The first supermassive black hole mass measurement in active galactic nuclei using the polarization of broad emission line Mg II

Spectropolarimetric efforts in the last few years have provided an efficient method that is based on the profiles of the polarization plane position angle of broad emission lines in active galactic nuclei (AGNs). Here we present black hole measurements of SBS 1419+538 using spectropolarimetric observations in the Mg II spectral band. The observations are performed by 6m telescope of SAO RAS using SCORPIO-2. We found a good agreement for the estimated supermassive black hole (SMBH) mass for this object using spectropolarimetry when compared with the mass obtained using other methods.


INTRODUCTION
According to the standard paradigm, AGNs are powered by an accretion of gas onto the SMBH which resides in the center (Salpeter 1964;Zel'dovich & Novikov 1964;Lynden-Bell 1969). Due to finite, but low viscosity of the gas, the gas temperature increases and the angular momentum is being transfered outwards, providing a slow, but steady inflow (Shakura & Sunyaev 1973). The gravitational binding energy is converted into enormous amount of radiation which ranks AGNs as the most luminous steady sources observed (Padovani 2017). SMBHs actively shape the environment in its vicinity, but also on kpc scales, through a process known as AGN feedback (Fabian 2012), which plays an important role in the host galaxy evolution (Kormendy & Ho 2013;Heckman & Best 2014). Therefore, reliable SMBH mass estimation is an important problem in modern astrophysics.
Many methods with different approach have been developed in the past few decades and extensively discussed in the literature (e.g. Peterson 2014; Popović 2020, and references therein). Broad emission lines in AGNs have been widely used for measuring SMBHs mass, most often in a long term reverberation mapping campaigns (Blandford & McKee 1982;Peterson 1993;Kaspi et al. 2000;Bentz et al. 2013;Du et al. 2016Du et al. , 2018Shapovalova et al. 2019, etc.).
When the polarized emission is taken into account, the broad line spectropolarimetry allows us to measure the SMBH mass with a single-epoch observations (Afanasiev & Popović 2015, hereafter AP15 method,). This method assumes that equatorial scattering of the inner side of the dusty torus is the dominant polarization mechanism (Smith et al. 2005;Savić et al. 2018Savić et al. , 2020Lira et al. 2020) and is in a good agreement with other methods (Afanasiev et al. 2019). In order to use the AP15 method, it is required that the distance (R sc ) between the SMBH and the scattering region is known. In the case of the Keplerian-like motion in combination with the equatorial scattering of the BLR light, the relation between velocities and polarization angle (tan ϕ) across the broad line is: where c is the speed of light, and constant a depends on the BH mass M bh as where G is the gravitational constant and θ is the angle between the BLR disk and the scattering region is assumed to be θ ∼ 0 in the case of equatorial scattering and therefore, the BH mass estimates is independent from the inclination. The constant b is close to 0.5 for the dominant Keplerian-like motion. One of the advantages is that this method can be applied to Mg II, C III], C IV broad lines, which would correspond to distant objects at high redshift, if these lines are observed in optical spectral range. In this work, we report the first spectropolarimetric observations of the Mg II line for a distant AGN SBS 1419+538 and we compare the SMBH mass estimated using the AP15 method with the estimates provided by different authors using other methods (most notably reverberation mapping).

POLARIMETRIC OBSERVATIONS OF Mg II SPECTRAL LINE
In order to test the model of polarization changes for the Mg II line, in February 2019 we carried out the spetropolarimetric observations of the quasar SBS 1419+538. SBS 1419+538 (RA 14 21 06.9 Dec +53 37 45.2, J2000) is a bright quasar (16.8 mag in the gsdss band) at the redshift z = 1.862 determined for the first time via the Second Byurakan Survey (Stepanian et al. 1993). The SDSS spectra of the quasar show broad (FWHM ∼ 5000 km s −1 ) components of the MgII and CIII] in the optical range (Schneider et al. 2005;Shen et al. 2011).
SBS 1419+538 was observed with the 6-m telescope BTA of SAO RAS with the focal reducer SCORPIO-2 (Afanasiev & Moiseev 2011). We used a 1 slit and a volume phase holographic grating covering the 5800-9500 Å range with a maximum at 7350 Å to obtain the spectrum images. Double Wollaston prism divided the image of the entrance pupil according to four polarization directions -0°and 90°, 45°and 135°. Then the parameters of the linear polarization and intensity -the Stokes parameters Q, U and I were obtained simultane-ously and are equal: where K Q and K U are the coefficients of the channel transmission, I 0 , I 90 , I 45 , I 135 correspond to the different polarization directions. Using K Q and K U coefficients one can minimize the influence of variable atmospheric depolarization (see Afanasiev & Amirkhanyan 2012, for more details). Then the polarization degree P and polarization angle ϕ are obtained from the following relations: where ϕ 0 is the zero point of polarization angle. To correct the device spectral sensitivity and to find ϕ 0 the non-polarized spectrophotometric and polarized standards were observed before the object. The polarimetric accuracy was up to variations of the atmospheric depolarization. Due to the high galactic latitude of the quasar (b ∼ 60°) the ISM polarization is neglected. The observations of the object were performed in a series of 16 frames with 300 s exposure times in order to make robust statistical estimations. The observational techniques and analysis method have been described in more details in several papers (see e.g. Afanasiev & Amirkhanyan 2012;Afanasiev et al. 2014;Afanasiev & Popović 2015;Afanasiev et al. 2019) and will not be repeated.

