Measuring black hole mass of type I active galactic nuclei by spectropolarimetry

Measuring black hole mass of type I active galactic nuclei by spectropolarimetry Abstract Black hole (BH) mass of type I active galactic nuclei (AGNs) can be measured or estimated through either reverberation mapping or empirical R–L relation, however, both of them suffer from uncertainties of the virial factor ($$f_{\rm _{BLR}}$$), thus limiting the measurement accuracy. In this letter, we make an effort to investigate $$f_{\rm _{BLR}}$$ through polarized spectra of the broad-line regions (BLR) arisen from electrons in the equatorial plane. Given the BLR composed of discrete clouds with Keplerian velocity around the central BH, we simulate a large number of spectra of total and polarized flux with wide ranges of parameters of the BLR model and equatorial scatters. We find that the $$f_{\rm _{BLR}}$$-distribution of polarized spectra is much narrower than that of total ones. This provides a way of accurately estimating BH mass from single spectropolarimetric observations of type I AGN whose equatorial scatters are identified. polarization, galaxies: active, quasars: supermassive black holes 1 INTRODUCTION Reverberation mapping (RM) is nowadays the most common technique of measuring black hole (BH) mass of type I active galactic nuclei (AGNs), except for a few local AGNs spatially resolved (Peterson 1993; Peterson 2014). RM measures time lags Δt of broad emission lines with respect to varying continuum as ionizing photons, allowing us to obtain the emissivity-averaged distance from the broad-line regions (BLR) to the central BH. Assuming fully random orbits of the BLR clouds with Keplerian velocity, we have the BH mass as   \begin{equation} M_{\bullet }= f_{\rm _{BLR}}\frac{c\Delta tV_{_{\rm FWHM}}^2}{G}, \end{equation} (1)where $$f_{\rm _{BLR}}$$ is the virial factor, $$V_{_{\rm FWHM}}$$ is the full width at half-maximum (FWHM) of the broad emission line profiles, c is the light speed and G is the gravity constant. The total error budget on the BH mass can be simply estimated by $$\delta M_{\bullet }/M_{\bullet }\approx [(\delta \ln f_{\rm _{BLR}})^2+0.08]^{1/2}$$, where Δt and $$V_{_{\rm FWHM}}$$ are usually of 20 per cent and 10 per cent for a typical measurement of RM observations, respectively. Obviously, the major uncertainty on the BH mass is due to $$f_{\rm _{BLR}}$$; however, it could be different by a factor of more than one order of magnitude (Pancoast et al. 2014b). Its dependence on kinematics, geometry and inclination of the BLR is poorly understood (Krolik 2001; Collin et al. 2006). For those AGNs with measured stellar velocity dispersions σ* of bulges and RM data, $$f_{\rm _{BLR}}$$ can be calibrated by the M•–σ* relation found in inactive galaxies (Onken et al. 2004; Woo et al. 2010). $$\langle f_{\rm _{BLR}}\rangle$$ as an averaged one can only remove the systematic bias between the virial product $$c\Delta tV_{\rm _{FWHM}}^2 / G$$ and M• for a large sample, however, $$f_{\rm _{BLR}}$$ is poorly understood individually. Furthermore, the zero-point and scatters of the M•–σ* relation depend on bulge types of the host galaxy, and virial factors of classical bulges and pseudo-bulges can differ by a factor of ∼2 (Ho & Kim 2014). The calibrated values of $$f_{\rm _{BLR}}$$ lead to δM•/M• ∼ 2, yielding only rough estimations of BH mass in AGNs. Recently, a motivated idea to test the validity of $$f_{\rm _{BLR}}$$ factor has been suggested by Du, Wang & Zhang (2017) in type II AGNs through the polarized spectra. In principle, the polarized spectra are viewed with highly face-on orientation to observers, and $$f_{\rm _{BLR}}$$ in type II AGNs should be the same as with type I AGNs. They reach a conclusion of $$f_{\rm _{BLR}}\sim 1$$ from a limited sample. For type I AGNs, the polarized spectra received by a remote observer correspond to ones viewed by an edge-on observer, lending an opportunity to measure BH mass similar to cases of NGC 4258 through water maser (Miyoshi et al. 1995) or others through CO line (Barth et al. 2016). In this letter, we investigate $$f_{\rm _{BLR}}$$ in type I AGNs through modelling the scattering polarized spectra quantitatively. In Section 2, we build a dynamical model for BLR and scattering region of type I AGNs for polarized spectra. In Section 3, we simulate a large number of spectra for a large range of model parameters to get the distribution of $$f_{\rm _{BLR}}$$ for both total and polarized spectra. We find that $$f_{\rm _{BLR}}$$ is in a very narrow range for polarized spectra. In Section 4, we draw conclusions and discuss potential ways of improving the accuracy of BH mass determination. 2 POLARIZED SPECTRA FROM EQUATORIAL SCATTERS Optical spectropolarimetric observations of type II AGNs discover that there is a broad component of emission line in polarized spectra, indicating the appearance of (1) a BLR hidden by the torus; and (2) at least one scattering region outside the torus (Antonucci & Miller 1985; Miller, Goodrich & Mathews 1991; Tran, Miller & Kay 1992). Radio observations also show that radio axes of most type II AGNs are nearly perpendicular to the position angle of polarization (Antonucci 1983; Brindle et al. 1990), showing that scattering regions of Type II AGNs situated outside the torus but aligned with the axes of the AGNs, called polar scattering region. In contrast, observations of type I AGNs reveal that position angles of the polarization are more often aligned with radio axes (Antonucci 1983, 1984; Smith et al. 2002). Equatorial scattering regions may exist, which are hidden by the torus, but can be seen in the polarized spectra of type I AGNs (Smith et al. 2005). We follow the geometry of equatorial scatters as in Smith et al. (2005). The geometry of the scattering regions and BLR are shown in Fig. 1(a). The details of the geometric relations are provided in the appendix. If the half opening angle of the scattering region is $$\Theta _{\rm _{P}}$$, the inner and outer radii of the scattering region are $$r_{\rm _{P,i}} \mbox{and } r_{\rm _{P,o}}$$, then we have $$r_{\rm _{P}}\in [r_{\rm _{P,i}},r_{\rm _{P,o}}],\theta _{\rm _{P}}\in [{\pi} /2-\Theta _{\rm _{P}},\pi /2+\Theta _{\rm _{P}}]$$. Scatterings caused by intercloud electrons in the BLR have been estimated as $$\tau _{_{\rm BLR}}\approx 0.04R_{\rm 0.1pc}$$ (see equation 5.13 in Krolik, McKee & Tarter 1981), where R0.1pc = RBLR/0.1 pc is a typical size of the BLR (Bentz et al. 2013), which can be totally neglected for polarized spectra. We thus assume that the scattering region is composed of free cold electrons beyond the BLR, and the whole region is optically thin, i.e. the optical depth $$\tau _{\rm es} \sim \bar{n}\ell \sigma _{\rm T} < 1$$, where $$\bar{n}$$ is the average number density of the electrons, ℓ is the typical scale of the region and σT is the Thomson cross-section. The distribution of the number density of electrons is assumed to be a power law as $$n_{\rm _{P}}(r_{\rm _{P}}, \theta _{\rm _{P}},\phi _{\rm _{P}}) = n_{\rm _{P0}} (r_{\rm _{P}}/r_{\rm _{P,i}})^{-\alpha }$$, where $$n_{\rm _{P0}}$$ is the number density at inner radius $$r_{\rm _{P,i}}$$. We assume BLR is composed of a large quantity of independent clouds rotating around the BH, and has a geometry indicated by Fig. 1(b) (Pancoast, Brewer & Treu 2011; Li et al. 2013; Pancoast, Brewer & Treu 2014a). The detailed geometric relations of BLR are given in the appendix. Suppose that the unit of length is Rg ≡ GM•/c2 and orbits of the clouds are circular. The velocity of the cloud is   \begin{equation} {\boldsymbol v}_{\rm cloud} = V_{\rm K} \left( \begin{array}{c}-\sin \phi _{\rm _{B}} \cos \phi _{\rm _{C}} - \cos \phi _{\rm _{B}}\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}}\\ -\sin \phi _{\rm _{B}} \sin \phi _{\rm _{C}} + \cos \phi _{\rm _{B}}\cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}}\\ \cos \phi _{\rm _{B}}\sin \theta _{\rm _{C}} \end{array} \right), \end{equation} (2)where $$V_{\rm K} = cr_{_{\rm B}}^{-1/2}$$. The half opening angle of the BLR is $$\Theta _{\rm _{BLR}}$$, the inner and outer radii of the BLR are $$r_{\rm _{B,i}}$$ and $$r_{\rm _{B,o}}$$, respectively. Then