The lens equation for a multiple lens system is a straight forward generalization of the single lens equation, eq. 3.4, but with more than one lens mass, we can no longer assume that the source, observer and lens system all lie in a single plane. However, as long as the distances to the lens and source (DL and DS) are much larger than the extent of the lens system, we can assume that the lens system resides a single distance, and define the Einstein radius of the total lens system mass using eq. 3.3. So, as before, we will rescale all the length variables with RE.
Because we can no longer define a source-lens-observer plane, we must now define the lens and source positions in the 2-dimensional "lens-plane" perpendicular to the line-of-sight and projected to the distance of the lens (or equivalently, we can just use angular variables for the positions of the source and lenses on the sky). The double-lens system was first solved using two real coordinates for the lens plane (Schneider & Weiss, 1986), but the algebra is much simpler if we describe the lens plane with complex coordinates following Bourassa et al. (1973) and Rhie (1997). The generalization of eq. 3.4 is where w and z are the complex positions of the source and image, respectively, and xi are the complex positions of the lens masses. The individual lens masses are represented by ei, which is the mass fraction of the ith lens mass, so that ^i ei = 1. The appearance of the complex conjugates in the denominator in the sum on the right side of eq. 3.9 is simply a reflection of the fact that the lens deflection is the in the direction from the source to the lens with a magnitude of the inverse of that distance. With real coordinates, we would express this as the vector difference of the positions divided by this vector squared, but with complex coordinates, we can divide through by this vector leaving only its complex conjugate in the denominator.
If we knew the position of the images, z, in eq. 3.9, then it would be trivial to solve for the position of the source. But this is the inverse of the problem that we will usually want to solve, which is to find the positions of the images based on a known position for the source. However, the solution of this "inverse" problem is the basis of the brute-force, ray-shooting method (Schneider & Weiss, 1987) for solving eq. 3.9. This method involves taking a large grid of points in the "image plane" and propagating them back to the source plane using eq. 3.9. This method has the advantage that it can handle very complicated lens mass distributions, but it is usually not the method of choice for the analysis of microlensing events.
The most successful method for calculating multi-lens microlensing light curves (Bennett & Rhie, 1996) involves solving eq. 3.9 for the positions of the point-source images and invoking the ray-shooting method only in the vicinity of images that are affected by finite-source size effects. For the majority of the light curve, the finite-source calculations are not needed, and we can use the point source magnification formula. This formula can be derived from the Jacobian determinant of the lens equation (and its complex conjugate):
Because eq. 3.10 gives the Jacobian determinant of the inverse mapping from the image plane to the source plane, the magnification of each image is given by
evaluated at the position of each image.
The solution of the lens equation, 3.9, is non-trivial. For the case of two lens masses, this equation can be embedded into a fifth order polynomial equation in z, which can be solved numerically. This equation has either 3 or 5 solutions (Rhie, 1997) that correspond to solutions of eq. 3.9, which means that a double lens system must have either 3 or 5 images depending on the configuration of the lens system and the location of the source. For the triple lens case (which is relevant for at least one planetary microlensing event), the lens equation can be embedded in a rather complicated tenth order polynomial that has 4, 6, 8, or 10 solutions that correspond to physical images (Rhie, 2002). This tenth order polynomial equation can be solved numerically, although it may require extended precision numerical calculations in order to avoid serious round-off errors (Bennett et al., in preparation). The case of 4 lens masses, has also been investigated (Rhie, 2001), but the lens equation has not been converted to a polynomial.
The most important feature of lensing by multiple masses occurs at the locations where J = 0. From eq. 3.12, this implies infinite magnification for a point source. (The magnification is always finite for the realistic case of a source of finite angular
size.) For a single lens, J = 0 only occurs at a single point in the source plane, the location of the lens mass, but for lens systems with more than one mass, there are a set of one or more closed curves with J = 0, known as critical curves. The source positions corresponding to the critical curves are obtained by applying the lens equation, 3.9, and they are referred to as caustic curves. When the source passes to the interior of a caustic curve, two new images are created, and it is these new images that have infinite magnification for the (unphysical) case of a point source. The shape of the light curve of a (point) source crossing a caustic has a characteristic form:
Vx Xc where Fc gives the amplitude of the caustic, Anc gives the magnification of the images that are not associated with the caustic and O(x) = 1 for x > 0 and O(x) = 0 otherwise. x is the distance perpendicular to the direction of the caustic curve, and xc is the location of the caustic curve. Eq. 3.13 is a good approximation to the magnification for a point source when the curvature of the caustic curve can be neglected. Note, that the singularity in eq. 3.13 is weak enough so that the integral of this formula will yield a finite magnification for a finite size source star.
Caustic crossings that follow the form of eq. 3.13 are often referred to as fold caustic crossings, and they have the feature that there is essentially no warning that the caustic crossing is imminent when the caustic curve is approached from the outside (i.e. x < xc). This is because the magnification pattern for a fold caustic extends only to the interior of the caustic since it involves the magnification of images that only exist inside the caustic curve. However, each caustic curve also has at least three sharp pointy features, known as cusps, and the magnification pattern extends outward from the cusps on a caustic curve. The magnification scales as the inverse of the distance to the cusp, just as in the single lens case, eq. 3.6.
The path of the source with respect to the caustic curves provides the basic characteristics of a multiple lens microlensing light curve. Multiple lens light curves frequently have features which match the expected A ~ 0(x)x-1/2 shape of a caustic crossing or the A ~ r-1 shape of a cusp approach. But, there are additional complications, as the strength of a caustic crossing (Fc in eq. 3.13) can vary and the angular size of the source star can sometimes be larger than the entire caustic curve for a planetary microlensing event.
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