© The Authors 2025

1. Introduction

Observations indicate that winds or outflows are commonly present in various objects, including active galactic nuclei (AGNs), X-ray binaries, and early and late stars. In luminous AGNs, blueshifted absorption lines are often detected in their ultraviolet (UV) and X-ray spectra (Churchill et al. 1999; Narayanan et al. 2004; Tombesi et al. 2010, 2012; Rodríguez Hidalgo et al. 2011; Gofford et al. 2013). For example, Misawa et al. (2007) found from 37 optically bright quasars that 10%–17% of C IV narrow absorption lines in the UV band are blueshifted at 5000–70 000 km s⁻¹. Tombesi et al. (2012, 2014) reported that >35% of radio-quiet AGNs have highly ionized Fe XXV and Fe XXVI absorption lines in the hard X-ray band, which blueshifted at ∼0.03–0.3 c (c is the speed of light). It was suggested that the Fe XXV and Fe XXVI absorption lines are generated by a highly ionized absorber from the accretion disc around the central black hole (BH Tombesi et al. 2012). Tombesi et al. (2010) referred to the highly ionized absorbers with a velocity above 10⁴ km s⁻¹ as UFOs. Based on photoionization modeling, Tombesi et al. (2011) derived the basic properties of ultra-fast outflows (UFOs). They found that the ionization parameter (ξ) and column density (N_H) of UFOs are distributed in the range of log(ξ/(erg s⁻¹cm)) ∼3–6 and log(N_H/(cm⁻²)) ∼22–24, respectively.

The formation of UFOs is still an open issue. Two kinds of models that produce UFOs from the accretion disc have been proposed. One is the line-force-driven model, and the other is the magnetic-driven model (Proga & Kallman 2004; Fukumura et al. 2014, 2018; Nomura et al. 2016; Nomura & Ohsuga 2017; Yang et al. 2021; Mizumoto et al. 2021). For the magnetic-driven model, it was proposed that when a poloidal component (B_p) is weaker than the toroidal component (B_ϕ), i.e. |B_ϕ/B_p|≲1, the magneto-centrifugally driven mechanism drives winds (Blandford & Payne 1982). When the toroidal magnetic field is much stronger than the poloidal component, i.e. |B_ϕ/B_p|≫1, the magnetic pressure can drive winds. The magnetic-driven model has been developed in previous works (e.g., Lovelace et al. 1987; Sakurai 1987; Cao & Spruit 1994; Fukumura et al. 2014, 2018). Fukumura et al. (2018) employed the magnetic-driving model to model the Fe K UFOs observed in PDS 456.

The line-force-driven model has also been employed to understand the formation of UFOs (Murray & Chiang 1995; Proga et al. 2000; Nomura et al. 2016; Yang et al. 2021). In luminous AGNs, a thin disc around a BH irradiates the gases above the disc, causing the photons from the disc surface to be scattered or absorbed by the gases, and then the Compton-scattering force and the spectral line force are exerted on the gas. For the weakly ionized gases, the line force caused by the absorption of UV photons becomes important and is the main force driving winds (Castor et al. 1975). For the highly ionized gas, the line force is negligible and the Compton-scattering force is the main radiation force. In the luminous AGNs, the Compton-scattering force is not strong enough to drive the winds with a mildly relativistic velocity, and then the line force is an important driving force because the thin disc can emit strong UV radiation (Murray & Chiang 1995). When the ionization parameter of gases is higher than 100 erg s⁻¹ cm, the line force becomes unimportant. However, the ionization parameter of UFOs is much higher than 100 erg s⁻¹ cm. Numerical simulations found that the line force can intermittently accelerate winds and then form the observed UFOs (Nomura & Ohsuga 2017; Mizumoto et al. 2021; Yang 2021; Yang et al. 2021). When the dense gases in the inner region of the disc shield the X-ray photons from the central corona, the gases are weakly ionized, and then the line force becomes effective and accelerates winds. When the winds driven by the line force are exposed to the X-ray photons, the winds are highly ionized and become UFOs. Therefore, UFOs may come of the line-force-driven winds (Nomura & Ohsuga 2017; Mizumoto et al. 2021; Yang 2021; Yang et al. 2021). Higginbottom et al. (2014) employed the post-processing approach to implement Monte Carlo simulations of radiation transfer. They found that the ‘shielding’ effect is ineffective because scattered X-ray photons can contribute to ionizing the gas. In the work of Higginbottom et al. (2014), the simulation of radiation transfer and hydrodynamics was decoupled. The dynamical properties of winds would not change due to alterations in the radiation field. In reality, winds could adjust themselves to ensure that the scattered photons were not sufficiently strong, thereby maintaining the effectiveness of the shielding effect and allowing the line force to efficiently drive winds.

When the line-force-driven winds move at mildly relativistic velocities in the radiation field, special relativistic effects, such as the radiation-drag effect, stop being negligible. When winds move at relativistic velocities within a radiation field, a radiation-drag force acts on them due to relativistic effects. This effect is referred to as the radiation-drag effect. We also note another relativistic effect, namely the Poynting-Robertson effect. This effect arises when the radiation emitted by atoms in winds is collimated in the direction of motion of the atoms due to relativistic corrections, resulting in a drag force acting on the emitting atoms. Marzi et al. (2023) have investigated the Poynting-Robertson effect on the black-hole-driven winds using an analytical approach. In this paper, we focus on the radiation-drag effect based on numerical simulations.

Previous works found that the wind speed is significantly reduced when the relativistic effects are taken into account (Luminari et al. 2020, 2021; Wang et al. 2022; Quera-Bofarull et al. 2023). The previous works employed radiation hydrodynamic simulations to study the line-force-driven winds in AGNs (e.g., Proga et al. 2000; Proga & Kallman 2004; Nomura & Ohsuga 2017; Yang 2021; Yang et al. 2021). However, the simulations ignored relativistic effects, such as the radiation-drag effect and the influence of the disc rotation on the radiation field. Therefore, our motivation in this paper is to study the influence of the relativistic effects on the line-force-driven winds.

The paper is organized as follows. In Section 2, we describe basic equations and the model set-up. In Sections 3 and 4, we give our results and application. In Section 5, we give conclusions.

