Chinese Physics C Vol. 45, No. 1 (2021) 015102 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes* 1,2† 2,3‡ 2,3§ 1♮ 2,3♯ Sen Yang(杨森) Cheng Liu(刘诚) Tao Zhu(朱涛) Li Zhao(赵力) Qiang Wu(武强) 4¶ 2,3,5♭ Ke Yang(杨科) Mubasher Jamil 1Institute of theoretical physics, Lanzhou University, Lanzhou 730000, China 2Institute for theoretical physics and Cosmology, Zhejiang University of Technology, Hangzhou 310032, China 3United center for gravitational wave physics (UCGWP), Zhejiang University of Technology, Hangzhou 310032, China 4School of Physical Science and Technology, Southwest University, Chongqing 400715, China 5School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan Abstract: The transonic phenomenon of black hole accretion and the existence of the photon sphere characterize strong gravitational fields near a black hole horizon. Here, we study the spherical accretion flow onto general para- metrized spherically symmetric black hole spacetimes. We analyze the accretion process for various perfect fluids, such as the isothermal fluids of ultra-stiff, ultra-relativistic, and sub-relativistic types, and the polytropic fluid. The influences of additional parameters, beyond the Schwarzschild black hole in the framework of general parameter- ized spherically symmetric black holes, on the flow behavior of the above-mentioned test fluids are studied in detail. In addition, by studying the accretion of the ideal photon gas, we further discuss the correspondence between the sonic radius of the accreting photon gas and the photon sphere for general parameterized spherically symmetric black holes. Possible extensions of our analysis are also discussed. Keywords: spherical accretion, black hole, RZ parametrization, photon sphere DOI: 10.1088/1674-1137/abc066 I. INTRODUCTION Bondi [4], where an infinitely large homogeneous gas cloud, steadily accreting onto a central gravitational ob- Accretion around a massive gravitational object is a ject, was considered. Bondi's treatment was formulated in basic phenomenon in astrophysics and has been essential the framework of Newtonian gravity. Later, in the frame- to the understanding of various astrophysical processes work of general relativity (GR), the steady-state spheric- and observations, including the growth of stars, the form- ally symmetric flow of a test fluid onto a Schwarzschild ation of supermassive black holes, quasar luminosity, and black hole was investigated by Michel [5]. Since then, X-ray emission from compact star binaries [1-3]. The ac- spherical accretion has been considered for various spher- cretion of matter in a realistic astrophysical process is ically symmetric black holes in GR and modified gravit- rather complicated, since it involves many challenging ies; see [6-30] and references therein for examples. aspects of general relativistic magnetohydrodynamics, in- One important feature of spherical accretion onto cluding turbulence, radiation processes, and nuclear burn- black holes is the phenomenon of transonic accretion and ing. To understand these accretion processes, it is useful the existence of a sonic point (or a critical point). At the to simplify the problem by making assumptions and/or sonic point, the accretion flow transits from the subsonic considering simple scenarios. to the supersonic state. Normally, the locations of the The simplest accretion scenario describes a stationary, sonic points in a given black hole spacetime are not far spherically symmetric solution, as first discussed by from its horizon. What is important and intriguing is that Received 2 August 2020; Published online 25 September 2020 * C.L., T.Z., and Q.W. are supported by National Natural Science Foundation of China (11675143), the Zhejiang Provincial Natural Science Foundation of China (LY20A050002), and the Fundamental Research Funds for the Provincial Universities of Zhejiang in China (RF- A2019015) † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected], corresponding author ♮ E-mail: [email protected] ♯ E-mail: [email protected] ¶ E-mail: [email protected] ♭ E-mail: [email protected] ©2021 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 015102-1 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021) the narrow region around the sonic point is closely re- phenomena not only for specific theories of gravity but lated to some ongoing observations about the spectra of also for analysis in a unified way by exploring the influ- electromagnetic and gravitational waves. Therefore, ence of different black hole parameters on the spherical studying the spherical accretion problem can not only accretion process [40]. Specifically, we focus our atten- help us to understand accretion processes in different tion on the perfect fluid accretion onto general parameter- black holes but, importantly, also provide us with an al- ized spherically symmetric black hole spacetimes and fur- ternative