*Communication* **Determination of the Magnetic Field Strength and Geometry in the Accretion Disks of AGNs by Optical Spectropolarimetry**

**Mikhail Piotrovich 1,2,\*, Stanislava Buliga 1,2 and Tinatin Natsvlishvili <sup>1</sup>**


**Abstract:** Based on the spectropolarimetric data of 33 Seyfert type 1 galaxies observed with the BTA-6m telescope of the Special Astrophysical Observatory, we estimated the magnetic field values at the event horizon of the supermassive black hole *B*<sup>H</sup> and the exponents of the power-law dependence *s* of the magnetic field on the radius. We used the model of optically thick geometrically thin Shakura– Sunyaev accretion disk. The average value of log *B*H[G] was found to be ∼4, which is in good agreement with the results obtained by other methods. The average value of *s* is *s* ≈ 1.7, and its distribution maximum span is in the range od 1.85 < *s* < 2.0. This is a rather interesting result, since *s* = 5/4 is usually adopted in calculations for Shakura–Sunyaev accretion disks. In addition, for two objects PG 1545+210 and 2MASX J06021107+2828382, the measured degree of polarization is greater than the maximum possible value at the angle between the line of sight and the axis of the accretion disk *i* = 45◦. It was concluded that for these objects the angle should be closer to *i* = 60◦.

**Keywords:** accretion disks; magnetic fields; polarization; active galactic nuclei; supermassive black holes

#### **1. Introduction**

According to modern concepts, accretion disks of active galactic nuclei (AGNs) should have an intense magnetic field [1,2]. It is assumed that the magnetic field is formed as a result of the interaction of accreting matter with a rotating supermassive black hole (SMBH) [3–8]. The presence of a magnetic field should have a noticeable effect on the spectropolarimetric characteristics of the accretion disk radiation. The polarimetric observations demonstrate that AGNs have polarized radiation in different wavelength ranges, from ultraviolet to radio waves [9–17]. Several mechanisms for the origin of the observed polarization are discussed, for example, the light scattering in accretion disks or synchrotron radiation of charged particles. These mechanisms can act in different structures, such as the plane and warped accretion disks, toroidal rings near the accretion disks and relativistic jets. It happens that different models are proposed to explain the same source. There are several models of accretion disks (see for example Pariev et al. [18]). For objects of the type under study, the most popular and simple model is the optically thick geometrically thin Shakura–Sunyaev disk [19]. In this work, we assume that for our sample of objects (Seyfert type 1) accretion disk is the main source of polarized radiation in optical range and we use Shakura–Sunyaev disk model.

It should be noted that accurate determination of the dependence of the magnetic field on the radius in the disk is rather difficult task [16,18,20,21], consisting of intricate spectropolarimetric observations of distant and faint objects and complex and time consuming numerical simulations. Our goal was to estimate the magnitude of the magnetic field at the event horizon of the SMBH and probe the dependence of the magnetic field intensity on the radius in the AGN accretion disks using our spectropolarimetric observations with

**Citation:** Piotrovich, M.; Buliga, S.; Natsvlishvili, T. Determination of the Magnetic Field Strength and Geometry in the Accretion Disks of AGNs by Optical Spectropolarimetry. *Universe* **2021** , *7* , 202. https:// doi.org/10.3390/universe7060202

Academic Editors: Nazar R. Ikhsanov, Galina L. Klimchitskaya and Vladimir M. Mostepanenko

Received: 29 May 2021 Accepted: 16 June 2021 Published: 18 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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the BTA-6m telescope and our relatively simple model. The methodology described in Silant'ev et al. [22] was taken as a basis for this work.

#### **2. Basic Equations**

#### *2.1. Stokes Parameters*

When considering radiation from an axially symmetric accretion disk with a magnetic field, its integral Stokes parameters can be written in the following form [22]:

$$\begin{array}{l} \langle Q \rangle = Q(0,\mu) \frac{2}{\pi} \int\_0^{\pi/2} d\Phi \, \frac{1+a^2+b^2\cos^2\Phi}{(1+a^2+b^2\cos^2\Phi)^2 - (2ab\cos\Phi)^2} \\\langle \langle \mathbf{U} \rangle = a \, Q(0,\mu) \frac{2}{\pi} \int\_0^{\pi/2} d\Phi \, \frac{1+a^2-b^2\cos^2\Phi}{(1+a^2+b^2\cos^2\Phi)^2 - (2ab\cos\Phi)^2} \end{array} \tag{1}$$

where *μ* = cos *i*, where *i* is the angle between the line of sight and the axis of the disk, *Q*(0, *μ*) is the value of the Stokes parameter without a magnetic field. The parameters *a* and *b* are expressed, in turn, as follows:

$$\begin{array}{l} a = 0.8\mu\lambda^2(\mu m)B\_{\parallel}(G), \\ b = 0.8\sqrt{1 - \mu^2}\lambda^2(\mu m)B\_{\perp}(G), \end{array} \tag{2}$$

where *λ* is the wavelength, *B* and *B*<sup>⊥</sup> are, respectively, the component of the magnetic field in the disk parallel and perpendicular to the disk axis (see Figure 1 from Silant'ev et al. [22]). Taking into account that in the Milne problem (multiple scattering of light in optically thick flattened atmospheres [23,24]) without a magnetic field, the Stokes parameter *U*(0, *μ*) ≡ 0, we obtained the following value of the relative polarization and positional angle *χ*:

$$\begin{array}{l} P\_{\rm rel} = P(B,\mu)/P(0,\mu) = \sqrt{\langle Q \rangle^2 + \langle \mathcal{U} \rangle^2/Q(0,\mu)}. \\ \chi = 0.5 \arctan(\langle \mathcal{U} \rangle/\langle Q \rangle). \end{array} \tag{3}$$

Note that *P*rel depends only on the dimensionless parameters *a* and *b* and does not depend on *Q*(0, *μ*).

The polarization value *P*(0, *μ*) without a magnetic field was previously calculated by us numerically using the Sobolev–Chandrasekhar model [23,24] and is tabulated in Gnedin et al. [25]. In addition, we note that the polarization has a small effect on the radiation intensity [23]. Thus, we have the opportunity to accurately calculate the polarization value *P* = *P*rel*P*(0, *μ*), using numerical integration for the parameter *P*rel and tabular values for *P*(0, *μ*).

