---
abstract: |
    In this paper the geodesics motion and accretion process around a subclass of Horndeski/Galileon black holes are investigated. Firstly, we present spherically symmetric geometries in a Horndeski/Galileon black hole spacetime by considering an isothermal fluid around the black hole. Then we focus on three main issues: in the first step circular orbits of test particles and their stability in equatorial plane are examined in details. Then, by treating perturbations via restoring forces, oscillations of particles around the central object are studied. Finally, the accretion process, the critical speed of the flow and accretion rate are investigated in this setup properly.\
    [**PACS**]{}: 97.10.Gz, 04.50.Kd\
    [**Key Words**]{}: Horndeski/Galileon Black Holes, Accretion Disks, Orbital Motion, Radiation Efficiency
author:
- 'K. Salahshoor$^{a,}$[^1] and K. Nozari$^{a,b,}$[^2]'
title: 'Circular orbits and accretion process in a class of Horndeski/Galileon black holes'
---

Introduction
============

Recent observations indicate that General Relativity might indeed be modified at large distances. Scalar-tensor theories are a prototype alternative and also they are most probably the simplest, consistent and nontrivial modification of the General Relativity (Babichev *et al.* 2016). Gregory Horndeski (Horndeski 1974) proposed the most general action of the scalar-tensor gravity. The same results were formulated in terms of Galileons interactions (Deffayet 2011, Kobayashi 2011, Deffayet 2013). Therefore, Horndeski/Galileons is the most general class of scalar-tensor field models with second-order field equations and it may be considered as a proper generalization of General Relativity in high energy regime (Latosh 2016). We consider the following action which is a subclass of general Horndeski/Galileons class

$$\label{a1}
S=\int d^{4}x\sqrt{-g}[\zeta R-\eta(\partial\phi)^{2}+\beta
G^{\mu\nu}\partial_{\mu}\phi \partial_{\nu}\phi-2\Lambda].$$

Here $R$ is the Einstein-Hilbert term, $G^{\mu\nu}$ is the Einstein tensor, $\phi$ is the scalar field, $\Lambda$ is a cosmological constant term, $\zeta>0$, $\eta$ and $\beta$ are model parameters. On astrophysical scales, which covers also static and spherically symmetric solutions, this scalar-tensor theory may play a crucial role. On the other hand, any modification of General Relativity must be consistent with astrophysical observations. So, it is important to see how astrophysical processes such as accretion onto black holes work in this scalar-tensor framework and can be used also as a probe to see viability of these theories from experimental viewpoint. The issue of black hole accretion disk is studied in some subclasses of scalar-tensor theories. However, there is a gap in literature since black hole accretion disk has not been studied in the mentioned scalar-tensor theory as a subclass of the general Horndeski/Galileon scenario. This is the motivation of the present study and we are going to fill this gap in this paper.

Accretion disks are constructed by rotating gaseous materials that move in bounded orbits because of the gravitational force of central mass, such as Young Stellar Objects (YSO), main-sequence stars (MSs), neutron stars (NSs), and supermassive black holes in Active Galactic Nuclei (AGN). In such systems, particles orbits are stable, but when the orbits of these materials become unstable, following it, accretion will be happened. Accretion is the process by which a massive central object such as a black hole captures particles from a fluid in its vicinity. The particles which accelerate from rest must be passed through a critical point, the point where the velocity of the gas matches its local sound speed. Then the gas falls onto the central mass at supersonic velocities. This process leads to increase in mass of the black hole (Martnez 2014). In addition, extra energy would be released in this process where this energy can be source of some astrophysical phenomena, such as the production of powerful jets, high-energy radiation, and quasars (Kato *et al.* 2008). Therefore, the study of the geodesic structure of particles in the vicinity of black holes and specially investigation about some characteristic radii such as marginally bound orbits $(r_{mb})$ and innermost stable circular orbits $(r_{isco})$ are interesting issues for a careful study of the subject matter. These radii are very important in the study of black hole accretion disks. For example, in thin accretion disks, the inner edge of the disk coincides with the innermost stable circular orbit (ISCO) and the efficiency of the energy released, which describes the significance of converting rest-mass energy into radiative energy (Xie *et al.* 2012), can be determined from this radii.

The location of unstable or stable circular orbits is consistent with the maximum or minimum of the effective potential respectively. In Newtonian theory, for any value of the angular momentum, the effective potential has a minimum and then stable circular orbit is free to have arbitrary radius, that is, there is no minimum radius of stable circular orbit, (ISCO) (Kaplane 1949). But this situation is different when the effective potential has a complicated form depending on the particle angular momentum and other parameters or when one incorporates general relativistic effects. For example, in General Relativity and for particles moving around the Schwarzschild black hole, for any value of the angular momentum, the effective potential has two extrema (minimum or maximum). But, only for a specific value of the angular momentum the two points coincide. This point introduces ISCO where is located at $r=3r_{g}$ (Landau 1993, Kaplane 1949) where $r_{g}$ is the Schwarzschild radius. In different metrics, the properties of spacetime affects the locations of these radii and some parameters such as: specific energy, angular momentum and angular velocity are important in the position of these points. A lot of research programs are devoted to study these radii and their physical significance. Ruffini *et al.* (1971) and Bardeen *et al.* (1972) studied the properties of innermost stable circular orbits around the Kerr black hole. Even Hobson *et al.* (2006) described these features in details in their textbook on General Relativity. The radiation efficiency of accretion disks, $\eta$, for Schwarzschild and Kerr black holes was obtained by Novikov and Thorne (1973) which its value lies in the range of $0.057$ $-$ $0.43$ depending on the black hole spin. The Kerr-like metric was constructed by Johannsen and Psaltis (2011) and then Johannsen (2013) has studied the accretion disks around such black holes. The study of the geodesic motion and the circular orbits of charged particles around weakly magnetized rotating black holes are carried out by Tursunov *et al.* (2016).

In an accretion disk particles move in stable orbits but when a perturbations, as a result of restoring forces, act on the particles, oscillations around the circular orbit can take place in vertical and radial directions with epicyclic frequencies. Happening the oscillations (in response to perturbations) in the inner region of an accretion disk is another important characteristic of these regions. Oscillations can be source of strong and chaotic time variations in spectrum of such systems. Therefore, study about orbital and epicyclic frequencies (radial and vertical) play an important role in the physics of relativistic accretion disks around the black holes. Isper (1994, 1996), Wagoner (1999), Kato (2001) and Ortega-Rodriges *et al.* (2006) have studied in this field. Resonance between such a frequency modes which proposed by Kluzniak and Abromowicz (2000) can be a physical mechanism for existing Quasi-periodic oscillations (QPOs). QPOs in the X-ray fluxes of some astrophysical objects such as a neutron star and black hole sources have been reviewed by many researches including van der Klis (2000) and McClintock *et al.* (2003). Johannsen (2013) has examined the radial and vertical epicyclic frequencies in the Kerr-like metric.

With these preliminaries, in this paper we study nonrotating black hole solutions with accretion disk in a subclass of Horndeski/Galileons spacetime general class. For simplicity we restrict our study to equatorial plane in a polar coordinates system. Firstly, the singularity and event horizon in this spacetime geometry are presented. Then, in order to investigate the circular orbits, effective potential is obtained in this setup. We study the locations of several characteristic radii, such as: marginally stable circular orbits $r_{isco}$, marginally bounded circular orbits $r_{mb}$ and photon orbits $r_{ph}$ in equatorial plane. Also, the ISCO binding energy, the maximum radiation efficiency, the emission and temperature in equatorial epicyclic frequencies are computed. Finally, some dynamical parameters and critical accretion of isothermal fluid are investigated in details.

This paper is organized as follows: In section 2 we introduce Horndeski/Galileon spacetime. The general formalism of a test particle’s motion is discussed in section 3 where circular motion, stable circular orbits and oscillations are examined in subsections 3.1, 3.2 and 3.3. In section 4 and it’s subsections, the general form of some dynamical parameters such as critical speed of the flow, accretion rate and the time of accretion for an isothermal fluid are obtained. In section 5 we have explained physically all of these results for a subclass of general solution of the Horndeski/Galileon black hole. Finally, section 6 is devoted to summary and discussion.

Horndeski/Galileon Spacetime
=============================

We study static and spherically symmetric limit of black hole solutions in a subclass of general Horndeski/Galileon theories (see Maselli (2015) for the case of slowly rotating black holes in Horndeski theory and also Babichev et. al. (2016) for black hole and star solutions for Horndeski theory). The general form of the line-element for such systems with the metric signature $(+, -, -, -)$ is described by $$\label{a1}
ds^{2}=h(r)dt^{2}-\frac{1}{f(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}).$$ The metric functions, $f(r)$ and $h(r)$, depend only on the radial coordinate $r$ and are given as follows (Tretyakova 2016) $$\begin{aligned}
\label{a2}
f(r)=\frac{(\beta+\eta r^2)h(r)}{\beta (r h(r))'},\end{aligned}$$ $$\label{a1}
h(r)=-\frac{\mu}{r}+\frac{1}{r}\int \frac{k(r)}{(\beta+\eta r^2)}dr,$$ $$\phi(r)=qt+\psi(r)\,.$$ In these relations, $\mu$ plays the role of the mass term and $k$ is obtained from the following constraint equation $$\label{a1}
q^{2}\beta(\beta+\eta r^2)^{2}-[2\zeta \beta+(2\zeta
\eta-\lambda)r^{2}]k+C_{0}k^{\frac{3}{2}}=0\,,$$ where $C_{0}$ is a constant of integration. It is important to note that the static metric (2) has the time rescaling symmetry. So, if $h(r)$ is a solution then $c h(r)$ should be also a solution where $c$ is a constant. This means that $c$ should be set for $t$ to describe the proper time in the relevant region, namely $h \rightarrow 1$. We note also that equation (5) is not a solution, but rather an ansatz on the scalar field. This type of scalar configuration was firstly considered by Babichev and Charmousis (2014) in which the solution for $\psi(r)$ is also presented. About the stability of solutions, Ogawa *et al.* (2016) and Takahashi and Suyama (2017) showed that solutions with nonzero $q$ are generically plagued by ghost or gradient instability. However, for solutions with $q=0$, there exist some stable solutions (see Kobayashi *et al.* (2012), Takahashi and Suyama (2017), and Tretyakova and Takahashi (2017)). We note that recently it has been pointed out by (Babichev et. al. (2017) and (2018)) that the conclusion of Ogawa et al. (2016) and Takahashi and Suyama (2017) is incorrect. Now from the recent relations, various solutions for different values of $C_{0}$ and $q$ can be obtained. We are going to discuss a common expression in this paper.

Now we study the properties of a subclass of black hole solutions in Horndeski/Galileon gravity as has been introduced above. The mentioned Horndeski spacetime contains a singularity at the location where the following condition holds $$\begin{aligned}
\label{a2}
f(r)=0\,.\end{aligned}$$ The event horizon is a null surface. A surface that is defined as $f(x^{\mu})=0$ will be null if $$\label{a1}
g^{\mu\nu}n_{\mu} n_{\nu}=0,$$ where $n_{\mu}$ is the normal 4-vector to the surface and it is defined as $n_{\mu}=\nabla_{\mu}f$. Since we are interested in to study the problem in the equatorial plane, then the relation (8) can be written as $g^{rr}(\partial_{r}f)^{2}=0$. Therefore, in the radial distance that $g^{rr}=0$ or equivalently $f(r)=0$, we would have an event horizon. The location of the event horizon in the geometry is a radial distance from the center of the core where the metric is singular, except the intrinsic singularity which cannot be removed via coordinate transformation.

