# Choked accretion onto Kerr-Sen black holes in Einstein-Maxwell-Dilation-
Axion gravity

Haiyuan Feng  Email address<EMAIL_ADDRESS>Department of Physics, Southern
University of Science and Technology, Shenzhen 518055, Guangdong, China
Yingdong Wu  Email address<EMAIL_ADDRESS>Department of
Physics, Southern University of Science and Technology, Shenzhen 518055,
Guangdong, China  Rong-Jia Yang  Email address<EMAIL_ADDRESS>College of Physical Science and Technology, Hebei University, Baoding 071002,
China  Leonardo Modesto  Email address<EMAIL_ADDRESS>Department of
Physics, Southern University of Science and Technology, Shenzhen 518055,
Guangdong, China

###### Abstract

We study the choked accretion process of an ultrarelativistic fluid onto
axisymmetric Kerr-Sen black holes in Einstein-Maxwell-Dilation-Axion theory.
We calculate the solution describing the velocity potential $\Phi$ of a
stationary, irrotational fluid, which satisfies the stiff equation of state
and draw the streamlined diagram of the quadrupolar flow solution. We
investigate how parameters affect the solution’s coefficient and stagnation
point. The injection rate, ejection rate, and critical angle are discussed in
detail at the end of the article. If the inner and outer event horizons of the
black hole are satisfied, then we can find that the ratio of the ejection rate
to the setting rate increases with an increase in the dilation parameter.

##  I Introduction

The most probable scenario for explaining the high energy output from active
galactic nuclei and quasars is matter accretion onto a black hole, which is a
significant phenomenon for astrophysicists. The theory of black hole accretion
also provides the most credible exposition for the high energy outflow from
X-ray Binaries and Gamma Ray Bursts Abramowicz:2011xu . The investigations of
the accretion processes onto celestial objects were initiated by Hoyle and
Lyttleton hoyle_lyttleton_1939 ; hoyle_lyttleton_1940_1 ;
hoyle_lyttleton_1940_2 ; 1941MNRAS.101..227H , and later studies were carried
out by Bondi and Hoyle for a pressure-less gas falling onto a moving star
Bondi:1944jm . Subsequently, the theory of stationary, spherically symmetric,
and transonic hydrodynamic accretion of adiabatic fluid onto a gravitating
astrophysical body at rest was formulated in a seminal paper by Bondi
Bondi:1952ni , such a model was perfectly generalized by Michel to the
Schwarzschild black hole of general relativity Michel1972 . Further research
was constructed for distinct black hole backgrounds, various accreted
constituents, and several alternative models see for example Malec:1999dd ;
Babichev:2004yx ; Jiao:2016iwp ; Ganguly:2014cqa ; Yang:2015sfa ; Font:1998sc
; Yang:2019qru . Accretion onto Charged or rotating black holes were explored
in Parev:1995me ; Font:1998sc ; Medvedev:2002wn ; Fragile:2004gp ;
Bhadra:2011me ; JimenezMadrid:2005rk ; Feng:2022bst  which used the same
processing approach. In order to be as close as possible to real astrophysical
processes, a precise completely relativistic solution describing stationary
accretion of matter onto a moving Schwarzschild or Kerr black hole with an
adiabatic equation of state was derived in Petrich:1988zz , which was
generalized to Reissner-Nordström and charged dilaton black hole Jiao:2016uiv
; Yang:2021opo . Recently, the accretion of Vlasov gas onto a moving
Schwarzschild black hole was proposed in Mach:2021zqe ; Mach:2020wtm ;
Cai:2022fdu , it is discussed analytically by using the Hamiltonian form
without considering the reaction of matter. Besides these analytical works,
more realistic situations were considered in background of astronomical
environments DiMatteo:2002hif ; Silich:2008xz ; Gil-Merino:2011yhj ;
Kuo:2014pqa ; He:2022lrc .

On the other hand, the astrophysical jet is an astronomical phenomenon in
which plasma matter is emitted as an extended beam along the rotation axis.
Blandford, Payne (BP), and Blandford, Znajek (BZ) proposed the jet mechanism
most widely accepted by previous works 10.1093/mnras/199.4.883 ;
10.1093/mnras/179.3.433 . BP mechanism utilizes magnetic centrifugation to
extract energy and angular momentum from a black hole accretion disc, whereas
the BZ mechanism involves rotational energy from a Kerr black hole. These
methods were commonly used in magneto-hydrodynamics and astrophysics
exploration Semenov:2004ib ; McKinney:2006tf ; Qian:2018qpg ; Liska:2018ayk .
Afterward, a new hydrodynamic accretion mechanism was proposed that explains
the effect of the steady-state axisymmetric partial fluid flowing from the
equatorial plane being ejected along poles by the cytaster’s gravitational
influence https://doi.org/10.48550/arxiv.1103.0250 . This process were also
generalized to the choked accretion scenario of Schwarzschild and Kerr black
holes in general relativity Tejeda:2019fwr ; Aguayo-Ortiz:2020qro .

In this paper, we will discuss the choked accretion process of
ultrarelativistic perfect fluid onto rotating Kerr-Sen black holes in
Einstein-Maxwell-Dilation-Axion (EMDA) theory, where an irrotational, steady-
state hydrodynamic flow with finite boundary conditions at the horizon is
considered. In particular, we will present a streamlined diagram of choked
accretion based on the stiff equation of the state of perfect fluid and will
investigate how the parameters affect the position of stagnation point and the
relative density disparity.

The article is organized as follows: In Section II, the EMDA model and the
Kerr-Sen black hole will be briefly reviewed. In Section III, based on the
assumption of irrotational fluids, we will derive an analytical solution for
the velocity potential $\Phi$ in Boyer-Lindquist coordinate system. In Section
IV, The quadrupolar flow solution will be used to analyze the variation of the
coefficients with parameters and will be displayed the streamline and
temperature diagrams in the ZAMO frame. Finally, we will present in the
mechanism of choked accretion the density ratio, the injection rate, the
ejection rate, and as well as the value range of the reference point. For
convenience, we will use geometrical units $c=G=1$ and signature convention
$(-,+,+,+)$ for spacetime metric throughout the article.

##  II Kerr-Sen black hole in Einstein-Maxwell-Dilaton-Axion gravity

The Einstein-Maxwell-Dilaton-Axion gravity (EMDA) model is derived from the
low energy limit behavior of heterotic string theory, it is composed of
dilaton field $\chi$, gauge vector field $A_{\mu}$, metric $g_{\mu\nu}$, and
pseudo-scalar axion field $\xi$ Sen:1992ua ; Rogatko:2002qe ; Campbell:1992hc
. The action of the EMDA model could be formed by the coupling of supergravity
and super-Yang Mills theory, it can be described by the following form,

$S=\frac{1}{16\pi}\int\sqrt{-g}d^{4}x\left[\tilde{R}-2\partial_{\mu}\chi\partial^{\mu}\chi-\frac{1}{2}\mathrm{e}^{4\chi}\partial_{\mu}\xi\partial^{\mu}\xi+\mathrm{e}^{-2\chi}F_{\mu\nu}F^{\mu\nu}+\xi
F_{\mu\nu}\widetilde{F}^{\mu\nu}\right],$ (1)

where $\tilde{R}$ is the Ricci scalar and $F_{\mu\nu}$ is the second-order
antisymmetric Maxwell field strength tensor with
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, $\widetilde{F}^{\mu\nu}$
is the dual tensor of the field strength. The variation of the aforementioned
four fields yields the following motion equations:

$\left\\{\begin{split}&\Box\chi-\frac{1}{2}\mathrm{e}^{4\chi}\nabla_{\mu}\xi\nabla^{\mu}\xi+\frac{1}{2}\mathrm{e}^{-2\chi}F_{\mu\nu}F^{\mu\nu}=0\\\
&\Box\xi+4\nabla_{\mu}\xi\nabla^{\mu}\xi-\mathrm{e}^{-4\chi}F_{\mu\nu}\widetilde{F}^{\mu\nu}=0\\\
&\nabla_{\mu}\widetilde{F}^{\mu\nu}=0\\\
&\nabla_{\mu}(\mathrm{e}^{-2\chi}F^{\mu\nu}+\xi\widetilde{F}^{\mu\nu})=0\\\
&G_{\mu\nu}=\mathrm{e}^{2\chi}(4F_{\mu\rho}F^{\rho}_{\nu}-g_{\mu\nu}F^{2})-g_{\mu\nu}(2\nabla_{\mu}\chi\nabla^{\mu}\chi+\frac{1}{2}\mathrm{e}^{4\chi}\nabla_{\mu}\xi\nabla^{\mu}\xi)\\\
&+\nabla_{\mu}\chi\nabla_{\nu}\chi+\mathrm{e}^{4\chi}\nabla_{\mu}{\xi}\nabla_{\nu}{\xi}.\end{split}\right.$
(2)

It shows that the dilaton field, axion field, electromagnetic, and
gravitational field are observed to be coupled. After more complex
calculations, the axisymmetric classical solution which is known as the Kerr-
Sen solution can be produced, which in the Boyer-Lindquist coordinates takes
the form Garcia:1995qz ; Ghezelbash:2012qn ; Bernard:2016wqo .

$\begin{split}\,\mathrm{d}s^{2}&=-\left(1-\frac{2M}{\tilde{\Sigma}}\right)\,\mathrm{d}t^{2}+\frac{\tilde{\Sigma}}{\tilde{\Delta}}\,\mathrm{d}r^{2}+\tilde{\Sigma}\,\mathrm{d}\theta^{2}-\frac{4aMr}{\tilde{\Sigma}}\sin^{2}{\theta}\,\mathrm{d}t\,\mathrm{d}\phi\\\
&+\sin^{2}{\theta}\,\mathrm{d}\phi^{2}\left(r(r+r_{2})+a^{2}+\frac{2Mra^{2}\sin^{2}{\theta}}{\tilde{\Sigma}}\right),\end{split}$
(3)

with

$\left\\{\begin{split}&\tilde{\Sigma}=r(r+r_{2})+a^{2}\cos^{2}{\theta}\\\
&\tilde{\Delta}=r(r+r_{2})-2Mr+a^{2}.\end{split}\right.$ (4)

where $M$ is the mass parameter of the black hole and the dilation parameter
$r_{2}=\frac{Q^{2}}{M}$ is made up of the electric charge $Q$, while $a$
refers to the black hole’s angular momentum per unit mass. Equation (3) shows
that when the black hole’s rotation parameter is removed, it only leaves a
spherically symmetric dilaton black hole composed of mass, electric charge,
and asymptotic value Garfinkle:1990qj . When the dilaton parameter $r_{2}$
vanishes, the Kerr-Sen solution returns to the Kerr black hole.

The Kerr-Sen black hole’s event horizon $r_{\pm}$ is specified by
$\tilde{\Delta}=0$, so we can get

$\left\\{\begin{split}&r_{+}=M-\frac{r_{2}}{2}+\sqrt{\left(M-\frac{r_{2}}{2}\right)^{2}-a^{2}}\\\
&r_{-}=M-\frac{r_{2}}{2}-\sqrt{\left(M-\frac{r_{2}}{2}\right)^{2}-a^{2}}.\end{split}\right.$
(5)

According to equation (LABEL:5), there are two event horizon within
$0\leqslant\frac{r_{2}}{M}\leqslant 2$. In the next section, we will consider
non-relativistic, steady-state, irrotational fluid accretion process in the
spactime of Kerr-Sen black hole and within the $r_{2}$-interval range.

##  III An accretion solutions for ultrarelativistic prefect fluid

The ultrarelativistic perfect fluid accretion solution to Kerr-Sen black holes
is the main goal of this section. The exact, steady-state fluid with an
ultrarelativistic stiff equation of state can be described as

$P=K\rho^{2},$ (6)

where $K$ is the constant, $\rho$ and $P$ are rest mass density and pressure
respectively. Since perfect fluids satisfy the first law of thermodynamics
$\,\mathrm{d}h=\frac{\,\mathrm{d}P}{\rho}$ Font_2007  where $h$ represents the
specific enthalpy which is defined as $\frac{u+P}{\rho}$ with $u$ the total
energy density. We substitute it into the formula above to get

$h=2K\rho.$ (7)

In hydromechanics, the speed of sound is a critical physical quantity that can
be used to analyze the velocity of the flow element. We derive the fluid’s
sound speed $c^{2}_{s}\equiv\sqrt{\frac{\partial\ln{h}}{\partial\ln{\rho}}}=1$
that indicates the fluid’s velocity is subsonic.

The basic equations for investigating fluid evolution are continuity equation
and energy-momentum tensor conservation equation, which have the following
forms:

$\left\\{\begin{split}&\nabla_{\mu}J^{\mu}=\nabla_{\mu}\left(\rho
U^{\mu}\right)=0\\\ &\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}\left(\rho
hU^{\mu}U^{\nu}+P\delta^{\mu}_{\nu}\right)=0.\end{split}\right.$ (8)

If we combine the above formula with the first law, we can get the Euler
equation of perfect fluid

$U^{\mu}\nabla_{\mu}\left(hU_{\nu}\right)+\nabla_{\nu}h=0,$ (9)

where $U^{\mu}=\frac{\,\mathrm{d}x^{\mu}}{\,\mathrm{d}\tau}$ is the four-
velocity for fluid and it satisfies $U^{\mu}U_{\mu}=-1$. The relativistic
vorticity tensor is Petrich:1988zz .

$\omega_{\mu\nu}=\nabla_{\nu}\left(hU_{\mu}\right)-\nabla_{\mu}\left(hU_{\nu}\right),$
(10)

Using the projection operator
$P^{\mu}_{\nu}=\delta^{\mu}_{\nu}+U^{\mu}U_{\nu}$, we can project the
vorticity tensor into space part.

$\tilde{\omega}_{\mu\nu}=P^{\alpha}_{\mu}P^{\beta}_{\nu}\left[\nabla_{\beta}\left(hU_{\alpha}\right)-\nabla_{\alpha}\left(hU_{\beta}\right)\right],$
(11)

we can derive the result consistent with the definition in (10) by
substituting (9) into the expression of (11). It shows that for a irrotational
fluid with zero vortex tensor, $hU_{\mu}$ could be expressed as the gradient
of velocity potential $\Phi$.

$hU_{\mu}=\nabla_{\mu}\Phi,$ (12)

Four-velocity normalization conditions require
$h=\sqrt{-\nabla_{\mu}\Phi\nabla^{\mu}\Phi}$. Using equations (12) and
(LABEL:8), we can get

$\nabla_{\mu}\left(\frac{\rho}{h}\nabla^{\mu}\Phi\right)=0,$ (13)

it is nonlinear differential equation and can be rewritten as in the case of
ultrarelativistic fluids with stiff equations of state (7).