#### *2.2. Magnetic Field*

When considering the magnetic field in the accretion disk, it is usually assumed (see, for example, Pariev et al. [18]) that its dependence on the radius has a power-law form:

$$B(R) = B\_{\rm H}(R\_{\rm H}/R)^{s} \,\,\,\,\,\,\tag{4}$$

where *B*<sup>H</sup> is the value of the magnetic field intensity at the event horizon of SMBH in AGN, *<sup>R</sup>*<sup>H</sup> = *GM*BH(<sup>1</sup> + <sup>1</sup> − *<sup>a</sup>*<sup>2</sup> <sup>∗</sup>)/*c*<sup>2</sup> is the radius of the event horizon, *<sup>M</sup>*BH is the mass of the SMBH, *<sup>G</sup>* is the gravitational constant, *<sup>c</sup>* is the speed of light, *<sup>a</sup>*<sup>∗</sup> <sup>=</sup> *c J*/*GM*<sup>2</sup> BH is the dimensionless spin of the SMBH, *J* is the angular momentum of the SMBH rotation. As for the *s* parameter, there are models with different values of this parameter [18], but for the Shakura–Sunyaev disk, the most often adopted value is *s* = 5/4 [19]. In our work, we decided to investigate in more detail the influence of this parameter on the model and therefore we tried a number of values within 0.5 < *s* < 2.

#### *2.3. Dependence of the Polarization Degree on the Wavelength*

Since the polarization degree depends on the magnetic field, and the magnetic field depends on the radius, then in order to obtain the dependence of the polarization on the wavelength, we need the dependence of the radius on the wavelength. For the Shakura– Sunyaev accretion disk, we have [26]:

$$R\_{\lambda}(cm) = 0.97 \times 10^{10} \lambda \, [\mu\text{m}]^{4/3} \left(\frac{M\_{\text{BH}}}{M\_{\odot}}\right)^{2/3} \left(\frac{l\_{\text{E}}}{\varepsilon}\right)^{1/3}.\tag{5}$$

where *R<sup>λ</sup>* is the distance in the accretion disk, which corresponds to wavelength *λ*, *l*<sup>E</sup> = *L*bol/*L*Edd is the Eddington ratio, *L*bol is the bolometric luminosity, *L*Edd = 1.3 × 1038*M*BH/*Merg*/*<sup>s</sup>* is the Eddington luminosity, *<sup>ε</sup>* <sup>=</sup> *<sup>L</sup>*bol/(*Mc* ˙ <sup>2</sup>) is the radiative efficiency, *M*˙ is the accretion rate.

#### **3. Results of Theoretical Calculations**

We estimated dependencies of radius *Rλ*, magnetic field strength *B*, polarization degree *<sup>P</sup>* and position angle *<sup>χ</sup>* on wavelength *<sup>λ</sup>* (in optical range) for *<sup>M</sup>*BH/*M* <sup>=</sup> <sup>10</sup>8, *<sup>B</sup>*<sup>H</sup> <sup>=</sup> 104 G, *<sup>a</sup>*<sup>∗</sup> <sup>=</sup> 0.9, *<sup>ε</sup>* <sup>=</sup> 0.155, *<sup>l</sup>*<sup>E</sup> <sup>=</sup> 0.2 and various values of *<sup>s</sup>*. Results are presented in Table 1. In Inoue and Doi [27], the authors obtained *B* ≈10 G at *R* ≈ 40*R*<sup>H</sup> for AGNs with *<sup>M</sup>*BH <sup>∼</sup> <sup>10</sup>8*M*. These data are in good agreement with our model, in which, for *<sup>B</sup>*<sup>H</sup> = <sup>10</sup><sup>4</sup> G and *<sup>s</sup>* = 1.85, *<sup>B</sup>*(40*R*H) ≈ 10.7 G.

**Table 1.** Dependencies of radius *Rλ*, magnetic field strength *B*, polarization degree *P* and position angle *χ* on wavelength *λ* for *<sup>M</sup>*BH/*M* <sup>=</sup> <sup>10</sup>8, *<sup>B</sup>*<sup>H</sup> <sup>=</sup> 104 G, *<sup>a</sup>*<sup>∗</sup> <sup>=</sup> 0.9, *<sup>ε</sup>* <sup>=</sup> 0.155, *<sup>l</sup>*<sup>E</sup> <sup>=</sup> 0.2 and various values of *<sup>s</sup>*.


We calculated the value of the polarization and the positional angle as a function of the value of the magnetic field intensity at the event horizon *B*<sup>H</sup> and the parameter *s* in the visible range. For this purpose we adopted the following parameter values characteristic of Seyfert type 1 AGN: *<sup>M</sup>*BH <sup>=</sup> 108*M*, *<sup>a</sup>*<sup>∗</sup> <sup>=</sup> 0.9 [17,28], radiative efficiency *<sup>ε</sup>* <sup>=</sup> 0.155, Eddington ratio *l*<sup>E</sup> = 0.2.

As for the angle *i*, since it is rather difficult to determine this angle from the observables, research usually adopts *i* ≈ 45◦ in the calculations. In our work, we used a more complex and arguably a more accurate method. Initially, the values were calculated for all angles from 0 to 90 degrees, and then the resulting data were convolved with a Gaussian centered at 45 degrees. If an angle other than 45 degrees was required for calculations, then a Gaussian with a center at this angle was taken.

Based on the generally assumed dipole nature of the magnetic field, and also based, for example, on the conclusions of Piotrovich et al. [29], the ratio between the components of the magnetic field parallel and perpendicular to the disk axis was taken as *<sup>B</sup>*<sup>⊥</sup> = 0.1*B*.

