Ordered magnetic fields around the 3C 84 central black hole

3C84 is a nearby radio source with a complex total intensity structure, showing linear polarisation and spectral patterns. A detailed investigation of the central engine region necessitates the use of VLBI above the hitherto available maximum frequency of 86GHz. Using ultrahigh resolution VLBI observations at the highest available frequency of 228GHz, we aim to directly detect compact structures and understand the physical conditions in the compact region of 3C84. We used EHT 228GHz observations and, given the limited (u,v)-coverage, applied geometric model fitting to the data. We also employed quasi-simultaneously observed, multi-frequency VLBI data for the source in order to carry out a comprehensive analysis of the core structure. We report the detection of a highly ordered, strong magnetic field around the central, SMBH of 3C84. The brightness temperature analysis suggests that the system is in equipartition. We determined a turnover frequency of $\nu_m=(113\pm4)$GHz, a corresponding synchrotron self-absorbed magnetic field of $B_{SSA}=(2.9\pm1.6)$G, and an equipartition magnetic field of $B_{eq}=(5.2\pm0.6)$G. Three components are resolved with the highest fractional polarisation detected for this object ($m_\textrm{net}=(17.0\pm3.9)$%). The positions of the components are compatible with those seen in low-frequency VLBI observations since 2017-2018. We report a steeply negative slope of the spectrum at 228GHz. We used these findings to test models of jet formation, propagation, and Faraday rotation in 3C84. The findings of our investigation into different flow geometries and black hole spins support an advection-dominated accretion flow in a magnetically arrested state around a rapidly rotating supermassive black hole as a model of the jet-launching system in the core of 3C84. However, systematic uncertainties due to the limited (u,v)-coverage, however, cannot be ignored.


Introduction
The formation of relativistic astrophysical jets is a manifestation of the activity of accreting supermassive black holes residing in the nuclei of galaxies.Such jets can have an immense impact on their surroundings, either by stunting or enhancing the evolution of their host galaxy.Despite substantial efforts dedicated to understanding the physics governing jets, a number of open questions remain, including questions relating to the launching mechanism of these jets.The radio source 3C 84 (NGC 1275; D L = 78.9± 2.4 Mpc, z = 0.0176, Strauss et al. 1992, corresponding to a conversion factor ψ = 0.36 pc/mas; see also Sect.2.1) is a nearby active galactic nucleus (AGN) and one of a handful of objects for which the jet formation zone can be resolved and probed with very-long-baseline interferometry (VLBI).Thus, 3C 84 is an ideal test bed for distinguishing between jet-launching models based on the resulting predictions for observables such as magnetic field strength.Using the unique polarimetric 1.3 mm VLBI observations of 3C 84, conducted with the Event Horizon Telescope (EHT; see EHTC et al. 2019aEHTC et al. , 2022a)), we are now able to distinguish between such models.
According to the current understanding, the linear polarisation is present in both the downstream jet (Nagai et al. 2017) and the compact region (Kim et al. 2019) of 3C 84, although its amplitude is low.A quantitative characterisation of the location of the 1.3 mm polarisation within the jet flow is crucial in order to distinguish between the different jet-launching models.To illustrate this, an interesting comparison can be made between the jet collimation near the jet base in M 87 (exhibiting a narrower opening angle, as seen, e.g. in Kim et al. 2018) and 3C 84 (featuring instead a wide structure as seen by RadioAstron and reported in Giovannini et al. 2018).Given this elongated structure, a disc-launched jet (Blandford & Payne 1982) threaded by toroidal magnetic field lines is a possible explanation.The alternative scenario is the more commonly invoked black hole launched jet (Blandford & Znajek 1977) associated with poloidal magnetic field lines.Polarimetry at 1.3 mm is less affected by opacity effects and can therefore be used to test the necessary conditions for different jet-launching scenarios, as presented in this work.We therefore employed high-resolution millimetre VLBI to investigate how the substantial increase in polarisation with frequency in 3C 84 can be explained by the prevalent magnetic field.

