title: Deflection angle of a light ray reflected by a general marginally unstable photon sphere in a strong deflection limit
authors: Naoki Tsukamoto
text: 
      
  
  tsukamoto@rikkyo.ac.jp
  
  ^1Department of General Science and Education, National Institute of Technology, Hachinohe College, Aomori 039-1192, Japan 
  
  
  
  We investigate the deflection angle in a strong deflection limit for a marginally unstable photon sphere 
  in a general asymptotically flat, static and spherically symmetric spacetime under some assumptions to calculate observables.
  The deflection angle of a light ray reflected by the marginally unstable photon sphere diverges nonlogarithmically 
  while the one reflected by a photon sphere diverges logarithmically.
  We apply our formula to a Reissner-Nordström spacetime and Hayward spacetime.
  
  
  Deflection angle of a light ray reflected by a general marginally unstable photon sphere in a strong deflection limit
      Naoki Tsukamoto^1
      Received XXX 2019 / Accepted XXX 2020
  =====================================================================================================================
  
  
  
  
  
  § INTRODUCTION
  
  Recently, the direct detection of 
  gravitational waves emitted by black holes has been reported by
  LIGO and VIRGO Collaborations <cit.> and 
  the detection of shadow of black hole candidate at center of a giant elliptical galaxy M87 
  has been reported by Event Horizon Telescope Collaboration <cit.>.
  From further observations,   
  we will check more details of physics in strong gravitational fields in nature.
  
  Static, spherically symmetric spacetimes, which describe strong gravitational fields caused by compact objects, have circular photon orbits and
  the set of the photon orbits is called photon (antiphoton) sphere if it is unstable (stable) <cit.>.
  The property of the (anti)photon sphere such as the upper bound of the radius <cit.> and number <cit.> has been investigated 
  since the (anti)photon sphere is related in phenomena in the strong gravitational field such as
  the high-frequency behavior of the photon absorption cross section <cit.>,
  stability of thin-shell wormholes <cit.>, 
  the high-frequency spectrum of quasinormal modes of compact objects <cit.>,
  a centrifugal force and gyroscopic precession <cit.>,
  Bondi's sonic horizon of a radial fluid <cit.>,
  and an apparent shape during a collapsing star to be a black hole <cit.>.
  Extensions and alternatives of the photon sphere have been investigated <cit.> 
  and instability of the compact objects caused by the slow decay of linear waves near stable photon rings has been considered <cit.>.
  
  Gravitational lensing not only in weak gravitational fields <cit.> 
  but also in the strong gravitational field has been investigated eagerly.
  Infinite number of dim images of light rays reflected by the photon sphere have been investigated many times <cit.>, 
  which are named relativistic images in Ref. <cit.>,
  to get the information of compact objects with the strong gravitational fields.
  
  In 2002, Bozza has investigated a semianalytic formula to calculate the deflection angle of light rays reflected by a photon sphere, its image angle, and magnification 
  in a general asymptotically flat, static and spherically symmetric spacetime <cit.>. 
  The deflection angle of the light, in a strong deflection limit b → b_m, where b and b_m are the impact parameter and the critical impact parameter of the light, respectively,
  is given by
  
      α_def   =   -a̅log( b/b_m-1 ) + b̅
            + O( ( b/b_m-1 )  log( b/b_m-1 ) ),
  
  where a̅ and b̅ are parameters.
  The formalism in the strong deflection limit has been extend by several authors <cit.>.
  Relationships between the strong deflection limit, quasinormal modes <cit.>, and high-energy absorption cross section <cit.> have been investigated.
  
  In this paper, we construct the deflection angle in the strong deflection limit for a marginally unstable photon sphere 
  in a general asymptotically flat, static and spherically symmetric spacetime under some assumptions.
  In the marginally unstable photon sphere case,  Eq. (<ref>) is not suitable because the parameter a̅ diverges.
  We show that the deflection angle α_def(b) is given by
  
      α_def(b)=c̅/(b/b_m-1)^1/6 +d̅ +O( ( b/b_m-1)^1/6),
  
  where c̅ and d̅ are constant.
  We apply the formula to the Reissner-Nordström spacetime and Hayward spacetime <cit.>.
  
  Very recently, the deflection angle of a light reflected 
  by a marginally unstable photon sphere at a wormhole throat in the Damour-Solodukhin wormhole spacetime <cit.> 
   in the strong deflection limit has been obtained as, in Ref. <cit.>, 
  
      α_def(b)=c̅_[87]/(b/b_m-1)^1/4 +d̅_[87] +O( ( b/b_m-1)^3/4),
  
  where c̅_[87] and d̅_[87] are constant, 
  and the deflection angle of a light ray in a spacetime with naked singularity, which is suggested by Joshi et al. <cit.>, has been obtained as, in Ref. <cit.>, 
  
      α_def(b)=c̅_[89]/(b/b_cr-1)^3/2 +d̅_[89] +O( ( b/b_cr-1) ),
  
  where c̅_[89], d̅_[89], and b_cr are constant.
  We notice that results on this paper are compatible with ones in Refs. <cit.> 
  since the spacetimes are not satisfied our assumptions.
  Note that we cannot apply our formula to light rays near a marginally unstable photon sphere if the marginally unstable photon sphere is correspond to a wormhole throat.
  
