Title: Conservative evaluation of the uncertainty in the LAGEOS-LAGEOS II Lense-Thirring test
ArXiv ID: 0710.1022
Date: 2009-11-06
Authors: ** - Lorenzo Iorio (주요 저자, 이탈리아 국립우주연구소 – INAF) - (공동 저자 명단은 원문에 명시되지 않아 확인이 필요함) **
📝 Abstract
We deal with the test of the general relativistic gravitomagnetic Lense-Thirring effect currently ongoing in the Earth's gravitational field with the combined nodes \Omega of the laser-ranged geodetic satellites LAGEOS and LAGEOS II. One of the most important source of systematic uncertainty on the orbits of the LAGEOS satellites, with respect to the Lense-Thirring signature, is the bias due to the even zonal harmonic coefficients J_L of the multipolar expansion of the Earth's geopotential which account for the departures from sphericity of the terrestrial gravitational potential induced by the centrifugal effects of its diurnal rotation. The issue addressed here is: are the so far published evaluations of such a systematic error reliable and realistic? The answer is negative. Indeed, if the difference \Delta J_L among the even zonals estimated in different global solutions (EIGEN-GRACE02S, EIGEN-CG03C, GGM02S, GGM03S, ITG-Grace02, ITG-Grace03s, JEM01-RL03B, EGM2008, AIUB-GRACE01S) is assumed for the uncertainties \delta J_L instead of using their more or less calibrated covariance sigmas \sigma_{J_L}, it turns out that the systematic error \delta\mu in the Lense-Thirring measurement is about 3 to 4 times larger than in the evaluations so far published based on the use of the sigmas of one model at a time separately, amounting up to 37% for the pair EIGEN-GRACE02S/ITG-Grace03s. The comparison among the other recent GRACE-based models yields bias as large as about 25-30%. The major discrepancies still occur for J_4, J_6 and J_8, which are just the zonals the combined LAGEOS/LAGOES II nodes are most sensitive to.
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In the weak-field and slow motion approximation, the Einstein field equations of general relativity get linearized to a form resembling Maxwell's equations of electromagnetism. Thus, a gravitomagnetic field, induced by the off-diagonal components g0i, i = 1, 2, 3 of the space-time metric tensor related to the mass-energy currents of the source of the gravitational field, arises [1]. It affects in several ways the motion of, e.g., test particles and electromagnetic waves [2]. Perhaps the most famous gravitomagnetic effects are gyroscope precession [3,4] and the Lense-Thirring1 precessions [6] of the orbit of a test particle, both occurring in the field of a central slowly rotating mass like a planet.
The measurement of gyroscope precession in the Earth’s gravitational field has been the goal of the dedicated space-based GP-B mission2 [7,8] launched in 2004; its data analysis is still in progress.
In this paper we critically discuss some issues concerning the test of the Lense-Thirring effect performed with the LAGEOS and LAGEOS II terrestrial artificial satellites [9] tracked with the Satellite Laser Ranging (SLR) technique [10]. [11,12] proposed measuring the Lense-Thirring nodal precession of a pair of counter-orbiting spacecraft in terrestrial polar orbits and equipped with drag-free apparatus. A somewhat equivalent, cheaper version of such an idea was put forth in Ref. [13] whose author proposed to launch a passive, geodetic satellite in an orbit identical to that of the LAGEOS satellite apart from the orbital planes which should have been displaced by 180 deg apart. 3The measurable quantity was, in this case, the sum of the nodes of LAGEOS and of the new spacecraft, later named LAGEOS III, LARES, WEBER-SAT, in order to cancel the confounding effects of the multipoles of the Newtonian part of the terrestrial gravitational potential (see below). Although extensively studied by various groups [14][15][16], such an idea has not been implemented for a long time. Recently, the Italian Space Agency (ASI) has approved this project and should launch a VEGA rocket for this purpose at the end of 2009-beginning of 2010 (http://www.asi.it/en/activity/cosmology/lares)
. For recent updates of the LARES/WEBER-SAT mission, including recently added additional goals in fundamental physics and related criticisms, see Refs. [17][18][19][20][21][22][23][24].
Among scenarios involving existing orbiting bodies, the idea of measuring the Lense-Thirring node rate with the just launched LAGEOS satellite, along with the other SLR targets orbiting at that time, was proposed in Ref. [25]. Tests have been effectively performed using the LAGEOS and LAGEOS II satellites [26], according to a strategy [27] involving a suitable combination of the nodes of both satellites and the perigee ω of LAGEOS II.
This was done to reduce the impact of the most relevant source of systematic bias, viz. the mismodelling in the even ( = 2, 4, 6 . . .) zonal (m = 0) harmonics J of the multipolar expansion of the Newtonian part of the terrestrial gravitational potential:4 they account for non-sphericity of the terrestrial gravitational field induced by centrifugal effects of the Earth’s diurnal rotation. The even zonals affect the node and the perigee of a terrestrial satellite with secular precessions which may mimic the Lense-Thirring signature. The three-elements combination used allowed for removing the uncertainties in J2 and J4. In [28] a ≈ 20% test was reported by using the5 EGM96
[29] Earth gravity model; subsequent analyses showed that such an evaluation of the total error budget was overly optimistic in view of the likely unreliable computation of the total bias due to the even zonals [30][31][32]. An analogous, huge underestimation turned out to hold also for the effect of non-gravitational perturbations [33] like direct solar radiation pressure, the Earth’s albedo, various subtle thermal effects depending on the the physical properties of the satellites’ surfaces and their rotational state [31,[34][35][36][37][38][39][40], which the perigees of LAGEOS-like satellites are particularly sensitive to. As a result, the realistic total error budget in the test reported in Ref. [28] might be as large as 60 -90% or even more (by considering EGM96 only).
The observable used in Ref. [9] with the GRACE-only EIGEN-GRACE02S model [41] and in Ref. [42] with other global terrestrial gravity solutions was the following linear combination6 of the nodes of LAGEOS and LAGEOS II, explicitly computed in Ref. [44] following the approach proposed in Ref. [27]:
where c1 ≡ -
The coefficients Ω. of the aliasing classical node precessions [45] Ωclass = P Ω. J induced by even zonals have been analytically worked out in e.g. [30]; a, e, i are the satellite’s semimajor axis, eccentricity and inclination, respectively and yield c1 = 0.544 for eq. ( 2). The Lense-Thirring signature of eq. ( 1) amounts to 47.8 milliarcseconds per year (mas yr -1 ). The combination eq.