Test Particle’s Motion: General Formalism
=========================================

The motion of a test particle is governed by the geodesic structure of the underlying spacetime manifold. In this section, we study general form of timelike geodesics around a subclass of Horndeski/Galileon black hole. Spacetime around this black hole is static and symmetric with two Killing vectors $\xi_{t}
=\partial_{t} $ and $\xi_{\varphi}=\partial_{\varphi} $ which imply two constants of motion $E$ and $L$ (conserved energy and angular momentum per unit mass) along the trajectory as follows $$\begin{aligned}
\label{a2}
&E=-g_{\mu\nu}\xi^{\mu}_{t} u^{\nu}\equiv -u_{t}\,,\nonumber\\
&L=g_{\mu\nu}\xi^{\mu}_{\varphi} u^{\nu}\equiv u_{\varphi}\,,\end{aligned}$$ where $u^{\mu}=(u^{t}, u^{r}, u^{\theta}, u^{\varphi})$ is the four-velocity of the test particle. Using the normalization condition for four-velocity, that is $u^{\mu}u_{\mu}=1$, we have $$\label{a3}
[g_{rr}(u^{r})^{2}+g_{\theta\theta}(u^{\theta})^{2}]=[1-g^{tt}(u_{t})^{2}-g^{\varphi\varphi}(u_{\varphi})^{2}]\,.$$ From equations (9) and (10) and in equatorial plane with $\theta=\frac{\pi}{2}$, four-velocity will be given by the following components $$\begin{aligned}
\label{a6}
&u^{t}=-\frac{E}{h(r)}\nonumber\\
&u^{\theta}=0\nonumber\\
&u^{\varphi}=-\frac{L}{r^{2}}\nonumber\\
&u^{r}=[-f(r)(1-\frac{E^{2}}{h(r)}+\frac{L^{2}}{r^{2}})]^{\frac{1}{2}}\,.\end{aligned}$$ Also the following equation can be derived easily $$\label{a4}
\frac{h(r)}{f(r)}(u^{r})^{2}+V_{eff}=E^{2}\,.$$ In this equation $V_{eff}$ is the effective potential for the test particle motion that is given by $$\label{a5}
V_{eff}=h(r)\Big[1+\frac{L^{2}}{r^{2}}\Big]\,.$$ It is clear that effective potential depends on the particle’s specific angular momentum radial distribution and the spacetime parameter via $h(r)$. The study of effective potential is very useful in geodesic motion. For example, the local exterma of the effective potential determine the location of the circular orbits.

Circular Motion
----------------

For circular motion in the equatorial plane, radial component $r$ must be constant and so $u^{r}=\dot{u}^{r}=0$ must be satisfied. Therefore, from equation (12) we would have $V_{eff}=E^{2}$ and $\frac{d}{dr}V_{eff}=0$. From these relations the specific energy $E$, the specific angular momentum $L$, the angular velocity $\Omega_{\varphi}$ and angular momentum $l$ can be obtained by the following relations respectively $$\label{a7}
E^{2}=\frac{2h^{2}(r)}{2h(r)-r h'(r)}\,,$$ $$\label{a8}
L^{2}=\frac{r^{3}h'(r)}{2h(r)-r h'(r)}\,,$$ $$\label{a9}
\Omega_{\varphi}=\frac{d\varphi}{dt}\equiv
\frac{u^{\varphi}}{u^{t}}\Rightarrow
\Omega_{\varphi}^{2}=\frac{1}{2r}h'(r)\,,$$ $$\label{a10}
l^{2}=\frac{L^{2}}{E^{2}}=\frac{r^{3}}{2h^{2}(r)}h'(r)\,.$$ In order the energy and angular momentum to be real, the following condition must be satisfied $$\begin{aligned}
\label{a11}
2h(r)-r h'(r)>0\,.\end{aligned}$$ By solving this inequality, the limited area of circular orbit can be obtained. Therefore, this is the condition for existence of the circular orbits. For bound orbit the relation $E^{2}<1$ must be hold and in marginally bound orbits we have $E^{2}=1$. Then from equation (14) we fined $$\begin{aligned}
\label{a12}
rh'(r)+2h(r)[h(r)-1]=0\end{aligned}$$ By solving this equation, marginally bound orbits can be obtained easily. From equations (14) and (15) it is seen that the energy and angular momentum diverge at the radius where the following relation holds $$\begin{aligned}
\label{a11}
2h(r)-r h'(r)=0\,.\end{aligned}$$ Photon sphere can be obtained by solving this relation. In a photon sphere, photon moves on circular orbits. This region plays a crucial role in the study of gravitational lensing, since lensing effect cannot be observed below this region.

Stable Circular Orbits and Radiant Energy Flux
----------------------------------------------

The local minima of the effective potential correspond to the stable circular orbits. Thus a stable circular orbit exists if $\frac{d^{2}}{dr^{2}}V_{eff}>0$ and in addition to this condition, in marginally stable circular orbits, $r_{isco}$, the condition $\frac{d^{2}}{dr^{2}}V_{eff}=0$ must be satisfied. From equation (13) we have $$\begin{aligned}
\label{a11}
\frac{d^{2}}{dr^{2}}V_{eff}=h''(r)(1+\frac{L^{2}}{r^{2}})-4h'(r)\frac{L^{2}}{r^{3}}+6h(r)\frac{L^{2}}{r^{4}}\,.\end{aligned}$$ Accretion process is possible in $r<r_{isco}$. When falling particles from rest at infinity accrete onto the central mass, the released gravitational energy of falling particles can convert into radiation where this energy is the source of the most energetic phenomena in astrophysics. The flux of the radiant energy over the disk can be expressed in terms of the specific angular momentum $L$, the specific energy $E$ and the angular velocity $\Omega_{\varphi}$ by the following relation (see for instance Kato et al., (2008)) $$\begin{aligned}
\label{a13}
K=-\frac{\dot{M}\Omega_{\varphi,r}}{4\pi \sqrt{-g}(E-L
\Omega_{\varphi})^{2}}\int_{r_{ms}}^{r}(E-L
\Omega_{\varphi})L_{,r}dr\,,\end{aligned}$$ where $\dot{M}$ is the accretion rate, $\Omega_{\varphi,r}\equiv
\frac{d\Omega_{\varphi}}{dr}$ and the parameter $g$ is determinant of $g_{\mu\nu}$ given by $$\begin{aligned}
\label{a13}
g=det(g_{\mu\nu})=-\frac{h(r)}{f(r)}r^{4}\sin^{2}\theta\,.\end{aligned}$$ We set $\sin\theta=1$, since we restrict our studies in equatorial plan. From relations (14)-(16) we would have $$\begin{aligned}
\label{a13}
K(r)=-\frac{\dot{M}}{4\pi r^{4}}\sqrt{\frac{r f(r)}{2
h(r)h'(r)}}\Big(\frac{[2 h(r)-r h'(r)][r h''(r)-h'(r)]}{[2 h(r)+r
h'(r)]^{2}}\Big)\int_{r_{ms}}^{r}\mathcal{F}(r)dr\,,\end{aligned}$$ where by definition $$\begin{aligned}
\label{a13}
 \mathcal{F}(r)\equiv\sqrt{\frac{r}{2h'(r)}}\frac{[2 h(r)+r
h'(r)][- h''(r)r h(r)+2r h'^{2}(r)-3h'(r) h(r)]}{[2 h(r)-r
h'(r)]^{2}}\,.\end{aligned}$$

The steady-state accretion disk model is supposed to be in thermodynamical equilibrium. Then the radiation emitted from the surface of the disk can be as a black body radiation. So, the relation $K(r)=\sigma T^{4}(r)$ can be hold between energy flux emitted at the surface of the disk and effective temperature of the disk ($\sigma$ is the Stefan-Boltzman constant). Using this relation, temperature distribution on the disk by assuming thermal black body radiation can be obtained easily and then we can compute the luminosity $L(\nu)$ of the disk. The observed luminosity at the distance $d$ to the source with the disk inclination angle $\gamma $ has the following form (Torres 2002) $$\begin{aligned}
\label{a58}
L(\nu)=4 \pi d^{2} I(\nu)=\frac{8}{\pi}(\cos \gamma)\int
_{r_{i}}^{r_{f}}\int_{0}^{2\pi} \frac{\nu_{e}^{3}\,r\, d\varphi\,\,
dr}{\exp(\frac{\nu_{e}}{T})-1}\,,\end{aligned}$$ where $I(\nu)$ is the thermal energy flux. In this relation $r_{i}$ indicates the position of the inner edge and we take $r_{i}=r_{ms}$. Also $r_{f}$ indicates the outer edge of the disk. Since for any kind of general relativistic compact object the flux over the disk surface could be vanishing at $r\rightarrow \infty$, we take $r_{f}\rightarrow \infty$. The emitted frequency is given by $\nu_{e}=\nu(1+z)$ where the redshift factor $z$, by neglecting the light bending, can be written as follows $$\begin{aligned}
\label{a58}
z=\frac{1+\Omega_{\varphi} r \sin \varphi\, \sin
\gamma}{\sqrt{-g_{tt}-\Omega_{\varphi}^{2}g_{\varphi\varphi}}}-1\,.\end{aligned}$$

The efficiency of the accreting flow is another important characteristic of the mass accretion process. The maximum efficiency of transforming gravitational energy into radiative flux of such particles between innermost circular orbit and infinity, $\eta^{*}$, is defined as the ratio of the specific binding energy of the innermost circular orbit to the specific rest mass energy which is given by the following relation $$\begin{aligned}
\label{a58}
\eta^{*}=1-E_{isco},\end{aligned}$$ where $E_{isco}$ is the specific energy of a particle rotating in an innermost stable circular orbit. This relation is valid for the case where all the emitted photons can escape to infinity.

Now we focus on perturbations. If a perturbation acts on the fluid element, the motion of a test particle will be nearly circular orbit in the equatorial plane and the particle will oscillate around the circular orbit with three components of motion, the issue which is discussed in the next section.

Oscillations
-------------

In an accretion disk, various types of oscillatory motions as a result of restoring forces are expected. Restoring forces act on perturbations in the accretion disks resulting Horizontal and Vertical oscillations. Some of these restoring forces in accretion disks are resulting from rotation of the disk and from a vertical gravitational filed. When a fluid element is displaced in the radial direction, it will return to its equilibrium position due to a restoring force resulting from rotation of the fluid. In accretion disks, because of existence of central object, centrifugal force is balanced by the gravitational force. When the former dominates over the latter or the reverse happens, the element of flow will be pushed inward or outward to return to the original radius with epicyclic frequency $\Omega_{r}$. On the other hand, when a fluid element is perturbed in the vertical direction, the vertical component of the gravitational field returns the perturbed element toward equilibrium position, that is, the equatorial plane. As a result of this restoring force, the element of the fluid makes harmonic oscillation around the equatorial plane with vertical epicyclic oscillations $\Omega_{\theta}$ (Kato *at al.* 2008).

