# Effect of radiation drag on the line-force-driven winds

Bei-Chuan Wang1, Xiao-Hong Yang1, De-Fu Bu2, and Shu-Su Huang1
1 Department of Physics and Chongqing Key Laboratory for Strongly Coupled
Physic, Chongqing University, Chongqing 400044, China
2 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, China Corresponding author 1:
<EMAIL_ADDRESS>(XH)Corresponding author 2<EMAIL_ADDRESS>(DF)

###### Abstract

Ultra-fast outflows (UFOs) with mildly relativistic velocities are measured
using the X-ray spectra of radio-quiet and -loud active galactic nuclei
(AGNs). In general, UFOs are believed to be generated from the accretion disk
around a black hole (BH). A line-force driving model is suggested to be the
mechanism to drive UFOs from the accretion disk. In this paper, we use the
non-hydrodynamic approach to examine the influences of radiation-drag effects
on the line-force-driven winds generated from the accretion disk. We find that
the radiation-drag effects can significantly weaken the line-force-driven
winds. Compared with the case without the radiation-drag effects, when the
radiation-drag effects are considered, the maximum speed of winds is reduced
by $\sim$60%–70%, the mass outflow rate is reduced by $\sim$50%–80%, and the
kinetic power is reduced by about an order of magnitude. The radiation-drag
effects narrow the area where the winds are generated.

###### keywords:

accretion, accretion discs–galacties: active–quasars: general–ISM: jets and
outflows.

††pagerange: Effect of radiation drag on the line-force-driven
winds–LABEL:lastpage††pubyear: 20**

##  1 INTRODUCTION

Blueshifted absorption lines are frequently detected on the ultraviolet (UV)
and X-ray spectra of luminous AGNs (Churchill et al. 1999; Narayanan et al.
2004; Hamann et al. 1997; Rodríguez Hidalgo et al. 2011; Tombesi et al. 2010,
2012; Gofford et al. 2013, 2015). For example, it is reported that 10%–17% of
the CIV absorption lines on the AGN UV spectra are blueshifted at 5000–70,000
km s-1 (Misawa et al. 2007), and the Fe XXV and Fe XXVI absorption lines on
the XMM-Newton spectra are blueshifted at 0.03–0.3 times the speed ($c$) of
light for $>$35 % of radio-quiet AGNs (Tombesi et al. 2011). This implies that
fast outflows/winds are common in AGNs and some winds can reach mildly
relativistic velocities.

The highly blueshifted Fe XXV and Fe XXVI absorption lines imply that the
highly ionized absorbers move fast outward from their nuclei with mildly
relativistic velocities. Such outflows are called UFOs (Tombesi et al. 2010,
2012). Tombesi et al. (2010, 2011) first define UFOs to describe the highly
ionized absorbers with velocities higher than 104 km s-1 and derive the basic
properties of UFOs. The ionization parameter ($\xi$) and column density
($N_{\rm H}$) of UFOs are in the range log($\xi$/(erg s-1 cm))$\sim$3–6 and
log($N_{\rm H}$/(cm-2))$\sim$22-24, respectively (Tombesi et al. 2011). The
UFOs are observed not only in a significant fraction ($>35\%$) of radio-quiet
AGNs (Tombesi et al. 2011) but also in a small sample of radio-loud AGNs
(Tombesi et al. 2014). In general, the UFOs are estimated to be formed in the
interval of $10^{2}$–$10^{4}$ Schwarzschild radius ($r_{\rm s}$) away from the
center BH (Tombesi et al. 2012; Gofford et al. 2015). This implies that the
UFOs may be the winds generated from the accretion disk around the center BH
(e.g. Proga & Kallman 2004; Fukumura et al. 2010; Nomura et al. 2016; Nomura &
Ohsuga 2017; Yang et al. 2021; Yang 2021).

Two kinds of models, i.e. a magnetic-driving model and a line-force driving
model, are proposed to produce UFOs from the accretion disk. The mechanism of
the magnetic-driven winds and jets, called the Blandford & Payne mechanism,
has been developed in previous works (e.g. Blandford & Payne 1982; sakurai
1987; Lovelace et al. 1987; Cao & Spruit 1994; Fukmura et al. 2014, 2018; Chen
et al. 2021). Fukmura et al. (2018) have used a model of magnetic-driven winds
to model the UFOs observed in PDS 456. In the magnetic-driving model, a large-
scale magnetic field is required. However, the formation of the large-scale
magnetic field is still an open issue (e.g. Cao 2011; Cao & Spruit 2013). The
magnetic structure is often taken as a free parameter in the model of
magnetic-driven winds.

A line-force driving model is the other promising model to understand UFOs.
For luminous AGNs, such as quasars, a geometrically thin and optically thick
disk is often proposed to be around the center BH (Shakura & Sunyaev 1973).
The thin disk is luminous and irradiates gases above the thin disk. The
photons emitted from the thin disk are scattered or absorbed by the gases and
therefore the Compton-scattering force and the spectral line force are
produced and exerted on the gases. For the highly ionized gases, the Compton-
scattering force is the main radiation force. When the effective barrier of
gravity and centrifugal force of gases is small, the Compton-scattering force
may drive winds (e.g. Cao 2014; Yang et al. 2018; 2019). For the weakly
ionized gases, the line force caused by the absorption of UV photons becomes
an important radiation force to drive winds (Castor et al. 1975; Murray et al.
1995). The Compton-scattering force cannot drive the winds with mildly
relativistic velocities. The line force cannot drive the highly ionized winds,
because the line force is negligible for the gases with the ionization
parameter higher than 100 erg s-1 cm. For UFOs, their ionization parameter is
much higher than 100 erg s-1 cm. Yang et al. (2021) pointed out that UFOs can
be intermittently accelerated by the line force. When the X-ray photons from
the center corona are shielded by dense gases in the inner region, the degree
of ionization of gases becomes so low that the line force can drive the gases
to form winds. When the line-force driven winds are exposed to the X-ray
photons, the winds become highly ionized and then form the observed UFOs.
Therefore, the line-force driven winds may be the origin of UFOs (Nomura &
Ohsuga 2017; Mizumoto et al. 2021; Yang et al. 2021; Yang 2021)

For the radiation-driven winds, such as the line-force-driven winds, if they
move at mildly-relativistic velocity in the radiation field, the special
relativistic effects (radiation drag) become important. When the relativistic
effects are taken into account, the wind speed is reduced up to 50% with
respect to the non-relativistic case (Luminari et al. 2020, 2021). Quera-
Bofarull et al. (2021) further found that the line-force-driven winds is
mildly relativistic, with velocity 0.1–0.3 c, when the relativistic effects
are included. In Quera-Bofarull et al.’ work (2021), they took into account a
correction to the radiation flux seen by the high-velocity winds, while they
ignored the influence of the disk rotation on the radiation field in the rest
frame standing at the infinity. Therefore, our motivation in this paper is to
continue to study the influence of the radiation-drag effects on the line-
force-driven winds.

