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The current LAGEOS–LARES 2 experiment aims to accurately measure the general relativistic Lense–Thirring effect in the gravitomagnetic field of the spinning Earth generated by the latter’s angular momentum J . The key quantity to a priori analytically assess the overall systematic uncertainty is the ratio R J 2 of the sum of the classical precessions of the satellites’ nodes Ω induced by the Earth’s oblateness J 2 to the sum of their post-Newtonian counterparts. In principle , if the sum of the inclinations I of both satellites were exactly 180 ∘ , the semimajor axes a and the eccentricities e being identical , R J 2 would exactly vanish. Actually, it is not so by a large amount because of the departures of the real satellites’ orbital configurations from the ideal ones. Thus, J 2 impacts not only directly through its own uncertainty, but also indirectly through the errors in all the other physical and orbital parameters entering R J 2 . The consequences of this fact are examined in greater details than done so far in the literature. The Van Patten and Everitt’s proposal in 1976 of looking at the sum of the node precessions of two counter-orbiting spacecraft in (low-altitude) circular polar orbits is revamped rebranding it POLAr RElativity Satellites (POLARES). Regardless the specific type of satellite and tracking technologies that may be eventually adopted, it might be conceptually superior to the LAGEOS–LARES 2 one from the point of view of the orbital characteristics since, given the same semimajor axes and eccentricities of the existing laser-ranged cousins, its R J 2 is less sensitive to the impact of the deviations from its ideal orbital configuration.

To the first post-Newtonian order, if two test particles revolve in opposite directions about a massive, spinning body along two circular and equatorial orbits with the same radius, they take different times to return to the reference direction relative to which their motion is measured: it is the so-called gravitomagnetic clock effect. The satellite moving in the same sense of the rotation of the primary is slower, and experiences a retardation with respect to the case when the latter does not spin, while the one circling in the opposite sense of the rotation of the source is faster, and its orbital period is shorter than it would be in the static case. The resulting time difference due to the stationary gravitomagnetic field of the central spinning body is proportional to the angular momentum per unit mass of the latter through a numerical factor which so far has been found to be 4 π . A numerical integration of the equations of motion of a fictitious test particle moving along a circular path lying in the equatorial plane of a hypothetical rotating object by including the gravitomagnetic acceleration to the first post-Newtonian order shows that, actually, the gravitomagnetic corrections to the orbital periods are larger by a factor of 4 in both the prograde and retrograde cases. Such an outcome, which makes the proportionality coefficient of the gravitomagnetic difference in the orbital periods of the two counter-revolving orbiters equal to 16 π , confirms an analytical calculation recently published in the literature by the present author. It is an important result in view of the astrophysical implications of the gravitomagnetic clock effect around Kerr black holes.

We develop a general approach to analytically calculate the perturbations Δ δ τ p of the orbital component of the change δ τ p of the times of arrival of the pulses emitted by a binary pulsar p induced by the post-Keplerian accelerations due to the mass quadrupole Q 2 , and the post-Newtonian gravitoelectric (GE) and Lense–Thirring (LT) fields. We apply our results to the so-far still hypothetical scenario involving a pulsar orbiting the supermassive black hole in the galactic center at Sgr A ∗ . We also evaluate the gravitomagnetic and quadrupolar Shapiro-like propagation delays δ τ prop . By assuming the orbit of the existing main sequence star S2 and a time span as long as its orbital period P b , we obtain Δ δ τ p GE ≲ 10 3 s , Δ δ τ p LT ≲ 0.6 s , Δ δ τ p Q 2 ≲ 0.04 s . Faster P b = 5 years and more eccentric e = 0.97 orbits would imply net shifts per revolution as large as Δ δ τ p GE ≲ 10 Ms , Δ δ τ p LT ≲ 400 s , Δ δ τ p Q 2 ≲ 10 3 s , depending on the other orbital parameters and the initial epoch. For the propagation delays, we have δ τ prop LT ≲ 0.02 s , δ τ prop Q 2 ≲ 1 μ s . The results for the mass quadrupole and the Lense–Thirring field depend, among other things, on the spatial orientation of the spin axis of the Black Hole. The expected precision in pulsar timing in Sgr A ∗ is of the order of 100 μ s , or, perhaps, even 1–10  μ s . Our method is, in principle, neither limited just to some particular orbital configuration nor to the dynamical effects considered in the present study.

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 ϖ , 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 I ˙ 1 pN X = - 4.3 microarcseconds per century μ as cty - 1 , Ω ˙ 1 pN X = 18.2 μ as cty - 1 , ϖ ˙ 1 pN X = 30.4 μ as cty - 1 , ϵ ˙ 1 pN X = 271.4 μ as cty - 1 , respectively. Such values, which are due to the added actions of the other planets from Venus to Saturn, are essentially at the same level of, or larger by one order of magnitude than, the latest formal errors in the Hermean orbital precessions calculated with the EPM2017 ephemerides. The perihelion precession ϖ ˙ 1 pN X turns out to be smaller than some values recently appeared in the literature in view of a possible measurement with the ongoing BepiColombo mission. Linear combinations of the supplementary advances of the Keplerian orbital elements for several planets, if determined experimentally by the astronomers, could be set up in order to disentangle the 1pN N -body effects of interest from the competing larger precessions like those due to the Sun’s quadrupole moment J 2 and angular momentum S .