Data description and analysis
In this work, we examined the first total intensity and polarimetric VLBI observations of 3C 84 at 228 GHz taken with the Event Horizon Telescope (EHT) and compared them with quasisimultaneous VLBI observations at lower frequencies.3C 84 was observed during the EHT 2017 campaign (EHTC et al. 2019a(EHTC et al. , 2022a) ) at 228 GHz on April 7 between 18:30 and 19:40 UTC, with six scans each around 5 mins in length.Five telescopes at three geographical sites participated in the observation: Atacama Large Millimeter/submillimeter Array (ALMA, observing as a phased array; see Goddi et al. 2019) and the Atacama Pathfinder Experiment (APEX) telescope in Chile; the Submillimeter Telescope (SMT) in Arizona; and the James Clerk Maxwell Telescope (JCMT) and the Submillimeter Array (SMA) in Hawai'i.Following the correlation, observations were subjected to the standard EHT data reduction path (EHTC et al. 2019b(EHTC et al. ,c, 2022b)), including the EHT-HOPS fringe-fitting and post-processing pipeline (Blackburn et al. 2019, see also Janssen et al. (2019) for an alternative pipeline used with the EHT data).Additional comments on the data reduction are given in Appendix A. The single-dish data used in this paper were observed by the POLAMI (Thum et al. 2008;Agudo et al. 2018) and QUIVER (Myserlis et al. 2018;Kraus et al. 2003) programmes on April 4 and April 8, 2017, respectively.
As 3C 84 exhibits a low jet expansion velocity inside the submilliarcsecond (submas) region, we are able to use quasisimultaneous VLBI observations of 3C 84 taken in March and April, 2017, at 15, 43, and 86 GHz to complement our analysis and assist our interpretation of the underlying jet physics without suffering from time-variability effects.Here, we define as compact region the entire region probed by the long EHT baselines, with an angular size of smaller than 200 µas.Specifically, we used the publicly available Very Long Baseline Array (VLBA) epochs from April 22, 2017 at 15 GHz (MOJAVE monitoring program; see Lister et al. 2018, for details regarding the calibration and imaging procedures) and April 16, 2017 at 43 GHz (VLBA-BU-BLAZAR monitoring program; see Jorstad et al. 2017, for details regarding the calibration and imaging procedures).As both monitoring programs publish fully calibrated and imaged data sets, we opted to use them as provided.
At 86 GHz, we used the Global Millimeter VLBI Array (GMVA) epoch from March 30, 2017 (see Paraschos et al. 2022b, for details regarding the calibration and imaging procedures).The antenna instrumental polarisation calibration (D-terms) was performed using the software polsolve (Martí-Vidal et al. 2021) and the data were imaged using the CLEAN algorithm (see e.g.Shepherd et al. 1994Shepherd et al. , 1995)).The combined (u, v)-coverage of our multi-wavelength observations is shown in Fig. 1.

Results
We find evidence of a highly ordered, strong magnetic field in the submas compact region of 3C 84.This region is best fitted by three circular Gaussian components, labelled core ('C'), east ('E'), and west ('W'), as shown in Fig. 2 (the method we used is described in Appendix B).The extended flux density detected on the short ALMA-APEX and JCMT-SMA baselines, while resolved out on all long EHT baselines, was fitted by a ∼ 5000 microarcseconds (µas) circular Gaussian component with a flux density of S 228 GHz core ∼ 6.4 Jy.Furthermore, by averaging1 the linear fractional polarisation measurements of these three components, we determined the net linear fractional polarisation in the compact region to be m net = (17.0± 3.9)%.The short baseline between ALMA and APEX yielded an estimate for the linear fractional polarisation on larger arcsecond scales, of ∼ 6% (denoted in the bottom panel of Fig. 3 with the grey marker).
