Center of Light Curves for Whitney Fold and Cusp

where Mag(x;y) = 1/| det[Jacη](x)| is the magnification of the lensed image x of y. Suppose that the light source at y moves along a path y(t). Then the shifted center-of-light curve relative to the source’s trajectory is defined by Xcl(t) = xcl(y(t)) − y(t). It is an important fact that locally stable k-plane lensing maps η are generic and each is differentiably equivalent about every critical point to either a Whitney fold or cusp (Petters, Levine & Wambsganss ). Whitney folds and cusps can then be used to study the generic, qualitative, local behavior of Xcl. A Whitney fold is a mapping ηF : R 2 → R of the form ηF (u, v) = (u, v). Let (s1, s2) denote rectangular coordinates on the target space of ηF . The critical curve of ηF is the u-axis, while the caustic is the s1-axis. For a light source at position s = (s1, s2), we have η (s) = ∅ for s2 < 0 and η(s) = {(s1, √ s2), (s1,− √ s2)} for s2 ≥ 0. The lensed images have magnification Mag((s1,± √ s2); s) = 1/(2 √ s2). By (1), there is no center-of-light at s for s2 < 0, while for s2 ≥ 0 we have xcl(s) = (s1, 0). In other words, the center-of-light follows moves along the the critical curve (Fig. 1). Suppose that the source follows a straight line s(t) = t exp(θ0), where −∞ < t < ∞ and θ0 is fixed with 0 ≤ θ0 < 2π. Without loss of generality, assume that the s(t) is transverse to the caustic curve. Then Xcl(t) moves along the v axis, i.e., Xcl(t) = (0, s2(t)) for 0 ≤ t < ∞ and Xcl(t) has no values for −∞ < t < 0. A Whitney cusp (or pleat) is a mapping ηF : R 2 → R of the form ηF (u, v) = (u,−uv + v). The critical curve is a parabola, u = 3v, while the caustic is the cusped curve C(s1, s2) ≡ −(s1/3) + (s2/2) = 0 with the origin a positive cusp. Outside the caustic curve (i.e., the region determined by C(s1, s2) > 0), a light source at s = (s1, s2) has one lensed image, namely, (s1, v(s1, s2)), where v(s1, s2) = (s2/2 + √