Accretion on high derivative asymptotically safe black holes

  • Asymptotically safe gravity is an effective approach to quantum gravity. It is important to differentiate modified gravity, which is inspired by asymptotically safe gravity. In this study, we examine particle dynamics near the improved version of a Schwarzschild black hole. We assume that in the context of an asymptotically safe gravity scenario, the ambient matter surrounding the black hole is of isothermal nature, and we investigate the spherical accretion of matter by deriving solutions at critical points. The analysis of various values of the state parameter for isothermal test fluids, viz., k = 1, 1/2, 1/3, 1/4 show the possibility of accretion onto an asymptotically safe black hole. We formulate the accretion problem as Hamiltonian dynamical system and explain its phase flow in detail, which reveals interesting results in the asymptotically safe gravity theory.
  • For thulium (Tm), the creation of (n, 2n) reaction products is especially sensitive to high-energy neutrons above the (n, 2n) threshold, such as neutrons around 14 MeV; therefore, thulium is an important radiochemical diagnostic element for determining neutron fluencies, and the 169Tm(n, 2n)168Tm reaction cross sections are important data for neutron diagnostics [1, 2].

    Several studies have provided data for the 169Tm(n, 2n)168Tm reaction from the threshold to 28 MeV [313]. Bayhurst et al. [4] measured the data at incident neutron energies between 8.65 and 28 MeV with an uncertainty of 5%−10%. Gamma-rays of the product were measured using an NaI detector; however, they did not measure the data for 9.5−13 MeV and 14.2−16.0 MeV. Veeser et al. [6] measured the data from 14.7 to 24 MeV with an uncertainty of 5%−47% using large liquid scintillators, and Luo et al. [11] measured the data from 13.5 to 14.8 MeV with an uncertainty of 4%−5%. These data are approximately 10% lower than most other measurements around 14 MeV. In 2016, Champine et al. [12] reported new data between 17 and 21 MeV. They used quasimonoenergetic neutrons produced by the 2H(d, n)3He reaction; however, the deuterons induced break-up reactions on the structural materials of the deuterium gas cell and the deuterium gas itself, thus creating a substantial number of contamination neutrons. This resulted in large corrections to the cross-section data, and hence, the uncertainty given in Ref. [12] ranged from 7% to 60%. In 2021 Finch et al. [13] reported new data between 14.8 and 21.1 MeV; they used three HPGe detectors, and the uncertainty ranged from 5.6% to 6.6%. There are obvious discrepancies among these experimental data, especially at incident energies of 13 to 18 MeV. There are also discrepancies among the evaluated data from ENDF/B-VIII.0 [14], JENDL-4.0 [15], and JEFF-3.3 [16] in the energy range 12-20 MeV. In particular, data from JEFF-3.3 are approximately 10%−20% lower than those from ENDF/B-VIII.0 and JENDL-4.0 between 16 and 20 MeV.

    Therefore, more precise measurements in this energy range are needed to clarify the discrepancies among the existing data and guide evaluations. The purpose of this study is to precisely determine the 169Tm(n, 2n)168Tm reaction cross sections in the 12−19.8 MeV energy range.

    For the precision of 169Tm(n, 2n)168Tm cross section measurement, two-ring orientation assembly was designed and successfully used to measure the (n, 2n) reaction cross section data for several other nuclei [10, 17]. Neutron flux was monitored using the BF3 detector. The radioactivity of the products was measured using a Ge detector (GEM60P type). The measurements were performed relative to the 93Nb(n, 2n)92mNb cross section [18]. The cross sections of the 169Tm(n, 2n)168Tm reaction were measured from 12 to 19.8 MeV and compared with previous experimental data and evaluated data. Model calculations were also performed with the UNF code [19].

    This paper is organized as follows. The experimental procedure is presented in Sec. II, the data processing procedure is described in Sec. III, the theoretical calculation is briefly presented in Sec. IV, results and discussion are provided in Sec. V, and conclusions are given in Sec. VI.

    12−19.8 MeV neutrons were produced via the D-T reaction on the target assembly at the 5SDH-2 1.7 MV Tandem accelerator. The incident deuteron beam energy and intensity were 3.276 MeV and approximately 7 μA, respectively. The target assembly was designed to be a single-tube construction to reduce the scattering neutrons caused by target assembly. The diameter of the titanium-tritide (TiT) target active area was 12 mm. The air-cooled device was used to cool the TiT target. The distance from the TiT target to the ground was approximately 3 m, and the distance to the wall and ceiling was greater than 5 m, so that the scattering neutrons from the environment were reduced.

    The sample assembly is shown in Fig. 1. It was a two-ring orientation assembly. The neutron source was surrounded by the two-ring orientation assembly with a radius of 5 cm. It was jointed with the target tube using a stainless steel sleeve. The sample assembly was slightly adjusted through two center orientation poles to ensure that the line crossing the two ring centers threaded the TiT target center. The rings (as shown in Fig. 1) were scaled from 0° to 180°.

    Figure 1

    Figure 1.  (color online) Sample assembly.

    The sample position in the experiment is shown in Fig. 2. Every sample was sandwiched between two niobium foils with a thickness of 0.5 mm. The samples were fixed at locations with angles of 0−161 degrees with respect to the deuteron beam direction, so that simultaneous irradiations could be performed in the range 12−19.8 MeV, and the distance from the TiT target to the samples was 5 cm, where the neutron beam could practically be considered monoenergetic.

    Figure 2

    Figure 2.  Sample setting.

    After deducting the half-target loss, the induced deuteron particle average energy was calculated at a high accelerator voltage and TiT target thickness. The neutron energy and energy resolution of the 0° direction were calculated using the TARGET program based on the geometry parameters of the target tube in this experiment. The neutron energy-angle distribution for 0°−180° with the 5 cm distance from the TiT target to the samples was provided using the NEUYIE program in the DROSG-200 program package. The calculated neutron energy-angle distribution from 0° to 180° was the same as that in Ref. [17].

    The two-ring orientation assembly ensured the samples' precise orientation and reduced the scattering neutron background produced by the assembly's material.

    Neutron flux was obtained by monitoring neutrons with the BF3 detector located in the 0 degree direction and at a distance of 4.5 m, as shown in Fig. 3. An electronic block diagram of the neutron flux monitor is shown in Fig. 4. The relative neutron flux data were acquired on-line using the 6612 counter. The neutron irradiation time was 112 h for the 169Tm samples. Irradiation history could be divided into any number of separate parts, each with a relative neutron flux given by the counts. The total neutron flux measured by the 93Nb(n, 2n)92mNb monitor reaction was apportioned into each irradiation step.

    Figure 3

    Figure 3.  Photograph of the BF3 detector.