In a general relativistic discussion about the motion of the fluid in an accretion disk, three frequencies around the central object are important. Circular motion at the orbital frequency $\Omega_{\varphi}$, harmonic radial motion at the radial frequency $\Omega_{r}$ and the harmonic vertical motion at the vertical frequency $\Omega_{\theta}$. As we have stated, resonance between such frequencies can be source of quasi-periodic oscillations which leads to chaotic and quasi-periodic variability in X-ray fluxes from many galactic black holes. Study in this field is important in some sense. For this purpose, radial and vertical motions around a circular equatorial plane are discussed in this section.

Radial and vertical motions can be explained by $\frac{1}{2}(\frac{dr}{dt})^{2}=V^{(r)}_{eff}$ and $\frac{1}{2}(\frac{d\theta}{dt})^{2}=V^{(\theta)}_{eff}$ where from equation (10), to describe radial motion $u^{\theta}=0$, and also for describing the vertical motion we have $u^{r}=0$. By setting $u^{r}=\frac{dr}{d\tau}=\frac{dr}{dt}u^{t}$ and $u^{\theta}=\frac{d\theta}{d\tau}=\frac{d\theta}{dt}u^{t}$, we find $$\begin{aligned}
\label{a13}
&\frac{1}{2}(\frac{dr}{dt})^{2}=-\frac{1}{2}\frac{f(r)h^{2}(r)}{E^{2}}[1-\frac{E^{2}}{h(r)}+\frac{L^{2}}{r^{2}sin^{2}\theta}]=V^{(r)}_{eff}\,,\nonumber\\
&\frac{1}{2}(\frac{d\theta}{dt})^{2}=-\frac{1}{2}\frac{h^{2}(r)}{r^{2}E^{2}}[1-\frac{E^{2}}{h(r)}+\frac{L^{2}}{r^{2}sin^{2}\theta}]=V^{(\theta)}_{eff}\,.\end{aligned}$$ In order to investigate the radial and vertical epicyclic frequencies, small perturbations $\delta r$ and $\delta \theta$ around the circular orbit in equatorial plane are considered. By taking the time-derivative of the first equation in (29), equation describing the radial oscillations can be obtained as follows $$\begin{aligned}
\label{a14}
\frac{d^{2}r}{dt^{2}}=\frac{dV_{eff}^{(r)}}{dr}.\end{aligned}$$ For a particle which is perturbed from its original radius at $r=r_{0}$ by a deviation $\delta r=r-r_{0}$, the perturbed equation of motion is given by $$\begin{aligned}
\label{a14}
\frac{d^{2}(\delta
r)}{dt^{2}}=\frac{d^{2}V_{eff}^{(r)}}{dr^{2}}(\delta r)\Rightarrow
(\delta {\ddot r})+\Omega^{2}_{r}(\delta r)=0\,,\end{aligned}$$ where a dote denotes differential with respect to time coordinate $t$ and $\Omega_{r}^{2}\equiv-\frac{d^{2}V_{eff}^{(r)}}{dr^{2}}$. By the same procedure, for a perturbation in the vertical direction by a deviation given as $\delta \theta=\theta-\theta_{0}$ we find $$\begin{aligned}
\label{a14}
\frac{d^{2}(\delta
\theta)}{dt^{2}}=\frac{d^{2}V_{eff}^{(\theta)}}{dr^{2}}\delta
\theta\Rightarrow (\delta \ddot{\theta})+\Omega^{2}_{\theta}(\delta
\theta)=0\,,\end{aligned}$$ where $\Omega^{2}_{\theta}=-\frac{d^{2}}{d\theta^{2}}V^{(\theta)}_{eff}$. Then from equations (29) in equatorial plane we would have respectively $$\begin{aligned}
\label{a15}
&\Omega^{2}_{r}=\frac{1}{2r^{4}E^{2}}\Big\{\Big[(r^{2}+L^{2})r^{2}h^{2}(r)-h(r)r^{4}E^{2}\Big]f''(r)+\Big[(r^{2}+L^{2})2h(r)-r^{2}E^{2}\Big]r^{2}f(r)h''(r)\nonumber\\
&+2r^{2}f(r)h'^{2}(r)(r^{2}+L^{2})-2r\Big[[-(r^{2}+L^{2})2h(r)+r^{2}E^{2}]r f'(r)+4f(r)h(r)L^{2}\Big]h'(r)\nonumber\\
&-4h^{2}(r)L^{2}(-\frac{3}{2}f(r)+r f'(r)) \Big\}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a16}
\Omega^{2}_{\theta}=\frac{h^{2}(r)L^{2}}{r^{4}E^{2}}\,.\end{aligned}$$ In these equations, a prime denotes differential with respect to the radial coordinate, $r$. To proceed further, now we present basic dynamical equations in this subclass of general Horndeski/Galileons black hole spacetime.

Basic dynamical equations
=========================

In this section we provide the basic dynamical equations for our forthcoming arguments (we refer to Babichev et al., (2005) and (2013) for more details). Here we consider a perfect fluid which is specified by the following energy-momentum tensor $$\begin{aligned}
\label{a18}
T^{\mu\nu}=(p+\rho)u^{\mu}u^{\nu}-p g^{\mu\nu},\end{aligned}$$ where $p$ and $\rho$ are pressure and energy density of the fluid respectively. In this relation, $u^{\mu}$ is the fluid elements four-velocity. Because of background symmetries, in relation (35) all of the components are functions of only the radial coordinate, $r$. Since we are assuming the fluid is flowing radially in the equatorial plane ($\theta=\frac{\pi}{2}$), the general form of the four-velocity will be as follows $$\begin{aligned}
\label{a18}
u^{\mu}=\frac{dx^{\mu}}{d\tau}=(u^{t},u^{r},0,0)\,,\end{aligned}$$ where $\tau$ is the proper time along the geodesic. From this relation and under the normalization condition $u^{\mu}u_{\mu}=1$, we obtain $$\begin{aligned}
\label{a18}
u^{t}=\sqrt{\frac{f(r)+(u^{r})^{2}}{h(r)f(r)}}\,,\end{aligned}$$ where for forward flow in time, $u^{t}$ must be positive and for accretion (inward flow), $u^{r}<0$. By deriving the energy-momentum and also particle-number conservation equations, all of the required equations for studying the accretion are obtained. Conservation of the energy- momentum tensor is given by $$\begin{aligned}
\label{a18}
T^{\mu \nu}_{;\mu}=0\Rightarrow T^{\mu
\nu}_{;\mu}=\frac{1}{\sqrt{-g}}(\sqrt{-g}T^{\mu
\nu})_{,\mu}+\Gamma^{\nu}_{\alpha\mu}T^{\alpha \mu }=0\,,\end{aligned}$$ where in this relation $(;)$ shows the covariant differentiation, $\sqrt{-g}=r^{2}\sin \theta \sqrt{\frac{h(r)}{f(r)}}$ and $\Gamma$ is the second kind Christoffel symbol (affine connection) where its non-zero components are as follows $$\begin{aligned}
\label{a18}
&\Gamma^{0}_{01}=\Gamma^{0}_{10}=\frac{1}{2}\frac{h'(r)}{h(r)}\nonumber\\
&\Gamma^{1}_{00}=\frac{1}{2}h'(r)f(r),\
\Gamma^{1}_{11}=-\frac{1}{2}\frac{f'(r)}{f(r)},\ \Gamma^{1}_{22}=-r
f(r), \    \Gamma^{1}_{33}=-r f(r)\sin \theta\nonumber\\
&\Gamma^{2}_{12}=\Gamma^{2}_{21}=\frac{1}{r}\nonumber\\
&\Gamma^{3}_{13}=\Gamma^{3}_{31}=\frac{1}{r}\,.\end{aligned}$$ From these relations, equation (38) yields $$\begin{aligned}
\label{a18}
T^{10}_{,r}+\frac{1}{\sqrt{-g}}T^{10}(\sqrt{-g})_{,r}+2\Gamma^{0}_{01}T^{10}=0\,,\end{aligned}$$ where after some manipulations we obtain $$\begin{aligned}
\label{a18}
(p+\rho)u^{r}r^{2}\sqrt{(u^{r})^{2}+f(r)}\frac{h(r)}{f(r)}=A_{0}\,,\end{aligned}$$ with $A_{0}$ as an integration constant. Projecting the energy-momentum conservation law onto the four-velocity via $u_{\mu}T^{\mu \nu}_{;\nu}=0$, yields $$\begin{aligned}
\label{a18}
(p+\rho)_{,\nu}
u_{\mu}u^{\mu}u^{\nu}+(p+\rho)u^{\mu}_{;\nu}u_{\mu}u^{\nu}+(p+\rho)u_{\mu}u^{\mu}u^{\nu}_{;\nu}+p_{,\nu}g^{\mu\nu}u_{\mu}+pu_{\mu}
g^{\mu\nu}_{;\nu}=0\,.\end{aligned}$$ By considering the normalization conditions as $u_{\mu}u^{\mu}=1$ and since $g^{\mu\nu}_{;\nu}=0$, this relation reduces to $$\begin{aligned}
\label{a18}
(p+\rho)u^{\nu}_{;\nu}+u^{\nu}\rho_{,\nu}=0\,.\end{aligned}$$ Since $A^{b}_{;a}=\partial_{a}A^{b}+\Gamma^{b}_{ac}A^{c}$, we find $$\begin{aligned}
\label{a18}
u^{r}\rho_{,r}+(p+\rho)\Big[\Gamma^{0}_{0c}u^{c}+(u^{r}_{,r}+\Gamma^{1}_{1c}u^{c})+\Gamma^{2}_{2c}u^{c}+\Gamma^{3}_{3c}u^{c}\Big]=0\,.\end{aligned}$$ By using the non-zero components of the connection, this relation after some simplification yields $$\begin{aligned}
\label{a18}
\frac{\rho'}{(p+\rho)}+\frac{1}{2}\frac{h'(r)}{h(r)}-\frac{1}{2}\frac{f'(r)}{f(r)}+\frac{u'}{u}+\frac{2}{r}=0\,,\end{aligned}$$ which after integration, we would have $$\begin{aligned}
\label{a18}
r^{2}u^{r}\sqrt{\frac{h(r)}{f(r)}} \exp(\int
\frac{d\rho}{p+\rho})=-A_{1}\,,\end{aligned}$$ where $A_{1}$ is an integration constant. Since in the left hand side $u^{r}<0$, the right hand side takes a minus sign too. So we find finally $$\begin{aligned}
\label{a18}
(p+\rho)\sqrt{h(r)\Big[\frac{(u^{r})^{2}}{f(r)}+1\Big]}\exp(-\int
\frac{d\rho}{p+\rho})=A_{2}\,,\end{aligned}$$ where $A_{2}$ is an integration constant. The equation of mass flux in this setup which is given by $$\begin{aligned}
\label{a18}
(\rho u^{\mu})_{;}\equiv \frac{1}{\sqrt{-g}}(\sqrt{-g} \rho
u^{\mu})_{,\mu}=0\end{aligned}$$ can be rewritten as $$\begin{aligned}
\label{a18}
\frac{1}{\sqrt{-g}}(\sqrt{-g} \rho
u^{\mu})_{,r}+\frac{1}{\sqrt{-g}}(\sqrt{-g} \rho
u^{\theta})_{,\theta}=0\,.\end{aligned}$$ Since we are interested in to study just in equatorial plane, the second term in equation (49) vanishes. Therefore, $\sqrt{-g} \rho
u^{\mu}$ would be as a constant, that is $$\begin{aligned}
\label{a18}
\rho u^{r}r^{2}\sqrt{\frac{h(r)}{f(r)}}=A_{3}\,,\end{aligned}$$ where $A_{3}$ is an integration constant. We note that while we have restricted our study to equatorial plane by symmetry considerations, the general case is not so complicated in essence. Because of symmetry all important characteristics of the model can be obtained in $\theta=\frac{\pi}{2}$ plane as well. Now we are going to determine dynamical parameter, critical accretion and accretion rate in this setup.