2. Radiation field

A geometrically thin and optically thick disc is usually employed to describe luminous AGNs such as quasars (Shakura & Sunyaev 1973). Observations also indicate the presence of a hot corona within a Schwarzschild radius (r_s) of 10 (Reis & Miller 2013; Uttley et al. 2014). We followed Proga et al. (2000) in assuming a model of a thin disc with a spherical hot corona to describe the luminous AGNs. The gases above the thin disc are irradiated by the radiation from the disc surface and the hot corona. On the thin disc, the local isotropic intensity, I_d(r_d), is written as

$$I_{\mathrm{d}}(r_{\mathrm{d}})=\frac{9ϵL_{\mathrm{Edd}}{r}_{\mathrm{s}}}{4\pi r_{\mathrm{d}}^3}\left[1-(\frac{3r_{\mathrm{s}}}{r_{\mathrm{d}}}{)}^{\frac{1}{2}}\right],$$(1)

where r_d is the radial position on the disc surface, L_Edd is the Eddington luminosity, and ϵ is the Eddington ratio of disc luminosity. The disc luminosity is given by L_d=ϵL_Edd. The disc can emit a large number of UV photons that contribute to the line force through the bound-bound transition. According to Proga et al. (2000) and Nomura et al. (2016), we roughly assume that the disc with a local temperature higher than 3×10³ K emits all disc photons that can contribute to the line force. The hot corona emits the X-ray photons, which play an important role in ionizing gases. We define the corona luminosity as L_cor=f_xL_d, where f_x is the ratio of the corona luminosity to the disc luminosity. In our models, the corona luminosity (L_cor) is determined by the ratio (f_x) of the corona luminosity to the disc luminosity. We evaluated f_x according to an observed non-linear relation between the X-ray (2 keV) and UV (2500 Å ) emissions in quasars (Lusso & Risaliti 2016). This relation implies that in optically brighter AGNs the X-ray luminosity is fainter, i.e. f_x is less. According to this relation, Yang et al. (2021) evaluated the f_x. The calculation of f_X is referred to in Yang et al. (2021).

The line-force-driven winds from the disc have mildly relativistic velocities and the special relativistic effects cannot be ignored. When we calculate the radiation flux from the thin disc, the transformation of the frame needs to be considered. There are two different frames here: one is the co-moving frame moving with gases and the other is the rest frame standing at infinity. In the rest frame, the radiation intensity from the disc surface is given by

$$I_0=\frac{I_{\mathrm{d}}}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4},$$(2)

where $\overrightarrow{\mathit{l}}$ is the direction-cosine vector pointing toward the rest observer from the disc surface and $\overrightarrow{{\mathit{v}}_{\mathrm{d}}}$ is the disc velocity. The denominator on the right side of equation (2) reflects the special relativistic effect caused by disc rotation. The radiation energy density (E₀), the radiative flux (F_i,0), and the radiation stress tensor (P_ij,0) in the rest frame read

$$E_0=\frac{1}{c}{\oint }_{\mathrm{\Omega }}{I}_0\mathrm{d}\mathrm{\Omega },$$(3)

$$F_{i,0}={\oint }_{\mathrm{\Omega }}{I}_0{l}_i\mathrm{d}\mathrm{\Omega },$$(4)

and

$$P_{\mathit{ij},0}=\frac{1}{c}{\oint }_{\mathrm{\Omega }}{I}_0{l}_i{l}_j\mathrm{d}\mathrm{\Omega },$$(5)

where c is the speed of light and Ω is the solid angle of the disc toward the rest observer. When we calculate the radiation force exerted on the gases, we need to transform the radiation flux generated by the disc from the rest frame into the co-moving frame. On the order of v/c, the transformation of the two frames is given by

$$F_i=F_{i,0}-E_0{\mathit{v}}_i-{\displaystyle \sum _j}{\mathit{v}}_j{P}_{\mathit{ij},0},$$(6)

where F_i is the radiative flux in the co-moving frame and v_i is the ith component of the gas velocities. In the above equations, the subscript ‘0’ denotes the quantities in the rest frame. In equation (6), the (−E₀v_i−∑_jv_jP_ij,0) term reflects the special relativistic effect caused by the motion of the gas, and means that the radiation flux measured by the gas moving in the direction of the radiation flux decreases and then the radiation force exerted on the moving gas weakens. This effect is called the radiation-drag effect.

2.1. Basic equations

To simulate the irradiated gases above the disc surface, we placed the computation region above the disc surface and implemented our simulations in spherical polar co-ordinates (r,θ,ϕ). We placed the θ=π/2 plane at the disc surface and assumed that the disc scale height is constant with radius. Following Nomura et al. (2016), we set the scale height of the thin disc to be H₀ = 3.1ϵr_s. We axis-symmetrically solved the hydrodynamical (HD) equations as follows:

$$\frac{\mathrm{d}\rho }{\mathrm{d}t}+\rho \nabla ·\overrightarrow{v}=0,$$(7)

$$\rho \frac{\mathrm{d}\overrightarrow{v}}{\mathrm{d}t}=-\nabla P+\rho \nabla \psi +\rho {\text{g}}^{\mathrm{rad}},$$(8)

$$\rho \frac{\mathrm{d}}{\mathrm{d}t}(\frac{e}{\rho })=-P\nabla ·\overrightarrow{v}+\rho \mathcal{L},$$(9)

where d/dt($\equiv \partial /\partial +\overrightarrow{v}·▽$) denotes the Lagrangian derivative. The ρ, P, $\overrightarrow{v}$, and e in equations (7)–(9) are density, pressure, velocity, and internal energy, respectively. An equation of state of ideal gases, P=(γ−1)/e, was applied and γ = 5/3 was set. ψ is the BH pseudo-Newtonian potential (Paczyńsky & Wiita 1980) and then ψ=−GM_BH/(r_c−r_s), where G and M_BH are the gravitational constant and the BH mass, respectively, and $r_{\mathrm{c}}=\sqrt{r^2+H_0^2-2{\mathit{rH}}_{0}cos(\pi -\theta )}$ is the distance from the co-ordinate origin (the centre of the BH) to the gases. In this paper, we assume M_BH = 10⁸ M_⊙.

The ℒ in equation (9) is the net cooling rate per unit mass, which is referred to equations (18)–(21) in Proga et al. (2000). The equations given by Proga et al. (2000) include Compton heating or cooling, X-ray ionization and recombination cooling, Bremsstrahlung cooling, and line cooling. The net cooling rate depends on the gas temperature (T), the ionization parameter (ξ), and the characteristic temperature (T_x) of X-ray radiation. The gas temperature is given by T=μm_pP/k_Bρ, where m_p, k_B, and μ are the proton mass, the Boltzmann constant, the mean molecular weight, respectively. The ionization parameter, ξ, is defined as ξ=e^−τ_xL_cor/(r²n), where τ_x is the optical depth of the X-ray and n is number density (n=ρ/μm_p) of the gases. We followed Proga et al. (2000) in setting μ = 1 and T_x = 10⁸ K.