  This paper is organized as follows. 
  In Sec. II, we investigate the deflection angle of a light in a general asymptotically flat, static and spherically symmetric spacetime with a marginally unstable photon sphere in the strong deflection limit under some assumptions.
  In Sec. III, we obtain a formula of observables in the strong deflection limit. 
  We apply the formula to a Reissner-Nordström spacetime and Hayward spacetime in Sec. IV 
  and we discuss and summarize our result in Sec. V. 
  On this paper we use the units in which a light speed and Newton's constant are unity.
  
  
  
  
  
  
  
  § DEFLECTION ANGLE IN THE STRONG DEFLECTION ANGLE
  
  The line element of an asymptotically flat, static and spherically symmetric spacetime is given by
  
      ds^2=-A(r)dt^2+B(r)dr^2+C(r)(dϑ^2+sin^2 ϑ dφ^2),
  
  where an asymptotically flat condition 
  
      lim_r→∞ A(r)=lim_r→∞ B(r)=1,
            lim_r→∞ C(r)=r^2
  
  is assumed. 
  We assume an marginal unstable photon sphere at r=r_m, which satisfies 
  
      D_m=D^'_m=0,
            D^''_m=C^'''_m/C_m-A^'''_m/A_m>0,
  
  where 
  
      D(r)≡C^'/C-A^'/A,
  
  and where ' denotes a differentiation with respect to r.
  Here D^'_m is given by
  
      D^'_m=C^''_m/C_m-A^''_m/A_m=0.
  
  Here and hereafter, a function with the subtract m denotes the function at r=r_m.
  We also assume that r=r_m is the largest positive solution of D(r)=0 and A(r), B(r), and C(r) are positive and finite in a domain r≥ r_m.
  Time translational and axial Killing vectors t^μ∂_μ=∂_t and φ^μ∂_μ=∂_φ exist 
  because of stationarity and axial symmetry, respectively.
  
  By using the wave number of a photon k^μ≡ẋ^μ, where x^μ is the coordinates and where  denotes a differentiation with respect to an affine parameter along a trajectory of a light ray,
  the trajectory of the light ray is described by k^μ k_μ=0.
  We assume ϑ=π/2 without loss of generality because of spherical symmetry.
  The trajectory of the light is rewritten as
  
      -A(r)ṫ^2+B(r)ṙ^2+C(r)φ̇^2=0.
  
  By using an effective potential 
  
      V(r)≡ -L^2 F(r)/B(r)C(r),
  
  where F(r) is defined by
  
      F(r) ≡C(r)/A(r)b^2-1,
  
  b≡ L/E is the impact parameter of the light, and where E≡ -g_μνt^μk^ν and L≡ g_μνφ^μ k^ν are the conserved energy and angular momentum of the light, respectively, 
  the trajectory of the light is expressed by
  
      ṙ^2+V(r)=0.
  
  For simplicity, we concentrate on the positive impact parameter b>0.
  [
  Given the position of a point source, the lensed images with negative impact parameters can also be formed in the static, spherical symmetric spacetime. See Sec. V.]
  The light can be in a region with V(r)≤ 0.
  From Eqs. (<ref>) and (<ref>),
  we obtain lim_r→∞ V(r)=E^2>0.
  Thus, the light can be at a spatial infinity.
  The first, second, and third derivatives of V(r) with respective to r are given by
  
      V^'   =   -L^2/BC( FG+CD/Ab^2), 
      
      V^''   =   L^2/BC[ F(G^2-G^')-C(D^2+D^')/Ab^2],
  
  and 
  
      V^'''   =   L^2/BC[ F ( -G^3+3GG^'-G^'')  . 
            +CD ( -D^2+DG-3D^'+G^2-G^')/Ab^2
            . +C( D^'G-D^'')/Ab^2],
  
  respectively,
  where G=G(r) is defined by
  
      G(r)≡B^'/B+C^'/C
  
  and we have used a relation
  
      F^'=CD/Ab^2.
  
  
  We assume that the light ray comes from the spatial infinity, it is reflected at a closest distance r=r_0 from a lensing object, and it goes to the spatial infinity.
  Note that 
  
      ṙ_0 ≡. ṙ|_r=r_0=0, 
            V_0 ≡ V(r_0)=0, 
            F_0 ≡ F(r_0)=0
  
  at the closest distance r=r_0. 
  Here and hereafter, a function with the subtract 0 denotes the function at the closest distance r=r_0.
  From Eq. (<ref>), we obtain
  
      A_0ṫ_0^2=C_0φ̇^2_0.
  
  From Eq. (<ref>), the impact factor is expressed as 
  
      b(r_0)=L/E=C_0 φ̇_0/A_0 ṫ_0= √(C_0/A_0)
  
  since it is constant along the trajectory of the light.
  Recall that we have assumed that the impact parameter is positive.
  
  We name a limit r_0 → r_m or b → b_m strong deflection limit,
  where b_m is the critical impact parameter of the light ray defined as
  
      b_m≡√(C_m/A_m).
  
  In the strong deflection limit to the marginal unstable photon sphere r_0 → r_m or b → b_m,
  we obtain 
  
      lim_r_0 → r_mV_0^'=0, 
            lim_r_0 → r_mV_0^''=0,
            lim_r_0 → r_mV_0^'''=-L^2D^''_m/B_mC_m<0.
  
  
  We can rewrite the trajectory equation as 
  
      ( dr/dφ)^2 = CF/B
  
  and we obtain the deflection angle α_def(r_0) of the light as
  
      α_def(r_0)≡ I_0-π,
  
  where I_0=I(r_0) is defined by
  
      I_0≡ 2 ∫^∞_r_0dr/√(CF/B).
  
