4.1 Solar systems constraints

Monitoring the orbits of the various bodies of the solar system offers a possibility to constrain deviations from general relativity, and in particular the time variation of G. This accounts for comparing a gravitational time scale (related to the orbital motion) and an atomic time scale and it is thus assumed that the variation of atomic constants is negligible over the time of the experiment.

The first constraint arises from the Earth-Moon system. A time variation of G is then related to a variation of the mean motion (n = 2π∕P) of the orbit of the Moon around the Earth. A decrease in G would induce both the Lunar mean distance and period to increase. As long as the gravitational binding energy is negligible, one has

P˙ G˙ P = − 2G . (133 )
Earlier constraints rely on paleontological data and ancient eclipses observations (see Section IV.B.1 of FVC [500Jump To The Next Citation Point]) and none of them are very reliable. A main difficulty arises from tidal dissipation that also causes the mean distance and orbital period to increase (for tidal changes 2˙n∕n + 3a˙∕a = 0), but not as in the same ratio as for G˙.

The Lunar Laser Ranging (LLR) experiment has measured the relative position of the Moon with respect to the Earth with an accuracy of the order of 1 cm over 3 decades. An early analysis of this data [544] assuming a Brans–Dicke theory of gravitation gave that |G˙∕G | ≤ 3 × 10 −11 yr− 1. It was improved [365] by using 20 years of observation to get | ˙G∕G | ≤ 1.04 × 10−11 yr−1, the main uncertainty arising from Lunar tidal acceleration. With, 24 years of data, one reached [542] |G ˙∕G | ≤ 6 × 10 −12 yr− 1 and finally, the latest analysis of the Lunar laser ranging experiment [543] increased the constraint to

| ˙G || --| = (4 ± 9) × 10−13 yr−1. (134 ) G |0

Similarly, Shapiro et al. [458] compared radar-echo time delays between Earth, Venus and Mercury with a caesium atomic clock between 1964 and 1969. The data were fitted to the theoretical equation of motion for the bodies in a Schwarzschild spacetime, taking into account the perturbations from the Moon and other planets. They concluded that |G ˙∕G | < 4 × 10 −10 yr− 1. The data concerning Venus cannot be used due to imprecision in the determination of the portion of the planet reflecting the radar. This was improved to |G˙∕G | < 1.5 × 10−10 yr−1 by including Mariner 9 and Mars orbiter data [429]. The analysis was further extended [457] to give G˙∕G = (− 2 ± 10) × 10−12 yr−1. The combination of Mariner 10 an Mercury and Venus ranging data gives [12]

| G˙| --|| = (0.0 ± 2.0) × 10−12 yr−1. (135 ) G |0

Reasenberg et al. [430] considered the 14 months data obtained from the ranging of the Viking spacecraft and deduced, assuming a Brans–Dicke theory, |G˙∕G | < 10− 12 yr−1. Hellings et al. [249Jump To The Next Citation Point] using all available astrometric data and in particular the ranging data from Viking landers on Mars deduced that

| ˙ | G-|| = (2 ± 4) × 10−12 yr−1. (136 ) G | 0
The major contribution to the uncertainty is due to the modeling of the dynamics of the asteroids on the Earth-Mars range. Hellings et al. [249] also tried to attribute their result to a time variation of the atomic constants. Using the same data but a different modeling of the asteroids, Reasenberg [428] got |G ˙∕G | < 3 × 10−11 yr−1, which was then improved by Chandler et al. [93] to ˙ − 11 −1 |G ∕G | < 10 yr.
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