The g^rad in equation (8) is the acceleration produced by both the line force and the Compton-scattering force. The ith component of radiation force (g^rad) was evaluated with

$$g_i^{\mathrm{rad}}=e^{-{\tau }_{\mathrm{d}}}\stackrel{}{{\oint }_{\mathrm{\Omega }}}\frac{{\sigma }_{\text{e}}}{\mathit{c}}d{F}_{\mathit{i}}+e^{-{\tau }_{\mathrm{d}}}\stackrel{}{{\oint }_{\mathrm{\Omega }}}\mathcal{M}\frac{{\sigma }_{\mathit{e}}}{\mathit{c}}\mathrm{d}{F}_{\mathit{i}}^{\mathrm{UV}},$$(10)

where the first term on the right side is the Compton-scattering force ($g_{\mathit{i}}^{\mathrm{rad},\mathrm{e}}$) produced by the all the radiation from the disc surface and the second term is the line force ($g_{\mathit{i}}^{\mathrm{rad},\mathrm{l}}$) produced by the all the radiation from the hot region (≥3000 K) of the disc surface. In the equation (10), τ_d is the optical depth of the radiation from the disc, σ_e is the mass-scattering coefficient of electrons, and ℳ is the force multiplier of line force. dF_i and $\mathrm{d}{F}_i^{\mathrm{UV}}$ are the fluxes measured in the comoving frame with gases. According to equation (6), the Compton-scattering force is written as

$$g_{\mathit{i}}^{\mathrm{rad},\mathrm{e}}=e^{-{\tau }_{\mathrm{d}}}\frac{{\sigma }_{\text{e}}}{c}(F_{i,0}-E_0{v}_i-{\displaystyle \sum _j}{v}_j{P}_{\mathit{ij},0}).$$(11)

The details of the calculations about $g_{\mathit{i}}^{\mathrm{rad},\mathrm{l}}$ are referred to in Appendix A. To evaluate τ_x and τ_d, we set the attenuation of the disc radiation to be 0.4 g⁻¹ cm² and the X-ray attenuation to be 0.4 g⁻¹ cm² for ξ≥10⁵ erg s⁻¹ cm but 40 g⁻¹ cm² for ξ<10⁵ erg s⁻¹ cm.

The numerical procedure of solving Equations (7)–(9) was divided into the following steps: (1) solving HD terms, (2) implicitly updating the gas temperature according to the net cooling rate, (3) explicitly updating the gas velocity according to the gravity and the radiation force. In the first step, a computational HD code, called Virginia Hydrodynamics One (VH1), was employed to evaluate HD terms. The VH1 code employs the piecewise parabolic method (Colella & Woodward 1984).

2.2. Model set-up

The computational domain covers the radial range of r_in( = 30r_s)≤r≤r_out( = 1500r_s) and the angular range of 0≤θ≤π/2. We divided the computational domain into 144 × 160 zones. In the r direction, we used 144 zones that were non-uniformly distributed in the range of r_in≤r≤r_out and set the radial size ratio to be (Δr)_i+1/(Δr)_i = 1.04. In the θ direction, we used the 16 uniform zones in an angular range of 0°–15° and the 144 non-uniform zones in an angular range of 15°–90° and set the angular size ratio to be (Δθ)_j+1/(Δθ)_j = 0.970072. The minimum angular size is then Δθ = 0.0117947° at the plane of θ=π/2.

Initially, we followed Proga et al. (1998) in assuming that the gases are in a state of hydrostatic equilibrium in the vertical direction. The initial density distribution is then given by

$$\rho (r,\theta )={\rho }_{\mathrm{d}}\text{exp}(-\frac{{\mathit{GM}}_{\mathrm{bh}}}{2c_{\mathrm{s},\mathrm{d}}^{2}r(1-\frac{r_{\mathrm{s}}}{r}{)}^2{\text{tan}}^2(\theta )}),$$(12)

where c_s,d and ρ_d are the isothermal sound speed and the density at the disc surface, respectively. We used the local effective temperature (T_eff(r_d) = (πI_d/σ)^1/4) at the disc surface to determine the isothermal sound speed. The initial temperature (T(r,θ)) of gases was simply set to be T(r,θ) = T_eff(r_d). The initial radial and polar velocities were set to zero. The rotational velocity is given by $v_{\varphi }(r,\theta )=\sqrt{{\mathit{GM}}_{\mathrm{bh}}/r}\text{sin}\theta r/(r-r_{\mathrm{s}})$ according to force balance.

For the boundary conditions, we applied the outflow boundary condition at the inner and outer radial boundaries and employed the axially symmetric boundary condition at the pole. At the plane of θ=π/2, we fixed density and temperature during the simulations. We set the density to be ρ_d = 10⁻¹² g cm⁻³ and the temperature to be T_eff(r_d).

3. Result

Table 1 lists the basic parameters of models and Table 2 summarizes our runs. In Table 2, columns (1) and (2) give run names and the disc luminosity in units of Eddington luminosity, respectively, columns (3)–(6) give the time-average quantities of winds at the outer boundary, such as the mass outflow rate (Ṁ_w), the momentum flux (P_m), the kinetic energy flux (P_k), and the thermal energy flux P_th, and columns (7)–(8) give the azimuth range of winds and the maximum velocity, respectively. We divided our runs into three groups, i.e. A1–A8, B1–B8, and C1–C8. The A-group runs (runs A1-A8) ignore the special relativity effects caused by disc rotation and gas motion. When we set the denominator on the right of equation (2) to be 1, we ignored the special relativity effect caused by disc rotation. When we ignored the radiation-drag term (−E₀v_i−∑_jv_jP_ij,0) in equation (6), we ignored the special relativity effects caused by gas motion. Therefore, the A-group runs are identical to the models proposed by Proga et al. (2000). The B-group runs (runs B1–B8) take into account the special relativity effects caused by disc rotation, while ignoring the radiation-drag term in equation (6). The C-group runs (runs C1-C8) take into account all special relativity effects, i.e. the special relativity effects caused by disc rotation and the radiation-drag effect.

3.1. Fiducial runs

Figure 1 shows the time evolution of the mass outflow rate (Ṁ_w) at the outer boundary for runs A6, B6, and C6. From Figure 1, a quasi-steady state can quickly be reached. Therefore, time averaging is implemented over the time interval of 1.0–1.5 T_orb, where T_orb is the Keplerian orbital period at the outer boundary.

Fig. 1. Time evolution of the mass outflow rate (Ṁw) at the outer boundary for runs A6, B6, and C6. The time unit (Torb) is the Keplerian orbital period at the outer boundary.

