Bondi-Hoyle Accretion around the Non-rotating Black Hole in 4D Einstein-Gauss-Bonnet Gravity
aa r X i v : . [ g r- q c ] J a n Bondi-Hoyle Accretion around the Non-rotating Black Hole in 4D Einstein-Gauss-BonnetGravity
O. Donmez College of Engineering and Technology, American University of the Middle East (AUM), Kuwait
ABSTRACT
In this paper, the numerical investigation of a Bondi-Hoyle accretion around a non-rotatingblack hole in a novel four dimensional Einstein-Gauss-Bonnet gravity is investigated by solv-ing the general relativistic hydrodynamical equations using the high resolution shock capturingscheme. For this purpose, the accreated matter from the wind-accreating X − ray binaries fallstowards the black hole from the far upstream side of the domain, suphersonically. We study theeffects of Gauss-Bonnet coupling constant α in D EGB gravity on the accreated matter andshock cones created in the downstream region in detail. The required time having the shockcone in downstream region is getting smaller for α > while it is increasing for α < . Itis found that increases in α leads violent oscillations inside the shock cone and increases theaccretion efficiency. The violent oscillations would cause increase in the energy flux, tem-perature, and spectrum of X − rays. So the quasi-periodic oscillations (QPOs) are naturallyproduced inside the shock cone when − ≤ α ≤ . . It is also confirmed that EGB blackhole solution converges to the Schwarzschild one in general relativity when α → . Besides,the negative coupling constants also give reasonable physical solutions and increase of α innegative directions suppresses the possible oscillation observed in the shock cone. Subject headings:
Relativistic hydrodynamics: Bondi-Hoyle accretion: 4D Einstein-Gauss-Bonnet black hole: shock cones, QPOs
1. Introduction
Einstein’s general theory of relativity has successfully passed many observational tests during the lastdecades (Akiyama1 et. al. 2019; Akiyama2 et. al. 2019; Akiyama3 et. al. 2019; Abbott1 et. al. 2016;Abbott2 et. al. 2016). The existence of super-massive black hole at the center of galaxy M87 was oneof the most impressive discoveries. The observed M87 black hole shadow indicates that the observed datawell consistent with the prediction of the general theory of relativity. The observed gravitational wavescreated during the coalescing of two stellar mass black holes made by LIGO detectors was another discov-ery to see the ripples of the space time predicted by Einstein’s general theory of relativity. The observed College of Engineering and Technology, American University of the Middle East (AUM), Kuwait α → α/ ( D − in the limit D → and bypasses the Lovelocks theorem.As a consequence of this limit, the Gauss-Bonnet coupling constant gives non-trivial contributions to thegravitational dynamics. It is also shown that the theory is free from Ostrogradsky instability and preservesthe number of degrees of freedom. And the static spherically symmetric black hole, which has two horizonsinstead of one compared to the Schwarzschild black hole, was discovered(Cheng et. al. 2020). The horizonis rescaled with the Gauss-Bonnet coupling constant α and the causal structure of the black hole radically isaltered with a repulsive effect. Although this D EGB theory is currently under debate (Julio et. al. 2020;Gurses et. al. 2020; Ai 2020), it is worthy and meaningful to study the spherical symmetric black holesolution of D EGB gravity.The newly discovered four-dimensional static and spherically symmetric Gauss-Bonnet black holewas used to reveal many interesting features of some astrophysical problems. Lots of work have beendone the different astrophysical phenomena the viability of the solution. The properties of the black holeand its shadow (Konoplya & Zinhailo 2020; Roy & Chakrabarti 2020), the innermost stable circular orbitfor massive particles and photons (Guo & Li 2020; Zhang et. al. 2020), generating and radiating blackhole (Ghosh & Maharaj 2020; Ghosh & Kumar 2020), week cosmic censorship conjecture (Yang et. al.2020), the gravitational lensing by black holes (Jin et. al. 2020; Islam et. al. 2020), observational con-straints on the Gauss-Bonnet constant (Feng et. al. 2020; Clifton et. al. 2020), the growth rate of non-relativistic matter perturbations (Zahra 2020), and Greybody factor and power spectra of the Hawkingradiation (Konoplya & Zinhailo 2020; Zhang et. al. 2020) were studied with all details.The wind accretion scenario onto the black hole is one of the important physical phenomena to explainsoft and hard X − rays observed by different X − ray telescopes. The wind accretion process can be ex-plained by using the Bondi-Hoyle model (Edgar 2004). A black hole moving inside the gas cloud capturesmass which causes either acceleration or deceleration of it. As a result of this the shock cone or bow shockwould be formed around the black hole. This is the one of hot topics studied by different researchers usingthe analytical and the numerical techniques in the last