Elliptically Symmetric Lenses and Violation of Burke's Theorem
EElliptically Symmetric Lenses and Violation of Burke’s Theorem
Sun Hong Rhie
Physics Department, University of Notre Dame, IN 46556 [email protected]
ABSTRACT
We show that the outside equation of a bounded elliptically symmetric lens(ESL) exhibits a pseudo-caustic that arises from a branch cut. A pseudo-causticis a curve in the source plane across which the number of images changes by one.The inside lens equation of a bounded ESL is free of a pseudo-caustic. Thusthe total parity of the images of a point source lensed by a bounded ellipticallysymmetric mass is not an invariant in violation of the Burke’s theorem. A smoothmass density function does not guarantee the validity of the Burke’s theorem.Pseudo-caustics of various lens equations are discussed. In the Appendix,Bourassa and Kantowski’s deflection angle formula for an elliptically symmetriclens is reproduced using the Schwarz function of the ellipse for an easy access; theoutside and inside lens equations of an arbitrary set of truncated circularly or el-liptically symmetric lenses, represented as points, sticks, and disks, are presentedas a reasonable approximation of the realistic galaxy or cluster lenses.One may consider smooth density functions that are not bounded but fallsufficiently fast asymptotically to preserve the total parity invariance. Anybounded function may be sufficiently closely approximated by an unboundedsmooth function obtained by truncating its Fourier integral at a high frequencymode. Whether to use a bounded function or an unbounded smooth function foran ESL lens mass density, whereby whether to observe the total parity invarianceor not, incurs philosophical questions. For example, is it sensible to insist thatthe elliptical symmetry of an elliptic lens galaxy be valid in the entire sky? Howa pseudo-caustic close to or intersecting with a caustic must be withered awayduring a smoothing process and what it means will be investigated in a separatework.
Subject headings: gravitational lensing a r X i v : . [ a s t r o - ph . E P ] J un
1. Introduction
Burke’s theorem (Burke 1981) is known as odd number theorem (MacKenzie 1985;Schneider et al. 1992), and is made of two pieces of claims. For a bounded mass distributionlens, 1) the total parity of the images is an invariant and 2) the number of images of a sourcesufficiently far away from the lens system is one. Here we will only focus on the invarianceof the total parity.Thirty five years ago, Bourassa and Kantowski (1975) found that the deflection angleof an elliptically symmetric lens (ESL) can be expressed as an integral over the interval[0 , ω = z − (cid:90) t cut ˜ σ ( t ) tdt (¯ z − c t ) / (1)where ω and z are the positions of the source and an image, c is the focal length of the massdistribution ellipse, and t is the elliptic radial parameter such that the position variable ofthe mass distribution is given by ξ = t ( a sin θ + ib cos θ ) : t ∈ [0 ,
1] where a and b arethe semi-major and semi-minor axes of the mass ellipse. The mass in the mass ellipse isused for the normalization of the lens equation. ˜ σ ( t ) is the projected mass density functionre-normalized such that 1 = (cid:90) ˜ σ ( t ) tdt . (2)If the image position z is inside the mass ellipse, then t cut = η where η is the elliptic radialparameter of z : z = η ( a sin θ + ib cos θ ), and if the image position z is outside the massellipse, then t cut = 1.A pseudo-caustic refers to a curve in the source plane across which the number of imageschanges by one and was considered a curious phenomenon (Evans and Wilkinson 1998). Theappearance of a pseudo-caustic is an indicator that the total parity of the images is not aninvariant in violation of the Burke’s theorem. The integrand of eq.(1) includes a square rootfunction, and we will see that that is the source of the pseudo-caustic of an ESL outside lensequation.We adopted the terminology pseudo-caustic because 1) a pseudo-caustic curve can formcusps in conjunction with open caustic curves and 2) we can continue to use the term “causticdomains” to describe the domains of the source plane with definite number of images thatare defined not only by caustics but also by pseudo-caustics.In section 2 we discuss the pseudo-caustics of various lens equations. The outsideequation of a generic elliptically symmetric lens is shown to have a pseudo-caustic thatarises from a branch cut. The inside equation of an elliptically symmetric finite density lens 3 –does not have a pseudo-caustic. Thus a bounded elliptically symmetric lens violates theinvariance of the total parity of the images; the arguments of smooth density functions forBurke’s theorem are invalidated. In section 3 we give a summary and raise a possibility toapproximate a bounded mass density by a smooth function with infinite extension to preservethe invariance of the total parity of the images. We leave the details to a separate work. Itis a natural wonder if other types of singularities than point singularities and branch cutsmay be to be found in gravitational lensing. In the Appendix, we reproduce the deflectionangle formula of Bourassa and Kantowski (1975) using the Schwarz function of the ellipse(Fassnacht et al. 2007; Khavinson and Lundberg 2009) and derive various lens equationsused in section 2; we also spell out the inside and outside lens equations of an arbitrary setof circularly symmetric lenses and elliptically symmetric lenses as a reasonable model forgalaxy and cluster lenses; they are represented as points, sticks, and disks.