Dynamical parameters
--------------------

To proceed further, we assume isothermal fluids. These fluids flow at a constant temperature. Therefore, $p\propto \rho$ and then the speed of sound throughout the accretion process remains constant. For such fluids, the equation of state is of the form $p=k \rho$ where $k$ is the equation of state parameter. Then equations (46), (47) and (50) yield $$\begin{aligned}
\label{a18}
\frac{p+\rho}{\rho}\sqrt{h(r)\Big[\frac{(u^{r})^{2}}{f(r)}+1\Big]}=A_{4}\,,\end{aligned}$$ where $A_{4}$ is an integration constant. By substituting $p=k
\rho$, we can obtain $u$ as follows $$\begin{aligned}
\label{a18}
u=\Big(\frac{1}{k+1}\Big)\sqrt{f(r)\Big[\frac{A_{4}^{2}}{h(r)}-(k+1)^{2}\Big]}\,.\end{aligned}$$ Then from eq. (50) the density of the fluid can be obtained as $$\begin{aligned}
\label{a18}
\rho=\frac{A_{3}}{r^{2}}\frac{(k+1)}{\sqrt{A_{4}^{2}-(k+1)^{2}h(r)}}\,.\end{aligned}$$ Finally, from the relation $p=k \rho$, the pressure can be obtained easily.

Mass evolution
--------------

In realistic astrophysical cases, the mass of the black hole is not fixed in essence. By some processes such as accreting of mass from accretion disk onto black hole and also Hawking radiation, its mass will be changed gradually. The rate of change of mass can be obtained by integrating the flux of the fluid over the surface of the black hole, that is $\dot{M}\equiv\frac{dM}{dt}=-\int T^{r}_{t}
ds$ where a dot denotes the time derivative, $ds=\sqrt{-g}\,
d\theta\, d\varphi$ and $T^{r}_{t}=(p+\rho)u_{t}u^{r}$. By substituting these relations, $\dot{M}$ can be obtained as follows $$\begin{aligned}
\label{a18}
\dot{M}=-4\pi r^{2}u(p+\rho)\frac{h(r)}{f(r)}\sqrt{u^{2}+f(r)}\equiv
-4\pi A_{0}\,,\end{aligned}$$ where $A_{0}=-A_{1}A_{2}$, and $A_{2}=(p_{\infty}+\rho_{\infty})\sqrt{h(r_{\infty})}$. Therefore, we obtain $$\begin{aligned}
\label{a18}
\dot{M}=4\pi
A_{1}(p_{\infty}+\rho_{\infty})\sqrt{h(r_{\infty})}M^{2}.\end{aligned}$$

Now, time evolution of mass of the black hole with initial mass $M_{i}$ can be obtained by integration of equation (55) that can be rewritten as $$\begin{aligned}
\label{a18}
\frac{d M}{M^{2}}=\mathcal{F}t\end{aligned}$$ where $\mathcal{F}\equiv4\pi A_{1}(p+\rho)\sqrt{h(r_{\infty})}$. By integrating from equation (56) we obtain $$\begin{aligned}
\label{a18}
M_{t}=\frac{M_{i}}{1-\mathcal{F}M_{i}t}\equiv
\frac{M_{i}}{1-\frac{t}{t_{cr}}}\end{aligned}$$ where $t_{cr}= \Big [4\pi A_{1}(p+\rho)\sqrt{h(r_{\infty})}M_{i} \Big ]^{-1}$ is the critical accretion time. In the case $t=t_{cr}$ the denominator of equation (57) vanishes and the black hole mass grows up to infinity in a finite time. After determining the time evolution of the disk and black hole mass, now we study critical accretion in this setup.

Critical Accretion
------------------

Very far from the black hole, the flow is at rest but gravitational field of black hole tends to accelerate it inwards. When flow moves inwards, it must pass through critical point (sonic point) where in this point $r=r_{c}$, the four-velocity of the fluid matches the local speed of sound, $u=c_{s}$. In order to obtain sonic point, an expression for the radial velocity gradient with no other derivatives is required. From equations (50) and (51), the following two equations are obtained $$\begin{aligned}
\label{a18}
\frac{\rho'}{\rho}+\frac{u'}{u}+\frac{1}{2}\Big[\frac{h'(r)}{h(r)}-\frac{f'(r)}{f(r)}\Big]+\frac{2}{r}=0\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
\frac{\rho'}{\rho}\Big[\frac{d\ln(p+\rho)}{d\ln
\rho}-1\Big]+\frac{1}{2}\Big[\frac{h'(r)}{h(r)}-\frac{f'(r)}{f(r)}\Big]+\frac{u
u'}{u^{2}+f(r)}+\frac{1}{2}\frac{f'(r)}{u^{2}+f(r)}=0\,.\end{aligned}$$ From these equations we obtain $$\begin{aligned}
\label{a18}
\frac{d\ln u}{d\ln r}=\frac{\mathcal{D}_{1}}{\mathcal{D}_{2}}\,,\end{aligned}$$ where by definition $$\begin{aligned}
\label{a18}
\mathcal{D}_{1}=-\Big[\frac{r}{2}(V^{2}-1)\Big(\frac{h'(r)}{h(r)}-\frac{f'(r)}{f(r)}\Big)+2V^{2}-\frac{r
f'(r)}{2(u^{2}+f(r))}\Big]\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
\mathcal{D}_{2}=\Big[V^{2}-\frac{u^{2}}{u^{2}+f(r)}\Big]\,.\end{aligned}$$ With these two equations, the following relation can be obtained $$\begin{aligned}
\label{a18}
V^{2}=\frac{d\ln (p+\rho) }{d\ln \rho}-1.\end{aligned}$$ The condition for critical points is $\mathcal{D}_{1}=\mathcal{D}_{2}=0$. This condition yields $$\begin{aligned}
\label{a18}
V_{c}^{2}=\frac{r h'(r)}{4h(r)+rh'(r)}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
u_{c}^{2}=\frac{1}{4}\frac{r f(r)h'(r)}{h(r)}\,,\end{aligned}$$ where index $c$ refers to the critical point. Since the right hand side of the relation (64) must be positive, then if dependence of $h(r)$ is known, by solving the following inequality the range of critical radius can be obtained $$\begin{aligned}
\label{a18}
4h(r)+rh'(r)>0.\end{aligned}$$ Finally the sound speed $c_{s}^{2}=\frac{dp}{d\rho}$ can be obtained from (52) as $$\begin{aligned}
\label{a18}
c_{s}^{2}=A_{4}\sqrt{\frac{f(r)}{h(r)}\Big[u^{2}+f(r)\Big]}-1\,.\end{aligned}$$

 A Subclass of Horndeski/Galileon Black Hole Solutions
=======================================================

As we have stated in section 2, a huge variety of solutions can be obtained from equations (3)-(6). Here we present a solution of this subclass of the Horndeski/Galileon setup that are characterized by the parameters $A, B, C$ and $\gamma$ (see for instance Tretyakova 2016). The solutions are as follows $$\label{a1}
h(r)=C-\frac{\mu}{r}+Ar^{2}+\Delta\,,$$ and $$\label{a1}
f(r)=(1+\frac{\eta}{\beta}r^{2})\Big[\frac{\Delta+A
r^{2}+C-\frac{\mu}{r}}{3A
r^{2}+C+\frac{B}{1+r^{2}\gamma^{2}}}\Big]\,,$$ where we have set $\Delta\equiv B\frac{\tan^{-1}( r\gamma)}{r\gamma}$. The coefficients $C$, $A$, $B$ and $\gamma$ are defined as follows $$\begin{aligned}
\label{a2}
&A=-\frac{\eta}{3\beta}\,,\,\,\, B=\frac{2(1+\gamma^{2})\epsilon}{\zeta+y}\,,\,\,\, \epsilon\ll\mid y-1 \mid\nonumber\\
&\gamma=\sqrt{\frac{\eta}{\mid \beta\mid}}\frac{\zeta
+y}{\zeta-3y}\,,\,\,\, C=1-\frac{2\epsilon}{\zeta+y}\,,\,\,\,\,  y=\frac{\Lambda
\beta}{\eta}\,.\end{aligned}$$ Here $\epsilon$ is a small parameter which marks the deviation from the inherent de Sitter solution. Also the scalar field in this case is given by $$\psi^{2}(r)=-\frac{2(2\eta)^{2}r^{2}(2\eta+2\Lambda\beta)[(2\eta-2\Lambda\beta)r^{2}+4\beta]^{2}}{2\beta(2\eta-2\Lambda\beta)^{2}(2\eta r^{2}+2\beta)^{3}f(r)}\,,$$ where $\psi=\frac{d\phi}{dr}$. As we have said, singular points can be obtained by solving the relation $f(r) = 0$. From this relation singular point is located at $r_{sing}=\frac{\mu}{B+C}$.

An important issue should be stressed here: about the behavior of the scalar field at the black hole horizon (say, $r_{h}$), the scalar field seems to be divergent at the horizon from Eq. (71). But since $f(r_h )=0$ and $\frac{df}{dr}(r_h)\neq 0$, one can expand $f(r)$ in Taylor series to find $f(r)=f_0+f_{1}(r-r_{h} )+f_{2}(r-r_{h})^{2}+...$ . For $\psi(r)$ this approximates to $\frac{1}{\sqrt{f(r)}}$ in near horizon which gives for the scalar field $\phi(r)=\phi_{0}+\phi_{1}(r-r_h )^{1/2}+\phi_{2}(r-r_h )^{3/2}+...$ . Therefore, the scalar field remains finite in the near horizon region (a similar analysis can be found in the paper by Miao and Xu (2016)). We note that although the scalar field itself does not diverge at the horizon, its derivative does. However, there is no physical divergence since all invariants, such as $g^{\mu\nu}\partial_{\mu}\phi \partial_{\nu}\phi$ remain finite. For more discussion on this issue we refer to Feng et al. (2015) (see also Hadar and Reall (2017) and Caceres et al. (2017)).

We note that for nonzero $\eta$ the model given by Eqs. (68)-(70) admits solutions in which the $\Lambda$-term in action (1) is totally screened. The metric then *is not asymptotically flat* but rather it is de Sitter with the effective cosmological constant proportional to $\eta$/$\beta$ since the scalar kinetic term tends to a constant around the present time (Gubitosi and Linder, 2011). It is important to note that as has been shown by Tretyakova (2016), for $1-(B+C) < 3 \times 10^{-4}$ this metric must be equivalent to the Schwarzschild metric in the sense that it matches with the observations of the gravitational light deflection and perihelion precession. This feature guarantees the existence of bounds orbits for $(B+C)>0$ (for more discussion on this issue see Tretyakova (2016)).