$\nabla_{\mu}\nabla^{\mu}\Phi=0.$ (14)

The issue of thermodynamics is completely transformed into the solution of a
massless scalar field with boundary conditions. However, not all solutions to
the equations are viable; a fluid’s four-velocity must be required timelike.
Within the acceptable ranges, pressure and specific enthalpy can be obtained
theoretically by calculating the solution of the field $\Phi$. In the
following subsection, we will explore, the analytical expression of this
solution in the Kerr-Sen metric with Boyer-Lindquist coordinates.

###  III.1 The solution in Kerr-Sen black hole

Petrich, Shapiro, and Teukolsky first investigated the solution of the
equation (14) in Kerr black hole under the assumption that the boundary
conditions are fulfilled. We’ll look for its solution to the case of Kerr-Sen
black hole. Equation (3) gives the Kerr-Sen black hole’s metric, and its
determinant is the $g=-\tilde{\Sigma}^{2}\sin^{2}{\theta}$. The analytical
expression of equation (14) is expressed in the Kerr-Sen background as

$-\frac{\tilde{A}}{\tilde{\Delta}\tilde{\Sigma}}\partial^{2}_{t}\Phi+\frac{1}{\tilde{\Sigma}}\partial_{r}\left(\tilde{\Delta}\partial_{r}\Phi\right)+\frac{1}{\tilde{\Sigma}\sin{\theta}}\partial_{\theta}\left(\sin{\theta}\partial_{\theta}\Phi\right)+\frac{\tilde{\Delta}-a^{2}\sin^{2}{\theta}}{\tilde{\Sigma}\tilde{\Delta}\sin^{2}{\theta}}\partial_{\phi}^{2}\Phi-\frac{4Mra}{\tilde{\Delta}\tilde{\Sigma}}\partial_{t}\partial_{\phi}\Phi=0,$
(15)

where $\tilde{A}$ is defined as

$\tilde{A}=\left(r(r+r_{2})+a^{2}\right)^{2}-a^{2}\sin^{2}{\theta}\tilde{\Delta}.$
(16)

According to the steady-state fluid accretion procedure, the solution of (3)
can be written as

$\Phi=e\left(-t+\sum_{lm}R_{lm}(r)Y_{lm}(\theta,\phi)\right),$ (17)

The general solution has decomposed by the standard spherical harmonics as a
set of complete basis. A positive $e$ is related to the Bernoulli constant
(per unit mass) which can be solved by

$e=-hU_{\mu}\left(\frac{\partial}{\partial t}\right)^{\mu}=-\partial_{t}\Phi,$
(18)

with the timelike Killing vector field $\left(\frac{\partial}{\partial
t}\right)^{\mu}=(1,0,0,0)$. Substituting (17) back into (15), we can obtain

$\frac{\,\mathrm{d}}{\,\mathrm{d}r}\left(\tilde{\Delta}\partial_{r}R_{lm}(r)\right)-l(l+1)R_{lm}(r)+\frac{m^{2}a^{2}}{\tilde{\Delta}}R_{lm}(r)=0.$
(19)

To simplify the radial component of (19), we introduce a new variable $z$ so
that the equation for $R_{lm}(r)$ becomes the Legendre equation.

$\left\\{\begin{split}&z\equiv\frac{r-M+\frac{r_{2}}{2}}{\sqrt{(M-\frac{r_{2}}{2})^{2}-a^{2}}}\\\
&(1-z^{2})\frac{\,\mathrm{d}^{2}}{\,\mathrm{d}z^{2}}R_{lm}(z)-2z\frac{\,\mathrm{d}}{\,\mathrm{d}z}R_{lm}(z)+l(l+1)R_{lm}(z)-\frac{(i\alpha
m)^{2}}{1-z^{2}}R_{lm}(z)=0.\end{split}\right.$ (20)

Then we can solve the aforementioned equation in general form

$\Phi=e\left(-t+\sum_{l}\left(A_{l}P_{l}(z)+B_{l}Q_{l}(z)\right)Y_{l0}+\sum_{lm}\left(A^{+}_{lm}P^{im\alpha}_{l}(z)+A^{-}_{lm}P^{-im\alpha}_{l}(z)\right)Y_{lm}(\theta,\phi)\right),$
(21)

where $\alpha\equiv\frac{a}{\sqrt{\left(M-\frac{r_{2}}{2}\right)^{2}-a^{2}}}$,
$m$ is a positive integer, and coefficients $A_{l}$, $B_{l}$, $A^{+}_{lm}$,
and $A^{-}_{lm}$ could be determined by boundary conditions. $P_{l}(z)$,
$P^{im\alpha}_{l}(z)$, and $Q_{l}(z)$ are Legendre functions of first and
second kinds respectively. $P^{im\alpha}_{l}(z)$ can be described as a
hypergeometric function in the following form Babichev:2008dy ,

$\left\\{\begin{split}&P^{im\alpha}_{l}(z)\propto\mathrm{e}^{imX}F\left(-l,l+1,1-im\alpha;\frac{1-z}{2}\right)\\\
&X\equiv\frac{\alpha}{2}\ln{\frac{z+1}{z-1}}=\frac{a}{2\sqrt{\left(M-\frac{r_{2}}{2}\right)^{2}-a^{2}}}\ln{\frac{r-r_{-}}{r-r_{+}}}.\end{split}\right.$
(22)

Using solution (12) and the four-velocity normalization condition, we obtain

$\left\\{\begin{split}&hU_{t}=-e\\\
&hU_{r}=\frac{e\left[\sum_{l}\left(A_{l}P^{\prime}_{l}(z)+B_{l}Q^{\prime}_{l}(z)\right)Y_{l0}(\theta,\phi)+\sum_{lm}\left(A^{+}_{lm}P^{\prime
im\alpha}_{l}(z)+A^{-}_{lm}P^{\prime-
im\alpha}_{l}(z)\right)Y_{lm}(\theta,\phi)\right]}{\sqrt{(M-\frac{r_{2}}{2})^{2}-a^{2}}}\\\
&hU_{\theta}=e\left[\sum_{l}\left(A_{l}P_{l}(z)+B_{l}Q_{l}(z)\right)\frac{\partial
Y_{l0}(\theta,\phi)}{\partial\theta}+\sum_{lm}\left(A^{+}_{lm}P^{im\alpha}_{l}(z)+A^{-}_{lm}P^{-im\alpha}_{l}(z)\right)\frac{\partial
Y_{lm}(\theta,\phi)}{\partial\theta}\right]\\\
&hU_{\phi}=e\left[\sum_{lm}\left(A^{+}_{lm}P^{im\alpha}_{l}(z)+A^{-}_{lm}P^{-im\alpha}_{l}(z)\right)\frac{\partial
Y_{lm}(\theta,\phi)}{\partial\phi}\right],\\\ \end{split}\right.$ (23)

with

$h^{2}=\frac{1}{\tilde{\Delta}\tilde{\Sigma}}\left[e^{2}\tilde{A}-\tilde{\Delta}^{2}(hU_{r})^{2}-\tilde{\Delta}(hU_{\theta})^{2}-\frac{\tilde{\Delta}-a^{2}\sin^{2}{\theta}}{\sin^{2}{\theta}}(hU_{\phi})^{2}+4Mrae(hU_{\phi})\right].$
(24)

Noticing that the prime represents the derivative of the variable $z$. Due to
the analytical solution we found a physical constraint, it must be finite at
$r_{+}$, which requires $A^{+}_{lm}$ to be 0 Using the limiting behavior
$z\rightarrow 1$ $(r\rightarrow r_{+})$, we have