2. Pseudo-caustics of Elliptically Symmetric Lens Equations
We start with examining the singular isothermal lens in which the pseudo-caustic arisesfrom a point mass density singularity where the Burke’s vector field is ill defined and move onto discussing the pseudo-caustics of various elliptically symmetric lenses (ESL). We will seethat the outside lens equation of any bounded ESL (mass density vanishes outside a certainfinite radius) has a pseudo-caustic that arises from the double-valuedness of the c-cut that isfrom a branch cut. The inside equation of a bounded ESL does not have a pseudo-caustic.Thus a bounded ESL violates the invariance of the total parity.
The “curious behavior” of a pseudo-caustic was noted first in an analysis of the isother-mal sphere of an infinite extension, whose projected mass density is inversely proportional tothe radius (referred to as SIS: singular isothermal sphere), and was attributed to the densitysingularity at the center (Kovner 1987). The nature of the pseudo-caustic can be readilyread off from the lens equation. ω = z − η ¯ z (3)where η is the radial parameter of z : z = ηae iθ and a is a reference radius the mass withinwhich is used to normalize the lens equation. Eq.(3) can be obtained by using the Newton’stheorem of a spherical symmetric mass or by calculating the deflection angle as in theAppendix. 4 –If we consider a monopole lens, for which η should be replaced by 1, an (cid:15) -circle aroundthe singularity at the origin is mapped to a large circle at infinity and the lens equation isa double-covering mapping. In other words, the image plane is divided into two regions bythe critical curve, which is mapped to the origin of the source plane, and each of the tworegions is mapped to the entire source plane.Now z = 0 is a singularity of eq.(3) where infinitely many finite-size vectors clash.Speaking in terms of the vector field considered by Burke (1981), the vector field is not welldefined at z = 0. If we cut out an (cid:15) -hole, the vector field behaves well in the neighborhoodof the punctured origin, and the (cid:15) -boundary around the singularity is mapped to a circleof radius 1 /a in the source plane. The z -plane outside the critical curve covers the whole ω -plane, but the punctured disk inside the critical curve covers only the disk of radius 1 /a centered at the origin of the ω -plane. In other words, the disk of radius 1 /a of the ω -planehas two images and and the rest of the ω -plane has only one image; the number of imageschanges as the source crosses the circle of radius 1 /a or the pseudo-caustic. As the sourcecrosses from inside the disk of radius 1 /a , one of the two images “disappears into the (cid:15) -hole.”The disappearing image is the negative image because the (cid:15) -hole is inside the critical curve.In summary, the pseudo-caustic is an indicator that the lensing equation is not a full doublecovering but a “one and half” covering of the source plane and that the total parity of theimages is not an invariant.If we truncate the isothermal sphere cylindrically such that the truncated mass is in-finitely elongated along the line of sight, the projected mass density is inversely proportionalto the radius, and the inside lens equation is given by the same equation as eq.(3) where a isthe radius of the mass disk. The reason why the lens equations are the same is that only themass inside the probing circle affects the two-dimensional gravitational field at the probingpoint z as is well known as the Newton’s theorem. The deflection angle is essentially, apartfrom some constants including the distance factor, the two-dimensional gravitational field. The deflection angle of the ESL can be expressed as an elementary function when thedensity is constant (Fassnacht et al. 2007). The (normalized) lens equation inside the massis a linear equation ω = (cid:18) − ab (cid:19) z + (cid:18) ab a − ba + b (cid:19) ¯ z (4)where a and b are the semi-major and semi-minor axes of the mass ellipse. The lens equa-tion is well defined everywhere., and there is no pseudo-caustic. The equation has always 5 –one solution and needs to be screened by hand or by studying the mass boundary and itssecondary curves (map the mass boundary by the lens equation into a curve in the sourceplane and study the pre-image curves in the image plane that are mapped to the curve inthe source plane; one of the pre-image curves is the mass boundary) if the solution is insidethe mass and hence is an actual image. ω = 0 has a solution at z = 0, and ω = ∞ has asolution at z = ∞ . Therefore a sufficiently large ω will fail to form an image inside the masswhere the equation is valid.The same equation applies to an infinite mass constant density ESL with an infiniteextension, which is due to the well known Newton’s theorem for the gravitational field of anelliptically symmetric mass. The deflection angle of the ESL can be expressed as an arcsine function when the pro-jected mass density is inversely proportional to the elliptic radius (Khavinson and Lundberg2009). The mass model is commonly referred to as a SIE (singular isothermal ellipsoid)since it was named as such in Kormann et al. (1994) even though it was neither provednor demonstrated to be an isothermal state . We shall call it arcsine lens. The inside lensequation of an arcsine lens is given as follows as can be found in the Appendix. ω = z − c arcsin (cid:16) cη ¯ z (cid:17) (5)where c is the focal length and η is the elliptic radial parameter of z .As in the case of the SIS, the elliptic version eq.(5) is ill-defined at z = 0, and an (cid:15) -circlearound the singularity is mapped to a finite loop in the source plane as (cid:15) →