It is necessary to mention that for calculations, the assumption $A\simeq 0$ is considered. Then the components of metric (2) will be as $h(r)=C-\frac{\mu}{r}+\Delta$ and $f(r)=\frac{h(r)}{B+C}$. On the other hand, since $\gamma\simeq \sqrt{A}$, due to smallness of $\gamma$, it is more suitable to substitute $\tan^{-1}(r\gamma)\simeq
r\gamma$ and therefore $\Delta\simeq B$. All calculations are done with these approximations and also with assumption $C=1$ and $0<B+C<1.125$ (Tretyakova 2016). By these assumptions, we focus mainly on the role of parameter $B$ as Horndeski/Galileon correction factor in our forthcoming treatment.

Circular Equatorial Geodesics
-----------------------------

In order to investigate circular geodesics in equatorial plane, we need the explicit form of the effective potential which is governed by equation (12) as $$\begin{aligned}
\label{a17}
V_{eff}=\Big(C-\frac{\mu}{r}+Ar^{2}+\Delta\Big)(1+\frac{L^{2}}{r^{2}})\,,\end{aligned}$$ where $\Delta\equiv B\frac{\tan^{-1} (r\gamma)}{r\gamma}$. From the condition $\frac{d^{2}}{dr^{2}}V_{eff}>0$ for existence of the stable circular orbits, we see that for $r<\frac{3}{2}\frac{\mu}{B+C}$ and $r>3\frac{\mu}{B+C}$ this condition holds, which with regard to equation (20), the location of the stable circular orbits would be at $r\geq 3\frac{\mu}{B+C}$. Then $$\begin{aligned}
\label{a18}
r_{isco}= 3\frac{\mu}{B+C}\,,\end{aligned}$$ is introduced as the radius of the innermost stable circular orbit. The left panel of figure 1 represents the effective potential versus $r$ for several values of the angular momentum $L$ in the case with $B=-0.1$. We see that for $L<2.2 \sqrt{3}$ no extremum can be observed and the first extremum is observed at $L=2.2 \sqrt{3}$ (solid circle in the figure). This point represents the location of the innermost stable circular orbit located at $r=6.6$. For larger values of the angular momentum, $V_{eff}$ has two extremum where the maximum one denotes the location of the unstable circular orbit and the minimum one denotes the stable circular orbit. By increasing the angular momentum, $V_{eff}$ will be larger and the maximum point turns to the smaller radii whereas the minimum point goes to larger radii. From the right panel of figure 1 the effect of Horndeski/Galileon correction factor $B$ on the effective potential can be seen. The effective potential achieves larger values for larger values of $B$ and by increasing this parameter, the loci of unstable circular orbit becomes closer to the central mass and stable orbits will be located farther from the central mass. Also the enhancement of the distance between these points by increasing $B$ is obvious.

\[a1\]

In addition to innermost stable circular orbit which is very important in studying the accretion around the black hole, there are other special radii where considering them is necessary. As we have stated previously, the circular orbits exist only for radii larger than the photon radius $r_{ph}$. For $r_{ph}<r<r_{ms}$, the motion of the particle will be unstable against the small perturbations. This means that particle falls into the black hole or flee away to infinity. In the region $r>r_{ms}$ the particle moves on stable circular orbits.

From equations (18)-(20), other characteristic radii including the photon sphere $r_{ph}$, circular orbit $r_{circ}$ and marginally bound orbit $r_{mb}$ can be obtained respectively as $$\begin{aligned}
\label{a18}
r_{ph}=\frac{3}{2}\frac{\mu}{B+C}\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
r_{circ}>\frac{3}{2}\frac{\mu}{B+C}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
r_{mb}=\frac{1}{4}\mu \frac{4(B+C)-3\pm
\sqrt{9-8(B+C)}}{(B+C)(B+C-1)}\,.\end{aligned}$$

We have plotted the characteristic radii versus the Horndeski/Galileon correction factor $B$ in figure 2. The value of this parameter affects the location of the characteristic radii in the vicinity of the black hole. For larger values of $B$, the location of $r_{ph}$, $r_{isco}$ and $r_{sing}$ will be closer to the black hole, whereas the behavior of $r_{mb}$ for negative and positive values of $B=0$ is different. For $B<0$, marginally bound radius decreases by increasing $B$, but for $B>0$ this behavior becomes reverse as it finally matches the innermost stable circular orbit. Since $r_{isco}$ represents the inner edge of the accretion disk, we see that in larger values of $B$, the disk will be extended close to the central mass.

\[a2\]

For a particle which moves in a circular orbit, the specific energy, specific angular momentum, angular velocity and angular momentum in equatorial plane can be derived as $$\begin{aligned}
\label{a18}
E^{2}=\frac{2\Big(C-\frac{\mu}{r}+Ar^{2}+\Delta\Big)^{2}}{\Big(2C-3\frac{\mu}{r}+3\Delta-\frac{B}{1+r^{2}\gamma^{2}}\Big)}\,,\end{aligned}$$ $$\begin{aligned}
\label{a17}
L^{2}=r^{2}\frac{\Big(\frac{\mu}{r}+2Ar^{2}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}\Big)}{\Big(2C-3\frac{\mu}{r}+3\Delta-\frac{B}{1+r^{2}\gamma^{2}}\Big)}\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
\Omega_{\varphi}^{2}=\frac{1}{2r^{2}}\Big(\frac{\mu}{r}+2Ar^{2}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}\Big)\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
l^{2}=\frac{r^{2}}{2}\frac{\Big(\frac{\mu}{r}+2Ar^{2}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}\Big)}{\Big(-\frac{\mu}{r}+Ar^{2}+\Delta+C\Big)^{2}}\,,\end{aligned}$$ respectively. For the innermost stable circular orbit, these relations reduce to $$\begin{aligned}
\label{a18}
E^{2}_{isco}= \frac{8}{9}(B+C)\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
L^{2}_{isco}= \frac{3\mu ^{2}}{(B+C)^{2}}\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
\Omega^{2}_{isco}=\frac{1}{54}\frac{(B+C)^{3}}{\mu^{2}}\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
l^{2}=\frac{27}{8}\frac{\mu^{2}}{(B+C)}\,,\end{aligned}$$ respectively.

\[a3\]

\[a4\]

\[a5\]

\[a6\]

\[a7\]

\[a7\]

In figure 3, the behavior of specific energy and angular momentum versus the radius are shown and the effect of the Horndeski/Galileon correction factor $B$ is studied. The upper left panel represents variation of the specific energy versus $r$. In the upper right panel the location of this radius in innermost stable circular orbits is shown by the solid circle. The empty circle denotes the limit of the bound orbit. Increasing the Horndeski/Galileon correction factor enhances the energy and decreases the range of bound orbit radius. The lower left panel represents the angular momentum where it gets smaller values by increasing $B$. The loci of these parameters in innermost stable circular orbits are represented by circles on the lower right panel of the figure.

Now by knowing $E$ in ISCO, we are able to determine the radiation energy efficiency of accretion. As we have said previously, the efficiency of accretion is $1-E$, where the maximum efficiency of accretion given by $1-E_{isco}$ in innermost stable orbit in this setup is $1-\sqrt{\frac{8}{9}(B+C)}$. In figure 4, the efficiency of accretion is plotted versus $r$ for different values of the Horndeski/Galileon correction factor $B$ where we have denoted the maximum with a circle in this figure. In the case $B=0$ where coincides with the Schwarzschild black hole (dashed line), the efficiency equals to $0.057$ at $r=6 r_{g}$ as usual. For negative values of the Horndeski/Galileon parameter, efficiency goes up and becomes greater than the Schwarzschild black holes ones. For example, in the case with $B=-0.1$, the efficiency equals to $0.105$ at $r=6.66 r_{g}$ where is tangible. On the other hand, positive values of $B$ is accompanied with decreasing of the efficiency.

Now we can study radiation flux from the surface of the accretion disk by knowing $E$, $L$ and $\Omega_{\varphi}$ in equatorial plane and $r_{ms}$. The Flux of the radiation energy of the accretion disk is obtained from equations (24) and (25) as $$\begin{aligned}
\label{a18}
&K(r)=-\frac{1}{8}\dot{M}\sqrt{\frac{2r^{2}(1+\frac{\eta}{\beta}r^{2})}{(3Ar^{2}+C+\frac{B}{1+r^{2}\gamma^{2}})(2Ar^{2}+\frac{\mu}{r}+
\frac{B}{1+r^{2}\gamma^{2}}-\Delta)}} \times \nonumber\\
&\frac{1}{r}\Big\{3\Delta+2C-3\frac{\mu}{r}-\frac{B}{1+r^{2}\gamma^{2}}\Big\}
\Big\{3\Delta-3\frac{\mu}{r}-3\frac{B}{1+r^{2}\gamma^{2}}-2\frac{Br^{2}\gamma^{2}}{(1+r^{2}\gamma^{2})^{2}}\Big\}\times \nonumber\\
&\Big\{\pi
r^{4}(\Delta+4Ar^{2}+2C-\frac{\mu}{r}+\frac{B}{1+r^{2}\gamma^{2}})^{2}\Big\}^{-1}\int_{r_{_{m_{s}}}}^{r}\mathcal{F}(r)
dr\,,\end{aligned}$$ where by definition $$\begin{aligned}
\label{a18}
&\mathcal{F}(r)=\frac{1}{2}\Big\{\sqrt{\frac{2r^{2}}{2Ar^{2}+\frac{\mu}{r}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}}}(4Ar^{2}-\frac{\mu}{r}+
\Delta+\frac{B}{1+r^{2}\gamma^{2}}+2C)\times\nonumber\\
&\Big(\frac{1}{r}(\Delta+Ar^{2}+C-\frac{\mu}{r})(\Delta-8Ar^{2}-\frac{\mu}{r}-\frac{B}{1+r^{2}\gamma^{2}}+\frac{2Br^{2}\gamma^{2}}{(1+r^{2}\gamma^{2})^{2}})\times\nonumber\\
&+2(\frac{B}{1+r^{2}\gamma^{2}}-\Delta+2Ar^{2}+\frac{\mu}{r})\Big)\Big\}\Big\{-3\frac{\mu}{r}+3\Delta+2C-\frac{B}{1+r^{2}\gamma^{2}}\Big\}^{-2}\,.\end{aligned}$$ Then the temperature can be obtained by using the equation $K=\sigma T^{4}$. In figure 5, the relation between the radiation flux and temperature is shown. The radiation flux has a maximum in the vicinity of the black hole and decreases at the smaller radii. By increasing the Horndeski/Galileon correction factor $B$, the energy flux raises and the maximum of emission flux tends to the smaller radii but the reverse happens after this point. The dependence on this parameter is very considerable in the vicinity of the black hole, but it is relatively weak far from the black hole. These behaviors are the same for the temperature. As we have said in subsection 3.2, the luminosity can be compute from equation (26), but because of complexity of the required equations, solving this equation is not possible analytically.