Figures 1 and 2 show the results of these calculations in the form of three-dimensional graphs. In Figure 1, one can see that the average value of the polarization in the visible range depends rather strongly on the parameters *BH* and *s*. The rapid decrease in polarization with increasing field and decreasing *s* is due to Faraday depolarization (see Equation (2) from Silant'ev et al. [22]). It can be seen in Figure 2 that the gradient of the positional angle depends on the parameters in a rather complex way, which theoretically can make it possible to accurately determine the parameters from the observed gradient. However, the amplitude of this dependence, unfortunately, is rather small.

**Figure 1.** Wavelength-averaged degree of polarization *P* depending on the value of the magnetic field intensity at the event horizon *B*<sup>H</sup> and the exponent of the power-law dependence *s* of the magnetic field on the radius in the disk.

**Figure 2.** The difference between the positional angles of polarized radiation Δ*χ* at a wavelength of 0.70 μm and 0.45 μm depending on the value of the magnetic field intensity at the event horizon *B*<sup>H</sup> and the exponent of the power-law dependence *s* of the magnetic field on the radius in the disk.

In addition, we also plotted the dependencies of the degree of polarization and the positional angle on the wavelength for the above parameter values for different values of *B*<sup>H</sup> and *s* (see Figures 3–6).

**Figure 3.** Dependence of the degree of polarization *P* on the wavelength *λ* for different values of the parameter *s*.

**Figure 4.** Dependence of the positional angle *χ* on the wavelength *λ* for different values of the parameter *s*.

**Figure 5.** Dependence of the degree of polarization *P* on the wavelength *λ* for different values of the parameter *B*H.

**Figure 6.** Dependence of the positional angle *χ* on the wavelength *λ* for different values of the parameter *B*H.

It should be noted that the dependence of the degree of polarization and positional angle on the wavelength in our model was found to be relatively weak. It is practically impossible to reliably measure such a gradient of polarization and position angle in real astronomical observations with the existing signal-to-noise level. Therefore, in further calculations, we used the average value of the polarization, making sure that the dependence of the observed polarization on the wavelength was not too strong and had a monotonic form. As for the observed gradient of the position angle, it was evaluated only qualitatively; in particular, objects with a pronounced non-monotonic dependence of the position angle on the wavelength were discarded, for example, in the presence of the so-called "S-shaped" feature, which appears to be due to other physical mechanisms (for example, scattering by a spherical optically thin envelope with a magnetic field [30]).

### **4. Estimations of** *B***<sup>H</sup> and** *s* **Based on Optical Spectropolarimetry of AGNs**

In our work, we used the already published data of spectropolarimetric observations of a sample of 33 AGN in type 1 Seyfert galaxies, carried out on the BTA-6m telescope with the participation of the authors [16,17,31]. First round of observations were performed in 2008–2009. The observations were carried out with the SCORPIO focal reducer in the spectropolarimetric mode mounted at the prime focus. We used an EEV42-40 2048 × 2048 pixel CCD array with a pixel size of 13.5 × 13.5 μm as the detector and a VPHG550g volume holographic phase grating from the SCORPIO kit operating in the range 3500–7200 Å as the dispersing element. The reciprocal linear dispersion in the detector plane was 1.8 Å/pixel. In the spectrograph, we used a set of five circular diaphragms 4".5 in diameter arranged in the form of a pseudoslit with a step of 9.7-arcsec. A Savart plate placed behind the diaphragms was used as the polarization analyzer. We used the central diaphragm to take the spectra of an object in perpendicular polarization planes and the remaining diaphragms to take the night-sky spectra. The actual spectral resolution of our data was determined by the monochromatic image of the diaphragms and was 40–42 Å. The seeing in all sets of observations was at least 2". The technique of polarization observations and calculations was described by Afanasiev and Moiseev [32]. To calibrate the wavelengths and the relative transmission of the diaphragms, we used an Ar–Ne–He filled line-spectrum lamp and a quartz lamp. To calibrate the spectropolarimetric channel of the spectrograph, we observed standards from Turnshek et al. [33]. Second round of observations were performed in 2012–2016. The observations were carried out with the SCORPIO-2 spectrograph [34]. The spectra were taken in two ranges: 4200–7500 Å for redshifts z < 0.1 (VPHG940 grating) and 5700–9500 Å for z > 0.1 (VPHG940 grating). The spectral resolution for a working 2" slit was 14 and 12 Å, respectively. For objects at Galactic latitudes < 30◦ we took into account the interstellar polarization that was determined from the observations of stars around the object. The technique of observations and data reduction is described in detail in Afanasiev and Amirkhanyan [35].

We have formed a sample of 33 sources with published mass estimates for their central SMBHs. Since our model assumes a geometrically thin, optically thick disk [19], we considered only objects with Eddington ratio in the 0.01 < *l*<sup>E</sup> < 0.3 range [36].

Note that in this work we used the average values of the parameters *M*BH, *a*<sup>∗</sup> and *l*E, neglecting the errors. This is explained by the following arguments. The parameters *a*<sup>∗</sup> and *l*<sup>E</sup> themselves have little effect on the polarization value. The *M*BH parameter has a more noticeable effect, but this effect can be neglected in comparison with the error of the observational spectropolarimetric data.

For each object, the dependence of the polarization and the gradient of the positional angle on *B*<sup>H</sup> and *s* was constructed as it was shown in Section 3. Then, this dependence was compared with observational data. The result is a set of values for *B*<sup>H</sup> and *s* that satisfies these conditions. After that, the *B*<sup>H</sup> values were additionally subject to the condition that they must fall within the limits obtained for these objects by independent methods [37,38]. In the paper Daly [38], errors in the determination of *B*<sup>H</sup> are not indicated, so we took them as ±0.3 in a logarithmic scale. For those objects for which the magnetic field strength was not previously estimated, we adopted the value log *BH*[*G*] = 4.0 ± 1.0, since the results of [37,38] estimates of the magnetic field for type 1 Seyfert nuclei give this characteristic range. The values of *B*<sup>H</sup> and *s* were averaged to obtain the average value and its associated dispersion.

For the objects PG 1545+210 and 2MASX J06021107+2828382, the measured polarization value was found to be greater than the maximum possible value with this calculation method. Since the polarization increases with the inclination angle, the angle value *i* = 60◦ was used for these objects. It is believed that for objects of the type under study (Seyfert type 1 galaxies) this angle usually lies within 20◦ ≤ *i* ≤ 60◦ (see, for example, Wu and Han [39]).

