^{*}

Edited by: Martin Anthony Hendry, University of Glasgow, UK

Reviewed by: Zurab Silagadze, Novosibirsk State University, Russia; Lee Samuel Finn, The Pennsylvania State University, USA

*Correspondence: Lorenzo Iorio, Ministero dell'Istruzione, dell'Università e della Ricerca, Istruzione, Viale Unità di Italia 68, Bari 70125, Italy e-mail:

This article was submitted to Cosmology, a section of the journal Frontiers in Astronomy and Space Sciences.

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The direct long-term changes occurring in the orbital dynamics of a local gravitationally bound binary system _{b} an astronomical body of mass _{b}′ « _{b} is considered. The characteristic frequencies of the non-Keplerian orbital variations of _{b} and _{b}′. General expressions for the resulting Newtonian and post-Newtonian tidal orbital shifts of ^{−1}), occur for the Ganymede orbiter of the JUICE mission. Although future improvements in spacecraft tracking and orbit determination might, perhaps, reach the required sensitivity, the systematic bias represented by the other known orbital perturbations of both Newtonian and post-Newtonian origin would be overwhelming. The realization of a dedicated artificial mini-planetary system to be carried onboard and Earth-orbiting spacecraft is considered as well. Post-Newtonian tidal precessions as large as ≈ 1−10^{2} mas yr^{−1} could be obtained, but the quite larger Newtonian tidal effects would be a major source of systematic bias because of the present-day percent uncertainty in the product of the Earth's mass times the Newtonian gravitational parameter.

Gravitation is one of the known fundamental interactions of physics, and the General Theory of Relativity (GTR) is, at present, its best theoretical description (Will,

The internal dynamics of a gravitationally bound binary system immersed in the external gravitational field of a massive rotating body is tidally affected at both the Newtonian and the post-Newtonian level (Mashhoon,

The paper is organized as follows. In Section 2, the long-term rates of change of the orbital parameters of the test particle of the restricted two-body system are calculated by keeping the elements of a generic tidal matrix constant over the orbital period of the particle around its primary. In Section 3, the direct orbital effects due to both the gravitoelectric and the gravitomagnetic tidal matrices are obtained by averaging their elements over the orbital period of the motion around the distant body. Section 4 is devoted to exploring some experimental possibilities offered by forthcoming spacecraft-based missions to astronomical bodies. Section 5 summarizes our findings.

Let us consider an isolated rotating body of mass _{2}′ of

Let a local inertial frame ^{1}_{b}′ and _{b}; for a critical discussion tidal phenomena occurring in the Sun-Earth-Moon system over timescales comparable to or larger than the de Sitter-Fokker and Pugh-Schiff ones, see (Gill et al., _{b} = 2π^{−1}_{b} and _{b} = 2π_{b}′^{−1} of the three-body system considered. As a further assumption, we will consider the local motion of _{b}′ « _{b}. In general, the internal dynamics of _{b}, _{b}, i.e., _{b}′, where Ψ denotes a generic precessing osculating Keplerian orbital element of

At both the Newtonian and the post-Newtonian level, the internal dynamics of

where the elements of the tidal matrix

are the tetrad components of the curvature Riemann tensor evaluated onto the geodesic of the observer in ^{−2}. It is

with the Newtonian (N), gravitoelectric (GE) and gravitomagnetic (GM) tidal matrices given by (Mashhoon et al.,

In Equations (4–6), which are symmetric and traceless, _{ij} is the Kronecker symbol,

The tidal acceleration of Equation (1) can be considered as a small perturbation _{pert} of the Newtonian monopole of _{b}′ « _{b}, the elements of the tidal matrix _{b}. Thus, by evaluating the right-hand-sides of the Gauss equations (Burns,

onto the unperturbed Keplerian ellipse

where ^{2}) is the semilatus rectum and _{R}_{T}_{N}_{b} can be calculated. To this aim, let us note that, in principle, the average should be made by means of (Brumberg,

where ^{−2}), and by neglecting small mixed terms of order _{2}c^{−2}) arising from the interplay between the external tidal and the local Newtonian perturbations due to _{2}, the approximate expression

can be used by integrating over

and the velocity

it is possible to compute from Equations (15–20) the unit vector

Then, the radial, transverse and normal components of Equation (1) turn out to be

where K is the dimensional scaling factor of the tidal matrix considered having dimensions of T^{−2}, while the dimensionless coefficients _{ij}_{b} with Equation (14), one finally has