few decades. Analytic representation of the windaccretion for a perfect fluid onto a Schwarzschild black hole has been found by Tejeda & Aguayo-Ortiz(2019) and numerical treatments from Newtonian and relativistic perspective were extensively studied byDonmez et. al. (2011); Donmez (2012); Cruz-Osorio & Lora-Clavijo (2016); Cruz-Osorio et. al. (2017).In the present paper, building onto the previous findings, we would like to study the creation of theshock cones and their dynamical evolution in case of Bondi-Hoyle accretion in the background of the D X − ray telescope. The companion star in the X − ray binary system provide the fast wind which causes to accreatetowards to the black hole(Orosz et. al. 2008). The accelerating single black holes due to the other blackholes could have Bondi-Hoyle accretion(Lora et. al. 2013). Studying the QPOs on accretion disk in vicinityof a black hole is important to explore the physical parameters of the black hole such as its spin and mass.QPO arises in the inner accretion disk of the black hole binary and could be created in terms of an oscillating,precessing hot flow in the truncated-disk geometry due to the strong shocks. Therefore, it would be a furtherstep to know whether the Gauss-Bonnet coupling constant α can play a dominant role in the oscillation ofthe shock cone created during the Bondi-Hoyle accretion. To reveal all these details, we numerically modelthe Bondi-Hoyle accretion in the vicinity of D EGB black hole. We compute the shock cone structures fordifferent values of Gauss-Bonnet constant α and compare them with standard general relativistic solution,Schwarzschild black hole.The plan of the paper is as follows: In Section 2, we briefly give summary of the recently proposed non-rotating black hole solution in D EGB gravity and definition of the horizon of the black hole. In section3, we describe the conserved form of the general relativistic hydrodynamical equations, lapse function, andshift vectors in D EGB black hole necessary in our numerical simulations. In Section 4, we describe thetheoretical framework of the Bondi-Hoyle accretion for pressureless gas, and initial and boundary conditionsused in our numerical simulation. The numerical results and discussion are also given to show dependenciesof the disk dynamics, creation of shock cones, accretion rates, and mode power to Gauss-Bonnet couplingconstant α . In Section 5, we discuss the implications from our numerical results and draw the direction offuture work. Unless specified, we use geometrized units throughout the paper for the speed of light andgravitational constant, G = c = 1 .
2. Non-rotating Black Hole Solution of 4D EGB Gravity
The theory of the static and spherically symmetric solution of the gravity can be driven by summingEinstein-Hilbert action and higher order Lovelock invariants with the vanishing bare cosmological constantGlavan & Lin (2020). S EH + S GB = Z d D x √− g (cid:18) M P R + αG (cid:19) , (1)where M P , R , α , and G are the reduced Planck mass, the Ricci scalar of the space-time, the Gauss-Bonnetcoupling constant, and the Gauss-Bonnet invariant, respectively (Glavan & Lin 2020; Cheng et. al. 2020).The theory of this black hole was already established for D ≥ (Boulware & S. Deser 2020). But theGauss-Bonnet term G does not contribute to the D gravitational dynamics. The reason is that the equationof a motion contains term D − and it will disappear while D = 4 . After rescaling the coupling constant 4 – α → αD − , the factor D − would be removed. So when the Lovelock theorem is bypassed, it gives nontrivialdynamics, and the static and spherically symmetric black hole solution was discovered(Glavan & Lin 2020).The static and spherically symmetric black hole solution in four dimensional EGB gravity has thefollowing form ds = − f ( r ) dt + 1 f ( r ) dr + r dθ + r sin ( θ ) dφ , (2)where f ( r ) = 1 + r α − r αMr ! . (3)Here M is the mass of the black hole. The coupling constant parameter plays a dominant role defining theblack hole horizon. By solving f ( r ) = 0 , we can reach the following two horizons for non-rotating blackhole in D EGB gravity, r ± = M ± p M − α, (4)where, as seen in Fig.1, the Gauss-Bonnet coupling constant is restricted to α < . If α > , one has twohorizons. One degenerate horizon is seen when α = 0 . Otherwise, α < , the black hole has only onehorizon. It was belied that static and spherically symmetric ansatz in Eq.2 might not give the real solution(Glavan & Lin 2020) for a short radial distance r < − αM . As shown in Fig.1, the black hole in D EGB gravity exists when α < . Therefore we would like to numerically model Bondi-Hoyle accretionfor possible values of Gauss-Bonnet coupling constant either < α < or α < and to compare themwith the theory and Schwarzschild solution. As it was also noted in reference Guo & Li (2020), the InnerStable Circular Orbit (ISCO) in the novel D Gauss-Bonnet gravity can be greater or less than the one inSchwarzschild black hole solution depending on the coupling constant α .
