Millimeter-VLBI observations of low-luminosity active galactic nuclei with source-frequency phase-referencing

We report millimeter-VLBI results of low-luminosity active galactic nuclei (M 84 and M 87) up to 88 GHz with source-frequency phase-referencing observations. We detected the weak VLBI core and obtained the first image of M 84 at 88 GHz. The derived brightness temperature of M 84 core was about 7.2$\times$10$^9$ K, which could serve as a lower limit as the core down to 30 Schwarzschild radii was still un-resolved in our 88 GHz observations. We successfully determined the core-shifts of M 87 at 22-44 GHz and 44-88 GHz through source-frequency phase-referencing technique. The jet apex of M 87 could be deduced at about 46 $\mu$as upstream of the 43 GHz core from core-shift measurements. The estimated magnetic field strength of the 88 GHz core of M 87 is 4.8$\pm$2.4 G, which is at the same magnitude of 1-30 G near the event horizon probed by the Event Horizon Telescope.


INTRODUCTION
The low-luminosity active galactic nuclei (LLAGNs) classified by their low bolometric luminosities and sub-Eddington accretion rate commonly exist in nearby galaxies (Nagar et al. 2002;Ho 2008). Unlike their bright cousins, the broadband spectral energy distributions prefer the model of an inner advection-dominated accretion flow and an outer truncated thin disc (Yu et al. 2011;Nemmen et al. 2014). Meanwhile compact flat-spectrum radio cores were detected in LLAGNs (Nagar et al. 2002) and suggested to be scaled-down versions of AGN jets (Falcke & Biermann 1999). Hosting a large mass of central supermassive black hole and its proximity make LLAGN approachable to the launching and accelaration region of inner jet, even to its event horizon at millimeter or sub-millimeter wavelength. According to the inhomogeneous model of relativistic jet, the position of the VLBI core is frequency-dependent. This frequency-dependent shift in the location of the core (core-shift) can be used to estimate the magnetic field strength and electron number density of the jet (Lobanov 1998;Kovalev et al. 2008). However, only a few active galactic nuclei (AGNs) have reliable core-shift measurements at millimeter wavelengths (O'Sullivan & Gabuzda 2009). The coreshift measurements are mostly obtained at low frequencies due to limited sensitivity at high frequencies (Pushkarev et al. 2012). Although millimeter-VLBI can approach the inner region of jet as the plasma turns optically thin at high frequency, the core-shift is difficult to obtain due to the rapid phase flutuations of atmosphere and thus limited coherent intergration time for the conventional VLBI phase-referencing observations. Fortunately, a newly proposed VLBI phase-referencing technique called source-frequency phase-referencing (SFPR) can be used to measure the coreshift effect (Rioja & Dodson 2011), essentially being of great advantadge at millimeter wavelengths. We successfully obtained the first VLBI image of the LLAGN M 81* at 88 GHz and measured the shift between 7 mm and 3 mm wavelengths in the compact jet (Jiang et al. 2018). In this paper, we will present the applications of SFPR to two LLAGNs, M 84 and M 87. The observation summary and data reduction are presented in section 2. The results are in section 3, followed by the conclusion in section 4.

