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On the Mean Anomaly and the Mean Longitude in Tests of Post-Newtonian Gravity

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On the Mean Anomaly and the Mean Longitude in Tests of Post-Newtonian Gravity Eur. Phys. J. C (2019) 79:816 https://doi.org/10.1140/epjc/s10052-019-7337-8 Regular Article - Theoretical Physics On the mean anomaly and the mean longitude in tests of post-Newtonian gravity Lorenzo Iorioa Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.)-Istruzione, Viale Unità di Italia 68, 70125 Bari, BA, Italy Received: 2 July 2019 / Accepted: 21 September 2019 / Published online: 4 October 2019 © The Author(s) 2019 Abstract The distinction between the mean anomaly M(t) with, e.g., Earth’s artificial satellites, Solar system’s plan- and the mean anomaly at epoch η, and the mean longitude ets and other astrophysical binaries, there is a certain con- l(t) and the mean longitude at epoch is clarified in the fusion in the literature about the possible use of the mean context of a their possible use in post-Keplerian tests of anomaly M (t) as potential observable in addition to the gravity, both Newtonian and post-Newtonian. In particular, widely inspected argument of pericentre ω and, to a lesser the perturbations induced on M(t), η, l(t), by the post- extent, longitude of the ascending node . Indeed, it is as Newtonian Schwarzschild and Lense–Thirring fields, and the if some researchers, including the present author, who have classical accelerations due to the atmospheric drag and the tried to compute perturbatively the mean rate of change of oblateness J2 of the central body are calculated for an arbi- the mean anomaly in excess with respect to the Keplerian trary orbital configuration of the test particle and a generic case due to some pN accelerations were either unaware of orientation of the primary’s spin axis Sˆ. They provide us the fact that what they, actually, calculated was the secular with further observables which could be fruitfully used, e.g., precession of the mean anomaly at the epoch η, or they sys- in better characterizing astrophysical binary systems and in tematically neglected a potentially non-negligible contribu- more accurate satellite-based tests around major bodies of tion to the overall change of the mean anomaly induced indi- the Solar System. Some erroneous claims by Ciufolini and rectly by the semimajor axis a through the mean motion nb. Pavlis appeared in the literature are confuted. In particular, Such a confusion has produced so far some misunderstand- it is shown that there are no net perturbations of the Lense– ing which led, e.g., to unfounded criticisms about alleged Thirring acceleration on either the semimajor axis a and the proposals of using the mean anomaly, especially in the case mean motion nb. Furthermore, the quadratic signatures on of man-made spacecraft orbiting the Earth, or even uncor- M(t) and l(t) due to certain disturbing non-gravitational rect evaluations of the total pN effects sought. An example accelerations like the atmospheric drag can be effectively that sums up well the aforementioned confusion and mis- disentangled from the post-Newtonian linear trends of inter- understanding, even in the peer-reviewed literature, is the est provided that a sufficiently long temporal interval for the following one. Ciufolini and Pavlis [6] wrote “[…] one of data analysis is assumed. A possible use of η along with the the most profound mistakes and misunderstandings of Iorio longitudes of the ascending node in tests of general rela- (2005) is the proposed use of the mean anomaly of a satel- tivity with the existing LAGEOS and LAGEOS II satellites lite to measure the Lense–Thirring effect […] This is simply is suggested. a nonsense statement: let us, for example, consider a satel- lite at the LAGEOS altitude, the Lense–Thirring effect on its mean longitude is of the order of 2 m/year, however, the 1 Introduction mean longitude change is about 1.8 × 1011 m/year. Thus, from Kepler’s law, the Lense–Thirring effect corresponds In regard to possible tests of post-Newtonian (pN) features to a change of the LAGEOS semi-major of less than 0.009 of general relativity1 and of alternative models of gravity cm! Since, even a high altitude satellite such as LAGEOS showed a semimajor axis change of the order of 1 mm/day, 1 For a recent overview of the current status and challenges of the due to atmospheric drag and to the Yarkoski–Rubincam effect Einsteinian theory of gravitation, see, e.g., Debono and Smoot [8], and references therein. (because of atmospheric drag, the change of semimajor axis and mean motion is obviously much larger for lower altitude a e-mail: [email protected] 123 816 Page 2 of 14 Eur. Phys. J. C (2019) 79 :816 satellites), and since the present day precision of satellite anomaly f the