RESULTS
We extracted spectra and observed polarization parameters are shown in Fig. 1. The 1st panel shows the total flux in the spectral region near the broad Mg II line 7500-8500 Å with 2Å spectral resolution. The continuum emission is approximated here with a linear regression plotted with a dashed line. The 2nd and 3rd panels show the Stokes parameters Q and U , respectively. The polarization degree P and the polarization angle ϕ are given on panels 4th and 5th. The Stokes parameters Q and U , P and ϕ are binned over 10Å and depend on the wavelength. For each bin, the value was calculated as a robust average in the 2-dimensional array of a size 10Å by 16 exposures; the error bars are equal to the 1σ level as a robust standard deviation. A 2σ rejection threshold was used in order to avoid the influence of the outlier points (less than 1% mostly due to the cosmic rays hints). The average values of the parameters Q , U , P , ϕ are also given in the figure. As the measured value of polarization is small and is comparable with the errors, the value of polarization degree P is biased. The correction of P to the bias was made according to the formula given in (Simmons & Stewart 1985): where P obs is measured value of polarization and σ P is its error. Therefore, there are unbiased values of P given in Fig. 1.
The polarization profile of Mg II is single peaked and blue shifted may indicate some complex structure in the Mg II BLR, as e.g. outflowing/inflowing BLR Savić et al. 2020) or more complex as two component model (Popović et al. 2004) which can hide the expected two-peaks of the polarized profile in the case of disk-like motion (Savić et al. 2020). However in the case of pure disk-like motion, the single peaked polarized profile can be detected in the case of lower viewing inclinations (see It is well-known that there is a strong iron emission underlying the Mg II line that also arises from the BLR. Estimation iron emission is a non-trivial task and much effort has been invested for solving this problem , and references therein). We used an improved model by Kovačević-Dojčinović & Popović (2015) that covers the spectral range between 2650 − 3050 Å. Details regarding this model were extensively described by Popović et al. (2019). An illustration of the Mg II decomposition is shown in Fig. 2. A blue asymmetry is dominant after Fe II subtracting indicating outflow, which is also seen in the blueshifted polarized profile.
To obtain the SMBH mass according to the polarization properties of the equatorially scattered emission in Mg II line by the method given in (Afanasiev & Popović 2015) one should estimate the radius of the scattering region R sc . In Afanasiev et al. (2019) the dependency connecting R sc in AGN and the luminosity at 1516 Å was revealed: log R sc = −(15.60 ± 0.54 ) + (0.40 ± 0.01 ) log(λL 1516 ).
As far as the spectropolarimetric observations given here have relatively bad photometric bounding due to the slit loses more confident estimations of luminosity of SBS 1419+538 should be used. According to Shen et al. (2016) λL 1350 = 8.9·10 46 erg s −1 and λL 1700 = 7.2·10 46 erg s −1 . As the continuum spectra slope in the spectral range is not steep let us consider λL 1516 ≈ 8 · 10 46 erg s −1 and according to the dependency 9, R sc is equal to: The error of R sc was estimated by the bootstrapping method (Efron 1979) and includes the errors of the coefficients from the Eqn. 9. The asynchronism of the continuum luminosity taken from Shen et al. (2016) with respect to the spectropolarimetric observation from the given work and the λL 1516 uncertainty being smaller than the coefficients error were not taken into account.   We applied the AP15 method to find the black hole mass, and as it can be seen from Fig. 3, for Mg II line in the spectrum of SBS 1419+538 the observational data could be fitted with a linear function with the regression coefficient a = −1.95 ± 0.13. Note here that the computed slope of b coefficient is 0.46±0.11, so practically it was assumed identically equal to 0.5, which corresponds to the case of a Keplerian motion. Assuming that the BLR is co-planar with dust scattering region (cos 2 (θ) = 1, see AP15 for more details), we obtained that the SMBH mass is: To examine the result we also tried other methods of indirect mass measurements. The mass could be estimated from the virial theorem. To calculate the virial product one should estimate the velocity dispersion of the broad emission line. We measured the FWHM = 4791 ± 552 km s −1 and line dispersion σ = 2275 ± 263 km s −1 after subtracting the Fe II contribu-  tion to the Mg II line using the UV Fe II model given in Popović et al. (2019) 1 The size of the BLR region in the Mg II line is estimated using the empirical BLR radius -luminosity (R-L) relation (see Czerny et al. 2019;Popović 2020). We used an updated R-L relation at 3000 Å given by Zajaček et al. (2020). The estimation of the quasar luminosity was obtained from (Shen et al. 2016) λL 3000 = 6.1 × 10 46 erg s −1 . The BLR size was estimated as R BLR = 1195 +936 −541 light days. Therefore, the relation between the scattering region size and the BLR size is R sc /R BLR ≈ 1.7 ± 0.7. This value is in a good agreement with the mean ratio obtained by Afanasiev et al. (2019) as well as models by (Savić et al. 2018) for which the ratio is expected to be in range between 1.5 − 2.5.
The virial product could be calculated: From the profile of the polarization angle, it is possible to determine the BLR direction of rotation. A maximum of the polarization angle in the blue wing of the line followed by the minimum in the red wing corresponds to the anticlockwise rotation of the central engine (Savić et al. 2018).