Data reduction
The data calibration and reduction followed the procedures in Rioja & Dodson (2011) and Jiang et al. (2018). Firstly we performed standard amplitude and phase calibrations in AIPS for both M 84 and M 87 at the reference frequencies (22 GHz of 2019 epoch and 44 GHz of 2021 epoch), respectively. Their images were obtained by further clean and self-calibration in Difmap. Secondly the corresponding phase solutions of the AIPS task FRING after taking into account clean models of M 84 and M 87 were multiplied by a factor of two, while the delay and delay rate solutions were unchanged. These revised solutions were then applied to the target frequencies (44 GHz in 2019 epoch and 88 GHz in 2021 epoch, respectively), which is called the frequency phase transfer (FPT) calibration. The phase flutuations in proportion to the observing frequency such as the unmodeled tropspheric and geometric errors were eliminated in this step. Thirdly the SFPR-ed images of M 84 at target frequencies were obtained by further phase referencing to the FRING solutions of the FPT calibrated data of M 87 at the same target frequencies, which refined the unmodeled dispersive ionospheric and instrumental errors. The corresponding core-shift measurements were derived from the SFPR-ed images using JMFIT in AIPS. Finally high resolution VLBI images of M 84 were obtained by clean and self-calibration in Difmap (Jiang et al. 2018). Since the brightness-peak position of the image is usually referred as the core position, the prominent jet of M87 in right ascension (RA) direction would cause the peak position to be slightly offset from the "true core" position towards the downstream side, due to the blending of near-core jet emission within the finite beam size. To evaluate this effect, we used the method in Hada et al. (2014), the M 87 structure was convolved by different beam size with diameters ranging from the minor axis of the nominal synthesized beam (shown in Table 1), whose direction was almost in RA direction in our observations, to about four times larger. Then, we plotted systematic changes of the brightness-peak position as a function of beam size. In the case of 22 GHz, the image were restored with beam sizes ranging from 0.3 to 1.2 mas in incremental steps of 0.1 mas. We found a progressive position shift of the brightness peak toward the downstream to be 10 µas per 0.1 mas in RA direction. In the case of 44 GHz, the image was restored with beam sizes ranging from 0.2 to 0.8 mas in incremental steps of 0.1 mas. The position shift of the brightness peak toward the downstream was 6.8 µas per 0.1 mas in RA direction. In the case of 88 GHz, the image was restored with beam sizes ranging from 0.15 to 0.6 mas in incremental steps of 0.05 mas. The position shift of the brightness peak toward the downstream was 2.5 µas per 0.1 mas in RA direction. That means the "true core" position would be shifted to upstream, with respect to the brightness-peak position when convolved with a nominal beam. At 22 GHz, the upstream shift in RA direction would be about 10×0.39 mas/0.1 mas= 39 µas. At 44 GHz, it would be about 6.8 × 0.22 mas/0.1 mas =15 µas for the 2019 epoch, and about 6.8×0.24 mas/0.1 mas = 16 µas for the 2021 epoch. At 88 GHz, it was about 2.5×0.16 mas/0.1 mas =4 µas. Where, 0.39 mas, 0.22 mas, 0.24 mas and 0.16 mas were the nominal beam sizes in the RA direction (see Table  1). Consequently, the measured core shift in RA direction from the peak-positions would be about 39−15 = 24 µas larger than the "true core shift" in magnitude at 22-44 GHz and, 16−4 =12 µas larger at 44-88 GHz.

Error analysis
The error budgets of SFPR mainly include the dynamic tropospheric error, the core identification error of M 87 and the statistical error from images. These errors are independently to each other and the total errors can be calculated as the root-sum-square of them. We adopt 11 µas for the dynamic tropospheric error among frequencies under relatively stable weather conditions, assuming 0.01 m uncanceled error by the water vapor fluctuation as that in Hada et al. (2011). The absolute tropospheric position error for a single frequency can be significantly larger than this value, while most of the error can be canceled out by the FPT calibration in SFPR observations. We also performed the error analysis for the core identification error of M 87 as in Hada et al. (2011), using the core position differences between two methods. One defined the centroid of the elliptical Gaussian fitting M 87 core region as the core position, the other is the brightness-peak position of the images convolved with a circular Gaussian beam of about a half of synthesized beam in the core-jet direction. The uncertainties of core identification of M 87 in RA direction are 7 µas, The nearby elliptical galaxy M 84 is located in the center of Virgo Cluster at a distance of 18.5 Mpc (z = 0.00339) and has a central supermassive black hole weighing ∼ 8.5 × 10 8 M . The combination of its proximity and a large black hole mass yields a privileged linear resolution conversion factor down to 1 micro-arcsecond (µas) ∼ 1 Schwarzschild radii (R S ), allowing us to investigate its close vicinity of the supermassive black hole with VLBI. Two side jets are seen at a large viewing angle of ∼74 • (Meyer et al. 2018). The image of M 84 at 88 GHz (Figure 1) was obtained by performing further clean and self-calibration in phase only to the SFPR-ed visibility data, which were scan averaged. The MODELFIT task in Difmap was used to fit the calibrated visibility with circular Gaussian components. The VLBI core of M 84 could be fitted by a circular Gaussian with a flux density of 38.0±5.7 mJy and a diameter of 29±11 µas well. The apparent brightness temperature of the core (Kim et al. 2018), T b,app , can be calculated by where S core is the core flux density in Jy, ν is the observing frequency in GHz, θ core is the equivalent size in milliarcsecond (mas) and z is the redshift. The T b,app of M 84 core at 88 GHz is ∼ 7.2 × 10 9 K. Similar to other LLAGNs (Kim et al. 2018), the T b,app is generally quite low. The derived brightness temperature could serve as a lower limit as the core size of 29 µas (30 R S ) was only one-tenth of the beam size and still un-resolved in our 88 GHz observations.