    Figure 4

    Figure 4.  Electronic block diagram for measuring neutron flux.

    The samples were procured from the Beijing General Research Institute for Nonferrous Metals. The niobium purity was 99.999%. Table 1 lists the purity, isotopic composition, thickness, and diameter of each sample.

    Table 1

    Table 1.  Sample characteristics.
    SamplePurity (%)Isotopic composition (%)Thickness/mmDiameter/mm
    Niobium99.99993Nb 1000.520
    Thulium99.99169Tm 1000.520
    DownLoad: CSV
    Show Table

    After irradiation, a high resolution Ge detector (type: ORTEC GEM60P) with high efficiency (relative efficiency of 60%) was used to measure the radioactivity of the samples. The details of the radioactivity constants are given in Table 2 and taken from the NuDat database [20]. The efficiency calibration of Ge was performed carefully with a set of standard γ ray surface sources with a diameter of 18 mm, including 152Eu, 166mHo, 241Am, 137Cs, 133Ba, and 60Co, at a distance of 8.2 cm. The thulium sample γ ray is shown in Fig. 5. Corrections were made for self-absorption in the samples.

    Table 2

    Table 2.  Details of the radioactivity constants used in the analysis of experimental data.
    NucleusHalf-life/dEγ/keVIγ (%)
    92mNb10.15934.4499.15
    168Tm93.1198.25154.49
    DownLoad: CSV
    Show Table

    Figure 5

    Figure 5.  (color online) γ ray energy spectrum of the irradiated thulium foil measured using the HPGe detector.

    The measured cross section is given by

    σX=λXNXAXλNbNNbANbWNbPNbWXPXηNbfsNbfNbεNbηXfsXfXεXFϕ1eλNbtmNb1eλXtmXσNb,

    (1)

    where the subscripts X and Nb denote thulium and niobium, respectively, σ is the cross section, λ is the decay constant of the activity, N is the γ-ray peak counts, A is the atomic weight of the target nucleus, W is the weight of the sample, P is the purity of the sample, η is the abundance of the target nucleus, fs is the γ-ray self-absorption correction factor, f is the branching ratio of the γ-ray, ε is the Ge detector γ-ray efficiency, FΦ is the correction factor for the neutron flux fluctuation during irradiation, and tm is the duration of γ-ray counting. The standard cross sections of the 93Nb(n, 2n)92mNb reaction were cited from Ref. [18].

    It is assumed that induced radioactivities distribute uniformly in a sample. Because the distance between the sample and the detector was approximately 8.2 cm and the thickness of the sample was at most 0.5 mm, the one-dimensional treatment was reasonably accepted. The γ-ray self-absorption correction factor, fs, is given by

    fs=1eμμt,

    (2)

    where t is the sample thickness (mm), and μ is the absorption coefficient (mm–1). FΦ is given by

    Fϕ=li=1Nϕi(1eλNbTi)eλNbtil˙i=1Nϕi(1eλXTi)eλXt,

    (3)

    where NΦi is the relative neutron number within the ith irradiation time-interval, Ti is the time of the ith time-interval, ti is the cooling time of the ith irradiation, and l is the total number of time bins.

    The main uncertainty sources were due to the γ-ray detector efficiency, counting statistics, and standard cross section. The efficiency uncertainty of the γ-ray detector was assigned as 2.0%. The statistical uncertainty of γ-rays, depending mainly on activity levels and γ-ray emission probabilities, was approximately 1.0%–1.2%. The uncertainty of the standard reaction cross section, 93Nb(n, 2n)92mNb, was approximately 1.0%–1.6% for the entire energy range from 12.0 to 20.0 MeV. The γ-ray self-absorption correction factor was calculated using the MCNP5 program, and its uncertainty was approximately 0.5%. The component uncertainties in the cross sections are listed in Table 3. This study obtained high-quality data with the minimum uncertainty compared with previous measurements [313]. The main reasons behind such a higher precision include: (1) Better efficiency calibration of Ge was performed using a set of standard γ ray surface sources with a diameter of 18 mm, similar to the samples, so that the efficiency uncertainty of the γ-ray detector was 2.0%. (2) A long neutron-irradiation time of up to 112 h and precise irradiation history with a 10 s time interval were used. (3) New data on the radioactivity constants and better standard cross sections of the 93Nb(n, 2n)92mNb reaction in 2010 were adopted.

    Table 3

    Table 3.  Uncertainties in the cross section.
    ItemsEstimated error (%)
    Standard reaction cross section of 93Nb(n, 2n)92mNb1.0-1.6
    Detector efficiency for Nb γ-ray2
    Detector efficiency for Tm γ-ray2
    Statistics of γ-ray counts for γ-ray1.0-1.2
    γ-ray self-absorption correction factor0.5
    Correction factor for the neutron flux fluctuation1.0
    Total3.4-3.6
    DownLoad: CSV
    Show Table

    The unified Hauser-Feshbach and exciton model [19] was employed to calculate the (n, 2n) cross section, which entirely originates from the pre-equilibrium and equilibrium reaction processes. In the model, the parity and angular momentum conservations are obeyed intrinsically in the description of both the equilibrium and pre-equilibrium decay processes. The theoretical model code UNF [19] was used for the calculations. The Koning-Delaroche global nucleon optical potential [21] was applied to calculate the total and nonelastic cross sections, inverse cross sections, and transmission coefficients of the compound-nucleus emission processes. The Pauli exclusion effect and Fermi motion of nucleons were considered in the exciton state densities [22]. The continuum excited states of the compound and residual nuclei are described by the Gilbert-Comeron level density formula [23].

    The experimental results for the 169Tm(n, 2n)168Tm cross section are given in Table 4. Figure 6 shows the present experimental results along with existing measurements, present calculated result, and evaluated data from ENDF/B-VIII.0, JENDL-4.0, and JEFF-3.3.

    Table 4

    Table 4.  Measured cross sections.
    Incident Energy/MeV169Tm (n, 2n)168Tm σ/mb
    12.01773(62)
    13.01900(67)
    14.01937(69)
    15.01994(71)
    16.02012(70)
    17.01834(62)
    18.01445(48)
    19.01171(40)
    19.8937(33)
    DownLoad: CSV
    Show Table

    Figure 6

    Figure 6.  (color online) Present experimental results (blue circles) of the 169Tm(n, 2n)168Tm cross section compared with previous experimental results (symbols), present calculated result (blue line), and evaluated data from ENDF/B-VIII.0 (gray line), JEFF-3.3 (red line), and JENDL-4.0 (black line).