Figure 2 shows the poloidal distribution of wind density for runs A1, B1, C1, A6, B6, and C6. As is shown Figure 2, when the disc luminosity is equal to 0.1L_Edd, runs B1 and C1 cannot generate winds that reach the outer boundary. In comparison to the wind observed in run A1, the failed wind is closer to the surface of disc. When the disc luminosity is higher then 0.2L_Edd, all the runs generate high-velocity winds that can run away out the outer boundary. Observations have shown that UFOs are also detected in low-Eddington AGNs, such as NGC 2992, which has a luminosity of 0.02 L_Edd (Marinucci et al. 2018). Our results imply that for objects with a luminosity lower than 0.1L_Edd, the line force plays little role in launching powerful outflows and that magnetic driving is a potential mechanism.

Fig. 2. Two-dimensional structure of winds for runs A1, B1, C1, A6, B6, and C6. In panels A–F, colors mean the density distribution and arrows mean the poloidal velocity. The structure of winds is obtained by averaging over the time interval of 1.0–1.5 Torb.

From Figure 2, we can see that the winds in runs B6 and C6 are closer to the disc surface than the winds in run A6. To understand this result, we drew Figure 4, which shows the ratio of the radiation fluxes and the line-force multiplier. Panels (A) and (B) in Figure 4 imply that when the special relativity effects (i.e. the Doppler effect) caused by disc rotation are taken into account, the radiation flux strengthens. However, as is shown in panel (C), when the special relativity effects are taken into account, the line-force multiplier is reduced over the range of ∼83°–90°. As a result, the gases in runs B6 and C6 are not lifted up to a higher latitude. In comparing runs B6 and C6, the line-force multiplier for run C6 is significantly weaker than that for run B6 over the range of ∼87° to 90°. This results in the gases in run C6 being lifted up to a lower latitude than the ones in run B6.

Figure 3 shows the poloidal distribution of the ionization parameter of winds for runs A6, B6, and C6. As is shown in Figure 3, the winds are extremely ionized in the polar region due to a very low column density. In the mid- to low-latitude regions, the ionization parameter undergoes a sudden change, which implies that a corresponding abrupt variation also occurs in X-ray radiation. The reasons are as follows. When the X-ray photons emitted from the central corona are shielded by the dense gases in the inner region at a specific angular position, the gases at the corresponding angular position exhibit low ionization. When the gases in the inner region are not dense enough to effectively shield the X-ray photons, the gases in the outer region are effectively ionized. Due to the scattered X-ray photons being neglected in our simulations, the high- and low-ionization regions exhibit a filament structure in the mid- to low-latitude regions. The distribution of the dense gases in the inner region evolves over time, which causes the angular position of filaments to change with time. After time-averaging, the ionization parameter also exhibits discontinuous variations in the mid- to low-latitude regions.

Fig. 3. Two-dimensional distribution of the ionization parameter for runs A6, B6, and C6. The two-dimensional distribution is obtained by averaging over the time interval of 1.0–1.5 Torb.

Fig. 4. The ratio of the radiation fluxes and the line-force multiplier. Panel (A) shows the ratio of the radial radiation fluxes at the outer boundary. Panel (B) shows the ratio of the angular radiation fluxes at r = 80rs. Panel (C) shows the angular distribution of the line-force multiplier of runs A6, B6, and C6 at r = 80rs. The ratio in panels (A) and (B) is calculated by dividing the radiation flux that incorporates the Doppler effect due to disc rotation by the radiation flux that excludes the Doppler effect.

For runs A6, B6, and C6, we calculated the radial dependence of the mass outflow rate (Ṁ_w(r)), the momentum flux (P_m(r)) of winds, and the kinetic power (P_k(r)) of winds. According to Table 2, the power of winds is mainly determined by the kinetic power and then the thermal energy flux can be ignored. Here, we investigate the radial dependence of the thermal energy of winds. Ṁ_w(r), P_m(r), and P_k(r) are given by

$${\mathit{Ṁ}}_{\mathrm{w}}(r)=4\pi r^2{\displaystyle {{\displaystyle \int }}_{0^{°}}^{89.9^{°}}}\rho \hspace{0.5em}\text{max}({\mathit{v}}_r,0)\hspace{0.5em}\text{sin}\theta \hspace{0.5em}\mathrm{d}\theta ,$$(13)

$$P_{\mathrm{m}}(r)=4\pi r^2{\displaystyle {{\displaystyle \int }}_{0^{°}}^{89.9^{°}}}\rho \hspace{0.5em}\text{max}({\mathit{v}}_r^2,0)\hspace{0.5em}\text{sin}\theta \hspace{0.5em}\mathrm{d}\theta ,$$(14)

and

$$P_{\mathrm{k}}(r)=2\pi r^2{\displaystyle {{\displaystyle \int }}_{0^{°}}^{89.9^{°}}}\rho \hspace{0.5em}\text{max}({\mathit{v}}_r^3,0)\hspace{0.5em}\text{sin}\theta \hspace{0.5em}\mathrm{d}\theta .$$(15)

Figure 5 shows the time-averaged Ṁ_w(r), P_m(r), P_k(r). As is shown in Figure 5, Ṁ_w(r), P_m(r), P_k(r) rapidly increase inside ∼40 r_s, and then Ṁ_w(r) almost stays constant outside ∼100 r_s. This implies that winds originates from the disc within 100 r_s and most of the material in the wind comes from discs within 40 r_s. P_m(r) and P_k(r) slowly increase on the range of ∼40–300 r_s, and then slightly increase outside ∼300 r_s, which indicates that the winds will be continuously accelerated within the computation region, and the region within ∼300 r_s is an important region.

Fig. 5. Radial dependence of time-average quantities. Panel (A): the mass outflow rate; panel (B): the kinetic power of winds; panel (C): the momentum flux of winds.