We cross-referenced the submas compact region model fit components at lower frequencies following the method detailed in Savolainen et al. (2008)  228 GHz structure at 86 GHz and even at 43 GHz.The results are reported in Table A.1.We also measured both the total intensity I and linearly polarised emission P in the submas region of 3C 84 in the 15, 43, and 86 GHz images.The values of linear fractional polarisation at these three lower frequencies are considered as upper limit estimates.The results are shown in Fig. 3.The VLBI total intensity increases up to the 86 GHz measurement, and then decreases towards 228 GHz.
Close-in-time single-dish measurements at 8, 86, and 228 GHz are also shown in Fig. 3 (see also Table A.2).The 86 GHz flux density is higher than that at 228 GHz.However, the 8 GHz measurement is also higher than at 86 GHz, suggesting a significant contribution from the parsec-scale jet.Furthermore, at 228 GHz, the compact-scale VLBI flux density is significantly lower than the corresponding extended flux density, as long EHT baselines over-resolve the large-scale jet emission structure (similar to M 87, see e.g.EHTC et al. 2019d).In terms of fractional polarisation, it is evident that there is a significant increase at 228 GHz, indicating a transition in the accretion flow to the optically thin regime.

Insights from the synchrotron spectrum
Our analysis shows that the east-west elongated core structure (Giovannini et al. 2018) also persists at 1.3 mm, and in lower frequencies (as reported at 7 mm by Punsly et al. 2021 and3 mm by Oh et al. 2022).Interpretation of the nature of the components comprising this broad core structure heavily depends on the uncertain jet viewing angle (ξ).An upper limit of ξ ∼ 40 • was reported by Oh et al. (2022) based on a VLBI analysis of the compact region, but much lower values have also been found, for example based on γ-ray analysis (Abdo et al. 2009).The histori-cally subluminal jet component velocities in the compact region (Punsly et al. 2021;Hodgson et al. 2021;Paraschos et al. 2022b) point towards an increased viewing angle.Moreover, different parts of the jet have been reported to be moving with different velocities, which is related to the so-called 'Doppler crisis' phenomenon (e.g.Henri & Saugé 2006) and jet stratification (Nagai et al. 2014).
The high-resolution, high-frequency EHT observation enables a novel diagnosis of the state of plasma surrounding the central black hole via calculation of the turnover frequency ν m and the synchrotron self-absorption magnetic field strength B SSA .Assuming ν m to be 86 GHz, Hodgson et al. (2018) and Kim et al. (2019) computed the B SSA to be ∼ 21 G.Using additional EHT flux density measurements, we can directly measure ν m .While the different observations correspond to different (u, v) coverages, we fitted a focused Gaussian model to the high-signalto-noise ratio (S /N) data at 228 GHz, finding core diameters within the order of magnitude of the diffraction limit.We also fixed the sizes of the components for all the frequencies in order to mitigate the effects of the different (u, v) coverages (see Table A.1). Subsequently, fitting, then, Eq. 5.90 from Condon & Ransom (2016) (see also Rybicki & Lightman 1979 and Appendix D) to the data yields ν m = (113 ± 4) GHz (see also Türler et al. 2000).We computed a core brightness temperature of T B = (3.6 ± 1.5) × 10 11 K from ν m , assuming that the angular size of the components at ν m is the same as at 228 GHz (as the system is optically thin at both frequencies).Within the error budget, the system seems to be in equipartition (Singal 1986) between kinetic and magnetic energies (also reported by Paraschos et al. 2023, based on light-curve variability analysis).
Furthermore, we computed B SSA = (2.9 ± 1.6) G using Eq. 2 from Marscher (1983) (see also Appendix D).We also calculated an equipartition magnetic field strength of B eq = (5.2± 0.6) G.The uncertainties were calculated through standard error propagation.The two values agree with each other within the error budget.Our results also tentatively agree within the error budget with the magnetic field reported by Kim et al. (2019).The equipartition Doppler factor is δ eq = 1.5 ± 0.4, suggesting that the acceleration happens further downstream, which is in line with lower frequency observations of 3C 84 (e.g.Hodgson et al. 2018;Paraschos et al. 2022b, and references therein).