we have $$r_{\rm _{B}} \in [r_{\rm _{B,i}},r_{\rm _{B,o}}], \theta _{\rm _{C}} \in [0,\Theta _{\rm _{BLR}}]$$. The distribution of clouds can be modelled by power law as well. The number density of the clouds is $$n_{\rm _{B}}(r_{\rm _{B}}, \theta _{\rm _{C}},\phi _{\rm _{C}},\phi _{\rm _{B}}) = n_{\rm _{B0}} (r_{\rm _{B}}/r_{\rm _{B,i}})^{-\beta }$$, where $$n_{\rm _{B0}}$$ is the number density at $$r_{\rm _{B,i}}$$. A single scattering process is illustrated by Fig. 1(c). Expressions of the scattering angle Θ and rotation angle χ are derived in appendix. Assuming the ionizing source is isotropic and line intensity of one cloud at B is linearly proportional to the intensity of local ionizing fluxes, the line intensity at B is $$i_{\rm _{B}} = kr_{\rm _{B}}^{-2}$$, where k is a constant. We further assume that all clouds emit unpolarized Hβ photons isotropically and neglect multiple scatterings of optically thin regions. Thus the intensity at P is simply given by $$i_{\rm _{P}} = i_{\rm _{B}} {\cal S}/ 4{\pi} r_{\rm _{BP}}^2$$, where $${\cal S}$$ is the surface area of the cloud. Given incident photons with the Stokes parameters of $$(i_{\rm _{P}},0,0,0)$$, the Stokes parameters in the (n⊥–n∥) frame are (Chandrasekhar 1960).   \begin{eqnarray} {\displaystyle {\frac{3\sigma }{8{\pi} R^2}}\!\!\! \left(\begin{array}{c{@}{\quad}c{@}{\quad}c{@}{\quad}c}\displaystyle\frac{1}{2}(1+\cos ^2\Theta ) & \displaystyle\frac{1}{2}(1-\cos ^2\Theta ) & 0 & 0\\ \displaystyle\frac{1}{2}(1-\cos ^2\Theta ) & \displaystyle\frac{1}{2}(1+\cos ^2\Theta ) & 0 & 0\\ 0 & 0 & \cos \Theta & 0 \\ 0 & 0 & 0 & \cos \Theta \end{array} \right) \left(\begin{array}{c}i_{\rm _{P}} \\ 0 \\ 0 \\ 0 \end{array} \right)} \nonumber \\ {\quad = {\cal A}_0\left( \begin{array}{c}1+\cos ^2\Theta \\ 1-\cos ^2\Theta \\ 0 \\ 0 \end{array} \right), } \end{eqnarray} (3)where $${\cal A}_0=3\sigma i_{\rm _{P}}/16{\pi} R^2$$, R is the distance between the observer and AGN. Converting the Stokes parameter to the fixed coordinate $${\boldsymbol n}_{z^{\prime }}-{\boldsymbol n}_{x^{\prime }}$$ system, we have   \begin{eqnarray} \left(\begin{array}{c}i \\ q \\ u \\ v \end{array}\right) &=& {\cal A}_0 \left(\begin{array}{c{@}{\quad}c{@}{\quad}c{@}{\quad}c}1 & 0 & 0 & 0 \\ 0 & \cos 2\chi & \sin 2\chi & 0\\ 0 & -\sin 2\chi & \cos 2\chi & 0 \\ 0 & 0 & 0 & \cos \Theta \end{array}\right)\! \left( \begin{array}{c}1+\cos ^2\Theta \\ 1-\cos ^2\Theta \\ 0 \\ 0 \end{array}\right) \nonumber \\ &=&{\cal A}_0 \left( \begin{array}{c}1+\cos ^2\Theta \\ (1-\cos ^2\Theta )\cos 2\chi \\ -(1-\cos ^2\Theta )\sin 2\chi \\ 0 \end{array}\right). \end{eqnarray} (4)The velocity of the cloud $${\boldsymbol v}_{\rm cloud}$$ projected to the direction of incident light $${\boldsymbol n}_{\rm _{BP}}$$ is   \begin{equation} \frac{V_{\parallel }}{c} = \frac{1}{r_{\rm _{B}}^{1/2}} \frac{r_{\rm _{P}}(q_1\cos \phi _{\rm _{B}} + q_2\sin \phi _{\rm _{B}})}{\left[r_{\rm _{P}}^2 + r_{\rm _{B}}^2 + 2r_{\rm _{B}} r_{\rm _{P}}(q_2 \cos \phi _{\rm _{B}} - q_1 \sin \phi _{\rm _{B}})\right]^{1/2}}, \end{equation} (5)where $$q_1 = \cos \theta _{\rm _{P}} \sin \theta _{\rm _{C}} + \sin \theta _{\rm _{P}}\cos \theta _{\rm _{C}}\sin (\phi _{\rm _{P}} - \phi _{\rm _{C}})$$, $$q_2 = -\sin \theta _{\rm _{P}}$$$$\cos (\phi _{\rm _{P}} - \phi _{\rm _{C}})$$. If the intrinsic wavelength of the line is λ0, the wavelength after scattering is λ΄ = λ0(1 − V∥/c) due to Doppler shifts. Integrating over all the clouds and electrons, we have the total polarized spectrum,   \begin{eqnarray} \left(\begin{array}{c}I_{\lambda } \\ Q_{\lambda } \\ U_{\lambda } \\ V_{\lambda } \end{array}\right) = \int _{V_{\rm _{P}}} \text{d}V_{\rm _{P}}n_{\rm _{P}} \int _{V_{\rm _{BLR}}} \text{d}V_{\rm _{BLR}} n_{\rm _{B}} \int _{0}^{2{\pi} } {\cal L}(\lambda ,\lambda ^{\prime }) \text{d}\phi _{\rm _{B}} \left( \begin{array}{c}i \\ q \\ u \\ v \end{array} \right), \nonumber\\ \end{eqnarray} (6)where the intrinsic profile of Hβ line is assumed to be a Lorentzian function of $${\cal L}(\lambda ,\lambda ^{\prime })\propto \Gamma /[(\lambda -\lambda^{\prime})^2+\Gamma ^2]$$, Γ is the intrinsic width much smaller than the broadening due to rotation of clouds (the Lorentzian profiles are a very good approximation in the present context). Similarly, we can calculate the spectrum of non-scattered photons. The velocity of the cloud $${\boldsymbol v}_{\rm cloud}$$ projected to the direction of observer $${\boldsymbol n}_{\rm obs}$$ is   \begin{equation} \frac{\tilde{V}_{\parallel }}{c} = \frac{1}{r_{\rm _{B}}^{1/2}} \left( \tilde{q}_1\cos \phi _{\rm _{B}} + \tilde{q}_2\sin \phi _{\rm _{B}} \right), \end{equation} (7)where $$\tilde{q}_1 = \sin \theta _{\rm _{C}} \cos i + \cos \theta _{\rm _{C}}\cos \phi _{\rm _{C}} \sin i$$, $$\tilde{q}_2 = -\sin \phi _{\rm _{C}}\sin i$$. The observed wavelength is $$\lambda ^{\prime \prime } = \lambda _0(1-\tilde{V}_{\parallel }/c)$$. Integrating over all the clouds, we have   \begin{equation} F_{\lambda } = \int _{V_{\rm _{BLR}}} \text{d}V_{\rm _{BLR}} n_{\rm _{B}} \int _{0}^{2{\pi} } {\cal L}(\lambda ,\lambda ^{\prime \prime })\,\text{d}\phi _{\rm _{B}} \frac{i_{\rm _{B}}S}{4{\pi} R^2}, \end{equation} (8)and the expression for polarization degree and position angle,   \begin{equation} P_{\lambda } = \frac{\sqrt{Q_{\lambda }^2 + U_{\lambda }^2}}{I_{\lambda } + F_{\lambda }}, \quad \theta _{\lambda } = \frac{1}{2} \arccos \left( \frac{Q_{\lambda }}{\sqrt{Q_{\lambda }^2+U_{\lambda }^2}} \right). \end{equation} (9)If Uλ > 0, θλ ∈ (0, π/2). If Uλ < 0, θλ ∈ ( − π/2, 0). Position angles represent the angle between the direction of maximum intensity and $$n_{z^{\prime }}$$. Table 1 summarises all the parameters of the present model. We calculate a series of profiles for different values of parameters and find that the profiles are only sensitive to $$\Theta _{_{\rm BLR}}$$ and i. Fig. 2 shows typical spectra of a type I AGN with an equatorial electron scattering region. The parameters of the model are $$r_{\rm _{P,i}} = 10^4$$, $$r_{\rm _{P,o}} = 2\times 10^4$$, $$\Theta _{\rm _{P}} = 30^{\circ }$$, $$r_{\rm _{B,i}} = 2\times 10^3$$, $$r_{\rm _{B,o}} = 6\times 10^3$$, α = 1, β = 1, $$\tau _{\rm es}=\sigma n_{\rm _{P0}}(r_{\rm _{P,o}}-r_{\rm _{P,i}}) = 1$$, i = (1°, 10°, 20°, 30°, 40°) and $$\Theta _{_{\rm BLR}} =(10^{\circ },20^{\circ },30^{\circ },40^{\circ },50^{\circ })$$. As shown in the left-hand panel of Fig. 2 , the total spectra get broader as i increases and show double-peaked profiles when i exceeds a critical inclination. By contrast, the width and the profile of the polarized spectra are not sensitive to i (can be found from normalized spectra). This interesting property results from the fact that the polarized spectra are equivalent to the ones seen by observers at edge-on orientations. Generally, the line centres have lowest polarization degrees. However, the polarization degrees become smaller with i. Figure 1. View largeDownload slide Panel (a) is a cartoon of a type I AGN with an equatorial scattering region. The blue points represent clouds in the BLR and the grey region the scatters on the equatorial plan. i is the inclination angle to a remote observer in the O–YZ plane. Panel (b) is the frame for the BLR geometry. O–XY is the equatorial plane. $$O\text{--}X_{\rm _{C}}Y_{\rm _{C}}$$ is the orbital plane of one specific cloud, which can be obtained by rotating X around Z by $$\phi _{\rm _{C}}$$ and then rotating Z around $$X_{\rm _{C}}$$ by $$\theta _{\rm _{C}}$$. The phase angle of the cloud relative to $$OX_{\rm _{C}}$$ is $$\phi _{\rm _{B}}$$. Panel (c) is the scattering geometry used here. BP is the incident light from point B on one orbit. $${\boldsymbol n}_{\rm obs}$$ is the direction of sight of the observer (i.e. the direction of scattered light). The vectors of $${\boldsymbol n}_{\rm obs}$$, $${\boldsymbol n}_{\bot }$$, $${\boldsymbol n}_{\parallel }$$, $${\boldsymbol n}_{z^{\prime }}$$ and $${\boldsymbol n}_{x^{\prime }}$$ are explained in the appendix. Figure 1. View largeDownload slide Panel (a) is a cartoon of a type I AGN with an equatorial scattering region. The blue points represent clouds in the BLR and the grey region the scatters on the equatorial plan. i is the