3. General Relativistic Hydrodynamical Equations
General Relativistic Hydrodynamics (GRH) is a necessary theory to understand the dynamic structuresof the accretion disks around the black holes especially inside the region, r < M where the stronggravitational field is encountered. Here, M is the mass of the black hole. The covariant forms of the GRHequations are in the following form (Donmez 2004, 2006), ▽ µ T µν = 0 , ▽ µ J µ = 0 , (5) 5 –where T µν and J µ are the stress-energy tensor and the matter current density, respectively. The latin indicesrun from to and Greek indices run from to . In order to define the characteristic waves of the GRHequation, we simplify them neglecting the magnetic field and any type of the viscosity. Therefore the stress-energy tensor for a perfect fluid is, T µν = ρhu µ u ν + P g µν , (6)where specific enthalpy h = 1 + ǫ + Pρ . ρ , P , u µ , and g µν are the rest-mass density, ideal gas equation ofstate (pressure), four-velocity, and four metric, respectively. The equation of state is P = (Γ − ρǫ . Here, Γ is adiabatic index which defines the compressibility of the fluid.The form of GRH equations given in Eq.5 is not suitable to use in High Resolution Shock Capturingscheme (HRSC). Hence, GRH equations are written in conservation form using formalism and we endup with the following first-order, flux-conservative hyperbolic system, ∂ ~U∂t + ∂ ~F i ∂x i = ~S, (7)where ~U , ~F i , and ~S are conserved quantities, fluxes, and sources, respectively. Detailed formulations ofthese three vectors are given as the functions of lapse function ˜ α , shift vectors β i , the determinant of thethree-metric γ , four-velocity u µ , three velocity v i , Lorentz factor W = ˜ αu = (1 − γ ij v i v j ) − / , rest-massdensity ρ , the components of the momentum vector S i , pressure P , enthalpy h , and the D Christoffelsymbol Γ αµν . The spectral decomposition of the Jacobian matrix of the system ∂ ~F i /∂ ~U is needed to useHRSC scheme and they are given in Donmez (2004) with the full details.The static and spherically symmetric solution of the black hole in D Gauss-Bonnet gravity metricgiven in Eq.2 is used to define the source term appearing in the right hand side of Eq.7. The four-metric,three-metric, their inverses, the D Christoffel symbol, and Lorentz factor can be easily driven by havingstraightforward calculations. The lapse function for the EGB black hole is ˜ α = r α − r αMr !! / , (8)and the shift vectors are, β r = 0 , β φ = 0 , β θ = 0 . (9) 6 –
4. Bondi-Hoyle Accretion onto 4D EGB Black Hole4.1. Theoretical Framework
A homogeneous supersonic flow that has a relative velocity v ∞ and density ρ ∞ at infinity is deflectedby point mass M due to the warped space-time (Hoyle & Lyttleton 1939). The Bondi-Hoyle accretioncondition for particles is defined with an impact parameter ζ . As it is seen in Fig.2, if the pressure effect isnegligible, the trajectory of the particle can be defined by conventional orbit. And then accretion conditionis, ζ < ζ HL = 2 Mv ∞ , (10)where ζ HL is commonly known as accretion radius or the stagnation point. The suggested accreated mattercould be computed by considering the trapped material inside the gravitational potential. The accretion ratesuggested by Hoyle & Lyttleton (1939) is, ˙ M HL = πζ HL ρ ∞ v ∞ = 4 πM ρ ∞ v ∞ . (11)The material falling towards the black hole that does not encounter to θ = 0 axis would not accreate and itwould escape from the system as seen in dashed (blue) lines in Fig.2.The theory given above does not exactly describe the motion of the gas flow accreated onto the blackhole in the presence of the gas pressure in the strong gravitation region but it helps us define what thephysical parameters are at infinity and how the accretion mechanism works.Bondi-Hoyle accretion is studied in the vicinity of the Schwarzschild and Kerr black hole and calledspherically symmetric accretion. During the accretion, the Bondi radius would be created and the accreatedflow outside the radius is subsonic, and the disk mass-density is almost uniform. But the gas inside theradius becomes supersonic and it will be accreated towards to the black hole (Donmez et. al. 2011; Donmez2012). The strong gravity focuses the material behind the black hole (down-stream region) and the materialwould be accreated. In this paper, we will numerically model the Bondi-Hoyle accretion in the vicinity ofthe static and spherically symmetric black hole in D Gauss-Bonnet gravity to reveal the effect of Gauss-Bonnet coupling constant, not only on the shock cone structures but also on the oscillation properties andQPOs of those shock cones.
