Advances in the measurement of the Lense-Thirring effect with planetary motions in the field of the Sun

E.V. Pitjeva, by processing more than 400,000 planetary observations of various types with the dynamical models of the EPM2006 ephemerides, recently estimated a correction to the canonical Newtonian-Einsteinian Venus' perihelion precession of -0.0004 +/- 0.0001 arcseconds per century. The prediction of general relativity for the Lense-Thirring precession of the perihelion of Venus is -0.0003 arcseconds per century. It turns out that neither other mismodelld/unmodelled standard Newtonian/Einsteinian effects nor exotic ones, postulated to, e.g., explain the Pioneer anomaly, may have caused the determined extra-precession of the Venus orbit which, thus, can be reasonably attributed to the gravitomagnetic field of the Sun, not modelled in the routines of the EPM2006 ephemerides. However, it must be noted that the quoted error is the formal, statistical one; the realistic uncertainty might be larger. Future improvements of the inner planets' ephemerides, with the inclusion of the Messenger and Venus-Express tracking data, should further improve the accuracy and the consistency of such a test of general relativity which would also benefit of the independent estimation of the extra-precessions of the perihelia (and the nodes) by other teams of astronomers.

In the weak-field and slow motion approximation the Einstein field equations of general relativity get linearized resembling the Maxwellian equations of electromagnetism. As a consequence, a gravitomagnetic field arises (Mashhoon 2001(Mashhoon , 2007; it is induced by the off-diagonal components g 0i , i = 1, 2, 3 of the space-time metric tensor related to the mass-energy currents of the source of the gravitational field. It affects orbiting test particles, precessing gyroscopes, moving clocks and atoms and propagating electromagnetic waves (Ruggiero and Tartaglia 2002;Schäfer 2004). The most famous gravitomagnetic effects are, perhaps, the precession of the axis of a gyroscope (Pugh 1959;Schiff 1960), whose detection in the gravitational field of the rotating Earth is the goal of the space-based GP-B experiment 1 (Everitt et al 1974), and the Lense-Thirring 2 precessions (Lense and Thirring 1918) of the orbit of a test particle for which some disputed satellite-based tests in the gravitational fields of the spinning Earth Pavlis 2004, 2005;Iorio 2005aIorio , 2006aIorio , 2007aLucchesi 2005) and Mars (Iorio 2006b;Krogh 2007;Iorio 2008a) have been reported.
We focus on the detection of the solar gravitomagnetic field through the Lense-Thirring planetary precessions of the longitudes of perihelia 3 ̟ = ω + cos i Ω where G is the Newtonian gravitational constant, S is the proper angular momentum of the Sun, c is the speed of light in vacuum, a and e are the semimajor axis and the eccentricity, 1 See on the WEB http://einstein.stanford.edu/ 2 According to an interesting historical analysis recently performed in (Pfister 2007), it would be more correct to speak about an Einstein-Thirring-Lense effect. 3 Here ω is the argument of pericentre, reckoned from the line of the nodes, i is the inclination of the orbital plane to the equator of the central rotating mass and Ω is the longitude of the ascending node. respectively, of the planet's orbit. It may be interesting to know that in (Haas and Ross 1975) it was proposed to measure the solar gravitomagnetic field through the Schiff effect with a drag-free gyroscope orbiting the Sun in a polar orbit.
The impact of the Sun's rotation on the Mercury's longitude of perihelion was calculated for the first time with general relativity by de Sitter (1916) who, by assuming a homogenous and uniformly rotating Sun, found a secular rate of −0.01 arcseconds per century ( ′′ cy −1 in the following). This value is also quoted at pag. 111 of Soffel (1989). Cugusi and Proverbio (1978) yield −0.02 ′′ cy −1 for the argument of perihelion of Mercury. Instead, recent determinations of the Sun's proper angular momentum 4 S ⊙ = (190.0 ± 1.5) × 10 39 kg m 2 s −1 from helioseismology (Pijpers 1998(Pijpers , 2003, accurate to 0.8%, yield a precessional effect one order of magnitude smaller. The predicted gravitomagnetic precessions of the four inner planets, according to the recent value of the Sun's angular momentum, are reported in Table 1; they are of the order of 10 −3 − 10 −5 ′′ cy −1 . Due to their extreme smallness it has been believed for a long time, until recently, that the planetary Lense-Thirring effect would have been undetectable; see, e.g., p. 23 of Soffel (1989). A preliminary analysis showing that recent advances in the ephemerides field are making the situation more favorable was carried out in (Iorio 2005b). Pitjeva (2005a) processed more than 317,000 planetary observations of various kinds collected from 1917 to 2003 with the dynamical 4 It could me measured also with a different approach (Ni 2008).  (Iorio 2007b) to perform a first test.