inclination angle to a remote observer in the O–YZ plane. Panel (b) is the frame for the BLR geometry. O–XY is the equatorial plane. $$O\text{--}X_{\rm _{C}}Y_{\rm _{C}}$$ is the orbital plane of one specific cloud, which can be obtained by rotating X around Z by $$\phi _{\rm _{C}}$$ and then rotating Z around $$X_{\rm _{C}}$$ by $$\theta _{\rm _{C}}$$. The phase angle of the cloud relative to $$OX_{\rm _{C}}$$ is $$\phi _{\rm _{B}}$$. Panel (c) is the scattering geometry used here. BP is the incident light from point B on one orbit. $${\boldsymbol n}_{\rm obs}$$ is the direction of sight of the observer (i.e. the direction of scattered light). The vectors of $${\boldsymbol n}_{\rm obs}$$, $${\boldsymbol n}_{\bot }$$, $${\boldsymbol n}_{\parallel }$$, $${\boldsymbol n}_{z^{\prime }}$$ and $${\boldsymbol n}_{x^{\prime }}$$ are explained in the appendix. Figure 2. View largeDownload slide Total, polarized spectra and polarization degrees of a type I AGN with an equatorial electron scattering region. The left-hand column is spectra for different inclinations, whereas the right-hand column is for different BLR opening angles. The tops of the total spectra become flatter and polarization decreases with increasing ΘBLR (tends to 90°). Figure 2. View largeDownload slide Total, polarized spectra and polarization degrees of a type I AGN with an equatorial electron scattering region. The left-hand column is spectra for different inclinations, whereas the right-hand column is for different BLR opening angles. The tops of the total spectra become flatter and polarization decreases with increasing ΘBLR (tends to 90°). Table 1. The dependence of $$f_{\rm _{BLR}}$$ on model parameters. Parameter  Range  Meaning  $$\delta \ln f_{_{\rm BLR}}$$        Total  Polarized  $$r_{\rm _{P,i}}(10^4 R_{\rm g})$$  [1, 5]  SR inner radius  0.01  0  $$r_{\rm _{P,o}}(10^4 R_{\rm g})$$  [2, 10]  SR outer radius  0  0  $$\Theta _{\rm _{P}}(^\circ )$$  [20, 50]  SR opening angle  − 0.04  0  α  [0, 1.5]  Index of DF of electrons  0.01  0  $$r_{\rm _{B,i}}(10^3 R_{\rm g})$$  [1, 5]  Inner radius of BLR  − 0.01  0.01  $$r_{\rm _{B,o}}(10^3 R_{\rm g})$$  [2, 10]  Outer radius of BLR  0.01  − 0.01  $$\Theta _{\rm _{BLR}}(^\circ )$$  [20, 50]  Opening angle of BLR  − 0.38  0.03  β  [0, 1.5]  Index of DF of clouds  0  − 0.02  i(°)  [0, 45]  Inclination angle  − 0.64  0  Parameter  Range  Meaning  $$\delta \ln f_{_{\rm BLR}}$$        Total  Polarized  $$r_{\rm _{P,i}}(10^4 R_{\rm g})$$  [1, 5]  SR inner radius  0.01  0  $$r_{\rm _{P,o}}(10^4 R_{\rm g})$$  [2, 10]  SR outer radius  0  0  $$\Theta _{\rm _{P}}(^\circ )$$  [20, 50]  SR opening angle  − 0.04  0  α  [0, 1.5]  Index of DF of electrons  0.01  0  $$r_{\rm _{B,i}}(10^3 R_{\rm g})$$  [1, 5]  Inner radius of BLR  − 0.01  0.01  $$r_{\rm _{B,o}}(10^3 R_{\rm g})$$  [2, 10]  Outer radius of BLR  0.01  − 0.01  $$\Theta _{\rm _{BLR}}(^\circ )$$  [20, 50]  Opening angle of BLR  − 0.38  0.03  β  [0, 1.5]  Index of DF of clouds  0  − 0.02  i(°)  [0, 45]  Inclination angle  − 0.64  0  Note. SR: scattering region and DF: distribution function. $$\delta \ln f_{\rm _{BLR}}$$ describes the dependence on parameters of the BLRs. See details for its definition in the main text. It shows that $$f_{\rm _{BLR}}$$ is sensitive to ΘBLR and i for the total spectra, but it is almost a constant for polarized spectra. View Large Total spectra show strong dependence on ΘBLR. As shown in the right-hand panel of Fig. 2, large-ΘBLR BLRs show a broader width and get narrower with decreasing of ΘBLR until double-peaked profiles. However, $$\Theta _{_{\rm BLR}}$$ does not change the polarized profiles too much. A large $$\Theta _{_{\rm BLR}}$$ indicates that the system tends to be more spherically symmetric and to decrease the polarization degree. Comparing with the total spectra, the width of the polarized spectrum is insensitive to $$\Theta _{_{\rm BLR}}$$ and i. This property allows us to infer $$f_{\rm _{BLR}}$$ from the polarized spectrum and improve the accuracy of BH mass measurement, as shown in Section 3. 3 THE VIRIAL FACTOR With the BLR model, we have the emissivity-averaged time lag of a broad emission line as   \begin{eqnarray} \Delta t &=& \frac{\int \text{d}V_{\rm _{BLR}} \Delta r i_{\rm _{B}} n_{\rm _{B}}}{\int \text{d}V_{\rm _{BLR}} i_{\rm _{B}} n_{\rm _{B}}} \left(\frac{R_{\rm g}}{c}\right) \nonumber\\ &=& \frac{1-\beta }{2-\beta } \frac{1-q_r^{2-\beta }}{1-q_r^{1-\beta }} \left(\frac{r_{_{\rm B,i}}R_{\rm g}}{c}\right), \end{eqnarray} (10)where $$\Delta r = r_{\rm _{B}} - {r}_{\rm _{B}}\cdot {n}_{\rm obs}$$ and the corresponding virial factor from equation (1) can be written as   \begin{equation} f_{\rm _{BLR}}^{-1} = \left(\frac{1-\beta }{2-\beta }\right) \left(\frac{1-q_r^{1-\beta }}{1-q_r^{2-\beta }}\right) \left(\frac{V_{\rm _{FWHM}}}{c}\right)^2r_{_{\rm B,i}}, \end{equation} (11)where $$q_r=r_{_{\rm B,o}}/r_{_{\rm B,i}}$$, $$V_{_{\rm FWHM}}$$ is the FWHM of profiles either from the total or polarized spectra. We generate profiles according to parameters listed in Table 1 and measure $$V_{_{\rm FWHM}}$$ to show dependences of $$f_{\rm _{BLR}}$$ on each parameter. We did Monte Carlo simulations for all the parameters listed in Table 1 and then get the $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations, where Xi is any one of the parameters. The $$f_{\rm _{BLR}}$$ dependence on Xi can then be obtained by $$\delta \ln f_{\rm _{BLR}}=(\partial \ln f_{\rm _{BLR}}/\partial \ln X_i)\delta \ln X_i$$, where $$\partial \ln f_{\rm _{BLR}}/\partial \ln X_i$$ is the slope of the $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations and δln Xi is its range. The slope is estimated from the line regression of $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations from the Monte Carlo simulations. The dependence listed in Table 1 shows that only i and ΘBLR are the major drivers in the total spectra, but $$f_{\rm _{BLR}}$$ is insensitive to all the parameters (only slightly relies on ΘBLR). We estimate the entire uncertainties of $$f_{\rm _{BLR}}$$ due to all parameters as $$\Delta \log f_{\rm _{BLR}}=[\sum _{i=1}^9 (\partial \ln f_{\rm _{BLR}}/\partial \ln X_i)^2(\Delta \log X_i)^2]^{1/2}$$. We have $$\Delta \log f_{\rm _{BLR}}=(0.74,0.04)$$ for total and polarized spectra, respectively. We plot the $$\log f_{\rm _{BLR}}\text{-}$$distributions and contour maps versus ΘBLR and i in Fig. 3. It shows that the distribution from total spectra is much broader than that from polarized spectra. The 68  per cent confidence interval of the former is $$\log f_{\rm _{BLR}}\in [-0.41,0.08]$$ agreeing with values from detailed Markov chain Monte Carlo modelling (Pancoast et al. 2014b), but $$\log f_{\rm _{BLR}}\in [-0.65,-0.62]$$ from the polarized spectra. For a typical RM campaign, we have the uncertainties of BH mass δM•/M• ≈ (0.8, 0.3) from total and polarized spectra, respectively. Obviously, spectropolarimetry provides much better $$f_{\rm _{BLR}}$$ for BH mass from RM campaign. However, the polarized spectra as the prerequisites of applications should be identified as originating from the equatorial scatters. Actually, this can be done by checking if the position angles are parallel to radio axis in AGNs. Figure 3. View largeDownload slide The upper three panels display the distribution of $$\log f_{\rm _{BLR}}$$ obtained from total spectra, while the lower three panels display polarized spectra. In each row, the first panel is the probability density function of $$\log f_{\rm _{BLR}}$$ and the shadow area marks out the 68  per cent confidence interval, and the second and third indicate correlations of $$\log f_{\rm _{BLR}}$$ with model parameters of $$\Theta _{\rm _{BLR}}$$ and i, respectively. $$f_{\rm _{BLR}}$$ is sensitive to both ΘBLR and i for the total spectra, whereas $$f_{\rm _{BLR}}$$ only very weakly depends on ΘBLR. Figure 3. View largeDownload slide The upper three panels display the distribution of $$\log f_{\rm _{BLR}}$$ obtained from total spectra, while the lower three panels display polarized spectra. In each row, the first panel is the probability density function of $$\log f_{\rm _{BLR}}$$ and the shadow area marks out the 68  per cent confidence interval, and