Numerical simulation of the Bondi-Hoyle accretion around static and spherically symmetric non-rotating black hole defined in D EGB gravity gives us an opportunity to understand how the dynamics 7 – -15 -10 -5 0Gauss-Bonnet coupling constant-4-20246 B l ac k ho l e ho r i z on r + r - One Horizon Region Two Hor.
Fig. 1.— Black hole horizon radius r + and r − as a function of the Gauss-Bonnet coupling constant α .Fig. 2.— Artistic visualization of the Boyle Hoyle accretion around the black hole. BH ≡ Black Hole, OB ≡ Outer Boundary, and IB ≡ Inner Boundary 8 –of the accretion disk and shock cone would be effected by the Gauss-Bonnet coupling constant α . For thispurpose, we numerically solve the GRH equations in spherical coordinate on the equatorial plane ( r, φ ) , i.e. θ = π/ (see Eq.7) assuming spherical symmetry using the HRSC scheme based on approximate Riemannsolver, further described in Donmez (2004).As explained in section 3, we need to know pressure to evolve GRH equation depending on given initialvalues. The pressure is evaluated by using the perfect fluid equation of states P = (Γ − ρǫ with adiabaticindex Γ = 4 / . In order to perform the numerical simulation of the Bondi-Hoyle accretion on the equatorialplane, the gas is injected from an outer boundary of the upstream region with the following velocities, V r = √ γ rr V ∞ cos ( φ ) V φ = − p γ φφ V ∞ sin ( φ ) (12) V ∞ is called the asymptotic velocity of the gas at infinity. The radial and angular velocities of the gasinjected from the outer boundary are also given in terms of the components of the three-metric. The Eq.12guarantees that V = V i V i = V ∞ is valid everywhere along the computational domain.The computational domain is defined on the equatorial plane with r ∈ [ r in , M ] and φ ∈ [0 , π ] inspherical coordinate. The uniform grid is used in the all models along the radial and the angular directionswith N r = 1024 × N φ = 512 cells. So that the grid spacing is ( △ r, △ φ ) = (4 . × − rad, . × − M ) in geometrized unit. We used the dynamical time step size in order to satisfy the Courant-Friedrich-Lewystability condition. In order to extract the resolution dependencies of the numerical results, we monitor thethe mass-accretion rate result from one of our initial condition, changing grid resolution. We explore theresults from three different resolutions, N r = 512 × N φ = 256 , N r = 1024 × N φ = 512 , and N r = 2048 × N φ = 512 and found that the oscillation amplitude of the mass-accretion rate slightly decreases withincreasing employed resolution. But the observed trend is true for all initial conditions. So that the importantoutcome of this paper, effects of Gauss-Bonnet coupling constant α in D EGB gravity on the accreatedmatter and shock cones created around the black hole, would not be effected from the resolutions.In order to model the Bondi-Hoyle accretion numerically, a homogeneous supersonic flow is injectedfrom the upstream region [ π/ , π/ at the location of the outer boundary using the same analytic prescrip-tions given in Eq.12. The computational domain is extended from r min , reported in Table 1 for differentmodels, to outer boundary r max = 100 M which is fixed for all models. φ goes from to π . The soundspeed and the rest-mass density are chosen as c s, ∞ = 0 . and ρ ∞ = 1 , respectively at the boundary. Andthen the gas pressure is computed using the expression p = c s, ∞ ρ (Γ − / [Γ(Γ − − c s, ∞ Γ] accordingly(Donmez et. al. 2011). The asymptotic velocity is chosen as V ∞ = 0 . with the parameters mentionedabove and given in Table 1.The initial injected values of a wind flow are used at upstream region at the outer boundary. Thecontinuation is needed to avoid matter reflected back towards the black hole at the outer boundary of thedownstream region (from π/ to π/ ) along the radial coordinate. It is treated simply by using the zeroth-order extrapolation for all primitive variables. The inner boundary of computational domain along the radialdistance close to the black hole horizon is implemented by using outflow boundary condition, handled 9 –simply copying first values to ghost zones. Lastly, the periodic boundary condition is adopted along the φ direction. In order to reveal the effects of Gauss-Bonnet coupling constant α on the accreated material and