Issue 
A&A
Volume 673, May 2023



Article Number  A164  
Number of page(s)  13  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202245426  
Published online  24 May 2023 
Extreme amplification regimes of the Schwarzschild gravitational lens
^{1}
Research Centre for Computational Physics and Data Processing, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic
^{2}
M. R. Štefánik Observatory and Planetarium, Sládkovičova 41, 920 01 Hlohovec, Slovak Republic
^{3}
Institute of Physics of the Czech Academy of Sciences, Na Slovance, 1999/2 Prague, Czech Republic
^{4}
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Břehová 7, Prague, Czech Republic
^{5}
Dipartimento di Fisica, Università La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy
^{6}
INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Roma, Italy
email: luigi.stella@inaf.it
^{7}
Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
^{8}
International Space Science Institute (ISSI), Hallerstrasse 6, 3012 Bern, Switzerland
^{9}
Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Received:
10
November
2022
Accepted:
8
February
2023
We investigated a complete set of relativistic images of a small source located at an arbitrary distance from a Schwarzschild black hole gravitational lens. This paper offers a description of a simple and efficient fully relativistic method for calculating the bolometric intensity amplification. We focused our analysis primarily on sources located at small radii and close angular distance from the caustic line, both behind and in front of the compact lens. We term the corresponding large deflection regime ‘extreme lensing’. We approximated the regime of fullyrelativistic, extreme amplification of point sources by simple analytical formulae valid over a wide range of source distances. Using such approximations, we also derived formulae for the maximum amplification of extended sources close to or intercepted by the caustic line. Simple analytical approximations of the time delay between the brightest consecutive images in extreme amplification regimes are also presented.
Key words: gravitational lensing: strong / black hole physics / methods: analytical
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
The strong gravitational field of black holes, neutron stars, and white dwarfs can lead to an extreme amplification of the intensity of sources in their surroundings. Infinitesimally small point sources located at the caustic curves of lensing maps are projected to critical curves in the observer’s sky and attain a infinite amplification in the approximation of geometric optics. Naturally, the maximum intensity is limited by wave effects and the finite sizes of real sources. Nevertheless, the intensity amplification of images of real sources in the close vicinity of critical curves can attain extremely high values. The description of these effects in the linearised, weak deflection limit is a powerful tool for many astrophysical applications (see Schneider et al. 1992; Perlick 2010 for reviews).
A fully general relativistic (hereafter FGR) formalism is necessary to completely describe the case of a source or an observer located deep in the extreme gravitational field of a black hole including the formation of infinite sequences of relativistic images and Einstein rings (Darwin 1959). A general relativistic treatment of lensing in the Schwarzschild space time was used to investigate optical effects due to supermassive black holes in galactic nuclei (Virbhadra & Ellis 2000; Virbhadra & Keeton 2008). The optical appearance of the stellar sky as seen by an observer located close to a spherically symmetric black hole was also investigated in a series of studies (see e.g. Cunningham 1975; Nemiroff 1993; Bakala et al. 2007; Müller & Weiskopf 2010). Several simplifying approximations of light bending in the strong Schwarzschild field have been developed (Beloborodov 2002; Semerák 2015; De Falco et al. 2016; La Placa et al. 2019; Poutanen 2020). The intensity amplification of point sources near the caustics, as well as the light curves of orbiting stars in the strong field regime of an extreme Kerr black hole were analysed using a numerical approach (Rauch & Blandford 1994; Cunningham & Bardeen 1973). Other works have been devoted to perturbation and numerical analyses of the full caustic structure in the Kerr field (Bozza 2008) and to the higher order images by adopting strong deflection limit techniques (Bozza & Scarpetta 2007). An analytical study of higher order relativistic images in the distant point source and distant observer limit for was carried out by Ohanian (1987) with a Schwarzschild black hole acting as a gravitational lens.
Renewed interest in the study of highly bent rays in the strong gravitational field regime has been inspired in recent years by observations of of the supermassive black hole shadows in M 87 and the Galaxy (Sgr A*) with the Event Horizon Telescope observations (Event Horizon Telescope Collaboration 2019, 2022). Gralla & Lupsasca (2020a) introduced a new, complete, and explicit classification and solution of Kerr photon geodesics based on Legendre elliptic integrals and Jacobi elliptic functions. These authors also developed a general analytic framework to characterise key features of successive images of highly deflected photons giving rise to the photon ring (Gralla & Lupsasca 2020b). The emission of matter in circular equatorial orbit around a Kerr black hole as observed from the celestial sphere was studied in detail, with special emphasis in the case of very high black hole spins (e.g. Dokuchaev & Nazarova 2019; Gates et al. 2020, 2021). Other works investigated the general relativistic effects impacting polarised images of a synchrotronemitting ring orbiting a Schwarzschild black hole (Narayan et al. 2021) and of a source in a circular equatorial orbit around a Kerr black hole (Gelles et al. 2021).
The emphasis of the present paper is on the extreme amplification that occurs when (small) sources are located in the strong gravitational field regime and close to the caustic line of a Schwarzschild lens. We present an FGR analysis of the complete set of rays and relativistic images of all orders for static and orbiting sources at arbitrary distance from a Schwarzschild black hole and seen by an observer at infinity. The paper is structured as follows: in Sect. 2, we present a simple, computationally efficient FGR method to calculate the intensity amplification of images of all orders; in Sect. 3, we analyse the regimes of extreme intensity amplification in the strong field regime and derive approximate formulae for the first two images of a static source located near the caustic line behind or in front of the black hole; in Sect. 4, we extend our analysis of the regimes of extreme intensity amplification to the case of a source in Keplerian circular orbit. Section 5 is devoted to our conclusions. Throughout the article, we use relativistic units with c = G = 1.
