Motion Of Mercury Research Articles

Instantons for the Destabilization of the Inner Solar System.

For rare events, path probabilities often concentrate close to a predictable path, called instanton. First developed in statistical physics and field theory, instantons are action minimizers in a path integral representation. For chaotic deterministic systems, where no such action is known, shall we expect path probabilities to concentrate close to an instanton? We address this question for the dynamics of the terrestrial bodies of the Solar System. It is known that the destabilization of the inner Solar System might occur with a low probability, within a few hundred million years, or billion years, through a resonance between the motions of Mercury and Jupiter perihelia. In a simple deterministic model of Mercury dynamics, we show that the first exit time of such a resonance can be computed. We predict the related instanton and demonstrate that path probabilities actually concentrate close to this instanton, for events which occur within a few hundred million years. We discuss the possible implications for the actual Solar System.

Precession of the Orbit of Mercury

The influence of the planets of the Solar System on the precession of the orbit of Mercury is investigated in the framework of classical mechanics. The influence of each planet on the motion of Mercury is calculated in the context of a restricted problem of three bodies: the Sun, the planet in question, and Mercury. It is demonstrated that the average shift of the perihelion of the orbit of Mercury determined in the context of a planar circular restricted three-body problem is 556.5″ per century and agrees with the observed shift (570″) with a relative accuracy of 2.5%. It is also demonstrated that the observed perihelion shift of Mercury features oscillatory components with a total amplitude up to 20″ and periods from several years to several decades. Owing to the presence of these components, the perihelion shift rate calculated based on observations conducted for several tens or even hundreds of years may differ considerably from the true average rate. This is likely the reason why the calculated average perihelion shift rate is not completely consistent with observational data.

Concerning the Paper by A. Einstein “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”

It is shown that in A. Einstein’s 1915 paper “Explanation of the perihelion motion of Mercury from the general theory of relativity” an error was actually incurred in the integration of the equation $$ \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$ , and in the result instead of $$ \phi \approx \uppi \left[1+\frac{5}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$ (if we limit ourselves to first-order terms in the small quantity (α1 + α2)) the value $$ \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$ was obtained, where α1 and α2 are the inverse values of the maximum and minimum distances of Mercury from the Sun, $$ \upalpha =\frac{2G{m}_0}{c^2} $$ is the gravitational radius, G is the gravitational constant, m0 is the mass of the Sun, c is the velocity of light (i.e., the Chinese mathematician Hua Di was actually right). And, as a result, for the precession of the orbit of Mercury in the gravitational field of a spherical Sun in the general theory of relativity (after 100 years) one obtains not ~43”, as A. Einstein obtained, but ~71.63”. Exactly this latter result is obtained by direct numerical modeling of the precession of the perihelion of Mercury’s orbit in the gravitational field of the Sun within the framework of the general theory of relativity if the fitting coefficient α in the equation of motion of Mercury (not to be confused with α in the above equations) is set equal to zero. The result $$ \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$ , obtained by A. Einstein if it is obtained, happens only upon integration of either the equation $$ \phi =\left[1+\frac{\upalpha}{2}\left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$ (if we also limit ourselves to first-order terms in the small quantity (α1 + α2)), i.e., in front of the integral in brackets the coefficient 1/2 should stand in front of the quantity α, or the equation $$ \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$ (if we also limit ourselves to first-order terms in the small quantity (α1 + α2)), i.e., in the denominator under the differential in the integral, inside the square root\ a plus sign should stand in front of the quantity αx.

Cassini's motions of the Moon and Mercury and possible excitations of free librations

On the basis of conditionally-periodic solutions of Hamiltonian systems at resonance of main frequencies Cassini's motions, their stability, Cassini's angle and periods of free librations of the Moon and Mercury have been recently studied and determined. The generalized formulations of Cassini's laws for the motion of the Moon and Mercury, that are considered as absolutely rigid non-spherical bodies, have been determined. The study of the second approximation equations of the desired quasi-periodic solutions in the case of the Moon allows us to determine the constant components of the first order for six Andoyer variables and the constant component of the second order for the angular velocity of the Moon. These effects are caused by the influence of the third harmonic of selenopotential. In this paper, these effects are described by analytical formulas, the dynamic and geometric interpretations are given, and a new interpretation of Mercury's motion under the generalized Cassini's laws has been proposed. Predictions of the existence of free librations of significant amplitude in the Mercury longitude, that are confirmed by the radar measurements data of the Mercury angular velocity, and in its pole motion in the body and in space have been made. The mechanism describing free librations of celestial bodies and their pole oscillations has been proposed due to the forced relative oscillations and wobble of the core-mantle system of celestial bodies (Moon, Mercury, Earth and other bodies in the solar system) under gravitational action of the external celestial bodies. The paper shows that the ascending node of equator of Mercury (and the intermediate plane orthogonal to the angular momentum) of epoch 2000.0 on the ecliptic does not coincide with the ascending node of orbital plane of Mercury on the same plane, and is ahead of it at an angle 23º4’. Angular momentum vector of the rotational motion of Mercury forms a constant angle ρG=4'1±1'1 with normal to the moveable plane of its orbit. The observed inclination of the angular velocityρω=2'1±0'1, can be considered as a possible evidence of a significant amplitude of the poles free motion of the Mercury rotation axis (c amplitude of about 2′- 4′).