$h^{2}\rightarrow\frac{\left(r_{+}(r_{+}+r_{2})+a^{2}\right)^{2}e^{2}-\left(\sqrt{(M-\frac{r_{2}}{2})^{2}-a^{2}}\sum_{l}B_{l}Y_{lm}(\theta,\phi)\right)^{2}}{\tilde{\Delta}(r_{+})\tilde{\Sigma}(r_{+})}.$
(25)

By analyzing specific enthalpy from perspective of continuity at horizon, it
can conclude that only $B_{0}$ contributes, and other $B_{l}(l>0)$ disappears,
which implies that

$B_{0}=\frac{e(r_{+}(r_{+}+r_{2})+a^{2})}{\sqrt{(M-\frac{r_{2}}{2})^{2}-a^{2}}},$
(26)

where $Y_{00}$’s contribution is absorbed into $B_{0}$. By using the formula
$Q_{0}(x)=\frac{1}{2}\ln{\frac{1+x}{1-x}}$, the associated solution
degenerates to

$\left\\{\begin{split}&\Phi=e\left(-t+\psi(r,\theta,\phi)\right)\\\
&\psi(r,\theta,\phi)\equiv\frac{r_{+}(r_{+}+r_{2})+a^{2}}{2\sqrt{(M-\frac{r_{2}}{2})^{2}-a^{2}}}\ln{\frac{r-r_{-}}{r-r_{+}}}+\sum_{l,m\geqslant
0}A^{-}_{lm}\mathrm{e}^{-im\chi}F\left(-l,l+1;1+im\alpha;\frac{1-z}{2}\right)\\\
&\times Y_{lm}(\theta,\phi)\end{split}\right.$ (27)

Because $\Phi$ is a real scalar field, there is the constraint that
$A^{-}_{l-m}=(-1)^{m}(A^{-}_{lm})^{*}$. In studying steady-state accretion
issues, the axisymmetric distribution of fluid and the reflection symmetry on
the equatorial plane are usually assumed. These allow us to consider only
modes with $m=0$ and angular quantum number $l$ must be an even integer. The
coefficient $A^{-}_{lm}$ corresponds to distinctive structure of the fluid,
which will be determined later.

Formulas (12), (LABEL:23) and (LABEL:27) give the exact analytical formulas
for the $U^{\mu}$ and specific enthalpy, which yields

$\left\\{\begin{split}&\frac{hU^{t}}{e}=\frac{\tilde{A}}{\tilde{\Delta}\tilde{\Sigma}}-\frac{2Mra}{\tilde{\Delta}\tilde{\Sigma}}\partial_{\phi}\psi\\\
&\frac{hU^{r}}{e}=\frac{\tilde{\Delta}}{\tilde{\Sigma}}\partial_{r}\psi\\\
&\frac{hU^{\theta}}{e}=\frac{1}{\tilde{\Sigma}}\partial_{\theta}\psi\\\
&\frac{hU^{\phi}}{e}=\frac{2Mra}{\tilde{\Delta}\tilde{\Sigma}}+\frac{\tilde{\Delta}-a^{2}\sin^{2}{\theta}}{\tilde{\Delta}\tilde{\Sigma}\sin^{2}{\theta}}\partial_{\phi}\psi,\\\
\end{split}\right.$ (28)

and

$\frac{h^{2}}{e^{2}}=\frac{\tilde{A}}{\tilde{\Delta}\tilde{\Sigma}}-\frac{\tilde{\Delta}}{\tilde{\Sigma}}(\partial_{r}\psi)^{2}-\frac{1}{\tilde{\Sigma}}(\partial_{\theta}\psi)^{2}-\frac{\tilde{\Delta}-a^{2}\sin^{2}{\theta}}{\tilde{\Delta}\tilde{\Sigma}\sin^{2}{\theta}}(\partial_{\phi}\psi)^{2}-\frac{4Mra}{\tilde{\Delta}\tilde{\Sigma}}(\partial_{\phi}\psi).$
(29)

There are two inherent constraints in (LABEL:28) and (29): the first is that
the right-hand side of (29) needs to be positive since the four-velocity field
$U^{\mu}$ requires to be timelike, however, it can be seen that not every
point $r$ in spacetime can encounter the timelike criteria. Nevertheless, it
could be addressed by selecting $A^{-}_{lm}$ small enough such that ensure
that $h$ is well defined in the spherical shell’s interval $r_{+}\leqslant
r\leqslant R$ ($R$ is a spherical radius). The second is that $U^{r}(r_{+})<0$
for the fluid flows radially into the event horizon. However, it can prove
that this condition is automatically satisfied during the accretion solution
of the ultrarelativistic fluid (LABEL:27).

To summarise, we discussed the solution of $\Phi$ under boundary conditions at
$r=r_{+}$, obtained formulas for $U^{\mu}$, specific enthalpy, and provided
two physical constraints. The fluid accretion rate will be determined in the
next subsection.

###  III.2 Mass accretion rate in ultrarelativistic stiff fluid

The accretion rate of black hole is one of the most significant issues in
astronomy. It is the phenomenon derived from the mass conservation flow that
describes how rapidly the black hole absorbs the surrounding fluids. In
geometry, the number of Killing vector fields determine the conserved
quantities of the physical system. Because the Kerr-Sen black hole has the
symmetries of $t$ and $\phi$ coordinates, the conservation flow may be defined

$\left\\{\begin{split}&J^{\mu}_{\varepsilon}=-T^{\mu}_{\nu}\left(\frac{\partial}{\partial
t}\right)^{\nu}\\\
&J^{\mu}_{L}=T^{\mu}_{\nu}\left(\frac{\partial}{\partial\phi}\right)^{\nu},\end{split}\right.$
(30)

which correspond to the energy, the angular momentum, and the mass flow
$J^{\mu}=\rho U^{\mu}$. Combining equation (12) and the stiff equation of
state, conserved current can be deduced as

$\left\\{\begin{split}&J^{\mu}=\frac{\rho}{h}\nabla^{\mu}\Phi\\\
&T^{\mu}_{\nu}=\frac{\rho}{h}\left(\nabla^{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}\delta^{\mu}_{\nu}\nabla_{\sigma}\Phi\nabla^{\sigma}\Phi\right).\end{split}\right.$
(31)

The mass flow through the surface of sphere with a certain radius is referred
to mass accretion rate:

$\dot{\mathscr{M}}=-\int_{S}J^{r}\sqrt{-g}\,\mathrm{d}\theta\,\mathrm{d}\phi,$
(32)

with dot denoting the time derivative and $S$ meaning any sphere of radius
$r$. Particularly, it should be highlighted that $\dot{\mathscr{M}}$ is
independent of position chosen during the steady-state accretion process, so
we select the surface $r=r_{+}$.