2. Amplification of intensity in the Schwarzschild field
The bolometric intensity of radiation coming from an infinitesimally small source and propagating through a gravitational field is (Schneider et al. 1992; Misner et al. 1973)
where I_{0} is the bolometric intensity measured in the absence of gravity, g is the frequency shift factor, and μ is the magnification factor, that is, the ratio of the solid angles subtended on the observer sky by the lensed and the nonlensed image of the source, dΩ and dΩ_{∞}, respectively. We define the bolometric amplification
consisting of the geometric amplification given by the magnification factor μ and the energy amplification given by the factor of g^{4}. In the case of an isotropically radiating infinitesimally small source and spherically symmetric gravitational field, the magnification factor can be written in the following form:
where β and θ are the angular coordinates of the source and of the image in the observer sky as measured from the caustic line, which is the line of sight connecting the observer and the centre of the lens. These angles lie in the conserved plane of the photon motion and their ratio determines the magnification in the direction perpendicular to such plane. The magnification in the plane of photon motion is determined by the ratio of the angular width of the source image in the presence of lensing to that in the absence of it, dθ and dβ, respectively (e.g. Ohanian 1987; Schneider et al. 1992; Bozza 2010). Owing to the spherical symmetry of the Schwarzschild field, any plane of photon motion can be identified with the equatorial plane.
In the absence of gravity, the impact parameter of photons on straight trajectories connecting the source and the observer simply reads
where R is the radial coordinate of the source and ϕ is its angular coordinate measured in the plane of the photon motion (see left panel of Fig. 1). The related angular coordinate of the source on the observer sky is β = b_{∞}/D_{s}, where D_{s} is a distance between the source and the observer. The impact parameter of photons on the curved trajectories b(Δϕ, R) and the related emission angle α(b, R), which is the angle between the ray direction at the source and the radial line connecting the source and the lens, can be determined by the trajectory integral (Eq. (A.1)) solved in the way described in Appendix A. In the Schwarzschild space time, an infinite number of photon trajectories connecting the observer and the source arises corresponding to a infinite number of source images (Darwin 1959). Here, Δϕ denotes the change of the angular coordinate along the particular photon trajectory corresponding to the image of order n, and it reads
Fig. 1. Schematic lensing geometry related to the determination of the magnification factor μ. Left: sketch illustrating the angles and distances introduced in the main text. Right: first three rays of orders 1, −1, 2 connecting the source with a small angular separation from the caustic line ψ to the distant observer. Top right: source is located behind the lens (see Sect. 3.1). Bottom right: source is located in front of the lens (see Sect. 3.2). 
where ‘+’ and ‘−’ denote direct (n > 0) and indirect (n < 0) images, respectively (see right panels of Fig. 1)^{1}. The sign of the image is determined by the orientation of angles β and θ with respect to the caustic line. For direct images, these angles have the same orientation, while for indirect ones, their orientation is the opposite^{2}. The angular coordinate of the particular image on the observer sky is θ = b(Δϕ, R)/D_{d}, where D_{d} is a distance between the lens and the observer.
In the absence of lensing, the distant observer (D_{s} ≫ R) sees a constant angular width of the source Δβ related to the source diameter ω as Δβ = ω/D_{s}. The image angular width Δθ = Δb/D_{d} can be determined by two rays emitted from opposite edges of the source of the diameter ω that reach an observer screen with impact parameters b and b − Δb (see left panel of Fig. 1). Here, the ray on the right edge is emitted at the radial coordinate R′ and the angular coordinate ϕ − dϕ. These new coordinates can be expressed by
We assume here that the diameter of the source ω is very small, and space time geometry remains constant along it. Therefore, the source diameter can be considered as a coordinate distance, and consequently relations (6) can be constructed in the pseudoCartesian chart underlying the Schwarzschild metric. The outgoing photon trajectories depending on the value of the impact parameter, the related existence of the radial turning point and the location of the source can be classified into three types in the Schwarzschild space time. A detailed discussion of the classification and the integration of outgoing photon trajectories is given in Appendix A. Photons on type I and II trajectories do not pass through a radial turning point, while photons on type III trajectories reach it. In the case of trajectory types I and II (α ≤ π/2), the angle ξ is given as ξ = π/2 − α, while in the case of type III trajectories (α > π/2) it reads ξ = 3π/2 − α (see left panel of Fig. 1 and Appendix A for details). We considered a limit case of a static observer at infinity (D_{d} → ∞, D_{s} → ∞). Then, the magnification factor in Eq. (3) can be rewritten into the following form:
The term of magnification in the plane of photon motion is expressed in a finite difference form. Nevertheless, it requires only one additional determination of the impact parameter by the trajectory integral (Eq. (A.1)) and can be evaluated for an arbitrarily small source diameter ω (limited only by the computer representation of floating point numbers) on a wide range of R and Δϕ. Such a magnification formula is consistent with the formula describing the magnification of distant sources by black holes (see Eqs. (15)–(19) in Bozza 2010), but it is well defined for arbitrary source distances.
Figure 2 illustrates the resulting bolometric amplification of a static source,
Fig. 2. Bolometric amplification of a static source for selected distances from the lens seen by an observer at infinity as a function of Δϕ. Top: wide range of Δϕ covering the first 12 images and first six Einstein rings. Peaks at Δϕ = 1, 3, 5, … (2, 4, 6, …) arise when the a source located exactly on the caustic line behind (in front of) the lens. Bottom: bolometric amplification of sources at short distances for a smaller range of Δϕ angles, encompassing the first amplification peak. 