Moreover, the equipartition magnetic field strength B eq in the vicinity of the jet apex was computed to reach up to 4 G in a core shift analysis carried out by Paraschos et al. (2021).However, the magnetic field value mentioned by these latter authors was calculated at the distance between the extrapolated jet apex and the 86 GHz core, resulting in a slightly lower estimate than that found in this work.Nevertheless, it is important to exercise caution when interpreting both ν m and B SSA .3C 84 is a variable source (recently up to 20-30% variation in total intensity and linear polarised flux density at 43 GHz within a year based on the monitoring program VLBA-BU-BLAZAR), which means that these observables might be time dependent (compare with the spectrum shown in, e.g.Hodgson et al. 2018).Moreover, our models still contain large uncertainties due to the sparsity of the (u, v) coverage, which may not be fully accounted for.

Model interpretation
Possible interpretations of the physical mechanisms driving the wide core structure largely depend on the exact location of the central engine with regard to the observed core.The current understanding is that the central engine is located north or northwest of the 86 GHz VLBI core (Giovannini et al. 2018;Paraschos  As its exact location is still ambiguous, it is unclear whether or not some of the identified components in this work correspond to the core (Case I) or a counter-jet (Case II).Simulations of the radio jet of M 87 (Mościbrodzka et al. 2017) show that the linear polarisation is produced inside the approaching jet, while the dense accretion disc depolarises any radiation reaching us from the counter-jet.In 3C 84, circumnuclear free-free absorption has already been reported for example by Walker et al. (2000), who cite a possible connection to the accretion disc.It is thought that the presence of this disc obscures the counter-jet in the milliarcsecond (mas) region of 3C 84, which only becomes visible at a distance of > 2 mas at higher frequencies (as reported e.g. in Wajima et al. 2020 at 86 GHz).As both E and W in Fig. B.1 are highly linearly polarised (20-80%), this points towards Case I, meaning that the two components might be at the origin of the double-rail structure seen on larger scales, as opposed to a jet and counter-jet geometry.However, we note that this interpretation remains speculative, given the uncertainties.
This high fractional polarisation in E and W could be evidence for highly ordered magnetic field lines in the jet plasma with almost no Faraday depolarisation present.On the other hand, C has lower fractional polarisation and the synchrotron opacity should be nearly negligible at 228 GHz according to the Stokes I spectrum shown in Fig 3 .This may indicate that the main source of depolarisation in the compact region probed by the EHT is beam depolarisation of complex magnetic field patterns or mild Faraday depolarisation, rather than opacity effects.Consequently, a possible Faraday screen located in the compact region could be at most the size of C, which is ∼ 20 µas.However, it should be noted here that W is the most uncertain low-totalintensity component, hindering a reliable conclusion about its nature (see also Appendices A and B).
3C 84 is known to show high amounts of Faraday rotation (RM) and the presence of circular polarisation (see e.g. the PO-LAMI and QUIVER programmes as described in Agudo et al. 2018;Myserlis et al. 2018, respectively).Using the SMA and CARMA, Plambeck et al. (2014) reported an RM of as high as ∼ 9 × 10 5 rad/m 2 , indicative of the presence of a strong magnetic field.This places 3C 84 in a small group of known radio sources exhibiting similarly high RMs, such as Sgr A * (∼ 5 × 10 5 rad/m 2 ; Wielgus et al. 2022 and references therein), M 87 (∼ 10 5 rad/m 2 ; Goddi et al. 2021), and PKS 1830-211 (∼ 10 8 rad/m 2 ; Martí-Vidal et al. 2015).However, whether this RM occurs in the medium surrounding the jet (e.g. from a disc wind) or is connected to the accretion flow remains unknown.The origin of the RM can be explored by determining its dependence on the observing frequency (Plambeck et al. 2014;Goddi et al. 2021) or the distance from the central engine (Park et al. 2015).
The density of the accretion flow, which is a related quantity that can be estimated via the RM, is required in order to constrain the mass-accretion rate around BHs (see Nagai et al. 2017, for a relevant discussion about 3C 84) for different accretion flow models, such as advection-dominated accretion flows (ADAFs; see Narayan & Yi 1995) and convection-dominated accretion flows (CDAFs; see Narayan et al. 2000).
