EPSC Abstracts Vol. 5, EPSC2010-591, 2010 European Planetary Science Congress 2010 c Author(s) 2010

LLR and the Lense-Thirring Effect

L. Iorio F.R.A.S. Viale Unità di Italia 68, 70125, Bari (BA) Italy ([email protected])

Abstract Gabadadze and Porrati (DGP) to explain the observed acceleration of the Universe without resorting to dark We mainly explore the possibility of measuring the ac- energy; indeed, as we will see, the magnitude of such tion of the intrinsic gravitomagnetic field of the rotat- an effect is the same as the Lense-Thirring one for ing on the orbital motion of the with the the Moon. Several researchers argued that it might be LLR technique. Expected improvements in it should possible to measure the Lue-Starkman with push the precision in measuring the Earth-Moon range LLR in view of the expected improvements in such a to the mm level; the present-day rms accuracy in re- technique. constructing the radial component of the lunar orbit is about 2 cm; its harmonic terms can be determined at the mm level. The current uncertainty in measuring 2 The Lense-Thirring effect 1 the lunar precession rates is about 10− milliarcsec- on the lunar orbit and some onds per . The Lense-Thirring secular of the node and the perigee of the Moon induced by sources of error 3 the Earth’s spin angular momentum amount to 10− milliarcseconds per year yielding transverse and nor- By assuming a suitably constructed geocentric equato- 1 2 1 mal shifts of 10− 10− cm yr− . In the radial direc- rial frame, it turns out that the node and the perigee of − tion there is only a short-period, i.e. non-averaged over the Moon undergo the Lense-Thirring secular preces- 1 1 one orbital revolution, oscillation with an amplitude of sions of 0.001 mas yr− and 0.003 mas yr− , re- 5 − 10− m. Major limitations come also from some sys- spectively. The Lue-Starkman LS pericentre preces- 1 sion is just about 0.005 mas yr− . tematic errors induced by orbital perturbations of clas- ∓ sical origin like, e.g., the secular precessions induced Let us now examine some sources os systematic er- by the Sun and the oblateness of the Moon whose mis- rors. In regard to the potentially corrupting action of modelled parts are several times larger than the Lense- the mismodelling in the even (` = 2, 4, 6, ...) zonal Thirring signal. The present analysis holds also for the (m = 0) harmonic coefficients J` of the multipolar Lue-Starkman perigee precession due to the multidi- expansion of the Newtonian part of the Earth’s gravita- mensional braneworld model by Dvali, Gabadadze and tional potential, which is not the most important source 3 Porrati (DGP); indeed, it amounts to about 5 10− of aliasing precessions in the case of the Moon, only × milliarcseconds per year. δJ2 would be of some concern. Indeed, the mismod- elled secular precessions induced by it on the lunar 4 1 node and perigee amount to 2.67 10− mas yr− 4 1 − × and 5.3 10− mas yr− , respectively; the impact of 1 Introduction × the other higher degree even zonals is negligible being 8 1 We consider in some details the possibility of measur- 10− mas yr− . As in the case of the spins, also the ≤ ing with LLR a general relativistic effect induced by asphericity of the Moon has to be taken into account the gravitomagnetic field of the spinning Earth through according to a non-central, Lorentz-like acceleration on the orbital Moon 2 motion of the Moon around the Earth, i.e. the Lense- ˙ 3 nMoon cos FJ2 RMoon ΩJ Moon = , (1) Thirring precessions of the longitude of the ascending 2 −2 (1 e2)2 a   node Ω and the argument of pericentre ω. − 3 Our analysis is equally valid also for the anoma- where nMoon = GM(1 + µ)/a is the lunar mean lous Lue-Starkman perigee precession predicted in motion, µ is the Moon-Earth mass ratio, and F is the p the framework of the multi-dimensional braneworld angle between the orbital angular momentum and the model of modified gravity put forth by Dvali, Moon’s spin angular momentum SMoon; it is about 3.61 deg since the spin axis of the Moon is tilted by the centers-of-mass of the Earth and the Moon. Im- 1.54 deg to the and the orbital plane has an provements in the precision of the Earth-Moon rang- inclination of 5.15 deg to the ecliptic. The mismod- ing of the order of 1 mm are expected in the near future Moon elled node precession due