Our results are presented in Table 2.



Sources: (1) Vestergaard and Peterson [40]; (2) Peterson et al. [41]; (3) Satyapal et al. [42]; (4) Gnedin et al. [43]; (5) Afanasiev et al. [17]; (6) Piotrovich et al. [37]; (7) Afanasiev et al. [31]; (8) Piotrovich et al. [44]; (9) Piotrovich et al. [45]; (10) Our estimations based on the method from Piotrovich et al. [37]; (11) Marin [46]; (12) Afanasiev et al. [16]; (13) Devereux [47]; (14) Savi´c et al. [48].

#### **5. Analysis of the Estimated Parameters of the Magnetic Field**

As mentioned earlier, the value of the spin *a*<sup>∗</sup> has a rather weak effect on the results, especially taking into account the fact that this value itself varies within rather narrow boundaries. Therefore, the dependence of *B*<sup>H</sup> and *s* on *a* is negligible.

Figure 7 shows the obtained values of the magnetic field at the event horizon *B*<sup>H</sup> and the exponent of the power-law dependence *s* in graphical form. No pronounced dependence among these parameters on each other is observed. However, one can notice that the values are concentrated in the right side of the graph. It should be noted that all the values of *s* we obtained are greater than the 5/4 value usually adopted for accretion disks in type 1 Seyfert nuclei [19]. Figure 8 presents the dependence of the magnetic field strength at the event horizon of the SMBH on its mass. There is a linear dependence of the form log *B*H[G] ≈ (−0.69 ± 0.04)log(*M*BH/*M*)+(9.76 ± 0.35), which in a close agreement with a similar relation obtained by us in Piotrovich et al. [37]. Figure 9 depicts the dependence of the exponent of the power-law index *s* on the SMBH mass *M*BH. We find a linear dependence *s* ≈ (0.19 ± 0.04)log *M*BH/*M* + (0.18 ± 0.34). It should be noted here that the accuracy of this linear approximation is lower than that obtained for *B*H. Figure 10 displays the dependence of the magnetic field at the event horizon of the SMBH on its Eddington ratio *l*E. We derive a linear dependence of the form log *B*H[G] ≈ (1.05 ± 0.09)log *l*<sup>E</sup> + (5.38 ± 0.11) in agreement with the results obtained by Piotrovich et al. [37]. Figure 11 gives the dependence of the exponent of the power-law dependence of *s* on the Eddington ratio *l*E. A linear dependence of the form *s* ≈ (−0.25±0.06)log *l*<sup>E</sup> + (1.41±0.08) is visible, which, however, also has a lower accuracy than for *B*H.

**Figure 7.** The obtained values of the magnetic field intensity *B*<sup>H</sup> and the exponent of the power-law dependence *s*.

**Figure 8.** Dependence of the magnetic field intensity *B*<sup>H</sup> on the mass of the SMBH *M*BH.

**Figure 9.** Dependence of the exponent of the power-law dependence *s* on the SMBH mass *M*BH.

**Figure 10.** Dependence of the magnetic field intensity *B*<sup>H</sup> on the Eddington ratio *l*E.

**Figure 11.** Dependence of the exponent of the power-law dependence *s* on the Eddington ratio *l*E.

The histograms in Figures 12 and 13 show the distributions of objects by the values *B*<sup>H</sup> and *s*. It should be noted that these histograms cannot be regarded as a source of completely accurate statistical data due to the limited number of objects. Our results are only estimates. It can be seen that, for the magnetic field, the peak of the distribution falls on the region 4.0 < log *B*H[G] < 4.5. For comparison, we can mention that in our work Piotrovich et al. [37] the peak of the distribution was in the region of 3.5 < log *B*H[G] < 4.0. This statistical difference is most likely due to the low statistics in current study. In general, these results are consistent with results from Daly [38].

As for the parameter *s*, the peak of the distribution is in the region of 1.85 < *s* < 2.00, which is quite interesting, given that, as mentioned earlier, the standard value of *s* for accretion disks of the Shakura–Sunyaev type is usually considered to be 5/4 [19].

Table 3 presents the main statistical properties of the parameters of our model and the results of our calculations. The statistical properties of *B*<sup>H</sup> were found to be close to the values from Piotrovich et al. [37] and Daly [38].

**Table 3.** Basic statistical properties of the parameters. "Mean" indicates the arithmetic mean, "Median" the median value, "SD" the standard deviation.


**Figure 12.** A histogram showing the number of objects with a certain *B*<sup>H</sup> value.

**Figure 13.** A histogram showing the number of objects with a certain *s* value.

#### **6. Conclusions**

Based on the spectropolarimetric data of 33 Seyfert type 1 galaxies obtained with the BTA-6m telescope of the Special Astrophysical Observatory, estimates of the values of magnetic fields at the event horizon of the SMBHs *B*<sup>H</sup> and the values of the exponents of the power-law dependence *s* of the magnetic field on the radius *B*(*R*) = *B*H(*R*H/*R*)*<sup>s</sup>* , where *R*<sup>H</sup> is the radius of the event horizon.

The average value of log *B*H[G] was found to be ∼4, which is in good agreement with the results obtained by other methods [37,38], in which the magnetic field strength was estimated using the physical parameters of the relativistic jets. It was possible to reveal the dependence of the magnetic field on the SMBHs mass and the Eddington ratio of the form log *B*H[G] ≈ (−0.69 ± 0.04)log(*M*BH/*M*)+(9.76 ± 0.35) and log *B*H[G] ≈ (1.05 ± 0.09)log *l*<sup>E</sup> + (5.38 ± 0.11), which agree well with the results of Piotrovich et al. [37].