Different plausible depolarisation mechanisms have been proposed for 3C 84, that is, originating from such an accretion flow and the jet itself (Li et al. 2016;Kim et al. 2019).Combining the single-dish data presented in Fig. 3, which were taken quasi-simultaneously with the EHT observations, allows us to estimate an estimate of the RM present in 3C 84.We find that RM = (6.06± 0.01) × 10 6 rad/m 2 by determining the gradient of the EVPAs as a function of the wavelength squared (see also Kim et al. 2019).The nπ ambiguity was resolved beforehand, as described in Hovatta et al. (2012).Such large RM values could be produced by the presence of relativistic and thermal electrons in the boundary layer between the jet and the interstellar medium, as reported in Goddi et al. (2021) for the jet in M 87.

Physical consequences
The high fractional linear polarisation in the innermost region of 3C 84, revealed at 228 GHz, clearly indicates that we are probing a previously elusive region, as we are able to achieve higher resolution while being less affected by opacity effects.We are probing the innermost region of 3C 84 at ∼ 500 R s , which appears to be an optically thin region with an ordered magnetic field framing the core.
Furthermore, this region is so compact that an association between the broad jet of 3C 84 and the accretion disk can be ruled out.However, it should be pointed out that both a BH-driven jet and a disc-driven wind could coexist and the present EHT observations are a better probe of the former.In a BH-driven jet scenario, jet launching in 3C 84 might be attributed to a magnetically arrested disc (MAD; see similar simulations carried out for M 87 in Chael et al. 2019), as opposed to a thin, broad disc structure (Liska et al. 2019).Jets in MAD ADAF systems are likely launched by the Blandford-Znajek mechanism Blandford & Znajek (1977), which is the case where a powerful jet spine is powered directly by the energy extracted from the ergosphere of the BH.
Using our estimate of the RM at 228 GHz, it is possible to test whether the magnetic field reaches saturation strength; that is, whether the system is in the MAD state (Narayan et al. 2000;Tchekhovskoy et al. 2011).Under the assumptions described in Appendix E, we find that the dimensionless magnetic flux ϕ = 41 − 93 (Tchekhovskoy et al. 2011).Values above the saturation value ϕ max = 50 indicate that the jet is in a magnetically arrested state, and therefore our analysis suggests that jet launching in 3C 84 is MAD.As higher BH spin values and β = 1.5 produce values close to ϕ max , our result indicates a preference for a high BH spin and the ADAF model.3C 84 is also classified as a low-luminosity AGN for which ADAF models are commonly invoked (de Menezes et al. 2020), further strengthening our conclusion.The mass-accretion rate estimated in Appendix E corre-sponds to Ṁ ∼ 10 −5 − 10 −4 ṀEdd , which is somewhat larger than in the case of M 87 MAD models (see e.g.EHTC et al. 2021b).This suggests that a non-negligible dynamical impact of radiation is possible, which could challenge the applicability of the presented analysis.It should be pointed out here that it is unclear whether Faraday rotation takes place exclusively inside the accretion flow.Our analysis described in Appendix E is based on the assumption that the accretion flow is dominant.
If a spine-sheath geometry (Tavecchio & Ghisellini 2014) is present, manifested in the observations as a transverse velocity gradient, it could also be the underlying depolarising structure.In this case the rotation of the central BH leads to an inhomogeneous and twisted magnetic field topology (see for example Tchekhovskoy 2015).Furthermore, this scenario would also provide an explanation for the Doppler crisis.As discussed in Hodgson et al. ( 2021), so-called 'jet-in-jet' formations (Giannios et al. 2009) associated with velocity stratification in the bulk jet flow could be responsible for the enhanced γ-ray emission observed in 3C 84.Such a spine-sheath geometry has already been shown by the EHT to exist on small scales in the jet-launching region of Centaurus A (Janssen et al. 2021).