  
  We introduce a variable [Bozza defined a variable z_[54] as
  
      z_[54]≡A-A_0/1-A_0
  
  in Ref. <cit.>.
  See Refs. <cit.> for the discussion on the variable.]
  
      z≡ 1-r_0/r
  
  and rewrite I_0 as
  
      I_0=∫^1_0 f(z,r_0)dz,
  
  where f(z,r_0) is defined as
  
      f(z,r_0)≡2r_0/√(H(z,r_0)),
  
  and where H(z,r_0) is defined by
  
      H(z,r_0)≡CF/B(1-z)^4.
  
  By using the expansion of F in the power of z 
  
      F   =   D_0 r_0 z + [ r_0^2/2( C^''_0/C_0-A^''_0/A_0)+ (1- A^'_0 r_0/A_0) D_0 r_0 ] z^2 
            +[ D_0 r_0 ( 1-2r_0 A^'_0/A_0 +r_0^2 A_0^' 2/A_0^2 -r_0^2 A^''_0/2A_0)  . 
            +( C^''_0/C_0-A^''_0/A_0)r_0^2 ( 1- r_0 A^'_0/2A_0)  
            .+ r_0^3/6( C^'''_0/C_0-A^'''_0/A_0) ] z^3 +O( z^4 ),
  
  the expansion of H(z,r_0) in the power of z is obtained as
  
      H(z,r_0)=∑^∞_n=1 c_n (r_0) z^n,
  
  where c_1(r_0), c_2(r_0), and c_3(r_0) are given by
  
      c_1(r_0)=C_0D_0r_0/B_0,
  
  
      c_2(r_0)=C_0/B_0[ r_0^2/2( C^''_0/C_0-A^''_0/A_0)- (3 +A^'_0 r_0/A_0) D_0 r_0 ],
  
  and 
  
      c_3(r_0)   =   C_0/B_0[ D_0 r_0 ( 3+2r_0 A^'_0/A_0 +r_0^2 A_0^' 2/A_0^2 -r_0^2 A^''_0/2A_0)  . 
            -( C^''_0/C_0-A^''_0/A_0)r_0^2 ( 1 +r_0 A^'_0/2A_0) 
            . +r_0^3/6( C^'''_0/C_0-A^'''_0/A_0) ],
  
  respectively. [The expansions of a function J(r) and its inverse 1/J(r) in the power of z are shown in Appendix A.]
  In the strong deflection limit r_0 → r_m, c_1(r_0), c_2(r_0), and c_3(r_0)  become
  
      c_1m=c_2m=0
  
  and 
  
      c_3m=C_m/6b_m D^''_m r_m^3
  
  and H(z,r_0) becomes
  
      H(z,r_m)=c_3mz^3 +O( z^4 ).
  
  Therefore, f(z,r_0) diverges with the leading order of z^-3/2 and I_0 diverges with z^-1/2 in the strong deflection limit r_0 → r_m.
  
  We separate I_0 into a divergence part 
  
      I_D   ≡   ∫^1_0 f_D(z,r_m)dz 
         =   4r_m/√(c_3m)1/√(r_0/r_m-1 )-4r_m/√(c_3m) +O( √(r_0/r_m-1)),
  
  where f_D(z,r_m) is defined as
  
      f_D(z,r_m)   ≡   2r_m/√(c_1m z+c_2m z^2+c_3m z^3)
         =   2r_m/√(c_3m z^3),
  
  and a regular part 
  
      I_R(r_0)   ≡   ∫^1_0 f_R(z,r_0) dz,
  
  where f_R is defined as
  
      f_R(z,r_0)≡ f(z,r_0)-f_D(z,r_m).
  
  The regular part I_R is expanded in the power of r_0-r_m as
  
      I_R(r_0)=∑^∞_j=01/j!( r_0-r_m)^j ∫^1_0 . ∂^j f_R/∂ r_0^j|_r_0=r_m dz
  
  and we are interested in the term of j=0 given by
  
      I_R(r_0)= ∫^1_0 f_R(z,r_m) dz +O( √(r_0/r_m-1)).
  
  By using an equation
  
      b=b_m+ b_m D^''_m/12( r_0 -r_m)^3 + O( ( r_0 -r_m)^4 ),
  
  I_D is rewritten as
  
      I_D(b)   =   2^13/6 3^1/3 b_m^1/2/ C_m^1/2 D_m^''1/3( b/b_m-1)^1/6 
      -4√(6 b_m/C_m D^''_m r_m)
             +O( ( b/b_m-1 )^1/6).
  
  Therefore, we obtain the deflection angle α_def(b) as
  
      α_def(b)=c̅/(b/b_m-1)^1/6 +d̅ +O( (b/b_m-1)^1/6),
  
  where 
  
      c̅=2^13/6 3^1/3 b_m^1/2/C_m^1/2 D_m^''1/3, 
            d̅=-4√(6b_m/C_m D^''_m r_m) +I_R -π.
  
  
  
  
  
  
  § OBSERVABLES IN THE STRONG DEFLECTION LIMIT
  
  We consider that 
  a light ray with an impact parameter b 
  is emitted by a source S with a source angle ϕ 
  and that it is reflected by a lens L with an effective deflection angle 
  
      α̅_def≡α_def  mod   2 π
  
  to reach into an observer O.
  A relation between a deflection angle α_def and the effective deflection angle α̅_def can be expressed by
  
      α_def=α̅_def+ 2 π n,
  
  where n is the winding number of the light ray. 
  The observer O observes an image I with an image angle θ. 
  See Figure 1 for a lens configuration.
  