to δJ2 is about 0.006 with the . Recently, sub-centimeter 1 mas yr− , i.e. 6 times larger than the Lense-Thirring precision in determining range distances between a rate. Among the N-body gravitational perturbations, laser on the Earth’s surface and a retro-reflector on the the largest ones are due to the Sun’s attraction. In Moon has been achieved. However, it must be consid- order to get an order-of-magnitude evaluation of their ered that the rms accuracy in the T and N directions mismodelling, let us note that some of such effects are is likely worse than in R. 2 proportional to n /nMoon; e.g. the node rate, referred to the equator, is ⊕ 3 Conclusions 3GM cos I 3 2 7 mas Ω˙ = sin ε 1 5 10 , 4a3 n 2 − ≈ − × yr We have examined the possibility of measuring the ac- Moon   ⊕ (2) tion of the intrinsic gravitomagnetic field of the spin- where ε = 23.439 deg is the obliquity of the ecliptic. ning Earth on the lunar orbital motion with the LLR 10 3 2 5 Since δGM = 5 10 m s− and δGM = 8 10 technique. After showing that the Lense-Thirring ap- 3 2 × × 1 m s− , we can assume a bias of 0.07 mas yr− proximation is adequate for the Earth-Moon system, ≈ which is 70 times larger than the Lense-Thirring pre- we found that the Lense-Thirring secular precessions cession. of the Moon’s node and the perigee induced by the Let us, now, consider the precision of LLR in re- Earth’s spin angular momentum are of the order of 3 1 constructing the lunar orbit with respect to the Lense- 10− mas yr− corresponding to transverse and nor- 1 2 1 Thirring effect. Concerning the precision in measur- mal secular shifts of 10− 10− cm yr− . The in- − ing the lunar precession rates, it amounts to about 0.1 trinsic gravitomagnetic field of the Earth does not sec- 1 mas yr− , i.e. it is two orders of magnitude larger than ularly affect the radial component of the Moon’s orbit; the Lense-Thirring precessions. The orbital perturba- a short-period, i.e. not averaged over one orbital rev- tions experienced by a test particle are usually decom- olution, radial oscillation is present, but its amplitude 5 posed along three orthogonal directions of a frame co- is of the order of 10− m. The current rms accuracy moving with it; they are named radial R (along the in reconstructing the lunar orbit is of the order of cm radius vector), transverse T (orthogonal to the radius in the radial direction; the harmonic components can vector, in the osculating orbital plane) and normal N be determined at the mm level. Forthcoming expected (along the orbital angular momentum, out of the oscu- improvements in LLR should allow to reach the mm lating orbital plane). The R T N perturbations precision in the Earth-Moon ranging. The present-day − − can be expressed in terms of the shifts in the Keplerian accuracy in measuring the lunar precessional rate is of 1 1 . The lunar Lense-Thirring shifts af- the order of 10− mas yr− . Major limitations come ter one year are, thus, about 0.40 cm and 0.07 cm in also from some orbital perturbations of classical origin − the T and N directions, respectively. It is important to like, e.g., the secular node precessions induced by the note that there is no Lense-Thirring secular signature Sun and the oblateness of the Moon which act as sys- in the Earth-Moon radial motion on which all of the tematic errors and whose mismodelled parts are up to efforts of LLR community have been concentrated so 70 times larger than the Lense-Thirring effects. As a far. It can be shown that a short-period, i.e. not aver- consequence of our analysis, we are skeptical concern- aged over one orbital revolution, radial signal exists; it ing the possibility of measuring intrinsic gravitomag- is proportional to netism with LLR in a foreseeable future. The same conclusion holds also for the Lue-Starkman perigee 2GS 5 ∆r ⊕ = 2 10− m, (3) precession predicted in the framework of the multidi- ∝ c2na2 × mensional braneworld DGP model of modified grav- which is too small to be detected since the present-day ity; indeed, it is as large as the Lense-Thirring one for accuracy in estimating the amplitudes of radial har- the Moon. monic signals is of the order of mm. Major limitations come from the post-fit rms accuracy with which the lunar orbit can be reconstructed; the present-day ac- curacy is about 2 cm in the radial direction R along