0. The arcsinelens inside equation has a pseudo-caustic. The curve is a simple loop given by w = − c (cid:18) ca cos θ − ib sin θ (cid:19) . (6)The pseudo-caustic loop has the same sense with the (cid:15) -circle but has its phase lagged by π .Kormann et al. (1994) called the pseudo-caustic curve a cut following the conventionused by Kovner (1987) and observed that the pseudo-caustic can intersect with the causticcurve. The caustic is a symmetric astroid and is either enclosed inside or intersect with thepseudo-caustic curve.In order to calculate the critical curve and caustic curve, it is convenient to use theelliptic coordinates: z = η (cos θ + ib sin θ ) because the deflection angle is independent of the 6 –elliptic radius. The partial derivative with respect to z , ∂ z , can be expressed in terms of thepartial derivatives in elliptic coordinates, ∂ ≡ ∂ z = a sin θ + ib cos θ iab ∂ η + a cos θ − ib cos θ ηiab ∂ θ , (7)and the Jacobiant matrix components of the inside lens equation (5) are ∂ω = 1 − ηab ; ¯ ∂ω = 12 ηab z ¯ z (8)where ¯ ∂ is the complex conjugate of ∂ . The Jacobian determinant is J = | ∂ω | − | ¯ ∂ω | = 1 − ηab , (9)and on the critical curve η = 1 ab . (10)The critical curve is an ellipse. z = 1 ab ( a cos θ + ib sin θ ) . (11)The corresponding caustic is a quadroid (or astroid) with the four cusps on the real andimaginary axes (or symmetry axes). The precusps can be found using the same method asin Rhie and Bennett (2009). Since ∂ω is real, the precusp condition 0 = ∂ − J becomes0 = z∂J − ¯ z ¯ ∂J . (12)The equation leads to sin 2 θ = 0, hence the precusps are at θ = 0, π/ π , and 3 π/
2. Theprecusps are mapped to cusps on the positive real, negative imaginary, negative real, andpositive imaginary axis respectively.∆ ω real ≡ ω − ω π = 2 b − c arcsin( ca ) (13)∆ ω imag ≡ ω π/ − ω π/ = 2 ia − c arcsin( icb ) (14)∆ ω real > (cid:61) ∆ ω imag <
0, and hence the caustic curve has the opposite sense with thecritical curve.Figure 1 shows the cases of the caustic and pseudo-caustic. The number of the imagesof the caustic zones are indicated and clearly demonstrate that the total parity of the imagesis not an invariant. There is no case where the pseudo-caustic is enclosed inside the causticcurve because the cusps on the imaginary axes are always inside the pseudo-caustic. Thepoints of the pseudo-caustic and caustic on the positive imaginary axis (semi-minor axis of themass ellipse) are from θ = 3 π/
2, and it is easy to see that (cid:61) ( ω pseudo (3 π/ − ω caustic (3 π/ /a >
0. 7 –
We have discussed a few cases in which a pseudo-caustic arises because of a pointsingularity where Burke’s vector field is not well defined and around where the vector valuesare not all infinite. In this section we will introduce a line singularity where Burke’s vectorfield is not well defined and around where the vector values are not all infinite.Consider truncating the constant density ESL at t = 1, and the lens equation validoutside the mass is given by ω = z − c (¯ z − √ ¯ z − c ) . (15) z = ± c are the branch points of the square root function and we need to place branch cuts toestablish a convention where the Riemann sheets are sewed up. Mathematically a branch cutcan be placed wherever we are pleased to put it as far as it is connected to the branch point inconcern. Physically, however, we are constrained because two images in an (cid:15) -neighborhoodcannot come from two widely separated sources unless there is a physical reason. Placing thebranch cuts randomly will induce such an unphysical behavior. The outside lens equation(15) depends only on the focal length c and is valid for an infinite family of confocal lensesfor which the image zone can be everywhere except for an (cid:15) -ellipse around the line segmentconnecting the two focal points or the two branch points. Thus physically the line segmentconnecting the two focal points is the only option as the branch cut. The square root functionshould be understood as √ z − c = ( r + r − ) / e i ( θ + + θ − ) / where z ± c = r ± e iθ ± .Across the branch cut, the (complex) deflection angle changes the phase by π . In otherwords, Burke’s vector field is not well defined on the branch cut because it is double-valued.We can cut along the branch cut to remedy it. The boundary of the cut is mapped to afinite loop by the lens equation. The finite loop is the pseudo-caustic, and across which anegative image disappears into or appears from the branch cut. If we truncate the arcsine lens at t = 1, the outside lens equation is given by ω = z − c arcsin (cid:16) c ¯ z (cid:17) . (16)The deflection angle is double-valued on the line segment connecting the two focal points.Call it c-cut. Cut the image plane along the c-cut, and the boundary of the c-cut is mappedto a semi-infinite loop whose extension is finite in the direction of the c-cut. The semi-infiniteloop is the pseudo-caustic. 8 –Bergweiler and Eremenko (2010) (BE10 herefrom) studied the number of images of thelens equation (16) using a converted equation and found that the number of images canbe anywhere from 1 to 6. Since the functional BE10 used is different from that of (16),the caustic planes in BE10 should be reinterpreted. It turns out that the transformationis simple: ω (cid:48) = − cω where ω (cid:48) is the source plane variable of the associated equation, andthe morphology of its caustic plane can be read almost as if it were the caustic plane of theoriginal equation (16) if one overlooks the absolute value of the coordinates and takes intoaccount of the reversing of the parities. The pseudo-caustic in a caustic plane in BE10 arisesfrom the image plane boundary (thus the converted equation cannot be the lens equation ofa realistic lens), and it corresponds to the pseudo-caustic that arises from the c-cut that isessentially from a branch cut.Figure 2 shows the caustic domains for c = 1 .