The average value of *s* is *s* ≈ 1.7, and the maximum distribution over *s* is within 1.85 < *s* < 2.0. This is a rather interesting result, since *s* = 5/4 is usually taken in calculations for accretion disks in type 1 Seyfert nuclei. We also managed to estimate the dependence of *s* on the SMBHs mass and the Eddington ratio of the form *s* ≈ (0.19 ± 0.04)log *M*BH/*M* + (0.18 ± 0.34) and *s* ≈ (−0.25 ± 0.06)log *l*<sup>E</sup> + (1.41 ± 0.08). In addition, although these approximations have a larger error than in the case of *B*H, they are still of interest. In particular, it may indicate that the more complex disk models are required than the Shakura–Sunyaev model. This problem undoubtedly requires further study.

In addition, for two objects PG 1545+210 and 2MASX J06021107+2828382, the measured polarization value was found to be greater than the maximum possible value at the inclination angle between the line of sight and the axis of the accretion disk *i* = 45◦. Since the polarization increases with the angle, it was concluded that for these objects the angle should be closer to *i* = 60◦.

**Author Contributions:** Conceptualization, M.P.; methodology, M.P.; validation, M.P.; formal analysis, M.P., S.B. and T.N.; investigation, S.B. and T.N.; resources, S.B. and T.N.; data curation, S.B. and T.N.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization, M.P.; supervision, M.P.; project administration, M.P.; funding acquisition, M.P. and S.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the grant of Russian Science Foundation project number 20-12-00030 "Investigation of geometry and kinematics of ionized gas in active galactic nuclei by polarimetry methods". Observations with the SAO RAS telescope are supported by the Ministry of Science and Higher Education of the Russian Federation (including agreement No.05.619.21.0016, project IDRFMEFI61919X0016).

**Data Availability Statement:** The data underlying this article are available in the article.

**Acknowledgments:** The authors are grateful to the employees of the Special Astrophysical Observatory V.L. Afanasiev (deceased 21 December 2020) and E.S. Shablovinskaya for help in carrying out observations and processing the results and the employee of the Central Astronomical Observatory at Pulkovo N.A. Silant'ev for useful comments and advice. The authors are also grateful to the reviewers for useful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Review* **Scattered Radiation of Protoplanetary Disks**

**Vladimir P. Grinin 1,2,\* and Larisa V. Tambovtseva <sup>1</sup>**


**Abstract:** Scattered radiation of circumstellar (CS) dust plays an important role in the physics of young stars. Its observational manifestations are various but more often they are connected with the appearance of intrinsic polarization in young stars and their CS disks. In our brief review we consider two classes of astrophysical objects in which the participation of scattered radiation is key for understanding their nature. First of all, these are irregular variables (UX Ori type stars). The modern idea of their nature and the mechanism of their variability has been formed thanks to synchronous observations of their linear polarization and brightness. The second class of objects is the CS disks themselves. Their detailed investigation became possible due to observations in polarized light using a coronographic technique and large telescopes.

**Keywords:** protoplanetary disk; scattered radiation; linear polarization; UX Ori stars; RW Aur

#### **1. Introduction**

Scattered radiation of circumstellar disks, as a rule, makes a small contribution to the optical radiation of young stars. The exception is a subclass of irregular variable stars with UX Ori as a prototype [1], and a small number of highly embedded stars and stars with edge-on disks. The family of UX Ori stars includes mainly the stars of the Ae spectral type. For a long time broadband photometric observations were the only method for their investigation. However, such observations could not unambiguously determine the mechanism of their unusual variability, representing a sequence of stochastic brightness weakening with an amplitude up to 2–3*<sup>m</sup>* and duration from a few days to a few weeks. The same observations can often be explained in completely different ways. For example, the reddening of the star with a decrease in its brightness has been equally well interpreted with both an increase in the CS extinction [2], and an appearance of magnetic spots on the star [3].

The important role of scattered radiation in understanding the nature of the variability of these objects was first pointed out by one of the authors of this paper [4]. During the deep brightness minima caused by screening the star with the CS gas and dust clouds, the direct radiation of the star is blocked by the screen. At such moments, the scattered radiation of the CS dust dominates the observed radiation. This explains a range of properties of these objects, including the unusual behavior of color indices at brightness fading, restriction of the brightness amplitudes and increases in the linear polarization in the brightness minima. Based on these facts, Grinin et al. [5] determined the evolutionary status of the stars from this subclass: the UX Ori stars (or UXOrs) are usually young stars, namely, intermediatemass Herbig Ae stars surrounded by protoplanetary disks, and they differ from the usual photometrically inactive Herbig stars only with a small inclination of their CS disks to the line of sight. This conclusion has been supported by further investigations [6,7], including interferometric observations in the near-infrared region of the spectrum [8,9].

It should be noted that an addition of the photometrically active UX Ori type stars to the photometrically inactive ("classical") Herbig AeBe stars had a strong influence on the further development of our ideas about all classes of Herbig stars. It turned out that these stars are not surrounded by spherical gas and dust envelopes, as previously assumed (see,

**Citation:** Grinin, V.P.; Tambovtseva, L.V. Scattered Radiation of Protoplanetary Disks. *Universe* **2022** , *8* , 224. https://doi.org/10.3390/ universe8040224

Academic Editors: Galina L. Klimchitskaya, Vladimir M. Mostepanenko and Nazar R. Ikhsanov

Received: 14 February 2022 Accepted: 30 March 2022 Published: 2 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

e.g., [10]), but by circumstellar disks. It was also found that the Herbig stars demonstrate not only spectral signs of the matter outflow but also signs of accretion. It all depends on the angle between the disk plane and the line of sight [11,12].

#### **2. Coronographic Effect**

During the long-standing photopolarimetric observations of the UX Ori type stars, it turned out that the observed changing in the linear polarization parameters is well described with the model suggested in [4]. It claims that the CS dust clouds obscure the star from the observer but do not influence the optical properties of the scattered radiation of the CS disk. The latter means that the shadow zones on the disk created by the clouds are much smaller in comparison with its size. Herewith the dust cloud themselves also do not influence the polarization of the stellar radiation that passes through them. The latter means that the dust grains in the clouds are not lined up. In this condition, changes in the intensity of the observed radiation during the eclipse *Iobs* are described with the following simple relationship:

$$I\_{\rm obs} = I\_\* e^{-\tau\_{\lambda}} + I\_{\rm sc} \tag{1}$$

where *I*<sup>∗</sup> is the intensity of the stellar radiation out of the eclipse, *τλ* is the optical depth of the cloud screening the star at the wavelength *λ*, and *Isc* is the intensity of the scattered radiation, which is considered unchanged during the eclipse, as was mentioned above.