Ultimately, our detection of the exceptionally high fractional polarisation at 228 GHz, the peculiar jet morphology, and the detailed radio spectrum suggest that the jet in 3C 84 might be launched from both the central BH and the surrounding accretion disc (e.g.Blandford & Globus 2022).As shown by the present findings, millimetre VLBI observations pave the way towards probing the ultimate vicinity of BHs.Future 3C 84 EHT observations with added antennas on short and intermediate baselines will help to constrain the jet morphology and improve the fidelity of the model.

Conclusions
In this work we present the first detection of microarcsecondscale polarised structures with the EHT.Our findings can be summarised as follows: -We report the first ever 228 GHz VLBI model of 3C 84, which reveals that the compact region is made up of three components.
-The increased values of linear polarisation suggest that the observed structure is the approaching jet, which is consistent with the large opening angle.Such a geometry can be produced by a thick disc associated with a Blandford & Znajek (1977) jet-launching scenario.-We find indications of a preference for higher values of BH spin and the ADAF model in the context of the MAD jet launching prevalent in 3C 84.
The EHT is an excellent instrument for probing AGN cores in nearby radio galaxies.Combined with lower frequency VLBI arrays, such as the GMVA and the VLBA, the EHT makes it possible to conduct multi-frequency studies, which provide valuable insights into jet formation and jet launching.New EHT and GMVA observations have already been carried out, with 3C 84 as the main target.The increased sensitivity and (u, v) coverage will enable us to conduct follow-up studies with higher fidelity.Total intensity images of the compact region will shed more light on whether or not the components we were able to identify here correspond to the broad structure seen with RadioAstron (Giovannini et al. 2018).Spectral index maps of EHT and GMVA images observed quasi-simultaneously might also assist in pinpointing the exact location of the BH (see e.g.Fig. 4 in Paraschos et al. 2022a) and in discriminating between jet launching-scenarios.
between the Max-Planck-Institut für Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden).The SMA is a joint project between the SAO and ASIAA and is funded by the Smithsonian Institution and the Academia Sinica.The JCMT is operated by the East Asian Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the Ministry of Finance of China, Chinese Academy of Sciences, and the National Key Research and Development Program (No. 2017YFA0402700) of China and Natural Science Foundation of China grant 11873028.Additional funding support for the JCMT is provided by the Science and Technologies Facility Council (UK) and participating universities in the UK and Canada.The LMT is a project operated by the Instituto Nacional de Astrófisica, Óptica, y Electrónica (Mexico) and the University of Massachusetts at Amherst (USA).The IRAM 30-m telescope on Pico Veleta, Spain is operated by IRAM and supported by CNRS (Centre National de la Recherche Scientifique, France), MPG (Max-Planck-Gesellschaft, Germany), and IGN (Instituto Geográfico Nacional, Spain).The SMT is operated by the Arizona Radio Observatory, a part of the Steward Observatory of the University of Arizona, with financial support of operations from the State of Arizona and financial support for instrumentation development from the NSF.Support for SPT participation in the EHT is provided by the National Science Foundation through award OPP-1852617 to the University of Chicago.Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago.The SPT hydrogen maser was provided on loan from the GLT, courtesy of ASIAA.This work used the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF grant ACI-1548562, and CyVerse, supported by NSF grants DBI-0735191, DBI-1265383, and DBI-1743442.XSEDE Stampede2 resource at TACC was allocated through TG-AST170024 and TG-AST080026N.XSEDE JetStream resource at PTI and TACC was allocated through AST170028.This research is part of the Frontera computing project at the Texas Advanced Computing Center through the Frontera Large-Scale Community Partnerships allocation AST20023.Frontera is made possible by National