the second and third indicate correlations of $$\log f_{\rm _{BLR}}$$ with model parameters of $$\Theta _{\rm _{BLR}}$$ and i, respectively. $$f_{\rm _{BLR}}$$ is sensitive to both ΘBLR and i for the total spectra, whereas $$f_{\rm _{BLR}}$$ only very weakly depends on ΘBLR. We apply the current $$f_{\rm _{BLR}}$$ to the radio-loud narrow line Seyfert 1 galaxy PKS 2004−447 with VFWHM(H α) = 1500 km s − 1(z = 0.240). The BH mass estimated by single total spectra is about 5 × 106 M⊙ (Oshlack, Webster & Whiting 2001), which is much smaller than the critical mass (M• ∼ 108 M⊙) invariably associated with classical radio-load AGNs (Laor 2000). Fortunately, the polarized spectra of VLT observations show its H α FWHM of (280 ± 50)Å (Baldi et al. 2016). Since the 5100Å luminosity is L5100 = 1.25 × 1044  erg s − 1 (Gallo et al. 2006), R–L relation indicates that the average time lag between emission line and continuum is about 101.6 ± 0.14 d (Bentz et al. 2013) for sub-Eddington AGNs (Du et al. 2016). Taking $$\log f_{\rm _{BLR}}= -0.63$$ for the polarized spectra of H α line, we have M• = 108.45 ± 0.2 M⊙. Employing the standard accretion disc model, we have the dimensionless accretion rates of $$\dot{\mathscr {M}}=\dot{M}c^2/L_{\rm Edd}=20.1(L_{44}/\cos i)^{3/2}m_7^{-2}\approx 0.05$$, where $$\dot{M}$$ is the accretion rates, LEdd = 1.4 × 1038(M•/M⊙) erg s − 1, L44 = L5100/1044 erg s − 1, cos i = 0.75 (inclination) and m7 = M•/107 M⊙ (Du et al. 2014). Such a low accretion rate agrees with the radio-loudness and accretion rate relation (Sikora, Stawarz & Lasota 2007). Finally, we would like to point out the temporal properties of polarized spectra. The equatorial distributions of scatters lead to delays of polarized photons with different frequencies relative to the BLR, and such a delay may need to be considered for polarized spectra at different epochs. Such a kind of polarization campaigns will provide a new way of accurately measuring the BH mass in type 1 AGNs. 4 CONCLUSION AND DISCUSSION In this Letter, we show that the factor fBLR has a wide range for total spectra of the virialized BLR. We investigate the polarized spectra of the BLR arisen from the equatorial scatters for $$f_{\rm _{BLR}}$$ in determination of the BH mass in type I AGNs. It is found that $$\log f_{\rm _{BLR}}\in [-0.65,-0.62]$$ for polarized spectra. This arises from the fact that the electrons on the equatorial plane scatter the broad-line photons to observers, assembling to view the BLR as edge-on orientation. The polarized spectra provide a way of accurately measuring BH mass from single-epoch polarized spectra, which is much better than that from single-epoch total spectra. For an individual application, equatorial scatters must be checked for the validity of the polarized spectra. We note the work of Afanasiev & Popović (2015), who employed the angles of polarization arisen by scatters on the inner edge of the dusty torus in order to alleviate dependence of BH mass on inclinations. This is different from what we suggest in this Letter. We would like to point out the major assumptions used in this Letter that scattering region is equatorial, but static. The geometry of scatters is supported by observations but the dynamics could be more complicated. We also neglect scatters of dust particles. This simple model shows the potential functions of polarized spectra in measuring BH mass. Fitting the polarized spectra leads to a more accurate BH mass, but we will conduct it in a forthcoming paper. Acknowledgements The authors thanks the referee for a useful report. We acknowledge the support of the staff of the Lijiang 2.4-m telescope. Funding for the telescope has been provided by Chinese Academy of Sciences and the People's Government of Yunnan Province. This research is supported by the National Key Program for Science and Technology Research and Development (grant 2016YFA0400701), Natural Science Foundation of China grants through Natural Science Foundation of China-11503026, -11173023 and -11233003, and an Natural Science Foundation of China – Chinese Academy of Sciences joint key grant U1431228, by the Chinese Academy of Sciences Key Research Program through KJZD-EW-M06, and by the Key Research Program of Frontier Sciences, Chinese Academy of Sciences, grant No. QYZDJ-SSW-SLH007. REFERENCES Afanasiev V. L., Popović L. 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E., 1992, ApJ , 397, 452 CrossRef Search ADS   Woo J. -H.et al.  , 2010, ApJ , 716, 269 CrossRef Search ADS   APPENDIX For type I AGN, the polarized spectra observed by a remote observer are mostly caused by electron scattering of the equatorial regions. Suppose the coordinate of the electron in spherical system is $$(r_{\rm _{P}}$$, $$\theta _{\rm _{P}}$$, $$\phi _{\rm _{P}})$$, we have   \begin{equation} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{\rm _{P}} = r_{\rm _{P}}\sin \theta _{\rm _{P}}\cos \phi _{\rm _{P}}, \\ y_{\rm _{P}} = r_{\rm _{P}}\sin \theta _{\rm _{P}}\sin \phi _{\rm _{P}}, \\ z_{\rm _{P}} = r_{\rm _{P}}\cos \theta _{\rm _{P}}. \end{array}\right. \end{equation} (A.1)As for a cloud in BLR at $$(r_{\rm _{B}}\cos \phi _{\rm _{B}}, r_{\rm _{B}}\sin \phi _{\rm _{B}},0)$$, we use the rotation matrix   \begin{equation} \mathbf {R} = \left( \begin{array}{ccc}\cos \phi _{\rm _{C}} & -\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}} & \sin \phi _{\rm _{C}}\sin \theta _{\rm _{C}} \\ \sin \phi _{\rm _{C}} & \cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}} & -\cos \phi _{\rm _{C}}\sin \theta _{\rm _{C}} \\ 0 & \sin \theta _{\rm _{C}} & \cos \theta _{\rm _{C}} \end{array} \right), \end{equation} (A.2)for the position of the cloud   \begin{equation} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{\rm _{B}} = r_{\rm _{B}} (\cos \phi _{\rm _{B}}\cos \phi _{\rm _{C}} - \sin \phi _{\rm _{B}}\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}}),\\ y_{\rm _{B}} = r_{\rm _{B}} (\cos \phi _{\rm _{B}}\sin \phi _{\rm _{C}} - \sin \phi _{\rm _{B}}\cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}}),\\ z_{\rm _{B}} = r_{\rm _{B}}\sin \theta _{\rm _{C}}\sin \phi _{\rm _{B}}, \end{array}\right. \end{equation} (A.3)in the O − XYZ frame. One cloud is at $$(x_{\rm _{B}},y_{\rm _{B}},z_{\rm _{B}})$$ and one scattering electron is at $$(x_{\rm _{P}},y_{\rm _{P}},z_{\rm _{P}})$$, we have the direction vector of incident light   \begin{equation} {\boldsymbol n}_{\rm _{BP}} = \frac{1}{r_{\rm _{BP}}} \left[(x_{\rm _{P}}-x_{\rm _{B}}){\boldsymbol i} + (y_{\rm _{P}}-y_{\rm _{B}}){\boldsymbol j} + (z_{\rm _{P}}-z_{\rm _{B}}){\boldsymbol k}\right], \end{equation} (A.4)where $$r_{\rm _{BP}} = \sqrt{(x_{\rm _{P}}-x_{\rm _{B}})^2 + (y_{\rm _{P}}-y_{\rm _{B}})^2 + (z_{\rm _{P}}-z_{\rm _{B}})^2}$$. The direction of the observer at infinity is taken to be $${\boldsymbol n}_{\rm obs} = (0,\sin i, \cos i)$$. The scattering angle is given by   \begin{equation} \cos \Theta \!=\! {\boldsymbol n}_{\rm obs} \cdot {\boldsymbol n}_{\rm _{BP}} = \frac{1}{r_{\rm _{BP}}} \left[ (y_{\rm _{P}} - y_{\rm _{B}})\sin i + (z_{\rm _{P}} - z_{\rm _{B}})\cos i\right].\!\!\!\!\!\! \end{equation} (A.5)The unit vector perpendicular to the scattering plane is   \begin{equation} {\boldsymbol n}_{\bot } = \frac{{\boldsymbol n}_{\rm _{BP}} \times {\boldsymbol n}_{\rm obs}}{|{\boldsymbol n}_{\rm _{BP}} \times {\boldsymbol n}_{\rm obs}|}. \end{equation} (A.6)We take the fixed coordinate system at celestial sphere of the observer to be $${\boldsymbol n}_{z^{\prime }}-{\boldsymbol n}_{x^{\prime }}$$, $${\boldsymbol n}_{z^{\prime }} = (0,-\cos i, \sin i),{n}_{x^{\prime }} = (1,0, 0)$$. The angle between $${\boldsymbol n}_{\bot }$$ and $${\boldsymbol n}_{z^{\prime }}$$ satisfies that   \begin{eqnarray} \cos \chi = \frac{x_{_{\rm P}}-x_{_{\rm B}}}{\left\lbrace [(z_{\rm _{P}}-z_{\rm _{B}})\sin i - (y_{\rm _{P}}-y_{\rm _{B}})\cos i]^2 + (x_{\mathrm{P}}-x_{\mathrm{B}})^2\right\rbrace ^{1/2}}. \nonumber\\ \end{eqnarray} (A.7) © 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society: Letters Oxford University Press

Measuring black hole mass of type I active galactic nuclei by spectropolarimetry

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Oxford University Press