The long-term rates of Equations (25–29) are valid for any symmetric and traceless tidal-type perturbation of the form of Equation (1) whose coefficients can be considered as constant over the characteristic orbital frequency _{b} of the local binary system considered. As such, Equations (25–29) are not limited just to Equations (5,6). Moreover, Equations (25–29) hold for a general orbital configuration of the test particle

In general, Equations (25–29) may not be regarded as truly secular rates over timescales arbitrarily long because of the slow time dependence encoded in both the tidal matrix elements themselves and in the orbital elements of

Let us, now, assume that the characteristic timescales _{Ψ} of all the non-Keplerian orbital effects within _{b} of _{b} by keeping ^{2}

The direct effects of order ^{−2}) can be obtained by evaluating the post-Newtonian tidal matrices Equations (5,6) onto an unchanging Keplerian ellipse as reference unperturbed trajectory.

Below, the averaged tidal matrix elements of Equations (4–6), computed to order ^{−2}) and to zero order in _{2}′, are listed. They are to be inserted in Equations (25–29) to have the direct long-term rates of change of _{b}.

As far as the Newtonian tidal matrix of Equation (4) is concerned, its average, to the zero order in _{2}′, is

For the post-Newtonian gravitoelectric tidal field of

In the limit of

For the post-Newtonian gravitomagnetic tidal field of

Note that Equations (42–47) have a general validity since they are restricted neither to any specific spatial orientation of the spin axis of

It should be remarked that indirect, mixed effects of order ^{−2}) arise, in principle, also from the Newtonian tidal matrix of Equation (4) when the post-Newtonian effects of the field of _{2}′, accounting for possible deviations of

Forthcoming space-based missions to astronomical bodies orbiting large primaries such as the Sun and Jupiter, in conjunction with expected progresses in interplanetary tracking techniques (Iess and Asmar,

Let us consider the forthcoming BepiColombo^{3}^{4}^{5}_{MPO} = 90°), elliptical orbit (_{MPO} = 0.16) around Mercury with an orbital period of approximately 2.3 h (_{MPO} = 3,394 km). The nominal science duration is one year, with a possible extension of another year. Importantly, orbital maneuvers to change the attitude of the spacecraft are scheduled every about 44 d, so that long smooth orbital arcs should be available. The JUICE mission (Grasset et al., ^{6}^{−1} − 10^{−2} mas yr^{−1} level. In principle, a rate of ≈ 0.5 mas yr^{−1}, which naively corresponds to a range-rate as little as ≈ 0.05 mm s^{−1} at the distance of Jupiter from us, might be detectable with the expected improvement down to 0.01 mm s^{−1} at 60 s integration time in the Doppler range-rate techniques from the ASTRA study (Iess et al., _{J2}/J_{2} = 2× 10^{−2} (Anderson et al., ^{−2}) as well.

^{−1}, of the Sun-Mercury-Mercury Planetary Orbiter (MPO) and of the Jupiter-Ganymede-Jupiter Ganymede Orbiter (JGO) (Grasset et al.,