In order to have a explicit expression for the accretion rate, we solve
$\dot{\mathscr{M}}$ by substituting $J^{r}=\frac{\rho
e\tilde{\Delta}}{\Sigma}\partial_{r}\psi$ and (LABEL:27) into (32),

$\dot{\mathscr{M}}=-\frac{\rho
e}{h}\int_{r=r_{+}}\tilde{\Delta}\partial_{r}\psi\sin{\theta}\,\mathrm{d}\theta\,\mathrm{d}\phi=\frac{4\pi\rho
e}{h}\left[r_{+}(r_{+}+r_{2})+a^{2}\right].$ (33)

It seems that the ultrarelativistic fluid with $\frac{\rho}{h}=\frac{1}{2K}$
would be elevated outside of the integral. energy accretion rate $\dot{M}$ and
angular momentum accretion rate $\dot{J}$ could be rewritten as

$\left\\{\begin{split}&\dot{M}=-\int_{r=r_{+}}J^{r}_{\varepsilon}\tilde{\Sigma}\sin{\theta}\,\mathrm{d}\theta\,\mathrm{d}\phi=e\dot{\mathscr{M}}\\\
&\dot{J}=-\int_{r=r_{+}}J^{r}_{L}\tilde{\Sigma}\sin{\theta}\,\mathrm{d}\theta\,\mathrm{d}\phi=0.\end{split}\right.$
(34)

The formulas show mass accretion rate and energy accretion rate are both
constant. However, the transformation of angular momentum is zero. It means
that the steady-state fluid is axisymmetric $(m=0)$ in the Boyer-Lindquist
coordinates system. This exactly matches our expected axisymmetric fluid
distribution. So far we have given analytical expressions for conserved
quantities. We will mainly calculate axisymmetric quadrupolar flow and choked
accretion in the last two sections.

##  IV The quadrupolar flow Solution in ZAMO Framework

To investigate the relative velocity in three dimensions, it is more
convenient to apply the zero angular momentum observer (ZAMO) framework
Frolov_2014 ; Liao:2016mum . The observer’s four-velocity depends on
$\frac{\partial}{\partial t}+\Omega\frac{\partial}{\partial\phi}$
($\Omega=\frac{2Mar}{\tilde{A}}$ is angular velocity). On the background of
the Kerr-Sen black hole, four orthogonal basis vectors in this reference frame
could be expressed as

$\left\\{\begin{split}&e_{\hat{t}}=\sqrt{\frac{\tilde{A}}{\tilde{\Sigma}\tilde{\Delta}}}\left(1,0,0,\Omega\right),~{}e_{\hat{r}}=\sqrt{\frac{\tilde{\Delta}}{\tilde{\Sigma}}}\left(0,1,0,0\right)\\\
&e_{\hat{\theta}}=\frac{1}{\sqrt{\tilde{\Sigma}}}\left(0,0,1,0\right),~{}~{}~{}~{}e_{\hat{\phi}}=\sqrt{\frac{\tilde{\Sigma}}{\tilde{A}\sin^{2}{\theta}}}\left(0,0,0,1\right).\end{split}\right.$
(35)

According to the coordinate transformational relation
$U^{\hat{\mu}}=e^{\hat{\mu}}_{\beta}U^{\beta}$, four-velocity within ZAMO
framework is

$\left\\{\begin{split}&\frac{h}{e}U^{\hat{t}}=\sqrt{\frac{\tilde{A}}{\tilde{\Sigma}\tilde{\Delta}}}\left(1-\Omega\partial_{\phi}\psi\right),~{}\frac{h}{e}U^{\hat{r}}=\sqrt{\frac{\tilde{\Delta}}{\tilde{\Sigma}}}\partial_{r}\psi\\\
&\frac{h}{e}U^{\hat{\theta}}=\frac{1}{\sqrt{\tilde{\Sigma}}}\partial_{\theta}\psi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{h}{e}U^{\hat{\phi}}=\sqrt{\frac{\tilde{\Sigma}}{\tilde{A}\sin^{2}{\theta}}}\partial_{\phi}\psi.\\\
\end{split}\right.$ (36)

The Lorentz factor is $\Gamma\equiv U^{\hat{t}}=\frac{1}{\sqrt{1-V^{2}}}$ and
the additional three-velocity corresponding to formula (LABEL:36) could be
defined as

$\left\\{\begin{split}&V^{\hat{r}}=\frac{U^{\hat{r}}}{U^{\hat{t}}}=\frac{\tilde{\Delta}\partial_{r}\psi}{\sqrt{\tilde{A}}\left(1-\Omega\partial_{\phi}\psi\right)}\\\
&V^{\hat{\theta}}=\frac{U^{\hat{\theta}}}{U^{\hat{t}}}=\frac{\sqrt{\tilde{\Delta}}\partial_{\theta}\psi}{\sqrt{\tilde{A}}\left(1-\Omega\partial_{\phi}\psi\right)}\\\
&V^{\hat{\phi}}=\frac{U^{\hat{\phi}}}{U^{\hat{t}}}=\frac{\tilde{\Sigma}\sqrt{\tilde{\Delta}}\partial_{\phi}\psi}{\tilde{A}\sin{\theta}\left(1-\Omega\partial_{\phi}\psi\right)},\\\
\end{split}\right.$ (37)

with

$V=\sqrt{(V^{\hat{r}})^{2}+(V^{\hat{\theta}})^{2}+(V^{\hat{\phi}})^{2}}.$ (38)

Apparently, these representations of three- and four-velocities lead to
variety of relevant conclusions. The timelike characteristics of four velocity
require $V<1$ ($U^{\hat{t}}>0$), so that the three-velocity highlighted above
is limited by the following two conditions.

$\Omega\partial_{\phi}\psi<1,$ (39)

and

$V^{2}=\frac{1}{\tilde{A}\left(1-\Omega\partial_{\phi}\psi\right)^{2}}\left[(\tilde{\Delta}\partial_{r}\psi)^{2}+\tilde{\Delta}(\partial_{\theta}\psi)^{2}+\frac{\tilde{\Delta}\tilde{\Sigma}^{2}}{\tilde{A}\sin^{2}{\theta}}(\partial_{\phi}\psi)^{2}\right]<1\\\
.$ (40)

The case of $(l,m)=(2,0)$, which is the most essential solution we employ to
characterize the choked accretion, is also the solution of the so-called
axisymmetric quadrupolar flow. It shows that the quadrupole flow solution
automatically satisfies the constraint equation (39)
$\left(\partial_{\phi}\psi=0\right)$. The restriction imposed by (40) will be
explained in more detail further. Peculiarly, the Bondi-Michel-type accretion
corresponds to $l=0$ and the wind accretion is accompanied by $l=1$, which
have been explored in 1993AmJPh..61..825K ; Tejeda:2017xqt . We extend
hypergeometric series near the black hole’s horizon and obtain quadrupolar
flow’s velocity potential $\Phi$ which yields

$\left\\{\begin{split}&\Phi=e\left(-t+\frac{r_{+}\left(r_{+}+r_{2}\right)+a^{2}}{2\sqrt{\left(M-\frac{r_{2}}{2}\right)^{2}-a^{2}}}\ln{\frac{r-r_{-}}{r-r_{+}}}+NF\left(r,\theta,\phi\right)\right)\\\
&F\left(r,\theta,\phi\right)=(3\cos^{2}{\theta}-1)\left(3r^{2}-6Mr+2M^{2}+a^{2}-2Mr_{2}+3rr_{2}+\frac{1}{2}r^{2}_{2}\right),\end{split}\right.$
(41)

where $N$ represents the coefficient $A^{-}_{20}$. Inserting the conclusion of
(LABEL:41) into (LABEL:36) and (LABEL:37), we can derive