thecreation of shock cone, we first focus on the injected matter from the upstream region of the computationaldomain. Later, we wait some time to shock cone reaching the steady state and it is called t critical . Thecritical times are varying from M to M , shown in Table.1. The evolution of these models are, atleast, followed up to t = 16470 M which is sufficient to extract oscillation properties of the shock cone afterreaches the critical time. The critical time is getting smaller for α > while it is increasing for α < .The morphology of the accreated shock cones formed in downstream region are given for the variousvalues of Gauss-Bonnet coupling constant α , seen in Fig.3 where we report the logarithmic rest-mass densityof the accreated matter. The snapshots refer to t ∼ M , which is much later than the critical timeneeded to reach the steady state. The results due to different α show a qualitatively similar behavior, althoughquantitative differences appear in all models. The development of the shock cone and its strong shocklocations, also seen in Fig.6, could cause a chaotic non-linear phenomena inside the cone. The shock conesare attached to the numerical horizon, r in given in Table 1 for different values of α . It is know that thecertain amount of matter would pass across the cone and fall into the black hole.Besides, understanding the dynamic evolution of an accretion disk due to the Bondi-Hoyle accretionand calculating the mass accretion rates allow us to find out more important features of the shock cone. Themass accretion rate is, dMdt = − Z π ˜ α √ γρu r dφ, (13)where all the physical quantities are defined in Section 3.Here, we report the mass accretion rate at a fixed radial distance r = 6 . M for various values of α .The mass accretion onto the D EGB black hole is clearly sensitive to the α , as can be seen in Fig.4. It isclearly seen that increasing the value of α towards zero leads to more violent phenomena inside the shockcone and thus creates severe oscillations after the shock cone reaches the steady state. The high amplitudeoscillation is an indicator of instability fully developed during the evolution. We have also confidence thatthe inner boundary of the computational domain does not play any role on oscillation properties of the shockcone. The oscillating properties of the mass accretion rates are not the same although the inner boundariesfor α = − and α = − are in the same locations, On the other hand, oscillation inside the shock cone isconsiderably dissipated by the larger values of the negative α .The occurrence of the instability is the result of moving wave-like stretching into the medium which 10 –Table 1: The physical parameters and times: Gauss-Bonnet coupling constant α , the inner boundary locationof computational domain r in , time to need to create a fully formed shock cone t critial (required time for thesteady state), and total simulation time t total . M is the mass of the black hole. α ( M ) r in ( M ) t critical ( M ) t total ( M ) −
12 5 ∼ − ∼ − . ∼ − . . ∼ − .
01 2 . ∼ − . . ∼ . . ∼ SCHW . ∼ Fig. 3.— Close-up view of the snapshots of the logarithmic rest-mass density on the equatorial plane at t ∼ M much later than the shock cone reached the steady state t ∼ M for varying values ofGauss-Bonnet constant α and Schwarzschild black hole. The highlighted dynamic boundary is at [ x, y ] → [ − M, M ] . 11 –has a lower density. Overall, it is fair to say that, the instability inside the shock cone has been confirmednumerically, but it is still in debate whether the physical mechanisms driving the instabilities. The instabilitycould be depended on the physical nature of the wind, such as sound speed, matter velocity, and Bondiaccretion radius. A very detailed analysis of the unstable behavior of Bondi-Hoyle accretion flows wasinvestigate by Foglizzo et. al. (2005), suggesting that the instability might be of adjective-acoustic naturein the case of shocks cones around the black hole.After showing the behavior of the accretion rates in vicinity of D EGB black hole, we can now discussthe maximum value of oscillation strength using the mass accretion rate. Fig.5 represents the maximumoscillation amplitude (∆Φ) of the shock cone with various Gauss-Bonnet coupling constant α after theshock cone is fully formed. This is clear