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© 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
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Abstract

Abstract Black hole (BH) mass of type I active galactic nuclei (AGNs) can be measured or estimated through either reverberation mapping or empirical R–L relation, however, both of them suffer from uncertainties of the virial factor ($$f_{\rm _{BLR}}$$), thus limiting the measurement accuracy. In this letter, we make an effort to investigate $$f_{\rm _{BLR}}$$ through polarized spectra of the broad-line regions (BLR) arisen from electrons in the equatorial plane. Given the BLR composed of discrete clouds with Keplerian velocity around the central BH, we simulate a large number of spectra of total and polarized flux with wide ranges of parameters of the BLR model and equatorial scatters. We find that the $$f_{\rm _{BLR}}$$-distribution of polarized spectra is much narrower than that of total ones. This provides a way of accurately estimating BH mass from single spectropolarimetric observations of type I AGN whose equatorial scatters are identified. polarization, galaxies: active, quasars: supermassive black holes 1 INTRODUCTION Reverberation mapping (RM) is nowadays the most common technique of measuring black hole (BH) mass of type I active galactic nuclei (AGNs), except for a few local AGNs spatially resolved (Peterson 1993; Peterson 2014). RM measures time lags Δt of broad emission lines with respect to varying continuum as ionizing photons, allowing us to obtain the emissivity-averaged distance from the broad-line regions (BLR) to the central BH. Assuming fully random orbits of the BLR clouds with Keplerian velocity, we have the BH mass as   \begin{equation} M_{\bullet }= f_{\rm _{BLR}}\frac{c\Delta tV_{_{\rm FWHM}}^2}{G}, \end{equation} (1)where $$f_{\rm _{BLR}}$$ is the virial factor, $$V_{_{\rm FWHM}}$$ is the full width at half-maximum (FWHM) of the broad emission line profiles, c is the light speed and G is the gravity constant. The total error budget on the BH mass can be simply estimated by $$\delta M_{\bullet }/M_{\bullet }\approx [(\delta \ln f_{\rm _{BLR}})^2+0.08]^{1/2}$$, where Δt and $$V_{_{\rm FWHM}}$$ are usually of 20 per cent and 10 per cent for a typical measurement of RM observations, respectively. Obviously, the major uncertainty on the BH mass is due to $$f_{\rm _{BLR}}$$; however, it could be different by a factor of more than one order of magnitude (Pancoast et al. 2014b). Its dependence on kinematics, geometry and inclination of the BLR is poorly understood (Krolik 2001; Collin et al. 2006). For those AGNs with measured stellar velocity dispersions σ* of bulges and RM data, $$f_{\rm _{BLR}}$$ can be calibrated by the M•–σ* relation found in inactive galaxies (Onken et al. 2004; Woo et al. 2010). $$\langle f_{\rm _{BLR}}\rangle$$ as an averaged one can only remove the systematic bias between the virial product $$c\Delta tV_{\rm _{FWHM}}^2 / G$$ and M• for a large sample, however, $$f_{\rm _{BLR}}$$ is poorly understood individually. Furthermore, the zero-point and scatters of the M•–σ* relation depend on bulge types of the host galaxy, and virial factors of classical bulges and pseudo-bulges can differ by a factor of ∼2 (Ho & Kim 2014). The calibrated values of $$f_{\rm _{BLR}}$$ lead to δM•/M• ∼ 2, yielding only rough estimations of BH mass in AGNs. Recently, a motivated idea to test the validity of $$f_{\rm _{BLR}}$$ factor has been suggested by Du, Wang & Zhang (2017) in type II AGNs through the polarized spectra. In principle, the polarized spectra are viewed with highly face-on orientation to observers, and $$f_{\rm _{BLR}}$$ in type II AGNs should be the same as with type I AGNs. They reach a conclusion of $$f_{\rm _{BLR}}\sim 1$$ from a limited sample. For type I AGNs, the polarized spectra received by a remote observer correspond to ones viewed by an edge-on observer, lending an opportunity to measure BH mass similar to cases of NGC 4258 through water maser (Miyoshi et al. 1995) or others through CO line (Barth et al. 2016). In this letter, we investigate $$f_{\rm _{BLR}}$$ in type I AGNs through modelling the scattering polarized spectra quantitatively. In Section 2, we build a dynamical model for BLR and scattering region of type I AGNs for polarized spectra. In Section 3, we simulate a large number of spectra for a large range of model parameters to get the distribution of $$f_{\rm _{BLR}}$$ for both total and polarized spectra. We find that $$f_{\rm _{BLR}}$$ is in a very narrow range for polarized spectra. In Section 4, we draw conclusions and discuss potential ways of improving the accuracy of BH mass determination. 2 POLARIZED SPECTRA FROM EQUATORIAL SCATTERS Optical spectropolarimetric observations of type II AGNs discover that there is a broad component of emission line in polarized spectra, indicating the appearance of (1) a BLR hidden by the torus; and (2) at least one scattering region outside the torus (Antonucci & Miller 1985; Miller, Goodrich & Mathews 1991; Tran, Miller & Kay 1992). Radio observations also show that radio axes of most type II AGNs are nearly perpendicular to the position angle of polarization (Antonucci 1983; Brindle et al. 1990), showing that scattering regions of Type II AGNs situated outside the torus but aligned with the axes of the AGNs, called polar scattering region. In contrast, observations of type I AGNs reveal that position angles of the polarization are more often aligned with radio axes (Antonucci 1983, 1984; Smith et al. 2002). Equatorial scattering regions may exist, which are hidden by the torus, but can be seen in the polarized spectra of type I AGNs (Smith et al. 2005). We follow the geometry of equatorial scatters as in Smith et al. (2005). The geometry of the scattering regions and BLR are shown in Fig. 1(a). The details of the geometric relations are provided in the appendix. If the half opening angle of the scattering region is $$\Theta _{\rm _{P}}$$, the inner and outer radii of the scattering region are $$r_{\rm _{P,i}} \mbox{and } r_{\rm _{P,o}}$$, then we have $$r_{\rm _{P}}\in [r_{\rm _{P,i}},r_{\rm _{P,o}}],\theta _{\rm _{P}}\in [{\pi} /2-\Theta _{\rm _{P}},\pi /2+\Theta _{\rm _{P}}]$$. Scatterings caused by intercloud electrons in the BLR have been estimated as $$\tau _{_{\rm BLR}}\approx 0.04R_{\rm 0.1pc}$$ (see equation 5.13 in Krolik, McKee & Tarter 1981), where R0.1pc = RBLR/0.1 pc is a typical size of the BLR (Bentz et al. 2013), which can be totally neglected for polarized spectra. We thus assume that the scattering region is composed of free cold electrons beyond the BLR, and the whole region is optically thin, i.e. the optical depth $$\tau _{\rm es} \sim \bar{n}\ell \sigma _{\rm T} < 1$$, where $$\bar{n}$$ is the average number density of the electrons, ℓ is the typical scale of the region and σT is the Thomson cross-section. The distribution of the number density of electrons is assumed to be a power law as $$n_{\rm _{P}}(r_{\rm _{P}}, \theta _{\rm _{P}},\phi _{\rm _{P}}) = n_{\rm _{P0}} (r_{\rm _{P}}/r_{\rm _{P,i}})^{-\alpha }$$, where $$n_{\rm _{P0}}$$ is the number density at inner radius $$r_{\rm _{P,i}}$$. We assume BLR is composed of a large quantity of independent clouds rotating around the BH, and has a geometry indicated by Fig. 1(b) (Pancoast, Brewer & Treu 2011; Li et al. 2013; Pancoast, Brewer & Treu 2014a). The detailed geometric relations of BLR are given in the appendix. Suppose that the unit of length is Rg ≡ GM•/c2 and orbits of the clouds are circular. The velocity of the cloud is   \begin{equation} {\boldsymbol v}_{\rm cloud} = V_{\rm K} \left( \begin{array}{c}-\sin \phi _{\rm _{B}} \cos \phi _{\rm _{C}} - \cos \phi _{\rm _{B}}\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}}\\ -\sin \phi _{\rm _{B}} \sin \phi _{\rm _{C}} + \cos \phi _{\rm _{B}}\cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}}\\ \cos \phi _{\rm _{B}}\sin \theta _{\rm _{C}} \end{array} \right), \end{equation} (2)where $$V_{\rm K} = cr_{_{\rm B}}^{-1/2}$$. The half opening angle of the BLR is $$\Theta _{\rm _{BLR}}$$, the inner and outer radii of the BLR are $$r_{\rm _{B,i}}$$ and $$r_{\rm _{B,o}}$$, respectively. Then we have $$r_{\rm _{B}} \in [r_{\rm _{B,i}},r_{\rm _{B,o}}], \theta _{\rm _{C}} \in [0,\Theta _{\rm _{BLR}}]$$. The distribution of clouds can be modelled by power law as well. The number density of the clouds is $$n_{\rm _{B}}(r_{\rm _{B}}, \theta _{\rm _{C}},\phi _{\rm _{C}},\phi _{\rm _{B}}) = n_{\rm _{B0}} (r_{\rm _{B}}/r_{\rm _{B,i}})^{-\beta }$$, where $$n_{\rm _{B0}}$$ is the number density at $$r_{\rm _{B,i}}$$. A single scattering process is illustrated by Fig. 1(c). Expressions of the scattering angle Θ and rotation angle χ are derived in appendix. Assuming the ionizing source is isotropic and line intensity of one cloud at B is linearly proportional to the intensity of local ionizing fluxes, the line intensity at B is $$i_{\rm _{B}} = kr_{\rm _{B}}^{-2}$$, where k is a constant. We further assume that all clouds emit unpolarized Hβ photons isotropically and neglect multiple scatterings of optically thin regions. Thus the intensity at P is simply given by $$i_{\rm _{P}} = i_{\rm _{B}} {\cal S}/ 4{\pi} r_{\rm _{BP}}^2$$, where $${\cal S}$$ is the surface area of the cloud. Given incident photons with the Stokes parameters of $$(i_{\rm _{P}},0,0,0)$$, the Stokes parameters in the (n⊥–n∥) frame are (Chandrasekhar 1960).   \begin{eqnarray} {\displaystyle {\frac{3\sigma }{8{\pi} R^2}}\!\!\! \left(\begin{array}{c{@}{\quad}c{@}{\quad}c{@}{\quad}c}\displaystyle\frac{1}{2}(1+\cos ^2\Theta ) & \displaystyle\frac{1}{2}(1-\cos ^2\Theta ) & 0 & 0\\ \displaystyle\frac{1}{2}(1-\cos ^2\Theta ) & \displaystyle\frac{1}{2}(1+\cos ^2\Theta ) & 0 & 0\\ 0 & 0 & \cos \Theta & 0 \\ 0 & 0 & 0 & \cos \Theta \end{array} \right) \left(\begin{array}{c}i_{\rm _{P}} \\ 0 \\ 0 \\ 0 \end{array} \right)} \nonumber \\ {\quad = {\cal A}_0\left( \begin{array}{c}1+\cos ^2\Theta \\ 1-\cos ^2\Theta \\ 0 \\ 0 \end{array} \right), } \end{eqnarray} (3)where $${\cal A}_0=3\sigma i_{\rm _{P}}/16{\pi} R^2$$, R is the distance between the observer and AGN. Converting the Stokes parameter to the fixed coordinate $${\boldsymbol n}_{z^{\prime }}-{\boldsymbol n}_{x^{\prime }}$$ system, we have   \begin{eqnarray} \left(\begin{array}{c}i \\ q \\ u \\ v \end{array}\right) &=& {\cal A}_0 \left(\begin{array}{c{@}{\quad}c{@}{\quad}c{@}{\quad}c}1 & 0 & 0 & 0 \\ 0 & \cos 2\chi & \sin 2\chi & 0\\ 0 & -\sin 2\chi & \cos 2\chi & 0 \\ 0 & 0 & 0 & \cos \Theta \end{array}\right)\! \left( \begin{array}{c}1+\cos ^2\Theta \\ 1-\cos ^2\Theta \\ 0 \\ 0 \end{array}\right) \nonumber \\ &=&{\cal A}_0 \left( \begin{array}{c}1+\cos ^2\Theta \\ (1-\cos ^2\Theta )\cos 2\chi \\ -(1-\cos ^2\Theta )\sin 2\chi \\ 0 \end{array}\right). \end{eqnarray} (4)The velocity of the cloud $${\boldsymbol v}_{\rm cloud}$$ projected to the direction of incident light $${\boldsymbol n}_{\rm _{BP}}$$ is   \begin{equation} \frac{V_{\parallel }}{c} = \frac{1}{r_{\rm _{B}}^{1/2}} \frac{r_{\rm _{P}}(q_1\cos \phi _{\rm _{B}} + q_2\sin \phi _{\rm _{B}})}{\left[r_{\rm _{P}}^2 + r_{\rm _{B}}^2 + 2r_{\rm _{B}} r_{\rm _{P}}(q_2 \cos \phi _{\rm _{B}} - q_1 \sin \phi _{\rm _{B}})\right]^{1/2}}, \end{equation} (5)where $$q_1 = \cos \theta _{\rm _{P}} \sin \theta _{\rm _{C}} + \sin \theta _{\rm _{P}}\cos \theta _{\rm _{C}}\sin (\phi _{\rm _{P}} - \phi _{\rm _{C}})$$, $$q_2 = -\sin \theta _{\rm _{P}}$$$$\cos (\phi _{\rm _{P}} - \phi _{\rm _{C}})$$. If the intrinsic wavelength of the line is λ0, the wavelength after scattering is λ΄ = λ0(1 − V∥/c) due to Doppler shifts. Integrating over all the clouds and electrons, we have the total polarized spectrum,   \begin{eqnarray} \left(\begin{array}{c}I_{\lambda } \\ Q_{\lambda } \\ U_{\lambda } \\ V_{\lambda } \end{array}\right) = \int _{V_{\rm _{P}}} \text{d}V_{\rm _{P}}n_{\rm _{P}} \int _{V_{\rm _{BLR}}} \text{d}V_{\rm _{BLR}} n_{\rm _{B}} \int _{0}^{2{\pi} } {\cal L}(\lambda ,\lambda ^{\prime }) \text{d}\phi _{\rm _{B}} \left( \begin{array}{c}i \\ q \\ u \\ v \end{array} \right), \nonumber\\ \end{eqnarray} (6)where the intrinsic profile of Hβ line is assumed to be a Lorentzian function of $${\cal L}(\lambda ,\lambda ^{\prime })\propto \Gamma /[(\lambda -\lambda^{\prime})^2+\Gamma ^2]$$, Γ is the intrinsic width much smaller than the broadening due to rotation of clouds (the Lorentzian profiles are a very good approximation in the present context). Similarly, we can calculate the spectrum of non-scattered photons. The velocity of the cloud $${\boldsymbol v}_{\rm cloud}$$ projected to the direction of observer $${\boldsymbol n}_{\rm obs}$$ is   \begin{equation} \frac{\tilde{V}_{\parallel }}{c} = \frac{1}{r_{\rm _{B}}^{1/2}} \left( \tilde{q}_1\cos \phi _{\rm _{B}} + \tilde{q}_2\sin \phi _{\rm _{B}} \right), \end{equation} (7)where $$\tilde{q}_1 = \sin \theta _{\rm _{C}} \cos i + \cos \theta _{\rm _{C}}\cos \phi _{\rm _{C}} \sin i$$, $$\tilde{q}_2 = -\sin \phi _{\rm _{C}}\sin i$$. The observed wavelength is $$\lambda ^{\prime \prime } = \lambda _0(1-\tilde{V}_{\parallel }/c)$$. Integrating over all the clouds, we have   \begin{equation} F_{\lambda } = \int _{V_{\rm _{BLR}}} \text{d}V_{\rm _{BLR}} n_{\rm _{B}} \int _{0}^{2{\pi} } {\cal L}(\lambda ,\lambda ^{\prime \prime })\,\text{d}\phi _{\rm _{B}} \frac{i_{\rm _{B}}S}{4{\pi} R^2}, \end{equation} (8)and the expression for polarization degree and position angle,   \begin{equation} P_{\lambda } = \frac{\sqrt{Q_{\lambda }^2 + U_{\lambda }^2}}{I_{\lambda } + F_{\lambda }}, \quad \theta _{\lambda } = \frac{1}{2} \arccos \left( \frac{Q_{\lambda }}{\sqrt{Q_{\lambda }^2+U_{\lambda }^2}} \right). \end{equation} (9)If Uλ > 0, θλ ∈ (0, π/2). If Uλ < 0, θλ ∈ ( − π/2, 0). Position angles represent the angle between the direction of maximum intensity and $$n_{z^{\prime }}$$. Table 1 summarises all the parameters of the present model. We calculate a series of profiles for different values of parameters and find that the profiles are only sensitive to $$\Theta _{_{\rm BLR}}$$ and i. Fig. 2 shows typical spectra of a type I AGN with an equatorial electron scattering region. The parameters of the model are $$r_{\rm _{P,i}} = 10^4$$, $$r_{\rm _{P,o}} = 2\times 10^4$$, $$\Theta _{\rm _{P}} = 30^{\circ }$$, $$r_{\rm _{B,i}} = 2\times 10^3$$, $$r_{\rm _{B,o}} = 6\times 10^3$$, α = 1, β = 1, $$\tau _{\rm es}=\sigma n_{\rm _{P0}}(r_{\rm _{P,o}}-r_{\rm _{P,i}}) = 1$$, i = (1°, 10°, 20°, 30°, 40°) and $$\Theta _{_{\rm BLR}} =(10^{\circ },20^{\circ },30^{\circ },40^{\circ },50^{\circ })$$. As shown in the left-hand panel of Fig. 2 , the total spectra get broader as i increases and show double-peaked profiles when i exceeds a critical inclination. By contrast, the width and the profile of the polarized spectra are not sensitive to i (can be found from normalized spectra). This interesting property results from the fact that the polarized spectra are equivalent to the ones seen by observers at edge-on orientations. Generally, the line centres have lowest polarization degrees. However, the polarization degrees become smaller with i. Figure 1. View largeDownload slide Panel (a) is a cartoon of a type I AGN with an equatorial scattering region. The blue points represent clouds in the BLR and the grey region the scatters on the equatorial plan. i is the inclination angle to a remote observer in the O–YZ plane. Panel (b) is the frame for the BLR geometry. O–XY is the equatorial plane. $$O\text{--}X_{\rm _{C}}Y_{\rm _{C}}$$ is the orbital plane of one specific cloud, which can be obtained by rotating X around Z by $$\phi _{\rm _{C}}$$ and then rotating Z around $$X_{\rm _{C}}$$ by $$\theta _{\rm _{C}}$$. The phase angle of the cloud relative to $$OX_{\rm _{C}}$$ is $$\phi _{\rm _{B}}$$. Panel (c) is the scattering geometry used here. BP is the incident light from point B on one orbit. $${\boldsymbol n}_{\rm obs}$$ is the direction of sight of the observer (i.e. the direction of scattered light). The vectors of $${\boldsymbol n}_{\rm obs}$$, $${\boldsymbol n}_{\bot }$$, $${\boldsymbol n}_{\parallel }$$, $${\boldsymbol n}_{z^{\prime }}$$ and $${\boldsymbol n}_{x^{\prime }}$$ are explained in the appendix. Figure 1. View largeDownload slide Panel (a) is a cartoon of a type I AGN with an equatorial scattering region. The blue points represent clouds in the BLR and the grey region the scatters on the equatorial