In term of research methods, the line-force-driven winds in AGNs have been
studied not only in radiation hydrodynamic simulations (e.g. Proga et al.
2000; Proga & Kallman 2004; Nomura & Ohsuga 2017; Yang et al. 2021; Yang 2021)
but also in non-hydrodynamic approach (Risaliti & Elvis 2010; Nomura et al.
2013). In the non-hydrodynamic approach, the gas pressure is neglected, gases
are taken as particles, and then the Lagrangian method is employed to model
the line-force-driven winds. Compared with the radiation hydrodynamic
simulations, the non-hydrodynamic approach takes less time in calculation and
then can test more parameter space. Therefore, in this paper, we employ the
non-hydrodynamic approach to study the line-force-driven winds.

Table 1: Summary of Models

Models | $L_{\rm bol}$ | $v_{0}$ | $n_{0}$ | $v_{\rm max}$ | $\dot{M}_{\rm out}$ | $P_{\rm kin}$ | $\theta_{\rm out}$
---|---|---|---|---|---|---|---
| $(L_{\rm Edd})$ | $({\rm m/s})$ | $({\rm cm^{-3}})$ | $(c)$ |  $(M_{\odot}{\rm year}^{-1}$) |  $(L_{\rm Edd}$) | $(^{\circ})$
A0 | $0.01$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.035$ | 0.003 | $9.00\times{10^{-6}}$ | 9.4–10.4
A1 | $0.05$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.176$ | 0.035 | $9.43\times{10^{-4}}$ | 7.3–13.0
A2 | $0.1$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.228$ | 0.069 | $2.27\times{10^{-3}}$ | 5.8–12.6
A3 | $0.2$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.288$ | 0.113 | $4.58\times{10^{-3}}$ | 4.1–12.1
A4 | $0.3$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.316$ | 0.137 | $6.42\times{10^{-3}}$ | 6.8–11.8
A5 | $0.4$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.317$ | 0.153 | $7.74\times{10^{-3}}$ | 7.0–10.2
A6 | $0.5$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.331$ | 0.163 | $9.24\times{10^{-3}}$ | 7.1–11.2
A7 | $0.6$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.353$ | 0.173 | $1.00\times{10^{-2}}$ | 7.3–10.8
A8 | $0.7$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.335$ | 0.190 | $1.10\times{10^{-2}}$ | 7.5–10.9
A9 | $0.8$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.360$ | 0.183 | $1.15\times{10^{-2}}$ | 7.6–10.4
A10 | $0.2$ | $1\times{10^{4}}$ | $5\times{10^{7}}$ | $0.384$ | 0.019 | $8.89\times{10^{-4}}$ | 7.6–14.6
A11 | $0.2$ | $5\times{10^{4}}$ | $5\times{10^{7}}$ | $0.317$ | 0.056 | $2.96\times{10^{-3}}$ | 4.6–12.3
A12 | $0.2$ | $5\times{10^{5}}$ | $5\times{10^{7}}$ | $0.203$ | 0.435 | $9.40\times{10^{-3}}$ | 6.6–11.7
A13 | $0.2$ | $1\times{10^{6}}$ | $5\times{10^{7}}$ | $0.170$ | 0.870 | $1.42\times{10^{-2}}$ | 10.1–11.9
A14 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{7}}$ | $0$ | 0 | $0$ | 0
A15 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{8}}$ | $0.437$ | 0.112 | $1.02\times{10^{-2}}$ | 6.7–12.9
A16 | $0.2$ | $1\times{10^{5}}$ | $5\times{10^{8}}$ | $0.621$ | 0.046 | $2.18\times{10^{-2}}$ | 14.7–18.2
A17 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{9}}$ | $0.909$ | 0.063 | $5.02\times{10^{-2}}$ | 16.2–22.6
B0 | $0.01$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0$ | 0 | $0$ | 0
B1 | $0.05$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0$ | 0 | $0$ | 0
B2 | $0.1$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.08$ | 0.012 | $1.08\times{10^{-4}}$ | 7.4–9.2
B3 | $0.2$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.112$ | 0.029 | $4.36\times{10^{-4}}$ | 6.7–9.6
B3a | $0.2$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.118$ | 0.032 | $4.72\times{10^{-4}}$ | 7.1–10.6
B4 | $0.3$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.129$ | 0.048 | $7.26\times{10^{-4}}$ | 6.2–9.9
B5 | $0.4$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.138$ | 0.063 | $8.38\times{10^{-4}}$ | 5.4–10.1
B6 | $0.5$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.136$ | 0.068 | $1.16\times{10^{-3}}$ | 5.2–9.3
B7 | $0.6$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.150$ | 0.084 | $1.44\times{10^{-3}}$ | 4.5–8.6
B8 | $0.7$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.144$ | 0.087 | $1.46\times{10^{-3}}$ | 4.1–9.0
B9 | $0.8$ | $1\times{10^{5}}$ | $5\times{10^{7}}$ | $0.152$ | 0.087 | $1.57\times{10^{-3}}$ | 4.4–9.6
B10 | $0.2$ | $1\times{10^{4}}$ | $5\times{10^{7}}$ | $0.202$ | 0.010 | $2.88\times{10^{-4}}$ | 4.9–9.8
B11 | $0.2$ | $5\times{10^{4}}$ | $5\times{10^{7}}$ | $0.138$ | 0.024 | $4.67\times{10^{-4}}$ | 5.9–10.1
B12 | $0.2$ | $5\times{10^{5}}$ | $5\times{10^{7}}$ | $0.042$ | 0.031 | $1.25\times{10^{-4}}$ | 7.9–8.7
B13 | $0.2$ | $1\times{10^{6}}$ | $5\times{10^{7}}$ | $0$ | 0 | $0$ | 0
B14 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{7}}$ | $0$ | 0 | $0$ | 0
B15 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{8}}$ | $0.190$ | 0.028 | $1.19\times{10^{-3}}$ | 9.1–12
B16 | $0.2$ | $1\times{10^{5}}$ | $5\times{10^{8}}$ | $0.224$ | 0.019 | $2.18\times{10^{-3}}$ | 12.1–20.4
B17 | $0.2$ | $1\times{10^{5}}$ | $1\times{10^{9}}$ | $0.314$ | 0.024 | $5.40\times{10^{-3}}$ | 17.3–21.8

  * Note. Column (1): the model names; Column (2): the bolometric luminosity; Column (3): the vertical velocity at the streamline foot; Column (4): the initial particle number density of streamlines; Column (5): the fastest velocity along streamlines; Columns (6) and (7): the mass outflow rate and kinetic power of the escape particles, Column (8): the range of the velocity azimuth angles (relative to the disk surface) of the escape particles at the outer boundary, respectively. We set the black hole mass to be $10^{8}M_{\odot}$, and the rate ($f_{X}$) of the X-ray luminosity to the bolometric luminosity to be 0.1, respectively. Models A0–A17 donot include the radiation-drag effects, while models B0–B17 include the radiation-drag effects. For models A14, B0, B1, B13, and B14, their $v_{\rm max}$, $\dot{M}_{\rm out}$, and $P_{\rm kin}$ equal 0, which means that the escape particles are not generated.