New General Relativistic Contribution to Mercury’s Perihelion Advance

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 "crossterms" 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 relativistic precession of 42.98arcseconds per century, these effects are likely to be detectable by the BepiColombo mission to place and track two orbiters around Mercury, scheduled for launch around 2018.

Comparison and historical evolution of ancient Greek cosmological ideas and mathematical models

We present a comparative study of the cosmological ideas and mathematical models in ancient Greece. We show that the heliocentric system introduced by Aristarchus of Samos was the outcome of much intellectual activity. Many Greek philosophers, mathematicians and astronomers such as Anaximander, Philolaus, Hicetas, Ecphantus and Heraclides of Pontus contributed to this. Also, Ptolemy was influenced by the cosmological model of Heraclides of Pontus for the explanation of the apparent motions of Mercury and Venus. Apollonius, who wrote the definitive work on conic sections, introduced the theory of eccentric circles and implemented them together with epicycles instead of considering that the celestial bodies travel in elliptic orbits. This is due to the deeply rooted belief that the orbits of the celestial bodies were normal circular motions around the Earth, which was still. There was also a variety of important ideas which are relevant to modern science. We present the ideas of Plato that are consistent wi...

Dynamic structure and rotation of Mercury

Constant and periodic variations in the gravitational coefficients of the second harmonic of the gravitational potential of Mercury caused by the attraction of the Sun and its resonant rotation are studied on the basis of a planar model of translatory–rotary motion of Mercury as an elastic body on an elliptical orbit. The main variations in the coefficients C 20, C 22 and S 22 with periods that are a multiple of the orbit period have been obtained and evaluated for Mercury. Dynamic effects in the resonant rotation of Mercury considered as unchangeable non-spherical body (or as body with liquid core) are studied on the basis of two simple models: firstly, the plane motion on the unperturbed elliptical orbit; secondly, the rotation of Mercury on the precessing elliptical orbit. A few sets of possible values of Mercury's gravitational field parameters C 20 and C 22 are used in the paper for evaluations of the unperturbed, forced and resonant effects in Mercury's rotation (the Cassini positional parameters, a...

A Numerical Solution of the Inverse Problem in Classical Celestial Mechanics, with Application to Mercury's Motion

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 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 assess the accuracy to which a Newtonian calculation may reproduce the DE403 ephemeris that involves general-relativistic corrections. The parameter optimization allows one to reduce the norm and angular differences between the Newtonian calculation and DE403 by a factor 10 (Mercury, Pluto) to 100 (Venus). The maximum angular difference of the heliocentric positions of Mercury is ca. 220″ per century before the optimization, and ca. 20″ after it. The latter is still far above the observational accuracy. On the other hand, Mercury's longitude of the perihelion is not affected by the optimization: it keeps the linear advance of 43″ per century.

Mercury's sodium exosphere: Magnetospheric ion recycling

A three‐dimensional Monte Carlo model of Mercury's neutral sodium exosphere was used to describe correlation between the observed variations in the exospheric density and the variations in the surface concentration of sodium along Mercury's motion around the Sun [Leblanc and Johnson, 2003]. Four processes of ejection were taken into account: thermal and photon stimulated desorptions, solar wind sputtering, and micrometeoroid vaporization. Here a model of ion circulation in Mercury's magnetosphere developed by Delcourt et al. [2002, 2003] is used to examine the contribution from sodium magnetospheric ion recycling. The model is coupled to the exospheric model to track newly ionized sodium formed in Mercury's exosphere. The coupled Na and Na+ model is used to examine the sputtering by and implantation of the reimpacting ions. The magnetospheric ion sputtering does not contribute significantly to the total amount of Na atoms ejected into Mercury's exosphere because of its rather small flux; however, magnetospheric ion implantation in Mercury's surface is sufficiently concentrated inside narrow latitude bands to enhance the local surface concentration of sodium. This enhancement is seen to contribute to the peaks in Mercury's exospheric sodium emission observed at high latitudes when the implanted sodium is exposed to the solar flux near the dayside terminator.