65. The critical curve bifurcates at c = √
2, and at c = 1 .
65 it is made of three loops which enclose the origin and two focalpoints respectively. The caustic curves are open and closed by the pseudo-caustic curve.The transition from the domain 1/0 to the domain 2/0 is by an increase of a positive imageas is indicated in the notations, and the reason for the relevance of a positive image is thatthe segment of the c-cut is outside the critical curves. One of the four “positive” segmentsof the pseudo-caustic is indicated by the symbol ⊕ . We have examined the pseudo-caustics of a few algebraic lens equations. For an arbi-trary elliptically symmetric lens, the equation is an integro-algebraic equation. However theexistence or non-existence of the pseudo-caustic arising from the c-cut can be easily discernedby looking at the behavior of the c-cut at the origin, z = 0. Since the deflection angle iscontinuous on the (cid:15) -boundary of the c-cut, double-valuedness at z = is an indicator that thec-cut is mapped to a finite loop, i.e., a pseudo-caustic. The ESL outside lens equation is ingeneral, as can be found in the Appendix, ω = z − (cid:90) ˜ σ ( t ) tdt (¯ z − c t ) / (17)where c is the focal length of the mass boundary ellipse. At z = 0 ± i DAngle = (cid:90) ± ic ˜ σ ( t ) dt . (18)It is double-valued and finite. At the intersections of the (cid:15) -boundary of the branch cut andthe real line, z = ± ( aη + i ω = z − (cid:90) η ˜ σ ( t ) tdt (¯ z − c t ) / (19)where c is the focal length of the mass boundary ellipse and η is elliptic radial parameter of z . z = 0 ± i DAngle = 0 because the integral upper bound η = 0. Any other point z on the branch cut is also single-valued because it is always outsidethe branch cut connecting the branch points ± cη . Thus the ESL inside lens equation of anarbitrary bounded mass has no pseudo-caustic.The examples of the bounded ESL lenses discussed in the previous subsections havesharp cutoffs at t = 1. However, the class of bounded density functions include smoothfunctions. For example, a family of bounded smooth functions can be constructed as follows(Eremenko 2010). σ ( t ) ∝ e − α / (1 − t ) : t < t ≥
3. Summary and Discussion
Burke’s theorem invokes Poincar´e-Hopf index theorem of the vector field on the com-pactified complex plane as a sphere. The index of a loop is determined only by the zeros ofthe vector field enclosed in the loop. The behavior of Burke’s vector field is determined bythe behavior of the deflection angle where the latter is an integral of a mass density functionwith the kernel (¯ z − ¯ ξ ) − where z is the probing position and ξ is the position variable ofthe density function. If the vector field is continuous everywhere, then the index of the totalnumber of the images can be accounted for by drawing a large loop at infinity. The largeloop at infinity encloses one positive (unmagnified) image at infinity when the source is atinfinity, and hence the index of the large loop is 1 and Burke’s theorem follows.However, the deflection angle integral is not necessarily continuous. The best knownsingularity is that of a monopole or a monopole-equivalent (due to the Newton’s theorem),and it introduces a hole in the sphere of the complex plane. In other words, the underly-ing manifold is not compact, and strictly, the Poincar´e-Hopf index theorem does not apply.However, a pole respects the total parity invariance, and the hole due to the monopole sin-gularity only violates the odd-numberedness of the Burke’s theorem; the asymptotic number 10 –of the images is determined by the zero at infinity and the number of the poles due to themonopole lens positions; the total parity of the n -point lens is 1 − n .The cylindrically truncated isothermal lens is another example of the singularities andhas been known for causing a pseudo-caustic violating the total parity invariance. The massdensity singularity at the origin is responsible for the pseudo-caustic, and it has been arguedthat Burke’s theorem should hold if the mass density function is smooth.Here we have shown another type of singularity that is common in bounded ellipticallysymmetric lenses; the new type of singularity comes from a branch cut and a smooth ESLmass density function is not immune from it. In conclusion, a smooth mass density functionis not a guarantee for the validity of the Burke’s theorem.Morse theory studies (MacKenzie 1985; Petters 1992) had not turned up the branchcut singualrities that we have found in the bounded elliptically symmetric lenses that arephysically perfectly reasonable. It may be the time to reexamine the Morse theory in relationto gravitational lensing or null geodesics.Point masses are the foundation of the Galactic microlensing experiments includingsearches for microlensing planets. The point masses preserve the total parity invariance butdo