\begin{document}$ {\cal{Q}}<0 $\end{document} and oblate at \begin{document}$ {\cal{Q}}>0 $\end{document}, and the octopole term causes to the shadow stretch upward at the octopolar strength ngth \begin{document}$ {\cal{O}}<0 $\end{document} and downward at \begin{document}$ {\cal{O}}>0 $\end{document}. The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal direction at \begin{document}$ {\cal{Q}}<0 $\end{document} and \begin{document}$ {\cal{Q}}>0 $\end{document}, respectively. Meanwhile, the black hole shadow shifts upward at \begin{document}$ {\cal{O}}<0 $\end{document} and downward at \begin{document}$ {\cal{O}}>0 $\end{document}. We present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms. From the observable width W, height H, oblateness K, and distortion parameter \begin{document}$ \delta_{c} $\end{document} of the black hole shadow, we can determine the quadrupole and octopolar strengths of a Schwarzschild black hole with a halo. Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo."> Surface geometry and shadow of a Schwarzschild black hole with a halo -
  • [1]

    https://eventhorizontelescope.org/

  • [2]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L1 (2019)

  • [3]

    K. Akiyamaet al. ( Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L2 (2019)

  • [4]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L3 (2019)

  • [5]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L4 (2019)

  • [6]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L5 (2019)

  • [7]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.875, L6 (2019)

  • [8]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L12 (2022)

  • [9]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L13 (2022)

  • [10]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L14 (2022)

  • [11]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L15 (2022)

  • [12]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L16 (2022)

  • [13]

    K. Akiyamaet al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett.930, L17 (2022)

  • [14]

    J. L. Synge, Mon. Not. Roy. Astron. Soc.131, 463 (1966)

  • [15]

    J. M. Bardeen,Timelike and null geodesics in the Kerr metric, inBlack Holes (Les Astres Occlus), edited by C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1973), p. 215.

  • [16]

    J. P. Luminet, Astron. Astrophys.75, 228 (1979)

  • [17]

    S. Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, New York, 1992).

  • [18]

    K. Hioki and K. I. Maeda, Phys. Rev. D80, 024042 (2009), arXiv:0904.3575

  • [19]

    Z. Li and C. Bambi, J. Cosmol. Astropart. Phys.1401, 041 (2014), arXiv:1309.1606

  • [20]

    L. Yang and Z. Li, Int. J. Mod. Phys. D25, 2 1650026 (2016), arXiv:1511.00086v2

  • [21]

    S. Wei, Y. Liu, and R. B. Mann, Phys. Rev. D99, 041303 (2019), arXiv:1811.00047v3

  • [22]

    R. Kumar and S. G. Ghosh, Astro. phys. J.892, 78 (2020), arXiv:1811.01260v4

  • [23]

    Z. Chang and Q. Zhu, JCAP09, 003 (2021), arXiv:2104.14221v2

  • [24]

    S. Wei and Y. Zou,Constraining rotating black hole via curvature radius with observations of M87*, arXiv:2108.02415

  • [25]

    R. A. Konoplya, Phys. Lett. B795, 1 (2019), arXiv:1905.00064

  • [26]

    X. Hou, Z. Xu, M. Zhouet al., JCAP07, 015 (2018)

  • [27]

    K. Jusufi, M. Jamil, P. Salucciet al., Phys. Rev. D100, 044012 (2019)

  • [28]

    M. Jamil, K. Jusufi, K. Linet al., Phys. Rev. D99, 044015 (2019)

  • [29]

    K. Jusufi, M. Jamil, and T. Zhu, Eur. Phys. J. C80, 354 (2020)

  • [30]

    L. Amarilla, E. F. Eiroa, and G. Giribet, Phys. Rev. D81, 124045 (2010), arXiv:1005.0607

  • [31]

    S. Dastan, R. Saffari, and S. Soroushfar, Eur. Phys. J. Plus137, 1002 (2022)

  • [32]

    R. Kumara, B. P. Singha, and S. G. Ghosh, Annals Phys.420, 168252 (2020), arXiv:1904.07652v2

  • [33]

    F. Long, S. Chen, M. Wanget al., Eur. Phys. J. C80, 1180 (2020)

  • [34]