other two anomalies. In the unperturbed Kep- laser ranging is, even in the case of the best SLR stations, lerian case, the mean anomaly is defined as of several millimeters, it is a clear nonsense to propose a . test of the Lense–Thirring effect based on using the mean M(t) = η + nb (t − t0) , (1) anomaly of any satellite, mean anomaly largely affected by 2 non-conservative forces.” It is difficult to understand what where η is the mean anomaly at the reference epoch t0, and is the target of the arrows by Ciufolini and Pavlis [6] since the mean anomaly is not even mentioned in the published μ nb = (2) version of the criticized paper by the present author, not to a3 mention any explicitly detailed proposal to use it. Be that as . μ = it may, in the following, we will show that, actually, using is the Keplerian mean motion. In Eq. (2), GM is the the mean anomaly, or the mean longitude l (t),inpNtests gravitational parameter of the primary having mass M, while with artificial Earth’s satellites may be feasible, provided that G is the Newtonian constant of gravitation; in the following, μ = η certain non-gravitational perturbations are compensated by we will assume const. The mean anomaly at epoch some active drag-free mechanism. However, even in case of is one of the six Keplerian orbital elements parameterizing passive, geodetic satellites, we will show that, under certain the orbit of a test particle in space. In the unperturbed case, M( ) η conditions, it is possible to separate the relativistic linear t is a linear function of time t because both a and trends of interest from the unwanted parabolic signatures are constants of motion. If a relatively small perturbing pK η of non-conservative origin. Furthermore, the arguments pro- acceleration A is present, both a and are, in general, affected vided by Ciufolini and Pavlis [6] about the Lense–Thirring by it, becoming time-dependent. As a result, also the mean effect and the mean longitude are erroneous. Finally, the use motion is, in general, modified so that of the mean anomaly at epoch or of the mean longitude at → pert = + ( ). epoch is, in principle, possible even with passive, geodetic nb nb nb nb t (3) spacecraft like those of the LAGEOS family because they are, by construction, free from the aforementioned potential Thus, the perturbed mean anomaly is the sum of the now η( ) drawbacks exhibited by the mean anomaly and the mean lon- time-dependent mean anomaly at epoch t and a function ( ) gitude themselves, which was completely ignored or unrec- of time t whose derivative is equal to the (perturbed) mean ognized by Ciufolini and Pavlis [6]. motion, i.e., The paper is organized as follows. In Sect.2, we will t Mpert( ) = η( ) + ( ) = η( ) + pert review the basics of the mean anomaly, the mean anomaly at t t t t nb t dt epoch, the mean longitude, and the mean longitude at epoch t0 along with the calculation of their perturbations with respect = η + η(t) to the purely Keplerian case in presence of a generic disturb- t + ( − ) + . ing post-Keplerian (pK) acceleration. Section 3 is devoted nb t t0 nb t dt (4) t0 to the calculation of the effects of some well-known pN accelerations (Schwarzschild and Lense–Thirring), while the The resulting change M of the mean anomaly with respect impact of the atmospheric drag and the oblateness of the pri- to the unperturbed case is, thus, mary are treated in Sect.4. The potential of a possible use of the mean anomaly at epoch in the ongoing tests with the M(t) = Mpert(t) − M(t) = η(t) + (t), (5) satellites LAGEOS and LAGEOS II is discussed in Sect. 5. Section6 summarizes our findings, and offers our conclu- where we defined sions. t . (t) = nb t dt . (6) t0 2 The mean anomaly and the mean longitude as a function whose derivative yields the perturbation of the mean motion. In Eq. (6), the instantaneous shift of the mean 2.1 The mean anomaly motion due to the time-varying semimajor axis3 a is In the restricted two-body problem, the mean anomaly M(t) 2 The symbol η is used for the mean anomaly at epoch by Milani et al. is one of the three time-dependent fast angular variables [15]. In the notation by Brumberg [3], the mean anomaly is l,whilethe which, in celestial mechanics, can be used to characterize mean anomaly at epoch is l0. Kopeikin et al. [9] denote η as M0, while the instantaneous position of a test particle along its Kep- Bertotti et al. [2] adopt . lerian ellipse, being the eccentric anomaly E and the true 3 It should be recalled that we kept μ constant. 123 Eur. Phys. J. C (2019) 79 :816 Page 3 of 14 816 3 nb in non-trivial scenarios in which several perturbing accelera- nb(t) =− a(t) 2 a tions of different nature act simultaneously on the test particle f =−3 nb da dt inducing non-vanishing long-term effects on the semimajor d f 2 a f0 dt d f axis a.
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