We investigate the relativistic modeling of spacecraft motion in Mercury's post-Newtonian local coordinates. This investigation is motivated by the fact that Mercury's post-Newtonian gravitational field (as well as that of any other planet) admits an expansion in terms of multipole moments, which are most appropriately defined in the local reference system. The equations of motion in the Mercury-centric local frame include relativistic local perturbations, given by the Schwarzschild term, Lense-Thirring precession, and the acceleration due to the quadrupole moment, and relativistic third-body perturbations, which are the gravito-electric and gravito-magnetic accelerations, along with a coupling term between Mercury and other solar system bodies. The relativistic third-body perturbations are usually neglected in all practical applications. In this study, we analyze the magnitude of the post-Newtonian terms of the equations of motion formulated in the Mercury-centric frame, evaluating them along the trajectories of the two BepiColombo spacecrafts. Based on this analysis, we provide a practical approach for constructing a high-accuracy relativistic orbital model suitable for a Mercury orbiter.
We review the results from {\it Mariner 10} regarding Mercury's gravity field and the results from radar ranging regarding topography. We discuss the implications of improving these results, including a determination of the polar component, as well as the opportunity to perform relativistic gravity tests with a future {\it Mercury Orbiter}. With a spacecraft placed in orbit with periherm at 400 km altitude, apherm at 16,800 km, period 13.45 hr and latitude of periherm at +30 deg, one can expect a significant improvement in our knowledge of Mercury's gravity field and geophysical properties. Th…
Due to its relatively large eccentricity and proximity to the Sun, Mercury's orbital motion provides one of the best solar-system tests of relativistic gravity. We emphasize the number of feasible relativistic gravity tests that can be performed within the context of the parameterized weak field and slow motion approximation - a usefulframework for testing modern gravitational theories in the solar system. We discuss a new approximation method, which includes two Eddington parameters $(γ,β)$, proposed for construction of the relativistic equations of motion of extended bodies. Within the prese…
We present a new simple relativistic model for planetary motion describing accurately the anomalous precession of the perihelion of Mercury and its origin. The model is based on transforming Newton's classical equation for planetary motion from absolute to real spacetime influenced by the gravitational potential and introducing the concept of influenced direction.
We point out the existence of a new general relativistic contribution to the perihelion advance of Mercury that, while smaller than the contributions arising from the solar quadrupole moment and angular momentum, is 100 times larger than the second-post-Newtonian contribution. It arises in part from relativistic "cross-terms" in the post-Newtonian equations of motion between Mercury's interaction with the Sun and with the other planets, and in part from an interaction between Mercury's motion and the gravitomagnetic field of the moving planets. At a few parts in $10^6$ of the leading general r…
C. M. Will in a recent Letter [arXiv:1802.05304] predicts a new Mercury's precession rate which is $100$ times larger than the second-post-Newtonian contribution, our calculation is about $30$ times larger. Thus the new Mercury's precession rate is about $6.4\ast 10^{-8}$, instead of $6.2\ast 10^{-8}$, degrees per century.
The local curvature of the space produced by the Sun causes not only the perihelion precession of Mercury's elliptical orbit, but also the variations of the whole orbit, in comparison with those predicted by the Newtonian theory of gravitation. Calculations show that the gravitational major-axis contraction of the Mercury's elliptical orbit is 1.3 kilometers which can be confirmed by the present astronomical distance measurement technology.
It is attempted to obtain the masses of the celestial bodies, the initial conditions of their motion, and the constant of gravitation, by a global parameter optimization. First, a numerical solution of the N-bodies problem for mass points is described and its high accuracy is verified. The osculating elements are also accurately computed. This solution is implemented in the Gauss iterative algorithm for solving nonlinear least-squares problems. This algorithm is summarized and its efficiency for the inverse problem in celestial mechanics is checked on a 3-bodies problem. Then it is used to ass…
If the concepts underlying Effective Theory were appreciated from the earliest days of Newtonian gravity, Le Verrier's announcement in 1845 of the anomalous perihelion precession of Mercury would have been no surprise. Furthermore, the size of the effect could have been anticipated through "naturalness" arguments well before the definitive computation in General Relativity. Thus, we have an illustration of how Effective Theory concepts can guide us in extending our knowledge to "new physics", and not just in how to reduce larger theories to restricted (e.g., lower energy) domains.