as a function of Δϕ and for selected source distances covering a wide range. The peaks where the amplification diverges for integer odd multiples of π correspond to the Einstein rings generated when the source is located exactly on the caustic line behind the lens, while the peaks at even multiples correspond to the Einstein retrorings generated when the source is located on the caustic line in front of the lens. The bottom panel of Fig. 2 shows the amplification of static sources at selected low distances over a smaller range of Δϕ, which encompasses only the first peak. In general, the closer the source, the higher the amplification is in the vicinity of the peaks. The decrease in amplification due to the frequency shift factor (g^{4}) is responsible for the values < 1 for small Δϕ, an effect that is more apparent for the lowradius curves in the bottom panel of Fig. 2.
Figure 3 illustrates the behaviour of the emission angle, α, which, for higher order images (Δϕ > π) and a distant source at R > 1000M, rapidly approaches π. In such a case, the radial coordinates of the lensed ray emitted from the two edges of source become almost identical (R′→R), and the magnification in the plane of photon motion can be easily expressed as db/Rdϕ in terms of a derivative of one incomplete elliptic integral of the first kind (see Eqs. (18)–(20) in Ohanian 1987). In this limit the magnification formula (7) merges with the solution derived by Ohanian (see Fig. 6 and Eq. (18) in the limit case for D_{o} → ∞, D_{os} → ∞, D → R in Ohanian 1987). The differential time delay computed with respect to the pure radial ray with b = 0 (see Appendix B for details) rapidly grows with Δϕ > π for direct images, while for images of higher order its slow growth corresponds only to the time difference along the looping section of the photon trajectories (see Fig. 4).
Fig. 3. Emission angle α of a source at different distances from the lens as a function of Δϕ. 
Fig. 4. Differential time delay for a source at different distances from the lens as a function of Δϕ. 
3. Extreme amplification regime: Static source
3.1. Primary amplification peak
The primary amplification peak occurs when the source is located close to the caustic line behind the lens, where the maximum amplification is related to the formation of the first Einstein ring (see top right panel of Fig. 1). Extremely high amplifications are achieved by sources with a very small angular separation from the caustic line ψ = π − ϕ. A significant contribution to the resulting total intensity comes from the first two images with Δϕ_{±} = π ∓ ψ, while the contributions from higher order images are negligible (see Fig. 2).
In the linearised, weak deflection limit (WDL) approximation of the Schwarzschild point lens, the magnification factor of first two images reads
where ± denotes the direct and indirect image, respectively, and denotes the source angular coordinate, β, normalised to the characteristic angle of the lens, α_{0}, the apparent angular radius of first Einstein ring on the observer sky (Schneider et al. 1992). The angle α_{0} is
In the absence of gravity and for a small angular separation ψ, the impact parameter of the source is b_{∞} = ψ R, and a related angular coordinate of the source reads β = ψ R/D_{s}. In the considered case of an observer at infinity (D_{d} → ∞, D_{s} → ∞), the normalised angle can be written as
Only in the case of a source with the angular coordinate β smaller than the apparent angular radius of the first Einstein ring α_{0} (, strong lensing regime) do both direct and indirect images attain high magnifications: μ_{±} ≫ 1. In the case of a source with a larger angular coordinate (, weak lensing regime), the direct image is almost nonmagnified (μ_{s+} ∼ 1), and the indirect image becomes very faint (μ_{s−} ≪ 1). The transition radius, R_{ws}(ψ), at which the lensing regime switches from weak lensing to strong lensing (), can be determined from Eq. (11) as
In the WDL approximation, the source angular coordinate β is considered to be small. Moreover, very large distances of the source and the observer from the lens are assumed, and therefore the gravitational redshift does not affect the resulting amplification. On the contrary, our FGR approach described in the previous section works everywhere, in particular where WDL fails, namely in the highdeflection regime and also for the sources in close vicinity of the lens.
In Fig. 5, we compare the FGR bolometric amplification given by Eq. (8) with the corresponding WDL value given by Eq. (9) as functions of the source distance R at different fixed values of the angular separation ψ. Significant deviations (≳3%) of the FGR value from the WDL approximation occur at source distances of R ≲ 200M as a result of light deflection in close vicinity of the lens and gravitational redshift. Hereafter, we denote such an FGR regime outside of the WDL approximation validity as the extreme lensing regime and the transition radius as R_{e} = 200M. In the case of a small angular separation ψ ≤ 0.01, the FGR bolometric amplification of the first two images of a static source located behind the Schwarzschild point lens can be approximated to a better than 1.6% accuracy for R > 6M and 6% for R > 3M by the following relation:
Fig. 5. Primary amplification peak of a static source. FGR bolometric amplification compared with the WDL approximation and the FGR approximation. The vertical dashed line at R_{e} = 200M divides the extreme lensing region where an FGR approach must be used from the region, where WDL approximation works. The vertical arrows denote R_{ws}(ψ), the radius of transitions from the strong lensing regime to the weak lensing regime for angular separations ψ = 10^{−2} and ψ = 10^{−4}. Top: first direct image. Bottom: first indirect image. 
where we introduced the correction factor
In the relevant case of small β, this formula can be written as
where ± denotes the direct and indirect images, respectively. In the extreme lensing regime, the dependency of the bolometric amplification, ℳ, on the source distance, R, becomes nonmonotonic. The source distance R_{max}, at which the bolometric amplification of the primary amplification peak reaches its maximum is identical for both images, and it can be determined by the extremum condition dℳ_{±}(R, ψ)/dR = 0. The results of numerical calculations yielding the R_{max} and related values of ℳ_{a±}(R_{max}, ψ) are summarised in Table 1.