  We assume a small angle lens equation <cit.>
  
      D_LSα̅_def=D_OS(θ -ϕ),
  
  where D_LS and D_OS are angular distances between the lens L and the source S and between the observer O and the source S, respectively. 
  We have assumed that all the angles α̅_def, θ, ϕ≪ 1.
  The impact parameter b is described by 
  
      b=θ D_OL,
  
  where D_OL=D_OS-D_LS is an angular distance between the observer O and the lens L.
  The deflection angle is rewritten as
  
      α_def(θ)=c̅/( θ/θ_∞-1 )^1/6 +d̅ +O( ( θ/θ_∞-1 )^1/6),
  
  where θ_∞≡ b_m/D_OL.
  
  We define θ^0_n as, for n≥ 1,
  
      α_def(θ^0_n )=2π n.
  
  From Eqs. (<ref>) and (<ref>), we get
  
      θ^0_n = [ 1 + ( c̅/2π n - d̅)^6 ] θ_∞.
  
  The deflection angle α_def(θ) can be expanded around θ=θ^0_n as 
  
      α_def(θ) 
         =   α_def(θ^0_n )+.dα_def/dθ|_θ=θ^0_n(θ-θ^0_n ) 
            +O( ( θ-θ^0_n )^2 ),
  
  where, from Eq. (<ref>), 
  
      .dα_def/dθ|_θ=θ^0_n=-c̅/6θ_∞( θ^0_n/θ_∞-1 )^-7/6.
  
  From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), the effective deflection angle is given by
  
      α̅_def(θ_n )=c̅/6θ_∞( θ^0_n/θ_∞ -1 )^-7/6( θ^0_n - θ_n ),
  
  where θ_n is an image angle with the winding number n.
  
  From the lens equation (<ref>) and the effective deflection angle (<ref>),
  the image angle θ_n=θ_n(ϕ) with the winding number n is obtained as 
  
      θ_n(ϕ)∼θ^0_n +6 c̅^6 D_OSθ_∞ (ϕ - θ^0_n )/D_LS( 2 π n - d̅)^7
  
  and an Einstein ring angle with the winding number n is given by
  
      θ_En≡θ_n(0)∼θ^0_n [1-6 c̅^6 D_OSθ_∞/D_LS( 2 π n - d̅)^7].
  
  The magnification of the image with θ_n is obtained by
  
      μ_n(ϕ) ≡θ_n/ϕdθ_n/dϕ∼6c̅^6 D_OSθ_∞^2/D_LSϕ K_n,
  
  where 
  
      K_n≡(2π n-d̅)^6+c̅^6/(2π n-d̅)^13.
  
  The ratio of the magnification of the outermost image with n=1 to the sum of the other images with n≥ 2 is given by 
  
      r̅≡μ_1/∑^∞_n=2μ_n∼K_1/∑^∞_n=2 K_n
  
  and the difference of the image angles between the outermost image and the marginally unstable photon sphere is given by
  
      s̅≡θ_1 - θ_∞∼θ^0_1 - θ^0_∞ = ( c̅/2π - d̅)^6 θ_∞.
  
  
  
  
  
  § APPLICATION
  
  We apply the formula in Sections II and III to the Reissner-Nordström spacetime and the Hayward spacetime 
  with the marginally unstable photon spheres.
  
  
  
  
  
   §.§ Reissner-Nordström spacetime
  
  Eiroa, Romero, and Torres <cit.> have investigated deflection angle of a light in the Reissner-Nordström spacetime in the strong deflection limit
  by using a numerical fitting method to calculate it. They have considered the magnifications of images of not only a point source but also an extended source. 
  In Ref. <cit.>, Bozza has investigated a formula in the strong deflection limit in a general static, spherically symmetric spacetime 
  to apply it to the Reissner-Nordstrom black hole spacetime.
  The time delay of light rays <cit.> and retrolensing in the Reissner-Nordstrom spacetime <cit.> have been investigated. 
  In the overcharged cases, the size of the photon sphere or the shadow and its magnifications has been studied in Refs. <cit.>.
  The shadow of a reflection-asymmetric thin-shell wormhole, which is consists of Reissner-Nordström geometries, also has been considered in Ref. <cit.>. 
  
  The functions A(r), B(r), and C(r) in the Reissner-Nordström spacetime with a marginally unstable photon sphere are given by
  
      A(r)=1/B(r)=1-2M/r+9M^2/8r^2,
            C(r)=r^2.
  
  There is the marginally unstable photon sphere at r=r_m, 
  where r_m is given by 
  
      r_m=3M/2
  
  and the critical impact parameter is given by
  
      b_m=3√(6)M/2.
  
  
  Figure 2 shows the effective potential V/E^2 as a function of r/r_m.
  
  
  In the Reissner-Nordström spacetime, 
  f_R(z,r_m) has a simple form given by
  
      f_R(z,r_m) = 2√(6/z^3(4-3z))-√(6/z^3)
  
  and we can obtain I_R(r_m)=√(6) analytically. 
  Therefore, we obtain c̅ and d̅ analytically as
  
      c̅=2^5/33^1/2∼ 5.49892
  
  and
  
      d̅= -√(6) -π∼ -5.59108,
  
  respectively.
  
  Figure 3 shows the deflection angle α_def(b/b_m) in the strong deflection limit (2.50) and the one calculated by Eq. (2.27) numerically. 
  
  The observables in the strong deflection limit are summarized in Table I.
  