Figure 6 shows the angular dependence of wind properties at the outer boundary (r = 1500r_s). In Figure 6, panel (A) shows the angular distribution of the column density ($N_{\text{H}}={\int }_{30r_s}^{1500r_s}\frac{\rho (r,\theta )}{\mu m_{\mathrm{p}}}\mathrm{d}r$). Panel (A) shows that the column density is higher than 10²² cm⁻² when θ>∼48° for run A6, θ>∼60° for run B6, and θ>∼53° for run C6, respectively. At θ=∼90°, the sudden increase in the column density is caused by the accumulation of material on the disc surface. At θ=∼80°–90°, the matter blown off the disc surface significantly contributes to the column density, while its radial velocity remains relatively low. As is shown in panel (A), the matter distribution of run C6 is located closer to the disc surface compared to runs A6 and B6. panel (B) shows the angular distribution of the density at the outer boundary. As is shown in panel (B), the maximum density of winds in runs B6 and C6 is slightly lower than that in run A6. In run B6, the maximum density reaches approximately 85% of that in run A6, whereas in run C6, it is about 42% of run A6. When the special relativistic effects are taken into account, the density of winds is reduced. Panel (C) shows the angular distribution of the radial velocity at the outer boundary. For run A6, the radial velocity over 41°<θ<67° is higher then 10⁴ km s⁻¹. For run B6, the radial velocity higher then 10⁴ km s⁻¹ is located within the angle range of 48°<θ<79°. For run C6, the radial velocity higher then 10⁴ km s⁻¹ is located within the angle range of 58°<θ<73°. For run C6, the maximum radial velocity at the outer boundary reaches 0.14c, which is lower than that in runs A6 and B6. In run B6, the maximum radial velocity reaches approximately 70% of that in run A6, whereas in run C6, it is about 29% of run A6. Panel (D) shows the angular distribution of the mass flux density at the outer boundary. As is shown in panel (D), the angle range of winds at the outer boundary is about 10°–20°. In run B6, the maximum value of the mass flux density reaches approximately 75% of that in run A6, whereas in run C6, it is about 26% of run A6. In runs B6 and C6, the reduction in the mass flux density, as compared to run A6, is primarily attributed to a decrease in the radial velocity.

Fig. 6. Angular profiles of time-averaged variables. Panel (A): the angular profiles of the column density; Panel (B): the angular profiles of the density at r = 1500rs; Panel (C): the angular profiles of the radial velocity at r = 1500rs; Panel (D): the angular profiles of the mass flux density at r = 1500rs. In panel (A), the horizontal line means the column density of 1022 cm−2. In panel (C), the horizontal line means the velocity of 104 km s−1.

3.2. Luminosity dependence of wind properties

Figure 7 shows the influences of the disc luminosity on wind properties at the outer boundary (r = 1500r_s), such as the mass outflow rate, the maximum velocity, the kinetic power of winds, and the angle of the winds. When ϵ = 0.1 (i.e. L_d = 0.1L_Edd), the radiation force in run A1 can drive very weak winds, while runs B1 and C1 cannot generate winds that can reach the outer boundary, as is shown in Figure 2. Panel (A) shows that the higher the luminosity, the greater the outflow mass rate. The outflow mass rate in runs C2–C8 is significantly lower than that in runs A2–A8 and is about 10%–65% of that in runs A2–A8. Panel (B) shows that the maximum velocity of winds at the outer boundary is greater than 10⁴ km s⁻¹ and reaches the mildly relativistic velocity in all the runs. When the radiation-drag effect is considered in runs C2–C8, the maximum velocity of winds is significantly reduced, compared with runs A2–A8. In runs C2–C8, when ϵ < 0.4 (i.e. L_d<0.4L_Edd) the maximum velocity is lower than 0.1c; when 0.4≤ϵ<0.8 (i.e. 0.4L_Edd≤L_d<0.8L_Edd) the maximum velocity is not higher than 0.2 c. For run C8, the maximum velocity reaches 0.21 c. According to Equation (6), as the speed of the winds increases the radiation-drag term becomes important. Comparing runs B6–B8 with runs C6–C8 (also see Table 2), the wind maximum speed is substantially reduced when the radiation-drag effect is taken into account. Therefore, the higher the wind speed, the more significant the radiation-drag effect. Panel (D) shows that the incline angle of winds in runs B2–B8 and C2–C8 is smaller than that in A2–A8, indicating that the winds in runs B2–B8 and C2–C8 are closer to the disc surface than in A2–A8.

Fig. 7. The influence of the disc luminosity on the wind properties at r = 1500rs, such as the mass outflow rate (Ṁw; panel A), the maximum velocity (vmax; panel B), the kinetic power of winds(Pk; panel C), and the angle of winds (panel D).

4. Application to UFOs

Tombesi et al. (2010) first define UFOs as the highly ionized absorbers with a velocity above 10⁴ km s⁻¹. The ionization parameter (ξ) and column density of UFOs are distributed in the range of log(ξ/(erg s⁻¹ cm))∼3−6 and log(N_H/cm⁻²)∼22−24, respectively (Tombesi et al. 2010). To investigate whether UFOs could be detected from our models or not, we picked out the matter with 3≤log(ξ/(erg s⁻¹ cm))≤6 and v_r≥10⁴ km s⁻¹ from each snapshot, then calculated the column density of the matter, and finally time-averaged the column density. We calculated the angular distribution of the column density for runs A6 and C6 and plot it in Figure 8. Figure 8 shows the angular profile of the column density of the matter with 3≤log(ξ/(erg s⁻¹cm))≤6 and v_r≥10⁴ km s⁻¹ and also shows the angular profile of the maximum radial velocity of the matter with 3≤log(ξ/(erg s⁻¹cm))≤6. At a different angle position, the maximum radial velocity may occur at a different radius. The shaded areas in the Figure 8 indicate where UFOs could form. For run A6, the column density is higher than 10²² cm⁻² over ∼50°–72°, where the UFOs could form and the maximum radial velocity of UFOs is about 0.25c. For run C6, the UFOs could form over the range of ∼64°–73° and the maximum radial velocity of UFOs is about 0.14c. Therefore, the special relativistic effects are taken into account, the angular range of UFOs formation is reduced and their maximum radial velocity is also reduced.

Fig. 8. Angular profile of the column density and the maximum radial velocity in runs A6 (left panel) and C6 (right panel). The doted blue line means the column density of the matter with 3≤log(ξ/(erg s−1cm))≤6 and vr≥104 km s−1 and the dashed red line means the maximum radial velocity of the matter with 3≤log(ξ/(erg s−1cm))≤6. The red solid line and the blue solid line show the line of vr,max = 104 km s−1 and NH = 1022 cm−2, respectively. The shaded areas indicate where UFOs could form.