Calculation of So‐Called General Relativistic Phenomena by Advancing Newton's Theory of Gravitation, Maintaining Classical Conceptions of Space and Relativity

With the example of the motion of Mercury around the Sun it is shown how Newton’s theory of gravitation should be advanced by taking into consideration the finite velocity of gravitational expansion and the present concept of transference of forces by particles to be able to calculate so-called general relativistic phenomena such as the additional motion of Mercury’s perihelion, the curvature of a light beam at the surface of the Sun, and the phenomena observed at the binary pulsar PSR 1913+16, maintaining classical conceptions of a Euclidean space and the Galilean principle of relativity.

Anomalous magnetic field of the sun at the beginning of cycle 23

Abstract Measurements of the mean magnetic field (MMF) of the Sun from 1968–1999 showed that (1) the Sun's magnetic field has a predominance of S-polarity, (2) it changes with periods 1.04, 1.60 and 23 yr, (3) the yearly-mean index of MMF energy reached the peak value in 1991, and after that (4) a significant decrease of MMF was observed. It is supposed that (a) the magnetic asymmetry of the Sun is a fundamental property of solar magnetism, (b) there are near-resonances between the MMF and orbital motions of Mercury, Venus and Earth which arose at early stages of formation of the Solar system, and (c) cycle 23 will display an anomalously low magnetic and sunspot activity.

One month in the history of the discovery of general relativity theory

Albert Einstein had initiated his work on the formulation of a general theory of relativity soon after completing the special theory. From 1907 to summer 1915 he worked hard on the problem of ‘generalizing’ the relativity theory. His efforts bore fruit during November 1915, and he presented the new theory in four communications to the Prussian Academy of Sciences in Berlin in that month; he obtained the correct expression and value for the advance of the perihelion motion of Mercury and motivated the existence of the final form of the field equations of gravitation. After his second communication to the Academy on 11 November, he had the expression and value for the advance of the perihelion of Mercury by 18 November, and the final form of the field equations by 25 November. He noted down both of these major results in an autograph manuscript, the English translation of which is presented here for the first time since Einstein wrote it, and the background of this fundamental work is discussed.

Imaging Mercury's surface within the context of the Mercury Orbiter study of 1994

Imaging Mercury's surface would be an integral part of any mission to planet Mercury as only 45% of Mercury's surface is known from the Mariner 10 fly-bys in 1974. The requirements for an imaging experiment were worked out within the context of the ESA M3 Mercury Orbiter assessment study. It is shown that the orbit of the spacecraft, the motions of Mercury and the limited data rate of a deep space mission require careful planning if complete imaging of the surface at as good as possible resolution is to be accomplished early in a mission.

Possible new properties of gravity

If static gravity or spacetime curvature information is carried by classical propagating particles or waves, a modern Laplace experiment places a lower limit on their speed of 1010 c. The so-called Lorentzian modification of special relativity permits such speeds without need of tachyons. But there are other consequences. If ordinary gravity is carried by particles with finite collision cross-section, such collisions would progressively diminish its inverse square character, converting to inverse linear behavior on the largest scales. At scales greater than several kiloparsecs gravity can apparently be modeled, without need for dark matter, by an inverse linear law. The orbital motions of Mercury and Earth may also show traces of this effect. If gravity were carried by particles, a mass between two bodies could partially shield each of them from the gravity of the other. Anomalies are seen in the motions of certain artificial Earth satellites during eclipse seasons that behave like shielding of the Sun's gravity. Certain types of radiation pressure might cause a similar behavior but require many free parameters. Particle-gravity models would change our understanding of gravitation and our views of the nature of time in relativity theory.

The motion of major planets from observations 1769?1988 and some astronomical constants