  
  
  
  
  
   §.§ Hayward spacetime
  
  In Ref. <cit.>, the deflection angle of a light in a weak-field approximation and in the strong deflection limit in the Hayward black hole spacetime <cit.> was considered. 
  Chiba and Kimura <cit.> showed the shadow images of in the black hole and its over-charged case.
  
  The functions
  A(r), B(r), and C(r) in the Hayward spacetime with a marginally unstable photon sphere are given by
  
      A(r)=1/B(r)=1-2Mr^2/r^3+2q^2M,
            C(r)=r^2,
  
  where q=±25√(30)/144M.
  The marginal unstable photon sphere is at r=r_m=25M/12 
  and the critical impact parameter b_m is obtained as
  
      b_m=25√(5)M/12.
  
  
  The effective potential V/E^2 as a function of r/r_m is shown as Fig. 4.
  
  
  
  In the Hayward spacetime, f_R(z,r_m) is given by
  
      f_R(z,r_m) = 2√(6-3z+3z^2-z^3/z^3(5-5z+z^2)) -2√(6/5z^3).
  
  It is integrated numerically and it gives I_R(r_m)=1.8973. 
  By applying the formula in Sec. II to the Hayward spacetime, we obtain
  
      c̅=2^13/6 3^1/3/5^1/6∼ 4.95196
  
  and
  
      d̅= -4√(6/5)+I_R(r_m) -π∼ -5.62607.
  
  
  Chiba and Kimura <cit.> obtained the deflection angle α_def(b) as
  
      α_def(b)∼c_[94] M^1/6/b_m^1/6( b/b_m-1 )^1/6∼6.01316/( b/b_m-1 )^1/6,
  
  where 
  
      c_[94]=2^11/63^2/35^5/4∫^∞_0dy/√(432y+900y^2+625y^3)∼ 7.771.
  
  See Eqs. (21) and (22) in Ref. <cit.>.
  Our formula (2.50) recovers the form of Eq. (4.14).
  Figure 5 shows the deflection angle α_def(b/b_m) in the strong deflection limit (2.50), the deflection angle (2.27) calculated in numerical, and the deflection angle (4.14) shown in Ref. <cit.>.
  Equation (4.14) is well matched with the numerical calculation of Eq. (2.27) only near b=b_m since it has a divergent term only.
  The observables in the strong deflection limit are shown in Table I.
  
  
  
  
  
  
  
  § DISCUSSION AND CONCLUSION
  
  On this paper, we have obtained the formula of the deflection angle (<ref>) of the light ray, which is reflected by the marginally unstable photon sphere, in the strong deflection limit b → b_m+0
  in a general asymptotically flat, static and spherically symmetric spacetime under some assumptions to calculate observables.
  We apply the formula to the Reissner-Nordström spacetime and the Hayward spacetime with the marginally unstable photon spheres.
  The light ray in the strong deflection limit b → b_m+0 comes from a spatial infinity and it approaches to the marginally unstable photon sphere as r→ r_m+0.
  From the effective potentials V in Figs. 2 and 4 in the Reissner-Nordström and Hayward spacetimes, respectively, 
  a light ray in a limit b→ b_m-0, which comes from the spatial infinity and it is reflected by the effective potential, 
  also approaches to the marginally unstable photon sphere as r→ r_m-0.
  The observables of the light rays in the limit b→ b_m-0 lie outside the scope of this paper 
  but results in Ref. <cit.> imply that the images with b→ b_m-0 might be brighter than the ones b→ b_m+0 in the winding number n≥ 1.
   
  We comment on the case of the winding number n=0.
  In the case, we cannot use the strong deflection limit analysis since its error becomes large. 
  Usually we apply the weak-field approximation r_0 ≪ b to solve the lens equation as shown in textbooks <cit.>. 
  Note that there is an image with b ≪ b_m near the center of the lens object. It can be calculated numerically 
  without the strong deflection limit analysis such as the deflection angles calculated numerically in Figs. 3 and 5.
  
  Notice that there is the other pair with a negative impact parameter to each image with a positive impact parameter. 
  The magnitudes of the image angles θ_n and magnifications μ_n of the images with the winding number n≥ 1 with the negative impact parameters 
  are almost the same as the ones with the positive impact parameters.
  The difference of image angles of a couple of images and their total magnification are given by 2θ_n∼ 2θ_En and μ_n tot∼ 2 | μ_n|, respectively,
  for each n in the strong gravitational fields as shown Table I.  
  If an observer, a lensing object, and a source object are aligned, i.e., ϕ=0, a couple of images become a (relativistic) Einstein ring. 
  
  We compare the strong deflection limit analysis of the marginally unstable photon sphere with a numerical calculation 
  to confirm it.
  In Tables II and III, the image angle θ_SDL in the strong deflection limit by using Eq. (2.50) and the image angle θ_NUM solved numerically by using Eq. (2.27) 
  with the distances D_OS=16kpc and D_OL=D_LS=8kpc, the mass M=4× 10^6 M_⊙, and the source angle ϕ=0
  in the Reissner-Nordström spacetime and the Hayward spacetime, respectively, are shown for the winding numbers n=1, 2, and 3.
  The errors (θ_NUM-θ_SDL)/θ_NUM with winding numbers n=1, 2, and 3 are given by -10^-2, -10^-3, and -10^-4, respectively.
  