not respect the odd number theorem because an odd number of point masses produces aneven number of images asymptotically. Here we found a line singularity of bounded ellipticmasses that is related to a branch cut and causes a pseudo-caustic. A pseudo-caustic violatesthe total parity invariance. A bounded elliptically symmetric mass is a reasonable model forgalaxy lenses or cluster lenses. Are there other types of singularities yet to be discovered ingravitational lensing?One can consider a smooth density function that is unbounded but asymptotically fallsreasonably fast to avoid a pseudo-caustic and an infinite mass. For example, a boundedmass density function that reasonably fits a real lens may be reasonably approximated bytruncating the large frequency modes of its Fourier integral. The truncated Fourier integralshould be infinitely differentiable and infinitely extended. Then does the real lens respector violate the invariance of the total parity? If we insist on the preservation of the totalparity invariance, we need to assume that the lens mass is infinitely extended. Howeversmall a density it might be at a large distance, we need to assume the elliptic symmetry.Is it sensible to do so? One practical approach may be to investigate the behavior of thepseudo-caustics of bounded masses that are close to or intersect with their caustics in theprocess of increasing the differentiability. It will be explored in a separate work. 11 – A. SIS or Cylindrically Truncated Isothermal Lens
If we consider a self-gravitating isothermal sphere of gas mass in thermal equilibrium,the algebraically known solution of the Lane-Emden equation, the so-called SIS (singularisothermal sphere), has the density profile that is proportional to the inverse radius square.The total mass is infinite because of the indefinite extension of the constant mass shells. Inpractice, the tidal interaction with the neighboring systems are likely to define the edges andmass breaks of the self-gravitating objects. If we truncate the SIS at a certain (3-d) radius,say r = a , and integrate for the projected mass of the lens, the 2-d mass density functionhas the form Σ( t ) ∝ t arctan (cid:18) (1 − t ) / t (cid:19) ; t ≤ ξ = ta (cos θ + i sin θ ) and the mass trucation is made at t = 1. In order to find the lens equation, wecan use the Newton’s theorem that the gravitational field of a circularly symmetric mass isdetermined by the mass inside the circle of the radius of the probing point placed at thecenter, and hence we need to integrate the density function in eq. (A1) from the center toan arbitrary t less than 1. The integration doesn’t relent to a nice manageable algebraicfunction. So we cheat a bit (maybe a lot) as is customary and ignore the arctangent factorto obtain an easy-to-handle density function Σ( t ) ∝ /t and truncate it at t = 1. In otherwords, the mass being considered is the infinite isothermal sphere cut into an infinite cylinderthat is infinitely long along the line of sight. Thus the (2-d) mass distribution may be bestreferred to as a cylindrically truncated isothermal lens (CTIL).The gravitational lens equation is given by ω = z − GD (cid:90) Σ( t )¯ z − ¯ ξ d ξ (A2) ω and z are variables for a source and its image in the lens plane at the distance of the lensand D is the reduced distance: 1 /D = 1 /D + D where D and D are the distances fromthe lens to the observer and the source. The area element d ξ = dξ dξ = ( − i ) − dξd ¯ ξ .If the mass of the lens is M , introduce the normalized density function σ ( t ) = Σ( t ) /M and renormalize the position variables by the Einstein ring radius of the mass M , R E =(4 GM D ) / , so that the unit distance of the lens plane is given by R E . The lens equation(A2) is rewritten as ω = z − (cid:90) σ ( t )¯ z − ¯ ξ d ξ . (A3)Since σ ( t ) depends only on t , we can manipulate the deflection angle integral as follows.If ∂D t denotes the circle of radius ta and D t is the circular disk inside the boundary circle 12 – ∂D t , then the area of the annulus D t + dt − D t is of the order of dt and σ ( t ) can be consideredconstant in the annulus. (cid:34)(cid:90) D t + dt − (cid:90) D t (cid:35) (cid:20) σ ( t ) d ξ (¯ z − ¯ ξ ) (cid:21) = σ ( t ) (cid:34)(cid:90) D t + dt − (cid:90) D t (cid:35) (cid:20) d ξ ¯ z − ¯ ξ (cid:21) = σ ( t ) dt ∂∂t (cid:20)(cid:90) D t d ξ ¯ z − ¯ ξ (cid:21) . (A4)Therefore the deflection angle DAngle is give by DAngle = (cid:90) dt σ ( t ) ∂∂t (cid:20)(cid:90) D t d ξ ¯ z − ¯ ξ (cid:21) . (A5)We can calculate the integral inside the square bracket in eq(A5), which we shall label D t Integral, using the Schwarz function of the circle and three facts are useful to know: theStoke’s theorem or Green’s theorem in two dimensions, the Cauchy integral of an analyticfunction, and the Schwarz function of the circle. In this case of the circular