Science Foundation award OAC-1818253.This research was done using services provided by the OSG Consortium (Pordes et al. 2007;Sfiligoi et al. 2009), which is supported by the National Science Foundation award Nos.2030508 and 1836650.Additional work used ABACUS2.0, which is part of the eScience center at Southern Denmark University.Simulations were also performed on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen cluster at the HLRS in Stuttgart, and on the Pi2.0 and Siyuan Mark-I at Shanghai Jiao Tong University.The computer resources of the Finnish IT Center for Science (CSC) and the Finnish Computing Competence Infrastructure (FCCI) project are acknowledged.This research was enabled in part by support provided by Compute Ontario (http://computeontario.ca), Calcul Quebec (http://www.calculquebec.ca), and Compute Canada (http://www.computecanada.ca).The EHTC has received generous donations of FPGA chips from Xilinx Inc., under the Xilinx University Program.The EHTC has benefited from technology shared under open-source license by the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER).The EHT project is grateful to T4Science and Microsemi for their assistance with hydrogen masers.This research has made use of NASA's Astrophysics Data System.We gratefully acknowledge the support provided by the extended staff of the ALMA, from the inception of the ALMA Phasing Project through the observational campaigns of 2017 and 2018.Partly based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg.We would like to thank A. Deller and W. Brisken for EHT-specific support with the use of DiFX.At 15 GHz the image resolution is insufficient to confidently distinguish between the compact region components, and so we limit ourselves to reporting the integrated flux density and fractional polarisation values instead.The positional uncertainty is of the order of ≤ 2% for E in the east-west and north-south directions, and ≤ 7% in the east-west and ≤ 60% in north-south direction for W. Here, C is fixed at (0, 0).The uncertainties of the flux density measurements are of the order of 20% at 15 GHz, 30% at 43 GHz, 50% at 86 GHz, and 15% at 228 GHz (EHTC et al. 2019c).The large relative uncertainty in the fractional polarisation of W is related to its small total flux density.The FWHM and positions of the 43, 86, and 228 GHz components have been fixed in the multi-frequency template-matching framework.Error margins indicate the 68% confidence level.= 6 µas, respectively.Therefore, we were able to apply the high-frequency template, given that the separation between the components comprising the template was sufficiently large.It should be noted here, that these calculations are performed under the assumption that the source's morphology is a Gaussian.However, given the complex structure in the compact region of 3C 84 revealed here, the actual resolution may be worse.We therefore adopted the more conservative approach of restricting the resolution limit to the typical value of approximately one-fifth of the beam size (e.g.Oh et al. 2022).This was still possible in our case at 43 GHz (beam size ∼ 100 µas) and 86 GHz (beam size ∼ 50 µas).As in our work we investigate the overall spectral behaviour of the submas region, this approach is sufficient to get an estimate for the flux densities and fractional polarisations for each component.We also disregarded the core shift effects (see e.g.Paraschos et al. 2021;Oh et al. 2022;Paraschos et al. 2023) between the images at different frequencies, because their effect is negligible for our analysis (of the order of a few tens of µas).Our results are summarised in Table A.1.The uncertainties of the flux density measurements are on the order of 20% at 15 GHz, 30% at 43 GHz, 50% at 86 GHz, and 15% at 228 GHz (EHTC et al. 2019c).takes the following form (Condon & Ransom 2016): for a homogeneous and cylindrical source, where ν 1 is the frequency where the opacity reaches unity, τ = 1, and S 0 = 5.7 ± 0.3 Jy is a multiplication constant, determined from the fit.Subsequently, ν m is calculated by determining the peak of the fitted spectrum.Following Kim et al. (2019), we set α thick = 0.51 ± 0.10.The parameter p is the power-law slope of the electron energy distribution function and is set to p = 2 (Condon & Ransom 2016).