Epicyclic Frequencies
---------------------

If a perturbation acts on a particle moving on a circular orbit in the equatorial plane, the particle experiences small oscillations in the vertical and radial directions. Using the epicyclic frequencies given by equations (33) and (34), we derive the radial and vertical epicyclic frequencies as follows $$\begin{aligned}
\label{a18}
\Omega_{\theta}^{2}=\frac{2Ar^{2}+\frac{\mu}{r}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}}{2r^{2}}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
&\Omega_{r}^{2}=-\frac{3}{2}(1+\frac{\eta}{\beta}r^{2})\Big\{\Big[
-\frac{8}{9}A\gamma^{4}r^{8}(\frac{15}{8}\Delta+C)+\frac{5}{3}A\gamma^{4}r^{7}\mu+\gamma^{2}
r^{6}\Big(\frac{1}{9}\Delta\gamma^{2}(3\Delta+C)+\nonumber\\
&A(-\frac{16}{9}C-\frac{10}{3}\Delta+B)\Big)+\frac{10}{3}\mu
\gamma^{2}r^{5}\Big((-\frac{1}{30}C-\frac{1}{5}\Delta)\gamma^{2}+\nonumber\\
&A\Big)\Big(\frac{1}{3}\mu^{2}\gamma^{2}+\Big[\frac{1}{9}(2\Delta+B)C-\frac{1}{3}\Delta(-2\Delta+B)\Big]
\gamma^{2}+\frac{7}{9}A(B-\frac{15}{7}\Delta-\frac{8}{7}C)\Big)r^{4}\nonumber\\
&+\Big((\frac{1}{3}B-\frac{2}{9}C-\frac{4}{3}\Delta)\gamma^{2}+\frac{5}{3}A\Big)\mu
r^{3}\Big(\frac{2}{3}\mu^{2}\gamma^{2}+\frac{2}{9}(-\Delta+B)(B-\frac{3}{2}\Delta-\frac{1}{2}C)\Big)r^{2}+\nonumber\\
&\frac{5}{9}(-\frac{6}{5}\Delta-\frac{1}{5}C+B)\mu
r+\frac{1}{3}\mu^{2}\Big]\Big\}\times\nonumber\\
&\Big\{(1+r^{2}\gamma^{2})\Big[Ar^{4}\gamma^{2}+(\frac{1}{3}\gamma^{2}C+A)r^{2}+\frac{1}{3}B+\frac{1}{3}C\Big]r^{4}\Big\}^{-1}\,.\end{aligned}$$ respectively. From equations (78), (86) and (87) it is clear that $$\Omega_{\theta}^{2}=\Omega_{\varphi}^{2}$$ and $$\Omega_{r}^{2}=\Omega_{\theta}^{2}[\frac{C+3\Delta
-2B-3\frac{\mu}{r}}{(B+C)}].$$ In the left hand side of figure 6 we have plotted epicyclic frequencies versus the radius for several values of the Horndeski/Galileon correction factor $B$ in order to have a comparison. The angular and vertical epicyclic frequencies are shown in the left panel of this figure by the dashed lines which are coincide and will decrease by increasing $r$. We see explicitly that they are not dependent on parameter $B$. The radial epicyclic frequency is shown by the solid and dotted curves (Schwarzschild black hole) with a maximum where increasing the parameter $B$ shifts this maximum to the smaller radii. The radial oscillation frequency increases by increasing the values of the parameter $B$. Dependence on this parameter is significant close to the black hole, but far from the central mass this effect is weak. We see that $\Omega_{r}<\Omega_{\theta}$. The ratio of $\frac{\Omega_{\theta}}{\Omega_{r}}$ is plotted in the right hand side panel of figure 6 which is a decreasing function of $r$. It is clear that in the vicinity of the black hole, this ratio is very greater than unity but far from the black hole it turns to unity and decreases by increasing $B$. In the left panel of figure 7, the locations of three particular resonances such as the parametric resonance with condition $\frac{\Omega_{\theta}}{\Omega_{r}}=\frac{3}{2}$ and the forced resonance with ratio $3:1$ and $2:1$ are shown. The dependence of these characteristic radii to the metric parameter $B$ is shown in the right panel of figure 7 where such radii are monotonically decreasing functions of this parameter. On the other hand, resonance will be happened in smaller distance from the central mass for larger values of the metric parameter, $B$.

Mass Evolution and Critical Points
----------------------------------

In this Horndeski/Galileon accretion disk, the energy density and radial velocity for an isothermal fluid are given by $$\begin{aligned}
\label{a18}
\rho=\frac{A_{3}(k+1)}{r^{2}\sqrt{A_{4}^{2}-(k+1)^{2}\Big(\Delta+A
r^{2}+C-\frac{\mu}{r}\Big)}}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
u=\frac{1}{(k+1)}\sqrt{(1+\frac{\eta}{\beta}r^{2})\Big(\frac{A_{4}^{2}-\Big(\Delta+A
r^{2}+C-\frac{\mu}{r}\Big)(1+k)^{2}\Big)}{\Big(3 A
r^{2}+C+\frac{B}{1+r^{2}\gamma^{2}}\Big)}\Big)}\,,\end{aligned}$$ respectively. For critical points we find $$\begin{aligned}
\label{a18}
r_{c}=-\frac{3}{4}\frac{\mu (k+1)^{2}}{A_{4}^{2}-(B+C)(k+1)^{2}}\,,\end{aligned}$$ $$\begin{aligned}
\label{a18}
V_{c}^{2}=\frac{\mu}{4r(B+C)-3\mu}\,,\end{aligned}$$ and $$\begin{aligned}
\label{a18}
u_{c}=\frac{1}{4}(1+\frac{\eta}{\beta}r^{2})\frac{\Big(\frac{\mu}{r}+2Ar^{2}-\Delta+\frac{B}{1+r^{2}\gamma^{2}}\Big)}{\Big(3Ar^{2}+C+\frac{B}{1+r^{2}\gamma^{2}}\Big)}\,.\end{aligned}$$ It is clear that $r_{c}$ depends on the equation of state parameter $k$. This is means that the location of a critical point is not the same for all fluids. The profiles of radial velocity, density and accretion rate are presented in figures 8, 9 and 10 versus the dimensionless parameter $r$ for equation of state parameter $k=\frac{1}{2}$ and constants integration $A_{4} = 1.4$ and $A_{3} =
1$ respectively. The dotted, dashed and solid curves are corresponding to cases with $B=0$, $B=-0.05$ and $B=-0.1$ respectively.

The left panel of figure 8 represents the radial velocity versus the radius. The locations of the critical points are marked by solid circles and their coordinates are stated in the right panel. The fluid has zero radial velocity far from the black hole and flows at the sub-sonic speed before the critical points. In a critical point, the speed of flow matches the speed of sound. After passing this point, in the vicinity of the black hole the speed of flow increases and turns to the super-sonic domain because of strong gravity. It can be observed that velocity decreases by increasing the Horndeski/Galileon correction factor $B$ and the loci of the critical points get shifted to the black hole. Therefore, the speed of infalling particle reaches the speed of sound closer to the central mass.

The density profile of the fluid around the black hole for different values of $B$ is shown in figure 9. Increasing the value of the parameter $B$ increases the density. In addition, for such isothermal fluids, the mass of the black hole changes with time by the following relation

$$\begin{aligned}
\label{a18}
\dot{M}=4\pi A_{1}(p+\rho)\sqrt{B+C}M^{2}\,.\end{aligned}$$

\[a8\]

\[a9\]

We see that accretion rate for a general spherically symmetric static black hole in Horndeski/Galileon gravity is different from the case of a Schwarzschild black hole. In Horndeski/Galileon case, the accretion rate completely depends on the nature of the accreting fluid and also the metric parameter (here, parameter $B$). So $\dot{M}>0$ for a normal fluid which satisfies $(p+\rho)>0$. The change of accretion rate for different values of the parameter $B$ with the same equation of state parameter are shown in figure 10. It is seen that accretion rate is higher in the vicinity of the black hole because of strong gravitational effect. Also, increment of the parameter $B$ enhances the accretion rate. By using equation (57), the critical accretion time and the mass of the black hole are given by $$\begin{aligned}
\label{a18}
t_{cr}=[4\pi A_{1}(p+\rho)\sqrt{B+C}M_{i}]^{-1}\end{aligned}$$ and $$\begin{aligned}
\label{a18}
M_{t}=M_{i}\Big [1-4\pi A_{1}(p+\rho)\sqrt{B+C}M_{i}t \Big ]^{-1}\,,\end{aligned}$$ respectively. We see that for normal fluid, the black hole mass increases by accretion matter subject to Horndeski/Galileon gravity and increasing correction factor $B$ will increase the black hole mass further. This behavior is studied by Rodrigues et al. (2009) for a Schwarzschild black hole in the presence of a non-minimally coupled scalar filed. They found that for black hole with initial masses smaller than a certain critical value, the accretion of the scalar filed can led to mass decreasing even in the absence of Hawking radiation and phantom energy. Also the black holes with initial masses greater than critical value grow by accreting the scalar filed similar to the minimally coupled scalar case.

Summary and Conclusion
======================

In this paper, the geodesic motion and accretion process of a test particle in a subclass of the general Horndeski/Galileon gravity theories in the equatorial plane of a non-rotating black hole are investigated. In this framework, the circular geodesics, the stability of such orbits, oscillations under the action of small perturbations, unstable orbits and finally accretion process of the fluid flowing around the black hole have been investigated in a general form. Expressions for the effective potential, energy, momentum, characteristic radii, emission rate, epicyclic frequencies and dynamical parameters of the system and also the mass evolution of the black hole are derived in details. Then isothermal fluid with equation of state $p=k \rho$ is considered and some discussions are done for this subclass of solutions of the Horndeski/Galileon black holes. In this manner, the metric parameters with some approximations are obtained as $h(r)\approx
C-\frac{\mu}{r}+B$ and $f(r)\approx\frac{h(r)}{B+C}$, where by assumption we have set $C=1$. The effect of the Horndeski/Galileon correction factor $B$ is considered for each case and our solutions are compared with the Schwarzschild black hole solutions. Our analysis has revealed that these Horndeski/Galileon solutions have deviations from the Schwarzschild solutions (which is recovered where $B=0$) substantially. Our results show that Horndeski/Galileon correction factor affects the effective potential and as a result changes the loci of the stable and unstable circular orbits. For larger values of this parameter, $V_{eff}$ achieves larger value and unstable circular orbits will be located at smaller radii, whereas stable orbits will be located at farther distances from the central mass. When the metric parameter $B$ decreases, two points joint together at ISCO.

In this spacetime the location of the characteristic radii such as $r_{isco}$, $r_{ph}$, $r_{sing}$ and $r_{mb}$ have considerable deviation from the Schwarzschild solutions. These radii, except $r_{mb}$, are decreasing functions with respect to the Horndeski/Galileon correction factor $B$, that is, they will be closer to the central mass for larger values of $B$. As $r_{isco}$ represents the inner edge of the accretion disk, our results show that for larger deviations, the disk will be extended close to the central mass. The behavior of $r_{mb}$ is different and has a minima at the Schwarzschild case. This radii is a decreasing function for $B\subset(-1, 0)$ and it is a growing function for $B\subset (0,
1.125)$, and finally it coincides with the innermost stable circular orbit.