    The experimental data given by Nethaway [3], Greenwood et al. [9], Zhu et al. [10], and Champine et al. [12] were in good agreement with the present results, as shown in Fig. 6. The present data gave a peak around 16 MeV. The experimental data from Bayhurst et al. [4] were approximately 6% higher than the present ones between 16 and 17 MeV and were in agreement with the present data from 19 to 20 MeV. The experimental data from Frehaut et al. [5] were approximately 5% lower than the present ones between 14 and 15 MeV and were in good agreement with the present results from 12 to 13 MeV. The experimental data measured by Veeser et al. [6] were approximately 6%−9% lower than the present ones between 15 and 16 MeV and were in agreement with the present results from 17 to 20 MeV within the uncertainty. The experimental data from Lu et al. [8] were approximately 6% higher than the present ones at 13 MeV and were in good agreement with the present results from 14 to 18 MeV. The experimental data from Luo [11] were approximately 5% lower than the present ones between 13 and 15 MeV, and the experimental data from Finch et al. [13] were approximately 7% lower than the present ones near 18 MeV and 20 MeV.

    Good agreements were observed between the ENDF/B-VIII.0, JENDL-4.0, and present experimental data from 12 to 20 MeV. The evaluated data from JEFF-3.3 were larger than the present data by 6% at 13 MeV and lower than the present data by 7%−17% at energies of 17−20 MeV. The present calculated result was in good agreement with the present experimental data and was consistent with the evaluated data from ENDF/B-VIII.0 and JENDL-4.0.

    Cross sections of the 169Tm(n, 2n)168Tm reaction were measured at incident energies of 12 to 19.8 MeV using the activation technique relative to the 93Nb(n, 2n)92mNb reaction. Model calculations were then performed with the UNF code.

    The present experimental data had the lowest uncertainty compared with those of previous measurements. The new data were consistent with some previous experimental data and were in good agreement with the ENDF/B-VIII.0 and JENDL-4.0 data in the measured energy region of this study. The present calculated result was in good agreement with the present experimental data and was consistent with the evaluated data from ENDF/B-VIII.0 and JENDL-4.0. The present experimental data are given in a wide energy region of 12.0−19.8 MeV with low uncertainty, which simultaneously includes the rising part, peak, and falling part of the reaction cross section; therefore, they are useful for future improvement of nuclear data evaluation.

    We owe thanks to the accelerator staff at the 5SDH-2 1.7MV Tandem accelerator.