Figure 1: The trajectories of winds in models A3 (panel (A)), B3 (panel (B)),
A13 (panel (C)), and B13 (panel (D)).

Figure 2:  Comparison of terms in equation (7). Panel (A) shows the
$(E_{0}v^{R}+v_{j}P_{0}^{Rj})/{F^{R}_{0}}$ and
${E_{0}v^{R}}/{v_{j}P_{0}^{Rj}}$ values along the thick line in panel (B) of
figure 1. Panel (B) shows the $(E_{0}v^{z}+v_{j}P_{0}^{zj})/{F^{z}_{0}}$ and
${E_{0}v^{z}}/{v_{j}P_{0}^{zj}}$ values along the thick line in panel (B) of
figure 1. Panel (C) shows the $(E_{0}v^{\phi}+v_{j}P_{0}^{\phi
j})/{F^{\phi}_{0}}$ and ${E_{0}v^{\phi}}/{v_{j}P_{0}^{\phi j}}$ values along
the thick line in panel (B) of figure 1.

Figure 3:  The influences of the bolometric luminosity ($L_{\rm bol}$) on the
wind properties, such as the maximum speed ($v_{\rm max}$) along streamlines
(Panel (A)), the mass outflow rate ($\dot{M}_{\rm out}$, Panel (B)), the
kinetic power ($P_{\rm kin}$, Panel (C)), the angle of winds ($\theta_{\rm
out}$, Panel (D)).

Figure 4: The influences of the initial velocity ($v_{0}$) on the wind
properties. Panels (A)–(D) are the same as the panels of Figure 3.

Figure 5: The influences of the number density ($n_{0}$) of particles on the
wind properties. Panels (A)–(D) are the same as the panels of Figure 3.

The paper is organized as follows. In section 2, we describe our model and
method. In section 3, we present our results and related discussions. In
section 4, we give conclusions and discussions.

##  2 MODEL AND METHOD

It is believed that a supermassive BH locates at the center of galaxies. In
luminous AGNs, such as quasars, it is often believed that a standard thin disk
exists around the supermassive BH. The standard thin disk is geometrically
thin and optically thick (Shakura & Sunyaev 1973). Observations imply that
there is a hot corona within 10 Schwarzschild radius ($r_{\rm s}$) (Reis &
Miller 2013; Uttley et al. 2014). The gases above the thin disk are irradiated
by the UV photons generated in the thin disk and the X-ray photons generated
in the hot corona, respectively. We trace trajectories of the irradiated gases
in the cylindrical coordinates ($R$,$\phi$,$z$). The trajectory lines are used
to model the winds driven by the radiation emitted by the thin disk. For
simplicity, we assume that the scale height of the thin disk is constant with
the change in radius, set the disk surface to be the plane of $z=0$, and set
the hot corona to locate at the center (i.e. $R=z=0$).

For the line-force-driven winds, their kinetic energy is much stronger than
their thermal energy, which causes that the gas pressure is negligible in
dynamics. Therefore, the line-force-driven winds are taken as particles. We
can calculate the trajectory of a particle by solving the dynamic equations of
the particles as follows:

$\frac{dR}{dt}=v_{\textit{\tiny{R}}},$ (1)
$\frac{d\phi}{dt}=\frac{1}{R}{v_{\phi}},$ (2)
$\frac{dz}{dt}=v_{\textit{\tiny{z}}},$ (3)
$\frac{dv_{\textit{\tiny{R}}}}{dt}=-\frac{GM_{\rm
BH}R}{(R^{2}+z^{2})^{3/2}}+\frac{v_{\phi}^{2}}{R}+\frac{\sigma_{\rm T}}{m_{\rm
p}c}(1+\mathcal{M})F^{\textit{\tiny{R}}}e^{-\tau_{uv}},$ (4)

$\frac{1}{R}\frac{d(Rv_{\phi})}{dt}=\frac{\sigma_{\rm T}}{m_{\rm
p}c}(1+\mathcal{M})F^{\phi}e^{-\tau_{uv}},$ (5)

$\frac{dv_{z}}{dt}=-\frac{GM_{\rm BH}z}{(R^{2}+z^{2})^{3/2}}+\frac{\sigma_{\rm
T}}{m_{\rm p}c}(1+\mathcal{M})F^{z}e^{-\tau_{uv}},$ (6)

where $v_{i}\equiv(v_{\textit{\tiny{R}}},v_{\phi},\text{ and }v_{z})$ is the
velocity of particles, $F^{i}\equiv(F^{\textit{\tiny{R}}},F^{\phi},\text{ and
}F^{z})$ is the radiative flux measured in the comoving frame with the moving
particles, $\mathcal{M}$ is the line-force multiplier, $\tau_{\rm uv}$ is the
ultraviolet optical depth, and $\sigma_{\rm T}$, $m_{\rm p}$, $G$, and $M_{\rm
BH}$ are the Thomson-scattering cross section, the proton mass, the gravity
constant, and the BH mass, respectively. The calculation of $\mathcal{M}$ is
referred to Arnau Quera-Bofarull et al. (2020), we will discuss it in Appendix
A. In equations (4)–(6), we ignore the gravitational force of the thin disk
and the Compton-scattering force of the X-ray photons generated in the hot
corona. The radiation field is divided into two components, i.e the X-ray
radiation generated in the hot corona and the UV radiation generated in the
thin disk. The UV radiation force is an important force of driving gases to
form winds by the Compton-scattering force and the spectral line force. The
X-ray radiation flux from the hot corona is much smaller than the UV radiation
flux from the thin disk. The X-ray radiation is negligible in dynamics, while
the X-ray photons play an important role in ionizing gases. We set the disk
luminosity ($L_{\rm disk}$) to be $L_{\rm disk}=f_{\rm disk}L_{\rm bol}$ and
the X-ray luminosity to be $L_{\rm X}=f_{\rm X}L_{\rm bol}=(1-f_{\rm
disk})L_{\rm bol}$, where $L_{\rm bol}$ is the bolometric luminosity. To study
the effect of radiation drag on the line-force-driven winds, the special
relativistic effects is also included when the radiation flux from the thin
disk is calculated. We have two different frame: the comoving frame with the
moving particles and the rest frame standing at the infinity. In equations
(4)–(6), $F^{i}$ is measured in the comoving frame. The radiation field
generated in the thin disk is transferred into the comoving frame. On the
order of $v/c$, the $i$th component of radiative flux in the comoving frame is
given by