^{−1}) |
^{−1}) |
||
---|---|---|---|

^{(GM)}_{☿} |
1 × 10^{−18} |
^{(GM)}_{Gan} |
8× 10^{−16} |

^{(Jʘ2)}_{☿} |
8× 10^{−17} |
^{(J♃2)}_{Gan} |
3× 10^{−9} |

^{(GE)}_{☿} |
7× 10^{−14} |
^{(GE)}_{Gan} |
4× 10^{−14} |

_{☿} |
8× 10^{−7} |
_{Gan} |
1 × 10^{−5} |

^{(GM)}_{MPO} |
3× 10^{−17} |
^{(GM)}_{JGO} |
2× 10^{−16} |

^{(GE)}_{MPO} |
2× 10^{−13} |
^{(GE)}_{JGO} |
8× 10^{−14} |

6× 10^{−9} |
5× 10^{−8} |
||

_{MPO} |
7× 10^{−4} |
_{JGO} |
6× 10^{−4} |

_{2} and to the angular momentum J of the primaries were calculated in equatorial coordinate systems. Indeed, while both MPO and JGO will move along polar trajectories at ι = 90° to the equators of their primaries, the orbits of Mercury and Ganymede lie almost in the equatorial planes of the Sun (ι_{☿} = 3.38°) and of Jupiter (ι_{Gan} = 0.20°), respectively. For Ganymede, the value J^{Gan}_{2} = 1.27× 10^{−4} (Anderson et al., ^{Gan} = 3× 10^{30} kg m^{2} s^{−1} was inferred from the values of its mass, equatorial radius and normalized polar moment of inertia (Anderson et al., ^{☿}_{2} = 1.92× 10^{−5} (Smith et al., ^{☿} = 8.4× 10^{29} kg m^{2} s^{−1} was obtained from the latest determinations of its equatorial radius (Byrne et al., ^{ʘ} = 1.90× 10^{41} kg m^{2} s^{−1} (Pijpers, ^{ʘ}_{2} = 2.1× 10^{−7} (Folkner et al., ^{♃} = 6.9× 10^{38} kg m^{2} s^{−1} (Soffel et al., ^{♃}_{2} = 1.469× 10^{−2} (Jacobson,

_{b} and _{b}, induced by the post-Newtonian gravitoelectric and gravitomagnetic tidal field of