As Horndeski/Galileon correction factor $B$ grows, the energy raises whereas angular momentum decreases and one can see from the energy diagram that the range of bound orbit will be smaller for larger deviations. Increasing the parameter $B$ enhances the $E_{isco}$ and then the efficiency of accretion will be decreased accordingly. In the case of Schwarzschild black hole the efficiency equals to $0.057$ at $r = 6r_{g}$. For negative values of the correction factor $B$, efficiency grows up and becomes greater in the Schwarzschild black hole limit ($B=0$), where this behavior is reverse for positive values of the correction factor. The flux of the radiation energy has a maximum in the vicinity of the black hole and decreases at smaller radii. By increasing the parameter $B$, the flux raises and the maximum of the flux turns to the smaller radii but the reverse happens after this point. The dependence on this parameter is very considerable in the vicinity of the black hole, but it is weak far from the black hole. These behaviors are the same for temperature.

In this paper, in addition to investigation of the circular orbits and their properties, epicyclic frequencies are studied. Vertical epicyclic frequency is monotonically decreasing function of $r$ and has no extrema. Deviation from the Schwarzschild case in this Horndeski/Galileon setup has no effect on the vertical epicyclic frequency, while the radial epicyclic frequency always has a maximum and the effect of Horndeski/Galileon correction factor on it is considerable. Dependence on this parameter is significant close to the black hole, but far from the central mass this effect is weak. Increasing the parameter $B$ shifts the maximum to the smaller radii. We found that $\Omega_{r}<\Omega_{\theta}$ and the ratio of $\frac{\Omega_{\theta}}{\Omega_{r}}$ is decreasing function with respect to $r$. In the vicinity of the black hole, this ratio is very larger than unity but far from the black hole, it turns to the unity and it decreases by increasing $B$. The dependence of some important resonances such as parametric and forced resonances with respect to Horndeski/Galileon parameter $B$ is obtained, which this dependence is a monotonically decreasing function, that is, larger values of the Horndeski/Galileon correction factor lead to happening the resonance at the smaller radii.

Finally, the accretion process of the isothermal fluid is discussed for equation of state parameter $k=\frac{1}{2}$ and the behavior of the radial velocity and density are studied. Radial velocity is a decreasing function of $r$ as fluids have zero radial velocity in far from the black hole. When accretion happens, fluid passes at a critical point where in this point, the speed of flow matches the speed of sound. The fluid flows at sub-sonic speed before the critical point. After passing this point and in the vicinity of the black hole, because of strong gravitational field, the speed of flow increases and will be super-sonic then after. Our results show that velocity decreases by increasing $B$ and the loci of the critical point shifts towards the black hole. Therefore, the speed of infalling particles reaches the speed of sound closer to the central mass. Finally, the rate of accretion is discussed where we found that this rate depends on the nature of the fluid and also the metric parameter. For normal fluid, $\dot{M}>0$ and its value is larger in the vicinity of the black hole because of strong gravitational effect and positive deviation from the Schwarzschild case increases the accretion rate.

We note that in this paper a non-spinning particle is considered. When a particle has spin, this spin has considerable influence on the particle’s orbit. Also, a perfect fluid is considered and viscosity as well as magnetic field of the accretion disk are ignored for simplicity. However, these effects can affect the motion of the test particle and therefore they can affect the structure and the emission rate of the accretion disk. So, we are going to study the behavior of spinning particles and accretion of viscose fluids subject to the Horndeski/Galileon gravity in presence of a magnetic field in our future work.\
[**Acknowledgement**]{}\
We would like to thank the referee for insightful comments. The work of K. Nozari has been financially supported by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/5411-6.

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[^1]: k.salahshoor@stu.umz.ac.ir

[^2]: knozari@umz.ac.ir(Corresponding Author)


               IN THE SUPREME COURT OF IOWA
                              No. 13–0397

                         Filed October 25, 2013

IOWA SUPREME COURT ATTORNEY DISCIPLINARY BOARD,

      Complainant,

vs.

CURT N. DANIELS,

      Respondent.

      On review of the report of the Grievance Commission of the

Supreme Court of Iowa.

      Grievance commission reports that respondent committed ethical

misconduct     and   recommends   a   public   reprimand.    ATTORNEY

REPRIMANDED.

      Charles L. Harrington and Nicholas Trè Critelli III, Des Moines, for

complainant.

      Curt N. Daniels, Chariton, pro se.
                                       2

CADY, Chief Justice.

      The Iowa Supreme Court Attorney Disciplinary Board charged

Curt N. Daniels with three violations of the Iowa Rules of Professional

Conduct, asserting Daniels filed frivolous pleadings and made a false

statement to the court in a pleading. The Grievance Commission of the

Supreme Court of Iowa found Daniels violated a rule of professional

conduct by filing frivolous matters with the court on two occasions. The

commission recommended Daniels receive a public reprimand. On our

review, we find Daniels violated one rule of professional conduct on one
occasion, and we impose a public reprimand.

      I. Background Facts and Prior Proceedings.

      Curt N. Daniels is both a lawyer and veterinarian. He is seventy-

five years old. He obtained a degree in veterinary medicine in 1965. He

obtained a law degree in 1973. Although he was admitted to the practice

of law in Iowa that same year, Daniels continued to practice veterinary

medicine in Chariton.

      Daniels also operated a hog confinement facility in Jasper County.

The confinement facility was located on property owned by an Iowa

corporation owned by Daniels.         In 1998, a limited liability company

owned and operated by John Holtz purchased the property at a tax sale.

The property was sold after the corporation owned by Daniels stopped

paying the real estate taxes on it.

      In 2002, Daniels left the practice of veterinary medicine and began

practicing law in Chariton as a sole practitioner. In 2003, the company

owned by Holtz brought a replevin action against Daniels and his

corporation to recover equipment and property Daniels removed from the
Jasper County property following the tax sale.        Daniels represented

himself and his corporation in the lawsuit.       The action resulted in a
                                     3

monetary judgment against Daniels and his corporation in 2004, based

on the wrongful detention of property. The events that followed over the

next several years culminated in this disciplinary action against Daniels.

        Daniels filed an appeal from the judgment entered against him and

his corporation in favor of Holtz’s company. The appeal was filed after

the district court denied a motion for new trial filed by Daniels and after

Holtz accepted a remittitur of damages established by the district court.

        During the pendency of the appeal, Daniels initiated an action

against Holtz and his company in district court, alleging they engaged in
fraudulent practices during the tax sale of the Jasper County property.

After commencing the action, Daniels moved to amend his petition to

include numerous additional claims against the attorney and the law

firm that represented Holtz and his company.         Daniels claimed the

attorney engaged in deception and collusion with Holtz during the tax

sale.   Five of the claims against the attorney were based on alleged

violations of the Iowa Code of Professional Responsibility. The district

court denied the motion to amend and ultimately granted summary

judgment for Holtz and his company.

        In 2008, we affirmed the 2004 judgment entered by the district

court in favor of the company owned by Holtz on the condition that the

remittitur again be accepted. WSH Props., L.L.C. v. Daniels, 761 N.W.2d

45, 52–53 (Iowa 2008). However, we remanded the case to the district

court for a new trial in the event the remittitur was not accepted. Id. at

53.

        On remand, Daniels filed a motion for new trial in district court,

and the company accepted the remittitur. The motion for new trial, filed
in October 2008, generally alleged the existence of newly discovered

evidence. In an affidavit accompanying the motion, Daniels alleged the
                                          4

existence of perjury and discovery improprieties during the case.                  In

January 2009, the district court denied this motion on its merits. No

appeal was filed within the thirty-day period permitted by the rules of

appellate procedure. See Iowa R. App. P. 6.101(1)(b) (“A notice of appeal

must be filed within 30 days after the filing of the final order or

judgment.”).

       On April 9, 2009, Daniels filed a “Renewed Motion for New Trial.”

This motion presented more detailed claims of perjury and other

irregularities alleged to have been committed by the plaintiffs and others
during the case. Daniels, however, made no reference in this motion to

the prior motion for new trial filed in October 2008. On its face, this

motion appeared to urge more detailed grounds for relief than alleged in

the October motion. Daniels made no claim that he had not received a

copy of the ruling on the prior motion filed in October 2008. The district

court summarily denied the renewed motion for new trial on April 27,

2009, based on the reasons used in January to deny the prior motion for

new trial.

       Daniels filed an appeal from the order denying his renewed motion

for new trial on May 6, 2009. The notice of appeal specifically identified

the appeal as taken from the ruling on the renewed motion for new trial

filed in April. During the pendency of the appeal, Daniels filed a petition

with the district court captioned “Petition for Relief.” Daniels asserted

the petition was filed pursuant to Iowa Rules of Civil Procedure 1.1012

and 1.1013.1 In the petition, Daniels claimed he was seeking relief from

        1Rule 1.1012 allows a court to “correct, vacate or modify a final judgment or

order, or grant a new trial” for a number of reasons, including when there has been
“[m]istake, neglect or omission of the clerk,” when there was “[i]rregularity or fraud
practiced in obtaining [the judgment],” or when there is “[m]aterial evidence, newly
discovered, which could not with reasonable diligence have been discovered and
produced at the trial, and was not discovered within the time for moving for new trial
                                          5

the district court to “restart” the time to appeal from the January 2009

order that denied his motion for new trial filed in October 2008.                 He

claimed the district court clerk failed to serve him with a copy of the

order until April 27, 2009, which did not allow him to timely file a notice

of appeal within thirty days from the time the order was filed. He wanted

the district court to declare the time to appeal from the January 2009

order began to run on April 27, 2009, due to an oversight by the clerk of

court in failing to serve him with a copy of the order in January. The

district court dismissed the petition, and Daniels appealed.
       Both appeals were transferred to the court of appeals. The court of

appeals held the appeal from the denial of the renewed motion for new

trial was untimely because Daniels failed to file a notice of appeal within

thirty days after the ruling by the district court on the motion for new

trial in January 2009. WSH Props., L.L.C. v. Daniels, No. 09–0703, 2010

WL 796506, at *1 (Iowa Ct. App. Mar. 10, 2010). In a separate opinion,

the court of appeals held the appeal from the denial of the petition for

relief was untimely because it was not filed within one year after the

judgment. WSH Props., L.L.C. v. Daniels, No. 10–0229, 2010 WL 150500,

at *2 (Iowa Ct. App. Jan. 19, 2012).

       II. Board Complaint and Commission Decision.

       The Board eventually brought a complaint against Daniels for filing

frivolous matters and for making a false statement to the court. Count I

of the complaint alleged two separate violations of the rules of

professional conduct.        First, the Board alleged Daniels violated rule

32:3.3 (prohibiting false statements to the court) when he filed the

petition for relief alleging grounds to restart the time to appeal the
_______________________
under rule 1.1004.” See Iowa R. Civ. P. 1.1012(1), (2), (6). Rule 1.1013 describes the
requirements and procedure for moving for relief under rule 1.1012. Id. r. 1.1013.
                                     6

January 2009 district court decision denying his October 2008 motion

for new trial, without disclosing to the court the facts and circumstances

surrounding the renewed motion for new trial and the appeal taken from

the ruling on the renewed motion.        Second, count I alleged Daniels

violated rule 32:3.1 (prohibiting frivolous proceedings) by instituting an

action for the purpose of pursuing relief not available under the law. In

count II, the Board alleged Daniels violated rule 32:3.1 when he sought

to amend his petition in the lawsuit against the attorney who represented

Holtz and his company to assert five legal claims based entirely on
alleged violations of the rules of professional conduct.