Source distances, R_{max}, at which the first amplification peak reaches its maxima, ℳ_{a} ± (R_{max}, ψ), calculated for representative values of ψ.
The dependence of the bolometric amplification on the angular separation ψ is illustrated in Fig. 6. The bolometric amplifications of both images grow symmetrically as 1/ψ for a sufficiently small angular separation (ψ < 10^{−3}). The complete behaviour of the bolometric amplification of the first direct image, ℳ_{a+}, as a 3D function of the source distance, R, and the source angular separation, ψ, is illustrated in Fig. 7, showing regions of the three abovedescribed qualitatively different lensing regimes (weak, strong, extreme). The bolometric amplification of the first indirect image, ℳ_{a−}, differs significantly only in the weak lensing regime where the absolute term in Eq. (9) is not negligible.
Fig. 6. Primary amplification peak of a static source. The bolometric amplification of the first two images of a source at different distances R is shown as a function of angular separation ψ. The solid lines refer to the first direct image, and the dashed lines to the first indirect image. 
Fig. 7. Primary amplification peak of a static source. The bolometric amplification of the first direct image, ℳ_{a+}, is shown as a 3D function of the source distance, R, and the angular separation ψ. Different lensing regimes are indicated. The magenta dashed line denotes the source distance of the maximal amplification at fixed ψ, R_{max}. The red line at R_{e} = 200M separates the region of extreme gravitational lensing regime and the region of WDL validity. The blue line denotes the source distance, R_{ws}(ψ), separating the region of the strong lensing regime () and the region of the weak lensing regime (). This boundary between the two regimes corresponds to the constant value of the bolometric amplification, ℳ_{a+} = 1.17047, for ψ ≤ 10^{−2}. 
The angular separation corresponding to the apparent radius of the Einstein ring measured at the source distance can be expressed from the condition as . The related impact parameter for a nonlensed source reads . The solution of the WDL lens equation for a point mass Schwarzschild lens can be rewritten in terms of impact parameters into the following form (see Eq. (2.7b) in Schneider et al. 1992):
where ± denotes the direct and indirect images, respectively. In Fig. 8, we compare the FGR impact parameter with the value obtained by the WDL formula (16) for different values of the angular separation ψ. The significant deviation of the exact value from the WDL one occurs at source distances of R ≲ 1000M. In the case of small separations, ψ ≤ 0.01, the exact impact parameter of the first two images of a static source behind the lens can be approximated with accuracy better than 1% for R > 3M through the relation
Fig. 8. Primary amplification peak. The exact impact parameter compared with the WDL approximation and the FGR approximated relation. The vertical dashed line at R_{e} = 200M divides the extreme lensing region where an FGR approach must be used from the region, where WDL approximation works. The vertical arrows denote R_{ws}(ψ), that is, the radius of transitions from the strong lensing regime to the weak lensing regime for angular separations ψ = 10^{−2} and ψ = 10^{−4}. Top: first direct image. Bottom: first indirect image. 
The differential time delay between first two images forming the primary amplification peak is illustrated in Figs. 9 and 10. In the case of a small separation, ψ < 0.01, the differential time delay can be approximated with a level of accuracy better than 1% for R > 6M and 3% for R > 3M through the relation
Fig. 9. Primary amplification peak. The time delay between the first two images of a source at different distances R is plotted as a function of the separation from the caustic line, ψ. 
Fig. 10. Primary amplification peak. The exact time delay between the first two images compared with the approximated relation. 
3.2. Secondary amplification peak
The secondary amplification peak occurs for the source located close to the caustic line in front of the lens, where maximum amplification is related to the formation of the first Einstein retroring (see bottom right panel of Fig. 1). Since we define the angular separation, ψ, from the closest point on the caustic line, in this case ψ = ϕ. A significant contribution to the resulting total intensity of the peak now comes from the first indirect and the second direct images with Δϕ_{∓} = 2π ∓ ϕ, whose photon trajectories contain one loop. Clearly, this peak cannot be described by the WDL formulae (Eqs. (9) and (16))^{3}. We found that the FGR bolometric amplification of both images forming the peak can be approximated with a level of accuracy better than 5% for R ≥ 3M and ϕ ≤ 0.01 through the relation
where we introduced the following correction factor:
In Fig. 11, we compare the FGR solution with the approximate formula (19) for different values of the angular separation from the caustic line ψ. We illustrate the bolometric amplification of the first indirect and the second direct image of the source in front of the compact lens in Fig. 12. As a result of the much steeper (∝R^{−2}) radial dependence in Eq. (19) as compared to that in Eq. (15) (∝R^{−1/2}), the secondary peak attains large amplifications only for sources at small radii. Similarly to the primary peak, the bolometric amplification of both images forming the secondary peak grows symmetrically as 1/ψ for a sufficiently small angular separation, ψ < 10^{−3}, as we illustrate in Fig. 12. The ratio of the bolometric amplifications by the primary and the secondary amplification peak reaches the value of ∼15 for the source located at R = 6M, while for the source located at R = 10 000M such ratio reaches values of ∼6.
Fig. 11. Secondary amplification peak of a static source. The FGR amplification (identical for the first indirect and second direct images) compared with the approximated relation. 
Fig. 12. Secondary amplification peak of a static source. The bolometric amplification of the first indirect and the second direct images of a static point source located in the front of the compact lens at different distances, R, are shown as a function of the angular separation from the caustic line, ψ. The solid lines correspond to the first indirect images, and the dashed line corresponds to the second direct images. 
Analogously to the case of the primary peak, the source distance R_{2max}, at which the bolometric amplification ℳ_{2a} reaches its maximum is given by the extremum condition dℳ_{2a}(R, ψ)/dR = 0. Contrary to the case of the primary peak, this distance does not depend on the angular separation ψ. Therefore, the corresponding value of maximum amplification scales exactly with 1/ψ. Numerical calculation yields the following values:
