By processing more than 400 000 planetary observations of various types
with the dynamical models of the EPM2006 ephemerides, E.V. Pitjeva 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 mismodelled/unmodelled
standard Newtonian/Einsteinian effects nor exotic ones, postulated to, for example,
explain the Pioneer anomaly, may have caused the estimated 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 from 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 [1, 2]; it is 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. It affects orbiting
test particles, precessing gyroscopes, moving clocks and atoms, and propagating
electromagnetic waves [3, 4]. The most famous gravitomagnetic effects are,
perhaps, the precession of the axis of a gyroscope [5, 6], whose detection in
the gravitational field of the rotating Earth is the goal of the space-based
GP-B experiment [7] (see
http://einstein.stanford.edu/), and the
Lense-Thirring precessions [8] of the orbit of a test particle for which some
disputed satellite-based tests in the gravitational fields of the spinning
Earth [9–14]
and Mars [15–17] have been reported.
(According to an interesting historical analysis
recently performed in [18], it would be more correct to speak about
an Einstein-Thirring-Lense effect.)
We focus on the detection of the solar gravitomagnetic
field through the Lense-Thirring planetary precessions of the longitudes of
perihelia ϖ=ω+cosiΩ,dϖdt=−4GScosic2a3(1−e2)3/2,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,
respectively, of the planet's orbit. (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). It may be interesting to know that in [19]
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 [20] 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 in [21, page 111].
Cagusi and Proverbio [22] yield −0.02′′cy−1 for the argument of perihelion
of Mercury. Instead, recent determinations of the Sun's proper angular momentum S⊙=(190.0±1.5)×1039kgm2s−1 from helioseismology [23, 24], 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 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, for
example, [21, page 23]. A preliminary analysis
showing that recent advances in the ephemerides field are making the situation
more favorable was carried out in [25]. Pitjeva [26] processed more than 317 000 planetary observations of various kinds collected from 1917 to 2003 with
the dynamical force models of the EPM2004 ephemerides [27]. This
produced a global solution in which she also estimated, among
many other parameters, corrections Δϖ˙ to the canonical Newton-Einstein perihelion
precessions for all the inner planets. Since the gravitomagnetic force was not
modelled at all, contrary to the static, Schwarzschild-like component of the general relativistic
force of order 𝒪(c−2),
such corrections to the usual perihelia evolutions account, in principle, for
the Lense-Thirring effect as well, in addition to the mismodelled parts of the
standard Newtonian/Einsteinian precessions. Thus, the estimated corrections for
the perihelion rates of Mercury, the Earth, and Mars have been used in [28] to
perform a first test. The errors δ(Δϖ˙) released in [26] 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 dataset used in the estimation of its extra-precession whose value, indeed,
turned out to be too large 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.
Lense-Thirring precessions, in ′′cy−1, of the longitudes of the perihelion ϖ of the inner planets of the solar system
induced by the gravitomagnetic field of the Sun. The value S⊙=(190.0±1.5)×1039kgm2s−1 has been assumed for its angular momentum.
Mercury
Venus
Earth
Mars
−0.0020
−0.0003
−0.0001
−0.00003
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 [29, 30], in the effort of continuously
improving the planetary ephemerides, recently processed more than 400 000 data
points (1913–2006) with the EPM2006 ephemerides which encompass 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 [29]. In the
case of Venus, the inclusion of the radiometric data of Magellan [30] as well
allowed her to obtain Δϖ˙Venus=−0.0004±0.0001′′cy−1,in which the quoted uncertainty is the formal, statistical one (personal communication by Pitjeva to the author, June 2008). 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%.
It may be objected 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. Since her goal in
estimating Δϖ˙ 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 toward the expected outcome.
The main question to be asked is, at this point, the
following: can the result of (2) be explained by other unmodelled/mismodelled
canonical or nonconventional dynamical effects? Let us, first, examine some
standard candidates like the residual precession due to the still imperfect
knowledge of the Sun's quadrupole mass moment J2⊙ [31] whose action was, in fact, modelled by
Pitjeva [26] by keeping it fixed to J2⊙=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,
(for an oblate body J2>0) whatever magnitude J2 may have, is positive. Indeed, it is [28, 32]ϖ˙J2=32nJ2(1−e2)2(Ra)2(1−32sin2i),where n=GM/a3 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 3.4 deg only. (Indeed, the orbit of
Venus is tilted by 3.7 deg to the mean ecliptic of J2000
(http://ssd.jpl.nasa.gov/txt/aprx_pos_planets.pdf)), while the Carrington's angle between the Sun's equator and the
ecliptic is 7.15 deg [33]). For J2⊙=2×10−7,
the nominal value of the Venus' perihelion precession induced by the solar quadrupole mass moment amounts to +0.0026′′cy−1. By assuming an uncertainty of about δJ2≈10%. [34], if Δϖ˙Venus was due to such a mismodelled effect, it
should amount to +0.0002′′cy−1, which is, instead, ruled out at 6−σ level. Concerning the precession due to the
solar octupole mass moment J4⊙,
it is [32]ϖ˙J4=−1516nJ4(Ra)4[3(1−e2)3+7(1+(3/2)e2)(1−e2)4]×(74sin4i−2sin2i+25).For Venus, it amounts to −1.2J4⊙′′cy−1.