    [1] H. Bondi, Mon. Not. R. Astron. Soc., 112: 195 (1952) doi: 10.1093/mnras/112.2.195
    [2] F. C. Michel, Astrophys. Space Sci., 15: 153 (1972) doi: 10.1007/BF00649949
    [3] M. Begelman, Astron. Astrophys., 70: 583 (1978)
    [4] L. I. Petrich, S. L. Shapiro, and S. A. Teukolsky, Phys. Rev. Lett., 60: 1781 (1988) doi: 10.1103/PhysRevLett.60.1781
    [5] M. Jamil, M. A. Rashid, and A. Qadir, Eur. Phys. J. C, 58: 325 (2008) doi: 10.1140/epjc/s10052-008-0761-9
    [6] S. B. Giddings and M. L. Mangano, Phys. Rev. D, 78: 035009 (2008) doi: 10.1103/PhysRevD.78.035009
    [7] M. Sharif and G. Abbas, Mod. Phys. Lett. A, 26: 1731 (2011) doi: 10.1142/S0217732311036218
    [8] A. J. John, S. G. Ghosh, and S. D. Maharaj, Phys. Rev. D, 88: 104005 (2013) doi: 10.1103/PhysRevD.88.104005
    [9] U. Debnath, Accretions of Dark Matter and Dark Energy onto (n+2)-dimensional Schwarzschild Black Hole and Morris-Thorne Wormhole, arXiv:1502.02982
    [10] A. Ganguly, S. G. Ghosh, and S. D. Maharaj, Phys. Rev. D, 90: 064037 (2014) doi: 10.1103/PhysRevD.90.064037
    [11] E. Babichev, S. Chernov, V. Dokuchaev et al., Phys. Rev. D, 78: 104027 (2008) doi: 10.1103/PhysRevD.78.104027
    [12] J. A. Jimenez Madrid and P. F. Gonzalez-Diaz, Gravitation Cosmol., 14: 213 (2008) doi: 10.1134/S020228930803002X
    [13] J. Bhadra and U. Debnath, Eur. Phys. J. C, 72: 1912 (2012) doi: 10.1140/epjc/s10052-012-1912-6
    [14] P. Mach and E. Malec, Phys. Rev. D, 88: 084055 (2013) doi: 10.1103/PhysRevD.88.084055
    [15] P. Mach, E. Malec, and J. Karkowski, Phys. Rev. D, 88: 084056 (2013) doi: 10.1103/PhysRevD.88.084056
    [16] J. Karkowski and E. Malec, Phys. Rev. D, 87: 044007 (2013) doi: 10.1103/PhysRevD.87.044007
    [17] R. Yang, Phys. Rev. D, 92: 084011 (2015) doi: 10.1103/PhysRevD.92.084011
    [18] E. Babichev, V. Dokuchaev, and Y. Eroshenko, Phys. Rev. Lett., 93: 021102 (2004) doi: 10.1103/PhysRevLett.93.021102
    [19] E. Babichev, V. Dokuchaev, and Y. Eroshenko, J. Exp. Theor. Phys., 100: 528 (2005) doi: 10.1134/1.1901765
    [20] C. Gao, X. Chen, V. Faraoni et al., Phys. Rev. D, 78: 024008 (2008) doi: 10.1103/PhysRevD.78.024008
    [21] S. Weinberg, Critical Phenomena for Field Theorists, in Understanding the Fundamental Constituents of Matter, ed. A. Zichichi (Plenum Press, New York, 1977)
    [22] M. Reuter, Phys. Rev. D, 57: 971 (1998) doi: 10.1103/PhysRevD.57.971
    [23] Gustavo P. De Brito, Nobuyoshi Ohta, Antonio D. Pereira et al., Phys. Rev. D, 98: 026027 (2018) doi: 10.1103/PhysRevD.98.026027
    [24] Mark Hindmarsh, Daniel Litim, and Christoph Rahmede, JCAP, 1107: 019 (2011)
    [25] Rong-Jia Yang, Eur. Phys. J. C, 72: 1948 (2012) doi: 10.1140/epjc/s10052-012-1948-7
    [26] Alfio Bonanno and Martin Reuter, Phys. Rev. D, 62: 043008 (2000) doi: 10.1103/PhysRevD.62.043008
    [27] Yi-Fu Cai and D. A. Easson, JCAP, 09: 002 (2010)
    [28] E. Chaverra and O. Sarbach, Quantum Grav., 32: 155006 (2015) doi: 10.1088/0264-9381/32/15/155006
    [29] E. Chaverra and O. Sarbach, AIP Conf. Proc., 1473: 54 (2012)
    [30] F. Ficek, Class. Quantum Grav., 32: 235008 (2015) doi: 10.1088/0264-9381/32/23/235008
    [31] P. Hartman, Ordinary differential equations, (SIAM Classics in Applied Mathematics vol 38) 2nd Edn (Philadelphia: society for Industrial and Applied Mathematics, 2002)
    [32] L. Perko Differential equations and dynamical systems, 3rd Edn (New York: Springer, 2001)
    [33] A. K. Ahmed, M. Azreg-Ainou, M. Faizal et al., Eur. Phys. J. C, 76: 280 (2016) doi: 10.1140/epjc/s10052-016-4112-y
    [34] A. K. Ahmed, M. Azreg-Ainou, S. Bahamonde et al., Eur. Phys. J. C, 76: 269 (2016) doi: 10.1140/epjc/s10052-016-4118-5
    [35] A. K. Ahmed, U. Camci, and M. Jamil, Class. Quant. Grav., 33: 215012 (2016) doi: 10.1088/0264-9381/33/21/215012
    [36] G. Abbas, A. Ditta1, A. Jawad et al., Gen. Rel. Grav., 51: 136 (2019) doi: 10.1007/s10714-019-2620-4
    [37] A. Jawad and M. U.Shahzad, Eur. Phys. J. C., 77: 515 (2017) doi: 10.1140/epjc/s10052-017-5075-3
    [38] S. Ghosh and P. Banik, Int.J. M. Phys. D, 24: 1550084 (2015)
    [39] L. Jiao and R. Yang, Eur. Phys. J. C, 77: 356 (2017) doi: 10.1140/epjc/s10052-017-4918-2
    [40] J. Frank, A. King, and D. Raine, Accretion Power in Astrophysics, Third Edition (Cambridge, UK: Cambridge University Press, 2002)
  • [1] H. Bondi, Mon. Not. R. Astron. Soc., 112: 195 (1952) doi: 10.1093/mnras/112.2.195
    [2] F. C. Michel, Astrophys. Space Sci., 15: 153 (1972) doi: 10.1007/BF00649949
    [3] M. Begelman, Astron. Astrophys., 70: 583 (1978)
    [4] L. I. Petrich, S. L. Shapiro, and S. A. Teukolsky, Phys. Rev. Lett., 60: 1781 (1988) doi: 10.1103/PhysRevLett.60.1781
    [5] M. Jamil, M. A. Rashid, and A. Qadir, Eur. Phys. J. C, 58: 325 (2008) doi: 10.1140/epjc/s10052-008-0761-9
    [6] S. B. Giddings and M. L. Mangano, Phys. Rev. D, 78: 035009 (2008) doi: 10.1103/PhysRevD.78.035009
    [7] M. Sharif and G. Abbas, Mod. Phys. Lett. A, 26: 1731 (2011) doi: 10.1142/S0217732311036218
    [8] A. J. John, S. G. Ghosh, and S. D. Maharaj, Phys. Rev. D, 88: 104005 (2013) doi: 10.1103/PhysRevD.88.104005
    [9] U. Debnath, Accretions of Dark Matter and Dark Energy onto (n+2)-dimensional Schwarzschild Black Hole and Morris-Thorne Wormhole, arXiv:1502.02982
    [10] A. Ganguly, S. G. Ghosh, and S. D. Maharaj, Phys. Rev. D, 90: 064037 (2014) doi: 10.1103/PhysRevD.90.064037
    [11] E. Babichev, S. Chernov, V. Dokuchaev et al., Phys. Rev. D, 78: 104027 (2008) doi: 10.1103/PhysRevD.78.104027
    [12] J. A. Jimenez Madrid and P. F. Gonzalez-Diaz, Gravitation Cosmol., 14: 213 (2008) doi: 10.1134/S020228930803002X
    [13] J. Bhadra and U. Debnath, Eur. Phys. J. C, 72: 1912 (2012) doi: 10.1140/epjc/s10052-012-1912-6
    [14] P. Mach and E. Malec, Phys. Rev. D, 88: 084055 (2013) doi: 10.1103/PhysRevD.88.084055
    [15] P. Mach, E. Malec, and J. Karkowski, Phys. Rev. D, 88: 084056 (2013) doi: 10.1103/PhysRevD.88.084056
    [16] J. Karkowski and E. Malec, Phys. Rev. D, 87: 044007 (2013) doi: 10.1103/PhysRevD.87.044007
    [17] R. Yang, Phys. Rev. D, 92: 084011 (2015) doi: 10.1103/PhysRevD.92.084011
    [18] E. Babichev, V. Dokuchaev, and Y. Eroshenko, Phys. Rev. Lett., 93: 021102 (2004) doi: 10.1103/PhysRevLett.93.021102
    [19] E. Babichev, V. Dokuchaev, and Y. Eroshenko, J. Exp. Theor. Phys., 100: 528 (2005) doi: 10.1134/1.1901765
    [20] C. Gao, X. Chen, V. Faraoni et al., Phys. Rev. D, 78: 024008 (2008) doi: 10.1103/PhysRevD.78.024008
    [21] S. Weinberg, Critical Phenomena for Field Theorists, in Understanding the Fundamental Constituents of Matter, ed. A. Zichichi (Plenum Press, New York, 1977)
    [22] M. Reuter, Phys. Rev. D, 57: 971 (1998) doi: 10.1103/PhysRevD.57.971
    [23] Gustavo P. De Brito, Nobuyoshi Ohta, Antonio D. Pereira et al., Phys. Rev. D, 98: 026027 (2018) doi: 10.1103/PhysRevD.98.026027
    [24] Mark Hindmarsh, Daniel Litim, and Christoph Rahmede, JCAP, 1107: 019 (2011)
    [25] Rong-Jia Yang, Eur. Phys. J. C, 72: 1948 (2012) doi: 10.1140/epjc/s10052-012-1948-7
    [26] Alfio Bonanno and Martin Reuter, Phys. Rev. D, 62: 043008 (2000) doi: 10.1103/PhysRevD.62.043008
    [27] Yi-Fu Cai and D. A. Easson, JCAP, 09: 002 (2010)
    [28] E. Chaverra and O. Sarbach, Quantum Grav., 32: 155006 (2015) doi: 10.1088/0264-9381/32/15/155006
    [29] E. Chaverra and O. Sarbach, AIP Conf. Proc., 1473: 54 (2012)
    [30] F. Ficek, Class. Quantum Grav., 32: 235008 (2015) doi: 10.1088/0264-9381/32/23/235008
    [31] P. Hartman, Ordinary differential equations, (SIAM Classics in Applied Mathematics vol 38) 2nd Edn (Philadelphia: society for Industrial and Applied Mathematics, 2002)
    [32] L. Perko Differential equations and dynamical systems, 3rd Edn (New York: Springer, 2001)
    [33] A. K. Ahmed, M. Azreg-Ainou, M. Faizal et al., Eur. Phys. J. C, 76: 280 (2016) doi: 10.1140/epjc/s10052-016-4112-y
    [34] A. K. Ahmed, M. Azreg-Ainou, S. Bahamonde et al., Eur. Phys. J. C, 76: 269 (2016) doi: 10.1140/epjc/s10052-016-4118-5
    [35] A. K. Ahmed, U. Camci, and M. Jamil, Class. Quant. Grav., 33: 215012 (2016) doi: 10.1088/0264-9381/33/21/215012
    [36] G. Abbas, A. Ditta1, A. Jawad et al., Gen. Rel. Grav., 51: 136 (2019) doi: 10.1007/s10714-019-2620-4
    [37] A. Jawad and M. U.Shahzad, Eur. Phys. J. C., 77: 515 (2017) doi: 10.1140/epjc/s10052-017-5075-3
    [38] S. Ghosh and P. Banik, Int.J. M. Phys. D, 24: 1550084 (2015)
    [39] L. Jiao and R. Yang, Eur. Phys. J. C, 77: 356 (2017) doi: 10.1140/epjc/s10052-017-4918-2
    [40] J. Frank, A. King, and D. Raine, Accretion Power in Astrophysics, Third Edition (Cambridge, UK: Cambridge University Press, 2002)
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M. Umar Farooq, Ayyesha K. Ahmed, Rong-Jia Yang and Mubasher Jamil. Accretion on High Derivative Asymptotically Safe Black Holes[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/6/065102
M. Umar Farooq, Ayyesha K. Ahmed, Rong-Jia Yang and Mubasher Jamil. Accretion on High Derivative Asymptotically Safe Black Holes[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/6/065102 shu
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Received: 2019-11-11
Revised: 2020-01-29
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Accretion on high derivative asymptotically safe black holes

