A new laser-ranged satellite for General Relativity and space geodesy: III. De Sitter effect and the LARES 2 space experiment

Eur. Phys. J. C (2017) 77:819 https://doi.org/10.1140/epjc/s10052-017-5339-y Regular Article - Theoretical Physics A new laser-ranged satellite for General Relativity and space geodesy: III. De Sitter effect and the LARES 2 space experiment Ignazio Ciufolini1,2,a , Richard Matzner3 , Vahe Gurzadyan4 , Roger Penrose5 1 Dip. Ingegneria dell’Innovazione, Università del Salento, Lecce, Italy 2 Centro Fermi, Rome, Italy 3 Theory Group, University of Texas at Austin, Austin, USA 4 Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia 5 Mathematical Institute, University of Oxford, Oxford, UK Received: 10 August 2017 / Accepted: 28 October 2017 © The Author(s) 2017. This article is an open access publication Abstract In two previous papers we presented the LARES 2 space experiment aimed at a very accurate test of framedragging and at other tests of fundamental physics and measurements of space geodesy and geodynamics. We presented the error sources of the LARES 2 experiment, its error budget and Monte Carlo simulations and covariance analyses confirming an accuracy of a few parts in one thousand in the test of frame-dragging. Here we discuss the impact of the orbital perturbation known as the de Sitter effect, or geodetic precession, in the error budget of the LARES 2 frame-dragging experiment. We show that the uncertainty in the de Sitter effect has a negligible impact in the final error budget because of the very accurate results now available for the test of the de Sitter precession and because of its very nature. The total error budget in the LARES 2 test of frame-dragging remains at a level of the order of 0.2%, as determined in the first two papers of this series. 1 LARES 2 and an introduction to the de Sitter effect The LARES 2-LAGEOS space experiment is designed to achieve a new, accurate measurement of the General Relativistic frame-dragging due to the rotation of the Earth. Analytical estimates, covariance studies, and Monte Carlo simulations concur that the expected error level in this effect is of order 0.2%, as shown in Refs. [1,2]. The two LAGEOS (Laser GEOdynamics Satellite) and the two LARES (Laser RElativity Satellite) are laser-ranged satellites. Satellite Laser Ranging (SLR) is the most accurate technique for measuring distances to the Moon [3] and to artificial satellites such as the LAGEOS and LARES satellites [4–6]. Short-duration laser pulses are emitted, with different elevations, from lasers on the Earth towards a satellite and then reflected back to the emitting laser-ranging stations by the retro-reflectors on the satellite. The tracking data collected by the SLR network are analysed, organized and distributed by the International Laser Ranging Service (ILRS) [7]. By measuring the total round-trip travel time we are today able to determine the instantaneous distance of a retro-reflector on the LAGEOS and LARES satellites with a precision of a few millimetres [8]. Then, using orbital estimators, such as GEODYN, EPOSOC and UTOPIA, the orbit of the satellite is reconstructed and its six Keplerian orbital elements are determined with extremely high accuracy. For example the longitude of the ascending node can be determined with an uncertainty of a fraction of milliarcsecond that, over a long period of time, allows for extremely high accuracy in the measurement of the total nodal precession of a laser-ranged satellite. The LAGEOS satellites (LAGEOS and LAGEOS 2) [4] are spherical, made of heavy brass and aluminium, with a radius of 300 mm and about 406 kg in weight, completely passive and covered with retro-reflectors. LAGEOS and LAGEOS 2 have an essentially identical structure but they have different orbits. The semimajor axis of LAGEOS is a = 12270 km, the eccentricity e = 0.004 and the inclination I = 109.9◦ . The semimajor axis of LAGEOS 2 is a I I = 12163 km, the eccentricity e I I = 0.014 and the inclination I I I = 52.65◦ . The LARES satellite [5], launched in 2012 by the Italian Space Agency (ASI) and ESA with the VEGA launch vehicle of ASI, ESA, AVIO and ELV, is spherical with a radius of 182 mm and a total mass of 386.8 kg. It is a single spherical piece of a very dense tungsten alloy and it is covered with 92 retro-reflectors. The LARES orbital elements are semimajor axis a L A R E S = 7820 km, a e-mail: 123 819 Page 2 of 6 Eur. Phys. J. C (2017) 77:819 orbital eccentricity e L A R E S = 0.0008, and orbital inclination I L A R E S = 69.5◦ . The LARES 2 satellite is planned for launch in 2019 with the new VEGA C launch vehicle of ASI, ESA, AVIO and ELV. It will be spherical with a radius of about 200 mm and a total mass of about 300 kg. Its orbital elements will be semimajor axis a L A R E S 2 = 12270 km, orbital inclination I L A R E S 2 = 70.16◦ (supplementary to that of LAGEOS) and approximately null orbital eccentricity. In addition to the frame-dragging, gravitomagnetic effect, whose test is the main objective of the LARES 2-LAGEOS space experiment, there is another general relativistic perturbation of an orbiting gyroscope, relative to an asymptotic inertial frame: the de Sitter or geodetic (or geodesic) precession [9] (see also [10]). This precession is due to the coupling between the velocity of a gyroscope orbiting a central body and the static part of the field (Schwarzschild metric) generated by the central mass. The de Sitter precession can be derived by Fermi–Walker [11] transport along the worldline of a test-gyroscope. We first consider a spacelike spin four-vector S α at each point of a timelike curve x α (s) with tangent vector u α . We thus have S α u α = 0. According to special relativistic kinematics and to the medium strong equivalence principle (all the laws of physics are the laws of special relativity in a local inertial frame [12–14,21]), the spin vector S α obeys Fermi–Walker transport along the curve: S α ;β u β = u α (a β Sβ ) ≡ u α (u β ;γ u γ Sβ ), (1) where a β ≡ u β ;γ u γ is the four-acceleration of the testgyroscope and the semicolon denotes the covariant derivative. Suppose the timelike curve is a geodesic [12–15]. (Any test particle in the gravitational field of a massive body follows a timelike geodesic of the spacetime; a timelike geodesic path – world line – in spacetime’s Lorentzian geometry is one that locally maximizes proper time, in analogy with the length-minimizing property of Euclidean straight lines. This is the case for a small body in free fall, affected only by gravitational forces.) Since a geodesic has zero four-acceleration: u β ;γ u γ = 0, we then have S α ;β u β = 0. In this case the Fermi–Walker transport is just the parallel transport along the geodesic. Therefore, in General Relativity, the orbital angular momentum of a test particle orbiting around a central body, assuming that both the test particle and the body follow geodesic motion, is parallel-propagated in the spacetime. Since geodesic motion is at (...truncated)