Today, the motion of spacecraft is still described by the classical Newtonian equations of motion plus some relativistic corrections. This approach might become cumbersome due to the increasing precision required. We use the Relativistic Motion Integrator (RMI) approach to numerically integrate the native relativistic equations of motion for a spacecraft. The principle of RMI is presented. We compare the results obtained with the RMI method with those from the usual Newton plus correction approach for the orbit of the BepiColombo (around Mercury) and Mars-NEXT (around Mars) orbiters. Finally, …
Recently it has been shown that despite previous claims the cosmological constant affects light bending. In the present article we study light bending and the advance of Mercury's perihelion in the context of the Newtonian limit of Schwarzschild-de Sitter spacetime employing the special relativistic equivalence of mass and energy. In both cases, up to a constant factor, we find the same results as in the full general relativistic treatment of the same phenomena. These approximate and intuitive arguments demonstrate clearly what effects should have been expected from the presence of $Λ$ in the …
We study a general relativistic gravitomagnetic 3-body effect induced by the spin angular momentum ${\boldsymbol S}_\textrm{X}$ of a rotating mass $M_\textrm{X}$ orbited at distance $r_\textrm{X}$ by a local gravitationally bound restricted two-body system $\mathcal{S}$ of size $r\ll r_\textrm{X}$ consisting of a test particle revolving around a massive body $M$. At the lowest post-Newtonian order, we analytically work out the doubly averaged rates of change of the Keplerian orbital elements of the test particle by finding non-vanishing long-term effects for the inclination $I$, the node $Ω$ a…
The possibility of measuring the post-Newtonian gravitoelectric correction to the orbital period of a test particle freely orbiting a spherically symmetric mass in the Solar System is analyzed. It should be possible, in principle, to detect it for Mercury at a precision level of 10^-4. This level is mainly set by the unavoidable systematic errors due to the mismodelling in the Keplerian period which could not be reduced by accumulating a large number of orbital revolutions. Future missions like Messenger and BepiColombo should allow to improve it by increasing our knowledge of the Mercury's or…
We study the motion of a secondary celestial body under the influence of the logarithmic corrected gravitational force of a primary one. This kind of correction was introduced by Fabris et al. (2009). We derive two equations to compute the rate of change of the periastron w.r.t. the eccentric anomaly and its total variation over one revolution, In a kinematical sense, this influence produces an apsidal motion. We perform numerical estimations for Mercury and for the companion star of the pulsar PSR 1913+16. We also consider the case of the artificial Earth satellite GRACE-A, but the results pr…
We revisit the relativistic restricted two-body problem with spin employing a perturbation scheme based on Lie series. Starting from a post-Newtonian expansion of the field equations, we develop a first-order secular theory that reproduces well-known relativistic effects such as the precession of the pericentre and the Lense-Thirring and geodetic effects. Additionally, our theory takes into full account the complex interplay between the various relativistic effects, and provides a new explicit solution of the averaged equations of motion in terms of elliptic functions. Our analysis reveals the…
On November 23, 2021, the Einstein-Besso manuscript on the perihelion motion of Mercury will be auctioned at Christie's. Expected to fetch around $3M, it promises to be the most expensive scientific manuscript ever sold at auction. In this preprint, we present the parts of our forthcoming book, How Einstein Found His Field Equations. Sources and Interpretation (Springer, 2021) dealing with Einstein's attempts, in 1913 and in 1915, to account for the anomalous advance of Mercury's perihelion. In 1913, as documented in the Einstein-Besso manuscript, Einstein and his friend Michele Besso found th…