Since J4⊙≈−4×10−9 [35, 36], we conclude that the second even
zonal harmonic of the multipolar expansion of the solar gravitational potential
cannot be responsible for (2). More generally, it
does not represent a potentially relevant source of systematic error for the
measurement of the Lense-Thirring planetary precessions. Similar arguments hold
also for other potential sources of systematic errors, for example, the
asteroid ring and the Kuiper Belt objects, both modelled in EPM2006. The
precessions induced by them are positive. Indeed, a Sun-centered ring of mass mring and inner and outer radius Rmin/max≫a induces a perihelion precession [37]:ϖ˙ring=34Ga3(1−e2)MmringRminRmax(Rmin+Rmax)>0.According to (5), the precession
induced by the asteroids' ring on the Venus' perihelion amounts to +0.0007±0.0001′′cy−1 by using mring=(5±1)×10−10M⊙ [38]. The lowest value +0.0006′′cy−1 is incompatible with (2) at 10−σ level. In the case of the Kuiper belt objects,
(5) yields a precession of the order of +0.00006′′cy−1 with m=0.052m⊕ [37]. Thus, we can rule out such modelled
classical features of the Sun and the solar system as explanations of Δϖ˙Venus.
General relativistic terms of order 𝒪(c−4) were not modelled by Pitjeva.
However, the first correction of order 𝒪(c−4) to the perihelion precession [39] can be safely
neglected because for Venus, it isϖ˙c4∝n(GM)2c4a2(1−e2)2≈10−7′′cy−1.
Concerning possible exotic explanations, that is, 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 [40] which would be able to phenomenologically accommodate the Pioneer anomaly
[41]. All of such hypothetical new forces have not been modelled by Pitjeva, so
that if they existed in nature, they would affect Δϖ˙Venus.
A central acceleration quadratic in the radial component vr of the velocity of a test particle [40, 42]A=−vr2ℋ,ℋ=6.07×10−18m−1would induce a retrograde
perihelion precession according to [43]ϖ˙=ℋna1−e2e2(−2+e2+21−e2)<0.(The
quoted numerical value of ℋ allows to reproduce the Pioneer
anomaly). However, (8) predicts a precession of −0.0016′′cy−1 for Venus, which is ruled out by (2) at 12−σ level. Another possible candidate considered
in [40] is an acceleration linear in the radial velocityA=−|vr|𝒦,𝒦=7.3×10−14s−1,which yields a retrograde
perihelion precession [43]:ϖ˙=−𝒦1−e2π[2e−(1−e2)ln((1+e)/(1−e))e2]<0.The prediction of (10) for Venus
is −0.1′′cy−1, clearly incompatible with (2). Should one consider a central uniform
acceleration with the magnitude of the Pioneer anomalous one, that is, A=−8.74×10−10ms−2, the exotic precession induced by it [44, 45] on the perihelion of Venus would beϖ˙Ven=Aa(1−e2)GM=−16′′cy−1.Another nonconventional effect
which may be considered is the precession predicted by Lue
and Starkman [46] in the framework of
the DGP multidimensional braneworld model by Dvali et al. [47] which is proposed to explain the cosmic
acceleration without invoking dark energy. It isϖ˙LS=∓3c8r0+𝒪(e2)≈∓0.0005′′cy−1,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; r0≈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 (2), while the FLRW case is still
compatible with (2) (1−σ discrepancy). By the way, apart from the fact
that there are theoretical concerns with the DGP model (see, e.g., [48] 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 (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 Δϖ˙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
(2). Indeed, the sum of the latter ones is +0.0004′′cy−1; the sum of, for example, (8) and (12) (FLRW) is −0.0021′′cy−1, while the sum of (8) and (12) (self-accelerated branch) is −0.0011′′cy−1.
Thus, we conclude that the most likely explanation for
(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 on ϖ˙ are compatible with the observational
determinations for the unmodelled perihelion precessions, contrary to the case
of [28]. 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. It would
be of great significance 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.
Acknowledgements
The author would like to thank E. V. Pitjeva for useful and important communications. He is also grateful to the referees for their useful and relevant
remarks.
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