  
  
  
  
  In Ref. <cit.>, Tsukamoto has considered the deflection angle of a light ray reflected 
  by a marginally unstable photon sphere in the strong deflection limit in the Damour-Solodukhin wormhole spacetime <cit.> with
  
      A(r)=1-2M/r,
            B(r)=[ 1-3M/r]^-1,
            C(r)=r^2.
  
  The deflection angle in the strong deflection limit has a form of Eq. (<ref>).
  See. Eq. (3.61) in  Ref. <cit.>.
  We notice that the marginally unstable photon sphere r=r_m= 3M corresponds with a wormhole throat which is satisfied 1/B(r_m)=0
  and it violates our assumption that B(r) is positive and finite in a domain r≥ r_m.
  Therefore, results on this paper are compatible with ones in Ref. <cit.>. 
  
  
  
  In Ref. <cit.>, Paul has investigated the deflection angle of a light ray in a spacetime, which is suggested by Joshi et al. <cit.>, with 
  
      A(r)=1/B(r)=1/( 1+ M/r)^2,
            C(r)=r^2.
  
  The spacetime has a naked singularity at r=0.
  Light rays reach into the curvature singularity 
  if b<b_cr, where b_cr≡ M is a critical impact parameter, 
  while they are scattered if b>b_cr.
  The deflection angle in a limit b → b_cr or r_0 → 0 has a form of Eq. (<ref>).
  See Eq. (15) in Ref. <cit.>. 
  Since the spacetime does not have a photon sphere at r=0 and A(0)=1/B(0)=0 there
  and since the spacetime does not have a marginally unstable photon sphere,
  the spacetime is not satisfied our assumptions.
  Thus, results on this paper are compatible with ones in Ref. <cit.>. 
  
  
  
  
  
  
  
  
  
  
  
  § EXPANSIONS IN THE POWER OF Z
  
  The expansions of a function J(r) and its inverse 1/J(r) in the power of z are given by
  
      J   =   J_0 +J^'_0 r_0 z+ ( J^'_0 r_0 +1/2 J^''_0 r_0^2  ) z^2 
            + ( J^'r_0+ J^''_0 r_0^2 +1/6J^'''_0 r_0^3 ) z^3 +O( z^4 ).
  
  
      1/J   =   1/J_0 -J^'_0 r_0/J_0^2z
            +( -J^'_0 r_0/J_0^2 +J_0^' 2 r_0^2/J_0^3- J^''_0 r_0^2/2J_0^2) z^2 
            +( -J^'_0 r_0/J_0^2 +2J_0^' 2 r_0^2/J_0^3 -J_0^' 3 r_0^3/J_0^4 -J^''_0 r_0^2/J_0^2. 
            . +J^'_0 J^''_0 r_0^3/J_0^3 -J^'''_0 r_0^3/6 J_0^2)z^3 +O(z^4 ).
  
  
  
  
  99
  
  
  
  Abbott:2016blz
    B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],
    
    Phys. Rev. Lett.  116, 061102 (2016).
  
  
  LIGOScientific:2018mvr 
    B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],
    
    Phys. Rev. X 9, 031040 (2019).
  
    
  Akiyama:2019cqa 
    K. Akiyama et al. [Event Horizon Telescope Collaboration],
    
    Astrophys. J.  875, L1 (2019).
  
  Perlick_2004_Living_Rev
  V. Perlick,
  Living Rev. Relativity 7, 9 (2004).
  
  Hod:2017xkz 
    S. Hod,
    
    Phys. Lett. B 727, 345 (2013).
  
  
  Hod:2017zpi 
    S. Hod,
    
    Phys. Lett. B 776, 1 (2018).
  
  
  Sanchez:1977si 
    N. G. Sanchez,
    
    Phys. Rev. D 18, 1030 (1978).
  
  Decanini:2010fz 
    Y. Decanini, A. Folacci, and B. Raffaelli,
    
    Phys. Rev. D 81, 104039 (2010).
  
  
  Barcelo:2000ta
  C. Barcelo and M. Visser,
  
  Nucl. Phys. B 584, 415 (2000).
  
  
  Koga:2020gqd
  Y. Koga,
  
  Phys. Rev. D 101 104022 (2020).
  
  
    Press:1971wr 
    W. H. Press,
    
    Astrophys. J.  170, L105 (1971).
  
  Goebel_1972 
   C. J. Goebel, 
   
  Astrophys. J. 172, L95 (1972).
  
  Raffaelli:2014ola 
    B. Raffaelli,
    
    Gen. Rel. Grav.  48,  16 (2016).
  
  
  Abramowicz_Prasanna_1990 
  M. A. Abramowicz and A. R. Prasanna, 
  Mon. Not. Roy. Astr. Soc. 245, 720 (1990).
  
  Abramowicz:1990cb 
    M. A. Abramowicz,
    
  Mon. Not. Roy. Astr. Soc. 245, 733 (1990).
  
  Allen:1990ci 
    B. Allen,
    
    Nature 347, 615 (1990).
  
  
  Hasse_Perlick_2002
  W. Hasse and V. Perlick,
  Gen. Relativ. Gravit. 34, 415 (2002).
  
  Mach:2013gia 
    P. Mach, E. Malec, and J. Karkowski,
    
    Phys. Rev. D 88, 084056 (2013)
  
  
  Chaverra:2015bya 
    E. Chaverra and O. Sarbach,
    
    Class. Quant. Grav.  32, 155006 (2015)
  
  
  Cvetic:2016bxi 
    M. Cvetic, G. W. Gibbons, and C. N. Pope,
    
    Phys. Rev. D 94, 106005 (2016).
  
