Introduction

The direct images of supermassive black holes (SMBH) M87* and Sgr A* by the Event Horizon Telescope (EHT) have opened a new era of black hole studies [1,2]. In particular, the polarimetric images around the black holes have attracted attention as they can reflect the configuration of magnetic fields around the SMBH [3]. It has been established from theoretical approaches that the magnetic fields should have an important role in the creation and acceleration of the jets from active galactic nuclei (AGNs) hosting the SMBHs [4,5]. We can thus expect to shed new light on the long-standing question of the driving mechanism of AGN jets through these unprecedented high-resolution observations. However, one should also note that the polarized synchrotron emissions from a SMBH can experience Faraday effects on the way to Earth. Observational studies have detected traces of significant Faraday rotation of the linear polarization (LP) vectors in a range of millimeter and submillimeter wavelengths for many AGNs with/without jets [6][7][8][9]. It has been pointed out by theoretical studies that these should be attributed to internal Faraday rotation; that is, the Faraday rotation should occur in tandem with the emission near an SMBH, e.g., [10][11][12]. Furthermore, recent calculations have demonstrated that significant Faraday conversion from LP to circular polarization (CP) can occur near a SMBH [11,13,14]. In such cases, CP components of the combination between the intrinsic emission and conversion from LP components can be a good tool for investigating the magnetic field structure.

In this way, it has been established that the unified interpretation of both LP and CP is essential for understanding the magnetic structure and other plasma properties around an SMBH. In this context, we have thought of calculating the polarization images theoretically. To predict images around and near a BH, we must take general relativistic (GR) effects into account, and, here, perform the GR radiative transfer (GRRT) calculation based on the GR magnetohydrodynamics (GRMHD) models, e.g., [15][16][17].

In the previous work Tsunetoe et al. [18], we confirmed, on the basis of our moderately magnetized models with a hot jet and cold disk, a scenario where the polarized emissions produced in the jet experience the Faraday rotation and conversion effects in the disk. In this description of the emitting jet and Faraday-thick disk, we found that the LP (or CP) intensities are mainly distributed in the downstream (upstream) side of the approaching jet for nearly face-on observers. In this work, we surveyed polarimetric features for different inclination angles between the black hole's spin-axis and an observer, to expand and develop the discussions. Here, we bore in mind observations of a diverse range of AGN jets at 230 GHz by the EHT and other very long baseline interferometers (VLBIs). Furthermore, we thought of applying them to the interpretation of the disk precession around SMBHs [19,20].

Methods

We followed the model parameters adopted in the fiducial model in [18]. Here, we used a three-dimensional GRMHD model simulated with UWABAMI code [21,22], which is categorized into the intermediary area between a magnetically arrested disk (MAD; [23]) and standard and normal evolution (SANE; [24]), and is thus called semi-MAD. The Rβ model was adopted for the determination of electron temperature, where the proton-electron temperature ratio (T i /T e ) was given at each point in the fluid model by a function of plasma-β (≡ p gas /p mag ; gas-magnetic-pressure ratio) and two parameters R low and R high [25] as follows:

(1)

The model parameters are summarized in Table 1. GRRT calculation was performed by a code implemented in [11,18,26] with a sigma cutoff of σ cutoff = 1, and fast-light approximation was performed for a snapshot fluid model. Here, we used a snapshot at t = 9000t g (here, t g ≡ r g /c) in the quasi-steady state as a main model, while three other snapshots are surveyed and discussed in Section 4.2. The mass accretion rate onto the black hole was fixed so that we reproduced the 230 GHz observed flux of M87 in [1], ≈0.5 Jy, for the case with i = 160 • . Under these assumptions, we calculated the total, LP, and CP images, varying the observer's inclination (viewing) angle i = 0 • -180 • by 10 • as pictured in Figure 1. screen, by using the latter three cases. Here, we also confirm this tendency in the northern face-on-like cases, where the directions of separation are reversed for the north-side cases because the approaching (north-side) jet extends upward on the images.

These results demonstrate that the description of the synchrotron-emitting jet and Faraday-thick disk, as pictured in [18] and Figure 1, can be applied to the face-on cases both in north and south sides. In fact, it is shown in Figure 5 that typical optical depths for Faraday rotation (blue) and conversion (red) have a symmetric structure for the face-on-like observers in the north and south sides.

Oscillation of CP Signs

Next, we examined an oscillation feature of CP signs in edge-on-like cases. In Figure 4, we see a hint of flipping CP signs that oscillate for two cycles in a range of i ≈ 40 • -140 • . This can be interpreted with a combination of polarimetric features introduced so far. In the following descriptions, we distinguish two cases, observed from the north side and from the south side. Figure 2 shows the former cases (i.e., i ≤ 90 • ).

