# THE IMPORTANCE OF EINSTEIN RINGS1 C. S. KOCHANEK,2 C. R. KEETON,3 AND B. A. MCLEOD2 Received 2000 June 9; accepted 2000 September 27

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3 52 KOCHANEK, KEETON, & MCLEOD Vol. 547 FIG. 1.ÈIllustration of ring formation by an SIE lens. An ellipsoidal source (left gray scale) is lensed into an Einstein ring (right gray scale). The source plane is magniðed by a factor of 2.5 relative to the image plane. The tangential caustic (asteroid on left) and critical line (right) are superposed. The Einstein ring curve is found by looking for the peak brightness along radial spokes in the image plane. For example, the spoke in the illustration deðnes point A on the ring curve. The long line segment on the left is the projection of the spoke onto the source plane. Point A on the image plane corresponds to point on the source plane, where the projected spoke is tangential to the intensity contours of the source. The ring in the image plane projects into the four-lobed pattern on the source plane. Intensity maxima along the ring correspond to the center of the source. Intensity minima along the ring occur where the ring crosses the critical line (e.g., point B). The corresponding points on the source plane are where the asteroid caustic is tangential to the intensity contours. of the magniðcation tensor (eq. [2]), the criterion for the maximum becomes \ + f (u) Æ M~1 Æ dx u S dj. (6) Geometrically, we are Ðnding the point at which the tangent vector of the curve projected onto the source plane (M~1 Æ dx/dj) is perpendicular to the local gradient of the surface brightness [+ f (u)], as illustrated in Figure 1. While the structure u of S the lensed image is complicated, in many systems it is reasonable to assume that the structure of the source has some regularity and symmetry. We require a model for the surface brightness of the source over a very limited region about its center where almost all galaxies can be approximated by an ellipsoidal surface brightness proðle. We assume that the source has a surface brightness f (m2), which is a monotonically decreasing function, df S /dm2 \, of an ellipsoidal coordinate m2\*u Æ S Æ *u. The S source is centered at u, with *u \ u [ u, and its shape is described by the two-dimensional shape tensor S.4 A noncircular source is essential to explaining the observed rings. With these assumptions, the location of the ring 4 An ellipsoid is described by an axis ratio q \ 1 [ e \ 1 and major- axis position angle h. In the principal axis frame s the shape s tensor S is diagonal, with components s 1 and q~2. The shape tensor S \ R~1S R D is found by rotating the diagonal shape s tensor with the rotation matrix D R(h ), corresponding to the major-axis P.A. of the source h. s s depends only on the shape of the source and not on its radial structure. The position of the extremum is simply the solution of \ *u Æ S Æ M~1 Æ dx dj. (7) The ring curve traces a four (or two) lobed cloverleaf pattern when projected onto the source plane if there are four (or two) images of the center of the source (see Figure 1). These lobes touch the tangential caustic at their maximum ellipsoidal radius from the source center, and these cyclic variations in the ellipsoidal radius produce the brightness variations seen around the ring (see Fig. 3, 3.2). Two problems a ect using equation (7) to deðne the ring curve. First, this assumes that the image is a surface brightness map. In principal, f (x) \ B\f (x) D f (x), and the ring location determined from D the data I can be I distorted by the PSF. In practice, however, we Ðnd that the position of the ring curve is insensitive to the e ects of the PSF unless there is a strong Ñux gradient along the ring. We see this primarily when the ring contains much brighter point sources, and the problem can be mitigated by Ðtting and subtracting most of the Ñux from the point sources before measuring the ring position. Alternatively, we can use deconvolved images. Monte Carlo experiments with a model for the ring are necessary to determine where the ring curve is biased from the true ring curve by the e ects of the PSF, but the usual

4 No. 1, 21 IMPORTANCE OF EINSTEIN RINGS 53 C A C A2 D A1 B B PG B B A B FIG. 2.ÈCASTLES (Falco et al. 2) HST /NICMOS H-band images PG 1115]8 (top left), B168]656 (top right), and B1938]666 (bottom). We partially subtracted the bright point sources in PG 1115]8 and B168]656 and completely subtracted the lens galaxies for all three systems to better show the ring structure. signature is a ripple ÏÏ in the ring near bright point sources (as seen between the A1 and A2 images in Fig. 4). Second, we are assuming that the source is ellipsoidal, although the model could be generalized for more complicated source structures. Extinction in the lens galaxy will generally not be a problem, because enormous extinction gradients are needed to perturb the observed positions of infrared Einstein rings. We discuss extinction further for the B168]656 ring. We can illustrate the physics governing the shapes of Einstein rings using the limit of a singular isothermal ellipsoid (SIE) for the lens. The general analytic solution for the Einstein ring produced by an ellipsoidal source lensed by a singular isothermal galaxy in a misaligned external shear Ðeld is presented in the Appendix. We use coordinates (x \ r cos h, y \ r sin h) centered on the lens galaxy. The SIE (see 2.2) has a critical radius scale b, ellipticity e \ 1 [ q, and its major axis lies along the x-axis. We add l an external l shear Ðeld characterized by amplitude c and orientation h. The source is an ellipsoid with axis ratio q \ 1 [ e and c major-axis angle h located at a position s (o cos s h, o sin h ) from the lens s center. The tangential critical line of the model is r crit b \ 1 ] e l 2 cos 2h [ c cos 2(h [ h c ), (8)