DISCUSSION AND CONCLUSIONS
Magnesium lines are often associated with powerful outflows in addition to Keplerian motion (Laha et al. 2020). The outflows may be triggered by radiation pressure from the accretion disk and recently have been directly observed (Miyauchi & Kishimoto 2020). In our previous works (Savić et al. 2018(Savić et al. , 2020, we found that the AP15 method may be used with sufficient accuracy even if the outflows are present. The main uncertainty in the SMBH mass estimate is proportional to the the radius of the scattering region for which we lack direct measurements. Instead, we rely on various scaling relations that typically involve measured UV, optical or infrared luminosity at certain wavebands (Koshida et al. 2014;Afanasiev et al. 2019), which in principle increase the error of the estimated SMBH mass. Another difficulty that arises using the current observational technique is the upper magnitude limitation. Due to the high redshift of the observed object, we are prone to observe only the brightest quasars in the spectropolarimetry mode.
As follows from the description of the spectropolarimetric AP15 method, the two main advantages of the approach are the use of single-epoch observations and independence from the orientation of the AGN relative to the observer. Due to the accumulated data on AGN reverberation mapping and the relatively high statistical accuracy of the luminosity dependences on the BLR size, the mass estimate can also be obtained from single observations. In Table 1 we report previous measurements of SBS 1419+538 found in literature. Earlier SMBH estimates using data from SDSS campaign are close to log(M bh /M ) ≈ 10 (Shen et al. 2008(Shen et al. , 2011Rafiee & Hall 2011;Grier et al. 2019). Generally, our results are in good agreement with the SMBH estimates. However, this estimate will depend on an unknown dimensionless factor f ≈ 1 ∼ 10, depending on the system orientation and geometry (Kaspi et al. 2000;Onken et al. 2004;Bentz et al. 2013). Thus, the mass estimation error can reach 1 order of magnitude. A joint approach combining several types of mass estimation allows us to give more accurate and independent estimates of the masses.
A comparison of two independent estimates of the masses of the SMBH in Eqn. 10 and 12 allows us to estimate the dimensionless factor f , which in this case is equal to approximately 4. This value of the factor is close to the average value (f = 5.5 is usually assumed for the most AGNs, see Onken et al. 2004). Even if we assume that the M bh obtained by spectropolarimetry is overestimated by 35%, how is this it is assumed that according to the results of numerical modelling (Savić et al. 2020), the factor f is expected to be equal to approximately 3. Note here that the 2 times difference between the mass estimate given in this article and in Grier et al. (2019) can also be explained by an incorrect choice of f .
An additional difficulty of the AP15 method is the need to to estimate the inner radius of dusty torus where equatorial scattering is probably starting (see Fig. 1 in Shablovinskaya et al. 2020). Since there are no estimates of the radius of the dust torus in the IR range for the quasar SBS 1419+538 in the literature, and the results of the SDSS RM campaign have not been published yet. Therefore, we used the empirical relation 9. Undoubtedly, this worsens the accuracy of the estimate of the size of the R sc . However, the R sc /R BLR ratio is close to the theoretically predicted, which indicates that the error in determining R sc is not large than 30%, i.e. lies within the accuracy of the AP15 method for the Mg II line (Savić et al. 2020). In the future, the data of the SDSS-RM project or the results of the application of a new method for estimating the scattering region by reverberation mapping in polarized light ) will help to improve the accuracy of the mass estimate.
We apply for the first time the AP15 method for Mg II broad line. Future work using the existing facility will include additional spectropolarimic observations of distant quasars focusing on C III] and C IV emission lines. Current limitations will be largely surpassed with a next-generation instrument POLLUX (Muslimov et al. 2018) that will be aboard the mission LUVOIR (Large UV/Optical/IR Surveyor, The LUVOIR Team 2019). Table 1. Previous measurements of SBS 1419+538 found in literature. Columns from left to right: references, redshift, continuum luminosities at 1350; 1700; 3000, bolometric luminosity, full width at half maximum (FWHM) of the Mg II and C IV, the broad line region radius and estimated SMBH mass. We ignore the error bars that were not given by previous authors. All SMBH measurement were obtained by a single-epoch approach. project number №20-12-00030 "Investigation of geometry and kinematics of ionized gas in active galactic nuclei by polarimetry methods", which supported the spectropolarimetric data analysis.