Orbital evolution of a planet on an inclined orbit interacting with a disc

We study the dynamics of a planet on an orbit inclined with respect to a disc. If the initial inclination of the orbit is larger than some critical value, the gravitational force exerted by the disc on the planet leads to a Kozai cycle in which the eccentricity of the orbit is pumped up to large values and oscillates with time in antiphase with the inclination. On the other hand, both the inclination and the eccentricity are damped by the frictional force that the planet is subject to when it crosses the disc. We show that, by maintaining either the inclination or the eccentricity at large values, the Kozai effect provides a way of delaying alignment with the disc and circularization of the orbit. We find the critical value to be characteristically as small as about 20 degrees. Typically, Neptune or lower mass planets would remain on inclined and eccentric orbits over the disc lifetime, whereas orbits of Jupiter or higher mass planets would align and circularize. This could play a significant role in planet formation scenarios.Comment: 28 pages, 8 figures, accepted for publication in MNRA

Direction of light propagation to order G^2 in static, spherically symmetric spacetimes: a new derivation

A procedure avoiding any integration of the null geodesic equations is used to derive the direction of light propagation in a three-parameter family of static, spherically symmetric spacetimes within the post-post-Minkowskian approximation. Quasi-Cartesian isotropic coordinates adapted to the symmetries of spacetime are systematically used. It is found that the expression of the angle formed by two light rays as measured by a static observer staying at a given point is remarkably simple in these coordinates. The attention is mainly focused on the null geodesic paths that we call the "quasi-Minkowskian light rays". The vector-like functions characterizing the direction of propagation of such light rays at their points of emission and reception are firstly obtained in the generic case where these points are both located at a finite distance from the centre of symmetry. The direction of propagation of the quasi-Minkowskian light rays emitted at infinity is then straightforwardly deduced. An intrinsic definition of the gravitational deflection angle relative to a static observer located at a finite distance is proposed for these rays. The expression inferred from this definition extends the formula currently used in VLBI astrometry up to the second order in the gravitational constant G.Comment: 19 pages; revised introduction; added references for introduction; corrected typos; published in Class. Quantum Gra

Quantum phase shift and neutrino oscillations in a stationary, weak gravitational field

A new method based on Synge's world function is developed for determining within the WKB approximation the gravitationally induced quantum phase shift of a particle propagating in a stationary spacetime. This method avoids any calculation of geodesics. A detailed treatment is given for relativistic particles within the weak field, linear approximation of any metric theory. The method is applied to the calculation of the oscillation terms governing the interference of neutrinos considered as a superposition of two eigenstates having different masses. It is shown that the neutrino oscillations are not sensitive to the gravitomagnetic components of the metric as long as the spin contributions can be ignored. Explicit calculations are performed when the source of the field is a spherical, homogeneous body. A comparison is made with previous results obtained in Schwarzschild spacetime.Comment: 14 pages, no figure. Enlarged version; added references. In the Schwarzschild case, our results on the non-radial propagation are compared with the previous work

Time-Varying Gravitomagnetism

Time-varying gravitomagnetic fields are considered within the linear post-Newtonian approach to general relativity. A simple model is developed in which the gravitomagnetic field of a localized mass-energy current varies linearly with time. The implications of this temporal variation of the source for the precession of test gyroscopes and the motion of null rays are briefly discussed.Comment: 10 pages; v2: slightly expanded version accepted for publication in Class. Quantum Gra

Range, Doppler and astrometric observables computed from Time Transfer Functions: a survey

Determining range, Doppler and astrometric observables is of crucial interest for modelling and analyzing space observations. We recall how these observables can be computed when the travel time of a light ray is known as a function of the positions of the emitter and the receiver for a given instant of reception (or emission). For a long time, such a function--called a reception (or emission) time transfer function--has been almost exclusively calculated by integrating the null geodesic equations describing the light rays. However, other methods avoiding such an integration have been considerably developped in the last twelve years. We give a survey of the analytical results obtained with these new methods up to the third order in the gravitational constant $G$ for a mass monopole. We briefly discuss the case of quasi-conjunctions, where higher-order enhanced terms must be taken into account for correctly calculating the effects. We summarize the results obtained at the first order in $G$ when the multipole structure and the motion of an axisymmetric body is taken into account. We present some applications to on-going or future missions like Gaia and Juno. We give a short review of the recent works devoted to the numerical estimates of the time transfer functions and their derivatives.Comment: 6 pages, 2 figures, proceedings of the Conference "Journ\'ees 2014 Syst\`emes de r\'ef\'erence spatio-temporels (Recent developments and prospects in ground-based and space astrometry)", 22-24 September 2014, Pulkovo Observatory, Russi

A generalized lens equation for light deflection in weak gravitational fields

A generalized lens equation for weak gravitational fields in Schwarzschild metric and valid for finite distances of source and observer from the light deflecting body is suggested. The magnitude of neglected terms in the generalized lens equation is estimated to be smaller than or equal to 15 Pi/4 (m/d')^2, where m is the Schwarzschild radius of massive body and d' is Chandrasekhar's impact parameter. The main applications of this generalized lens equation are extreme astrometrical configurations, where 'Standard post-Newtonian approach' as well as 'Classical lens equation' cannot be applied. It is shown that in the appropriate limits the proposed lens equation yields the known post-Newtonian terms, 'enhanced' post-post-Newtonian terms and the Classical lens equation, thus provides a link between these both essential approaches for determining the light deflection.Comment: 11 pages, 3 figure

Relativistic theory for time and frequency transfer to order c^{-3}

This paper is motivated by the current development of several space missions (e.g. ACES on International Space Station) that will fly on Earth orbit laser cooled atomic clocks, providing a time-keeping accuracy of the order of 5~10^{-17} in fractional frequency. We show that to such accuracy, the theory of frequency transfer between Earth and Space must be extended from the currently known relativistic order 1/c^2 (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order 1/c^3. We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order 1/c^3, a mixture of these effects. As for the time transfer, it contains the standard Shapiro time delay, and we present an expression also including the first and second-order Sagnac corrections. Higher-order relativistic corrections, at least O(1/c^4), are numerically negligible for time and frequency transfers in these experiments, being for instance of order 10^{-20} in fractional frequency. Particular attention is paid to the problem of the frequency transfer in the two-way experimental configuration. In this case we find a simple theoretical expression which extends the previous formula (Vessot et al. 1980) to the next order 1/c^3. In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments.Comment: 11 pages, 2 figures, to appear in Astronomy & Astrophysic