Contrary to the case of the primary peak, the differential time delay between the images mainly arising on looping parts of photon trajectories is independent of the source distance, R (see Fig. 10), and it can be very precisely approximated by a simple relation:
3.3. Finitesized sources
Both the FGR and the WDL formula based on geometric optics predict an infinite amplification for a point source located on the caustic line (ψ = 0). In general, the amplification of a finitesized source with uniform brightness can be calculated by integrating a sum of the point source amplifications of both firstorder direct and indirect images over the solid angle subtended by the source and normalised to the solid angle itself (see Schneider et al. 1992; Witt & Mao 1994, for details). Using the results of Mao & Witt (1998) and the approximate relation (13), we can express the total bolometric amplification considering both significant images of the uniformbrightness circular source of radius ν (ν ≪ R), located behind the lens with the small normalised separation angle (Eq. (11)), in the following form:
where we introduced the normalised source angular radius . Here, the ‘elliptic’ factor reads
and F, E, and Π are, respectively, the elliptic integrals of the first, second, and third kind. Their parameters read
and the coefficients are defined as follows:
In the case of the source centred exactly on the caustic line behind the compact lens (), formula (23) takes the following simple form:
Comparing the approximate relations (13) and (19), the total bolometric amplification of a uniformbrightness source of the radius ν (ν ≪ R) located in front of the lens with the small normalised separation angle Eq. (11) can be written as
In the case of a source centred exactly on the caustic line in front of the compact lens (), formula (30) takes the following simple form:
For decreasing values of ψ, the amplification curves in Fig. 13 change their shape from a powerlawlike to a plateaulike behaviour at the inflection point, ψ_{i} = ν/R, where the source angular separation is equal to the source angular size (, that is, where the source rim just touches the caustic line). In such a specific case the total amplification for a source located behind and a source located in front of the lens can be expressed as
Fig. 13. Total bolometric magnification of uniformbrightness circular source with radius ν = 10^{−3}M, located behind the lens (upper panel) and in front of the lens (bottom panel), as a function of the angular separation ψ. 
respectively. Here, the normalised magnification integral over the source surface takes the following simple form (Mao & Witt 1998):
4. Extreme amplification regime: Source on circular Keplerian orbit
For a moving source, the bolometric amplification of the static source discussed in the previous sections must be multiplied by the fourth power of the relativistic Doppler shift factor, g_{d}. In the case of a circular Keplerian motion, the relativistic Doppler shift factor with respect to a distant static observer located on the orbital plane reads
where is the orbital angular velocity, b is the impact parameter, and the ± sign refers to the direct and indirect images, respectively. In Fig. 14, we show the light curves of first three images forming the primary and secondary amplification peaks for a point source in Keplerian circular orbit at a radii of R = 6M, 20M, and 100M, as seen by a distant static observer located in the orbital plane. These values were chosen so as to show the increasing orbital swing in the energy shifts for decreasing radii (most apparent in the first image light curves), as well as the increasing time shifts in the peaks of the higher order images resulting from light traveltime delays of photons circling around the black hole.
Fig. 14. Light curves of the first three images of the point source on a circular Keplerian orbit with radii of R = 6M (top), R = 20M (middle), and R = 100M (bottom) seen by an equatorial observer. 
Light curves are constructed with respect to the observer detection time expressed by the differential time delay (see Appendix B for details). The light curves are plotted in false colours representing the frequency shift factor, g, of the appropriate image^{4}. The frequency shift increases with the angular velocity, Ω_{K}, which grows with decreasing orbital radius. For closer orbits, the different time delay of individual rays causes the resulting time asymmetry of the light curves. Consequently, the contribution to the first magnification peak by the second direct image (forming the third Einstein ring) is significantly shifted forwards in time (see top and middle panels of Fig. 14). The Doppler effect is responsible for the asymmetric shape of the primary peaks as the source moves away from the observer before the maximum and towards the observer after it. On the contrary, in the case of secondary peaks, the source moves towards the observer before the maximum and away after it. The related Doppler boosting or suppression of the particular image amplification is apparent in Figs. 15 and 16.
Fig. 15. Primary amplification peak of a source in Keplerian circular orbit. The bolometric amplification of the first two images of a source approaching the observer (moving away from the caustic line) is plotted as a function of angular separation, ψ, at different source distances R. The solid lines correspond to the first direct images, and the dashed line corresponds to the first indirect images. 
Fig. 16. Secondary amplification peak of a source on a Keplerian circular orbit. The bolometric amplification of the first indirect and the second direct images of a source moving away from the observer (moving away from the caustic line) is plotted as a function of angular separation, ϕ, at different source distances, R. The solid lines correspond to the first direct images, and the dashed line corresponds to the first indirect images. 