mass boundary,it is simpler to apply the Newton’s theorem, but we will develop the machinery that can bealso used for an elliptic mass boundary.The Stoke’s theorem is well known in arbitrary dimensions because of their physicalrelevance in electrodynamics, fluid dynamics, etc. See for example Jackson (1975). In twodimensions, (cid:90) ∂ Ω udx + vdy = (cid:90) Ω ( ∂ x v − ∂ y u ) dxdy , (A6)in which it is implicit that u and v are differentiable and the differentials are continuous,and which can be rewritten in terms of the exterior differentiation d . (cid:90) ∂ Ω udx + vdy = (cid:90) Ω d ( udx + vdy ) (A7)(Sometimes d ∧ is used instead of d in order to make the exterior differentiation more explicit.)If u and v are x and y components of vector (cid:126)w , it can be written in the vector form. (cid:90) ∂ Ω (cid:126)w · d(cid:126)l = (cid:90) Ω ∇ × (cid:126)w dxdy . (A8)The real space Stoke’s theorem can be readily translated into the complex form in thecomplex plane using eq.(A7), and it states (Gong and Gong 2007) that if w = w dz + w d ¯ z is an exterior differential one form in a domain Ω and ∂ Ω is the boundary of Ω, where w ( z, ¯ z )and w ( z, ¯ z ) are differentiable once and the differentials are continuous, then (cid:90) ∂ Ω w = (cid:90) Ω dw (A9) 13 –(The orientation of the boundary curve ∂ Ω is assumed to be counterclockwise as usual. Ifone walks along the boundary, the left hand points to the domain Ω. Walking along a loopwith the right hand in is the negtaive of the left hand in.)The Cauchy integral of f ( ξ ) is defined as (Titchmarsh 1991)12 πi (cid:90) Γ f ( ξ ) ξ − z dξ (A10)where f ( ξ ) is analytic inside and on a simple closed curve Γ. The integral is equal to f ( z )if z is inside Γ and the equality is called Cauchy’s integral formula; it is equal to 0 if z isoutside Γ because the integrand is analytic (Cauchy’s theorem). The derivative of f ( z ) canbe expressed in terms of the integral of f ( ξ ) and a kernel. f (cid:48) ( z ) = 12 πi (cid:90) Γ f ( ξ )( ξ − z ) dξ (A11)The Schwarz reflection symmetry principle (Needham 2004) states that given a smoothenough arc in the complex plane, there is an analytic function (in a domain that includesthe arc) that maps the arc into its complex conjugate. The analytic function is called theSchwarz function of the arc and commonly denoted S ( ξ ). The complex conjugate S ( ξ )reflects the domain with respect to the arc. On the arc, ξ = S ( ξ ), or ¯ ξ = S ( ξ ). Here therelevance is that the Schwarz function of the circle is nothing but S ( ξ ) = t a /z becauseon the circle ξ ¯ ξ = t a where a position variable in the circular coordinate is written as ξ = ta (cos θ + i sin θ ).Now we are ready to integrate the deflection angle over the unit circular disk D . A.1. CTIL Outside Equation
If the probing point z is outside the mass disk, the integrand of the D t Integral does nothave a singularity and is analytic. The complex conjugate of the D t Integral is D t Integral = (cid:90) D t d ξz − ξ = 12 i (cid:90) D t d ¯ ξdξz − ξ = 12 i (cid:90) D t d (cid:18) ¯ ξdξz − ξ (cid:19) (A12)= 12 i (cid:90) ∂D t ¯ ξdξz − ξ = 12 i (cid:90) ∂D t t a dξξ ( z − ξ ) = πt a z (A13)where z is outside the boundary ∂D t . Therefore the deflection angle in eq.(A5) is DAngle = (cid:90) dtσ ( t ) ∂∂t (cid:18) πt a ¯ z (cid:19) = (cid:90) πa σ ( t ) tdt ¯ z = (cid:90) ˜ σ ( t ) tdt ¯ z (A14) 14 –where 1 = (cid:90) ˜ σ ( t ) t dt , (A15)and hence the outside lens equation is given by the monopole lens equation. ω = z − z . (A16)The outside lens equation of a circularly symmetric lens is independent of the mass densityfunction. A.2. CTIL Inside Equation
If the probing point z is inside the mass disk D , we need to mind the singularity at¯ ξ = ¯ z in D t Integral. If we cut out from the mass disk D a small disk of radius (cid:15) centered at z , D ( z, (cid:15) ), then we can apply the Green’s theorem to the D t Integral over D t − D ( z, (cid:15) ). (cid:90) D t − D ( z,(cid:15) ) d ξz − ξ = 12 i (cid:90) ∂D − ∂D ( z,(cid:15) ) ¯ ξdξz − ξ . (A17)Now the surface integral over D ( z, (cid:15) ) on the LHS vanishes, (cid:90) D ( z,(cid:15) ) d ξz − ξ = (cid:90) π (cid:90) (cid:15) rdrdθre iθ = (cid:15) (cid:90) π dθe iθ = 0 , (A18)hence the D t Integral is given by the RHS of eq.(A17). D t Integral = 12 i (cid:20)(cid:90) ∂D t − (cid:90) ∂D ( z,(cid:15) ) (cid:21) ¯ ξdξz − ξ . (A19)The first contour integral over ∂D t vanishes if z is inside D t because the integrand is analyticoutside the contour; if z is outside the contour, the integral picks up the contribution fromthe pole at ξ = 0 inside the contour or the pole ξ = z outside the contour which result inthe same value. 