    Corresponding author: M. Umar Farooq, m_ufarooq@yahoo.com
    Corresponding author: Ayyesha K. Ahmed, ayyesha.kanwal@sns.nust.edu.pk
    Corresponding author: Rong-Jia Yang, yangrongjia@tsinghua.org.cn
    Corresponding author: Mubasher Jamil, mjamil@zjut.edu.cn
  • 1. DBS and H, College of E and ME, National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan
  • 2. Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan
  • 3. College of Physical Science and Technology, Hebei University, Baoding 071002, China
  • 4. Institute for Theoretical Physics and Cosmology, Zhejiang University of Technology, Hangzhou 310023, China

Abstract: Asymptotically safe gravity is an effective approach to quantum gravity. It is important to differentiate modified gravity, which is inspired by asymptotically safe gravity. In this study, we examine particle dynamics near the improved version of a Schwarzschild black hole. We assume that in the context of an asymptotically safe gravity scenario, the ambient matter surrounding the black hole is of isothermal nature, and we investigate the spherical accretion of matter by deriving solutions at critical points. The analysis of various values of the state parameter for isothermal test fluids, viz., k = 1, 1/2, 1/3, 1/4 show the possibility of accretion onto an asymptotically safe black hole. We formulate the accretion problem as Hamiltonian dynamical system and explain its phase flow in detail, which reveals interesting results in the asymptotically safe gravity theory.

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    1.   Introduction
    • The process of accretion of matter onto a black hole is a hot topic in theoretical physics. This process is mainly responsible for the formation of Quasi-periodic oscillations and emission of gravitational waves. The accretion phenomena plays a role in the formation of astronomical objects, such as stars, planets, galaxies, quasars, etc. The most significant phenomena in the universe, including gamma-ray bursts, X-ray binaries, active glactic nuclei, tidal disruption events mainly occur because of the accretion of gas onto black holes. An accretion disk forms when gaseous matter rotates and accumulates around the black hole. In the past century, it was realized that gravity powers most luminous objects in the universe through accretion. Our main aim is to study accretion onto black holes, known to be the strongest gravitating objects responsible for emitting high-energy fluxes from astronomical objects. Moreover, black holes have an event horizons that act as borders that are traversed by fluids that enter them.

      Research on the accretion process in Newtonian gravity began in 1952 by Bondi [1] and later on by Michel [2] in 1972 in the context of general relativity. The difference between the Bondi and the relativistic accretion models is that the former allows stellar winds or ejecta (v>0) from the stellar surface, which is the opposite of the accretion (v<0), whereas the same process of ejecta does not directly apply to black holes, since black holes are not composed of gas and have no gaseous surface. Any kind of ejection, such as jets from the black holes, occurs only in the presence of charged plasma floating around the black hole under the effect of strong magnetic fields. We have disregarded these considerations in this study. According to Michel's approach, the discussion of critical points in relation to accretion is provided in Ref. [3]. A detail study of the accretion process onto spherically symmetric black holes in general relativity and other theories of gravities can be found in [4-17], and the references therein. Babichev et. al. [18, 19] have shown that phantom accretion onto a black hole decreases its mass. In contrast, if the black hole solutions are considered in the Friedmann-Robertson-Walker universe, it is observed that the black hole accretion may increase the mass of gravitational objects [20].

      Black holes that exist as a fundamental part of our universe provide the most intriguing solutions to Einstein's field equations. Einstein's general theory of relativity explains very well the exterior and horizons of black holes, whereas it fails to describe the physics of the deep central region, which is is strongly affected by quantum effects. In this context, Weinberg [21] proposed a theory of asymptotic safe gravity (ASG), which embeds gravity in the quantum field theory framework. As its central property, the effective average action satisfies a formally exact functional renormalization group equation. Presently, it has accumulated substantial evidence that the gravitational renormalization group flow possesses a nontrivial fixed point, which could provide the ultra-violet completion of gravity at trans-Planckian energies. The concept of an asymptotically safe gravitational theory has been applied to theories of gravity (such as the Einstein gravity [22] and f(R) gravity [23]), cosmology [24, 25], and black holes [26]. In this study, we investigate the relativistic accretion problem in the context of an infra-red limit of an asymptotically safe scenario. We follow the Hamiltonian dynamical formalism in the phase space (r,n), where r is the areal radius, and n is the particle density of the fluid. We assume that the improved version of the Schwarzschild black hole is surrounded by a special type of perfect fluid, namely the isothermal type. We then investigate the matter particle dynamics by deriving solutions at critical points. The solutions we obtain for the accretion problem describe the Michel flow and the critical point through which it passes. Hence, to discuss critical flows, we investigate the effect of ASG on the accretion process, which is the main purpose of this work.