The behaviour of the frequency shift in the vicinity of both amplification maxima is quite complex. In the case of the source approaching the primary amplification peak (the caustic line behind the lens), the first direct image is redshifted, while the first indirect image is blueshifted. After passing the caustic line, the two images exchange their frequency shift factors almost symmetrically, the first direct image becoming blueshifted and the first indirect one becoming redshifted. The result is an almost constantly blueshifted dominant fraction as well as an almost constantly redshifted secondary fraction of the radiation coming from the source passing through the primary amplification peak. The small time increase of the frequency shift of both fractions is caused by a small asymmetry of the angular coordinate change along the rays forming the peak (Δϕ_{±} = π ∓ ψ). Both rays are only exactly symmetrical for sources located exactly at the caustic line and forming the first Einstein ring. Therefore, the difference in the intensity of the blueshifted and the redshifted fraction is primarily caused by the Doppler effect associated with the source orbital motion. The behaviour of the first indirect image and the second direct image for the source in the vicinity of the secondary amplification peak (crossing the caustic line in front of the lens) is completely analogous. The second Einstein ring (the first retroring) arises when the source is located exactly at the caustic line. However, the blueshift of the dominant fraction is significantly lower than in the case of the primary peak. We also note that the first direct image with b ∼ 0 is not amplified by the second peak. In Figs. 17 and 18, we illustrate the details of the light curves around both amplification maxima for the case of Keplerian circular orbit at R = 6M.
Fig. 17. Detail of the light curves of a point source in Keplerian circular orbit at R = 6M around the primary amplification peak. The thick and thin continuous lines represent, respectively, the first direct and first indirect images. Top: light curves of images forming the peak. Bottom: frequency shift factor profiles. 
Fig. 18. Detail of light curves of the point source in Keplerian circular orbit at R = 6M around the secondary amplification peak. The dashed line represents the first direct image, while the thick and thin continuous lines represent, respectively, the first indirect and second direct images. Top: light curves of both images forming the peak as well as the light curve of the nonamplified first direct image. Bottom: frequency shift factor profiles. 
5. Discussion
We investigated the extreme amplification regime of pointlike and smallsized sources located in close vicinity of the caustic line at a short distance from a Schwarzschild black hole gravitational lens. The bolometric amplification of static sources, as well as that of sources in Keplerian circular orbit, were calculated based on an efficient fully relativistic method in the cases of both direct and indirect images. We also derived analytical formulae, which approximate the amplification in the extreme regime to an accuracy level of a few percent. Such formulae can provide an efficient and computationally cheap alternative to the usual numerical integration necessary to calculate the total amplification and the time delay between the first two images, especially in the regime in which their changes are steepest. Moreover, we provide a simple analytic expression for the maximum amplification as a function of the source position and size in the case of extended sources. These approximations may be particularly relevant together both when exploring the parameter space of such systems and in rapidly assessing their properties.
The lensing parameterisation that we adopted is based on the angle, ψ, separating the source from the caustic line as measured from the centre of the lens, and is thus especially convenient for applications in which the source is located close to the lens^{5}. For any given small value of the offset ψ ≪ 1, a pointlike source behind the lens attains increasingly large amplifications for decreasing radii. The characteristic amplification scaling of the WDL approximation, , holds for R ≥ 200M. For smaller radii, the extreme amplification regime sets in, leading to a gradual flattening of the amplification curve for decreasing R and to a reduction of the amplification below R ∼ 8.1M, where the maximum amplification is attained (see Eq. (15) and the upper panel of Fig. 5). At a given R, the amplification grows ∝ ψ^{−1}, diverging on the caustics behind the lens (ψ = 0).
The amplification of a source of finite radius ν ≪ M behaves in virtually the same way as a pointlike source as long as ν < ψR. Its amplification saturates at a value of when the source is intersected by the caustic behind the lens (ν ≥ ψR, see Eq. (29)). The maximum amplification that a finite source can attain at a given R is thus limited by the inverse of its size. Conversely, for a given ν the maximum amplification grows ∝ R^{1/2}, as the source angular size decreases relatively to the lens. However, the corresponding chances that the source and the lens attain the required degree of alignment scales as the solid angle subtended by the source at the lens, ∼(ν/2R)^{2}, which decreases rapidly with increasing R. In practice, for an astrophysical system consisting of a source of radius, ν, at a distance, R, from a Schwarzschild black hole this translates into the following tradeoff between maximum attainable amplification and the chances, 𝒫(ν, R), of the required degree of alignment:
For instance, a maximum amplification of ∼10^{3} at R = 100M corresponds to . Similar considerations would also apply to the instantaneous position of sources in Keplerian circular orbits seen edgeon at a 90° inclination, once the additional contribution to the amplification arising from relativistic Doppler and transverse shifts is taken into account. It is worth noting that for a source of size ν the chances of attaining the alignment required for maximum amplification during an orbit are a factor of 4R/(νπ) higher than the value in Eq. (36).
The above discussion applies to direct image amplification of sources close to the caustic behind the lens. It emphasises that in actual astrophysical systems large amplifications through gravitational lensing are possible, but the conditions that lead to them are difficult to attain (even more so for indirect images). Limitations may also arise from diffraction effects, especially in the radio. Schneider et al. (1992) estimated that the maximum amplification for an infinitesimally small source lensed by the Schwarzschild point lens in the WDL regime is
with f being the radiation frequency and M_{⊙} the solar mass. Detailed applications of the extreme Schwarzschild lensing regime presented here are beyond the scope of this paper and will be the subject of forthcoming studies.
This indexing can be related to the one in Gralla & Lupsasca (2020a), Sect. VI C, by noting that their even values correspond to the positive ones of our n (), and their odd values correspond to the negative ones of our n ().
