Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's "geometrical" method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.Comment: 12 pages; corrected typos, journal-ref adde

Lagrangian planetary equations in Schwarzschild space--time

We have developed a method to study the effects of a perturbation to the motion of a test point--like object in a Schwarzschild spacetime. Such a method is the extension of the Lagrangian planetary equations of classical celestial mechanics into the framework of the full theory of general relativity. The method provides a natural approach to account for relativistic effects in the unperturbed problem in an exact way.Comment: 7 pages; revtex; accepted for publication in Class. Quantum Gra

Constraining the orbit of the possible companion to Beta Pictoris: New deep imaging observations

We recently reported on the detection of a possible planetary-mass companion to Beta Pictoris at a projected separation of 8 AU from the star, using data taken in November 2003 with NaCo, the adaptive-optics system installed on the Very Large Telescope UT4. Eventhough no second epoch detection was available, there are strong arguments to favor a gravitationally bound companion rather than a background object. If confirmed and located at a physical separation of 8 AU, this young, hot (~1500 K), massive Jovian companion (~8 Mjup) would be the closest planet to its star ever imaged, could be formed via core-accretion, and could explain the main morphological and dynamical properties of the dust disk. Our goal was to return to Beta Pic five years later to obtain a second-epoch observation of the companion or, in case of a non-detection, constrain its orbit. Deep adaptive-optics L'-band direct images of Beta Pic and Ks-band Four-Quadrant-Phase-Mask (4QPM) coronagraphic images were recorded with NaCo in January and February 2009. We also use 4QPM data taken in November 2004. No point-like signal with the brightness of the companion candidate (apparent magnitudes L'=11.2 or Ks ~ 12.5) is detected at projected distances down to 6.5 AU from the star in the 2009 data. As expected, the non-detection does not allow to rule out a background object; however, we show that it is consistent with the orbital motion of a bound companion that got closer to the star since first observed in 2003 and that is just emerging from behind the star at the present epoch. We place strong constraints on the possible orbits of the companion and discuss future observing prospects.Comment: 8 pages, 8 figures, 1 table, accepted for publication in Astronomy and Astrophysic

Detecting local synchronization in coupled chaotic systems

We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally syncronized configurations, in a pair of signals with an arbitrary state of global syncronization. Application to a pair of interacting Rossler oscillators shows that our method is capable to quantify the number of dynamical configurations where a local prediction task is possible, also in absence of global synchronization features

Morphology of the very inclined debris disk around HD 32297

Direct imaging of circumstellar disks at high angular resolution is mandatory to provide morphological information that bring constraints on their properties, in particular the spatial distribution of dust. New techniques combining observing strategy and data processing now allow very high contrast imaging with 8-m class ground-based telescopes (10^-4 to 10^-5 at ~1") and complement space telescopes while improving angular resolution at near infrared wavelengths. We carried out a program at the VLT with NACO to image known debris disks with higher angular resolution in the near IR than ever before in order to study morphological properties and ultimately to detect signpost of planets. The observing method makes use of advanced techniques: Adaptive Optics, Coronagraphy and Differential Imaging, a combination designed to directly image exoplanets with the upcoming generation of "planet finders" like GPI (Gemini Planet Imager) and SPHERE (Spectro-Polarimetric High contrast Exoplanet REsearch). Applied to extended objects like circumstellar disks, the method is still successful but produces significant biases in terms of photometry and morphology. We developed a new model-matching procedure to correct for these biases and hence to bring constraints on the morphology of debris disks. From our program, we present new images of the disk around the star HD 32297 obtained in the H (1.6mic) and Ks (2.2mic) bands with an unprecedented angular resolution (~65 mas). The images show an inclined thin disk detected at separations larger than 0.5-0.6". The modeling stage confirms a very high inclination (i=88{\deg}) and the presence of an inner cavity inside r_0~110AU. We also found that the spine (line of maximum intensity along the midplane) of the disk is curved and we attributed this feature to a large anisotropic scattering factor (g~0.5, valid for an non-edge on disk). Abridged ...Comment: 12 pages, 10 figures, accepted for publication in Astronomy and Astrophysic

Sparse aperture masking at the VLT II. Detection limits for the eight debris disks stars β\beta Pic, AU Mic, 49 Cet, η\eta Tel, Fomalhaut, g Lup, HD181327 and HR8799

Context. The formation of planetary systems is a common, yet complex mechanism. Numerous stars have been identified to possess a debris disk, a proto-planetary disk or a planetary system. The understanding of such formation process requires the study of debris disks. These targets are substantial and particularly suitable for optical and infrared observations. Sparse Aperture masking (SAM) is a high angular resolution technique strongly contributing to probe the region from 30 to 200 mas around the stars. This area is usually unreachable with classical imaging, and the technique also remains highly competitive compared to vortex coronagraphy. Aims. We aim to study debris disks with aperture masking to probe the close environment of the stars. Our goal is either to find low mass companions, or to set detection limits. Methods. We observed eight stars presenting debris disks ( $\beta$ Pictoris, AU Microscopii, 49 Ceti, $\eta$ Telescopii, Fomalhaut, g Lupi, HD181327 and HR8799) with SAM technique on the NaCo instrument at the VLT. Results. No close companions were detected using closure phase information under 0.5 of separation from the parent stars. We obtained magnitude detection limits that we converted to Jupiter masses detection limits using theoretical isochrones from evolutionary models. Conclusions. We derived upper mass limits on the presence of companions in the area of few times the diffraction limit of the telescope around each target star.Comment: 7 pages, All magnitude detection limits maps are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5