plan. i is the inclination angle to a remote observer in the O–YZ plane. Panel (b) is the frame for the BLR geometry. O–XY is the equatorial plane. $$O\text{--}X_{\rm _{C}}Y_{\rm _{C}}$$ is the orbital plane of one specific cloud, which can be obtained by rotating X around Z by $$\phi _{\rm _{C}}$$ and then rotating Z around $$X_{\rm _{C}}$$ by $$\theta _{\rm _{C}}$$. The phase angle of the cloud relative to $$OX_{\rm _{C}}$$ is $$\phi _{\rm _{B}}$$. Panel (c) is the scattering geometry used here. BP is the incident light from point B on one orbit. $${\boldsymbol n}_{\rm obs}$$ is the direction of sight of the observer (i.e. the direction of scattered light). The vectors of $${\boldsymbol n}_{\rm obs}$$, $${\boldsymbol n}_{\bot }$$, $${\boldsymbol n}_{\parallel }$$, $${\boldsymbol n}_{z^{\prime }}$$ and $${\boldsymbol n}_{x^{\prime }}$$ are explained in the appendix. Figure 2. View largeDownload slide Total, polarized spectra and polarization degrees of a type I AGN with an equatorial electron scattering region. The left-hand column is spectra for different inclinations, whereas the right-hand column is for different BLR opening angles. The tops of the total spectra become flatter and polarization decreases with increasing ΘBLR (tends to 90°). Figure 2. View largeDownload slide Total, polarized spectra and polarization degrees of a type I AGN with an equatorial electron scattering region. The left-hand column is spectra for different inclinations, whereas the right-hand column is for different BLR opening angles. The tops of the total spectra become flatter and polarization decreases with increasing ΘBLR (tends to 90°). Table 1. The dependence of $$f_{\rm _{BLR}}$$ on model parameters. Parameter  Range  Meaning  $$\delta \ln f_{_{\rm BLR}}$$        Total  Polarized  $$r_{\rm _{P,i}}(10^4 R_{\rm g})$$  [1, 5]  SR inner radius  0.01  0  $$r_{\rm _{P,o}}(10^4 R_{\rm g})$$  [2, 10]  SR outer radius  0  0  $$\Theta _{\rm _{P}}(^\circ )$$  [20, 50]  SR opening angle  − 0.04  0  α  [0, 1.5]  Index of DF of electrons  0.01  0  $$r_{\rm _{B,i}}(10^3 R_{\rm g})$$  [1, 5]  Inner radius of BLR  − 0.01  0.01  $$r_{\rm _{B,o}}(10^3 R_{\rm g})$$  [2, 10]  Outer radius of BLR  0.01  − 0.01  $$\Theta _{\rm _{BLR}}(^\circ )$$  [20, 50]  Opening angle of BLR  − 0.38  0.03  β  [0, 1.5]  Index of DF of clouds  0  − 0.02  i(°)  [0, 45]  Inclination angle  − 0.64  0  Parameter  Range  Meaning  $$\delta \ln f_{_{\rm BLR}}$$        Total  Polarized  $$r_{\rm _{P,i}}(10^4 R_{\rm g})$$  [1, 5]  SR inner radius  0.01  0  $$r_{\rm _{P,o}}(10^4 R_{\rm g})$$  [2, 10]  SR outer radius  0  0  $$\Theta _{\rm _{P}}(^\circ )$$  [20, 50]  SR opening angle  − 0.04  0  α  [0, 1.5]  Index of DF of electrons  0.01  0  $$r_{\rm _{B,i}}(10^3 R_{\rm g})$$  [1, 5]  Inner radius of BLR  − 0.01  0.01  $$r_{\rm _{B,o}}(10^3 R_{\rm g})$$  [2, 10]  Outer radius of BLR  0.01  − 0.01  $$\Theta _{\rm _{BLR}}(^\circ )$$  [20, 50]  Opening angle of BLR  − 0.38  0.03  β  [0, 1.5]  Index of DF of clouds  0  − 0.02  i(°)  [0, 45]  Inclination angle  − 0.64  0  Note. SR: scattering region and DF: distribution function. $$\delta \ln f_{\rm _{BLR}}$$ describes the dependence on parameters of the BLRs. See details for its definition in the main text. It shows that $$f_{\rm _{BLR}}$$ is sensitive to ΘBLR and i for the total spectra, but it is almost a constant for polarized spectra. View Large Total spectra show strong dependence on ΘBLR. As shown in the right-hand panel of Fig. 2, large-ΘBLR BLRs show a broader width and get narrower with decreasing of ΘBLR until double-peaked profiles. However, $$\Theta _{_{\rm BLR}}$$ does not change the polarized profiles too much. A large $$\Theta _{_{\rm BLR}}$$ indicates that the system tends to be more spherically symmetric and to decrease the polarization degree. Comparing with the total spectra, the width of the polarized spectrum is insensitive to $$\Theta _{_{\rm BLR}}$$ and i. This property allows us to infer $$f_{\rm _{BLR}}$$ from the polarized spectrum and improve the accuracy of BH mass measurement, as shown in Section 3. 3 THE VIRIAL FACTOR With the BLR model, we have the emissivity-averaged time lag of a broad emission line as   \begin{eqnarray} \Delta t &=& \frac{\int \text{d}V_{\rm _{BLR}} \Delta r i_{\rm _{B}} n_{\rm _{B}}}{\int \text{d}V_{\rm _{BLR}} i_{\rm _{B}} n_{\rm _{B}}} \left(\frac{R_{\rm g}}{c}\right) \nonumber\\ &=& \frac{1-\beta }{2-\beta } \frac{1-q_r^{2-\beta }}{1-q_r^{1-\beta }} \left(\frac{r_{_{\rm B,i}}R_{\rm g}}{c}\right), \end{eqnarray} (10)where $$\Delta r = r_{\rm _{B}} - {r}_{\rm _{B}}\cdot {n}_{\rm obs}$$ and the corresponding virial factor from equation (1) can be written as   \begin{equation} f_{\rm _{BLR}}^{-1} = \left(\frac{1-\beta }{2-\beta }\right) \left(\frac{1-q_r^{1-\beta }}{1-q_r^{2-\beta }}\right) \left(\frac{V_{\rm _{FWHM}}}{c}\right)^2r_{_{\rm B,i}}, \end{equation} (11)where $$q_r=r_{_{\rm B,o}}/r_{_{\rm B,i}}$$, $$V_{_{\rm FWHM}}$$ is the FWHM of profiles either from the total or polarized spectra. We generate profiles according to parameters listed in Table 1 and measure $$V_{_{\rm FWHM}}$$ to show dependences of $$f_{\rm _{BLR}}$$ on each parameter. We did Monte Carlo simulations for all the parameters listed in Table 1 and then get the $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations, where Xi is any one of the parameters. The $$f_{\rm _{BLR}}$$ dependence on Xi can then be obtained by $$\delta \ln f_{\rm _{BLR}}=(\partial \ln f_{\rm _{BLR}}/\partial \ln X_i)\delta \ln X_i$$, where $$\partial \ln f_{\rm _{BLR}}/\partial \ln X_i$$ is the slope of the $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations and δln Xi is its range. The slope is estimated from the line regression of $$\ln f_{\rm _{BLR}}\text{--}\ln X_i$$ relations from the Monte Carlo simulations. The dependence listed in Table 1 shows that only i and ΘBLR are the major drivers in the total spectra, but $$f_{\rm _{BLR}}$$ is insensitive to all the parameters (only slightly relies on ΘBLR). We estimate the entire uncertainties of $$f_{\rm _{BLR}}$$ due to all parameters as $$\Delta \log f_{\rm _{BLR}}=[\sum _{i=1}^9 (\partial \ln f_{\rm _{BLR}}/\partial \ln X_i)^2(\Delta \log X_i)^2]^{1/2}$$. We have $$\Delta \log f_{\rm _{BLR}}=(0.74,0.04)$$ for total and polarized spectra, respectively. We plot the $$\log f_{\rm _{BLR}}\text{-}$$distributions and contour maps versus ΘBLR and i in Fig. 3. It shows that the distribution from total spectra is much broader than that from polarized spectra. The 68  per cent confidence interval of the former is $$\log f_{\rm _{BLR}}\in [-0.41,0.08]$$ agreeing with values from detailed Markov chain Monte Carlo modelling (Pancoast et al. 2014b), but $$\log f_{\rm _{BLR}}\in [-0.65,-0.62]$$ from the polarized spectra. For a typical RM campaign, we have the uncertainties of BH mass δM•/M• ≈ (0.8, 0.3) from total and polarized spectra, respectively. Obviously, spectropolarimetry provides much better $$f_{\rm _{BLR}}$$ for BH mass from RM campaign. However, the polarized spectra as the prerequisites of applications should be identified as originating from the equatorial scatters. Actually, this can be done by checking if the position angles are parallel to radio axis in AGNs. Figure 3. View largeDownload slide The upper three panels display the distribution of $$\log f_{\rm _{BLR}}$$ obtained from total spectra, while the lower three panels display polarized spectra. In each row, the first panel is the probability density function of $$\log f_{\rm _{BLR}}$$ and the shadow area marks out the 68  per cent confidence interval, and the second and third indicate correlations of $$\log f_{\rm _{BLR}}$$ with model parameters of $$\Theta _{\rm _{BLR}}$$ and i, respectively. $$f_{\rm _{BLR}}$$ is sensitive to both ΘBLR and i for the total spectra, whereas $$f_{\rm _{BLR}}$$ only very weakly depends on ΘBLR. Figure 3. View largeDownload slide The upper three panels display the distribution of $$\log f_{\rm _{BLR}}$$ obtained from total spectra, while the lower three panels display polarized spectra. In each row, the first panel is the probability density function of $$\log f_{\rm _{BLR}}$$ and the shadow area marks out the 68  per cent confidence interval, and