evidence that the significant oscillations happen when α is gettingcloser to zero as compared to high negative values. Together with the dependency of the accretion rate to α ,we have also investigated the convergence of EGB gravity solution to the general relativistic one. As it canbe seen in Fig.5, (∆Φ) around the D EGB black hole converges to Schwarzschild one when α → .To have deeper understanding of Fig.3, we extract the information along the angular direction ( φ ) at afixed radial coordinate r = 6 . M . The left panel of Fig.6 shows one-dimensional profiles of the rest-massdensity for different values of Gauss-Bonnet coupling constant α and Schwarzschild solution. The strongshocks are created at the border of the shock cone. Therefore, a sharp transition is seen in density. The sharptransition location has a lowest density throughout the disk since along this location gas falls towards to theblack hole along the streamlines, supersonically. The locations of streamlines seen in left side of the leftpanel of Fig.6 versus α are given in the right panel of the same figure. Increasing in α (going from − to . causes a slight exponential increase in the location of streamline, seen in the right panel of Fig.6. Hencethe shock opening angle appeared in the downstream side of the computational domain expands along theangular direction.In order to uncover the effects of Gauss-Bonnet coupling constant α on the instability created duringevolution and especially inside the shock cone after it reaches the steady state, we numerically study theFourier mode analyze to compute the saturation point and to analysis the characterization of the instability.The Fourier mode m = 1 is performed and the growth rates are obtained for the the rest-mass density powerusing the following equation (Donmez 2014), P m = 1 r out − r in Z r out r in ln ([ Re ( w m ( r ))] + [ Im ( w m ( r ))] ) dr, (14)where r out and r in are outer and inner boundary of the computational domain, respectively. The real andimaginary parts are Re ( w m ( r )) = R π ρ ( r, φ ) cos ( mφ ) dφ and Im ( w m ( r )) = R π ρ ( r, φ ) sin ( mφ ) dφ .As seen in Fig.7, the instability mode grows in the beginning of the simulation until t ∼ M for α = − but the time is slightly getting greater when α converges to . And then we have witnesseddecreasing in the mode for a short time scale ( t ∼ M ) . Later, the m = 1 mode creates an exponentialgrowth again until t ∼ M . The m = 1 mode power reaches a maximum for the α getting close to zero 12 – t/M d M / d t ( a r b it r a r y un it ) α = - 12 α = - 7 α = - 5 α = - 0.8 α = - 0.01 α = 0.8 Fig. 4.— Mass accretion rate as a function of time for varying values of α at r = 6 . M . The accretion ratesreach the steady state around t ≃ M in all models. -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 α /M ∆ Φ ( a r b it r a r y un it ) Schwarzschild α = -0.0001 α = - 0.01 Fig. 5.— The behavior of the maximum oscillation amplitude (∆Φ) of the shock cone with varying Gauss-Bonnet coupling constant α after the shock cone is fully formed. Star seen in figure represents the oscillationamplitude of the shock cone around the Schwarzschild black hole. 13 –and < α < , although the modes in all models saturate almost at the same time t ∼ M . On the otherhand, both α close to zero and Schwarzschild black hole power modes exhibit the same behavior during theevolution.Fourier analysis of the computed data allows us to extract a lot of information about emerged phe-nomenology and their connections with Gauss-Bonnet coupling constant α . In Fig.8, for instance, we obtainthe power spectra using the rest-mass accretion rate computed during the time evolution for different valuesof α and Schwarzschild black hole. We have also conducted an extra test to find the dependency of powerspectra to radial position and found that the mode of oscillation does not depend on the radial position. Itmeans that the modes are global eigenmodes of the oscillating shock cone. As it is seen in Fig.8, there aretwo genuine eigenmodes appearing in the power spectra and the rest are the results of nonlinear couplingsof these genuine modes. For Bondi-Hoyle accretion around the Schwarzschild solution, f = 9 . Hz and f = 17 Hz are genuine eigenmodes, while Hz , . Hz , Hz , and . Hz are the nonlinear