The dependence of *τ* on *λ* is assumed as *τλ* = *τ f*(*λ*), which suggests that within the stellar disk the screen is homogeneous (this permits us to treat the star as a point source of light), and dependence of its optical depth on the wavelength does not change during the eclipse. The latter assumption is based on a very important observational fact: the existence of the straight section on the color–magnitude diagram that is used to determine the CS extinction law (see, e.g., [13]).

From Formula (1) one can directly obtain a link between the intensity of the scattered radiation of the disk and the possible maximum amplitude of the stellar brightness reduction:

$$(\Delta m)\_{\text{max}} = 2.5 \log(1 + I\_{\ast}/I\_{\text{sc}}) \,\tag{2}$$

Knowing the magnitudes (Δ*m*)*max* from the photometric observations, one can immediately find for each star the contribution of the scattered radiation of the CS disk to the optical radiation of the star out of the eclipse. On average about 10% [4] is confirmed when modeling the interferometric observations [9].

This model explains changes in the color indices observed in UX Ori type stars during the brightness minima (Figure 1), and describes well the observational link between the brightness variations and parameters of the linear polarization of these stars. It permits us to distinguish with high accuracy between the intrinsic polarization of the star caused by the scattered radiation of the CS disk and the interstellar one:

$$\mathbf{P}\_{\rm obs} = \mathbf{P}\_{\rm IS} + \mathbf{P}\_{\rm in} (\Delta m) \,\prime \tag{3}$$

Here **P***obs*, **P***IS* and **P***in* are pseudo-vectors of the observed, interstellar and intrinsic polarization of the investigated star, and Δ*m* is the amplitude of the diminution of the star brightness counted from its brightest state.

$$\mathbf{P}\_{\rm in}(\Delta m) = \mathbf{P}\_{\rm in}(0) \, e^{-0.4 \, \Delta m}. \tag{4}$$

**Figure 1.** Color–magnitude diagram of the UX Ori type star WW Vul from observations of the deep brightness minimum in 1997 [14]. Open circles mark the ascending part of the minimum; triangles mark its descending part. Lines show theoretical dependencies calculated in [15] on the base of Equation (1).

After making up such equations for each *i*-th observation, we obtain a redundant system of equations for each photometric band. The solution of such a system with the least square method allows us to find two unknown quantities, **P***IS* and **P***in*(0). Each of them is a pseudo-vector that gives the degree of the interstellar and intrinsic polarization and their position angles. Thus, for each photometric band the model solution is obtained on the base of the observed photometry and polarimetry of the object investigated. Then, this information is used for modeling the physical parameters of the CS dust (including the chemical composition and the particle size distribution), and what is more important, parameters of the protoplanetary disks (see, e.g., [16–18]). The most important results obtained in this field are as follows: (1) the circumstellar dust in the surface layers of the protoplanetary disks is close to the dust in the interstellar medium with its chemical composition and differs from it in minimum grain size, which is about an order of magnitude larger than that in the interstellar medium; and (2) the best agreement with observations is provided by the model of the protoplanetary disk with thickening in the dust evaporation zone [18]. In particular, this model explains a non-trivial observational fact: the hopping change in the position angle of the linear polarization in some young stars with changing in the wavelength of the radiation [19].

As an example, in Figure 2 it is shown that the linear polarization degrees depend on the brightness of the UX Ori type star WW Vul in the *U* band from [14]. One can see that it corresponds rather well with the model curve calculated on the base of Equations (2) and (3). Generally, the results of the synchronous observations of the linear polarization and the brightness of the UX Ori type stars show that, like in T Tauri stars [20], the main source of the linear polarization in the Herbig Ae stars in the visible spectrum is the scattered radiation of the circumstellar dust. The role of the optical dichroism in the formation of intrinsic polarization in the young stars is negligible. The circumstellar clouds intersecting the line of sight play the role of a natural coronograph: in blocking the direct radiation of the star, they permit us to observe the weak scattered radiation of the disk. This radiation turned out highly polarized (5–8%) [5], leading to the conclusion about the small inclinations of the disks in UX Ori type stars to the line of sight.

**Figure 2.** Dependence of the linear polarization degree on the brightness of the UX Ori type star WW Vul in the *U* band from [14]. Lines show theoretical dependencies calculated using Equations (2) and (3).

#### *2.1. Simulations of Long-Lasting Eclipses*

Cases are known in which eclipses of the UX Ori type stars lasted several months. During such events an interesting phenomenon was observed: the change in the position angle of polarization was out of sync with the change in the star brightness [14,21]. Simulation of such events showed [22] that the reason of these anomalies was an obscuration of a noticeable part of the disk with the dust screen. This led to the appearance of vast shadows on the disk whose movement on the disk behind the screen generated changes in the parameters of the disk intrinsic polarization. Of course, in such cases the coronographic effect mentioned above cannot be realized completely.

Very seldom are deep eclipses lasting more than a year observed in young stars [23–27]. Their physics apparently differs from the simple model of the CS dust cloud transit across the stellar disk accepted for the UX Ori type stars. Such eclipses indicate the appearance of a large amount of matter in the nearest vicinity of the star, for example, as a result of the fall of massive gas and dust blobs from the remnant of the protostellar cloud onto the CS disk. Such a type of cloudy accretion onto the young objects was discussed in the literature according to FUORs outbursts [28]. Certain hopes were placed on this mechanism in connection with the discussion of the nature of the eclipses of UX Ori type stars [29]. However, this hypothesis did not receive support because of the significantly modest scale of the eclipses observed in these stars. This can be due to density fluctuations in the dusty atmosphere of the disk, or in the disk wind.