      Daniels denied the charges brought by the Board. At the hearing

before the commission on the charges, Daniels testified he filed the

renewed motion for new trial in April 2009 because at the time he had

not yet received a ruling on his October 2008 motion for new trial. He

believed the renewed motion was the best method to alert the trial judge

to the absence of a ruling.     He also testified he received the district

court’s ruling on the October 2008 motion for new trial and a ruling on

the April 2009 renewed motion at the same time on April 27, 2009.

Daniels believed the clerk of court must have inadvertently failed to mail

him a copy of the ruling in January 2009. He also testified he filed the

petition for relief, after filing an appeal from the order on the renewed

motion for new trial, after he realized his best hope for relief would be for

the district court to allow him to file a new motion for new trial due to the

alleged clerical error.

      The commission found the Board established Daniels violated rule

32:3.1, as alleged in counts I and II of the complaint. It found the Board
failed to establish Daniels violated rule 32:3.3, as alleged in count I. The

commission recommended that Daniels receive a public reprimand.
                                     7

      III. Scope and Standard of Review.

      We review attorney disciplinary matters de novo. Iowa Supreme Ct.

Bd. of Prof’l Ethics & Conduct v. Bernard, 653 N.W.2d 373, 375 (Iowa

2002).

      IV. Violations.

      Our rules of professional conduct prohibit a lawyer from asserting

a claim or issue to the court “unless there is a basis in law and fact for

doing so that is not frivolous.” Iowa R. Prof’l Conduct 32:3.1. Similarly,

our rule of candor towards courts prohibits a lawyer from knowingly
making a false statement of fact or law to the court. Id. r. 32:3.3. Both

rules are part of a group of rules governing advocacy by a lawyer. We

must first consider the scope of these rules to decide if they apply to

lawyers who represent themselves.

      In Iowa Supreme Court Attorney Disciplinary Board v. Rhinehart, we

observed that some of the rules of professional conduct governing

advocacy target only conduct of an attorney while serving as an advocate

representing a client. 827 N.W.2d 169, 176 (Iowa 2013). Rhinehart, who

was charged with violating rule 32:3.3 based on the lack of candor he

exhibited in his own dissolution-of-marriage case, was represented by an

attorney.   Id. at 173 (indicating Rhinehart’s attorney was unaware of

Rhinehart’s failure to disclose the information). We held rule 32:3.3 did

not apply to Rhinehart because he was not advocating as a lawyer when

he failed to disclose information in the dissolution case, but was

testifying as a witness. See id. at 173, 176.

      Although Daniels was representing a corporation as well as himself

during the course of the litigation that generated the ethics charges in
this case, Rhinehart does not limit the application of the rules of

professional conduct at issue here to Daniels’s conduct in representing
                                          8

the corporation.     While rule 32:3.3 may target the conduct of lawyers

while representing their client, it also extends to advocacy by lawyers

who have themselves as a client. The rationale and need for candor in

advocacy applies to self-represented lawyers with the same force as to

lawyers who represent others. See Comm. on Prof’l Ethics & Conduct v.

Ramey, 512 N.W.2d 569, 571 (Iowa 1994) (“The administration of justice

entrusted to our branch of government can be rendered only when our

officers can be counted upon for absolute reliability and an impeccable

reputation for honesty.”). Accordingly, we turn to consider each alleged
violation in this case.

       In addressing the charges against Daniels, we focus on the theory

alleged by the Board. The charges are not predicated on the veracity of

the overarching claim by Daniels that he did not timely receive a copy of

the ruling filed in January 2009 by the district court on the motion for

new trial.2 The charges by the Board also do not address the propriety of

pursuing a new trial following an appeal. Instead, the charge of making

a false statement to the court under count I was prosecuted by the Board

based on Daniels’s failure to present all of the relevant procedural

background of the request in his petition for relief. The Board claimed

the petition constituted a false statement because Daniels failed to

disclose he also filed the renewed motion for new trial. Additionally, the

       2On   our de novo review of the record, the allegations of clerical error as
explained by Daniels at the commission hearing appear questionable. It is unlikely that
a lawyer would renew a prior motion without mentioning the prior motion. It is equally
unlikely that a lawyer would file a renewed motion as a means to remind a judge to rule
on a prior motion by amplifying the allegations of the prior motion. It is also
improbable that a lawyer would renew a prior motion to seek a ruling on the motion
without first checking on the status of the motion with the clerk of court or opposing
counsel. Nevertheless, Daniels was not charged with making a false statement before
the commission, and Daniels has not been called upon to defend his allegations. We
only address the charges brought by the Board.
                                     9

charge of filing a frivolous pleading under count I was prosecuted on the

theory that a petition under rule of civil procedure 1.1012 cannot be

used to extend time to file an appeal from a final order or judgment.

Finally, the charge of filing a frivolous pleading under count II was based

solely on the proposition that a breach of an obligation under the rules of

professional conduct does not supply legal grounds for a claim of civil

liability.   Accordingly, we proceed to consider each charge under the

theory urged by the Board.

       We first consider count I. A lawyer is prohibited from knowingly
making a false statement of fact to the court.     Iowa R. Prof’l Conduct

32:3.3(a)(1).

       We agree with the Board that the omission of information by a

lawyer can give rise to a false statement to the court. See id. r. 32:3.3

cmt. 3 (“There are circumstances where failure to make a disclosure is

the equivalent of an affirmative misrepresentation.”).    However, within

the context of notice pleading, we cannot conclude by a convincing

preponderance of the evidence that the omission from the petition for

relief identified by the Board constituted an affirmative false statement of

fact. Daniels may not have captured the full background of the relief he

sought, but such background was not essential to pleading his claim,

and the absence of the information did not necessarily render his petition

untruthful.     Like the commission, we conclude Daniels did not violate

rule 32:3.3(a)(1).

       We next consider the charge that the petition for relief filed by

Daniels was frivolous. Advocacy by a lawyer that is frivolous violates rule

32:3.1. Stated in another way, lawyers have “a duty not to abuse legal
procedure.” Id. r. 32:3.1 cmt. 1. Additionally, claims brought by lawyers
                                      10

must be based both on law and fact or “a good faith argument for an

extension, modification, or reversal of existing law.” Id. r. 32:3.1.

      Daniels filed his petition for relief under the authority of rule of

civil procedure 1.1012. This rule allows the court to correct, vacate, or

modify a final judgment in an order or grant a new trial based on

grounds that include mistake, neglect, or omission of the clerk. Iowa R.

Civ. P. 1.1012(1). Although Daniels did not pursue his claim with a high

degree of clarity, his petition for relief sought to restart the time to appeal

from the January 2009 order denying the motion for new trial.
      Under our rules of appellate procedure, the supreme court is

authorized to extend the time for filing a notice of appeal in the event a

clerk of court fails to notify a prospective appellant of the filing of an

appealable order. See Iowa R. App. P. 6.101(5). Furthermore, a motion

to extend under rule 6.101(5) must be filed within sixty days after the

prescribed time for appeal.     Id.   The Board claims the clear absence

under the circumstances of the remedy requested by Daniels pursuant to

the rules of appellate procedure rendered his petition for relief under rule

1.1012 frivolous.

      Daniels had no available remedy under rule 6.101(5) to extend the

time to file an appeal after he failed to request such relief within the

sixty-day period provided under the rule. See id. Nevertheless, our court

of appeals has recognized that the consequences of failing to seek relief

under rule 6.101(5) within the sixty-day period may be addressed

indirectly by filing a motion to vacate under rules of civil procedure

1.1012 and 1.1013 with the district court.          See Hays v. Hays, 612

N.W.2d 817, 819 (Iowa Ct. App. 2000) (noting that “[a] litigant may avoid
the harsh consequences of [the predecessor to rule 6.101(5)] by filing a

motion to vacate or modify under [the predecessors of rules 1.1012 and
                                     11

1.1013]”).   This approach allows the district court to vacate an order

based on the omission by the clerk and then provide a new time period to

file an appeal by filing the order anew. See In re Estate of Young, 273

N.W.2d 388, 391 (Iowa 1978) (considering an appeal filed within thirty

days of an order denying a motion to modify a substantive ruling issued

seven months earlier as timely).

       Although Daniels did not clearly identify his objective in filing the

petition for relief, there is legal authority for the course of action he

pursued.     Accordingly, we conclude he did not violate rule 32:3.1 by
filing the petition for relief.

       Finally, we consider the charge in count II that Daniels asserted

claims of civil liability that were frivolous. In his motion to amend his

petition against the attorney for Holtz and the attorney’s law firm,

Daniels asserted claims against the attorney and his law firm that were

based solely on alleged violations of the rules of professional conduct.

Daniels v. Holtz, 794 N.W.2d 813, 825 (Iowa 2010).            The rules of

professional conduct do not form a basis for civil liability. See Iowa R.

Prof’l Conduct 32 pmbl. ¶ 20; Daniels, 794 N.W.2d at 825. Daniels had

no basis in the law to bring the claims and did not otherwise conform to

the duty to present at least arguably meritorious claims to the court.

Consequently, Daniels violated rule 32:3.1 by filing a motion to amend

his petition to assert claims not supported by the law or by a good-faith

argument for changing or extending the law. See Iowa R. Prof’l Conduct

32:3.1.

       V. Sanction.
       We discipline lawyers for professional misconduct to protect the
public and the courts and to ensure public confidence in the integrity of
our system of justice. Iowa Supreme Ct. Att’y Disciplinary Bd. v. Hearity,
                                         12

812 N.W.2d 614, 623 (Iowa 2012). In determining the proper discipline,
we primarily consider the nature and extent of the violation.                    Iowa
Supreme Ct. Att’y Disciplinary Bd. v. Weaver, 750 N.W.2d 71, 91 (Iowa
2008).
      Frivolous claims place unnecessary burdens on the court system
and diminish public confidence in our system of justice.                  Cf. Iowa
Supreme Ct. Bd. of Prof’l Ethics & Conduct v. Ronwin, 557 N.W.2d 515,
520 (Iowa 1997) (indicating lawyers have an obligation not to clog the
courts with frivolous claims). We have in the past suspended lawyers
from the practice of law for filing frivolous matters, although these cases
have been accompanied by other unethical conduct. Id. at 523 (revoking
attorney’s license to practice law for filing frivolous lawsuit against
individual and making repeated unsupported allegations of criminal
conduct against judges); Iowa Supreme Ct. Bd. of Prof’l Ethics & Conduct
v. Hohnbaum, 554 N.W.2d 550, 552 (Iowa 1996) (suspending license to
practice law for three months for making misleading statements and
persisting in patently frivolous position).
      Because Daniels’s conduct is not as severe as the conduct in
Ronwin    or    Hohnbaum, we         conclude    Daniels should       be publicly
reprimanded.      This discipline should serve to remind Daniels of his
professional obligation to refrain from filing frivolous matters and to
protect   the    integrity    of   the   profession.     It    also   reflects    the
recommendation       of      the   commission,   which        we   give   respectful
consideration.     Iowa Supreme Ct. Bd. of Prof’l Ethics & Conduct v.
Gottschalk, 729 N.W.2d 812, 815 (Iowa 2007).
      VI. Conclusion.
      We publicly reprimand Curt N. Daniels for asserting a frivolous
claim in the course of litigation. Costs of this action are taxed to Daniels.
      ATTORNEY REPRIMANDED.


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