12 i (cid:90) ∂D t ¯ ξdξz − ξ = 12 i (cid:90) ∂D t t a dξξ ( z − ξ ) = π t a z : η ≥ t . (A20)where z = ηa (cos θ + i sin θ ). The second contour integral over ∂D ( z, (cid:15) ) can be calculatedusing the Schwarz function of the circle: ( ¯ ξ − ¯ z )( ξ − z ) = (cid:15) .12 i (cid:90) ∂D ( z,(cid:15) ) ¯ ξdξz − ξ = 12 i (cid:90) ∂D ( z,(cid:15) ) (cid:18) ¯ zz − ξ − (cid:15) ( ξ − z ) (cid:19) dξ = − π ¯ z : η < t (A21) 15 –The contribution comes from the first term only. The second term vanishes because thederivative of the analytic function (cid:15) (constant) is zero. Since the result is independent of t ,it does not contribute to the deflection angle integral. Thus the deflection angle at z insidethe mass disk is given as follows. DAngle = (cid:90) η ˜ σ ( t ) t dt ¯ z . (A22)For a CTIL, ˜ σ ( t ) = 1 /t , and the inside lens equation is given by ω = z − η ¯ z . (A23) B. Elliptically Symmetric Lenses
For galaxy lenses that have elliptic shapes and light distributions, one can consider aclass of mass distributions whose projected densities depend on only the elliptical radius.For example, Σ( t ) ∝ /t where the position variable ξ = t ( a sin θ + ib cos θ ). It is currentlyunknown if a singular elliptic mass distribution with infinite or finite extension is exactlyor closely related to a self-gravitating gas mass in thermal equilibrium. However one cansuspect that it might be the case at least for small ellipticities because of the existence ofthe circularly symmetric system SIS and the fact that an angular momentum tends to addoblateness to the system.Consider the family of ellipses parameterized by t ( ≥
0) that fills the two-dimensionalspace. ξ = t ( a cos θ + ib sin θ ) , a ≥ b (B1)The curve t = constant is an ellipse that is converted to the familiar equation for the ellipsein real coordinates ξ and ξ where ξ = ξ + iξ . ξ a + ξ b = t , t ≥ t = 1, and the deflection angle of anelliptically symmetric lens (ESL) is given as follows similarly to eq.(A5). DAngle = (cid:90) dtσ ( t ) ∂∂t (cid:20)(cid:90) Ω t d ξ ¯ z − ¯ ξ (cid:21) (B3)where Ω t is the ellipse with the elliptic radial parameter t .In order to calculate the integral in the square bracket of eq.(B3), which we labelΩ t Integral, we need to calculate the Schwarz function of the ellipse. Note that an analytic 16 –function is uniquely determined in a domain if it is known in an arc in the domain. Thus itsuffices to find the Schwarz function on the ellipse where it is satisfied that ¯ z = S ( z ). Sincewe know the equation of the ellipse in real coordiates, eq.(B2), we only need to substitute x = ( z + ¯ z ) / y = ( z − ¯ z ) / (2 i ) and solve ¯ z in terms of z .¯ z = a + b c z ± abc (cid:0) z − c t (cid:1) / (B4)where c = a − b and z = ± ct are the focii of the ellipse. The square root function isintrinsically a double-valued function and here one branch is chosen so that it is a univalentanalytic function (except at the branch points z = ± ct ). If z ± ct = r ± e iθ ± , we choose( z − c t ) / = ( r + r − ) / e i ( θ + + θ − ) / . Then the Schwarz function of the ellipse is given by S ( z ) = a + b c z − abc (cid:0) z − c t (cid:1) / . (B5)The other branch is not a solution because the function does not return ¯ z . (The readers areencouraged to check numerically.)The Schwarz funciton can be decomposed into two parts: S ( z ) = S ( z ) + S ( z ) where S ( z ) = a − ba + b z ; S ( z ) = 2 abc (cid:0) z − ( z − c t ) / (cid:1) . (B6) S ( z ) diverges at infinity and S ( z ) is singular (branch points) at the focii. Thus S ( z ) isanalytic inside the ellipse ∂ Ω t and S ( z ) is analytic outside ∂ Ω t . (The decomposition is notessential for the integration even though one can say it is convenient. Handling S ( z ) directlywithout the decomposition by manipulating the pole at z = ∞ by an inverse transformationis also instructive.)Be reminded from the previous section that the singularity at ξ = z when the probingpoint z is inside the mass disk D t does not contribute to the deflection angle. Thus, theonly difference between the outside lens equation and inside lens equation is the range ofthe integration in t in eq.(B3). Now assuming that η > t , the Ω t Integral can be calculatedusing the machineries of Stoke’s theorem, Cauchy integral, and the Schwarz function of theellipse. (cid:90) Ω t d ξz − ξ = 12 i (cid:90) Ω t d ¯ ξdξz − ξ = 12 i (cid:90) Ω t d (cid:18) ¯ ξdξz − ξ (cid:19) = 12 i (cid:90) ∂ Ω t ¯ ξdξz − ξ = 12 i (cid:90) ∂ Ω t S ( ξ ) + S ( ξ ) z − ξ dξ = πS ( z ) = 2 πabc ( z − ( z − c t ) / )Thus the ESL outside lens equation is ω = z − (cid:90) ˜ σ ( t ) tdt (¯ z − c t ) / , (B7) 17 –and the ESL inside lens equation is ω = z − (cid:90) η ˜ σ ( t ) tdt (¯ z − c t ) / (B8)where c is the focal length of the mass boundary ellipse and η is elliptic radial parameter of z . B.1. Inverse Elliptic Radial Density: ˜ σ ( t ) = 1 /t If the mass density is proportional to the inverse of the elliptic radius, ˜ σ ( t ) = 1 /t , thenthe outside and inside lens equations are ω = z − c arcsin (cid:16) c ¯ z (cid:17) : z is outside the lens mass (B9)and ω = z − c arcsin (cid:16) cη ¯ z (cid:17) : z is inside the lens mass , z (cid:54) = 0 (B10)where η is the elliptic radius of z : z = η ( a cos θ + ib sin θ ). We call the ESL lens with˜ σ ( t ) = 1 /t an arcsin lens named after the arcsin functions of the deflection angles. B.2. Constant Density: ˜ σ ( t ) = 2If the projected mass density is constant, ˜ σ ( t ) = 2. The outside equation is ω = z − c (¯ z − (¯ z − c ) / ) , (B11)and the inside equation is ω = z − c (¯ z − (¯ z − c η ) / ) = z − ab (cid:18) z − a − ba + b ¯ z (cid:19) (B12)where the last equality holds because ¯ z = S ( z ) on the η -ellipse. C. Points and Sticks and Disks