In the case of eclipses lasting years, the cloudy accretion may well be a source of matter into which the young object is temporarily immersed. In favor of this suggestion two observational facts testify: (1) an increase in infrared fluxes at the wavelengths ≥2 μm observed during the optical minima in some T Tauri stars [30–32] (this means that during such events the CS dust blocks the large amount of stellar radiation and that this dust is fairly close to the star); (2) an increase in the linear polarization of one of them (RW Aur) up to 30% in the *I* band [33]. Such high polarization is typical for very young stars immersed in gas and dust cocoons. Its source is the radiation of the star scattered in the polar (optically semi-transparent) regions of the cocoon [34]. The position angle (P.A.) of linear polarization in such objects is parallel to the disk plane (see, e.g., [33]).

#### *2.2. Scattering by Moving Dust*

As is known, the main part of the thermal radiation of the protoplanetary disks in the near-infrared spectrum region originates near their inner boundary of the dust sublimation zone [35,36]. In the vicinity of this region, the main part of the scattered radiation is

also formed. In T Tauri stars this region is at a distance of 5–10 stellar radii and rotates with velocities of about 150–200 km/s. Therefore, in the scattering of stellar radiation by dust particles the frequency of the scattered radiation will change due to the Doppler effect [37,38]. For the broadband observations this effect has no meaning. However, when studying the spectra of the UX Ori type stars and relative objects it has to be taken into account. Such a problem was first solved in [39]. An example of the photospheric line transformation in the spectrum of the T Tauri type star during the deep brightness minimum is presented in Figure 3. It is seen that at first in the absorption line wide wings appear due to an increase in the contribution of the scattered radiation. In the deep minimum the photospheric line transforms into a shallow but wide absorption band. Keeping in mind that the spectrum of T Tauri type star is rich with photospheric lines, one should expect that in its spectrum of the scattered radiation wide bands will overlap because of blending and form a quasi-continuum [40]. This case is illustrated by the fragment of the synthetic spectrum of the typical T Tauri star in the vicinity of Ca I 6103 and 6122 Ålines shown in Figure 4 in the bright state and in the deep minimum. Namely, the same spectrum transformation was recently observed at the deep brightness minimum for RW Aur [41,42].

**Figure 3.** Photospheric line broadening in the T Tauri type star (solid line) during an eclipse due to scattering at the inner boundary of the protoplanetary disk. The dot-dashed line corresponds to the total eclipse.

**Figure 4.** A part of the synthetic spectrum of RW Aur in the vicinity of the Ca I 6103 and 6122 Å lines in the bright state (black) and the deep minimum (red).

The other source of the intrinsic polarization of young stars can be the dust component of the disk wind [18]. In this case the moving dust has two velocity components: tangential and poloidal ones. Therefore, the scattering of stellar radiation by the disk wind can lead to more complex transformation of the photospheric lines: to the line broadening due to rotation and redshift due to the poloidal motion. This issue is briefly discussed in [38,39] and deserves a more detailed quantitative analysis.

In UX Ori type stars the effect considered above is revealed in a significantly weaker form [38]. This is caused by two reasons: (1) in most of these stars the photospheric lines are broadened by the strong stellar rotation; (2) the dust sublimation zone is located much farther from the star due to the higher luminosity. Therefore, the Keplerian velocities of the disk in this region are significantly less compared to those in T Tauri stars. Figure 5 demonstrates the part of the synthetic spectrum of the UX Ori type star CQ Tau in the vicinity of the Fe II 5018 line in the bright state and the deep minimum. It is seen that in this case the spectrum of the scattered radiation differs little from the photospheric one.

**Figure 5.** A part of the synthetic spectrum of CQ Tau in the vicinity of the Fe II 5018 Å line in the bright state (black) and the deep minimum (red).

The same occurs in the emission lines in the spectra of young stars. Nevertheless, the influence of the scattered radiation on the emission lines is well known from the observations of the linear polarization (see [43] and cited papers therein). When traversing the H*α* line profile, both the value and position angle change, which reflects the contribution of various parts of the CS disk to the polarization of the different parts of the line profile. As shown in [43], this effect is sensitive to the parameters of the inner CS disk regions, and may shed additional light on their structure.

#### **3. Images of Circumstellar Disks in the Scattered Light**

The coronographic method has been very successful in observations of the scattered radiation of the CS debris disks. In particular, with its help Smith and Terrile [44] first observed the circumstellar disk of *β* Pictoris. This disk is seen nearly edge-on on the coronographic image. Therefore, its radiation (as well as the radiation of the UX Ori type stars) was found to be strongly polarized [45]. The coronographic observations with HST revealed that the inner region of the *β* Pic disk was slightly curved relative to its outer part [46]. The authors of the paper quoted above suggested that this curvature was caused by a disturbing body (a massive planet) whose orbit was slightly inclined relative to the plain of the outer disk. Fourteen years later this planet has been found [47], also with the coronagraphic technique. An important contribution to such observations has been made with the Hubble Space Telescope (see, e.g., [48,49] and references there).

At the present time the technique of observations of weak objects in the vicinity of stars based on the coronographic method is widely used in the different astrophysical fields, among them the study of the fine structure of the circumstellar disks in polarized light (see, e.g., [50–53] and the references therein). Of particular interest are the first and rather

successful attempts to monitor the protoplanetary disk images in the polarized light [51]. They showed that on the images of the CS disk of HD 135344B (observed nearly pole-on), the shadows were caused by absorption of the stellar radiation by local perturbations in the inner disk. These shadows are manifested as narrow radial bands of the variable brightness, and also as the wide bands caused by absorption by large scale structures. These observations point to the direct physical connection between eclipses of the UX Ori type stars and shadow formation on the circumstellar disks.

Another example of successful monitoring of a circumstellar disk in the polarized light is the long-term observations of CQ Tau [54]. This star is one of the most active members of the UXOrs family [55], having a very complex light curve [56]. According to interferometric observations in the near-IR [57] and millimeter wavelengths [58,59], the inner and outer parts of the CS disk of CQ Tau have different inclinations: *i* = 48.5 ± 5◦ and *i* = 35◦, correspondingly. In such cases, a narrow shadow from the inner disk can be observed on the outer one (see, e.g., [60–62]). The observations of CQ Tau have demonstrated that such a shadow exists on the peripheral region of its CS disk [54].