First, the CP images in edge-on-like cases are characterized by changing signs between the neighboring quadrants, as seen in the bottom-right panel in Figure 2. We confirm the validity of this description for edge-on "like" (not only i = 90 • ) cases. 1 In particular, the CP intensities in the second and third quadrants are brighter due to the relativistic beaming effect (see the bottom-right of Figure 2), and are dominant for the image-integrated, unresolved CP flux. Second, the unresolved CP fluxes in lower inclination cases are predominantly contributed from the counter-side jet (as shown in Section 4.3); the stronger CP intensities are found in the third quadrant (or second quadrant) side for the north (south) side observers. Third, the SSA effect becomes significant for a large inclination angle, as shown in Figure 5. Then, the CP emissions from the counter-side jet are suppressed because of large optical depths, and yield their dominance to those from the foregroundside jet (second quadrant for north and third quadrant for south). This is also shown by upward and downward I -|V| arrows for north and south side cases in Figure 6, respectively.

As a result, the dominant part for the CPs changes in order of the negative ring (face-on cases in the north), third quadrant (positive), second quadrant (negative), (i = 90 • ; crossing the equatorial plane,) third quadrant (positive), second quadrant (negative), and the positive ring (face-on in the south), if starting from i = 0 • to 180 • . In this way, the oscillation of the total CPs is explained with the monochromatic rings and quadranted images.

Ricarte et al. [14], using MAD and SANE models, also calculated profiles of unresolved CP fractions for inclination angles. Their profiles give negative and positive values for faceon-like cases in the north and south side, respectively, in a similar way to ours. Meanwhile, they show a sign-changing feature for edge-on cases but for one cycle (cf. two cycles in our case). The difference may be explained by removing our third sign-changing factor above: the change in the bright region. Where the emission, Faraday rotation/conversion, and SSA occur depends on many factors, such as the magnetic strength and electron temperature and density. Thus, the difference in the MAD-SANE regime and the electron temperature prescription can drastically affect the morphology of the images and integrated CP fractions.

The total CP fraction is a product from integrating an image consisting of the intrinsic emission and rotation-and twisted-field-driven conversion, which have different dependencies on the plasma and observational properties to each other. Thus, it may be difficult to access the characteristics of the system through this unresolved quantity alone [14]. One straightforward application is to combine it with the total LP, which we will discuss in the next subsection.

Combination of the Unresolved LP and CP Fractions

Finally, we focus on the relationship between the unresolved LP and CP fractions. The unresolved LP fractions in Figure 4 give a symmetric-like profile with high ( 1%) and low ( 1%) values in face-on and edge-on-like cases, respectively. This result is due to a larger optical depth for Faraday rotation for larger inclination angles. As pictured in Figure 1, the emitted LP vectors to the more edge-on-like observer experience a larger Faraday rotation in the outer cold disk. In fact, the typical optical depths for Faraday rotation are larger in the edge-on cases by approximately one order of magnitude than in the face-on cases.

If we combine this with the discussion in Section 4.2, we can conclude that the unresolved LP and CP fractions are characterized by relatively strong LPs and sign-persistent CPs in the face-on-like cases, and weak LPs and time-variable CPs in the edge-on cases. Precisely speaking, CP signs are time-varying in the edge-on-like cases (see Figure 4). M87(*) has been known to show strong LP and weak CP flux at radio wavelenths [2,6,34]. If we apply the model constraints for the total LP and CP fractions from Event Horizon Telescope Collaboration [35], the face-on like models are favored, which is consistent with the well-known large-scaled M87 jet. Future stimulating resolved/unresolved LPs and CPs will become a good tool for investigating the system of SMBH and plasma in M87 itself, and for applying knowledge of M87 to other AGN jets.

In contrast, M81* has been reported to show larger CP fractions than LP fractions [36,37]. These observation may be explained by our edge-on-like i ≈ 40 • -140 • cases with low LPs due to strong Faraday rotation, which is also consistent with high inclination angles referred for radio galaxies.

Whether and how source types, such as blazars, quasars, and radio galaxies, are related to the LP and CP fractions [29,38] has also been discussed. Coupled with more statistical data from the future resolved/unresolved spectro-polarimetry of various targets, a survey of inclination angles can give a clue for accessing a unified description of a diversity of AGNs.

Future Prospects

In this work, we focused our discussion on the diverse appearance of AGN jets, using the same fluid model, to demonstrate how jets are observed differently depending on the viewing angle. Meanwhile, it should be surveyed whether the results based on the semi-MAD model are common even for other fluid models, such as more SANE-or MAD-like models, and/or with different electron-temperature prescriptions, including the time-variability. More statistical surveys for various fluid models and long time durations is the scope of our future works.