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&

Note on Inversion Formula to Determine Binary Elements by Astrometry

Simplified solutions to determine binary elements by astrometry were obtained in terms of elementary functions (Asada et al. 2004), and therefore require neither iterative nor numerical methods. In the framework of the simplified solution, this paper discusses the remaining two parameters of the time of periastron passage and the longitude of ascending node in order to complete the solution. We thus clarify a difference between the simplified solution and other analytical methods.Comment: 12 pages, 4 figures, 2 tables; accepted for publication in PAS

Observation of Gravitational Lensing in the Clumpy Universe

We discuss how inhomogeneities of the universe affect observations of the gravitational lensing; (1) the bending angle, (2) the lensing statistics and (3) the time delay. In order to take account of the inhomogeneities, the Dyer-Roeder distance is used, which includes a parameter representing the clumpiness of the matter along the line of sight. It is shown analytically that all three combinations of distances appearing in the above observations (1)-(3) are monotonic with respect to the clumpiness in general, for any given set of the density parameter, cosmological constant and redshifts of the lens and the source. Some implications of this result for the observation are presented; the clumpiness decreases both the bending angle and the lensing event rate, while it increases the time delay. We also discuss cosmological tests using the gravitational lensing in the clumpy universe.Comment: 12 pages, LaTeX(aaspp4); accepted for publication in The Astrophysical Journa

The Light-cone Effect on the Shapiro Time Delay

We investigate the light-cone effect on the Shapiro time delay. The extra time delay caused by Jupiter on the 8th of September 2002 can be measured by advanced VLBI (very long baseline interferometry). Our expression for the delay is in complete agreement with that of Kopeikin (2001), in which he argued that the excess time delay was due to the propagation of gravity. The present letter, however, shows that the excess comes from nothing but the propagation of light, namely the light-cone effect. To make a robust confirmation of general relativity by the coming Jupiter event, it is important to take account of the light-cone effect on the Shapiro time delay.Comment: 5 pages; accepted for publication in ApJ Letter