Core-shift of M 87
M87 is the most prominent elliptical galaxy within the Virgo Cluster, located at a distance of 16.8±0.8 Mpc away. Its central supermassive black hole (∼ 6.5 × 10 9 M ) and jet have been well-studied in almost every wave band from radio to γ-rays (Event Horizon Telescope Collaboration et al. 2019; EHT MWL Science Working Group et al. 2021). The core-shift of M 87 up to 43 GHz has been measured through conventional phase-referencing to M 84 (Hada et al. 2011(Hada et al. , 2013. The core-shift in RA direction r RA (ν) followed ν −0.94 and indicated that the black hole was located at ∼41 µas eastwards of the 43 GHz core (Hada et al. 2011). The core-shifts at 22-44 GHz and 44-88 GHz could be obtained from the SFPR-ed images (Rioja & Dodson 2011;Jiang et al. 2018) as shown in Figure 2. Since the jet extended structure of M 84 is toward the north direction, the core-shift of M 84 in RA direction could be negligible. The core-shift of M 87 in RA direction could be obtained from the SFPR-ed images by Equation (2) in Jiang et al. (2018). As a result, we obtained a position shift in RA direction of −64 ± 8 µas at the 22 GHz core with regard to the 44 GHz core, and of −33 ± 11 µas at 44-88 GHz. The error in 1σ was given by JMFIT in AIPS. After taking into account of the blending effect of near-core jet emission in M 87 (see Section 2.2), the "true core shift" would be about −(64 − 24)=−40 µas at 22-44 GHz and −(33 − 12)=−21 µas at 44-88 GHz. Incorporating the uncertainties given by error analysis (see Section 2.3), the core positions relative to that of the 44 GHz core in RA was −40 ± 19 µas at 22 GHz for the 2019 epoch and 21 ± 18 µas at 88 GHz for the 2021 epoch, respectively. Using the same formula r RA (ν) = Aν −k + B in Hada et al. (2011), the above-mentioned two core-shifts in RA direction could be solved with solutions A = −1.45, k = 0.92 and B = 0.045. As presented in the bottom-right corner of Figure  3, assuming a jet position angle of −69 • with respect to north in M 87 (Kim et al. 2018), B value indicates that the jet apex is located at ∼48 µas upstream of the 43 GHz core. It was consistent with previous result of 44 ± 13 µas in Hada et al. (2011). Since there was no strong flare event to cause the core-shift variations (Plavin et al. 2019) during our observations as well as in Hada et al. (2011) session, it would be reasonable to align the 43 GHz core positions of M 87 among these sessions. The core-shift in RA at 22-43 GHz even during the elevated very high energy Gamma-ray state in 2012 was found to be also at a similar level (∼10 µas larger) (Hada et al. 2014). Furthermore, aligning the 43 GHz cores among Hada et al. (2011) and the two epochs of this work, we fitted these combined core-shift measurements with weighted least square method. It gave out A = −1.36 ± 0.15, k = 0.92 ± 0.06 and B = 0.043 ± 0.007 as shown in Table 2. The results are consistent and it implies the jet apex is 43±7 µas in RA (as indicated by black dashed line in Figure 3) or at ∼46 µas upstream of the 43 GHz core. Following the Equation (4) in Lobanov (1998), we could estimate the core-shift measure Ω r,22−44 = 0.13±0.06 [pc GHz] and Ω r,44−88 = 0.12 ± 0.10 [pc GHz], using the resultant power law index k r = 1.09. The magnetic field strength at the 88 GHz core is estimated to be 4.8 ± 2.4 G using the mean value of Ω r , by Equation (B.2) in Paraschos et al. (2021) with a spectral index of −0.5, a Doppler factor of 2, a jet viewing angle of 18 • and an intrinsic jet opening angle of 63 • .6 at 88 GHz (Kim et al. 2018). This is consistent with the estimated 1-30 G near event horizon by the Event Horizon Telescope at 230 GHz (Event Horizon Telescope Collaboration et al. 2021).

CONCLUSIONS
We have successfully demonstrated that the SFPR technique could be applied to the mm-VLBI observations of LLAGNs. It helps to overcome the limited coherent intergration time and has great advantages in detecting the weak VLBI core as well as measuring the core-shift at millimeter-wavelengths. The VLBI core of M 84 at 88 GHz was detected and the lower limit of its apparent brightness temperature ∼ 7.2 × 10 9 K was obtained. By means of SFPR to M 84, the core-shift of M 87 in RA direction was determined at a precision of ∼20 µas, which further constrained the jet apex at ∼46 µas upstream of the 43 GHz core. With the aid of simultaneous multi-frequency receiving system and more stations available (Zhao et al. 2019;Rioja & Dodson 2020), SFPR will be a very powerful tool to investigate the compactness of the jet base at high frequencies as well as physical parameters such as core structure, brightness temperature and magnetic field of the inner region of jet.