$F^{i}=F^{i}_{0}-E_{0}v^{i}-v_{j}P^{ij}_{0},$ (7)

where $F^{i}_{0}$, $E_{0}$, and $P^{ij}_{0}$ are the $i$th component of
radiative flux, the radiation energy density, and the radiation stress tensor
in the rest frame, respectively. In equation (7), $v^{i}\equiv v_{i}$ (i.e.
$v^{R}=v_{R}$, $v^{z}=v_{z}$, and $v^{\phi}=v_{\phi}$) and the Einstein
summation convention is applied to $v_{j}P^{ij}_{0}$. In the previous works
about the line-force-driven winds, $F^{i}_{0}$ in the rest frame does not
include the Doppler effect of rotating of the disk, while in our calculation
$F^{i}_{0}$, $E_{0}$, and $P^{ij}_{0}$ include the Doppler effect of rotating
of the disk. The calculation of these quantities is referred to Tajima & Fukue
(1998). For the completeness of this paper, we also give the introduction of
the calculation in Appendix B.

In equations (4)–(6), the line-force multiplier $\mathcal{M}$ is a function of
the ionization parameter ($\xi$) and the local optical depth parameter ($t$)
(Rybicki & Hummer 1978), according to the Sobolev approximation. The
ionization parameter $\xi$ is given by

$\xi=\frac{L_{X}\text{exp}(-\tau_{X})}{n(R^{2}+z^{2})},$ (8)

where $n$ is the number density and $\tau_{X}$ is the X-ray optical depth.
According to Quera-Bofarull et al. (2020), $\tau_{X}$ can be given by

$\tau_{X}\equiv\int_{R_{\rm
b}}^{R_{0}}\sec(\theta)m_{p}n_{0}\sigma_{X}dR^{\prime}+\int_{R_{0}}^{R}\sec(\theta)m_{p}n(r)\sigma_{X}dR^{\prime},$
(9)

where $R_{\rm b}$ is the radius at which we start the first streamline,
$R_{0}$ is the initial radius of the considered streamline, $\theta$ is the
angle between the line from the particle to the center and the disk surface,
$n_{0}$ is the particle number density on the disk surface, $n(r)$ is the
particle number density located at $r=(R^{2}+z^{2})^{\frac{1}{2}}$ in the
considered streamline, and $\sigma_{X}$ is the X-ray attenuation,
respectively. Following Proga et al. (2000), the X-ray attenuation is set to
be $\sigma_{X}=0.4\text{ g}^{-1}\text{cm}^{2}$ for $\xi\geq 10^{5}$ erg s-1 cm
and $\sigma_{X}=40\text{ g}^{-1}\text{cm}^{2}$ for $\xi<10^{5}$ erg s-1 cm.
Moreover, the ultraviolet optical depth $\sigma_{\rm uv}=0.4\text{
g}^{-1}\text{cm}^{2}$ for any $\xi$. Similarly, the UV optical depth is given
by

$\tau_{uv}\equiv\int_{R_{\rm b}}^{R_{0}}\sec(\theta)m_{p}n_{0}\sigma_{\rm
uv}dR^{\prime}+\int_{R_{0}}^{R}\sec(\theta)m_{p}n(r)\sigma_{\rm
uv}dR^{\prime},$ (10)

and we set $\sigma_{\rm uv}=0.4\text{ g}^{-1}\text{cm}^{2}$ for any $\xi$.
Here, for simplicity, we employ the simply method to calculate the optical
depth of UV and X-ray photons. In Quera-Bofarull et al.’s work (2021), the
treatment of the optical depth is more accurate.

AGNs in our models have three parameters, i.e. the black hole mass ($M_{\rm
BH}$), the bolometric luminosity ($L_{\rm bol}$), and the radio ($f_{\rm
disk}$) of the disk luminosity to the bolometric luminosity. According to the
disk luminosity, we can determine the accretion rate $\dot{M}=12L_{\rm
disk}/c^{2}$. Winds are launched from the disk surface. We use streamlines to
describe winds. We simulate $30$ streamlines at equal intervals from 10
$r_{\rm s}$ to 1500 $r_{\rm s}$. For comparison, we employ 40 streamlines to
calculate a model (model B3a in Table 1). We find that when the number of
streamlines is more than 30, the results donot significantly change (see Table
1). The positions of streamline foot are set to be ($R_{0}$, $\phi_{0}=0$,
$z_{0}$), where $R_{0}$ is the distance from the rotational axis and $z_{0}$
is the height from the disk surface, respectively. In the previous works, the
disk height is often assumed to be constant with the change of radius for
simplicity of calculation. We also assume the disk height to be constant and
set $z_{0}=1$ $r_{\rm s}$. Then, the position of streamline foot is determined
only by $R_{0}$. For calculating the streamlines of winds, besides $R_{0}$,
the vertical velocity ($v_{0}$) and particle number density ($n_{0}$) of
lunched winds need to be determined as the boundary conditions of streamlines.
For a streamline, we assume its initial width to be $\Delta L_{0}$ and assume
the streamline width $\Delta L\varpropto r$. According to the mass continuity
equation, along a streamline the mass outflow rate ($\dot{M}_{\rm line}$) of
winds is constant. Therefore, we have

$\dot{M}_{\rm line}=m_{\rm p}n(r)v2\pi r\Delta L=m_{\rm p}n_{0}v_{0}2\pi
r_{0}\Delta L_{0},$ (11)

where $r_{0}=R_{0}$ on the disk surface and $n(r)$ is the particle number
density at $r=\sqrt{R^{2}+z^{2}}$. For simplicity, we assume $\Delta L\propto
r$ along the streamlines. Then we have $\Delta L=\frac{r}{r_{0}}\Delta L_{0}$
and get $n(r)vr^{2}=n_{0}v_{0}r^{2}_{0}$. When the boundary conditions (i.e.
$R_{0}$, $v_{0}$, and $n_{0}$) are given, we integrate equations (1)–(6), then
get the position and velocity, and finally update the number density using
equation (9). This process is iterated until streamline is obtained.

