Eur. Phys. J. C (2018) 78:191 https://doi.org/10.1140/epjc/s10052-018-5684-5
Regular Article - Theoretical Physics
Weak deflection gravitational lensing for photons coupled to Weyl tensor in a Schwarzschild black hole Wei-Guang Cao1,2 , Yi Xie1,2,a 1 2
School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210093, China
Received: 12 February 2018 / Accepted: 27 February 2018 © The Author(s) 2018. This article is an open access publication
Abstract Beyond the Einstein–Maxwell model, electromagnetic field might couple with gravitational field through the Weyl tensor. In order to provide one of the missing puzzles of the whole physical picture, we investigate weak deflection lensing for photons coupled to the Weyl tensor in a Schwarzschild black hole under a unified framework that is valid for its two possible polarizations. We obtain its coordinate-independent expressions for all observables of the geometric optics lensing up to the second order in the terms of ε which is the ratio of the angular gravitational radius to angular Einstein radius of the lens. These observables include bending angle, image position, magnification, centroid and time delay. The contributions of such a coupling on some astrophysical scenarios are also studied. We find that, in the cases of weak deflection lensing on a star orbiting the Galactic Center Sgr A*, Galactic microlensing on a star in the bulge and astrometric microlensing by a nearby object, these effects are beyond the current limits of technology. However, measuring the variation of the total flux of two weak deflection lensing images caused by the Sgr A* might be a promising way for testing such a coupling in the future.
1 Introduction Gravitational lensing has become an invaluable tool in astronomy, cosmology and gravitational physics [1–4]. The intrinsic essence of gravitational lensing is the interaction between electromagnetic and gravitational fields. Beyond the standard Einstein–Maxwell theory, the effect of one-loop vacuum polarization on photons was considered under different spacetimes and found to be extremely small [5–15]. Extended models of the coupling were also investigated for various physical circumstances [16–30]. a e-mail:
[email protected]
The Weyl tenser can also play as a mediator to couple the electromagnetic and gravitational fields. Such a coupling has been widely investigated in several contexts [31–43]. Recently, strong deflection gravitational lensing for photons coupled to the Weyl tensor has received much attention due to the deployment of direct observation on the supermassive black hole at the Galactic center, Sgr A*, by the Event Horizon Telescope.1 The unique feature of the strong deflection lensing is the relativistic images of photons winding several loops around the lens [44,45], which can not be generated in a weak gravitational field. Focusing on photons coupled to the Weyl tensor in a Schwarzschild black hole, the strong deflection lensing was studied [46,47]. It was then extended to a more complicated background by considering a Kerr black hole [48]. However, the relativistic images of Sgr A* are extremely faint [49–51] and, therefore, exceedingly difficult to detect. As an alternative, the primary and secondary images of weak deflection gravitational lensings are much easier to observe and they have been extensively used in astronomy and cosmology [1–3]. Weak deflection lensings can also provide insights on modified theories of gravity [52–54] and clues on the interaction between electromagnetic and gravitational fields. In this work, we will study weak deflection lensing for photons coupled to Weyl tensor in a Schwarzschild black hole, which was absent in the literature. By focusing on coordinate-invariant quantities, we obtain all of its geometric optics lensing observables, which include bending angle, image position, magnification, centroid and time delay. These observables are worked out to the second order in the perturbation parameter ε which is the ratio of the angular gravitational radius to angular Einstein radius of the lens. The results are represented in a unified form which is valid for both of two polarization directions of the Weyl coupling.
1
http://www.eventhorizontelescope.org/.
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In Sect. 2, after the unified effective metric for photons coupled to the Weyl tensor in a Schwarzschild black hole with two polarization directions is briefly reviewed, we will derive its light bending angle that is expressed with invariant quantities. The lensing observables, including positions, magnifications and time delay of images, are obtained in Sect. 3 and relations between them are represents in Sect. 4. We work out practical observables of the lensing and investigate its observability for several astrophysical scenarios in Sect. 5. Finally, in Sect. 6, we summarize and discuss our results.
in Refs. [52,53], which is called “Keeton–Petters formalism” for short hereafter, cannot be applied to the spacetime (1) in this paper. Keeton–Petters formalism is valid for a static, spherically symmetric and asymptotically flat spacetime, whose metric is written in the standard Schwarzschild coordinates denoted by overbar with coefficients: [52,53] 2 3 φ φ φ AKP (¯r ) = 1 + 2a1 2 + 2a2 2 + 2a3 2 + ··· , c c c 2 3 φ φ φ BKP (¯r ) = 1 − 2b1 2 + 4b2 2 − 8b3 2 + ··· , c c c CKP (¯r ) = r¯ 2 .
2 Effective metric and light bending
We consider a Schwarzschild black hole with mass M• as the lens, and set the observer and the source in the asymptotically flat region of its spacetime. We assume that it is vacuum outside the lens. When a photon couples to the Weyl tensor in the background of the Schwarzschild black hole, its worldline will no longer follow the null geodesic. However, it was found [39,46] that the geodesic rule can be recovered by taking an effective metric for such a coupling. This metric can be written as − A(r )dt 2 + B(r )dr 2 + C(r )dΩ 2 ,
(1)
where r is the radial coordinate and dΩ 2 = dθ 2 + sin2 θ dϕ 2 . The functions A(r ), B(r ) and C(r ) are A(r ) = B(r )−1 = 1 − and C(r ) =
r 3 + 16α m • r 3 − 8α m •
2m • r
(2)
s r 2,
(3)
where α is a constant with dimension of [Length]2 characterising strength of the coupling between the photon and the Weyl tensor, m • = G M• /c2 is the gravitational radius of the Schwarzschild black hole, and s is an constant. We use s to unify the expression of C(r ) for two different polarizations of the photon respectively along lμ (PPL) and m μ (PPM) (see [39,46] for more details): + 1 for PPL, s= (4) − 1 for PPM. In fact, the results of weak deflection lensing that we obtain in the following parts are also valid when s takes other real numbers with different physical interpretations. Before we perform detailedly and lengthy calculation on the light bending and its resulting lensing observables, it is worth mentioning that the parameterized second-order postNewtonian formalism for weak deflection lensing established
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(6) (7)
Here, φ is the Newtonian potential with
2.1 Effective metric
ds 2 =
(5)
φ m• =− , 2 c r¯
(8)
and a1,2,3 and b1,2,3 are dimensionless and numerical parameters. However, after transforming (1) from its current coordinates to the standard Schwarzschild ones, we find that b1,2,3 depend on the radial coordinate rather than numerical ones, such as α (9) b1 (¯r ) = 1 + 24s 2 . r¯ The additional r¯ -dependent terms have to be taken into account when evaluating the integral of the light bending angle (see next subsection for details); however, they are absent in the Keeton–Petters formalism. This issue was also recognized for Solar System tests of a scalar-tensor gravity [55]. One exception is the trivial case of α = 0 in which the metric (1) reduce to the Schwarzschild black hole in the standard coordinates. Therefore, due to the fact that such a formalism cannot be directly employed, we stick to the metric (1) and perform all of the indispensable calculation for proceeding our investigation. 2.2 Light bending For a light ray propagating through spacetime (1), the distance of closet approach r0 and the impact parameter b of the light ray satisfy the relation as [56] C(r0 ) = b2 A(r0 ).