In general, the sign (parity) of the image is equal to the sign of the Jacobian determinant of the lens map (see e.g. Schneider et al. 1992 for details).
Light bending of the rays forming the secondary amplification peak can be calculated by the strong deflection limit (SDL) approximation (Bozza & Scarpetta 2007).
The case of R = 20M is in a very good quantitative agreement with the results obtained for an orbiting point source in the extreme Kerr space time by relatively complicated pure numerical methods, as the difference of spacetime geometries at this radial coordinate becomes small (compare Fig. 9a in Cunningham & Bardeen 1973).
A different algorithm of the reduction to elliptic functions can be found in Čadež & Kostić (2005).
Acknowledgments
The authors wish to thank the anonymous referee for the valuable suggestions. L.S. acknowledges financial contributions from ASIINAF agreements 201714H.O and I/037/12/0, from “iPeska” research grant (P.I.: Andrea Possenti) funded under the INAF call PRINSKA/CTA (resolution 70/2016), and from PRININAF 2019 no. 15.
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Appendix A: Light deflection
The change in the angular coordinate along the photon trajectory in the Schwarzschild space time can be written as (Darwin 1959; Misner et al. 1973)
where r is the radial coordinate, b is the photon impact parameter, and the ± sign corresponds to outward and inward photon motion. This sign changes if a photon passes through a radial turning point. The photon trajectories connecting a source located at a radial coordinate, R, larger than that of the unstable photon circular orbit (R > r_{pco} = 3M) and an observer at infinity can therefore be classified as follows. Photons on type I trajectories with can attain any value of the radial coordinate. The change of the angular coordinate along these rays growing with b is limited by the following relation (Hadrava et al. 1994):
Although no simple analytical expression can be found for the integral in Eq. (A.1) over type I trajectories, its numerical integration is easily carried out by using the variable u = 1/r, owing to the absence of improper values of the integrand. In the case of type I trajectories, the emission angle α, measured with respect to radial direction, is related to the impact parameter by
The minimum value of the radial coordinate, periastron P > 3M, of photons escaping to infinity with b > b_{crit} is given by the following formula (Misner et al. 1973):
The change of the angular coordinate for a photon moving along the null geodesics from the periastron, P, to the radial coordinate, R, can be expressed by the following relation (Chandrasekhar 1983)^{6}:
where K is the complete elliptic integral of the first kind, F is the incomplete elliptic integral of first kind, and
The maximum impact parameter, b_{max}, of photons emitted by the source at the radial coordinate R and reaching infinity corresponds to a trajectory with the periastron exactly at R and is given by the following formula:
The corresponding change of the angular coordinate Δϕ_{bmax} for P = R and the observer at infinity can be expressed using (A.5) in the following form:
where the asymptotic value of the elliptic integral amplitude (A.8) for r → ∞ reads
The trajectories of photons with an impact parameter in the interval b_{crit} < b < b_{max} and not reaching the periastron, P, are classified as type II. By using (A.5), the change of the angular coordinate along the type II trajectories can be expressed as
The change of the angular coordinate Δϕ_{crit} < Δϕ < Δϕ_{bmax} increases with b. Similarly to the type I case, the emission angle is related to the impact parameter by Eq. (A.3).
Finally, photons on type III trajectories have impact parameter in the b_{crit} < b < b_{max} interval, but they pass through periastron. Using the time symmetry of the Schwarzschild metric, we can express the change of the angular coordinate along type III trajectories through the following relation:
For these trajectories, the impact parameter decreases with growing Δϕ and asymptotically approaches b_{crit} from above. The limit case of photons with b → b_{crit} corresponds to the capture of photons in an unstable circular orbit with Δϕ → ∞. In the case of type III trajectories, the emission angle reads
For completeness, we remind the reader that in the case of a source located below a circular photon orbit with R < 3M, which emits only photons to infinity on type I trajectories, the impact parameter grows monotonically with Δϕ from zero and asymptotically approaches b_{crit} from below. The limiting case of photons with b → b_{crit} corresponds to the capture of photons along an unstable circular orbit with Δϕ → ∞.
We constructed the algorithm to calculate the impact parameter b as a function of Δϕ based on the above relations and the Brent numerical method of root finding. We used Romberg integration to integrate type I trajectories. Elliptic integrals in Eqs. (A.5, A.12, and A.13) can be evaluated by means of the Carlson elliptic function with high precision, limited only by the computer representation of floating point numbers (Press et al. 2002).
Appendix B: Time delay
Since the time delay for an observer at infinity diverges, we considered the differential time delay computed relatively to the time delay of the purely radial reference ray with b = 0 emitted at the caustic line in front of the lens and at the same radial coordinate where the source is located. Such a differential time delay along type I and II trajectories can be written in the following form (Hadrava et al. 1994):
The integral can easily be evaluated numerically by using u = 1/r. In the case of a type III photon trajectory, double the time spent by a photon moving from periastron, P, to the source radial coordinate, R, must be added. This part of the time delay can be expressed in a similar way to Eq. (A.13) as (Hadrava et al. 1994)
where E is an elliptic integral of the second kind, Π is an elliptic integral of the third kind, and
All Tables
Source distances, R_{max}, at which the first amplification peak reaches its maxima, ℳ_{a} ± (R_{max}, ψ), calculated for representative values of ψ.
All Figures
Fig. 1. Schematic lensing geometry related to the determination of the magnification factor μ. Left: sketch illustrating the angles and distances introduced in the main text. Right: first three rays of orders 1, −1, 2 connecting the source with a small angular separation from the caustic line ψ to the distant observer. Top right: source is located behind the lens (see Sect. 3.1). Bottom right: source is located in front of the lens (see Sect. 3.2). 

In the text 
Fig. 2. Bolometric amplification of a static source for selected distances from the lens seen by an observer at infinity as a function of Δϕ. Top: wide range of Δϕ covering the first 12 images and first six Einstein rings. Peaks at Δϕ = 1, 3, 5, … (2, 4, 6, …) arise when the a source located exactly on the caustic line behind (in front of) the lens. Bottom: bolometric amplification of sources at short distances for a smaller range of Δϕ angles, encompassing the first amplification peak. 

In the text 
Fig. 3. Emission angle α of a source at different distances from the lens as a function of Δϕ. 

In the text 
Fig. 4. Differential time delay for a source at different distances from the lens as a function of Δϕ. 

In the text 
Fig. 5. Primary amplification peak of a static source. FGR bolometric amplification compared with the WDL approximation and the FGR approximation. The vertical dashed line at R_{e} = 200M divides the extreme lensing region where an FGR approach must be used from the region, where WDL approximation works. The vertical arrows denote R_{ws}(ψ), the radius of transitions from the strong lensing regime to the weak lensing regime for angular separations ψ = 10^{−2} and ψ = 10^{−4}. Top: first direct image. Bottom: first indirect image. 

In the text 
Fig. 6. Primary amplification peak of a static source. The bolometric amplification of the first two images of a source at different distances R is shown as a function of angular separation ψ. The solid lines refer to the first direct image, and the dashed lines to the first indirect image. 

In the text 
Fig. 7. Primary amplification peak of a static source. The bolometric amplification of the first direct image, ℳ_{a+}, is shown as a 3D function of the source distance, R, and the angular separation ψ. Different lensing regimes are indicated. The magenta dashed line denotes the source distance of the maximal amplification at fixed ψ, R_{max}. The red line at R_{e} = 200M separates the region of extreme gravitational lensing regime and the region of WDL validity. The blue line denotes the source distance, R_{ws}(ψ), separating the region of the strong lensing regime () and the region of the weak lensing regime (). This boundary between the two regimes corresponds to the constant value of the bolometric amplification, ℳ_{a+} = 1.17047, for ψ ≤ 10^{−2}. 

In the text 
Fig. 8. Primary amplification peak. The exact impact parameter compared with the WDL approximation and the FGR approximated relation. The vertical dashed line at R_{e} = 200M divides the extreme lensing region where an FGR approach must be used from the region, where WDL approximation works. The vertical arrows denote R_{ws}(ψ), that is, the radius of transitions from the strong lensing regime to the weak lensing regime for angular separations ψ = 10^{−2} and ψ = 10^{−4}. Top: first direct image. Bottom: first indirect image. 

In the text 
Fig. 9. Primary amplification peak. The time delay between the first two images of a source at different distances R is plotted as a function of the separation from the caustic line, ψ. 

In the text 
Fig. 10. Primary amplification peak. The exact time delay between the first two images compared with the approximated relation. 

In the text 
Fig. 11. Secondary amplification peak of a static source. The FGR amplification (identical for the first indirect and second direct images) compared with the approximated relation. 

In the text 
Fig. 12. Secondary amplification peak of a static source. The bolometric amplification of the first indirect and the second direct images of a static point source located in the front of the compact lens at different distances, R, are shown as a function of the angular separation from the caustic line, ψ. The solid lines correspond to the first indirect images, and the dashed line corresponds to the second direct images. 

In the text 
Fig. 13. Total bolometric magnification of uniformbrightness circular source with radius ν = 10^{−3}M, located behind the lens (upper panel) and in front of the lens (bottom panel), as a function of the angular separation ψ. 

In the text 
Fig. 14. Light curves of the first three images of the point source on a circular Keplerian orbit with radii of R = 6M (top), R = 20M (middle), and R = 100M (bottom) seen by an equatorial observer. 

In the text 
Fig. 15. Primary amplification peak of a source in Keplerian circular orbit. The bolometric amplification of the first two images of a source approaching the observer (moving away from the caustic line) is plotted as a function of angular separation, ψ, at different source distances R. The solid lines correspond to the first direct images, and the dashed line corresponds to the first indirect images. 

In the text 
Fig. 16. Secondary amplification peak of a source on a Keplerian circular orbit. The bolometric amplification of the first indirect and the second direct images of a source moving away from the observer (moving away from the caustic line) is plotted as a function of angular separation, ϕ, at different source distances, R. The solid lines correspond to the first direct images, and the dashed line corresponds to the first indirect images. 

In the text 
Fig. 17. Detail of the light curves of a point source in Keplerian circular orbit at R = 6M around the primary amplification peak. The thick and thin continuous lines represent, respectively, the first direct and first indirect images. Top: light curves of images forming the peak. Bottom: frequency shift factor profiles. 

In the text 
Fig. 18. Detail of light curves of the point source in Keplerian circular orbit at R = 6M around the secondary amplification peak. The dashed line represents the first direct image, while the thick and thin continuous lines represent, respectively, the first indirect and second direct images. Top: light curves of both images forming the peak as well as the light curve of the nonamplified first direct image. Bottom: frequency shift factor profiles. 

In the text 
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