couplingsof those genuine eigenmodes with a Hz of error bar. These nonlinear couplings are f , f , f , and f + f , respectively. Similarly, for α = 0 . , f = 6 Hz and f = 13 . Hz are genuine eigenmodes, while Hz , Hz , and . Hz are the nonlinear couplings of those genuine eigenmodes with a Hz of errorbar. These nonlinear couplings are f + f , f , and f + f , respectively. For α = − . , f = 11 Hz and f = 15 . Hz are genuine eigenmodes, while Hz , Hz , and . Hz are the nonlinear couplingsof those genuine eigenmodes with a Hz of error bar. These nonlinear couplings are f − f , f + 2 f ,and f , respectively. For α = − , f = 30 Hz , f = 39 . Hz , and f = 63 Hz are genuine eigenmodes,while Hz , . Hz , Hz , and Hz are the nonlinear couplings of those genuine eigenmodes with a Hz of error bar. These nonlinear couplings are f − f , f − f , f , and f − f , respectively. Thisbehavior is expected behavior of the nonlinear equations in the physical system in the limit of small oscil-lations (Landau & Lifshitz 1976). It is also important to note that the genuine modes and their nonlinearcouplings show different behavior for α = − . There are three main differences between α close to zero(Schwarzschild solution) and negative value of α when α = − . First, there are two genuine modes for − . ≤ α ≤ . while there are three for α = − . Second, the frequency of the first genuine mode f isgetting greater when α goes from . to − . Third, the frequencies of these genuine eigenmodes are muchhigher in α = − than the other three cases. As it is expected, the amplitude of genuine modes are gettingsmaller when α is getting larger in negative direction. To illustrate the consistency of the results found from the static spherically symmetric black hole solu-tion in a novel D EGB gravity with the Schwarzschild one, the Schwarzschild solution of the Bondi-Hoyleaccretion using the same initial conditions is performed. Fig.9 shows the comparison of the accretion ratesbetween EGB and Schwarzschild solutions using certain values of the Gauss-Bonnet coupling constant α .The accretion rates are plotted after the initial phase of relaxation of the shock cone. As it is also seen in theprevious section, the computed accretion rate for the α = − . shows similar behavior (oscillation +amplitude) with the Schwarzschild solution. Obviously, tendency of α to reach zero would give the solution 14 – φ (rad) ρ ( a r b it r a r y un it ) α =-7 α =-5 α =-0.8 α =-0.01 α =0.8 α = Schwarzschild -6 -4 -2 0 α /M S t r ong s ho c k l o ca ti on o f s ho c k c on e (r a d ) Fig. 6.—
Left panel:
The rest-mass density as a function of the angular coordinate φ at r = 6 . M for thevarying values of the Gauss-Bonnet coupling constant α and Schwarzschild black hole. The embedded plotsindicates the left shock locations to see the differences for varying α . Right panel:
The variation of theopening angle of the shock cone versus α computed at φ = ∼ . rad in left panel. The black star representsthe location for the Schwarzschild black hole. t/M M od e P o w e r ( a r b it r a r y un it ) α = - 7 α = -5 α = - 0.8 α = - 0.01 α = 0.8Schwarzschild Fig. 7.— m = 1 mode growth evolution around the EGB black hole for different values of α andSchwarzschild black hole. The region of interested is zoomed to see the saturation points and oscillations ofthe mode powers after they reached the steady state. 15 –in the general relativity. But decreasing in the parameter α causes a significant change in the oscillationproperties of the accreated disk. So that a smaller α would cause to form a less luminous, cooler, and lessefficient shock cone around the black hole. This is in agreement with a thin accretion solution around the D EGB black hole (Cheng et. al. 2020).Power spectrum data has a clear signature about the creation of shock cone instability for various valuesof Gauss-Bonnet coupling constant α and Schwarzschild black hole. Fig.10 represents how the maximumvalue of the m = 1 power mode ( A mod ( max )) changes with α and Schwarzschild black hole. A mod ( max ) increases with an increasing α and it goes to the Schwarzschild solution when α → (e.g. see the star andthe square symbols at α ∼ in Fig.10). For the positive values of α , it is seen in Fig.10 that increasing in α produces higher values of the maximum power spectrum.