5 54 KOCHANEK, KEETON, & MCLEOD Vol. 547 when expanded to Ðrst order in the shear and ellipticity. If we expand the solution for the Einstein ring to Ðrst order in the shear, lens ellipticity, source ellipticity, and source position (o/b D c D v in a four-image lens), the Einstein ring is located at l r Eb \ 1 ] o b cos (h [ h ) [ e l 6 cos 2h ] c cos 2(h [ h g ). (9) -1 1 FIG. 3.ÈRing brightness proðles as a function of the spoke position angle. We show log f (r(m), m) for PG 1115]8 (top), B168]656 (middle), and B1938]666 D (bottom) after suppressing the point sources in PG 1115]8 and B168]656. The (labeled) maxima constrain the host center, and the minima constrain where the critical line crosses the Einstein ring. The properties of the extrema were determined using parabolic Ðts to the local extrema of the curves. The two local minima in the proðle between the B and D images of B168]656 are an artifact due to noise, extinction, or lens galaxy subtraction errors. At this order, the average radius of the Einstein ring is the same as that of the tangential critical line. The quadrupole of the ring has a major axis orthogonal to that of the critical line if the shear and ellipticity are aligned, e \ or c \, but can be misaligned in the general solution l because of the di erent coefficients for e in equations (8) and (9). For an external shear, the ellipticity l of the ring is the same as that of the critical line, while for the ellipsoid the ring is much rounder than the critical line. The ring has a dipole moment when expanded about the center of the lens, such that the average ring position is the source position. At this order, the ring shape appears not to depend on the source shape. The higher order multipoles of the ring are important, as can be seen from the very nonellipsoidal shapes of many observed Einstein rings (see Fig. 2). The even multipoles of the ring are dominated by the shape of the lens potential, while the odd multipoles are dominated by the shape of the source (see eq. [A5]). Their large amplitudes are driven by the ellipticity of the source, because the ellipticity of the lens potential is small (c D e /3 D.1) even if the lens is Ñattened l (e D.5), while the ellipticity of the source is not (e D.5). l s For example, in a circular lens (e \, c \ ) the ring is l located at r Eb \ 1 ] o b FIG. 4.ÈPG 1115]8. The quasar images A1, A2, B, and C are marked by Ðlled symbols. The light black lines show the ring centroid and its uncertainties, and the gray triangles mark the Ñux minima in the ring. The gray dashed line shows the best-ðt model of the ring, and the heavy solid line shows the tangential critical line of the best-ðt model. The model was not constrained to Ðt the critical line crossings (see text). [See the electronic edition of the Journal for a color version of this Ðgure.] D ] C(2 [ e s ) cos (h [ h ) ] e s cos (2m [ h [ h ), (1) 2 [ e ] e cos 2(m [ h) s s which has only odd terms in its multipole expansion and converges slowly for Ñattened sources. The ring shape is a weak function of the source shape only if the potential is nearly round and the source is almost centered on the lens Maxima and Minima in the Brightness of the Ring The other easily measured quantities for an Einstein ring are the locations of the maxima and minima in the brightness along the ring. Figure 3 shows the brightness proðles as a function of the spoke azimuth for the three lenses. The brightness proðle is f (r(m), m) for a spoke at azimuth m and radius r(m) determined I from equation (7), and it has an extremum when L f \. For the ellipsoidal model, there are maxima at the m images I of the center of the host galaxy (*u \ ), and minima when the ring crosses a critical line and the magniðcation tensor is singular ( o M~1 o \ ). In fact, these are general properties of Einstein rings and do not depend on our assumption of ellipsoidal symmetry (see Blandford, Surpi, & Kundic 2). The extrema at the critical line crossings are created by having a merging image pair on the critical line. The two merging images are created from the same source region and have the same surface brightness as the source, so the surface brightness of the ring must be continuous across the critical line. Alternatively,

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