(10)
In the scenario of weak deflection lensing, r0 and b are much larger than m • which leads to the Taylor expanded solution to (10) as r0 =1− bn q n + O(q 7 ) b 6
(11)
n=1
with small parameter q≡
m• . b
(12)
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The first two coefficients b1 and b2 are b1 = 1 + 12s α, ¯ 3 b2 = + 36s α¯ + 24s α¯ 2 (15s − 2), 2
where (13) (14)
where we define a dimensionless parameter as α α¯ ≡ 2 ; b
(15)
the details of other bn (n = 3, . . . , 6) can be found in Appendix A. When the coupling vanishes, i.e., α¯ = 0, Eq. (11) returns to the one for the Schwarzschild black hole in Einstein’s general relativity (GR) [52]. Following the standard procedure, e.g. [56], the bending angle can be obtained as [49,56] √ ∞ 2 B(r ) dr − π, (16) α(r ˆ 0) = √ C(r ) A(r0 ) r0 C(r ) C(r − 1 0 ) A(r ) which can be written in the form of a series for weak deflection lensing as [52] α(h) ˆ =
6
an h n + O(h 7 )
(17)
n=1
with small parameter h≡
m• . r0
(18)
The coefficients an for n = 1 and 2 are a1 = 4 + 32s α¯ 0 , 135 15 π − 4 + s α¯ 0 π − 144 a2 = 4 2 1125 π − 1152 s − 75π , + s α¯ 02 2 α . r02
6 n=1
(24)
and higher-order coefficients αˆ n (n = 3, . . . , 6) can be found in Appendix A. It can be easily checked that the gaugeinvariant deflection angle (22) can return to the one for the Schwarzschild black hole in GR [52] when the coupling vanishes.
3 Image positions, magnifications and time delay After the deflection angle has been obtained, we can determine the image positions, magnifications, and time delay of the weak deflection lensing for photons coupled to the Weyl tensor in the Schwarzschild black hole. Denoting dL , dS and dLS as angular diameter distances between the observer, lens and the source, we adopt the general lens equation as [49,50] tan B ≡ tan ϑ − D[tan ϑ + tan(αˆ − ϑ)],
(25)
where B is the angular position of the source, ϑ = arcsin(b/dL ) is the angular position of the image and D = dLS /dS . Following the convention of Refs. [52–54], angles of image positions are set to be positive so that the angular position of the source B is positive if the image is on the same side of the lens as the source while B is negative if the image is on the opposite side. We also define scaled variables [52–54] B , ϑE
ϑ , ϑE
τ , τE
ϑ• , ϑE
(20)
where ϑ• = arctan(m • /dL ) is the angular gravitational radius at distance dL , τ is the time delay between images, the angular Einstein ring radius is
4m • dLS ϑE = (27) dL dS
(21)
αˆ n q n + O(q 7 ),
(23)
β=
For testing photons coupled to the Weyl tensor in the Solar System, the leading term of α(h) ˆ ≈ a1 h was calculated previously [57]; and our result is in agreement with that. Other higher-order coefficients an (n = 3, . . . , 6) can be found in Appendix A. The deflection angle (17) can go back to the one for the Schwarzschild black hole in GR [52] when α¯ = 0. However, such an expression depends on coordinate of r0 and should be transformed into an gauge-invariant form. With the help of Eq. (11), we can replace the distance of closet approach r0 with the impact parameter b and obtain α(b) ˆ =
¯ αˆ 1 = 4 + 32s α, 15 135 75 + s α¯ + s α¯ 2 (15s − 2) , αˆ 2 = π 4 2 2
(19)
where the dimensionless parameter α¯ 0 is defined as α¯ 0 ≡
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(22)
θ=
τˆ =
and the time scale is m• τE = 4 . c
ε=
(26)
(28)
Then, it is assumed that solution to the lens equation (25) can be expressed in the form of a series as θ = θ0 + εθ1 + ε2 θ2 + O(ε3 ),
(29)
where θ0 , θ1 and θ2 are respectively the zeroth-order, firstorder and second-order terms for the image position in the weak deflection lensing. With that, the bending angle (22) can be written as
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ε ε2 15 1 π − 4θ1 + s α¯ 135π (1 + 8s α) ¯ + 2 θ0 2 θ0 4 75 ε3 128 − 64θ1 + π s α¯ 2 15s − 2 + 3 2 3 θ0 32 2 4 6144 15 D θ0 + s α¯ − π θ1 − 4θ0 θ2 + 4θ12 + 2 3 5 256 2 4 D θ0 − 135π θ1 + 32θ12 − 32θ0 θ2 + 3 663552 73728 s − 150π θ1 + − s α¯ 2 1125π θ1 − 35 35 524288 3 s α¯ (24s 2 − 6s + 1) + O(ε4 ), + (30) 105
αˆ = 4
and the lens equation (25) can be found as ε ε2 15 π 0 = 4D (βθ0 − θ02 + 1 + 8s α) ¯ + 2D θ0 4 θ0 1 − 4θ1 (θ02 + 1) + s α(135π ¯ − 64θ1 ) 2 75 ε3 15 + π s α¯ 2 (15s − 2) + 3 D 64 − π θ1 2 2 θ0 64 224 2 4 D θ0 − 64Dθ02 − 4θ0 θ2 + 4θ12 + β 3 D 2 θ03 + 3 3 64 2 6 8704 D θ0 + s α¯ − 135π θ1 − 4θ03 θ2 − 3 5 1792 2 4 2 2 D θ0 − 1024Dθ0 − 32θ0 θ2 + 32θ1 + 3 806912 2 2 + s α¯ s − 1125π θ1 − 4096Dθ0 35 32768 3 73728 2 + s α¯ (419s − 96s + 16) + 150π θ1 − 35 105 + O(ε ), 4
(31)
which will be used to work out the observables. 3.1 Image positions By making the coefficients of , 2 and 3 in (31) vanish, we can find out θn (n = 0, 1, 2). The first term gives that β = θ0 −
1 + 8s α¯ , θ0
(32)
which leads to the zeroth-order image position for the weak deflection lensing as 1 (β + η), 2 where η = β 2 + 4 + 32s α. ¯
θ0 =
(33)
(34)
It is worth mentioning that the negative solution of θ0 is neglected due to our convention that the angular position of
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an image is set to be positive. The positive- and negativeparity images can respectively be found by using β > 0 and β < 0 (see next section for details). If s α¯ > 0, it will make θ0 bigger than its corresponding values for the Schwarzschild black hole in the absence of such a coupling to the Weyl tensor; if s α¯ < 0, θ0 will become smaller. Additionally, Eq. (33) itself also imposes a bound on α. ¯ In order to ensure that η and resulting θ0 are real, it demands that 1 β2 s α¯ ≥ − − . 8 32
(35)
For the special case of β = 0 and s = − 1, it implies α¯ ≤ 1/8 so that α 6 × 1016 m2 where b is assumed to be the radius of the Sun by considering the Sun as the lens. Such a specific bound is consistent with but looser than the one obtained by the Solar System test on the deflection of light due to the Sun [57]. The coefficient of 2 in (31) does not depend on β explicitly so that, after substituting (33), we can obtain the firstorder correction to the image position as θ1 =
15 1 + 18s α¯ + 10s(15s − 2)α¯ 2 π . 16 θ02 + 1 + 8s α¯
(36)
It can be easily checked that when α¯ = 0, θ1 will return to its familiar value for the Schwarzschild black hole in GR [52]. Vanishing the coefficient of 3 in (31) yields the secondorder term as θ2 = p
5
pn α¯ n ,
(37)
n=0
where the factor p and first two coefficients p0 and p1 are 1 , θ0 (θ02 + 1 + 8s α) ¯ 3 8 2 8 64 D − 16 θ06 p0 = D θ0 + D 3 3 88 2 D − 32D + 16 θ04 + 3 225 2 2 16 2 D − 16D + 32 − π + θ0 3 128 225 2 16 π , − D 2 + 16 − 3 256 64 2 8 1024 D θ0 + D D − 256 θ06 p1 = s 3 3 2176 4 2 θ0 + 704D − 768D + 5 512 2 5632 2025 2 2 D − 512D + − π θ0 + 3 5 32 640 2 3456 2475 2 D + − π . − 3 5 64 p=
(38)
(39)
(40)
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The higher-order coefficients pn (n = 2, . . . , 5) can be found in Appendix B. If the coupling to the Weyl tensor is absent, θ2 will have its value as the same as the one for the Schwarzschild black hole [52].