      By expressing accreting matter by the isothermal equation of state, we provide a complete description of the fluid flow behavior near the black hole.

      The remainder of this paper is structured as follows: in Section 2 we provide a brief review of field equations to define the static, spherically symmetric black hole metric in ASG within the infra-red limit. We also present some fundamental equations related to accretion to explain a steady-state, radial perfect fluid flow by specifying our assumptions on the fluid equation of state and presenting our results. In Section 3, governing equations for improved Schwarzschild black hole accretion and conservation laws are presented. We then evalute our results at sonic points by considering the isothermal fluid and analyze its flow by choosing suitable values of the state parameter. We also formulate fluid equations as two-dimensional Hamiltonian dynamical systems on the phase space (r,n) by assuming that the Hamiltonian depends on the accretion rate in Section 4. We perform a detailed analysis by providing numerical plots for the phase flow of isothermal fluid on modified Schwarzschild background and discuss the effect of coupling parameter. Conclusions are presented in the last section.

    2.   Notation and equations for spherical accretion in ASG with in infra-red limit
    • Recently, Cai and Easson [27] found black hole solution in ASG scenario considering higher derivative terms in their investigation. They discuss how the inclusion of quantum corrections modifies the Schwarzschild black hole solution. According to Ref. [27], the geometry of a static spherically symmetric Schwarzschild (anti)-de Sitter black hole in ASG in the IR limit is given by

      ds2=(12GMr+2G2Mξr3)dt2+(12GMr+2G2Mξr3)1dr2+r2dθ2+r2sin2θdϕ2,

      (1)

      where G and M denotes the gravitational constant and mass of the black hole, respectively. The outer horizon, which is merely the null hypersurface of the modified version Eq. (1) of the Schwarzschild black hole taking quantum corrections into account can be written in approximate form as

      rh=2GM3[12cosh(13cosh1β)],

      (2)

      where β=27ξ8GM21. rh given in Eq. (2) is the only real root of 12GMr+2G2Mξr3=0, which can be calculated using the Weierstrass Polynomial r=z+2GM3. By expanding Eq. (2) to the leading order of ξ, we can approximate it as [27]

      rIR2GMξ2M.

      (3)

      If we insert the running coupling parameter ξ=0 into Eqs. (1) and (3), we can retrieve the classical Schwrazschild black hole metric and the corresponding event horizon, respectively. Here, we review some important equations describing the steady state Michel flow on a Schwrazschild (anti) de Sitter black hole in ASG. Further detail and generalization to an even more general static spherically symmetric black hole background is provided in Refs. [16, 28, 29].

      As described in the introduction, we model the flow of a perfect relativistic fluid, neglecting the effects related to viscosity or heat transport, and further assume that the fluid's energy density is sufficiently small, such that its self-gravity can be neglected. We assume that the flow of the perfect fluid onto the improved Schwrazschild black hole is steady-state flowing in the radial direction, described by the particle density n (also called baryonic number density), pressure p, and the energy density e by an observer moving along the fluid four-velocity uαuα=1. To investigate the accretion process onto high derivative black hole, as described above, we need to review the fundamental equations of accretion for the underlying geometry of the spacetime.

      The accretion dynamics of a perfect matter is governed by the following conservation laws

      αJα=0,

      (4)

      αTαβ=0,

      (5)

      where Jα=nuα is the particle current density, and Tαβ=nhuαuβ+pgαβ is the stress energy tensor. refers to the covariant derivative with respect to the spacetime metric. Here and onwards, we assume that h denotes the enthalpy per particle defined by h=p+en [30] where h=h(n) is a function of the particle density n only. In the spherical symmetry stationary case, the above Eqs. (4) and (5) reduce to

      r2nu=const=K,

      (6)

      h(12GMr+2G2Mξr3+u2)1/2=const=L,

      (7)

      which expresses the conservation of particle and energy flux through a sphere of constant areal radius r. We stress here that to analyze the perfect fluid flow, Eqs. (6) and (7) will play the main role in the background of improved Schawrzschild black hole, as they will be helpful to convert the present problem into a Hamiltonian dynamical system.

    3.   Flow behavior at critical point
    • Physically, a critical point r=rc describes the transition of the flow's radial velocity measured by the static observer from subsonic to supersonic. If we consider the barotropic fluid for which there is a constant pressure throughout (i.e. h=h(n)), then its equation of state can be expressed as [30]

      dhh=a2dnn,

      (8)

      where a denotes the local speed of sound.

      Differentiating Eqs. (6) and (7) with respect to r, we obtain

      dudr=2ur.c2s(12GMr+2G2Mξr3+u2)GM2r3G2Mξ2r3u2c2s(12GMr+2G2Mξr3+u2),

      (9)

      where c2s=k is the square of the speed of sound, and k is a state parameter for the isothermal equation of state (EoS) p=ke. The above Eq. (9) can be converted into a two-dimensional autonomous Hamiltonian dynamical system:

      f1(r,u)=drdl=r{u2c2s(12GMr+2G2Mξr3+u2)},

      (10)

      f2(r,u)=dudl=2u{c2s(12GMr+2G2Mξr3+u2)GM2r3G2Mξ2r3},

      (11)

      with an arbitrary parameter l, whose phase portraits consist of r versus s, indicate solutions of Eqs. (6) and (7). To obtain critical points, we set the right hand side of Eqs. (10) and (11) equal to zero, which after solving yield

      u2c=GM2rc+3G2Mξ2r3c,

      (12)

      c2s=GM2rc+3G2Mξ2r3c13GM2r2c+7G2Mξ2r3c.

      (13)

      By using Eqs. (12) and (13), we can obtain the sonic points, which refer to the critical points of the dynamical systems of Eqs. (10) and (11). From Eqs. (6) and (7), after performing several intermediate steps, one can arrive at very important equation, which is helpful to describe the critical flows of the fluid under consideration

      12GMr+2G2Mξr3+(u)2=Ar2kuk.

      (14)

      For the standard equation of state, the critical point (rc,uc) is the saddle point, and thus the solution must pass through it. The detailed discussion of this critical point will be presented in the forthcoming sections.

    4.   Hamiltonian analysis for isothermal test fluids
    • To analyze the perfect matter flow, it is useful to employ a dynamical system whose orbits consist of graphs of solutions of the system Eqs. (6) and (7). Such a system can be defined conveniently in terms of r and v (where v is the three-velocity of the fluid). We formulate our problem in terms of the Hamiltonian dynamical system on the phase space (r,n), where the vector field describing the dynamics is the Hamiltonian vector field associated with the function F(r,n). By assumption, F is constant along the trajectories of phase flow and thus meets the definition of level curves. The main usefulness of converting the accretion problem into a dynamical system is that the fluid behavior near the critical point of F can be analyzed using standard tools of the theory of dynamical systems [31, 32].