  
   Koga:2016jjq 
    Y. Koga and T. Harada,
    
    Phys. Rev. D 94, 044053 (2016).
  
  
  Koga:2018ybs 
    Y. Koga and T. Harada,
    
    Phys. Rev. D 98, 024018 (2018).
  
  
  Koga:2019teu 
    Y. Koga,
    
    Phys. Rev. D 99, 064034 (2019).
  
  
  Ames_1968 
  W. L. Ames and K. S. Thorne, 
  
  Astrophys. J. 151, 659 (1968).
  
  Synge:1966okc 
    J. L. Synge,
    
    Mon. Not. Roy. Astron. Soc.  131, no. 3, 463 (1966).
  
  Yoshino:2019qsh 
    H. Yoshino, K. Takahashi, and K. i. Nakao,
    
    Phys. Rev. D 100, 084062 (2019).
  
  
  Claudel:2000yi 
    C. M. Claudel, K. S. Virbhadra, and  G. F. R. Ellis,
    
    J. Math. Phys.  42, 818 (2001).
  
  
  Gibbons:2016isj
  G. W. Gibbons and C. M. Warnick,
  
  Phys. Lett. B 763, 169 (2016).
  
  
  Cunha:2017eoe 
    P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu,
    
    Phys. Rev. D 96, 024039 (2017).
  
  
  Shiromizu:2017ego
  T. Shiromizu, Y. Tomikawa, K. Izumi, and  H. Yoshino,
  
  PTEP 2017, 033E01 (2017).
  
  
  Yoshino:2017gqv
  H. Yoshino, K. Izumi, T. Shiromizu, and  Y. Tomikawa,
  
  PTEP 2017, 063E01 (2017).
  
  
  Galtsov:2019bty
  D. V. Gal'tsov and K. V. Kobialko,
  
  Phys. Rev. D 99, 084043 (2019).
  
  
  Galtsov:2019fzq
  D. V. Gal'tsov and K. V. Kobialko,
  
  Phys. Rev. D 100, 104005 (2019).
  
  
  Koga:2019uqd 
    Y. Koga and T. Harada,
    
    Phys. Rev. D 100, 064040 (2019).
  
  
  Siino:2019vxh
  M. Siino,
  
  [arXiv:1908.02921 [gr-qc]].
  
  Yoshino:2019dty
  H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa,
  
  PTEP 2020, 023E02 (2020).
  
  
  Cao:2019vlu
  L. M. Cao and Y. Song,
  
  [arXiv:1910.13758 [gr-qc]].
  
  Yoshino:2019mqw
  H. Yoshino, K. Izumi, T. Shiromizu, and  Y. Tomikawa,
  
  PTEP 2020, 053E01 (2020).
  
  
  Lee:2020pre
  K. Lee, T. Shiromizu, H. Yoshino, K. Izumi, and  Y. Tomikawa,
  
  [arXiv:2007.03139 [gr-qc]].
  
  Keir:2014oka 
    J. Keir,
    
    Class. Quant. Grav.  33, 135009 (2016).
  
  
  Cardoso:2014sna 
    V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani,
    
    Phys. Rev. D 90, 044069 (2014).
  
  
  Cunha:2017qtt 
    P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro,
    
    Phys. Rev. Lett.  119, 251102 (2017).
  
  
  Schneider_Ehlers_Falco_1992
  P. Schneider, J. Ehlers, and E. E. Falco,
  Gravitational Lenses (Springer-Verlag, Berlin, 1992).
  
  Schneider_Kochanek_Wambsganss_2006
  P. Schneider, C. S. Kochanek, and J. Wambsganss,
  Gravitational Lensing: Strong, Weak and Micro,
  Lecture Notes of the 33rd Saas-Fee Advanced Course,
  edited by G. Meylan, P. Jetzer, and P. North (Springer-Verlag, Berlin, 2006).
  
  Hagihara_1931 
  Y. Hagihara, 
  
  Jpn. J. Astron. Geophys., 8, 67 (1931).
  
  Darwin_1959
  C. Darwin,
  Proc. R. Soc. Lond. A 249, 180 (1959).
  
  Atkinson_1965
  R. d' E. Atkinson,
  
  Astron. J. 70, 517 (1965).
  
  Luminet_1979
  J.-P. Luminet,  Astron. Astrophys. 75, 228 (1979).
  
  Ohanian_1987
  H. C. Ohanian, Am. J. Phys. 55, 428 (1987).
  
  Nemiroff_1993
  R. J. Nemiroff,  Am. J. Phys. 61, 619 (1993).
  
  Frittelli_Kling_Newman_2000
  S. Frittelli, T. P. Kling, and E. T. Newman,
  Phys. Rev. D 61, 064021 (2000).
  
  Virbhadra_Ellis_2000
  K. S. Virbhadra and G. F. R. Ellis,
  Phys. Rev. D 62, 084003 (2000).
  
  Bozza_Capozziello_Iovane_Scarpetta_2001
  V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta,
  Gen. Relativ. Gravit. 33, 1535 (2001).
  
  Bozza:2002zj 
    V. Bozza,
    
    Phys. Rev. D 66, 103001 (2002).
  
  
  Perlick:2003vg
  V. Perlick,
  
  Phys. Rev. D 69, 064017 (2004).
  
  
  Nandi:2006ds
  K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov,
  
  Phys. Rev. D 74, 024020 (2006).
  
  
  Virbhadra:2008ws
  K. S. Virbhadra,
  
  Phys. Rev. D 79, 083004 (2009).
  
  
  Bozza_2010
  V. Bozza,
  Gen. Relativ. Gravit. 42, 2269 (2010).
  
  Tsukamoto:2016zdu
    N. Tsukamoto and T. Harada,
    
    Phys. Rev. D 95, 024030 (2017).
    
    
  Shaikh:2019jfr 
    R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar,
    
    JCAP 1907, 028 (2019).
  
  
  Wielgus:2020uqz
  M. Wielgus, J. Horak, F. Vincent, and M. Abramowicz,
  
  [arXiv:2008.10130 [gr-qc]].
  
    Bozza:2002af 
    V. Bozza,
    
    Phys. Rev. D 67, 103006 (2003).
  
  
  Eiroa:2002mk 
    E. F. Eiroa, G. E. Romero, and D. F. Torres,
    
    Phys. Rev. D 66, 024010 (2002).
  
  
  Petters:2002fa 
    A. O. Petters,
    
    Mon. Not. Roy. Astron. Soc.  338, 457 (2003).
  
  
  Eiroa:2003jf 
    E. F. Eiroa and D. F. Torres,
    
    Phys. Rev. D 69, 063004 (2004).
    
  
  Bozza:2004kq 
    V. Bozza and L. Mancini,
    
    Astrophys. J.  611, 1045 (2004).
  
  
  Bozza:2005tg 
    V. Bozza, F. De Luca, G. Scarpetta, and M. Sereno,
    
    Phys. Rev. D 72, 083003 (2005).
    
  
  Bozza:2006sn
  V. Bozza and M. Sereno,
  
  Phys. Rev. D 73, 103004 (2006).
  
  
  Bozza:2006nm 
    V. Bozza, F. De Luca, and G. Scarpetta,
    
    Phys. Rev. D 74, 063001 (2006).
  
  
  Iyer:2006cn 
    S. V. Iyer and A. O. Petters,
    
    Gen. Rel. Grav.  39, 1563 (2007).
  
  
  Bozza:2007gt 
  V. Bozza and G. Scarpetta,
  
  Phys. Rev. D 76, 083008 (2007).
  
  
  Tsukamoto:2016qro 
    N. Tsukamoto,
    
    Phys. Rev. D 94, 124001 (2016).
  
  
  Ishihara:2016sfv 
    A. Ishihara, Y. Suzuki, T. Ono, and H. Asada,
    
    Phys. Rev. D 95, 044017 (2017).
  
  
  Tsukamoto:2016oca
  N. Tsukamoto and Y. Gong,
  
  Phys. Rev. D 95, 064034 (2017).
  
  
  Tsukamoto:2016jzh 
    N. Tsukamoto,
    
    Phys. Rev. D 95, 064035 (2017).
  
  
  Tsukamoto:2017edq 
    N. Tsukamoto,
    
    Phys. Rev. D 95, 084021 (2017).
  
  
  Shaikh:2018oul 
    R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar,
    
    Phys. Lett. B 789, 270 (2019)
    Erratum: [Phys. Lett. B 791, 422 (2019)].
  
  
  Shaikh:2019itn 
    R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar,
    
    Phys. Rev. D 99, 104040 (2019).
  
  
  Stefanov:2010xz 
    I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev,
    
    Phys. Rev. Lett.  104, 251103 (2010).
  
  
  Wei:2011zw 
    S. W. Wei, Y. X. Liu, and H. Guo,
    
    Phys. Rev. D 84, 041501 (2011).
  
  
  Hayward:2005gi
  S. A. Hayward,
  
  Phys. Rev. Lett. 96, 031103 (2006).
  
  
  Damour:2007ap 
    T. Damour and S. N. Solodukhin,
    
    Phys. Rev. D 76, 024016 (2007)
  
  
  Nandi:2018mzm 
    K. K. Nandi, R. N. Izmailov, E. R. Zhdanov, and A. Bhattacharya,
    
    JCAP 1807, 027 (2018).
  
  
  Ovgun:2018fnk 
    A. Ovgun,
    
    Phys. Rev. D 98, 044033 (2018).
  
  
  Bhattacharya:2018leh 
    A. Bhattacharya and R. K. Karimov,
    
    arXiv:1811.00768 [gr-qc].
    
  Ovgun:2018swe 
    A. Ovgun,
    
    arXiv:1811.06870 [gr-qc].
  
  Tsukamoto:2020uay
  N. Tsukamoto,
  
  Phys. Rev. D 101, 104021 (2020).
  
  
  Joshi:2020tlq
  A. B. Joshi, D. Dey, P. S. Joshi, and P. Bambhaniya,
  
  Phys. Rev. D 102, 024022 (2020).
  
  
  Paul:2020ufc
  S. Paul,
  
  Phys. Rev. D 102, 064045 (2020).
  
  
  Bozza:2008ev 
    V. Bozza,
    
    Phys. Rev. D 78, 103005 (2008).
  
  
  Sereno:2003nd 
    M. Sereno,
    
    Phys. Rev. D 69, 023002 (2004).
  
  
  Zakharov:2014lqa 
    A. F. Zakharov,
    
    Phys. Rev. D 90, 062007 (2014).
  
  
  Wei:2015qca 
    S. W. Wei, Y. X. Liu, and  C. E. Fu,
    
    Adv. High Energy Phys.  2015, 454217 (2015).
  
  
  Chiba:2017nml
  T. Chiba and M. Kimura,
  
  PTEP 2017, 043E01 (2017).
  
  
  
  
  
  
  
  
  