A very famous ``test'' of the General Theory of Relativity (GTR) is the advance of Mercury's perihelion (and of other planets too). To be more precise, this is not a prediction of General Relativity, since the anomaly was known in the XIXth century, but no consistent explanation had been found yet at the time GTR was elaborated. Einstein came up with a solution to the problem in 1914.
In the case of Mercury, the closest planet to the Sun, the effect is more pronounced than for other planets, and observed from Earth; there is an advance of the perihelion of Mercury of about 5550~arc seconds per century (as/cy). Among these, about $5000$ are due to the equinox precession (the precise value is {$5025.645$}~as/cy) and about $500$ ({$531.54$}) to the influence of the external planets. The remaining, about $50$~as/cy ({$42.56$}), are not understood within Newtonian mechanics. Here, we revisit the problem in some detail for a presentation at the undergraduate level.
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 motion of Mercury using numerical methods in the framework of a model including only the non-relativistic Newtonian gravitational interactions of the solar system, 9 planets in translation (including Pluto) around the sun has been studied. Since the true trajectory of Mercury is an open, non-planar curve, we have paid special attention to the exact definition of the advance of Mercury's perihelion. For this purpose, we have introduced the notions of an extended and a geometrical perihelion. In addition, for each orbital period, a mean ellipse was fitted to the trajectory of Mercury. I have…
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…
In general relativity, the Mercury's orbit becomes approximately elliptical and the its perihelion has thus an additional advance. We demonstrate, meanwhile, that in comparison of those given by the Newton's theory of gravitation for the orbit of the Mercury, the circumference and the area are reduced by $40.39$ km and $2.35\ast10^{9}$ km$^{2}$, respectively, beside the major-axis contraction pointed out recently, and all are produced by the curved space within the Einstein's theory of gravitation. Since the resolution power of present astronomical distance measurement technology reaches $one$…
The Hermean average perihelion rate $\dotω^\mathrm{2PN}$, calculated to the second post-Newtonian (2PN) order with the Gauss perturbing equations and the osculating Keplerian orbital elements, ranges from $-18$ to $-4$ microarcseconds per century $\left(μ\mathrm{as\,cty}^{-1}\right)$, depending on the true anomaly at epoch $f_0$. It is the sum of four contributions: one of them is the direct consequence of the 2PN acceleration entering the equations of motion, while the other three are indirect effects of the 1PN component of the Sun's gravitational field. An evaluation of the merely formal un…
We find stringent upper bound on the momentum scale in noncommutative phase space of canonical type on the basis of studies of perihelion shift of the Mercury planet with taking into account features of description of motion of macroscopic body in the space with noncommutativity of coordinates and noncommutativity of momenta. Using results for precession of perihelion of the Mercury planet from ranging to the MESSENGER spacecraft we obtain upper bound for parameter of momentum noncommutativity $10^{-80}\textrm{kg}^2\textrm{m}^2/\textrm{s}^2$ which is many orders less than known in the literatu…
The predicted Lense-Thirring perihelion precession of Mercury induced by the Sun's angular momentum through its general relativistic gravitomagnetic field amounts to 2 milliarcseconds per century. It turned out to be compatible with the latest experimental determinations of the supplementary perihelion precession of Mercury with the INPOP15a ephemerides, whose accuracy level has nowadays reached the magnitude of the predicted relativistic effect itself thanks to the analysis of some years of tracking data of the MESSENGER spacecraft, which orbited Mercury from 2011 to 2015. A dedicated analysi…
An Exact solution of the Einstein-Maxwell field equations for a conformastatic metric with magnetized sources is study. In this context, effective potential are studied in order to understand the dynamics of the magnetic field in galaxies. We derive the equations of motion for neutral and charged particles in a spacetime background characterized by this class of solutions. In this particular case, we investigate the main physical properties of equatorial circular orbits and related effective potentials. In addition, we obtain an effective analytic expression for the perihelion advance of test …
Planetary ephemerides are a good tool for studying general relativity at the scale of our solar system. We present here new evaluations of advances of perihelia for Mercury and Saturn.
Mercury and frame-dragging in light of the MESSENGER flybys: conflict with general relativity, poor knowledge of the physical properties of the Sun, data reduction artifact, or still insufficient observations?