The errors δ(∆̟) released in (Pitjeva 2005a) were slightly larger than the gravitomagnetic precessions whose predicted values, however, were found compatible with the estimated corrections. Venus was not used because of the poor data set used in the estimation of its extra-precession whose value, indeed, turned out too large to be due to a physically plausible effect amounting to +0.53 ± 0.30 ′′ cy −1 : the Lense-Thirring prediction for the Venus perihelion precession was incompatible with such a result at about 2 − σ level. Now the situation for the second planet of the Solar System has remarkably improved allowing for a more stringent test of the Lense-Thirring effect. Indeed, Pitjeva ( , 2008, in the effort of continuously improving the planetary ephemerides, recently processed more than 400,000 data points  with the EPM2006 ephemerides which encompasses better dynamical models with the exception, again, of the gravitomagnetic force itself. Also in this case she estimated, among more than 230 parameters, the corrections to the usual perihelion precessions for some planets ). In the case of Venus the inclusion of the radiometric data of Magellan (Pitjeva 2008) as well allowed her to obtain 5 in which the quoted uncertainty is the formal, statistical one. By looking at Table 1 it turns out that such an extra-precession can be well accommodated by the general relativistic prediction for the Lense-Thirring rate of the Venus'perihelion whose existence would, thus, be confirmed at 25%. Somebody may object that the gravitomagnetic force should have been explicitly modelled and an ad-hoc parameter accounting for it should have been inserted in the set of parameters to be estimated. Certainly, it may be an alternative approach which could be implemented in future; in addition, we note that the procedure followed by Pitjeva may be regarded, in a certain sense, as safer for our purposes because it is truly model-independent and, since her goal in estimating ∆̟ was not the measurement of the Lense-Thirring effect, there is a priori no risk that, knowing in advance the desired answer, something was driven just towards the expected outcome.
The main question to be asked is, at this point, the following one: Can the result of eq. (2) be explained by other unmodelled/mismodelled canonical or non-conventional dynamical effects? Let us, first, examine some standard candidates like, e.g., the residual precession due to the still imperfect knowledge of the Sun's quadrupole mass moment J ⊙ 2 (Pireaux and Rozelot 2003) whose action was, in fact, modelled by Pitjeva (2005a) by keeping it fixed to J ⊙ 2 = 2 × 10 −7 in the global solution in which she estimated the corrections to the perihelion precessions. The answer is negative since the Newtonian secular precession due to the Sun's oblateness 6 , whatever magnitude J 2 may have, is positive. Indeed, it is (Capderou 2005;Iorio 2007b) where n = GM/a 3 is the Keplerian mean motion and R is the Sun's mean equatorial radius; the angle i between the Venus'orbit and the Sun's equator amounts to 7 3.4 deg only.
Concerning possible exotic explanations, i.e. due to some modifications of the currently known Newton-Einstein laws of gravity, it may have some interest to check some of the recently proposed extra-forces (Standish 2008) which would be able to phenomenologically accommodate the Pioneer anomaly (Nieto 2006). All of such hypothetical new forces have not been modelled by Pitjeva, so that if they existed in Nature they would affect ∆̟ Venus .
A central acceleration quadratic in the radial component v r of the velocity of a test particle 8 (Jaekel and Reynaud 2005;Standish 2008) would induce a retrograde perihelion precession according to (Iorio 2008b) However, eq. (8) predicts a precession of −0.0016 ′′ cy −1 for Venus, which is ruled out by eq.
Should one consider a central uniform acceleration with the magnitude of the Pioneer 8 The quoted numerical value of H allows to reproduce the Pioneer anomaly.
anomalous one, i.e. A = −8.74 × 10 −10 m s −2 , the exotic precession induced by it (Iorio and Giudice 2006;Sanders 2006) on the perihelion of of Venus would bė Another non-conventional effect which may be considered is the precession predicted by Lue and Starkman (2003) in the framework of the DGP multidimensional braneworld model by Dvali, Gabadadze and Porrati (Dvali et al 2000) proposed to explain the cosmic acceleration without invoking dark energy. It iṡ where the plus sign is related to the self-accelerated branch, while the minus sign is for the standard, Friedmann-Lemaître-Robertson-Walker (FLRW) branch; r 0 ≈ 5 Gpc is a threshold characteristic of the DGP model after which gravity would experience neat deviations from the Newtonian-Einsteinian behavior. As can be noted, the self-accelerated branch is ruled out at 9 − σ level by eq. (2), while the FLRW case is still compatible with eq. (2) (1 − σ discrepancy). By the way, apart from the fact that there are theoretical concerns with the DGP model (see, e.g., (Izumi et al 2007) and references therein), the existence of both the Lue-Starkman FLRW precession and the Lense-Thirring one, implying a total unmodelled effect of −0.0008 ′′ cy −1 , would be ruled out by eq. (2) at 4 − σ level.
As a consequence, we can conclude not only that the examined exotic modifications of the standard laws of gravity, not modelled by Pitjeva, are not responsible for the estimated ∆̟ Venus , but also that their existence in the inner regions of the Solar System is falsified by the observations. Moreover, given the magnitudes of the hypothetical effects with the negative sign, it is not possible that reciprocal cancelations with the positive classical mismodelled precessions can explain eq. (2). Indeed, the sum of the latter ones is +0.0004 ′′ cy −1 ; the sum of, e.g., eq. (8) and eq. (12) (FLRW) is −0.0021 ′′ cy −1 , while the sum of eq. (8) and eq. (12) (self-accelerated branch) is −0.0011 ′′ cy −1 .
Thus, we conclude that the most likely explanation for eq. (2) is just the general relativistic Lense-Thirring effect. However, caution is in order in assessing the realistic uncertainty in such a test because, as already stated, the released error of 0.0001 ′′ cy −1 is the formal, statistical one; the realistic uncertainty might be larger. By the way, we can at least firmly conclude that now also in the case of Venus the general relativistic predictions for the Lense-Thirring effect on̟ are compatible with the observational determinations for the unmodelled perihelion precessions, contrary to the case of (Iorio 2007b). Moreover, future modelling of planetary motions should take into account the relativistic effects of the rotation of the Sun as well. The steady improvement in the planetary ephemerides, which should hopefully benefit of the radiometric data from Messenger and Venus-Express as well, should allow for more accurate and stringent test in the near-mid future. Of great significance would be if also other teams of astronomers would estimate their own corrections to the canonical perihelion (and also node) precessions in order to enhance the statistical significance and robustness of this important direct test of general relativity.