The UFOs are detected not only in a substantial proportion of radio-quiet AGNs but also in a limited sample of radio-loud AGNs (Tombesi et al. 2012, 2014). In radio-loud AGNs, the UFOs are detected at inclination angles of ∼10°–70° relative to jets (Tombesi et al. 2014). According to our simulations, we do not expect to observe the line-force-driven UFOs in low-inclination galaxies, such as type-1 Seyfert galaxies. Therefore, there is a significant discrepancy between the UFOs in radio-loud AGNs and the line-force-driven UFOs. To understand the UFOs in radio-loud AGNs, a magnetic-driven model should be suggested (Fukumura et al. 2014, 2018; Yang 2021). Yang (2021) suggests that disc winds, driven by both the line force and the magnetic field, can be used to understand the UFOs in radio-loud AGNs. In Yang (2021) simulations, the UFOs are observed in the angular range of ∼10°–75° relative to jets. The simulations in the present work do not take into account the effect of magnetic field. When the effect of the magnetic field is further incorporated into our simulations, the angular range of visibility of the predicted UFOs is expected to increase. In the future, it will be necessary to study the influence of special relativity effects on the winds driven by the magnetic field and line force together. We also note that the special relativistic effects are essentially negligible in pure magnetically driven winds.

The UFOs are also known to be highly variable with an unknown cadence. To show the variability of UFOs, we present Figure 9, which shows the angular profile of the column density and the maximum radial velocity of UFOs in runs C6 at three time points. The shaded areas in Figure 9 indicates where UFOs could form. From Figure 9, we can see that there is slight change in the angle position of formation of the UFOs. This implies that the observed UFOs in our simulations slightly swing. The angular profile of the column density and the maximum radial velocity of UFOs exhibit subtle variations over a short timescale. This can result in the observational features of the theoretical UFOs remaining practically unchanged. Based on the current numerical simulations, a conjecture is that the changes in the observational UFOs may originate from other factors. For example, outflows may induce changes in the accretion flows, which in turn could lead to variation in the outflows themselves (Nomura et al. 2020). Alternatively, changes occur in the X-ray radiation due to scattering, absorption, and re-emission. The local X-ray radiation determines the gas ionization state and subsequently affects the strength of line force. In order to study the time dependence of the winds launched from the AGN disc, Dyda et al. (2024) treated the time-dependent radiation transfer of the X-ray radiation field in AGN winds by directly solving the grey (frequency-averaged) time-dependent radiation transport equation. They found that an episodic wind can form.

Fig. 9. Angular profile of the column density and the maximum radial velocity in runs C6 at t = 1.0 (panel A), 1.25 (panel B), and 1.5 Torb (panel C). The meanings of the various lines in the figure are the same as those in Figure 8. The shaded areas indicate where UFOs could form.

We also note that the angular range of the formation of UFOs corresponds to a Compton-thick regime, in which scattering can be the dominant emission mechanism. In that case, a significant number of scattered X-ray photons escape from the sides of UFOs, while a smaller quantity of scattered photons could fill in the absorption feature of UFOs. Therefore, a direct consequence would be the dilution of UFO features, i.e. the UFOs absorption line becomes shallower or weaker. The quantitative calculation of the UFO features needs to implement radiative transfer. Based on the post-processing approach, it is feasible to employ the Monte Carlo method to calculate the UFO features. However, this calculation is beyond the scope of the current work.

5. Conclusions

In AGNs, when the disc winds driven by line forces have mildly relativistic velocities, the special relativistic effects become non-negligible. The special relativistic effects are reflected as follows: (1) the influence of the disc rotation on the radiation field, and (2) the radiation-drag effect. We have performed a series of numerical simulations to investigate the influence of the special relativistic effects on the line-force-driven winds. Compared with the case without special relativistic effects, when special relativistic effects are considered in our simulations, the kinetic power of the line-force-driven winds is significantly reduced. Here, we summarise the details of our conclusions as follows:

(1) The special relativistic effects enhance the minimum disc luminosity required to launch line-force-driven winds, raising the luminosity threshold. When the special relativistic effects are not considered, the disc with L_d≥0.1 L_Edd can produce winds that escape from the gravity of the BH. However, when the influence of the disc rotation on the radiation field is considered, the disc with L_d≥0.2 L_Edd can produce the winds that escape from the gravity of the BH. The disc of L_d = 0.1 L_Edd cannot produce winds that reach an infinite distance.

(2) The influence of the disc rotation on the radiation field causes the line-force-driven winds to be closer to the disc surface and weakens the winds. The maximum speed of winds in the runs of L_d>0.2 L_Edd is reduced by ∼20 percent–70 percent when the radiation-drag effect is taken into account. The radiation-drag effect further weakens the winds. The higher the wind speed, the more significant the radiation-drag effect. For example, compared with the case without the radiation-drag effect, the maximum speed of winds in the runs of L_d>0.5 L_Edd is reduced by ∼60 percent–70 percent when the radiation-drag effect is considered.

(3) The mass outflow rate is significantly reduced due to the special relativistic effects.

Acknowledgments

This work is supported by Chongqing Natural Science Foundation (grant CSTB2023NSCQ-MSX0093) and the Natural Science Foundation of China (grant 12347101). D. Bu is supported by the Natural Science Foundation of China (grant 12173065, 12133008, 12192220, 12192223).

References

Appendix A: Calculation of the line force

Here, we give the calculation of the line force ($g_{\mathit{i}}^{\mathrm{rad},\mathrm{l}}$) in equation (10). The line force exerting on a unit mass is written as

$$g_{\mathit{i}}^{\mathrm{rad},\mathrm{l}}=\frac{e^{-{\tau }_{\mathrm{d}}}{\sigma }_{\text{e}}}{\mathit{c}}(f_{1,i}-v_i{f}_2-{\displaystyle \sum _j}{v}_j{f}_{3,\mathit{ij}}),$$(A.1)

where f_1,i, f₂, and f_3,ij is defined as:

$$f_{1,i}={\oint }_{\mathrm{\Omega }}\frac{\mathcal{M}{I}_{\mathrm{d}}{l}_i}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}d\mathrm{\Omega },$$(A.2)

$$f_2=\frac{1}{c}{\oint }_{\mathrm{\Omega }}\frac{\mathcal{M}{I}_{\mathrm{d}}}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}d\mathrm{\Omega },$$(A.3)

and

$$f_{3,\mathit{ij}}=\frac{1}{c}{\oint }_{\mathrm{\Omega }}\frac{\mathcal{M}{I}_{\mathrm{d}}{l}_i{l}_j}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}d\mathrm{\Omega }.$$(A.4)