Our calculation is based on the QWIND code (Quera-Bofarull et al. 2020). In
the QWIND code, the ASSIMULO simulation software package (Andersson et al.
2015) is employed and the IDA solver (Hindmarsh et al. 2004) is implemented to
solve equations (1)–(6). The details of the QWIND code are referred to Quera-
Bofarull et al. (2020). According to equation (7), we modified the QWIND code
for considering the radiation-drag effects. This updated code, Dwind, is now
available as a public release on
GitHub.111https://github.com/wangbeicuan/Dwind.git

##  3 RESULTS

###  3.1 Effects of radiation drag.

The radiation forces exerted on particles are determined by the radiation
fluxes ($F^{i}$) measured in the comoving frame with the moving particles.
According to equation (7), the measured radiation fluxes ($F^{i}$) in the
comoving frame include two terms: one is the radiation fluxes ($F^{i}_{0}$) in
the rest frame standing at the infinity, and other term
($-E_{0}v^{i}-v_{j}P^{ij}_{0}$) is the radiation-drag effect, which is
attributed to the relativistic effect due to that particles move in radiation
field at high speed. The radiation-drag force is proportional to the velocity
of particles moving and reduces the particle velocity.

To study the radiation-drag effect, we have calculated a series of models,
which are summarized in Table 1. For all the models, we set $M_{\rm
BH}={10^{8}}M_{\odot}$,and $f_{\rm disk}=0.9$ (or $f_{X}$=0.1), respectively.
Other parameters are listed in Table 1, where Col. (2) gives the bolometric
luminosity, Col. (3) gives the initial vertical velocity ($v_{0}$), Col. (4)
gives the initial particle number density ($n_{0}$), Col. (5) gives the
fastest velocity of particles along streamlines, Cols. (6) and (7) also give
the mass outflow rate and kinetic power of the escape particles, and Col. (8)
gives the range of the velocity azimuth angles (relative to the disk surface)
of the escape particles at the outer boundary, respectively. For comparing the
properties of radiation-dragged particle winds with that of non-dragged
particle winds, all the models are classified into A-group and B-group models,
respectively. For the A-group models (models A0–A17) without the radiation-
drag effects, we remove $-E_{0}v^{i}-\sum v_{j}P^{ij}_{0}$ from equation (7),
while all the terms are included in equation (7) for the B-group models
(models B0–B17) with the radiation-drag effects.

Figure 1 shows the trajectories of winds in models A3 and B3 (panels (A) and
(B)) and models A13 and B13 (panels (C) and (D)). Models A13 and B13 have
higher initial velocity than models A3 and B3. In the top panel, we can see
that the formation region of winds is different in models A3 and B3. The
escape winds of model B3 are generated in a smaller region than that of model
A3. This implies that the radiation-drag effect reduces the radiation force of
driving particles and then more particles cannot reach the escape velocity.
According to equations (A1) and (A3) in Appendix A, the force multiplier of
line force is positively correlated with the acceleration of particles and is
inversely correlated with the speed of particles. When the radiation flux in
the comoving frame becomes weak due to the radiation-drag effects, the
acceleration of particles becomes small due to the reduction of radiation
force and then the force multiplier of line force further becomes small so
that the line force is weakened. Therefore, the radiation-drag effect can
weaken the line force by directly reducing the radiation flux in the comoving
frame and weakening the line force multiplier. That is the reason the escape
winds in model B3 form within a smaller region and the opening angle of winds
becomes smaller, compared with the case of model A3. In panel (D), we can see
that the wind particles of model B13 cannot escape to infinity and eventually
they fall back on the disk surface. In models A13 and B13, we set a higher
initial velocity than that in models A3 and B3. The higher initial velocity
weakens the line force multiplier and therefore the line force becomes weaker
in models A13 and B13. Comparing models A3 and A13, the initial velocity in
model A13 is one order of magnitude higher than that in model A3 (see Table
1), but the maximum velocity of winds is slightly three quarters of that in
model A3, while the mass outflow rate exceeds seven times that in model A3 and
kinetic power are almost three times that in model A3. In model A13, the line
force becomes weak, which causes that the maximum velocity of winds becomes
slower than that in model A3. However, the higher initial velocity in model
A13 makes more particles reach the escape velocity and therefore the mass
outflow rate and kinetic power of winds are higher than that in model A3.
Furthermore, due to the radiation-drag effects, the radiation force in model
B13 is too weak to accelerate particles to the escape velocity.

The radiation drag term in equation (7) includes a component ($E_{0}v^{i}$)
related to the radiation energy density and a component ($v_{j}P_{0}^{ij}$)
related to the radiation pressure tensor. Figure 2 shows not only the ratio
($\frac{E_{0}v^{i}}{v_{j}P_{0}^{ij}}$) between the two components along a
streamline (i.e. the thick line in panel (B) of figure 1), but also the ratio
of the radiation-drag term to the radiation flux ($F^{i}_{0}$) in the rest
frame. The radiation-drag term in the radial and vertical direction is less
than 15% of the radiation flux in the rest frame, and the $E_{0}v^{i}$
component is stronger than the $v_{j}P_{0}^{ij}$ component on most areas of
the streamline while the $v_{j}P_{0}^{ij}$ component should not be ignored.
When the Doppler effect of rotating of the thin disk is ignored, the radiation
flux ($F^{\phi}_{0}$) in the azimuthal direction equals 0 since the radiation
field is axisymmetric. When the Doppler effect of rotating is included,
$F^{\phi}_{0}$ is also a small quantity. In the radiation-drag term, the
$E_{0}v^{\phi}$ component is dominant. When the $\phi$-direction radiation
drag term is ignored, the angular momentum of particles is conserved along
their streamlines. When the $\phi$-direction radiation drag term is included
in dynamics equations, the angular momentum of particles is slightly reduced
along their streamlines and then the centrifugal force of particles is
weakened slightly, compared with the case of angular momentum conservation.
However, the $\phi$-direction radiation drag term is still negligible in the
dynamics of particles.