      In Eqs. (6) and (7), we used two integrals of motion K and L. We stress here that any one of them , or any combination of these integrals can be utilized as Hamiltonian system for the fluid flow. Assuming that the Hamiltonian system is a function of two variables r and v and square of the left hand side of Eq. (7), i.e.,

      H(r,v)=h2(12GMr+2G2Mξr3+u2),

      (15)

      which in a more general form can be written as

      H(r,v)=h2(r,v)(f(r)+u2)1v2.

      (16)

      We introduce the following pair of a dynamical system

      ˙r=H,v˙v=H,r,

      (17)

      where the dots denote the ˜t derivatives, and the H,r and H,v denote partial derivatives of H with respect to r and v, respectively. By solving the right-hand-side of the above equation and subsequently equating to zero results in the desired critical point (rc,uc), we obtain the following fundamental pair of equations

      v2c=a2c,rc(1a2c)fc,rc=4fca2c,

      (18)

      which are thus helpful to derive the following important equations

      (a2c=k)[14rcfc,rc+f]=[14rcfc,rc],

      (19)

      (urc)2=14rcfc,rc.

      (20)

      We point out here that the above pair of Eqs. (19) and (20) is equivalent to Eqs. (12) and (13) and provide the critical radius and critical speed of the moving fluid. Thus, we shall use these equations to locate the position of the critical point (r,ur). We know that one of the most appreciative tools for energy conservation is the Hamiltonian. In the current study, the precise form of the general Hamiltonian Eq. (15) in terms of the variables r and v for the isothermal test fluid can be expressed as

      H(r,v)=(12GMr+2G2Mξr3)1k(1v2)1kv2kr4k,

      (21)

      where k is the state parameter, and v is the ordinary three-dimensional speed of the fluid, which is given by

      v2=u212GMr+2G2Mξr3+u2.

      (22)

      We remark here that u is well-defined everywhere, and the velocity v is defined outside the horizon. Complete derivation of these fundamental equations is provided in Refs. [33-35].

    • 4.1.   Isothermal fluids

    • Expressing accreting matter by the isothermal equation of state (EoS) P=ke (where k is a state parameter), we present a complete description of the fluid flow behavior near the black hole.

      1) The fluid at which isotropic pressure and the energy density of the fluid particles is same is referred to as ultra stiff fluid. In this case, the state parameter has the value k=1. This value of the state parameter reduces Eq. (19) to fc=0, which yields rc=rh, i.e., the critical radius and event horizon coincide. In this case, the Hamiltonian in Eq. (21) reduces to

      H=1v2r4.

      (23)

      As above, the Hamiltonian shows constant of motion i.e., H=H0, and we observe that v behaves as 1r. To explain the physical behavior of the fluid flow, we need to sketch contour plots of H(rc,vc)=Hc. From the Fig. 1 on top left, the black curve indicates the solution for H=Hc, the red curve indicates the solution for H=Hc+0.005, the green curve shows the solution for H=Hc+0.02999, the magenta curve depicts H=Hc0.0001, and the blue curve depicts H=Hc0.09. In summary, we observe that for v>0, there is particle emission, and v<0 depicts the fluid accretion.

      Figure 1.  (color online) Contour plot of Hamiltonian H(21) for ultra-stiff (k=1), ultra-relativistic (k=1/2), radiation (k=1/3) and sub-relativistic (k=1/4) fluids where M=1,G,ξ=0.5. The black curve in these graphs depicts the curve that passes through the critical saddle point i.e. H=H.

      2) If the isotropic pressure is less than the energy density, it has characteristics of an ultra-relativistic fluid. In this type of fluid, the EoS takes the form p=e/2. After setting k=1/2 in Eq. (19), we obtain the following expression for the critical radius

      rc52GM1425ξM.

      (24)

      The Hamiltonian in Eq. (21) in this case reduces to

      H=(12GMr+2G2Mξr3)r2v1v2.

      (25)

      We can observe that H in Eq. (25) is not defined for (r,v2)=(rh,1). However, for some constant values of H=H0, one can solve it for v2. The five trajectories of solutions to Eq. (25) in the phase space are shown in top right diagram of Fig. 1. Here, the black curve indicates the solution for H=Hc, the red curve depicts the solution for H=Hc0.01, the green curve for H=Hc0.005, the magenta curve for H=Hc+0.01, and the blue curve for H=Hc+0.005087. From the contour plots, we see that they are doubly-valued and show unphysical behavior, such we can say that there is no physical significance of such fluid in ASG.

      3) For the radiation fluid, we have the state parameter k=1/3. This fluid has the property to absorb the radiations emitted by the black hole. The insertion of k=1/3 in Eq. (19) results in the following real approximation of the critical radius

      rc3GM59ξM,

      (26)

      while the Hamiltonian in Eq. (21) takes the form

      H=(12GMr+2G2Mξr3)2/3r4/3v2/3(1v2)2/3.

      (27)

      From above Hamiltonian, we see that the point (r,v2)=(rh,1) is not a critical point of the dynamical Hamiltonian system. However, the expression for v2 can be obtained by fixing H=H0. The characteristics of solution curves are depicted in the left lower picture, where the black curve shows the solution for H=Hc, the red curve for H=Hc+0.00099, the green curve for H=Hc+0.0009, the magenta curve for H=Hc0.04, and the blue curve for H=Hc0.09.

      Here, for the radiation fluid, we find some surprising characteristics as it gets closer to the black hole. The black, magenta, and blue curves exhibit unphysical behavior, however, the green curves describe highly interesting behavior of the transonic type. The fluid has supersonic velocity before the critical point, but as soon as it approaches the critical point, the speed becomes subsonic.

      4) In sub-relativistic fluids, energy density exceeds the isotropic pressure, and the assigned value to the state parameter is k=1/4. Repeating previous steps, we obtain an approximation of the critical radius as

      rc72GM2649ξM.

      (28)

      The insertion of Eq. (28) into Eq. (20) provides the desired critical point (rc,uc).

      In this case of a sub-relativistic fluid, the Hamiltonian Eq. (24) takes the following form

      H=(12GMr+2G2Mξr3)3/4rv1/2(1v2)3/4.