广义相对论利用日震学中太阳的角动量S = 190 x 10^39 kg m^2 s^-1，预测水星近日点经度的兰斯-瑟林岁差为-2.0毫角秒/世纪（在天体赤道参考系中计算）。在4-σ水平上，它与标准牛顿/爱因斯坦岁差的0.4 +/- 0.6毫角秒/世纪修正值不一致，只要后者完全归因于标架拖曳。补充岁差是最近用INPOP 10a星历表与一个长行星数据记录（1914-2010年）进行全球拟合确定的，其中包括2008-2009年从信使号航天器收集的3个数据点。INPOP 10a模型根本不包括太阳重力磁场，因此它的特征可能在数据简化过程中被部分删除。另一方面，Lense-Thirring进动可能在一定程度上被竞争进动所抵消，这些竞争进动是由太阳四极质量矩和PPN参数β进入Schwarzschild类1 PN进动的小错误建模引起的，两者都是在INPOP 10a中建模的。相反，水星本身的扁率对其近日点的影响可以忽略不计。最大的单个小行星和小行星环的错误模型行为也是如此。未来对目前正在进行的信使号任务的更多观测结果的分析将进一步阐明这一问题，如果得到证实，可能会挑战我们目前对目前公认的引力定律和/或太阳物理特性的看法。
By using the higher-order geodesic deviation equations for charged particles, we apply the method described by Kerner et.al. to calculate the perihelion advance and trajectory of charged test particles in the Riessner-Nordstrom spacetime. The effect of charge on the perihelion advance is studied and compared the results with those obtained earlier via the perturbation method. The advantage of this approximation method is to provide a way to calculate the perihelion advance and orbit of planets in the vicinity of massive and compact objects without considering Newtonian and post-Newtonian appro…
Two different ways of generalizing Einstein's general theory of relativity with a cosmological constant to Brans-Dicke type scalar-tensor theories are investigated in the linearized field approximation. In the first case a cosmological constant term is coupled to a scalar field linearly whereas in the second case an arbitrary potential plays the role of a variable cosmological term. We see that the former configuration leads to a massless scalar field whereas the latter leads to a massive scalar field. General solutions of these linearized field equations for both cases are obtained correspond…
The standard model particles can be gauged in an anomaly free way by three possible gauge symmetries namely ${L_e-L_μ}$, ${L_e-L_τ}$, and ${L_μ-L_τ}$. Of these, ${L_e-L_μ}$ and ${L_e-L_τ}$ forces can mediate between the Sun and the planets and change the perihelion precession of planetary orbits. It is well known that a deviation from the $1/r^2$ Newtonian force can give rise to a perihelion advancement in the planetary orbit, for instance, as in the well known case of Einstein's gravity which was tested from the observation of the perihelion advancement of the Mercury. We consider the long ra…
Solar System motions and the cosmological constant: a new approach
我们利用一些行星近日点的牛顿-爱因斯坦长期岁差的修正（水星，地球，火星，木星，土星），俄罗斯天文学家E. V. Pitjeva在全球范围内将近一个世纪的数据与EPM 2004星历表相匹配，为了检验在Schwarzschild-de Sitter（或Kottler）时空框架中由均匀宇宙学常数λ引起的近日点进动的表达式。我们比较这样一个额外的速率估计修正的行星近日点岁差，采取他们的比率为不同的行星对，而不是使用一个近日点在一个时间为每个行星分别，到目前为止，在文献中。答案是完全否定的，即使进一步将皮特杰瓦公布的近日点额外岁差估计的误差重新缩放10倍（土星甚至100倍）。我们的结论也适用于任何其他的度规扰动具有相同的依赖于空间坐标，因为那些引起的其他广义相对论宇宙学的情况下，许多修改后的引力模型。目前正在进行的和计划中的行星际航天器飞行任务应能提高我们对行星轨道的认识，允许更严格的限制。
We consider how a nearly massless scalar field conformally and disformally coupled to matter can affect the dynamics of gravitationally interacting bodies. We focus on the case of two interacting objects and we obtain the effective metric driving the dynamics of the two body system when reduced to one body in the centre of mass frame. We then concentrate on the case of a light particle in the scalar and gravitational fields generated by a heavy object and find the effects of the conformal and disformal couplings on the body's trajectory such as the advance of perihelion and the Shapiro time de…
First preliminary tests of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawa-like fifth force from planetary data
Advances in the measurement of the Lense-Thirring effect with planetary motions in the field of the Sun
E. V. Pitjeva通过使用EPM 2006星历表的动力学模型处理超过400，000个不同类型的行星观测数据，最近估计了对牛顿-爱因斯坦金星近日点进动的修正为每世纪-0.0004 +/- 0.0001弧秒。根据广义相对论的预测，金星近日点的Lense-Thirring岁差为每世纪-0.0003角秒。事实证明，无论是其他错误建模/未建模的标准牛顿/爱因斯坦效应，还是外来效应，假设，例如，解释先锋异常，可能导致了确定的额外岁差的金星轨道，因此，可以合理地归因于太阳的重力磁场，而不是在EPM 2006星历的例程建模。然而，必须指出的是，引用的错误是正式的，统计的一个;现实的不确定性可能更大。未来对内行星星历表的改进，包括信使号和金星快车号的跟踪数据，应该会进一步提高这种广义相对论测试的准确性和一致性，这也将有利于其他天文学家团队对近日点（和节点）的额外岁差的独立估计。
