2 and m = k, so the condition (A11) is indeed fulfilled and has an underlying clear physical meaning, as described in Section 4. This relationship holds for any dependence of the Doppler factor δ on Γ, namely for any relationship between the viewing angle θ and Lorentz factor Γ, as long as expressions (7) and (10) hold.

To examine whether there are significant trends of Ω_rν with r, we analysed the very long baseline interferometry (VLBI) observations of 20 pre-selected sources carried out during four 24-h sessions between 2007 March and June (project code BK134) aimed at core-shift analysis (Sokolovsky et al. 2011). The observations were performed simultaneously at nine frequencies, ranging from 1.4 to 15.4 GHz, thus forming both narrow and wide frequency pairs and enabling us to probe different spatial scales of the inner jet regions of the target sources. In Fig. A1 (left), it can be seen that Ω_rν is quite stable for the relatively close z = 0.107 (Paiano et al. 2017), that is, a luminosity distance of D_L = 465 Mpc, BL Lacertae object 1219+285 (W Comae), assuming k_r = 1. Note that the inner jet shape of the source is quasi-parabolic (Fig. A1, right), as inferred from the observations at the highest frequency of 15.4 GHz by analysing profiles transverse to the total intensity ridgeline of the outflow that we constructed following the procedure described in Pushkarev et al. (2017) in detail. Among the other 19 sources characterized by different inner jet shapes, such as quasi-parabolic and quasi-conical, a similar flat frequency dependence on Ω_rν is observed.