      (29)

      From above equation, it is evident that the point (r,v2)=(rh,1) is not a critical point of the dynamical system. Now, we draw contour plots of H in the (r,v) plane by fixing H=Hc which describes the following behavior of the moving fluid. The black curve shows the solution for H=Hc, the red curve for H=Hc+0.03, the green curve for H=Hc0.0399, the magenta curve for H=Hc+0.0009, and the blue curve for H=Hc0.0317999. The curves shown in the lower right figure describe the behavior of the moving fluid as follows: in the blue and green curves, the fluid reaches near the critical point to show transonic behavior, but surprisingly fails to do so. Hence, in this scenario, we define this motion as unphysical behavior of the fluid (as they show velocity as a double valued function). However, blue and magenta curves show the supersonic accretion motion in the region v>vc and subsonic motion in the region where v<vc.

    • 4.2.   Critical analysis for isothermal fluids

    • Unlike the Schawrzschild black hole, the quantum gravity affects the accreting fluid near the improved version of Schawrzschild black hole. Furthermore, if we do not entertain the quantum gravity effects, the above presented results are easily reducible to what already been published in [2]. We discuss the asymptotic behavior of isothermal fluids with EoS p=ke, such that 0<k<1. Eq. (13) can be written in the following form

      GM2rc+3G2Mξ2r3c=k(13GM2r2c+7G2Mξ2r3c),

      (30)

      which can again be reduced to a depressed cubic equation by introducing the Wierstrass polynomial r=t+7GM6, which is equivalently expressed as

      t3cptcq=0,

      (31)

      where

      p=49G2M212,q=343G3M3108.

      (32)

      Here, Eq. (31) has three roots: one real root and the other two will be complex conjugates of each other. This follows directly from the Cardano formula

      rc=3q+ιW+3qιW,

      (33)

      where W=p3+q2. We can perform a detailed analysis by computing the Jacobian matrix for the Eqs. (10) and (11), for instance

      J=(f1rf1uf2rf1u).

      (34)

      With the help of the above Jacobian matrix, one can determine that either the critical values are center, saddle, or spiral. If both eigenvalues are real and have different signs, we have a saddle point. If the real part of the complex eigenvalues is negative, then we have a spiral and if the real part of the complex number is zero, we have a center.

      Hence, using Eqs. (19) and (20), one can obtain rc and urc, respectively (velocity of the fluid at sonic point). Then, after putting (rc,urc) in Eq. (14), we find the constant A to obtain u in an explicit form. Moreover, from the normalization of the four-velocity vector, one can also derive an expression for ut(r). Thus, after finding the explicit forms of u and ut, one can sketch (uut)2 along the r-axis to see whether for each case k = 1, 1/2, 1/3, 1/4, the fluid passes through the sonic point, as sketched in Fig. 2. In Fig. 1, we have discussed non-transonic solutions, but we have also plotted the transonic solutions for the isothermal fluid in Fig. 2. The transonic solutions yield a maximum accretion rate, because they pass through the critical point. In Fig. 2, we see that the fluid trajectories may form an orbit for k=1/4 near the Cauchy and event horizon.

      Figure 2.  (color online) Transonic solutions for isothermal fluid with equation of state p=ke with k=1,1/2,1/3,1/4. The value of coupling parameter ξ=0.5 and other constants are fixed to be M=G=1.

    5.   Conclusion
    • The model of spherical accretion is used generally to test various theories of modified gravity. To test these theories from the astrophysical perspective, we must investigate how the behaviour of fluids is modified under the change of parameters of modified gravity appearing in the metric of black holes. By varying these free parameters of modified gravity, the positions of critical points might shift, and the speed of fluid flow might enhance or decay near the black hole. Moreover, the fluid behavior might shift from supersonic to subsonic. In the literature, the spherical accretion on black holes has been studied under the frameworks of different modified gravities, like the braneworld gravity [36], Horava-Lifshitz gravity [37], f(R) gravity [33] and f(T) gravity [34], to list a few. In the present study, we were motivated to test another candidate theory of quantum gravity namely, the higher derivative asymptotic safe gravity in the infra-red limit. From Eq. (3) of our paper, the size of black holes in the ASG theory is smaller compared to the Schwarzschild BH. Furthermore, Eq. (26) suggests that the position of the critical point shifts further towards the BH, and it is smaller than the respective critical point for the Schwarzschild BH. Thus, one can compare and distinguish the relativistic accretion models from Schwarzschild BH from a ASG BH by changing the ξ parameter.

      In this study, we adapt the Hamiltonian method of Michel type accretion as developed by several authors of this work [33]. This method is more general than the original method of Michel. Here, we use the general equations for spherical accretion including conservation laws for the ASG BH static metric. The pressure of the perfect fluid for such spherically symmetric flows is, up to a sign, the Legendre transform of the energy density. This leads to a simple differential equation allowing the determination of the energy density, enthalpy, or pressure knowing one of the equations of state. Furthermore, Bondi's model of accretion on a normal star is the oldest model of spherical accretion employing Newtonian mechanics. In that model, the fluid is adiabatic and non-viscous, and the flow is always transonic, thus it allows the existence of critical point. The Bondi model also allows the outflows during accretion, which can explain the jet phenomenon from certain active galactic galaxies, see for details [38]. However, we found that the fluid flow can be more general than Bondi model. The fluid flow can have subsonic, supersonic, and transonic regimes. Also, because of relativistic treatment, more than one critical point might exist, which allows the heteroclinic flows as well.

      In our study, we investigated both adiabatic and isothermal fluid flows, since both of them have important astrophysical relevance, see Refs. [38]. Thus, there are astrophysical situations where either entropy is constant and temperature varying (adiabatic) or vice versa (isothermal). In the isothermal case, the sound speed of accretion flow at any radii is always equivalent to the sound speed at a sonic point. Hence, if the temperature of the flow is known, one can easily compute the critical point.

      In the Newtonian stellar accretion model, the size of critical radius is considerably larger than the relevant Schwarzschild radius of the star (several hundreds or thousands times the Schwarzschild radius of the star), while the corresponding critical radius for the ASG improved black hole is between 2M and 6M, or comparable to three times the Schwarzschild radius. Therefore, for accretion over black holes, the fluid experiences transonic or ultra-sonic flow just seconds before entering the horizon. The ASG black holes are predicted to be smaller in size compared to the Schwarzschild black hole, while both have same mass. Thus, the relevant transitions from sonic to supersonic to ultrasonic flows occur at a significantly faster rate for ASG black holes. In the study of accretion near the black hole, we found that no physical significance of the radiation fluid exists in asymptotic safe gravity. We observe that the effect of ASG parameter affects the fluid behavior for small values of radial parameter. Moreover, in the critical analysis of the isothermal test fluids, we observe that the fluid trajectories may form a closed orbit for k=1/4 near the Cauchy horizon. It is interesting to consider astronomical observation effects, such as accretion rate and temperature, as done in Refs. [17, 39, 40], which we aim to address our future research.

      The authors would like to thank the anonymous referee for providing insightful comments.

Reference (40)

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