The aforementioned interferometric observations of CQ Tau support the point of view according to which the extinction events in UX Ori stars take place in the innermost part of the CS disk where the NIR radiation is formed. Continuation of such observations is of undeniable interest for understanding the nature of the perturbations in the inner regions of CS disks, which can be caused by different reasons such as an azimuthal heterogeneity of the dusty disk wind, collisions of planetesimals or hydrodynamical fluctuations in the dust evaporation zone of the disk.

If the extinction events are driven by the disk wind, the UXOrs activity will depend not only on the disk inclination but also on the mass loss rate and the dusty wind loading area. The latter depends on the magnetic field in the disk and the star luminosity. The mass loss and accretion rates are closely connected. Therefore, in this case the UXOrs activity will be sensitive to the disk inclination, as well as the stellar luminosity and mass accretion rate.

#### *The Edge-On Disks*

Young stars with edge-on CS disks are also observed through scattered light. The prototype of such objects is the well-known T Tauri star HH 30. The first image of this object was obtained with high resolution by Burrows et al. [46] with the Hubble Space Telescope. It revealed a flared CS disk (in accordance with the prediction of the Shakura and Sunyaev [63] model (see Kenyon and Hartmann [64]) and the highly asymmetric jet. In addition to the jet, a conical molecular outflow is also observed in the CO lines [65]. Photometric and polarimetric observations have shown that HH 30 is a variable object [66–68]. It demonstrates the periodic variability of brightness and linear polarization with a period of 7.49 days [67]. The physical model of such a periodicity is not clear. It could be a hot spot on the rotating star or periodic variations of the CS extinction in the star's vicinity.

Variable illumination of the disk leads to changes in its shape [67]. These changes are one of the most interesting manifestations of the circumstellar activity of young stars reminiscent of the moving shadows on the disk images [50]. The other non-trivial special property of HH 30 is its almost one-side jet and molecular outflow [46,64,69]. The origin of such asymmetric outflows is discussed in [70]. An opposite case of the edge-on disk with a well-developed and almost symmetrical jet is HH 212 [71].

To date, more than ten young stars with edge-on disks are known (see, e.g., [72] and the references there). Most of them are T Tauri stars. Observations and modeling of such objects permit us to study in detail the internal structure of CS disks (see, e.g., [73,74]). It is obvious that the spectra of such objects are strongly distorted by the scattered radiation [37].

#### **4. Intrinsic Polarization of Young Stars and Orientation of Their CS Disks**

The position angle of linear polarization depends on the geometry of the scattering medium. Calculations show [75] that in models with the classical accretion disk (with a

flared surface) the P.A. is orthogonal to the disk plane. In the stars at earlier evolutionary stages surrounded by accreting envelopes the P.A. is parallel to the disk plane [76]. The average age of the UX Ori type stars is several million years [77]. Therefore, most of them belong to the first group of young stars. Keeping in mind what was mentioned above, one can use the position angle of the intrinsic polarization in order to determine the orientation of protoplanetary disks in the projection on the sky plane. Such a possibility was confirmed by results of interferometric observations of UX Ori in the near-infrared spectrum region: according to [9] the position angle of the symmetry axis of its circumstellar disk is equal to 127.5 ± 24.5◦. Polarimetric observations of UX Ori in the deep minima give P.A. = 125◦–129◦ [21,78]. Orientation of the circumstellar disk of another UX Ori type star VV Ser is determined with the position of the shadow formed by the disk on the reflective nebula behind the star. According to [79], the P.A. of the shadow is equal to 15 ± 5◦, which corresponds to the P.A. of the disk symmetry axis 105 ± 5◦. Observations of the intrinsic polarization of VV Ser gives the position angle in the close range: P.A. = 88◦–100◦ [80].

These two examples testify that the linear polarization of most UX Ori type stars in the deep brightness minima in fact characterizes the position of the symmetry axis of the circumstellar disk on the sky plane, and this position can be compared with the direction of the interstellar magnetic field determined with the help of polarization of the neighboring stars. Unfortunately, this is not always possible because of the complex structure of the interstellar magnetic fields in star formation regions. Figure 6 shows an example of the polarization map of the BM And, as well as the vicinity in which the polarization pseudovectors of BM And itself are shown in the bright stage and deep brightness minimum. It is seen that the interstellar magnetic field in the stellar vicinity is fairly homogeneous in direction, and the position angle of the stellar intrinsic polarization coincides with this direction with high accuracy [81]. This implies that the interstellar magnetic filed controlled the star formation process from the protostellar cloud, and that the circumstellar disk of the star "remembered" the direction of the magnetic lines.

Similar results have also been obtained during photopolarimetric observations for three other UX Ori type stars: WW Vul [14], BF Ori [82] and VV Ser [79,83,84]. Furthermore, the same method was used in [85] for a large group of Herbig AeBe stars. It was shown that subsamples of the more polarized stars from their list present a statistically significant tendency toward intrinsic polarization aligned with the interstellar magnetic field.

**Figure 6.** Polarization map of the BM and as well as the neighborhood from [81,84]. Two values of the star polarization are given: at the bright state and at the deep minimum. Credit: Grinin et al. 1995, A&AS, 112, 457, reproduced with permission © ESO.

#### **5. Conclusions**

Thus, although the scattered radiation of protoplanetary disks makes up only a very small part of the radiation of young stars, its existence provides us very important information about young stars and their circumstellar environment. Of great interest is the possibility to study the orientation of CS disks not resolved in a telescope with the help of polarimetric observations. Polarized radiation makes it possible to see fine details on the disk images and study their structure and variability. Photopolarimetric observations of UXOrs and their modeling permit us to investigate the perturbations in the innermost regions of the disks where planetary systems are formed.

**Author Contributions:** V.P.G.—conceptualization, data analysis and writing-editing; L.V.T.—numerical modelling and writing—editing; All authors have read and agreed to the published version of the manuscript.

**Funding:** We acknowledge the financial support of the Ministry of Science and Higher Education of the Russian Federation (grant no. 075-15-2020-780).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All necessary data are contained in this paper.

**Acknowledgments:** The authors are grateful to the reviewers for useful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