    A. Stepanian, Sh. Khlghatyan, and V. G. Gurzadyan, Eur. Phys. J. Plus136, 127 (2021)

  • [35]

    V. A. Prokopov, S. O. Alexeyev, and O. I. Zenin, J. Exp. Theor. Phys.135, 91 (2022)

  • [36]

    Z. Younsi, D. Psaltis, and F. Özel,Black Hole Images as Tests of General Relativity: Effects of Spacetime Geometry, arXiv:2111.01752

  • [37]

    P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D96, 024039 (2017)

  • [38]

    M. Wang, S. Chen, and J. Jing, J. Cosmol. Astropart. Phys.10, 051 (2017)

  • [39]

    P. V. P. Cunha, C. A. R. Herdeiro, E. Raduet al., Phys. Rev. Lett.115, 211102 (2015), arXiv:1509.00021

  • [40]

    M. Wang, S. Chen, and J. Jing, Phys. Rev. D97, 064029 (2018)

  • [41]

    A. Bohn, W. Throwe, F. Hbertet al., Class. Quantum Grav.32, 065002 (2015), arXiv:1410.7775

  • [42]

    P. V. P. Cunha, C. A. R. Herdeiro, E. Raduet al., Int. J. Mod. Phys. D25, 1641021 (2016), arXiv:1605.08293

  • [43]

    F. H. Vincent, E. Gourgoulhon, C. A. R. Herdeiroet al., Phys. Rev. D94, 084045 (2016), arXiv:1606.04246

  • [44]

    P. V. P. Cunha, J. Grover, C. A. R. Herdeiro,et al., Phys. Rev. D94, 104023 (2016)

  • [45]

    M. Wang, S. Chen, and J. Jing, Phys. Rev. D98, 104040 (2018)

  • [46]

    D. Nitta, T. Chiba, and N. Sugiyama, Phys. Rev. D84, 063008 (2011)

  • [47]

    J. O. Shipley and S. R. Dolan, Class. Quantum Grav.33, 175001 (2016)

  • [48]

    D. Nitta, T. Chiba, and N. Sugiyama, Phys. Rev. D98, 044053 (2018)

  • [49]

    M. Wang, S. Chen, J. Wanget al., Eur. Phys. J. C80, 110 (2020)

  • [50]

    P. V. P. Cunha, N. A. Eiro, C. A. R. Herdeiroet al., JCAP03, 035 (2020)

  • [51]

    M. Wang, S. Chen, and J. Jing, Phys. Rev. D104, 084021 (2021), arXiv:2104.12304v2

  • [52]

    H. C. D. L. Junior, P. V. P. Cunha, C. A. R. Herdeiroet al., Phys. Rev. D104, 044018 (2021)

  • [53]

    T. Johannsen, Astrophys. J.777, 170 (2013)

  • [54]

    M. Wang, S. Chen, J. Wanget al., Eur. Phys. J. C81, 509 (2021), arXiv:1908.04527

  • [55]

    M. Wang, S. Chen, and J. Jing, Commun. Theor. Phys.74, 097401 (2022), arXiv:2205.05855

  • [56]

    M. Wang, S. Chen, and J. Jing,Determination of the spin parameter and the inclination angle by the relativistic images in black hole image, (2021) arXiv:2208.10219

  • [57]

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D92, 104031 (2015)

  • [58]

    S. Chen, M. Wang, and J. Jing, JHEP07, 054 (2020), arXiv:2004.08857v3(2020)

  • [59]

    A. Grenzebach, V. Perlick, and C. Lammerzahl, Phys. Rev. D89, 124004 (2014)

  • [60]

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D97, 104062 (2018)

  • [61]

    P. C. Li, M. Y. Guo, and B. Chen,Shadow of a Spinning Black Hole in an Expanding Universe, arXiv:2001.04231

  • [62]

    P. G. Nedkova, V. Tinchev, and S. S. Yazadjiev, Phys. Rev. D88, 124019 (2013)

  • [63]

    G. Gyulchev, P. Nedkova, V. Tinchevet al., Eur. Phys. J. C78, 544 (2018)

  • [64]

    R. Shaikh, Phys. Rev. D98, 024044 (2018)

  • [65]