At angular position ϑ, the signed magnification μ is [58]
sin B(ϑ) dB(ϑ) μ(ϑ) = sin ϑ dϑ
−1
.
(41)
With scaled variables (26), we can have a series of μ expanded in terms of as μ = μ0 + εμ1 + ε μ2 + O(ε ), 2
3
(42)
θ04 − (1 + 8s α) ¯ 2
μ2 = m
,
15 3 1 + 18s α¯ + 10s(15s − 2)α¯ 2 , πθ 16 0 (θ02 + 1 + 8s α) ¯ 3 6
mn α¯ n .
(43) (44) (45)
n=0
In the expression of μ2 , the factor m and the first two coefficients m0 and m1 are m=
dS cos B
(49)
where Rsrc and Robs are the radial coordinates of the source and observer and they have the relations as [52] 2 + dS2 tan2 B)1/2 , Rsrc = (dLS
(50)
Robs = dL .
(51)
The function T (R) has the form as R dt dr T (R) = r0 dr
(52)
T (R) = T0 + r0
3
Tn h n + O(h 4 ),
(53)
n=1
θ04
μ1 = −
The time delay is the difference between the light travel time with and without the lens and it can be expressed as
and it can be integrated and expanded as
where the zeroth-order, first-order and second-order terms are μ0 =
3.3 Time delay
cτ = T (Rsrc ) + T (Robs ) −
3.2 Magnifications
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θ02
, (46) (θ02 + 1 + 8s α) ¯ 5 (θ02 − 1 − 8s α) ¯ 8 m0 = D 2 θ08 + (48D 2 − 32D − 32)θ06 3 272 2 675 2 D − 64D + π − 64 θ04 + 3 128 8 + (48D 2 − 32D − 32)θ02 + D 2 , (47) 3 128 2 8 4352 6 θ0 m1 = s D θ0 + 1152D 2 − 768D − 3 5 8704 2 6075 2 11264 4 D − 2048D + π − θ0 + 3 32 5 6912 2 θ0 + 128D 2 , (48) + 1920D 2 − 1280D − 5 and the higher-order coefficients mn (n = 2, . . . 6) can be found in Appendix B. It can be checked that μ will be divergent if s α¯ equals to either −(θ02 + 1)/8 or (θ02 − 1)/8. When α¯ = 0, μn (n = 0, 1, 2) can also return to their familiar values for the Schwarzschild black hole [52].
where the first two terms are T0 = R 2 − r02 ,
1 + 1 − ξ2 1 − ξ2 T1 = + 2 ln 1+ξ ξ
2 1−ξ (ξ + 2) + 12s α¯ 0 1+ξ
(54)
(55)
with ξ=
r0 R
(56)
and higher-order terms T3 and T4 can be found in Appendix B. It is obvious that T0 is not affected by the coupling to the Weyl tensor, but T1 has the Shapiro delay term with an additional correction proportional to s α¯ 0 which is consistent with the result obtained for the Solar System test [57]. When α¯ 0 = 0, Tn (n = 1, 2, 3) can return to their values for the Schwarzschild black hole [52]. After replacing r0 with b by using Eq. (11) and substituting (50) and (51), we can have the scaled time delay [see Eq. (26)] in a series as τˆ = τˆ0 + ετˆ1 + O(ε2 ),
(57)
where
dL θ02 ϑE2 1 2 2 + 24s α¯ , 1 + β − θ0 − ln τˆ0 = 2 4dLS s α¯ 15π 1 2 + τˆ1 = ¯ [7 + 3θ02 + α(15θ 0s 8θ0 2 θ02 + 1 + 8s α¯ 2 2 + 111s − 2θ0 − 2) + s α¯ (720s − 96)] .
(58) (59)
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Although it is also possible to obtain the O(ε2 ) term for τˆ , that is less vital than the O(ε2 ) corrections to θ and μ. If s α¯ > 0, it will make τˆ0 bigger than its corresponding value in the absence of the coupling to the Weyl tensor; if s α¯ < 0, τˆ0 will become smaller. When such a coupling vanishes, τˆ0 and τˆ1 can go back to their values in GR [52].
where the factors P± and the first two coefficients of Pn and Pn (n = 0, 1) are P± = P0 =
4 Relations between lensing observables Considering photons coupled to the Weyl tensor in the Schwarzschild black hole, we can find some relations between lensing observables given in the previous section. P1 =
4.1 Position relations With Eq. (33), we can respectively obtain the positive- and negative-parity images at the leading order by specifying β > 0 and β < 0 as 1 (η ± |β|) 2 which also leads to
θ0± =
(60) P0 =
θ0+ − θ0− = |β|,
(61)
and θ0+ θ0− = 1 + 8s α. ¯
(62)
It is clear that the value of θ0+ − θ0− is not affected by the coupling to the Weyl tensor and θ0+ θ0− is dependent on the coupling only. When such a coupling vanishes, θ0± and θ0+ θ0− will have their values in GR [52]. If s α¯ > 0, then it will make θ0± and θ0+ θ0− bigger than their corresponding values in the absence of such a coupling to the Weyl tensor; and vice versa. According to Eqs. (36) and (60), we can have the firstorder corrections to the image positions as θ1± =
15 1 + 18s α¯ + 10s α¯ 2 (15s − 2) π . 8 η(η ± |β|)
(63)
They generate two relations that one is − 2) 15 1 + 18s α¯ π , (64) = 16 1 + 8s α¯ which is independent on the angular position of source; and the other is
θ1+
+ 10s α¯ 2 (15s
+ θ1−
θ1+ − θ1−
1 + 18s α¯ + 10s α¯ 2 (15s − 2) 15 . = − π |β| 16 η(1 + 8s α) ¯
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n=0
and higher-order coefficients Pn (n = 2, . . . , 5) and Pn (n = 2, 3, 4) can be found in Appendix C. They yield a relation as θ2+ − θ2− = s
(65)
4
sn α¯ n ,
(72)
n=0
where the factor s and the first two coefficients s1 and s2 are |β| , (1 + 8s α) ¯ 3 225 2 s0 = 8D 2 + π − 16, 256 2025 2 2816 π − . s1 = s 256D 2 + 64 5 s=
which is source-dependent. Based (37), we can obtain the second-order corrections to the image positions as 5 4 ± n n (66) Pn α¯ ± η|β| Pn α¯ , θ2 = P± n=0
P1 =
1 , (67) η3 (η ± |β|)4 1024 64 2 8 D β +D D − 128 β 6 3 3 5056 2 + D − 1024D + 128 β 4 3 8576 2 225 2 2 D − 2304D + 768 − π β + 3 16 2560 2 675 2 D − 1024D + 1024 − π , + (68) 3 16 512 2 8 16384 D β +D D − 2048 β 6 s 3 3 17408 4 + 40448D 2 − 24576D + β 5 135168 2025 2 2 274432 2 D − 73728D + − π β + 3 5 4 102400 2 221184 7425 2 D − 40960D + − π , + (69) 3 5 4 64 2 6 896 D β +D D − 128 β 4 3 3 3392 2 D − 768D + 128 β 2 + 3 3328 2 225 2 D − 1024D + 512 − π , + (70) 16 3 512 2 6 14336 s D β +D D − 2048 β 4 3 3 17408 2 β + 27136D 2 − 18432D + 5 106496 2 90112 2025 2 + (71) D − 32768D + − π , 3 5 4
(73) (74) (75)
The higher-order coefficients sn (n = 2, 3, 4) can be found in Appendix C.
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4.2 Magnification relations
4.3 Total magnification and centroid
Using Eqs. (43)–(45) for magnifications and Eq. (60) for image positions, we can have
If the two images can not be separated, the observables are the total magnification and magnification-weighted centroid position. The relations about magnifications (82)–(84) leads to the total magnification as
μ± 0
1 |β|(β 2 + 4) ± (β 2 + 2)η = 2|β|η2 + 16s α(2|β| ¯ ± η) ,
μtot = |μ+ | + |μ− | (76)
− μ+ 1 = μ1
=−
15 1 + 18s α¯ + 10s α¯ 2 (15s − 2) π 16 η3
(77)
and 4
Mn α¯ n ,
(78)
n=0
where the factor M and the first two coefficients M0 and M1 are 1 , (79) |β|η5 8 176 2 M0 = D 2 β 4 + D − 32D − 32 β 2 3 3 675 2 − 128D + 192D 2 + (80) π − 128, 128 128 2 4 4352 2 D β + 1408D 2 − 768D − β M1 = s 3 5 6075 2 22528 + 6144D 2 − 4096D + π − . (81) 32 5 M=
The higher-order coefficients Mn (n = 2, 3, 4) can be found in Appendix C. They can give three simple magnification relations as + μ− 0 − μ− 1 + μ− 2
= 1,
(82)
= 0,
(83)
= 0.
(84)
Again, the sign of the magnification denotes the parity of an image so that the absolute value of μ indicate its brightness. At the zeroth-order, the difference between the fluxes of − images, |μ+ 0 | − |μ0 |, equals to the flux of the source without lensing. The relation (83) emerges because both images have the same μ1 . These relations are immune to the coupling to the Weyl tensor. By combining Eqs. (33), (36), (43) and (44), we can verify that + − − + + − − μ+ 0 θ1 + μ0 θ1 + μ1 θ0 + μ1 θ0 = 0.
(85)
(86)
− The exact cancellation between μ+ 1 and μ1 [see Eq. (83)] guarantee that μtot does not have the O( ) term. The magnification-weighted centroid position is defined by [52]
cent =
μ± 2 = ±M
μ+ 0 μ+ 1 μ+ 2
2 + 3 = (2μ+ 0 − 1) + 2 μ2 + O( ).
θ + μ+ + θ − μ− θ + |μ+ | − θ − |μ− | = . + − |μ | + |μ | μ+ − μ−
(87)
With the results obtained in the previous parts of this section, it can be expanded in the series of as cent = 0 + ε1 + ε2 2 + O(ε3 ),
(88)
where the zeroth-order, first-order and second-order terms are 0 = |β| 1 = 0, 2 = S
β 2 + 3 + 24s α¯ , β 2 + 2 + 16s α¯
(89) (90)
4
Sn α¯ n .
(91)
n=0
In the expression of 2 , the factor S and the first two coefficients S0 and S1 are |β| , + 2 + 16s α) ¯ 2 8 2 6 104 272 2 4 D − 16 Dβ + D S0 = D β + 3 3 3 64 2 675 2 D − π + 128, − 64D + 32 β 2 − 3 128 64 2 6 1664 D β + D − 256 Dβ 4 S1 = s 3 3 4352 2 2 β + 2176D − 1536D + 5 2048 2 6075 2 22528 D − π + , − 3 32 5 S=
η2 (β 2
(92)
(93)
(94)
and the higher-order coefficients Sn (n = 2, 3, 4) can be found in Appendix C. 4.4 Differential time delay The differential delay between the positive- and negativeparity images is Δτˆ = τˆ− − τˆ+ ,
(95)
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ΔF ≡ F + − F −
and it has a series form as Δτˆ = Δτˆ0 + εΔτˆ1 + O(ε2 ), where the zeroth-order and first-order terms are η + |β| 1 Δτˆ0 = η|β| + ln , 2 η − |β| 1 15 |β| Δτˆ1 = π + 11s α¯ 2 8 (1 + 8s α) ¯ 2 + s α¯ 2 (111s − 2) + 48s 2 α¯ 3 (15s − 2) .
(96)
(97)
(98)
Since both of them are affected by the coupling to the Weyl tensor, it is theoretically possible to test it by observing the differential time delay between two images (see next section for discussion).
Here, we define that ¯ E2 . E = B 2 + 4ϑE2 + 32s αϑ
5 Observational effects By using the lensing relations found in the previous section, we can obtain the practical observables for the photons coupling to the Weyl tensor in the Schwarzschild black hole. After that, like Ref. [53], we will consider and discuss several astrophysical scenarios. 5.1 Practical observables
+
+
(99)
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(100)
−
Ftot ≡ F + F B 2 + 2ϑE2 + 16s αϑ ¯ E2 + O(ε2 ), = Fsrc |B|E
δΔτ ≡ Δτ − Δτ (α¯ = 0),
(101)
(106) (107) (108) (109) (110) (111)
where the differences between fluxes are converted into magnitudes. Keeping the leading contributions, we can have their dominant terms as
15 |β|(5β 2 + 12) επ ϑE s α¯ + O(ε2 , α¯ 2 ), 8 (β 2 + 4)3/2 80 s α¯ + O(ε2 , α¯ 2 ), δrtot = 2 (ln 10)(β + 2)(β 2 + 4) 225 3β 2 + 4 δΔr = − επ 2 s α¯ + O(ε2 , α¯ 2 ), 8 ln 10 (β + 4)5/2 8|β|3 ϑE s α¯ + O(ε2 , α¯ 2 ), δScent = 2 (β + 2)2 45 dL dS επ |β|ϑE2 s α¯ + O(ε2 , α¯ 2 ). δΔτ = 8 cdLS δΔP = −
ΔP ≡ ϑ − ϑ 15 ϑE 1 + 18s α¯ + 10s α¯ 2 (15s − 2) = |B| 1 − επ 16 E 1 + 8s α¯ + O(ε2 ),
δΔP ≡ ΔP − ΔP(α¯ = 0), Ftot δrtot ≡ 2.5 log10 , Ftot (α¯ = 0) ΔF , δΔr ≡ 2.5 log10 ΔF(α¯ = 0) δScent ≡ Fcent − Fcent (α¯ = 0),
16ϑE s α¯ + O(ε, α¯ 2 ), δ Ptot = β2 + 4
−
(105)
These combinations of observables represent our results about weak deflection lensing for photons coupling to the Weyl tensor in the Schwarzschild black hole. We will investigate the effects of the coupling on these practical observables so that we define following indicators to demonstrate its contributions: δ Ptot ≡ Ptot − Ptot (α¯ = 0),
In order to proceed the investigation, we focus on the zerothorder and first-order lensing effects. The former ones are the observables of the weak deflection limits, and the latter ones might be able to be measured in the near future. To fulfill this purpose, we need convert the scaled variables (β, θ, μ, τˆ ) to practical observables (B, ϑ, F, τ ). Observables of lensing usually are the positions, fluxes and time delays of the images. The fluxes are connected to the magnifications through the flux of the source, i.e., Fi = |μi |Fsrc . Following the discussion in Ref. [53], we also construct some possibly measurable combinations of observables which are Ptot ≡ ϑ + + ϑ − 1 + 18s α¯ + 10s α¯ 2 (15s − 2) 15 = E + επ ϑE 16 1 + 8s α¯ + O(ε2 ),
15 ϑE3 επ 3 [1 + 18s α¯ + 10s α¯ 2 (15s − 2)] 8 E (102) + O(ε2 ), + − − − ϑ F −ϑ F Scent ≡ Ftot 2 ¯ E2 B + 3ϑE2 + 24s αϑ = |B| 2 + O(ε2 ), (103) B + 2ϑE2 + 16s αϑ ¯ E2 E + |B| dL dS 1 |B|E + ϑE2 ln Δτ = cdLS 2 E − |B| 15 1 ϑE |B| +ε π + 11s α¯ + s α¯ 2 (111s − 2) 8 (1 + 8s α) ¯ 2 2 (104) + 48s 2 α¯ 3 (15s − 2) + O(ε2 ) . = Fsrc − Fsrc
(112) (113) (114) (115) (116) (117)
Eur. Phys. J. C (2018) 78:191
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They imply that the polarization s and the strength α for photons coupling to the Weyl tensor cannot be simultaneously determined at least based on its leading observational effects. Before investigating some observational examples numerically, we need to specify the domain of the coupling constant α. In order to ensure that a photon can continuously propagate outside the event horizon of a Schwarzschild black hole, α has to satisfy two theoretical bounds [46,47] that th for s = +1, α < α+1
(118)
for s = − 1,
(119)
α>
th α−1
with
M• 2 2 m , M 2 1 2 6 M• = − m • = −1.1 × 10 m2 . 2 M
th = m 2• = 2.2 × 106 α+1
(120)
th α−1
(121)
where M• and M are the mass of the hole and the Sun. Based on experiments in the Solar System, an observational bound was found as [57] obs = 4 × 1011 m2 . |sα| α±1
(122)
Thus, if it is assumed that the strength of α is irrelevant to obs will be the mass of the lens, the observational bound α±1 th much tighter than the theoretical ones α±1 for a supermassive th will be much more system with M• 106 M and α±1 stringent for a stellar system with M• 10 M . Both of these two kinds of bounds will be adopted in the following parts. 5.2 The supermassive black hole in the Galactic Center By monitoring stellar orbits in the Galactic Center, the mass and distance to the supermassive black hole Sgr A* was determined as M• = 4.28 × 106 M and dL = 8.32 kpc [59]. Its gravitational radius is m • = 6.32 × 109 m = 2.05 × 10−7 pc, whose angular radius is 5.08 × 10−6 arcsecond (as) and whose time scale is τE = 84.3 s. We consider a source orbiting the Sgr A* with a distance dLS dL so that dS ≈ dL . We also define a scaled distance ∗ = d /(1 pc). From the perspective of the observer, dLS LS if the source can be close enough to Sgr A* in the sky, it can be strongly lensed. The angular Einstein radius is ϑE = ∗ )1/2 as and the perturbation parameter is = 0.0224 (dLS ∗ )−1/2 . −4 2.26 × 10 (dLS Therefore, based on Eq. (99), we can make a rough estimation, in which the first-order correction is dropped since it is smaller by 4 orders of magnitude, and find that the angular separation of the two lensed images is larger than √ ∗ 1/2 ) 1 + 32s α¯ as. (123) Ptot,min = 0.0448 (dLS
191
When the weak deflection lensing is considered that b m • , ∗ ∼ 10−3 (and resulting ε ∼ Ptot,min even for a source with dLS −3 7.16 × 10 ) is still larger than the current resolution as low as 50 microarcsecond (μas) achieved by phase referencing optical/infrared interferometry [60]. It means that Ptot , ΔP, Ftot , ΔF and Δτ can be obtained according to the positions, flux and timing of two separated images. As a case study, we consider a source at dLS = 10−3 pc with its β ranging from 10−3 to 10. We take a wider domain of α belonging to [105 , 1020 ] m2 which can cover obs and |α th | for Sgr A*. It can be checked that α ¯ both α±1 ±1 1.2 × 10−4 for both s = 1 and − 1 by using the relation of sin ϑ = b/dL and the series solutions to the image positions. Our interest is the contributions on the lensing observables due to α. ¯ Estimations [see Eqs. (112)–(117)] suggest when α¯ is sufficiently small that is satisfied for this case, s solely changes the sign of the leading effects of α. ¯ Therefore, we focus on the leading contributions for s = 1, which can nicely approximate the results for s = −1 after changing their signs. Figure 1 shows color-indexed δ Ptot , δΔP, δrtot , δΔrtot and δΔτ against α and β in logarithmic scales. The coupling constant α can make the total separation of two images larger th , than its value in GR, i.e., δ Ptot > 0. When α is around α+1 δ Ptot can reach the level of 0.1 μas, which is still beyond the obs , current ability. δ Ptot decreases when α drops; and if α ∼ α±1 −9 δ Ptot will be down to ∼ 10 μas far beyond the capability today. The existence of α can cause the angular difference between two images become smaller than the one in GR, i.e., δΔP < 0. The sub-figure of −δΔP has a similar pattern with th . The its most significant contribution at 10−3 μas at α ∼ α+1 effects of α on δ Ptot and δΔP for Sgr A* are too small to detect in the near future. The total brightness of two images can be enhanced under the coupling to the Weyl tensor with respect to the one in GR and its difference δrtot can reach the level of ∼ 5 × 10−4 mag, corresponding to relative flux of about 500 parts per million. Although it is within the current photometric accuracy of a dedicated space telescope, such as the Kepler mission for searching transiting exoplanets [61], the emission of Sgr A* is constantly changing and its variable flux might easily overwhelm this variation of the total flux of two images due to the coupling. But it might still be a promising way to constrain α by measuring δrtot in the future if this noise can be well understood and separated. Additionally, the coupling can reduce the brightness difference between two images, but such as reduction is less than ∼ 10−6 mag below the current threshold. Finally, effects of α on the differential time delay th , δΔτ between two images are represented. When α ∼ α+1 −4 can reach the level of ∼ 10 s which is practically inaccessible because the exposure time for astronomical imaging is usually much longer than it.
123
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Eur. Phys. J. C (2018) 78:191 th
10
10
−2
10
−9
β
−1
10
15
10
10
−2
10
10−19
10−17 10
10−11
−1
10
10−3 5 10
20
10
10
10
20
10
2
α (m )
α (m )
obs
αth +1
α±1
10
15
10
2
obs
αth +1
α±1
10 −5
−7
10
10
1
1
10−11
−15
10 −2
β
−1
10
δrtot (mag)
β
10−10 −1
10
−15
10
10−19
−2
10
−δΔr (mag)
10−3 5 10
10
−15
−13
10
10−3
−7
1
−5
10
10
th
α+1
α±1
10
−1
1
β
obs
α+1
−δΔP (μas)
obs
α±1
10
δPtot (μas)
191
10 −20
10 10−3 5 10
10
15
10
10
10−3 5 10
20
10
2
10
15
10
10
20
10
2
α (m )
α (m )
obs
th
α+1
α±1
10
10−23
10−5 1 −10
δΔτ (s)
β
10 10−1
−15
10 −2
10
10−20
−3
10
5
10
10
15
10
10
20
10
α (m2)
Fig. 1 Estimated contributions in observables of the weak deflection gravitational lensing caused by photons coupling to the Weyl tensor in the Schwarzschild black hole. We consider Sgr A* as the lens with the mass and distance as M• = 4.28 × 106 M and dL = 8.32 kpc [59] and assume a source at a distance of 10−3 pc from Sgr A*. From top to bottom panels and from left to right, color-indexed δ Ptot , δΔP, δrtot ,
123
δΔrtot and δΔτ against α and β are respectively presented in logarithmic scales. In each sub-figure, the dashed line denotes the observational bound on α and the dash-dot one shows the theoretical bound. Here, the polarization s = 1 is taken; the contributions of s = − 1 can be sufficiently approximated by changing the signs of the ones shown in each sub-figures
Galactic Microlensing
obs
α±1
10
−12
10
2
−22
10
10
αth +1
Astrometric Microlensing
191 obs
α±1
10
10
10−27 1
1 −32
10 10−1 6 10
8
10
10
10
10
−6
10
−9
10
−12
10
−15
2
10−17
10
β
3
β
10
αth +1
δrtot (mag)
3
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12
10
2
α (m )
δScent (μas)
Eur. Phys. J. C (2018) 78:191
10−18 10−1
10
6
8
10
10
10
12
10
2
α (m )
Fig. 2 Estimated contributions in microlensing caused by photons coupling to the Weyl tensor in the Schwarzschild black hole. Left: δrtot is shown for a Galactic lensing with M• = M and 2dL = dS = 8 kpc. Right: δScent is represented for an astrometric lensing with M• = 0.676
M , dL = 5.55 pc and dS = 2 kpc. In each panels, the dashed line denotes the observational bound on α and the dash-dot one shows the theoretical bound, where the polarization s = 1 is taken
In a summary, based on the current limit of the technology and the specific circumstance of the Sgr A*, the impact of the photons coupling to the Weyl tensor in the Schwarzschild black hole is unable to be detected in the observables of weak deflection lensing, while measuring on the variation of the total flux of two images might be a promising way for testing such a coupling in the future.
1.0 × 10−9 (M•∗ /dL∗ ) as, ϑE ∼ 2.9 × 10−2 (M•∗ /dL∗ )1/2 as and ε ∼ 3.5 × 10−8 (M•∗ /dL∗ )1/2 . Astrometric lensing by a nearby star has an Einstein ring radius almost 30 times larger then the one of Galactic lensing. The centroid position Scent is the observables, in which δScent indicates the contributions of α. The left panel of Fig. 2 shows color-indexed δrtot for a specific case of Galactic lensing, where we assume that M• = M and 2dL = dS = 8 kpc. The coupling constant α is set belonging to the domain [106 , 1012 ] m2 , covering both obs (dashed line) and α th (dash-dot line) with s = 1 for α±1 +1 the lens. The right panel of Fig. 2 represents color-indexed δScent for a case of astrometric lensing with M• = 0.676 M , dL = 5.55 pc and dS = 2 kpc based on the microlensing event caused by a near white dwarf [73]. We take α ∈ [105.5 , 1012 ] with s = 1 in order to contain the observational and theoretical bounds. However, the contributions of α in these two microlensing δrtot and δScent are far beyond the current observational limits. The contributions from the other polarization for the coupling, i.e. s = −1, can be sufficiently approximated by changing the signs of the ones shown in this figure; and they are too small to detect as well.
5.3 Galactic and astrometric microlensing In a scenario of microlensing, a foreground gravitational object lenses a more distant background star. Plenty of projects [62–67] on Galactic microlensing measure change of the total fluxes of stars in the bulge with time, while astrometric microlensing on centroid shifts of a remote source by a nearby lens is also discussed [68–70] and practiced [71–73]. For Galactic lensing, masses and distances of the lens and source are respectively scaled by the solar mass and by 8 kpc as M•∗ = M• /M and dS∗ = dS /(8 kpc). We suppose a situation that the lens is located at rough midpoint between the observer and the source, i.e. dL ∼ dLS ∼ dS /2. We can, therefore, have that ϑ• ∼ 2.5 × 10−12 (M•∗ /dS∗ ) as, ϑE ∼ 10−3 (M•∗ /dS∗ )1/2 as and ε ∼ 2.4 × 10−9 (M•∗ /dS∗ )1/2 . We concentrate on the contribution of the coupling to the Weyl tensor on Ftot which is indicated by δrtot . For astrometric lensing, masses and distances of the lens and source are respectively scaled by the solar mass and by 2 kpc as M•∗ = M• /M and dS∗ = dS /(2 kpc). We assume that a nearby lens is located at 10 pc from the observer, i.e. dL∗ = dL /(10 pc), so that dS ≈ dLS . We can have that ϑ• ∼
6 Conclusions and discussion In order to provide one of the missing puzzles of the whole physical picture of photons coupled to Weyl tensor in a Schwarzschild black hole, we investigate its weak deflection lensing, as an extension of the previous works on its
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Eur. Phys. J. C (2018) 78:191
strong deflection lensing [46,47]. Under a unified framework valid for both two polarization directions of the coupling, we obtain its bending angle, image position, magnification, centroid and time delay in the coordinate-invariant forms upon to the second order in the perturbation parameter of the ratio of the angular gravitational radius to angular Einstein radius of the lens. The contributions of such a coupling on some astrophysical scenarios are also studied. We find that, in the weak deflection lensing on a star orbiting the Sgr A*, Galactic microlensing on a star in the bulge and astrometric microlensing by a nearby lens, these effects caused by coupling are beyond the current limits of technology. However, measuring the variation of the total flux of two images caused by the Sgr A* might be a promising way for testing such a coupling in the future. In this work, as an astrophysically important ingredient, the spin of a black hole has not been taken into account. A self-consistent treatment should move from the present model we considered to photons coupled to the Weyl tensor in a Kerr black hole, in which the strong deflection lensing was studied [48]. As expected, the spin can make the light propagation more complicated and effectively causes the caustic shifted and distorted [74]. We will leave the detailed investigation on its weak deflection lensing for future works. Acknowledgements This work is funded by the National Natural Science Foundation of China (Grant No. 11573015). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3 .
Appendix A: Higher-order coefficients of bn , an and αˆ n In Eq. (11) for relation between r0 and b, the higher-order coefficients bn (n = 3, . . . , 6) are b3 = 4 + 144s α¯ + 288s α¯ 2 (9s − 1) + 768s α¯ 3 (24s 2 − 6s + 1), 105 b4 = + 630s α¯ + 756s α¯ 2 (21s − 2) 8 + 864s α¯ 3 (243s 2 − 54s + 8) + 96s α¯ 4 (11979s 3 − 4356s 2 + 1188s − 80),
(A.1)
(A.2)
b5 = 48 + 2880s α¯ + 7680s α¯ (12s − 1) 2
+ 23040s α¯ 3 (75s 2 − 15s + 2) + 18432s α¯ 4 (972s 3 − 324s 2 + 81s − 5) 6144 5 + s α¯ (64827s 4 − 30870s 3 + 11025s 2 5 − 1470s + 88),
123
(A.3)
b6 =
3003 27027 + s α¯ + 19305s α¯ 2 (27s − 2) 16 2 + 34320s α¯ 3 (363s 2 − 66s + 8) + 9360s α¯ 4 (19773s 3 − 6084s 2 + 1404s − 80) + 1152s α¯ 5 (1366875s 4 − 607500s 3 + 202500s 2 − 25200s + 1408) 768 6 s α¯ (38336139s 5 − 22550670s 4 + 5 + 9727740s 3 − 1907400s 2 + 217056s − 8960). (A.4)
In Eq. (17) for the deflection angle, the higher-order coefficients an (n = 3, . . . , 6) are 122 15 6624 − π − s α¯ 0 360π − 3 2 5 67008 794592 s − 450π + − s α¯ 02 6615π − 35 35 4920064 − s α¯ 03 40500π − s2 35 887296 524288 + (A.5) − 5400π s − , 35 105 20925 3465 a4 = π − 130 + s α¯ 0 π − 7640 64 8 2193345 1042272 72768 π− s+ + s α¯ 02 32 5 5 7112853 81405 95704704 3 − π + s α¯ 0 π− s2 16 8 35 1680384 15065856 582957 − π s + 18711π − + 35 4 35 137971917 461693952 + s α¯ 04 π− s3 32 35 107440128 8560647 2091231 2 + − π s + π 35 8 8 38115 21454848 s− π , (A.6) − 35 2 31185 7783 3465 a5 = − π − s α¯ 0 π − 50724 10 16 2 2230245 354592 2 π − 1775920 s + − s α¯ 0 4 3 72525 1927205376 2 44293365 − π − s α¯ 03 π− s 2 4 55 299910144 3407805 + − π s + 192870π 55 2 899497791 35034112 − s α¯ 04 π − 55 8 433882040832 1777650959232 3 s + − 5005 5005 11438253 54360801 − π s2 + π 2 2 a3 =
Eur. Phys. J. C (2018) 78:191
93408532992 121436160 s+ − 228690π 5005 143 891827253 35310191717376 5 s4 − s α¯ 0 π− 2 25025 2292283060224 − 142427646π s 3 + 5005 140029464576 2 + 41789358π − s 1001 66077966336 − 3089880π s + 5005 939524096 − (A.7) , 975 25394985 310695 21397 a6 = π− + s α¯ 0 π − 306390 256 6 256 1107098685 2 π − 13502016 s + s α¯ 0 256 7260774525 33315975 + 804768 − π + s α¯ 03 π 128 64 530614935 π s − 355394880 s 2 + 51735936 − 32 14503095 π − 5601792 + 8 224621915265 60538960512 3 π− s + s α¯ 04 128 11 71747670528 13344763365 + − π s2 55 32 14582128128 2726718795 π− s + 32 55 125491200 30018825 + − π 11 8 3713261122953 4140725045760 4 π− s + s α¯ 05 256 91 14750750029824 301384679553 − π s3 + 1001 64 86997944619 4236163098624 2 + π− s 64 1001 368270204928 1933875405 + − π s 1001 16 193527808 10830105 π− + 2 13 12488303405007 765525411606528 5 π− s + s α¯ 06 256 5005 307851253530624 2521647706203 + − π s4 5005 128 111457297907712 3 460902199923 π− s + 64 5005 14186020405248 30733018821 + − π s2 5005 32
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191
725498163 99346808832 π− s 8 455 − 4339335π .
−
+
(A.8)
In Eq. (22) for the gauge-invariant form of the deflection angle, the higher-order coefficients αˆ n (n = 3, . . . , 6) are 73728 128 6144 + s α¯ + s(9s − 1)α¯ 2 3 5 35 524288 + (A.9) s(24s 2 − 6s + 1)α¯ 3 , 105 3465 17325 72765 2 αˆ 4 = π + s α¯ + s α¯ (21s − 2) 64 8 32 7623 4 18711 3 s α¯ (243s 2 − 54s + 8) + s α¯ (11979s 3 + 8 32 αˆ 3 =
− 4356s 2 + 1188s − 80) ,
(A.10)
262144 2 3584 + 36864s α¯ + s α¯ (12s − 1) 5 3 2621440 3 + s α¯ (75s 2 − 15s + 2) 11 25165824 4 + s α¯ (972s 3 − 324s 2 + 81s − 5) 143 117440512 5 + s α¯ (64827s 4 − 30870s 3 + 11025s 2 10725 − 1470s + 88), (A.11) 558242685 255255 16081065 + s α¯ + s α¯ 2 s αˆ 6 = π 256 256 256 20675655 3057699645 2 277972695 − + s α¯ 3 s − s 128 64 32 8423415 84359626065 3 + + s α¯ 4 s 8 128 21332025 6489202005 2 1497508155 s + s− − 32 32 8 1345763615625 4 149529290625 3 5 s − s + s α¯ 256 64 10830105 49843096875 2 1550674125 s − s+ + 64 16 2 4752952849359 5 1397927308635 4 s − s + s α¯ 6 256 128 301513733235 3 29560169925 2 s − s + 64 32 840963123 s − 4339335 . + (A.12) 8 αˆ 5 =
Appendix B: Higher-order coefficients of pn , mn and Tn In Eq. (37) for the second-order correction to the image position, the higher-order coefficients pn (n = 2, . . . , 5) are
123
191
p2 =
p3 =
p4 =
p5 =
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4096 D − 1024 θ06 s s D 3 201728 4 + 5632D 2 − 6144D + θ0 35 647168 8775 2 2 + 2048D 2 − 6144D + − π θ0 35 8 96256 10240 2 12825 2 − D − π + 7 3 16 1125 2 36864 2 18432 4 θ0 + π − θ0 − 35 16 35 18432 1125 2 − (B.13) + π , 35 32 45056 2 3432448 4 D − 16384D + θ0 s s2 3 105 32768 2 3309568 151875 2 2 + D − 32768D + − π θ0 3 21 16 81920 2 16039936 292275 2 − D + − π 3 105 32 262144 4 10125 2 163840 2 θ0 + π − θ0 +s − 35 8 7 14625 2 557056 + π − 16 35 131072 4 262144 2 131072 + θ0 + θ0 + , (B.14) 105 105 105 65536 2 54919168 s2 s2 D − 65536D + 3 105 1265625 2 2 327680 2 93650944 − π θ0 − D + 32 3 105 3695625 2 84375 2 4194304 2 − π +s π − θ0 64 8 35 2097152 165375 2 5373952 + π − + 16 35 105 5625 2 2 2097152 5625 2 − (B.15) π θ0 + − π , 8 105 16 524288 2 219676672 1265625 2 D + − π s3 s2 − 3 105 8 84375 2 16777216 8388608 π − + +s 2 35 105 5625 2 − (B.16) π . 2
In Eq. (45) for the second-order correction to the magnification, the higher-order coefficients mn (n = 2, . . . , 6) are 512 2 8 m2 = s s D θ0 + 9216D 2 − 6144D 3 403456 6 θ0 + 34816D 2 − 24576D − 35 26325 2 1294336 4 + π − θ0 + 30720D 2 8 35
123
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192512 2 36864 6 − 20480D − θ0 + 2560D 2 + θ 7 35 0 73728 3375 2 4 36864 2 + (B.17) − π θ0 + θ , 35 16 35 0 6864896 6 θ0 m3 = s s 2 24576D 2 − 16384D − 105 557056 2 455625 2 + D − 131072D + π 3 16 6619136 4 θ0 + 245760D 2 − 163840D − 21 524288 4 32079872 2 81920 2 − θ0 + D + sθ02 θ0 105 3 35 327680 30375 2 2 1114112 + − π θ0 + 7 8 35 262144 6 524288 4 262144 2 θ − θ − θ , (B.18) − 105 0 105 0 105 0 1114112 2 3796875 2 m4 = s 2 s 2 D − 262144D + π 3 32 109838336 4 − θ0 + 983040D 2 − 655360D 105 8388608 187301888 2 2 2 θ0 + 163840D + sθ0 − 105 35 16875 2 253125 2 2 10747904 − π θ0 + + π 8 35 8 4194304 4 4194304 2 − (B.19) θ0 − θ0 , 105 105 439353344 2 3 2 2 1572864D − 1048576D − θ0 m5 = s s 105 33554432 2 16777216 2 + 524288D 2 + s θ0 − θ0 , 35 105
(B.20) 2097152 6 2 m6 = s D . 3
(B.21)
In Eq. (53) for the time delay, the higher-order coefficients Tn (n = 2, 3) are 15 ξ 15 1 − ξ2 5 π− arctan ξ +2 − T2 = 4 2 (1 + ξ )2 2 1 − ξ2 ξ + s α¯ 0 45π − 90 arctan 1 − ξ2
1 − ξ2 3 2 (3ξ + 4ξ − 13ξ − 12) +6 (1 + ξ )2 ξ 675 π − 675 arctan + s α¯ 02 s 2 1 − ξ2
2 1−ξ 5 4 3 2 (2ξ + 4ξ + 21ξ + 22ξ − 69ξ − 64) +9 (1 + ξ )2 ξ − 45π + 90 arctan 1 − ξ2
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− 6ξ(2ξ 2 + 7) 1 − ξ 2 (B.22) 15 15 ξ T3 = − π + arctan 4 2 1 − ξ2
1 1 − ξ2 (35ξ 3 + 133ξ 2 + 157ξ + 60) + 2 (1 + ξ )3
1 − ξ2 ξ + s α¯ 0 − 180π + 360 arctan +6 (1 + ξ )3 1 − ξ2 5 4 3 2 × (5ξ + 12ξ + 60ξ + 244ξ + 316ξ + 128) ξ 6615 + s α¯ 02 s − π + 6615 arctan 2 1 − ξ2
9 1 − ξ2 + (24ξ 7 + 62ξ 6 + 354ξ 5 + 535ξ 4 5 (1 + ξ )3 3 2 + 2605ξ + 12239ξ + 16517ξ + 6784) + 225π
6 1 − ξ2 ξ (24ξ 5 + 14ξ 4 − − 450 arctan 5 (1 + ξ ) 1 − ξ2 + 102ξ 3 + 17ξ 2 + 489ξ + 944) ξ 3 2 + s α¯ 0 s − 20250π + 40500 arctan 1 − ξ2
36 1 − ξ 2 + (40ξ 9 + 120ξ 8 + 896ξ 7 + 2158ξ 6 35 (1 + ξ )3 + 6466ξ 5 + 8675ξ 4 + 30105ξ 3 + 131411ξ 2 + 176273ξ + 72256) + s 2700π
ξ 72 1 − ξ 2 − 5400 arctan − (40ξ 7 35 (1 + ξ ) 1 − ξ2 + 40ξ 6 + 272ξ 5 + 202ξ 4 + 946ξ 3 + 351ξ 2
768 1 − ξ 2 + 3887ξ + 7072) + (5ξ 7 + 5ξ 6 35 (1 + ξ ) 5 4 3 2 + 13ξ + 13ξ + 29ξ + 29ξ + 93ξ + 128)
(B.23)
Appendix C: Higher-order coefficients of sn , Pn , Pn , Mn and Sn In Eq. (72) for the relation about the second-order corrections to the image positions, the higher-order coefficients sn (n = 2, 3, 4) are 8775 2 323584 2 π − s2 = s s 3072D + 16 35 18432 1125 2 (C.24) − π , + 35 32
191
151875 2 1654784 π − s3 = s s 2 16384D 2 + 32 21 131072 81920 10125 2 − π − , (C.25) +s 7 16 105 1265625 2 27459584 π − s4 = s 2 s 2 32768D 2 + 64 105 5625 2 2097152 84375 2 − π + π +s 35 16 16 1048576 . (C.26) − 105 In Eq. (66) for the second-order correction to the positions of the positive- and negative-parity images, the higher-order coefficients Pn (n = 2, . . . , 5) and Pn (n = 2, 3, 4) are 65536 D − 8192 β 6 + 323584D 2 P2 = s s D 3 1613824 4 β + 1097728D 2 − 196608D + 35 15532032 − 8775π 2 β 2 − 884736D + 35 1638400 2 6160384 D − 655360D + − 38475π 2 + 3 7 147456 4 1125 2 884736 2 β + π − β − 35 2 35 3375 2 1179648 + π − , (C.27) 2 35 2588672 2 27459584 4 β D − 524288D + P3 = s s 2 3 105 17563648 2 26476544 D − 4718592D + + 3 7 151875 2 2 13107200 2 π β + D − 5242880D − 2 3 2097152 4 1026555904 876825 2 − π +s − β + 105 2 35 3932160 2 β + 43875π 2 + 10125π 2 − 7 1048576 4 2097152 2 35651584 + β + β − 35 105 35 8388608 , (C.28) + 105 35127296 2 439353344 D − 9437184D + P4 = s 2 s 2 3 35 1265625 2 2 52428800 2 − π β + D − 20971520D 4 3 5993660416 11086875 2 − π + s 84375π 2 + 105 4 100663296 2 343932928 2 β − + 496125π − 35 35
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16777216 134217728 − 5625π 2 β 2 + 35 105 (C.29) − 16875π 2 , 83886080 2 D − 33554432D − 7593750π 2 = s3 s2 3 14059307008 + s 2025000π 2 + 105 536870912 1073741824 + − 135000π 2 , − 35 105 (C.30) 57344 D − 8192 β 4 =s s D 3 1613824 2 + 217088D 2 − 147456D + β 35 10354688 − 8775π 2 + 425984D 2 − 393216D + 35 147456 2 1125 2 589824 β + π − , (C.31) − 35 2 35 1736704 2 27459584 2 D − 393216D + β = s s2 3 105 6815744 2 52953088 D − 2097152D + + 3 21 2097152 2 151875 2 π +s − β + 10125π 2 − 2 35 1048576 2 4194304 2621440 + β + , (C.32) − 7 105 105 13631488 2 878706688 = s2 s2 D − 4194304D + 3 105 1265625 2 67108864 π + s 84375π 2 − − 4 35 33554432 (C.33) − 5625π 2 + 105
+
P5
P2
P3
P4
In Eq. (78) for the second-order correction to the magnifications of the positive- and negative-parity images, the higher-order coefficientsMn (n = 2, 3, 4) are 512 2 4 403456 2 2 M2 = s s D β + 11264D − 6144D − β 3 35 26325 2 2588672 π − + 73728D 2 − 49152D + 8 35 36864 2 3375 2 147456 + (C.34) β − π + , 35 16 35 90112 2 6864896 2 M3 = s s 2 D − 16384D − β 3 105 455625 2 13238272 π − + 393216D 2 − 262144D + 16 21 524288 2 655360 30375 2 +s β + − π 35 7 8
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262144 2 1048576 (C.35) β − , 105 105 3796875 2 M4 = s 2 s 2 786432D 2 − 524288D + π 32 16777216 253125 2 219676672 +s − π − 105 35 8 16875 2 8388608 (C.36) π − . + 8 105 −
In Eq. (91) for the second-order correction to the magnification-weighted centroid position, the higher-order coefficients Sn (n = 2, 3, 4) are 6656 2 4 S2 = s s D β − 1024Dβ 4 + 17408D 2 β 2 3 403456 2 2588672 − 12288Dβ 2 + β + − 8192D 2 35 35 26325 2 36864 2 147456 − π − β − 8 35 35 3375 2 + (C.37) π , 16 139264 2 2 6864896 2 D β − 32768Dβ 2 + β S3 = s s 2 3 105 131072 2 455625 2 13238272 − D − π + 3 16 21 524288 2 30375 2 655360 +s − β + π − 35 8 7 262144 2 1048576 + β + , (C.38) 105 105 262144 2 219676672 3796875 2 2 2 S4 = s s − D + − π 3 105 32 253125 2 16777216 8388608 π − + +s 8 35 105 16875 2 − π (C.39) 8
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