In equation (A.2)–(A.4), $\overrightarrow{\mathit{l}}$ and $\overrightarrow{{\mathit{v}}_{\mathrm{d}}}$ are the direction-cosine vector pointing toward the rest observer (point W) from the disc surface (point D) and the disc velocity, respectively. In spherical co-ordinates, we are given by

$$\overrightarrow{l}=(\frac{r-r_{\mathrm{d}}\text{sin}\theta \text{cos}{\varphi }_{\mathrm{D}}}{d_{\mathrm{D}}},\frac{-r_{\mathrm{d}}\text{cos}\theta \text{cos}{\varphi }_{\mathrm{D}}}{d_{\mathrm{D}}},\frac{r_{\mathrm{d}}\text{sin}{\varphi }_{\mathrm{D}}}{d_{\mathrm{D}}}),$$(A.5)

and

$$\overrightarrow{{\mathit{v}}_{\mathrm{d}}}=(v_{\mathrm{k}}\text{sin}\theta \text{sin}{\varphi }_{\mathrm{D}},v_{\mathrm{k}}\text{cos}\theta \text{sin}{\varphi }_{\mathrm{D}},v_{\mathrm{k}}\text{cos}{\varphi }_{\mathrm{D}}),$$(A.6)

where $d_{\mathrm{D}}=\left|\mathrm{DW}\right|=(r_{\mathrm{d}}^2+r^2-2{\mathit{rr}}_{\mathrm{d}}\text{sin}\theta \text{cos}{\varphi }_{\mathrm{D}}{)}^{\frac{1}{2}}$ (see figure A.1).

Fig. A.1. Schematic diagram for calculating the radiation field.

Based on the Sobolev approximation, the line-force multiplier (ℳ(t)) in equation (A.2)–(A.4) is a function of the optical depth parameter $t=\frac{{\sigma }_e\rho v_{\mathrm{th}}}{\left|{\mathit{dv}}_l/\mathit{dl}\right|}$, where v_th and $\frac{{\mathit{dv}}_l}{\mathit{dl}}$ are the thermal velocity and the velocity gradient along the line of sight, respectively. We set v_th = 20km/s, which corresponds to the gas temperature of 2.5 ×10⁴K. We employ the CAK approach (Castor et al. 1975, CAK) modified by Owocki et al. (1988, CAK) to calculate the line-force multiplier, that is given by

$$\mathcal{M}(t)=k{t}^{-\alpha }\left[\frac{(1+{\tau }_{\max }{)}^{1-\alpha }-1}{{\tau }_{\max }^{1-\alpha }}\right],$$(A.7)

where α is the ratio of optically thick to optically thin spectral lines, k is a parameter proportional to the total number of lines, τ_max=tη_max and η_max is a parameter that determines the maximum value of the line force multiplier (ℳ_max). In all our simulations, we set α = 0.6. According to Stevens & Kallman (1990), t and η_max are respectively give by

$$k=0.03+0.385\exp (-1.4{\xi }^{0.6})$$(A.8)

and

$${\text{log}}_{10}{\eta }_{\max }=\{\begin{array}{cc}\hfill 6.9\exp (0.16{\xi }^{0.4})\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\mathrm{for}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}{\log }_{10}(\xi )\le 0.5& \hfill \\ \hfill 9.1\exp (-7.96\times {10}^{-3}\xi )\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\mathrm{for}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}{\log }_{10}(\xi )>0.5& \hfill \end{array}\hspace{0.5em}\hspace{0.5em}.$$(A.9)

For dv_l/dl, we can write as (Rybicki & Hummer 1978)

$$\frac{{\mathit{dv}}_l}{\mathit{dl}}=Q\equiv {\displaystyle \sum _{i,j}}\frac{1}{2}(\frac{\partial v_i}{\partial r_j}+\frac{\partial v_j}{\partial r_i})l_i{l}_j,$$(A.10)

where r_i is the components of the position vector (r) and l_i is the components of the unit vector ($\hat{\text{l}}$) of the direction D toward W (see figure A.1), respectively. We assume that dv_l/dl is dominated by the gradient of the velocity along the vertical direction and then dv_l/d_l is given by

$$\frac{{\mathit{dv}}_l}{\mathit{dl}}=\frac{{\mathit{dv}}_{\mathrm{z}}}{d\mathrm{z}}{n}_{\mathrm{z}}^2=\left[{\text{cos}}^2\theta \frac{\partial v_r}{\partial r}+{\text{sin}}^2\theta \frac{1}{r}(v_r+\frac{\partial v_{\theta }}{\partial \theta })+\text{sin}\theta \text{cos}\theta (\frac{v_{\theta }}{r}-\frac{\partial v_{\theta }}{\partial r}-\frac{\partial v_r}{r\partial \theta })\right]n_{\mathrm{z}}^2,$$(A.11)

where $n_{\mathrm{z}}=\frac{r\text{cos}\theta }{d_{\mathrm{D}}}$. For the element D, its area dS=r_ddr_ddϕ_D. The solid angle subtended by the element D towards the observer standing at the point W is $d\mathrm{\Omega }=\frac{r\text{cos}\theta \mathit{dS}}{d_{\mathrm{D}}^3}$. Therefore, f_1,i can be written

$$f_{1,i}=\frac{9ϵL_{\mathrm{Edd}}{r}_{\mathrm{s}}}{4{\pi }^2}k({\sigma }_{\mathrm{e}}\rho v_{\mathit{th}}|\frac{{\mathit{dv}}_{\mathrm{z}}}{d\mathrm{z}}{|}^{-1}{)}^{-\alpha }{\displaystyle {\int }_{3r_{\text{s}}}^{1500r_{\text{s}}}}{\displaystyle {\int }_0^{2\pi }}\frac{r{\text{cos}}_{\theta }}{r_{\mathrm{d}}^2{d}_{\mathrm{D}}^3}\left(1-\sqrt{\frac{3r_{\mathrm{s}}}{r_{\mathrm{d}}}}\hspace{0.5em}\right)\left[\frac{(1+{\tau }_{\max }{)}^{(1-\alpha )}-1}{{\tau }_{\max }}\right](\frac{r{\text{cos}}_{\theta }}{d_{\mathrm{D}}}{)}^{2\alpha }\frac{l_i}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}{\mathit{dr}}_{\mathrm{d}}d{\varphi }_{\mathrm{D}},$$(A.12)