The distinction between the mean anomaly \\[{\\mathcal {M}}(t)\\] and the mean anomaly at epoch \\[\\eta \\], and the mean longitude l(t) and the mean longitude at epoch \\[\\epsilon \\] is clarified in the context of a their possible use in post-Keplerian tests of gravity, both Newtonian and post-Newtonian. In particular, the perturbations induced on \\[{\\mathcal {M}}(t),\\,\\eta ,\\,l(t),\\,\\epsilon \\] by the post-Newtonian Schwarzschild and Lense–Thirring fields, and the classical accelerations due to the atmospheric drag and the oblateness \\[J_2\\] of the central body are calculated for an arbitrary orbital configuration of the test particle and a generic orientation of the primary’s spin axis \\[\\varvec{{\\hat{S}}}\\]. They provide us with further observables which could be fruitfully used, e.g., in better characterizing astrophysical binary systems and in more accurate satellite-based tests around major bodies of the Solar System. Some erroneous claims by Ciufolini and Pavlis appeared in the literature are confuted. In particular, it is shown that there are no net perturbations of the Lense–Thirring acceleration on either the semimajor axis a and the mean motion \\[n_{\\mathrm{b}}\\]. Furthermore, the quadratic signatures on \\[{\\mathcal {M}}(t)\\] and l(t) due to certain disturbing non-gravitational accelerations like the atmospheric drag can be effectively disentangled from the post-Newtonian linear trends of interest provided that a sufficiently long temporal interval for the data analysis is assumed. A possible use of \\[\\eta \\] along with the longitudes of the ascending node \\[\\Omega \\] in tests of general relativity with the existing LAGEOS and LAGEOS II satellites is suggested.

The need to choose appropriate and meaningful names for the objects of scientific inquiry, in the spirit of Michael Faraday and, on a different level, of the ancient Chinese doctrine of rectification of names (正名, Zhèngmíng), is illustrated here in the case of the so-called Planet Nine. Since before the discovery of Neptune, the fascinating hypothesis of the possible existence of a new, distant planet in the solar system, yet to be discovered, has regularly surfaced in the pages of astronomy journals in various guises. Its most recent incarnations have been tentatively given names such as Planet X, Planet Y, and, most famously, Planet Nine. Such labels are unsatisfactory because they reveal no significant physical or orbital properties of the object which they are attributed to. I propose here the name Telisto, from the ancient Greek word τήλɩστoς for ‘farthest, most remote’ which captures a feature common to all versions of this scenario that seems destined to remain at the forefront of astronomical research for a long time to come: its supposedly great heliocentric distance, estimated at several hundred astronomical units. By exploring the history of astronomy, I also respond to some criticisms that might be leveled at this proposal. Among other things, I also draw a comparison with the naming of the so-called axions, which are hypothetical elementary particles proposed almost fifty years ago and which continue to be an active object of research.

The Planet Nine hypothesis encompasses a body of about 5–8 Earth’s masses whose orbital plane would be inclined to the ecliptic by one or two tens of degrees and whose perihelion distance would be as large as about 240–385 astronomical units. Recently, a couple of his epigones have appeared: Planet X and Planet Y. The former is similar to a minor version of Planet Nine in that all its physical and orbital parameters would be smaller. Instead, the latter would have a mass ranging from that of Mercury to Earth’s and a semimajor axis within 100–200 astronomical units. By using realistic upper bounds for the orbital precessions of Saturn, one can obtain insights on their position which, for Planet Nine, appears approximately confined around its aphelion. Planet Y can only be a Mercury-sized object at no less than about 125 astronomical units, while Planet X appears to be ruled out. Dedicated data reductions by modeling such perturber(s) are required to check the present conclusions, to be intended as hints of what might be detectable should planetary ephemerides include them. A probe on the same route of Voyager 1 would be perturbed by Planet Nine by about 20–40 km after some decades.

‘Black hole’ is the denomination of the most extreme prediction of the General Theory of Relativity made popular by J. A. Wheeler in the late sixties of the twentieth century, having now entered widely into the collective imagination. Nonetheless, the term is somewhat misleading since there is nothing that tears apart in black holes, which, furthermore, are not even black. Thus, the new name pyknons, from the ancient Greek word for ‘compact; constricted; close-packed’, is proposed for them since it captures a key distinctive feature of theirs. In deference to the objects thus renamed, it also has the merit of introducing a greater compactness in the terms denoting them.

The adventure of the journal Universe began about ten years ago, when I received an invitation from Dr [...]