$\left\\{\begin{split}&V^{\hat{r}}=\frac{\tilde{\Delta}NF_{,r}-2Mr_{+}}{\sqrt{\tilde{A}}}\\\
&V^{\hat{\theta}}=\sqrt{\frac{\tilde{\Delta}}{\tilde{A}}}NF_{,\theta}\\\
&V^{\hat{\phi}}=0,\\\ \end{split}\right.$ (42)

and

$\frac{h^{2}}{e^{2}}=\frac{\tilde{A}(1-V^{2})}{\tilde{\Sigma}\tilde{\Delta}}=\frac{1}{\tilde{\Sigma}}\left(\frac{\tilde{A}}{\tilde{\Delta}}-\tilde{\Delta}\left(NF_{,r}-\frac{2Mr_{+}}{\tilde{\Delta}}\right)^{2}-N^{2}F^{2}_{,\theta}\right),$
(43)

with

$\left\\{\begin{split}&F_{,r}=(3\cos^{2}{\theta}-1)\left(6r-6M+3r_{2}\right)\\\
&F_{,\theta}=-6\cos{\theta}\sin{\theta}\left(3r^{2}-6Mr+2M^{2}+a^{2}-2Mr_{2}+3rr_{2}+\frac{1}{2}r^{2}_{2}\right).\\\
\end{split}\right.$ (44)

The distribution of fluid is entirely defined by equation (LABEL:42). By
resolving the stagnation point, we may figure out its morphological structure.
The stagnation related to $V^{\hat{\theta}}=0$ could be resolved with
$\theta=0,\frac{\pi}{2}$, and $\pi$, while $V^{\hat{r}}=0$ will eventually fix
the location of the stagnation point. Since the flow is symmetric in the
equatorial plane, the preceding describes two scenarios.

Case 1:$(\theta=0,\pi)$ $N>0$ with inflow traveling through the equatorial
plane $(\theta=\frac{\pi}{2})$ and outflow along the polar axis. On the polar
axis, the stagnation point $r=r_{s}$ is symmetrical distribution and fulfills

$N=\frac{Mr_{+}}{(6r_{s}-6M+3r_{2})(r_{s}-r_{+})(r_{s}-r_{-})}.$ (45)

Case 2:$(\theta=\frac{\pi}{2})$ $N<0$ is associated with inflow entering at
both ends of the polar axis and outflow along the equatorial plane, in this
case two stagnation points exist symmetrically in the equatorial plane.

$N=-\frac{2Mr_{+}}{(6r_{s}-6M+3r_{2})(r_{s}-r_{+})(r_{s}-r_{-})}.$ (46)

We showed that, while the black hole’s parameters are fixed, the choice of the
potential function’s coefficients totally determines where the stationary
point will be located. In quadrupolar flow situation, the restriction of (40)
is equivalent to the right-hand side of the last term of equation (43) that is
positive, it follows that

$G\equiv g_{0}(r)\cos^{4}{\theta}+g_{1}(r)\cos^{2}{\theta}+g_{2}(r)>0,$ (47)

with

$\left\\{\begin{split}&g_{0}(r)=9N^{2}\left(2a^{2}+4M^{2}+6r^{2}+6rr_{2}+r^{2}_{2}-4M(3r+r_{2})\right)^{2}-9N^{2}(6r-6M+3r_{2})^{2}\tilde{\Delta}\\\
&g_{1}(r)=12NMr_{+}(6r-6M+3r_{2})+6N^{2}(6r-6M+3r_{2})^{2}\tilde{\Delta}+a^{2}\\\
&-9N^{2}\left(2a^{2}+4M^{2}+6r^{2}+6rr_{2}+r^{2}_{2}-4M(3r+r_{2})\right)^{2}\\\
&g_{2}(r)=-N^{2}(6r-6M+3r_{2})^{2}\tilde{\Delta}-4NMr_{+}(6r-6M+3r_{2})+r^{2}+rr_{2}+2Mr+4M^{2}\frac{r+r_{+}}{r-r_{-}}.\\\
\end{split}\right.$ (48)

To ensure that there are two event horizons, the range of $r_{2}$ is
constrained as $0<r_{2}<M$ when $a=0.5M$. The physical solution depends on
four-velocity’s timelike restrictions shown in the isothermal diagram figure
1, where the horizontal axis indicates angle $\theta$ and the vertical axis
represents dimensionless value of $r$. It illustrates that coefficient $N$ has
an effect in the confined physical regions. The value of $N$ is the most
essential factor in determining the height of the diagram. The diagram becomes
narrower as $N$ increases, until there is no longer a desirable region.

Figure 1:  The above contour map representing G. Both figures assume $a=0.5M$,
$r_{2}=0.8M$, and different $N$. The physical areas associated with $NM$ =
0.01 and 0.02. The transition from bottom to the top region contact with the
value range of $G$.

Figure 2:  Two figures correspond to the streamline (solid arrow) and the
isocontour (dotted line). The change in colour entirely determines the
magnitudes of the three-velocity. The diagram use cylindrical-like coordinate
transformation $\overline{r}=\sqrt{r^{2}+a^{2}}\sin{\theta}$,
$z=r\cos{\theta}$ to make the diagram more distinct.

The streamline and isocontour diagram of the non-relativistic stiff fluid
determined by equation (LABEL:42) is given by Figure 2. The isocontours of
three-velocity are represented by the dashed line, the streamline is shown by
the black arrow, and the stagnation point is symbolized by a blue area, so
that part of the fluid flows into the black hole and others flow to the poles
or the equatorial plane. From Figure 1, we chose $R=10M$ as the outer radius
for investigation in the following discussion, since the timelike
characteristic of velocity is assured in the interval $(r_{+},R)$. It is worth
noticing that the black hole domain is symbolised by the hole in the middle,
which also highlights that $\Phi$ is not well defined in the ZAMO coordinate
system.

##  V Choked accretion

The choked accretion is a hydrodynamic mechanism that was first proposed by
Hernandez, and applied to the discussion of Schwarzschild black holes and Kerr
black holes Aguayo-Ortiz:2020qro ; Tejeda:2019fwr ; Aguayo-Ortiz:2019fap . The
mechanism describes that perfect fluid is injected radially along the
equatorial plane. If the injection velocity is too large, the anisotropic
density distribution fluctuates dramatically so that part of the fluid
deviates from the original orbit and is ejected along poles. This process is
reversible, as seen in cases 1 and 2, respectively. We’ll focus on $N>0$ (case
1) in this section.

It was mentioned previously that to fulfill the requirements of four-
velocity’s timelike character, a spherical surface with a radius of $R$ must
be supplied. Physical values in the spherical shell are well-defined. At the
same time, one reference point should always be established for physical
quantity to be measurable. The injection rate $V_{0}$ is set at
$\left(R,\frac{\pi}{2}\right)$ in equatorial plane and ejection rate $V_{ej}$
is determined at $(R,0)$ can be represented as

$\left\\{\begin{split}&V_{0}=V^{\hat{r}}\left(R,\frac{\pi}{2}\right)=\frac{\tilde{\Delta}_{0}\left(6R-6M+3r_{2}\right)N+2Mr_{+}}{\sqrt{\tilde{A_{0}}}}\\\
&V_{ej}=V^{\hat{r}}\left(R,0\right)=\frac{2V_{0}\sqrt{\tilde{A_{0}}}-6Mr_{+}}{R(R+r_{2})+a^{2}},\\\
\end{split}\right.$ (49)

with

$\left\\{\begin{split}&\tilde{\Delta}_{0}=(R-r_{+})(R-r_{-})\\\
&\tilde{A}_{0}=R(R+r_{2})(a^{2}+R^{2}+Rr_{2})+2MRa^{2}.\\\ \end{split}\right.$
(50)

Since the fluid is subluminal and need to ensure that flows out along both
ends of the polar axis in a spherical shell, we must require $(0<V_{ej}<1)$ to
acquire the range of the initial velocity $V_{0}$.