approach to explore the nature of the black hole ther investigate transonic phenomena for different fluids, spacetimes in the regime of strong gravity. including isothermal fluids and polytropic fluids. By On the other hand, the EHT collaboration recently re- studying the accretion of the ideal photon gas, we further ported their first image concerning the detection of the reveal the correspondence between the sonic points of the shadow of a supermassive black hole at the center of a accreting photon gas and the photon sphere, for general neighboring elliptical M87 galaxy [31-36]. This image re- parameterized spherically symmetric black holes. vealed that the diameter of the center black hole shadow Our paper is organized as follows. In Sec. 2, we is (42 3) μ, leading to the measured center mass present a very brief introduction to general parameter- M = (6:5 0:7) × 109 M⊙ [31]. The outer edge of the shad- ized spherically symmetric black holes. Then, in Sec. 3, ow image, if one considers a Schwarschild black hole, we derive the basic equations for subsequent discussions forms a photon sphere near the black hole horizon, at on the spherical accretion of various fluids and present which the trajectories of photons create a closed circular several useful quantities. Sec. 4 is devoted to performing orbit. Within astrophysical observations, the existence of a dynamical systems analysis of the accretion process and a photon sphere is related to the electromagnetic observa- finding the critical points of the system. In Sec. 5, we ap- tions of black holes via the background electromagnetic ply these results to several known fluids and further in- emission and the frequencies of quasi-normal modes. The vestigate the transonic phenomena of the accretion of latter is determined by the parameters of null geodesic these fluids onto general parameterized spherically sym- motions on and near the photon sphere of a given black metric black holes. In Sec. 6, by studying the spherical hole spacetime. accretion of the ideal photon gas and photon sphere of Recently, it was shown that there is a correspondence general parameterized spherically symmetric black holes, between the sonic points of an accreting ideal photon gas we establish the correspondence between the sonic points and the photon sphere, for static spherically symmetric of the ideal photon gas and its photon sphere. The conclu- spacetimes [37]. This important result is valid not only sion of this paper is presented in Sec. 7. for spherical accretion of the ideal photon gas but also for rotating accretion in static spherically symmetric space- II. PARAMETERIZED SPHERICALLY SYMMET- times [38, 39]. In an observational viewpoint, as men- RIC BLACK HOLE SPACETIME tioned in [39], this correspondence connects two inde- pendent observations, the observation of light coming In this section, we present a brief introduction to the from sources behind a black hole and the observation of parameterization by L. Rezzolla and A. Zhidenko's (RZ) emission from the accreted radiation fluid onto the black [40], for generic spherically symmetric black hole space- hole. This is because the size of the hole's shadow is de- times. First, let us consider the line element of any spher- termined by the radius of the photon sphere, and the ac- ically symmetric stationary configuration in the spherical creted fluid can signal the sonic point. polar coordinate system (t;r;θ;ϕ), which can be written as In light of the above studies, it is interesting to ex- plore spherical accretion flows in different black hole B2(r) spacetimes. The additional parameters, beyond the ds2 = −N2(r)dt2 + dr2 + r2(dθ2 + sin2 θdϕ2); (1) N2(r) Schwarzschild black hole in these spacetimes, may affect the accretion flow behavior; thus, we have a potentially important approach for studying the strong gravity beha- where N(r) and B(r) are two functions of the radial co- vior of black holes in many alternative theories of grav- ordinate r alone. In the RZ parameterization, N(r) is ex- ity. Instead of finding the exact solution and studying the pressed as spherical accretion case by case for each given theory, a reasonable strategy is to consider a model-independent N2(x) = xA(x); (2) framework that parameterizes the most generic black- hole geometry through a finite number of adjustable where A(x) > 0 for 0 < x < 1 with x = 1 − r0=r. It is obvi- quantities. For this purpose, in this paper, we consider ous that x = 0 represents the location of the event horizon spherical accretion flows in general parameterized spher- of the black hole, and x = 1 is the spatial infinity. Then, ically symmetric black hole spacetimes [40]. This para- the functions A(x) and B(x) can be further parameterized meterized description allows one to consider accretion in terms of the parameters ϵ, ai , and bi as 015102-2 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin.