We work out the impact that the recently determined time-dependent component of the Pioneer Anomaly (PA), interpreted as an additional exotic acceleration of gravitational origin with respect to the well known PA-like constant one, may have on the orbital motions of some planets of the solar system. By assuming that it points towards the Sun, it turns out that both the semi-major axis a and the eccentricity e of the orbit of a test particle experience secular variations. For Saturn and Uranus, for which modern data records cover at least one full orbital revolution, such predicted anomalies are up to 2-3 orders of magnitude larger than the present-day accuracies in empirical determinations their orbital parameters from the usual orbit determination procedures in which the PA was not modeled. Given the predicted huge sizes of such hypothetical signatures, it is unlikely that their absence from the presently available processed data can be attributable to an "absorption" for them in the estimated parameters caused by the fact that they were not explicitly modeled. The magnitude of a constant PA-type acceleration at 9.5 au cannot be larger than 9 10^-15 m s^-2 according to the latest observational results for the perihelion precession of Saturn.
Deep Dive into Orbital effects of the time-dependent component of the Pioneer anomaly.
We work out the impact that the recently determined time-dependent component of the Pioneer Anomaly (PA), interpreted as an additional exotic acceleration of gravitational origin with respect to the well known PA-like constant one, may have on the orbital motions of some planets of the solar system. By assuming that it points towards the Sun, it turns out that both the semi-major axis a and the eccentricity e of the orbit of a test particle experience secular variations. For Saturn and Uranus, for which modern data records cover at least one full orbital revolution, such predicted anomalies are up to 2-3 orders of magnitude larger than the present-day accuracies in empirical determinations their orbital parameters from the usual orbit determination procedures in which the PA was not modeled. Given the predicted huge sizes of such hypothetical signatures, it is unlikely that their absence from the presently available processed data can be attributable to an “absorption” for them in the
According to the latest analysis 1 of extended data records of the Pioneer 10/11 spacecraft, the small frequency drift 2,3 (blue-shift) observed analyzing the navigational data of both the spacecraft, known as Pioneer Anomaly (PA), may present a further time-dependent component in addition to the well known constant one. Both linear and exponential models were proposed 1 for the PA; according to the authors of Ref. 1, the exponential one is directly connected to non-gravitational effects 4 since it takes into account the possible role of the on-board power generators suffering a radioactive decay.
In this letter we work out the orbital effects of such a new term in the hypothesis that the time-dependent PA component is due to some sort of long-range modification of the known laws of gravitation resulting in an additional anomalous
in terms of which the constant part of the PA has often been interpreted. Indeed, in this case it should act on the major bodies of the solar system as well, especially those whose orbits lie in the regions in which the PA manifested itself in its presently known form. In this respect, we will not consider the exponential model. Recent studies 5,6,7,8,9,10,11 , partly preceding the one in Ref. 1, pointed towards a mundane explanation of a large part of the PA in terms of non-gravitational effects pertaining the spacecraft themselves.
Since the anomalous acceleration is 1
the time-dependent linear component of the postulated PA-type acceleration 1
can be treated as a small perturbation of the dominant Newtonian monopole A N over timescales of the order of an orbital period P b for all the planets of the solar system. Table 1 explicitly shows this fact for Saturn, Uranus, Neptune and Pluto which move just in the spatial regions in which the PA perhaps started to appear (Saturn), or fully manifested itself (Uranus, Neptune, Pluto) in its presently known form. Thus, the Gauss equations for the variation of the osculating Keplerian orbital elements 12 , which are valid for any kind of disturbing acceleration A, independently of its physical origin, can be safely used for working out the orbital effects of eq.
(3). In particular, the Gauss equations for the semi-major axis a and eccentricity e November
Orbital effects of a time-dependent Pioneer-like acceleration 3 of the orbit of a test particle moving around a central body of mass M are
they allow one to work out the rates of changes of a and e averaged over one orbital period P b as
In eq. ( 5) (dΨ/dt) K are the right-hand-sides of eq. ( 4) evaluated onto the unperturbed Keplerian ellipse. In eq. ( 4) A R , A T are the radial and transverse components of a the generic disturbing acceleration A, p . = a(1 -e 2 ) is the semilatus rectum, n . = GM/a 3 is the unperturbed Keplerian mean motion related to the orbital period by n = 2π/P b , G is the Newtonian constant of gravitation, and f is the true anomaly. Since the new data analysis 1 does not rule out the line joining the Sun and the spacecrafts as a direction for the PA, we will assume that eq. ( 3) is entirely radial, so that A R = A, A T = 0. Using the eccentric anomaly E as a fast variable of integration turns out to be computationally more convenient. To this aim, useful relations are
As a result, a and e experience non-vanishing secular variations
Notice that eq. ( 7) are exact in the sense that no approximations in e were assumed. Moreover, they do not depend on t 0 .
In order to make a meaningful comparison of eq. ( 7) with the latest empirical results from planetary orbit determinations, we recall that modern data records cover at least one full orbital revolution for all the planets with the exception of Neptune and Pluto. The author of Ref. 13, in producing the EPM2006 ephemerides, made a global fit of a complete suite of standard dynamical force models acting on the solar system’s major bodies to more than 400,000 observations of various kinds ranging over ∆t = 93 yr (1913 -2006). Among the about 230 estimated parameters, there are the planetary orbital elements as well. According to
so that
can naively be inferred for their rates by simply dividing eq. ( 8) by ∆t. The PA was not modeled in the EPM2006. It is important to remark that the figure for σ a quoted in eq. ( 8) was obtained without processing the radiotechnical observations of the Cassini spacecraft. According to eq. ( 7), the putative PA-induced secular changes of the semi-major axes of Saturn and Uranus are ȧ(Pio) = 42, 505 m yr -1 ȧ(Pio) = 290, 581 m yr -1 .
These are about 3 orders of magnitude larger than eq. ( 9): even by re-scaling the formal uncertainties of eq. ( 9) by a factor of 10, the PA-type anomalous rates of eq. ( 10) would still be about 2 orders of magnitude too large to have escaped from a detection. Such conclusions are confirmed, and even enforced, by using the latest results published in Ref. 14
which is 4 orders of magnitude smaller than the predicted value of eq. ( 10). Inc
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