###  3.2 Dependence on related parameters: $L_{\rm bol}$, $v_{0}$, and $n_{0}$

In this section, we test the effects of three parameters (such as $L_{\rm
bol}$, $v_{0}$, and $n_{0}$) on the wind properties. For models A0–A9 and
models B0–B9 in Table 1, $L_{\rm bol}$ varies from 0.01 to 0.8 $L_{\rm Edd}$
while $v_{0}$ and $n_{0}$ are set to be $1\times 10^{5}$ m/s and $5\times
10^{7}$ cm-3, respectively. Figure 3 shows the influences of $L_{\rm bol}$ on
the maximum speed along streamlines, the mass outflow rate, and the kinetic
power of winds. When a particle reaches the escape velocity, we take into
account the contribution of the particle to the mass outflow rate and the
kinetic power of winds. Due to that the mass of wind particles is constant
along streamlines, the mass outflow rate and the kinetic power can be given by

$\dot{M}_{\rm out}=4\pi\sum_{i}{\rho_{0,i}v_{0,i}R_{0,i}\Delta L_{0,i}},$ (12)

and

$P_{\rm kin}=2\pi\sum_{i}{\rho_{0,i}v_{0,i}R_{0,i}\Delta L_{0,i}}v_{i}^{2},$
(13)

respectively, where $i$ represents the $i$-th escape particle, $v_{i}$ is the
particle velocity at the outer boundary, and $\rho$ ($\equiv m_{\rm p}n$)is
the density of particles, respectively. The emitting angles ($\theta_{\rm
out}$) of winds are given by

$\theta_{\rm out}=Arctan(\frac{v^{z}_{\rm out}}{v^{R}_{\rm out}}),$ (14)

where $v^{z}_{\rm out}$ and $v^{R}_{\rm out}$ are the vertical and radial
velocities of escape particle at the outer boundary, respectively.

Panel (A) of figure 3 shows the dependence of the maximum speed along
streamlines on the bolometric luminosity. In the A-group models without the
radiation-drag effects, winds are very weak when $L_{\rm bol}<$0.05 $L_{\rm
Edd}$ and winds cannot be generated when $L_{\rm bol}<$0.01 $L_{\rm Edd}$. In
the B-group models including the radiation-drag effects, winds cannot be
generated when $L_{\rm bol}<$0.05 $L_{\rm Edd}$. For models A2 and B2, whose
luminosity is 0.1 $L_{\rm Edd}$, the maximum speed along streamlines can reach
0.228 $c$ and 0.08 $c$, respectively. In the A-group models, as $L_{\rm bol}$
increases from 0.01 to 0.3 $L_{\rm Edd}$, the maximum speed increases rapidly,
while when $L_{\rm bol}>$ 0.3 $L_{\rm Edd}$ the maximum speed increase
slightly with the increase of bolometric luminosity. In the B-group models,
the maximum speed is significantly lower than that in the A-group models.
According to equation (12), the mass outflow rate is determined by the disk
surface area emitting the escaped particles. In the A-group models, the
emission region of winds becomes wider and farther away from the central BH
with the increase of $L_{\rm bol}$. In the B-group models, the emission region
of winds almost does not change when $L_{\rm bol}>$0.6 $L_{\rm Edd}$.
Comparing the A-group and B-group models, the emission region of winds in the
B-group models is narrower than that of A-group models, so that the mass
outflow rate in the B-group models is significantly lower than that in the
A-group models, as shown in Panel (B).

Panel (C) shows the dependence of the wind power ($L_{\rm kin}$) on the
bolometric luminosity. According to equation (13), the wind power depends on
the areas where winds are generated and the velocity at the outer boundary.
The higher the disk luminosity, the stronger the winds, as shown in panel (C).
Due to the radiation-drag effects, the winds in the B-group models are
significantly weaker than that in the A-group models. Panel (D) shows that the
radiation-drag effects make winds closer to the disk surface when $L_{\rm
bol}>$0.3 $L_{\rm Edd}$, while when $L_{\rm bol}<$0.3 $L_{\rm Edd}$ the
opening angles of winds in the B-group models become narrower.

Figure 4 shows the influences of the initial velocity ($v_{0}$) of particles
on the properties of winds. For models A3 and A10–A13 and models B3 and
B10–B13, that shown in Figure 4, $L_{\rm bol}$ and $n_{0}$ are set to be 0.2
$L_{\rm Edd}$ and $5\times 10^{7}$ $\text{cm}^{-3}$, respectively, while
$v_{0}$ varies from $10^{4}$ to $10^{6}$ m/s, as given in Table 1. Panel (A)
shows that the maximum speed along streamlines is significantly reduced due to
the increase of the initial velocity. For model A10, the maximum speed reaches
0.384 $c$, while in model A13 the maximum speed is 0.170 $c$. The reason is
that line force is weaken due to the increase of particle speed according to
equations (A1) and (A3) in Appendix A. Therefore, the wind particles can not
be accelerated to higher velocity, when the initial velocity increases. When
the radiation-drag effects are considered, the maximum speed is reduced by
$50\%\sim 100\%$, because the decrease of acceleration is caused by radiation
drag. For models A3 and A10–A13, the mass outflow rate and the kinetic power
increase rapidly with the increase of the initial velocity, because the
increase of initial velocity causes that more particles are emitted per unit
time from the disk surface. However, when the radiation-drag effects are
considered, the kinetic power in the B-group models decreases with the
increase of $v_{0}$ when $v_{0}>5\times 10^{4}$ m/s, and the mass outflow rate
increases slightly with the increase of $v_{0}$. For model B13 ($v_{0}=10^{6}$
m/s), the mass outflow rate and the kinetic power decrease to 0, because the
particles cannot escape to the infinity because line force is significantly
weaken. Compared with the A-group models, more particles in the B-group models
are not accelerated to the escape velocity to form winds. As the initial
velocity increases, the opening angles of winds decrease, as shown in Panel
(D).

Figure 5 shows the influences of the particle number density ($n_{0}$) of wind
particles on the properties of winds. For models A2 and A14–A17 and models B2
and B14–B17, $L_{\rm bol}$ and $v_{0}$ are set to be 0.2 $L_{\rm Edd}$ and
$10^{5}$ m/s, respectively, while $n_{0}$ varies from $10^{7}$ to
$10^{9}\text{cm}^{-3}$, as given in Table 1. The maximum speed along
streamlines and the kinetic power of winds increase with the number density of
streamlines, as shown in Panel (A) and (C). Panel (B) shows that the mass loss
rate increases with the number density when $n_{0}<5\times
10^{7}\text{cm}^{-3}$, keeps almost constant when $n_{0}\sim 5\times
10^{7}$–$1\times 10^{8}\text{cm}^{-3}$, and then decreases when
$n_{0}>10^{8}\text{cm}^{-3}$. This reason is as follows. As pointed out in
previous discussion, the mass loss rate depends on area of the disk surface
from which the escape winds are emitted. The wider the area, the higher the
mass loss rate. Higher density is helpful to shield the X-ray photons from the
central corona and then the ionization degree of gases becomes low, so that
line force becomes stronger in driving particles. When the gases become dense,
the region from which the escape particles are emitted extends inward and
becomes wide. However, when the gases become very dense, the UV photons
produced in the inner region of disk are also shielded to some extent, which
can weaken line force. The region from which the escape particles are emitted
becomes narrow to some extent. In general, with the increase of number
density, the particles are accelerated by line force to higher velocity and
then stronger winds are generated. As the particle number density increases,
the azimuth angles (relative the disk surface) of winds increases, as shown in
Panel (D).