5. Conclusion
We have performed a systematic investigation of Gauss-Bonnet coupling constant α during the Bondi-Hoyle accretion onto a non-rotating black hole in D EGB gravity. The effect of α on the shock cone andits oscillation properties have been extensively studied when − ≤ α < . The numerical simulation ofBondi-accretion onto Schwarzschild black hole using the same initial condition is also performed to comparethe general relativistic solution with D EGB gravity one.We investigate how the physical features of the shock cone and its oscillation properties depend on α and find the following outcomes summarized in Table 2.• We have found that the Bondi-Hoyle accretion around D EGB black hole with a negative Gauss-Bonnet coupling constant produces physical solution even it is very close to the black hole horizon.The same was also confirmed by Guo & Li (2020) for the geodesic motions of time-like and nullparticles.• The numerical simulations reveal that moving matter from upstream side of the computational domaincreates a steady state shock cone in a short time scale. The required time having the shock cone indownstream region is called critical time which decreases for α > while increases for α < .• The key indicator of having instability inside the shock cone is a high amplitude oscillation. Theincrease in the α that is getting close to zero, would lead to more violent phenomena inside the shockcone. Therefore the severe oscillations are created after the shock cone reaches the steady state. Inaddition, the oscillation inside the shock cone is considerably dissipated by the larger values of thenegative α .• The shock opening angle slightly depends on the variation of α . Going from α = − to α = 0 . would lead an exponential growth in the location of the shock cone.• The power mode m = 1 reaches a saturation point almost at the same time for all models. The m = 1 mode power gets the maximum value when α is close to zero and < α < . In addition, the power 16 – PS D ( a r b it r a r y un it ) Schwarzschild 0123 α = 0.820 40 60 f (Hz) PS D ( a r b it r a r y un it ) α = -0.8 20 40 60 80 100 f (Hz) α = - 5f f + f f f f +f +f f f - f f + 2f f f -f -2f f -f Fig. 8.— Power spectra of the mass accretion rate for different values of Gauss-Bonnet coupling constant α and Schwarzschild solution computed at r = 6 . M . The mass of the black hole was assumed to be M = 10 M ⊙ . t/M d M / d t ( a r b it r a r y un it ) α = - 7 α = - 0.0001Schwarzschild Fig. 9.— Comparison of the mass accretion rate for α = − and − . with Schwarzschild solution as afunction of time. The mass accretion rates are plotted after the shock cone reaches to the steady state. 17 –modes calculations for the Schwarzschild black hole and for α close to zero show the same behaviorduring the evolution.In addition to recovering many important features of the shock cones for varying α , the novel featuresof the shock cones which traps the pressure modes inside the high density are also extracted by using Fouriertransform. QPOs are computed for the Schwarzschild and different values of α , i.e. α = 0 . , α = − . , and α = − . The global genuine modes and their nonlinear oscillation counterparts are obtained in all cases. α influences not only the amplitude of modes but also their absolute frequencies of the trapped modes.There are three genuine modes found in α = − while it is two for α = 0 . and α = − . including theSchwarzschild one. It is also important to note that the higher absolute frequencies are found in case of α = − . As a result, having three genuine modes need to be confirmed by doing more accurate simulationsin the negative α direction.Possible applications of the numerical results discussed here could be Sagittarius A ∗ (Sgr A ∗ ) and HighMass X − ray Binaries (HMXBs). It is possible to have a direct comparison between numerically computedQPOs and QPOs observed in the X − ray spectra of these sources. Sgr A ∗ black hole is located at the centerof our own Galaxy and believed to be a supermassive black hole with a mass . ± . × M ⊙ whichis the one of target of Event Horizon Telescope (EHT) (Akiyama1 et. al. 2019; Akiyama2 et. al. 2019;Akiyama3 et. al. 2019; Abbott1 et. al. 2016; Abbott2 et. al. 2016). HMXBs are composed of black holeand of an OB star. We are planing to do more direct comparisons in the forthcoming paper. In this paper,we will investigate the effects of the black hole rotation parameter and Gauss-Bonnet coupling constant α on the shock cone dynamics around the rotating EGB black hole.Finally, we have carried out a comparison of D EGB solution with Schwarzschild one. Obviously, itis seen from our numerical simulations that the tendency of α to reach zero would yield solution in generalrelativity.Table 2: The effects of Gauss-Bonnet coupling constant α to the accretion mechanism, shock cones, andphysical interpretation are summarized. D represents decreasing while I represents increasing. SS , SC , P D stand for Steady State, Shock Cone , and Power Mode, respectively.
Time to reach the SS α = − ⇐ = ( I ) α = 0 ⇐ = ( I ) α < Density inside the SC α = − ⇐ = ( D ) α = 0 ⇐ = ( I ) α < Max. PD after saturation α = − ⇐ = ( D ) α = 0 ⇐ = ( D ) α < The shock location α = − ⇐ = ( D ) α = 0 ⇐ = ( D ) α <
18 –
Acknowledgments
The author thanks to the anonymous referee for constructive comments on the original manuscript.All simulations were performed using the Phoenix High Performance Computing facility at the AmericanUniversity of the Middle East (AUM), Kuwait.
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This preprint was prepared with the AAS L A TEX macros v5.2.
20 – -12 -8 -4 0 α /M A m od ( m a x ) Fig. 10.— The maximum value of the m = 1 power mode ( A mod ( max )) versus Gauss-Bonnet couplingconstant αα