In the above integrand, the τ_max term is time-dependent and then strictly this term cannot be removed from the integrand. For removing the τ_max term from the integrand, we refer to Proga et al. (1998) and then make an approximation that n_z = 1 for τ_max. We introduce a new variable ${\tau }_{\max }^{\prime }={\sigma }_{\mathrm{e}}\rho v_{\mathit{th}}|{\mathit{dv}}_{\mathrm{z}}/d\mathrm{z}{|}^{-1}{\eta }_{\max }$ and rewrite equation (A.12) as

$$f_{1,i}=\frac{9ϵL_{\mathrm{Edd}}{r}_{\mathrm{s}}}{4{\pi }^2}k({\sigma }_{\mathrm{e}}\rho v_{\mathit{th}}|\frac{{\mathit{dv}}_{\mathrm{z}}}{d\mathrm{z}}{|}^{-1}{)}^{-\alpha }\left[\frac{(1+{\tau }_{\max }^{\prime }{)}^{(1-\alpha )}-1}{{\tau }_{\max }^{\prime }}\right]{\displaystyle {\int }_{3r_{\text{s}}}^{1500r_{\text{s}}}}{\displaystyle {\int }_0^{2\pi }}\frac{r{\text{cos}}_{\theta }}{r_{\mathrm{d}}^2{d}_{\mathrm{D}}^3}\left(1-\sqrt{\frac{3r_{\mathrm{s}}}{r_{\mathrm{d}}}}\hspace{0.5em}\right)(\frac{r{\text{cos}}_{\theta }}{d_{\mathrm{D}}}{)}^{2\alpha }\frac{l_i}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}{\mathit{dr}}_{\mathrm{d}}d{\varphi }_{\mathrm{D}}.$$(A.13)

Similarly, f₂ and f_3,ij can be expressed as

$$f_2=\frac{9ϵL_{\mathrm{Edd}}{r}_{\mathrm{s}}}{4{\pi }^{2}c}k({\sigma }_{\mathrm{e}}\rho v_{\mathit{th}}|\frac{{\mathit{dv}}_{\mathrm{z}}}{d\mathrm{z}}{|}^{-1}{)}^{-\alpha }\left[\frac{(1+{\tau }_{\max }^{\prime }{)}^{(1-\alpha )}-1}{{\tau }_{\max }^{\prime }}\right]{\displaystyle {\int }_{3r_{\text{s}}}^{1500r_{\text{s}}}}{\displaystyle {\int }_0^{2\pi }}\frac{r{\text{cos}}_{\theta }}{r_{\mathrm{d}}^2{d}_{\mathrm{D}}^3}\left(1-\sqrt{\frac{3r_{\mathrm{s}}}{r_{\mathrm{d}}}}\hspace{0.5em}\right)(\frac{r{\text{cos}}_{\theta }}{d_{\mathrm{D}}}{)}^{2\alpha }\frac{1}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}{\mathit{dr}}_{\mathrm{d}}d{\varphi }_{\mathrm{D}},$$(A.14)

and

$$f_{3,\mathit{ij}}=\frac{9ϵL_{\mathrm{Edd}}{r}_{\mathrm{s}}}{4{\pi }^{2}c}k({\sigma }_{\mathrm{e}}\rho v_{\mathit{th}}|\frac{{\mathit{dv}}_{\mathrm{z}}}{d\mathrm{z}}{|}^{-1}{)}^{-\alpha }\left[\frac{(1+{\tau }_{\max }^{\prime }{)}^{(1-\alpha )}-1}{{\tau }_{\max }^{\prime }}\right]{\displaystyle {\int }_{3r_{\text{s}}}^{1500r_{\text{s}}}}{\displaystyle {\int }_0^{2\pi }}\frac{r{\text{cos}}_{\theta }}{r_{\mathrm{d}}^2{d}_{\mathrm{D}}^3}\left(1-\sqrt{\frac{3r_{\mathrm{s}}}{r_{\mathrm{d}}}}\hspace{0.5em}\right)(\frac{r{\text{cos}}_{\theta }}{d_{\mathrm{D}}}{)}^{2\alpha }\frac{l_i{l}_j}{(1-\frac{\overrightarrow{{\mathit{v}}_{\mathrm{d}}}·\overrightarrow{\mathit{l}}}{c}{)}^4}{\mathit{dr}}_{\mathrm{d}}d{\varphi }_{\mathrm{D}}.$$(A.15)

All Tables

All Figures

	Fig. 1. Time evolution of the mass outflow rate (Ṁw) at the outer boundary for runs A6, B6, and C6. The time unit (Torb) is the Keplerian orbital period at the outer boundary.
In the text

	Fig. 2. Two-dimensional structure of winds for runs A1, B1, C1, A6, B6, and C6. In panels A–F, colors mean the density distribution and arrows mean the poloidal velocity. The structure of winds is obtained by averaging over the time interval of 1.0–1.5 Torb.
In the text

	Fig. 3. Two-dimensional distribution of the ionization parameter for runs A6, B6, and C6. The two-dimensional distribution is obtained by averaging over the time interval of 1.0–1.5 Torb.
In the text

Fig. 4. The ratio of the radiation fluxes and the line-force multiplier. Panel (A) shows the ratio of the radial radiation fluxes at the outer boundary. Panel (B) shows the ratio of the angular radiation fluxes at r = 80rs. Panel (C) shows the angular distribution of the line-force multiplier of runs A6, B6, and C6 at r = 80rs. The ratio in panels (A) and (B) is calculated by dividing the radiation flux that incorporates the Doppler effect due to disc rotation by the radiation flux that excludes the Doppler effect.

In the text

	Fig. 5. Radial dependence of time-average quantities. Panel (A): the mass outflow rate; panel (B): the kinetic power of winds; panel (C): the momentum flux of winds.
In the text

Fig. 6. Angular profiles of time-averaged variables. Panel (A): the angular profiles of the column density; Panel (B): the angular profiles of the density at r = 1500rs; Panel (C): the angular profiles of the radial velocity at r = 1500rs; Panel (D): the angular profiles of the mass flux density at r = 1500rs. In panel (A), the horizontal line means the column density of 1022 cm−2. In panel (C), the horizontal line means the velocity of 104 km s−1.

In the text

	Fig. 7. The influence of the disc luminosity on the wind properties at r = 1500rs, such as the mass outflow rate (Ṁw; panel A), the maximum velocity (vmax; panel B), the kinetic power of winds(Pk; panel C), and the angle of winds (panel D).
In the text

Fig. 8. Angular profile of the column density and the maximum radial velocity in runs A6 (left panel) and C6 (right panel). The doted blue line means the column density of the matter with 3≤log(ξ/(erg s−1cm))≤6 and vr≥104 km s−1 and the dashed red line means the maximum radial velocity of the matter with 3≤log(ξ/(erg s−1cm))≤6. The red solid line and the blue solid line show the line of vr,max = 104 km s−1 and NH = 1022 cm−2, respectively. The shaded areas indicate where UFOs could form.

In the text

	Fig. 9. Angular profile of the column density and the maximum radial velocity in runs C6 at t = 1.0 (panel A), 1.25 (panel B), and 1.5 Torb (panel C). The meanings of the various lines in the figure are the same as those in Figure 8. The shaded areas indicate where UFOs could form.
In the text

	Fig. A.1. Schematic diagram for calculating the radiation field.
In the text