##  4 Conclusions and Discussions

In AGNs, when outflows/winds with mildly relativistic velocities move in the
radiation field produced by the thin disk around a BH, the outflows are
affected by the radiation-drag force. In this paper, we study the effects of
the radiation-drag force on the properties of the line-force-driven winds,
based on a non-hydrodynamic approach. In the approach, we neglect the gas
pressure and take gases as particles. Compared with the case without the
radiation-drag force, when the radiation-drag force is included in dynamics,
the line-force-driven winds are significantly weakened. For example, the
maximum speed of winds, the mass outflow rate, and the kinetic power of winds
are significantly weakened. The details of our conclusions are summarized as
follows:

1) The radiation-drag effects increase the lower limit of the bolometric
luminosity that can drive winds. In A-group models without the radiation-drag
effects, winds become very weak when $L_{\rm bol}<$0.05 $L_{\rm Edd}$. In the
B-group models with the radiation-drag effects, the lower limit of the
bolometric luminosity becomes 0.1 $L_{\rm Edd}$.

2) The radiation-drag force narrows the area where the winds are generated and
then increases the area where the failed winds are formed.

3) The radiation-drag force makes the maximum speed of winds $\sim$60%-70%
lower than that in the case without the radiation-drag force. In models A1–A9,
the maximum speed of winds can reach $\sim$0.17–0.36 $c$, while in models
B0–B9 the maximum speed can reach $\sim$0.11–0.15 $c$. When the density of
winds increases, the line force is strengthened, and then the wind velocities
increase.

The observation of the Fe XXV and Fe XXVI lines implies that UFOs move at
$\sim$0.03–0.3 $c$, with a mean value of $\sim$0.14 $c$ (Tombesi et al. 2012).
When the radiation-drag effects are considered, the line-force-driven winds
are closer to the velocity range of UFOs. However, UFOs are highly ionized.
The line-force-driven winds are low-ionization. Therefore, our results donot
seem to be employed to explain the properties of UFOs. However, previous
numerical simulations have implied that the line-force-driven winds may become
highly-ionized winds (i.e. UFOs) when the winds are exposed to the X-ray
photons (Normua & Ohsuga 2017; Yang et al. 2021; Yang 2021). The density of
the inner shielding gases varies with time. When the inner shielding gases
become dense, the line force is efficient. When the inner shielding gases
become thin, the line-force-driven winds become highly ionized. Therefore, the
winds are intermittently accelerated by the line force to the mildly
relativistic velocity. The time evolution of winds is not studied in present
work. In the future, we will implement numerical simulations to examine the
effects of the radiation-drag force on the line-force-driven winds. Another
possibility is the effect of geometry. When winds are relatively close to the
disk surface, X-ray photons are shielded by the inner dense gases above the
thin disk, the ionization degree of winds is low, and then the winds are
accelerated by the line force to the mildly relativistic velocity. When the
winds leave the shielding area, they are ionized by X-rays photons and then
form the observed UFOs.

There are some approximations in our calculations. For example, we ignore the
frequency dependence of the line-force driving. It is often thought that the
200–3200$\AA$ UV radiation mainly contributes to the line-force driving (Proga
& Kallman 2004). In our calculations, there are three factors that may affect
the strength of the line force. 1) In equation (7), the radiative flux
(F${}^{i}_{0}$), the radiation energy density (E0), and the radiation stress
tensor (P${}^{i}j_{0}$) in the rest frame are the quantities integrated by the
frequency throughout the entire range. This causes an overestimation of the
$200-3200\AA$ UV radiation. For the disk of $L_{\rm disk}=0.6L_{Edd}$, the
200–3200$\AA$ UV luminosity is overestimated by $\sim$14%. 2) The Doppler
effects caused by the rotation of the accretion disk can affect the spectral
shape and intensity measured in the rest frame. In our calculations, equation
(B2) has taken into account the influence of Doppler effects on the intensity
on the order of $v/c$. The Doppler shift of spectral shape can affect the
integrated intensity over the 200–3200$\AA$ UV bands in the rest frame, which
is ignored in our calculations. In the rest frame, when the Doppler shift of
spectral shape is not included, the 200–3200$\AA$ UV flux in the rest frame is
overestimated, compared to the case that includes the Doppler effects caused
by disk rotation. For example, for the disk of $L_{\rm disk}=0.6L_{Edd}$, the
200–3200$\AA$ UV flux in the rest frame is overestimated by $\sim$5% at most.
3) The Doppler effects caused by the motion of particles can further affect
the spectral shape and intensity measured in the comoving frame with the
moving particles, which changes the intensity integrated over the
200–3200$\AA$ UV bands in the comoving frame. Equation (7) has taken into
account the correction of the Doppler effects on the radiative flux in the
comoving frame on the order of $v/c$. For most B-group models, the maximum
speed of particles is about $\sim$0.15$c$ and the particles move at $<0.15c$
over most regions along the streamlines. The first-order accuracy of $v/c$ is
acceptable in our calculations. Equation (7) ignores the Doppler shift of the
spectral shape in the comoving frame. For simply estimating the correction, we
set the disk to be the radiation source as a point and assume that a particle
moves away from the radiation source at 0.15$c$. Then, when only the Doppler
shift of the spectral shape is taken into account, the intensity integrated
over the 200–3200$\AA$ UV bands in the rest frame is close to the integrated
intensity in the comoving frame. Therefore, our calculations overestimate the
200–3200$\AA$ UV radiation of the line-force driving. However, it is
acceptable that the frequency dependence of the line-force driving is ignored
in our calculations.

##  5 ACKNOWLEDGMENTS

This work is supported in part by the Natural Science Foundation of China
(grant 11973018 and 12147102). D. Bu is supported by the Natural Science
Foundation of China (grant 12173065).