The ESA/JAXA BepiColombo mission, launched on 20 October 2018, is currently in cruise toward Mercury. The Mercury Orbiter Radio-science Experiment (MORE), one of the 16 experiments of the mission, will exploit range and range-rate measurements collected during superior solar conjunctions to better constrain the post-Newtonian parameter $γ$. The MORE radio tracking system is capable of establishing a 5-leg link in X- and Ka-band to obtain 2-way range-rate measurements with an accuracy of $0.01 mm/s^-1$ @ 60 s sampling time and 2-way range measurements at centimeter level after a few seconds of …
We numerically and analytically work out the first-order post-Newtonian (1pN) orbital effects induced on the semimajor axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $Ω$, the longitude of perihelion $\varpi$, and the mean longitude at epoch $ε$ of a test particle orbiting its primary, assumed static and spherically symmetric, by a distant massive third body X. For Mercury, the rates of change of the linear trends found are $\dot I_\mathrm{1pN}^\mathrm{X} = -4.3\,\mathrm{microarcseconds\,per\,century}\,\left(μ\mathrm{as\,cty}^{-1}\right)$, $\dotΩ_\mathr…
We recently suggested a nonlocal modification of Einstein's field equations in which Newton's constant $G$ was promoted to a covariant differential operator $G_Λ(\Box_g)$. The latter contains two independent contributions, which operate respectively in the infrared (IR) and ultraviolet (UV) energy regimes. In the light of the recent direct gravitational radiation measurements we aim to determine the UV-modified 1.5 post-Newtonian radiative quadrupole moment of a generic $n$-body system. We eventually use these initial results in the particular context of a binary system and observe that in the…
The direct long-term changes occurring in the orbital dynamics of a local gravitationally bound binary system $S$ due to the post-Newtonian tidal acceleration caused by an external massive source are investigated. A class of systems made of a test particle $m$ rapidly orbiting with orbital frequency $n_{\rm b}$ an astronomical body of mass $M$ which, in turn, slowly revolves around a distant object of mass $M^{'}$ with orbital frequency $n_{\rm b}^{'}\ll n_{\rm b}$ is considered. The characteristic frequencies of the non-Keplerian orbital variations of $m$ and of $M$ itself are assumed to be n…
Recently, Will calculated an additional contribution to the Mercury's precession of the longitude of perihelion $\varpi$ of the order of $\dot\varpi_\textrm{W}\simeq 0.22$ $\textrm{milliarcseconds per century}$ ($\textrm{mas cty}^{-1}$). It is partly a direct consequence of certain 1pN third-body accelerations entering the planetary equations of motion, and partly an indirect, mixed effect due to the simultaneous interplay of the standard 1pN pointlike acceleration of the primary with the Newtonian $N$-body acceleration, to the quadrupole order, in the analytical calculation of the secular per…
In this paper we first calculate the post-Newtonian gravitoelectric secular rate of the mean anomaly of a test particle freely orbiting a spherically symmetric central mass. Then, we propose a novel approach to suitably combine the presently available planetary ranging data to Mercury, Venus and Mars in order to determine, simultaneously and independently of each other, the Sun's quadrupole mass moment J_2 and the secular advances of the perihelion and the mean anomaly. This would also allow to obtain the PPN parameters gamma and beta independently. We propose to analyze the time series of thr…
On incorporating special relativity theory into an extended equivalence principle, post-Newtonian gravitational phenomena beyond that originally predicted by Einstein are predicted (required), such as geodetic and gravitomagnetic precessions of local inertial frames, and precession of Mercury's orbital perihelion. Why were not these phenomena predicted in the years 1907-1911? The unique 1/c^2 order dynamical equations for clock rates and motion of both bodies and light in local gravity are derived which guarantee fulfillment of the special relativistic equivalence principle.