the second and third indicate correlations of $$\log f_{\rm _{BLR}}$$ with model parameters of $$\Theta _{\rm _{BLR}}$$ and i, respectively. $$f_{\rm _{BLR}}$$ is sensitive to both ΘBLR and i for the total spectra, whereas $$f_{\rm _{BLR}}$$ only very weakly depends on ΘBLR. We apply the current $$f_{\rm _{BLR}}$$ to the radio-loud narrow line Seyfert 1 galaxy PKS 2004−447 with VFWHM(H α) = 1500 km s − 1(z = 0.240). The BH mass estimated by single total spectra is about 5 × 106 M⊙ (Oshlack, Webster & Whiting 2001), which is much smaller than the critical mass (M• ∼ 108 M⊙) invariably associated with classical radio-load AGNs (Laor 2000). Fortunately, the polarized spectra of VLT observations show its H α FWHM of (280 ± 50)Å (Baldi et al. 2016). Since the 5100Å luminosity is L5100 = 1.25 × 1044  erg s − 1 (Gallo et al. 2006), R–L relation indicates that the average time lag between emission line and continuum is about 101.6 ± 0.14 d (Bentz et al. 2013) for sub-Eddington AGNs (Du et al. 2016). Taking $$\log f_{\rm _{BLR}}= -0.63$$ for the polarized spectra of H α line, we have M• = 108.45 ± 0.2 M⊙. Employing the standard accretion disc model, we have the dimensionless accretion rates of $$\dot{\mathscr {M}}=\dot{M}c^2/L_{\rm Edd}=20.1(L_{44}/\cos i)^{3/2}m_7^{-2}\approx 0.05$$, where $$\dot{M}$$ is the accretion rates, LEdd = 1.4 × 1038(M•/M⊙) erg s − 1, L44 = L5100/1044 erg s − 1, cos i = 0.75 (inclination) and m7 = M•/107 M⊙ (Du et al. 2014). Such a low accretion rate agrees with the radio-loudness and accretion rate relation (Sikora, Stawarz & Lasota 2007). Finally, we would like to point out the temporal properties of polarized spectra. The equatorial distributions of scatters lead to delays of polarized photons with different frequencies relative to the BLR, and such a delay may need to be considered for polarized spectra at different epochs. Such a kind of polarization campaigns will provide a new way of accurately measuring the BH mass in type 1 AGNs. 4 CONCLUSION AND DISCUSSION In this Letter, we show that the factor fBLR has a wide range for total spectra of the virialized BLR. We investigate the polarized spectra of the BLR arisen from the equatorial scatters for $$f_{\rm _{BLR}}$$ in determination of the BH mass in type I AGNs. It is found that $$\log f_{\rm _{BLR}}\in [-0.65,-0.62]$$ for polarized spectra. This arises from the fact that the electrons on the equatorial plane scatter the broad-line photons to observers, assembling to view the BLR as edge-on orientation. The polarized spectra provide a way of accurately measuring BH mass from single-epoch polarized spectra, which is much better than that from single-epoch total spectra. For an individual application, equatorial scatters must be checked for the validity of the polarized spectra. We note the work of Afanasiev & Popović (2015), who employed the angles of polarization arisen by scatters on the inner edge of the dusty torus in order to alleviate dependence of BH mass on inclinations. This is different from what we suggest in this Letter. We would like to point out the major assumptions used in this Letter that scattering region is equatorial, but static. The geometry of scatters is supported by observations but the dynamics could be more complicated. We also neglect scatters of dust particles. This simple model shows the potential functions of polarized spectra in measuring BH mass. Fitting the polarized spectra leads to a more accurate BH mass, but we will conduct it in a forthcoming paper. Acknowledgements The authors thanks the referee for a useful report. We acknowledge the support of the staff of the Lijiang 2.4-m telescope. Funding for the telescope has been provided by Chinese Academy of Sciences and the People's Government of Yunnan Province. This research is supported by the National Key Program for Science and Technology Research and Development (grant 2016YFA0400701), Natural Science Foundation of China grants through Natural Science Foundation of China-11503026, -11173023 and -11233003, and an Natural Science Foundation of China – Chinese Academy of Sciences joint key grant U1431228, by the Chinese Academy of Sciences Key Research Program through KJZD-EW-M06, and by the Key Research Program of Frontier Sciences, Chinese Academy of Sciences, grant No. QYZDJ-SSW-SLH007. REFERENCES Afanasiev V. L., Popović L. 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E., 1992, ApJ , 397, 452 CrossRef Search ADS   Woo J. -H.et al.  , 2010, ApJ , 716, 269 CrossRef Search ADS   APPENDIX For type I AGN, the polarized spectra observed by a remote observer are mostly caused by electron scattering of the equatorial regions. Suppose the coordinate of the electron in spherical system is $$(r_{\rm _{P}}$$, $$\theta _{\rm _{P}}$$, $$\phi _{\rm _{P}})$$, we have   \begin{equation} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{\rm _{P}} = r_{\rm _{P}}\sin \theta _{\rm _{P}}\cos \phi _{\rm _{P}}, \\ y_{\rm _{P}} = r_{\rm _{P}}\sin \theta _{\rm _{P}}\sin \phi _{\rm _{P}}, \\ z_{\rm _{P}} = r_{\rm _{P}}\cos \theta _{\rm _{P}}. \end{array}\right. \end{equation} (A.1)As for a cloud in BLR at $$(r_{\rm _{B}}\cos \phi _{\rm _{B}}, r_{\rm _{B}}\sin \phi _{\rm _{B}},0)$$, we use the rotation matrix   \begin{equation} \mathbf {R} = \left( \begin{array}{ccc}\cos \phi _{\rm _{C}} & -\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}} & \sin \phi _{\rm _{C}}\sin \theta _{\rm _{C}} \\ \sin \phi _{\rm _{C}} & \cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}} & -\cos \phi _{\rm _{C}}\sin \theta _{\rm _{C}} \\ 0 & \sin \theta _{\rm _{C}} & \cos \theta _{\rm _{C}} \end{array} \right), \end{equation} (A.2)for the position of the cloud   \begin{equation} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{\rm _{B}} = r_{\rm _{B}} (\cos \phi _{\rm _{B}}\cos \phi _{\rm _{C}} - \sin \phi _{\rm _{B}}\sin \phi _{\rm _{C}}\cos \theta _{\rm _{C}}),\\ y_{\rm _{B}} = r_{\rm _{B}} (\cos \phi _{\rm _{B}}\sin \phi _{\rm _{C}} - \sin \phi _{\rm _{B}}\cos \phi _{\rm _{C}}\cos \theta _{\rm _{C}}),\\ z_{\rm _{B}} = r_{\rm _{B}}\sin \theta _{\rm _{C}}\sin \phi _{\rm _{B}}, \end{array}\right. \end{equation} (A.3)in the O − XYZ frame. One cloud is at $$(x_{\rm _{B}},y_{\rm _{B}},z_{\rm _{B}})$$ and one scattering electron is at $$(x_{\rm _{P}},y_{\rm _{P}},z_{\rm _{P}})$$, we have the direction vector of incident light   \begin{equation} {\boldsymbol n}_{\rm _{BP}} = \frac{1}{r_{\rm _{BP}}} \left[(x_{\rm _{P}}-x_{\rm _{B}}){\boldsymbol i} + (y_{\rm _{P}}-y_{\rm _{B}}){\boldsymbol j} + (z_{\rm _{P}}-z_{\rm _{B}}){\boldsymbol k}\right], \end{equation} (A.4)where $$r_{\rm _{BP}} = \sqrt{(x_{\rm _{P}}-x_{\rm _{B}})^2 + (y_{\rm _{P}}-y_{\rm _{B}})^2 + (z_{\rm _{P}}-z_{\rm _{B}})^2}$$. The direction of the observer at infinity is taken to be $${\boldsymbol n}_{\rm obs} = (0,\sin i, \cos i)$$. The scattering angle is given by   \begin{equation} \cos \Theta \!=\! {\boldsymbol n}_{\rm obs} \cdot {\boldsymbol n}_{\rm _{BP}} = \frac{1}{r_{\rm _{BP}}} \left[ (y_{\rm _{P}} - y_{\rm _{B}})\sin i + (z_{\rm _{P}} - z_{\rm _{B}})\cos i\right].\!\!\!\!\!\! \end{equation} (A.5)The unit vector perpendicular to the scattering plane is   \begin{equation} {\boldsymbol n}_{\bot } = \frac{{\boldsymbol n}_{\rm _{BP}} \times {\boldsymbol n}_{\rm obs}}{|{\boldsymbol n}_{\rm _{BP}} \times {\boldsymbol n}_{\rm obs}|}. \end{equation} (A.6)We take the fixed coordinate system at celestial sphere of the observer to be $${\boldsymbol n}_{z^{\prime }}-{\boldsymbol n}_{x^{\prime }}$$, $${\boldsymbol n}_{z^{\prime }} = (0,-\cos i, \sin i),{n}_{x^{\prime }} = (1,0, 0)$$. The angle between $${\boldsymbol n}_{\bot }$$ and $${\boldsymbol n}_{z^{\prime }}$$ satisfies that   \begin{eqnarray} \cos \chi = \frac{x_{_{\rm P}}-x_{_{\rm B}}}{\left\lbrace [(z_{\rm _{P}}-z_{\rm _{B}})\sin i - (y_{\rm _{P}}-y_{\rm _{B}})\cos i]^2 + (x_{\mathrm{P}}-x_{\mathrm{B}})^2\right\rbrace ^{1/2}}. \nonumber\\ \end{eqnarray} (A.7) © 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

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Monthly Notices of the Royal Astronomical Society: LettersOxford University Press

Published: Jan 1, 2018

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