$\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}<V_{0}<\frac{R(R+r_{2})+a^{2}+6Mr_{+}}{2\sqrt{\tilde{A}_{0}}}.$
(51)

Up to now, we haven’t solved the stagnation point or the value of the
coefficient $N$ in principle, thus boundary constraints must be imposed. It
can be concluded from (LABEL:37) that

$e=h\Gamma\sqrt{\frac{\tilde{\Sigma}\tilde{\Delta}}{\tilde{A}}}=h_{0}\Gamma_{0}\sqrt{\frac{\tilde{\Sigma}_{0}\tilde{\Delta}_{0}}{\tilde{A_{0}}}},$
(52)

where

$\left\\{\begin{split}&\Gamma_{0}=\frac{1}{\sqrt{1-V^{2}_{0}}}\\\
&\tilde{\Sigma}_{0}=R(R+r_{2}).\\\ \end{split}\right.$ (53)

Using the (7) and substituting (52) into (LABEL:37), we get

$\frac{h}{h_{0}}=\frac{\rho}{\rho_{0}}=\frac{\Gamma_{0}\sqrt{\tilde{\Sigma}_{0}\tilde{\Delta}_{0}\tilde{A}}}{\Gamma\sqrt{\tilde{\Sigma}\tilde{\Delta}\tilde{A}_{0}}},$
(54)

where density $\rho_{0}=\rho\left(R,\frac{\pi}{2}\right)$ and specific
enthalpy $h_{0}=h\left(R,\frac{\pi}{2}\right)$ are determined by parameters
$M$, $a$ and $r_{2}$ as the value at reference point. According to (LABEL:49),
it follows that

$N=\frac{V_{0}\sqrt{\tilde{A}}_{0}-2Mr_{+}}{\tilde{\Delta}_{0}(6R-6M+3r_{2})},$
(55)

Simultaneously, these two equivalent conditions (45) and (55) give the
location of stagnation point.

$r_{s}=M-\frac{r_{2}}{2}+\left(\lambda-\sqrt{\lambda^{2}-\frac{\left(M^{2}-a^{2}-Mr_{2}+\frac{1}{4}r^{2}_{2}\right)^{3}}{27}}\right)^{\frac{1}{3}}+\left(\lambda+\sqrt{\lambda^{2}-\frac{\left(M^{2}-a^{2}-Mr_{2}+\frac{1}{4}r^{2}_{2}\right)^{3}}{27}}\right)^{\frac{1}{3}},$
(56)

with

$\lambda\equiv\frac{Mr_{+}(R-r_{+})(R-r_{-})(R-M+\frac{r_{2}}{2})}{2(V_{0}\sqrt{\tilde{A_{0}}}-2Mr_{+})}.$
(57)

The correlation between $r_{s}$ and $r_{2}$ is depicted in Figure 3 (left)
with $V_{0}=0.3,0.4,0.5$ . It is worth noting that the range selected by
$V_{0}$ satisfies the constraints of (51). As the value of $V_{0}$ increases,
the initial value will become smaller, and the stagnation point will show a
downward trend with the increase of $r_{2}$.

Figure 3: The stagnation point is depicted by the real line on the left with
$a=0.5M$, $R=10M$ and $V_{0}=0.3,0.4,0.5$. The density perturbation colour
temperature map is determined by the right.

###  V.1 Distribution of the density and the streamlines

Through (54) we define a quantity $\delta$ that describes the relative
deviation of the ejection’s value at $\left(R,0\right)$ from the injection’s
value at $\left(R,\frac{\pi}{2}\right)$, which yield

$\delta_{\rho}\equiv
1-\frac{\rho\left(R,0\right)}{\rho\left(R,\frac{\pi}{2}\right)}=1-\sqrt{\frac{\left(1-V^{2}_{ej}\right)R(R+r_{2})\left(R(R+r_{2})+a^{2}\right)}{\left(1-V^{2}_{0}\right)\left(R(R+r_{2})(a^{2}+R^{2}+Rr_{2})+2MRa^{2}\right)}}.$
(58)

As can be shown, the conclusion is that when $\delta_{\rho}=1$, the density of
location $(R,0)$ tends to zero. When $\delta_{\rho}=0$, the density of the
ejection point is equivalent to incoming point. The distribution of
$\delta_{\rho}$ is given in figure 3 (right). It illustrates that the maximum
value of $\delta_{\rho}$ is not greater than 1, and the entire color is
homogeneous. The orange area at the bottom reveals that the injection point’s
density is extremely smaller than the ejection point’s density. The density
ratio approaches zero as $\delta_{\rho}$ increases in the orange area
suggesting a reduced ejection rate.

Finally, we obtain a streamlined diagram that can depict the fluid’s
trajectory. Applying equations (LABEL:42) to cancel time $t$ can result in

$\alpha=\cos{\theta}\left[1+\frac{\sin^{2}{\theta}\left(r-r_{+}\right)\left(r-r_{-}\right)\left(r-M+\frac{r_{2}}{2}\right)}{2\left(r_{s}-r_{+}\right)\left(r_{s}-r_{-}\right)\left(r_{s}-M+\frac{r_{2}}{2}\right)}\right],$
(59)

where $\alpha$ is the constant of integration. Streamlines with $|{\alpha}|=1$
are connected to the stagnation point, whereas those with $|\alpha|>1$ escape
via the bipolar outflow, and the $|\alpha|<1$ area is absorbed into black
hole. The streamline is described in figure 4, and the diagram(left) shows
that the complete diagram is symmetrically distributed along
$\theta=\frac{\pi}{2}$ (the middle solid line). The dashed line depicts the
streamline’s trajectory, which differs from $|{\alpha}|$ values and represents
several pathways. It is straightforward to notice that the dividing line
($|{\alpha}|=1$) explains whether fluid is absorbed into or ejected from the
black hole. The streamlined diagram (right) in the cylindrical-like coordinate
system ($\overline{r},z$) will be shown. Similarly, the boundary is
$|{\alpha}|=1$ of the blue region generated by the dotted line.

Figure 4:  The parameters $r_{2}=0.8M$, $a=0.5M$ and $r_{s}=4M$ for two
figures. The solid line (left) denotes $\theta=\frac{\pi}{2}$, while the
dashed line reflects the isocontour line. The right side employs the
coordinate transformation $\overline{r}=\sqrt{r^{2}+a^{2}}\sin{\theta}$,
$z=r\cos{\theta}$, and the centre empty region represents the interior of
Kerr-Sen black hole.

