Effects of the core radius of an isothermal ellipsoidal gravitational lens on the caustics and the critical curves

We study the effect of the core radius of an isothermal ellipsoidal gravitational lens on the caustics and the critical curves. We derive an analytic expression of the caustics for an isothermal ellipsoidal gravitational lens via a sixth-order algebraic equation. Since the expression is too long, by using another method we obtain a parametric representation of the critical curves in order to show analytically that there exist three cases: There are two curves for a small core radius, one for a quite large one, and no curves appear for an extremely large one, though the latter two cases are not realistic. The caustics are represented also by the same parameter.Comment: 4 pages; accepted for publication in A&

Effects of a deformation of a star on the gravitational lensing

We study analytically a gravitational lens due to a deformed star, which is modeled by using a monopole and a quadrupole moment. Positions of the images are discussed for a source on the principal axis. We present explicit expressions for the lens equation for this gravitational lens as a single real tenth-order algebraic equation. Furthermore, we compute an expression for the caustics as a discriminant for the polynomial. Another simple parametric representation of the caustics is also presented in a more tractable form. A simple expression for the critical curves is obtained to clarify a topological feature of the critical curves; the curves are simply connected if and only if the distortion is sufficiently large.Comment: 8 pages; accepted for publication in MNRA

Images for an Isothermal Ellipsoidal Gravitational Lens from a Single Real Algebraic Equation

We present explicit expressions for the lens equation for a cored isothermal ellipsoidal gravitational lens as a single real sixth-order algebraic equation in two approaches; 2-dimensional Cartesian coordinates and 3-dimensional polar ones. We find a condition for physical solutions which correspond to at most five images. For a singular isothermal ellipsoid, the sixth-order equation is reduced to fourth-order one for which analytic solutions are well-known. Furthermore, we derive analytic criteria for determining the number of images for the singular lens, which give us simple expressions for the caustics and critical curves. The present formulation offers a useful way for studying galaxy lenses frequently modeled as isothermal ellipsoids.Comment: 5 pages; accepted for publication in A&

On the Speed of Gravity and the v/cv/c Corrections to the Shapiro Time Delay

Using a relatively simple method, I compute the v/c correction to the gravitational time delay for light passing by a massive object moving with speed v. It turns out that the v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845 that was used to measure the speed of gravity.Comment: 8 pages, LaTeX (or Latex, etc), one figure, which is also available at http://www-theory.lbl.gov/~samuel/sog_figure.pdf; Revised version is the one to appear in Phys. Rev. Lett

Algebraic Properties of the Real Quintic Equation for a Binary Gravitational Lens

It has been recently shown that the lens equation for a binary gravitational lens, which is apparently a coupled system, can be reduced to a real fifth-order (quintic) algebraic equation. Some algebraic properties of the real quintic equation are revealed. We find that the number of images on each side of the separation axis is independent of the mass ratio and separation unless the source crosses the caustics. Furthermore, the discriminant of the quintic equation enables us to study changes in the number of solutions, namely in the number of images. It is shown that this discriminant can be factorized into two parts: One represents the condition that the lens equation can be reduced to a single quintic equation, while the other corresponds to the caustics.Comment: 7 pages (PTPTeX); accepted for publication in Prog. Theor. Phy

Properties of Planetary Caustics in Gravitational Microlensing

Although some of the properties of the caustics in planetary microlensing have been known, our understanding of them is mostly from scattered information based on numerical approaches. In this paper, we conduct a comprehensive and analytic analysis of the properties of the planetary caustics, which are one of the two sets of caustics in planetary microlensing, those located away from the central star. Under the perturbative approximation, we derive analytic expressions for the location, size, and shape of the planetary caustic as a function of the star-planet separation and the planet/star mass ratio. Based on these expressions combined with those for the central caustic, which is the other set of caustics located close to the central star, we compare the similarities and differences between the planetary and central caustics. We also present the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics. These analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signalsComment: total 6 pages, including 6 figures, ApJ, submitte

Simplified solution to determination of a binary orbit

We present a simplified solution to orbit determination of a binary system from astrometric observations. An exact solution was found by Asada, Akasaka and Kasai by assuming no observational errors. We extend the solution considering observational data. The generalized solution is expressed in terms of elementary functions, and therefore requires neither iterative nor numerical methods.Comment: 15 pages; text improved, Accepted for publication in the Astronomical Journa

Analysis of Microlensing Light Curves Induced by Multiple-Planet Systems

To maximize the number of planet detections by increasing efficiency, current microlensing follow-up observation experiments are focusing on high-magnification events to search for planet-induced perturbations near the peak of lensing light curves. It was known that by monitoring high-magnification events, it is possible to detect multiplicity signatures of planetary systems. However, it was believed that the interpretation of the signals and the characterization of the detected multiple-planet systems would be difficult due to the complexity of the magnification pattern in the central region combined with the large number of lensing parameters required to model multiple-planet systems. In this paper, we demonstrate that in many cases the central planetary perturbations induced by multiple planets can be well approximated by the superposition of the single planetary perturbations where the individual planet-primary pairs act as independent binary lens systems (binary superposition). The validity of the binary-superposition approximation implies that the analysis of perturbations induced by multiple planets can be greatly simplified because the anomalies produced by the individual planet components can be investigated separately by using relatively much simpler single-planetary analysis, and thus enables better characterization of these systems.Comment: Manuscript with high-resolution figures are available at http://astroph.chungbuk.ac.kr/~cheongho/preprint.htm

A Parametric Representation of Critical Curves and Caustics for a Binary Gravitational Lens

We find a simple expression for critical curves of a binary gravitational lens. On the basis of this, we present a parametric representation of such curves. The caustics can also be expressed with the same parameterization. The present result is helpful for efficiently constructing many templates of light curves due to binary systems, particularly extrasolar planets, which cause spikes in the light curves when a source crosses the caustics.Comment: 8 pages (PTPTeX); accepted for publication in Prog. Theor. Phy