##  6 DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A the line-force multiplier $\mathcal{M}$

Based on the Sobolev approximation, the line-force multiplier ($\mathcal{M}$)
is a function of the ionization parameter ($\xi$) and the local optical depth
parameter ($t$) (Rybicki & Hummer 1978). the line-force multiplier is given by

$\mathcal{M}(t,\xi)=k(\xi)t^{-0.6}[\frac{(1+t\eta_{\rm
max}(\xi))^{0.4}-1}{(t\eta_{\rm max}(\xi))^{0.4}}],$ (15)

where $k(\xi)$ and $\eta_{\rm max}(\xi)$ are a parameter proportional to the
total number of lines and a parameter determining the maximum value of
$\mathcal{M}$, respectively. The two parameters are a function of the
ionization parameter. The local optical depth parameter ($t$) can be obtained
by

$t=\sigma_{T}nv_{\rm th}|\frac{dv_{\textit{l}}}{d\textit{l}}|^{-1},$ (16)

where $\frac{dv_{\textit{l}}}{d\textit{l}}$ is the velocity gradient along the
light of sight, $v_{\rm th}$ is the thermal velocity, and $\sigma_{T}$ is the
Thomson cross section, respectively. For simplicity, the velocity gradient is
written as

$\frac{dv_{\textit{l}}}{d\textit{l}}=\frac{dv}{dt}\frac{dt}{d\textit{l}}=\frac{a}{v}=\frac{\sqrt{a_{\textit{\tiny{R}}}^{2}+a_{z}^{2}}}{\sqrt{v_{\textit{\tiny{R}}}^{2}+v_{z}^{2}}}.$
(17)

We set the gas thermal velocity $v_{\rm th}=20$ $\rm km\rm s^{-1}$, which
corresponds to the gas temperature of $2.5\times 10^{4}$ K (Stevens & Kallman
1990; Proga 2007). For $k(\xi)$ and $\eta_{\rm max}(\xi)$, we use the data
obtained from Stevens and Kallman (1990) to obtain the interpolation
functions.

##  Appendix B the calculation of the radiation field in the rest frame

Figure 6: Schematic diagram for the calculation of the radiation field in the
rest frame.

Figure B1 is a schematic diagram to display the calculation of the radiation
field. In a cylindrical coordinate system ($R$, $\phi$, $z$), the position of
an observer (B point) is given by $(R,0,z)$ while the position of an element
(A point) on the disk is given by ($R_{d},\phi_{d},0$).

According to a theory of the standard thin disk, the local isotropic intensity
on the disk surface of A is written as

$I_{d}=\frac{3GM_{\rm BH}\dot{M}}{8\pi^{2}R_{\rm d}^{3}}(1-\sqrt{\frac{R_{\rm
in}}{R_{\rm d}}}),$ (18)

where $\dot{M}$ is the mass accretion rate, $R_{\rm in}$ ($R_{\rm in}=3$
$r_{\rm s}$) the inner radius of the disk, and $R_{\rm d}$ the disk radius.
The accretion disk rotates with the Keplerian velocity of
$v_{K}=\sqrt{{GM_{BH}}/{R_{\rm d}}}$ and then the relativistic Doppler effect
changes the radiation field in the rest frame standing at infinity. The
radiation intensity measured by an observer rotating with the disk ($I_{\rm
d}$) can be transformed to that measured by the rest observer ($I_{0}$) as
follows:

$I_{0}=\frac{I_{\rm d}}{(1-\frac{\overrightarrow{\textit{v}_{\rm
d}}\cdot\overrightarrow{\textit{l}}}{c})^{4}},$ (19)

where $\overrightarrow{\textit{l}}$ is the direction-cosine vector pointing
toward the rest observer from the location of the disk radiation and
$\overrightarrow{\textit{v}_{\rm d}}$ is the disk velocity. In Cartesian
coordinates, we can get

$\overrightarrow{\textit{l}}=(\frac{R-R_{\rm d}{\rm cos}(\phi_{\rm
d})}{D},\frac{-R_{\rm d}{\rm sin}(\phi_{\rm d})}{D},\frac{z}{D}),$ (20)

and

$\overrightarrow{\textit{v}_{\rm d}}=(-v_{\rm K}\sin(\phi_{\rm d}),v_{\rm
K}\cos(\phi_{\rm d}),0),$ (21)

where $D=(R^{2}+z^{2}+R_{d}^{2}-2RR_{\rm d}\cos(\phi_{\rm d}))^{\frac{1}{2}}$
is the distance from the rest observer to the element A on the disk surface,
as shown in figure B1.

For the element A, its area $dS=R_{\rm d}d\phi_{\rm d}dR_{\rm d}$. The solid
angle subtended by the element A towards the observer standing at the B point
is $d\Omega=\frac{zdS}{D^{3}}$. The $E_{0}$, $F^{i}_{0}$, and $P^{ij}_{0}$
measured in the rest frame are expressed as (Tajima $\&$ Fukue 1998)

$\begin{split}E_{0}=\frac{1}{c}\int
I_{0}d\Omega=\frac{3GM\dot{M}}{8c\pi^{2}r_{\rm
s}^{3}}\int_{3}^{1500}\int_{0}^{2\pi}\frac{z}{R_{\rm
d}^{2}}(1-\sqrt{\frac{3}{R_{\rm d}}})\\\
\times\frac{1}{D^{3}(1-\frac{\overrightarrow{\textit{v}_{\rm
d}}\cdot\overrightarrow{\textit{l}}}{c})^{4}}d\phi_{\rm d}dR_{\rm
d},\end{split}$ (22) $\begin{split}F^{i}_{0}=\int
I_{0}l^{i}d\Omega=\frac{3GM\dot{M}}{8\pi^{2}r_{\rm
s}^{3}}\int_{3}^{1500}\int_{0}^{2\pi}\frac{z}{R_{\rm
d}^{2}}(1-\sqrt{\frac{3}{R_{\rm d}}})\\\
\times\frac{l^{i}}{D^{3}(1-\frac{\overrightarrow{\textit{v}_{\rm
d}}\cdot\overrightarrow{\textit{l}}}{c})^{4}}d\phi_{\rm d}dR_{\rm
d},\end{split}$ (23) $\begin{split}P^{ij}_{0}=\frac{1}{c}\int
I_{0}l^{i}l^{j}d\Omega=\frac{3GM\dot{M}}{8c\pi^{2}r_{\rm
s}^{3}}\int_{3}^{1500}\int_{0}^{2\pi}\frac{z}{R_{\rm
d}^{2}}(1-\sqrt{\frac{3}{R_{\rm d}}})\\\
\times\frac{l^{i}l^{j}}{D^{3}(1-\frac{\overrightarrow{\textit{v}_{\rm
d}}\cdot\overrightarrow{\textit{l}}}{c})^{4}}d\phi_{\rm d}dR_{\rm
d}.\end{split}$ (24)