    N. Ortiz, O. Sarbach, and T. Zannias, Phys. Rev. D92, 044035 (2015)

  • [66]

    D. Dey, R. Shaikh, and P. S. Joshi, Phys. Rev. D103, 024015 (2021)

  • [67]

    H. Falcke, F. Melia, and E. Agol, Astrophys. J.528, L13 (2000), arXiv:9912263

  • [68]

    A. de Vries, Class. Quant. Grav.17, 123 (2000)

  • [69]

    R. Takahashi, Astrophys. J.611, 996 (2004), arXiv:0405099

  • [70]

    C. Bambi and K. Freese, Phys. Rev. D79, 043002 (2009), arXiv:0812.1328

  • [71]

    L. Amarilla and E. F. Eiroa, Phys. Rev. D85, 064019 (2012), arXiv:1112.6349

  • [72]

    E. F. Eiroa and, C. M. Sendra, Eur. Phys. J. C78, 91 (2018), arXiv:1711.08380

  • [73]

    A. Yumoto, D. Nitta, T. Chibaet al., Phys. Rev. D86, 103001 (2012), arXiv:1208.0635

  • [74]

    L. Amarilla and E. F. Eiroa, Phys. Rev. D87, 044057 (2013), arXiv:1301.0532

  • [75]

    V. K. Tinchev and S. S. Yazadjiev, Int. J. Mod. Phys. D23, 1450060 (2014), arXiv:1311.1353

  • [76]

    S. W. Wei and Y. X. Liu, JCAP11, 063 (2013)

  • [77]

    Y. Huang, S. Chen, and J. Jing, Eur. Phys. J. C76, 594 (2016)

  • [78]

    Z. Younsi, A. Zhidenko, L. Rezzollaet al., Phys. Rev. D94, 084025 (2016), arXiv:1607.05767

  • [79]

    A. Abdujabbarov, M. Amir, B. Ahmedovet al., Phys. Rev. D93, 104004 (2016)

  • [80]

    A. Abdujabbarov, L. Rezzolla, and B. Ahmedov, Mon. Not. R. Astron. Soc.454, 2423 (2015)

  • [81]

    F. Atamurotov, B. Ahmedov, and A. Abdujabbarov, Phys. Rev. D92, 084005 (2015)

  • [82]

    A. Abdujabbarov, F. Atamurotov, N. Dadhichet al., Eur. Phys. J. C75, 399 (2015)

  • [83]

    F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D88, 064004 (2013)

  • [84]

    N. Tsukamoto, Phys. Rev. D97, 064021 (2018), arXiv:1708.07427

  • [85]

    P. V. P. Cunhal and C. A. R. Herdeiro, Gen.Rel.Grav.50(4), 42 (2018), arXiv:1801.00860

  • [86]

    S. W. Wei, Y. C. Zou, Y. X. Liuet al., JCAP1908, 030 (2019), arXiv:1904.07710

  • [87]

    R. Roy and U. A. Yajnik, Phys. Lett. B803, 135284 (2020), arXiv:1906.03190

  • [88]

    F. Long, J. Wang, S. Chen, and J. Jing, JHEP10, 269 (2019)

  • [89]

    X. X. Zeng and H. Q. Zhang, Eur. Phys. J. C80(11), 1058 (2020)

  • [90]

    X. X. Zeng, H. Q. Zhang, and H. Zhang, Eur. Phys. J. C80(9), 872 (2020)

  • [91]

    J. Grover and A. Wittig, Phys. Rev. D96, 024045 (2017)

  • [92]

    https://blackholecam.org/

  • [93]

    W. M. Vieira and P. S. Letelier, Phys. Rev. Lett.76, 1409 (1996), arXiv:gr-qc/9604037

  • [94]

    W. M. Vieira and P. S. Letelier, Phys. Rev. Lett76, 4098 (1996)

  • [95]

    F. L. Dubeibe, T. Saeed, and E. E. Zotos, APJ.908, 74 (2021)

  • [96]

    M. Basovník and O. Semerák, Phys. Rev. D94, 044007 (2016), arXiv:1608.05961v1

  • [97]

    L. Smarr, Phys. Rev. D7, 289 (1973)

  • [98]

    W. J. Wild and R. M. Kerns, Phys. Rev. D21, 332 (1980)

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