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&

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

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

Enhanced Polarized Emission from the One-Parsec-Scale Hotspot of 3C 84 as a Result of the Interaction with Clumpy Ambient Medium

We present Very Long Baseline Array polarimetric observations of the innermost jet of 3C$\sim$84 (NGC$\sim$1275) at 43$\sim$GHz. A significant polarized emission is detected at the hotspot of the innermost re-started jet, which is located $\sim$1 pc south from the radio core. While the previous report presented a hotspot at the southern end of the western limb, the hotspot location has been moved to the southern end of the eastern limb. Faraday rotation is detected within an entire bandwidth of the 43-GHz band. The measured rotation measure (RM) is at most (6.3$\pm$1.9)$\times10^{5}$$\sim$rad$\sim$m$^{-2}$ and might be slightly time variable on the timescale of a month by a factor of a few. Our measured RM and the RM previously reported by the CARMA and SMA observations cannot be consistently explained by the spherical accretion flow with a power-law profile. We propose that a clumpy/inhomogeneous ambient medium is responsible for the observed rotation measure. Using equipartition magnetic field, we derive the electron density of $2\times10^{4}$$\sim$cm$^{-3}$. Such an electron density is consistent with the cloud of narrow line emission region around the central engine. We also discuss the magnetic field configuration from black hole scale to pc scale and the origin of low polarization.Comment: 8 pages, 8 figures, accepted for publication in Ap

Surprising Evolution of the Parsec-scale Faraday Rotation Gradients in the Jet of the BL Lac Object B1803+784

Several multi-frequency polarization studies have shown the presence of systematic Faraday Rotation gradients across the parsec-scale jets of Active Galactic Nuclei (AGN), taken to be due to the systematic variation of the line-of-sight component of a helical magnetic (B) field across the jet. Other studies have confirmed the presence and sense of these gradients in several sources, thus providing evidence that these gradients persist over time and over large distances from the core. However, we find surprising new evidence for a reversal in the direction of the Faraday Rotation gradient across the jet of B1803+784, for which multi-frequency polarization observations are available at four epochs. At our three epochs and the epoch of Zavala & Taylor (2003), we observe transverse Rotation Measure (RM) gradients across the jet, consistent with the presence of a helical magnetic field wrapped around the jet. However, we also observe a "flip" in the direction of the gradient between June 2000 and August 2002. Although the origins of this phenomena are not entirely clear, possibly explanations include (i) the sense of rotation of the central supermassive black hole and accretion disc has remained the same, but the dominant magnetic pole facing the Earth has changed from North to South; (ii) a change in the direction of the azimuthal B field component as a result of torsional oscillations of the jet; and (iii) a change in the relative contributions to the observed rotation measures of the "inner" and "outer" helical fields in a magnetic-tower model. Although we cannot entirely rule out the possibility that the observed changes in the RM distribution are associated instead with changes in the thermal-electron distribution in the vicinity of the jet, we argue that this explanation is unlikely.Comment: 21 pages, 10 figures. Accepted for publication in MNRA

Choreographic solution to the general relativistic three-body problem

We revisit the three-body problem in the framework of general relativity. The Newtonian N-body problem admits choreographic solutions, where a solution is called choreographic if every massive particles move periodically in a single closed orbit. One is a stable figure-eight orbit for a three-body system, which was found first by Moore (1993) and re-discovered with its existence proof by Chenciner and Montgomery (2000). In general relativity, however, the periastron shift prohibits a binary system from orbiting in a single closed curve. Therefore, it is unclear whether general relativistic effects admit a choreographic solution such as the figure eight. We carefully examine general relativistic corrections to initial conditions so that an orbit for a three-body system can be closed and a figure eight. This solution is still choreographic. This illustration suggests that the general relativistic N-body problem also may admit a certain class of choreographic solutions.Comment: 10 pages, 4 figures, text improved, accepted for publication in PR