Modern planetary theories may be considered as a realisation of a four-dimensional dynamical reference frame. The existence of secular trends between the dynamical system and the adopted system of the Fundamental Catalogue (as well as between time scales involved) has been studied by discussing planetary observations of different types and by comparison with a numerical theory constructed for the time span 1769–1988. Parameters of the theory were fitted to radar ranging data for 1961–1988 for inner planets and to meridian observations of 18th–20th centuries for outer planets. Then a set of the inner planet optical observations, which includes USNO meridian observations, transits through the solar disk and occultations of fundamental stars are discussed. The main results are the following: 1. Radar data were used to estimate the time derivativeĠ of the gravitational constantG (in another interpretation, the secular trend between the atomic and dynamic time scales): $$\dot G/G = (0.37 \pm 0.45) \times 10^{ - 11} /y.$$ This estimation, being statistically insignificant, gives some physically meaningful restriction toĠ. 2. From the same data a new estimation of relativistic effects in the motion of Mercury was obtained, which has confirmed the Einstein value of the perihelion advance with the error 0″.06/cy. So in the frame of Einstein's theory the value of solar dynamic oblateness cannot be larger than 2×10−6. 3. The analysis of time behavior of residuals in the inner planet longitudes shows secular trends. It is demonstrated that these trends may be explained by combined action of a linear trenddT of Brouwer's time scale (which is adopted as a standard for reduction of observations before 1959) and the error in Newcomb's value of the constant of precession. From USNO meridian observations fordT the following estimate was obtained:dT=−14.5±2.1 sec/cy with the corresponding correction,dp, to Newcomb's precessiondp=0″.46±0″.13/cy. The estimate ofdT is in good agreement with the value ofdT determined from transits of Mercury and Venus through the solar diskdT=−12.9±1.3 sec/cy which does not depend on any precession error. 4. As a by-product, new accurate ephemerides of the outer planets are obtained over the time interval 1769–1988, the average residuals being presented.

A scalar Euclidean theory of gravitation: motion in a static spherically symmetric field

The motion of a particle in a static, spherically symmetric gravitational field is investigated in Euclidean space. The gravitational effects are described as due to a scalar field: To every point in space there is assigned a refractive index deciding the velocity of light in that point. The motion of light in the vacuum is described by the equation of classical optics. An equation of motion for material test particles is then derived by employing the usual Lagrangian formalism. The motion of the planets around the sun is explained, in particular the perihelion motion of Mercury. The present theory fully explains the four “classical” tests of general relativity in a mathematically far simpler way, and it can be equivalent to the Schwarzschild solution. It is also found that the effect of gravitation depends on the velocity of the particle, becoming repulsive for radial velocities larger thanc/\(\sqrt 3 \) (c is the velocity of light). This seemingly odd result can also be obtained from the equations of general relativity, as was shown by Cavalleri and Spinelli.

On the existence and geometry of invariant tori in the method of averaging

In this paper some results on the Method of Averaging are presented. We give new conditions for the existence of attractive invariant tori together with some information on their domain of attraction. The results are based on invariant manifold theory and error estimation. An application to the rotational motion of Mercury is given.

A note on the tidal expansion of the orbit of Mercury

Under the assumption that the secular decrease in the sidereal mean motion of Mercury is due to the Sun-Mercury tidal torque, the expansion of the orbit of Mercury and the tidal phase-lag angle of the tidal bulge are estimated.

The real value of Mercury's perihelion advance

The perihelion advance of the orbit of Mercury has long been one of the observational cornerstones of general relativity. After the effects of the gravitational perturbations of the other planets have been accounted for, the remaining advance fits accurately to predictions according to the theory of general relativity, that is, 43 arc s per 100 yr. In fact, radar determinations of Mercury's motion since the mid-1960s have produced agreement with general relativity at the level of 0.5%, or to ±0.2arcs per 100 yr. We have recently been puzzled by an apparent uncertainty in the value of the theoretical prediction for the advance within general relativity, as cited in various sources. The origin of this discrepancy is a 1947 article by Clemence1, who used unconventional values for the astronomical unit (A) and the speed of light (c) in his calculation of the predicted advance. Since that time, virtually everyone has followed suit. Although the current value of the prediction, using the best accepted values for the astronomical constants and for the orbital elements of Mercury, is 42.98 arc s per 100 yr, there is a preponderance of citations over the past 25 years of Clemence's value 43.03 arc s per 100 yr. Here we derive the accurate value and uncover the source of Clemence's value.

Electrodynamical origins of Einstein's theory of general relativity

The failure of the Newtonian theory of gravitation to satisfactorily account for the motion of Mercury's perihelion cannot be held to have justified the development of general relativity. This paper shows how the origins of general relativity were firmly embedded in contemporary attempts to introduce the new mechanics of special relativity into gravitational theory. These new theories of gravitation took as their basis the electrodynamical equations as formulated by Minkowski and attempted to represent the gravitational potential first by a vector and then by a scalar (in the four-dimensional sense). That Einstein chose the symmetric fundamental tensorg ij as his gravitational potential is seen to have been both a natural and necessary development. With this viewpoint the full theory of general relativity can be seen to be remarkably similar to those theories of gravitation that preceded it. The paper also contains a previously unpublished letter written by Einstein to H. A. Lorentz.