New tools for determining the light travel time in static, spherically symmetric spacetimes beyond the order G2G^2

This paper is mainly devoted to the determination of the travel time of a photon as a function of the positions of the emitter and the receiver in a large class of static, spherically symmetric spacetimes. Such a function - often called time transfer function - is of crucial interest for testing metric theories of gravity in the solar system. Until very recently, this function was known only up to the second order in the Newtonian gravitational constant $G$ for a 3-parameter family of static, spherically symmetric metrics generalizing the Schwarzschild metric. We present here two procedures enabling to determine - at least in principle - the time transfer function at any order of approximation when the components of the metric are expressible in power series of the Schwarzschild radius of the central body divided by the radial coordinate. These procedures exclusively work for light rays which may be described as perturbations in power series in $G$ of a Minkowskian null geodesic passing through the positions of the emitter and the receiver. It is shown that the two methodologies lead to the same expression for the time transfer function up to the third order in $G$. The second procedure presents the advantage of exclusively needing elementary integrations which may be performed with any symbolic computer program whatever the order of approximation. The vector functions characterizing the direction of light propagation at the points of emission and reception are derived up to the third order in $G$. The relevance of the third order terms in the time transfer function is briefly discussed for some solar system experiments.Comment: 37 pages; published in "Frontiers in Relativistic Celestial Mechanics", vol. 2, ed. by S. M. Kopeikin, Series "De Gruyter Studies in Mathematical Physics 22", 2014. arXiv admin note: substantial text overlap with arXiv:1304.368

Time transfer functions in Schwarzschild-like metrics in the weak-field limit: A unified description of Shapiro and lensing effects

We present a complete analysis of the light rays within the linearized, weak-field approximation of a Schwarzschild-like metric describing the gravitational field of an isolated, spherically symmetric body. We prove in this context the existence of two time transfer functions and we obtain these functions in an exact closed-form. We are led to distinguish two regimes. In the first regime, the two time transfer functions correspond to rays which are confined in regions of spacetime where the weak-field approximation is valid. Such a regime occurs in gravitational lensing configurations with double images of a given source. We find the general expressions of the angular separation and the difference in light travel time between the two images. In the second regime, there exists only one time transfer function corresponding to a light ray remaining in a region of weak field. Performing a Taylor expansion of this function with respect to the gravitational constant, we obtain the Shapiro time delay completed by a series of so-called "enhanced terms". The enhanced terms beyond the third order are new.Comment: 12 pages, added one figure in section 3; a paragraph in Introduction rewritten without changing the argument; corrected typos; one reference added for section 2; Eq. (84) rewritten in a more elegant form; slightly revised argument in section 9, results unchange

Evolution of eccentricity and orbital inclination of migrating planets in 2:1 mean motion resonance

We determine, analytically and numerically, the conditions needed for a system of two migrating planets trapped in a 2:1 mean motion resonance to enter an inclination-type resonance. We provide an expression for the asymptotic equilibrium value that the eccentricity $e_{\rm i}$ of the inner planet reaches under the combined effects of migration and eccentricity damping. We also show that, for a ratio $q$ of inner to outer masses below unity, $e_{\rm i}$ has to pass through a value $e_{\rm i,res}$ of order 0.3 for the system to enter an inclination-type resonance. Numerically, we confirm that such a resonance may also be excited at another, larger, value $e_{\rm i, res} \simeq 0.6$, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of $e_{\rm i}$ is larger than $e_{\rm i,res}$. We find that, for $q \le 1$, the system cannot enter an inclination-type resonance if the ratio of eccentricity to semimajor axis damping timescales $t_e/t_a$ is smaller than 0.2. This result still holds if only the eccentricity of the outer planet is damped and $q \lesssim 1$. As the disc/planet interaction is characterized by $t_e/t_a \sim 10^{-2}$, we conclude that excitation of inclination through the type of resonance described here is very unlikely to happen in a system of two planets migrating in a disc.Comment: 22 pages, 10 figures, accepted for publication in MNRA

The Early Aurignacian in Central Europe and its Place in a European Perspective

This paper places the current research on the Aurignacian of the Upper and Middle Danube region in a broader European context. Technological and typological studies show that the Swabian Aurignacian, particularly as documented in the well-dated deposits from Geißenklösterle, closely resemble the assemblages of Peyrony'sAurignacian I. We use the term Early Aurignacian in this context to distinguish the well documented Swabian assemblages including Geißenklösterle, Hohle Fels, and Vogelherd from other early Upper Paleolithic cultural groups including the Proto-Aurignacian of southern Europe. Although the assemblage from Willendorf II,3 is very small, it also appears to belong to the Early Aurignacian. The early phases of the Aurignacian date to about 35,000 radiocarbon years ago and about 40,000 calendar years ago based on TL measurements. These dates indicate a great antiquity of the upper and middle Danubian Early Aurignacian, but similar radiocarbon ages are also known from the Early Aurignacian of the Aquitain region. Thus, for now, questions about the poly- or monocentric origin of the Aurignacian remain open. The available data, however, do not support the claims for an origin of the Aurignacian in the Balkans or other regions of Eastern Europe