HERO (Highly Eccentric Relativity Orbiter) is a space-based mission concept aimed to perform several tests of post-Newtonian gravity around the Earth with a preferably drag-free spacecraft moving along a highly elliptical path fixed in its plane undergoing a relatively fast secular precession. We considered two possible scenarios: a fast, 4-hr orbit with high perigee height of $1,047\,\mathrm{km}$, and a slow, 21-hr path with a low perigee height of $642\,\mathrm{km}$. HERO may detect, for the first time, the post-Newtonian orbital effects induced by the mass quadrupole moment $J_2$ of the Ear…
We analytically work out the orbital effects induced by the gravitomagnetic spin-octupole moment of an extended spheroidal rotating body endowed with angular momentum $\boldsymbol{S}$ and quadrupole mass moment $J_2$. Our results, proportional to $S J_2 c^{-2}$, hold for an arbitrary orientation of the body's symmetry axis $\boldsymbol{\hat{S}}$ and a generic orbital configuration of the test particle. Such effects may be measurable, in principle, with a dedicated spacecraft-based mission to Jupiter. For a moderately eccentric and fast path, the gravitomagnetic precessions of the node and the …
Here we explore the possibility of extending to the Solar System scenario the combined residual approach employed for the deterination of the Lense-Thirring effect in the Earth-LAGEOS satellites system. After calculating the secular gravitoelectric advance of the mean anomaly, we propose several linear combinations of the residuals of the nodes, the perihelia and the mean anomalies of some inner planets in order to measure the gravitoelectric advance, the Lense-Thirring effect and the solar quadrupole mass moment. The obtainable accuracies are comparable to those of new complex and expensive m…
The asymptotic method of post-Newtonian (PN) expansion for weak gravitational fields, recently developed, is compared with the standard method of PN expansion, in the particular case of a massive test particle moving along a geodesic line of a weak Schwarzschild field. First, the expression of the active mass in Schwarzschild's solution is given for a barotropic perfect fluid, both for general relativity (GR) and for an alternative, scalar theory. The principle of the asymptotic method is then recalled and the PN expansion of the active mass is derived. The PN correction to the active mass is …
The dynamical system studied in previous papers of this series, namely a bound time-like geodesic motion in the exact static and axially symmetric space-time of an (originally) Schwarzschild black hole surrounded by a thin disc or ring, is considered to test whether the often employed "pseudo-Newtonian" approach (resorting to Newtonian dynamics in gravitational potentials modified to mimic the black-hole field) can reproduce phase-space properties observed in the relativistic treatment. By plotting Poincaré surfaces of section and using two recurrence methods for similar situations as in the r…
Recently it has been suggested to use the perigee of the proposed LARES/WEBER-SAT satellite in order to measure the secular precession which would be induced on such Keplerian orbital element by a weak-field modification of gravity occurring in some Brane-World scenarios put forth by Dvali, Gabadadze and Porrati. This precession, derived for the first time by Lue and Starkman, amounts to 4\times 10-3 milliarcseconds per year for an Earth orbiting satellite. In this paper we show that, according to the recently released EIGEN-CG01C Earth gravity model the quite larger systematic errors due to t…
The 2PN Point-Mass N-Body Equations of Motion in Harmonic Gauge: A Computable Formulation
School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院）；Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University（兰州大学理论物理研究所与引力研究中心）；Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University（教育部量子理论与应用重点实验室，兰州大学）；Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University（兰州大学兰州理论物理中心与甘肃省理论物理重点实验室）；Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology (APM), Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院武汉物理与数学研究所）；International Centre for Theoretical Physics Asia-Pacific, University of Chinese Academy of Sciences（中国科学院大学亚太理论物理中心）
A two-body system hypothetically affected by an additional radial acceleration H v_r, where v_r is the radial velocity of the binary's proper orbital motion, would experience long-term temporal changes of both its semimajor axis a and the eccentricity e qualitatively different from any other standard competing effect for them. Contrary to what one might reasonably expect, the analytical expressions of such rates do not vanish in the limit M--> 0, where M is the mass of the primary, being independent of it. This is a general requirement that any potentially viable physical mechanism able to pro…