###  V.2 Injection rate, ejection rate and critical angle

The inflow and outflow should be accompanied by injection rate and ejection
rate, respectively. When matched with the black hole mass accretion rate, the
whole concept of choked accretion is presented. Those three rates are
connected in the following way:

$\dot{\mathscr{M}}_{in}=\dot{\mathscr{M}}+\dot{\mathscr{M}}_{out},$ (60)

where the mass accretion rate is decided by formula (32). It can be described
as

$\dot{\mathscr{M}}=8\pi
Mr_{+}\rho_{0}\Gamma_{0}\sqrt{\frac{R(R+r_{2})\tilde{\Delta}_{0}}{\tilde{A}_{0}}}.$
(61)

It’s obvious that the accretion rate is restricted by the parameters chosen
and the bounding sphere radius $R$. Now we must identify the critical angle
which appears to correspond to fluid moving towards the black hole in the
interval $(\theta_{c},\pi-\theta_{c})$. In this interval, the projection of
the tangent vector of the streamline along the radial direction of the black
hole is negative. The critical angle $\theta_{c}$ could be determined by using
$V^{\hat{r}}$ in (LABEL:42). The flow is absorbed by the black hole related to
$V^{\hat{r}}<0$, it follows that

$\theta_{c}=\arccos{\left[3\left(1-\frac{2Mr_{+}}{V_{0}\sqrt{\tilde{A}_{0}}}\right)\right]^{-\frac{1}{2}}},$
(62)

where $\dot{\mathscr{M}}_{in}$ associated with the critical angle is

$\left\\{\begin{split}&\dot{\mathscr{M}}_{in}=-2\int^{\frac{\pi}{2}}_{\theta_{c}}\rho
U^{r}\tilde{\Sigma}\sin{\theta}\,\mathrm{d}\theta\,\mathrm{d}\phi=\frac{8\pi
q\rho eMr_{+}}{h}=q\dot{\mathscr{M}}\\\
&q\equiv\frac{2\cos^{3}{\theta_{c}}}{3\cos^{2}{\theta_{c}}-1}=\frac{\sqrt{\tilde{A}_{0}}V_{0}}{3\sqrt{3}Mr_{+}}\left(1-\frac{2Mr_{+}}{V_{0}\sqrt{\tilde{A}_{0}}}\right)^{-\frac{1}{2}},\end{split}\right.$
(63)

Similarly, the ejection rate could be obtained
$\dot{\mathscr{M}}_{out}=(q-1)\dot{\mathscr{M}}$ by using (60). The physics
relating to $\dot{\mathscr{M}}_{in}=\dot{\mathscr{M}}$ is that all fluids are
dragged into black hole while the critical angle $\theta_{c}=0$ $(q=1)$. The
condition $\theta_{c}=0$ provides $V_{0}=\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}$
by (62). It shows that this is consistent with (51) restriction, it follows.

$\dot{\mathscr{M}}_{in}=\left\\{\begin{aligned} &\dot{\mathscr{M}},\quad
V_{0}\leqslant\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}\\\
&q\dot{\mathscr{M}},\quad\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}\leqslant
V_{0}\leqslant\frac{R(R+r_{2})+a^{2}+6Mr_{+}}{2\sqrt{\tilde{A}_{0}}},\\\
\end{aligned}\right.$ (64)

and

$\dot{\mathscr{M}}_{out}=\left\\{\begin{aligned} &0,\quad
V_{0}\leqslant\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}\\\
&(q-1)\dot{\mathscr{M}},\quad\frac{3Mr_{+}}{\sqrt{\tilde{A}_{0}}}\leqslant
V_{0}\leqslant\frac{R(R+r_{2})+a^{2}+6Mr_{+}}{2\sqrt{\tilde{A}_{0}}}.\\\
\end{aligned}\right.$ (65)

As $V_{0}$ range changes, $\dot{\mathscr{M}}_{in}$ steadily increases that
results in a maximum value $\theta_{max}$ calculated by the formula below

$\frac{1}{3\cos{\theta^{2}_{max}}}=\frac{R(R+r_{2})+a^{2}+2Mr_{+}}{R(R+r_{2})+a^{2}+6Mr_{+}}$
(66)

with maximum value of injection accretion rate
$\dot{\mathscr{M}}_{inmax}=q_{\max}\dot{\mathscr{M}}$, where

$q_{\max}=\frac{2\cos^{3}{\theta_{max}}}{3\cos^{2}{\theta_{max}}-1}.$ (67)

According to the above formula, when $R$ is large, the maximum angle
$\theta_{max}\approx 54.7^{\circ}$, $q_{\max}\rightarrow\infty$ corresponds to
the boundary case. To understand the transformation of accretion rate ratio,
we depicts the ratio of accretion rate
$\frac{\dot{\mathscr{M}}_{out}}{\dot{\mathscr{M}}_{in}}\equiv\eta$ in figure
5.

Figure 5:  The left figure depicts the ratio of accretion rate as the
parameter $r_{2}$ changes when $V_{0}=0.2,0.3,0.4,0.5$ are respectively fixed.
The figure on the right depicts a contour map of ratios, where the upper and
lower boundaries represent the range of values for $V_{0}$.

The figure above shows how $r_{2}$ influences the accretion rate ratio. When
$V_{0}$ is fixed, we see that the ratio $\eta$ grows as $x$ increases. When
the value of parameter $x$ is fixed in figure 5 (right), $V_{0}$ varies from
the bottom boundary to the upper, and the variation range of $\eta$ is
$40\%-70\%$. It can be shown that as the parameter $r_{2}$ in the Kerr-Sen
black hole increases, it amplifies the ratio of ejection rate to incident
rate. Consequently, this leads to a scenario where the majority of the fluid
escapes from the black hole, while the inhaled fluid becomes a relatively
minor component.

##  VI Conclusion and discussion

In this work, we first studied the choked accretion process of
ultrarelativistic fluid onto axisymmetric Kerr-Sen black holes in Einstein-
Maxwell-Dilation-Axion gravity. Based on solving procedure mentioned by
Petrich, Shapiro, and Teukolsky, we calculated the solution describing the
velocity potential field $\Phi$ of a stationary, irrotational
ultrarelativistic fluid in the Boyer-Lindquist coordinates system. We
discussed the analytical expression of four-velocity and converted it to the
ZAMO framework to give three-velocity. The value of mass accretion rate, the
energy accretion rate, and the angular momentum increase rate was also
finished. We found that the angular momentum increase rate is zero, indicating
that the fluid has an axisymmetric distribution $(m=0)$.

Secondly, as a result of the axial symmetry of perfect fluid and the
requirement of reflection symmetry in the equatorial plane, we gave the
lowest-order (2,0) solution, commonly known as quadrupolar flow. We
investigated the character of quadrupolar flow and showed that one of the two
constraints described is automatically satisfied, and timelike requirement of
four-velocity is also solved once an appropriate region is chosen.
Subsequently, we examined the correlation between dilation parameters $r_{2}$
as a function of coefficients $N$ and stagnation point $r_{s}$ in the
solution, as well as the color temperature and streamlined diagrams of these
regions with timelike properties.

Finally, we introduced the choked accretion model and restricted the physical
region to $(r_{+},R)$ to ensure that the solution is reasonable. We also
calculated the stagnation point analytic formulas based on the boundary values
and drew isocontour. Within the permissible range, the initial velocity
$V_{0}$ at the reference point allows us to draw a picture of its dependency
on the parameters according to different colors. The connection between the
density ratio of endpoint and initial point is also evaluated. We also gave
the streamline diagram in cylindrical-like coordinates and discovered that the
streamline associated with $\alpha\leqslant 1$ indicates that fluid is
absorbed into the black hole, whereas the streamline with $\alpha>1$
represents flow ejected along poles. The injection rate and ejection rate were
discussed in detail at the end of the article. The conditions for the
existence of the injection rate were given and the critical angle $\theta_{c}$
during the accretion process was determined so that flow in the interval
$(\theta_{c},\pi-\theta_{c})$ is sucked into black hole.

In future work, we will try to solve the choked accretion issue of curled
fluid numerically and provide numerical results by merging it with the
equation of motion with boundary via space-time and matter field coupling. We
will consider the reaction of matter, and explore the full accretion issue
using dark energy and dark matter as carriers.

###### Acknowledgements.

This study is supported in part by National Natural Science Foundation of
China (Grant No.12333008) and Hebei Provincial Natural Science Foundation of
China (Grant No. A2021201034).

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