A set of lens masses that are either spherically symmetric or ellliptically symmetric canbe represented by a set of points and sticks for the images outside the lensing masses. If we 18 –normalize the lens equation by the total mass, then the normalized outside lens equation isgiven by ω = z − Σ j (cid:15) cj ¯ z − ¯ x cj − Σ k (cid:90) (cid:15) ek ˜ σ ( t ) tdt ((¯ z − ¯ x ek ) − e i θ k c k t ) / ; (C1)Σ j (cid:15) cj + Σ k (cid:15) ek = 1 . (C2) (cid:15) cj and x cj are the fractional mass and position of the center of mass of the j -th point(circularly symmetric mass); (cid:15) ek , x ek , c k and θ k are the fractional mass, position of thecenter of mass, focal length and direction angle of the k -th stick (elliptically symmetricmass). The focal points of the k -th elliptic mass are at z = x ek ± e iθ k c k .For an image inside the disk of a circularly symmetric lens j , the lens equation isobtained by replacing the j -th point mass lens deflection angle in eq.(C1) by its insidedeflection angle. (cid:15) cj ¯ z − ¯ x cj ⇒ (cid:15) cj η [ z − x cj ]¯ z − ¯ x cj (C3)where η [ z − x cj ] denotes the elliptic radial parameter of z − x cj .For an image inside the disk of an elliptically symmetric lens k , the lens equation isobtained by replacing the k -th elliptical mass lens deflection angle in eq.(C1) by its insidedeflection angle by changing the upper bound of the integral from 1 to η [ z − c ek ].We thank Dmitri Khavinson for discussions, and Alexandre Eremenko for a criticalcomment that helps to delineate our finding. We appreciate the Rhie family for their financialsupport. REFERENCES
Bergweiler, W., and Eremenko, A. 2010, Comput. Methods Funct. Theory, 10, 303.Bourassa, R.R., and Kantowski, R. 1975, ApJ, 195, 13.Burke, W.L. 1981, ApJ, 244, 1.Eremenko, A. 2010, Private Communication.Evans, N.W., and Wilkinson, M.I., 1998, MNRAS, 296, 800.Fassnacht, C.D., Keeton, C.R., and Khavinson, D. 2009, Gravitational lensing by ellipticalgalaxies, and the Schwarz function, Analysis and Mathematical Physics, Birkh¨auserVerlag. 19 –Gong, S. and Gong, Y., Concise Complex Analysis. World Scientific Publishing, 2007.Jackson, D., Electrodynamics, John Wiley and Sons, 1975.Khavinson, D., and Lundberg, E. 2009, Transcendental harmonic mappings and gravitationallensing by isothermal galaxies, arXiv:0908.3310 [math-ph].Kormann, R., Schneider, P., and Bartelmann, M., A&A, 284, 285.Kovner, I. 1987, ApJ, 312, 22.MacKenzie, R.H. 1985, J. Math. Phys. 26, 1592.Needham, T., Visual Complex Analysis, Oxford University Press, 2004.Petters, A.O. 1992, J. Math. Phys., 33, 1915.Rhie, S. 1997, ApJ, 484, 63.Rhie, S. 2001, Can a gravitational quadruple lens produce 17 images?, arXive:astro-ph/0103463.Rhie, S., and Bennett, C.S. 2009, Perturbation of Gravitational Lenses, submitted to MN-RAS, arXiv:0911.3050[astro-ph.GA].Schneider, P., Ehlers, J., and Falco, E. 1992, Gravitational Lensing, Springer, New York.Titchmarsh, E.C., The Theory of Functions, 2nd ed., Oxford University Press, 1991.
This preprint was prepared with the AAS L A TEX macros v5.2.
20 – -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-1.5-1-0.50.511.5 a=0.8c=0.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-1.5-1-0.50.511.5 a=1.55c=1.5
Fig. 1.— There are two morphologies of the caustic domains of the Arcsin lens insideequation: 1) the caustic is enclosed inside the pseudo-caustic: the images are of 1 /
0, 1 / / m and n of m/n are the numbers of positive and negative images. 2) thecaustic and pseudo-caustic intersect: the images are of 1/0, 1/1, 2/1, and 2/2.The causticsare cuspy and the pseudo-caustics are smooth. The numbers in the caustic domains indicatethe total number of images of each caustic domain. The total parity is 0 or1 for both cases. 21 – -2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4-1.6-1.2-0.8-0.40.40.81.21.6 Arcsine lens c=1.65red: (open) causticblue: pseudo-caustic1/0 1/1 2/1 3/1 2/02/1 + Fig. 2.— The caustic domains of the arcsine lens outside equation with c = 1 .
65. Thered is the (open) caustic and the blue is the pseudo-caustic. At the joining points, thecaustic and pseudo-caustic are tangential. m/n stands for ⊕⊕