We used two different prescriptions for the magnetic field strength.First we calculated the equipartition magnetic field B eq using Pacholczyk (1970) with the following form2 : We note that the exponent 2/7 only holds for α thin = 0.5 (see Beck & Krause 2005, for a relevant discussion).Here, k u is a ratio that provides an estimate of the energy in relativistic protons compared to electrons and f is a factor denoting the fraction of the total volume of the emitting region occupied by the plasma and magnetic field in equipartition.Under the assumption of an electron-positron pair plasma (see Paraschos et al. 2023, for a discussion about electron-positron pair plasma in the vicinity of the SMBH in 3C 84), which is volume filling, k u = 0 and f = 1.The uncertainties for k u and f are difficult to constrain; their impact on the magnetic field strength computation is discussed below.The constant c 12 (in cgs units) is given by the following expression: where c 1 = 3e Our choice of α thin = −0.5 results in b(α thin ) = 3.2.The resulting estimates for the magnetic field are B SSA = (2.9 ± 1.6) G and B eq = (5.2± 0.6) G for the core component C at 228 GHz.We point out that the B SSA calculation is strongly impacted by the value of ν m .An increase or decrease of a few GHz would vary the value of B SSA by two orders of magnitude.Similarly, the B eq calculation strongly depends on the assumption of k u , that is, the particle composition of the jet.Alternative assumptions of the jet composition resulting in an increase in k u (diffusive shock acceleration would result in values of k u ≤ 50, see e.g.Bell 1978) would increase the value of B eq by up to a factor of 3. Likewise, decreasing the value of f (assuming a clumpier medium, filling only half of the total emitting region for example), would result in an increase in B eq by a factor of 1.2.However, we note that the good agreement between the two magnetic field estimates indicates that the choice of k u = 0 and f = 1 is reasonable.The equipartition Doppler factor required for B SSA to match B eq is δ eq = 1.5 ± 0.4.Finally, we can compare B eq and B SSA to the strength of the coherent field based on the observed RM.Using Eq. 15 from Gardner & Whiteoak (1966), written: RM = 8.1 × 10 5 n e B tot ∥ dl, (D.6) we can compute the lower limit of the strength of the ordered field, B tot ∥ .Here, n e is the number density of the thermal electrons, which we set to n e = 3 × 10 4 cm −3 (Scharwächter et al. 2013).The path length of integration through the plasma is dl and can be approximated by ψ × θ.Using these values, B tot ∥ = 4.7 ± 0.6 mG, which is consistent as a lower limit to our calculations of B eq and B SSA .
Our assumption is physically motivated; the jet viewing angle used in our computations (ξ ∼ 40 • ) is larger than half of the intrinsic jet opening angle (i ≲ 20 • ; see e.g.Paraschos et al. 2021), suggesting that we are peering through the jet sheath and boundary layer (see also Plambeck et al. 2014;Nagai et al. 2017).Furthermore, at 1.3 mm, we are able to directly examine the environment of the central engine, because the opacity effects become comparatively minor.
Finally, the total luminosity of the jet in 3C 84 (Rafferty et al. 2006) is P jet = 1.5 × 10 44 erg s −1 .Thus, setting a * = 1, we computed a range of ϕ = 41 − 93.Values of ϕ ≳ 50 refer to MAD models (Tchekhovskoy et al. 2011;Zamaninasab et al. 2014).We note here that the RM would need to be underestimated by more than an order of magnitude (e.g.due to the Faraday screen not being external) for ϕ to equal its MAD saturation value.Our investigation of different flow geometries and black hole spins supports an advection-dominated accretion flow in a magnetically arrested state as a model of the jet launching system in the core of 3C 84.
Fig. A.1.Best-fit model of 3C 84 compared to the data.Presented here from left to right are the data points (denoted with round blue markers) and models (denoted with dark orange crosses) of the visibility amplitudes, closure phases, and fractional polarisation as a function of the (u, v) distance.The combined (u, v) distance used in the middle panel is defined as the square root of the sum of squared lengths of all three baselines forming a triangle.Error bars in all panels indicate the 68% confidence level.
Fig. A.2. Closure phases as a function of the time of observation, detected on the APEX-SMT-JCMT non-trivial triangle, compared with the predictions of the model presented in this paper.Error bars indicate the 68% confidence level.The best-fit model 2g with two Gaussian components representing the compact emission region fails to adequately capture the trend in the data, unlike the 3g model with three components.The reported χ 2 corresponds to the non-trivial subset of all measured closure phases.