^{(tid GM)} |
1 × 10^{−21} s^{−1} |
^{(tid GM)} |
0 s^{−1} |

^{(tid GM)} |
4 × 10^{−6} mas yr^{−1} |
^{(tid GM)} |
0.01 mas yr^{−1} |

^{(tid GM)} |
1 × 10^{−5} mas yr^{−1} |
^{(tid GM)} |
0.07 mas yr^{−1} |

^{(tid GM)} |
8 × 10^{−5} mas yr^{−1} |
^{(tid GM)} |
0.54 mas yr^{−1} |

^{(tid GE)} |
1 × 10^{−18} s^{−1} |
^{(tid GE)} |
0 s^{−1} |

^{(tid GE)} |
0.0025 mas yr^{−1} |
^{(tid GE)} |
8 × 10^{−7} mas yr^{−1} |

^{(tid GE)} |
0.0084 mas yr^{−1} |
^{(tid GE)} |
2 × 10^{−6} mas yr^{−1} |

^{(tid GE)} |
0.0458 mas yr^{−1} |
^{(tid GE)} |
1 × 10^{−5} mas yr^{−1} |

^{(GM)} |
0.1038 mas yr^{−1} |
^{(GM)} |
0.59 mas yr^{−1} |

^{(GM)} |
0.1907 mas yr^{−1} |
^{(GM)} |
1.24 mas yr^{−1} |

^{(GM)} |
0.2075 mas yr^{−1} |
^{(GM)} |
1.18 mas yr^{−1} |

^{(GE)} |
1087.78 mas yr^{−1} |
^{(GE)} |
499.6 mas yr^{−1} |

^{(tid N)} |
2.9 × 10^{−10} s^{−1} |
^{(tid N)} |
0s^{−1} |

^{(tid N)} |
6.7 × 10^{5} mas yr^{−1} |
^{(tid N)} |
7 × 10^{7} mas yr^{−1} |

^{(tid N)} |
2 × 10^{6} mas yr^{−1} |
^{(tid N)} |
3 × 10^{8} mas yr^{−1} |

^{(tid N)} |
1.3 × 10^{7} mas yr^{−1} |
^{(tid N)} |
2 × 10^{9} mas yr^{−1} |

^{(J2)} |
8 × 10^{6} mas yr^{−1} |
^{(J2)} |
6 × 10^{7} mas yr^{−1} |

^{(J2)} |
3 × 10^{7} mas yr^{−1} |
^{(J2)} |
2.7 × 10^{8} mas yr^{−1} |

^{(J2)} |
4 × 10^{7} mas yr^{−1} |
^{(J2)} |
3.5 × 10^{8} mas yr^{−1} |

_{max} and ω_{max}, which are different for each orbital effect considered, are not reported. The units for the precessions are milliarcseconds per year (mas yr^{−1}), apart from the eccentricity ^{−1}. The Newtonian _{2} and the post-Newtonian

In principle, a rather unconventional possibility could be the realization of an artificial mini-planetary system to be carried onboard a drag-free spacecraft orbiting, say, the Earth; such an idea was already proposed in the past to accurately measure the Newtonian constant of gravitation _{W} = 19.6 g cm^{−3} and

Moreover, the local dynamics of such a spaceborne artificial planetary system would be practically free from systematic non-Keplerian gravitational perturbations due to ^{−12} mas yr^{−1}. Table ^{−9} relative accuracy (Petit and Luzum,

_{b} and _{b}, induced by the post-Newtonian gravitoelectric and gravitomagnetic tidal field of the Earth and by its Newtonian tidal field

^{(tid GM)} |
0s^{−1} |

^{(tid GM)} |
0 mas yr^{−1} |

^{(tid GM)} |
3.2 mas yr^{−1} |

^{(tid GE)} |
0s^{−1} |

^{(tid GE)} |
32.8 mas yr^{−1} |

^{(tid GE)} |
189.6 mas yr^{−1} |

^{(tid N)} |
0s^{−1} |

^{(tid N)} |
2.6 × 10^{10} mas yr^{−1} |

^{(tid N)} |
1.5 × 10^{11} mas yr^{−1} |

_{max} and ω_{max}, which are different for each orbital effect considered, are not reported. The units for the precessions are milliarcseconds per year (mas yr^{−1}), apart from the eccentricity e whose rate of change is expressed in s^{−1}. The mean equinox and the mean equatorial plane of the Earth at the epoch J2000.0 of the International Celestial Reference Frame (ICRF) were adopted for both

We looked at the direct long-term orbital rates of change occurring within a local gravitationally bound two-body system as gradiometers to potentially detect post-Newtonian tidal effects due to its slow motion in the external field of a distant third body. We also assumed that the characteristic orbital frequencies of the internal dynamics of the local binary are quite smaller than the frequency of its orbital motion around the external source. We obtained general analytical expressions valid for arbitrary orbital configurations and for a generic orientation of the spin axis of the external body. Future work should be devoted to the calculation of the indirect, mixed post-Newtonian effects arising from the interplay between the Newtonian tidal matrix and the post-Newtonian orbital motion of the binary in the external field.

We applied our results to the future BepiColombo and JUICE man-made missions to Mercury and Ganymede, respectively. It turned out that that, although the expected improvements in interplanetary tracking may, perhaps, allow for a detection of the tidal effects we are interested in, especially for JUICE, the impact of several competing orbital effects of Newtonian and post-Newtonian origin, acting as sources of potential systematic errors, should be carefully considered.

Another possibility which, in principle, may be further pursued is the realization of an artificial mini-planetary system to be carried onboard an Earth-orbiting drag-free spacecraft. If, on the one hand, the post-Newtonian tidal precessions occurring in such a system may be relatively large, amounting to about 1 −10^{2} mas yr^{−1}, on the other hand, the product of the Earth's mass times the Newtonian gravitational constant is currently known with insufficient accuracy to allow for an effective subtraction of the competing Newtonian tidal precessions.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{*}for a generic orientation of its spin axis

^{1}A comoving coordinate system is said to be kinematically nonrotating if it is corrected for the post-Newtonian precessions of its axes.

^{2}It turned out computationally more convenient to adopt the true anomaly

^{3}See also

^{4}See also

^{5}See

^{6}See