4 Inverse Square Law

The inverse square law (ISL) of gravity has been meaningfully tested over length scales spanning 20 orders of magnitude, eliminating Yukawa-like couplings competitive with the strength of gravity from 10–4 to 1016 meter length scales. The deepest probe of the ISL is from LLR at a scale of ∼ 108 meters, where any new force must be weaker than gravity by more than ten orders-of-magnitude [38Jump To The Next Citation Point]. Short-range tests of the ISL have recently been prompted by the energy scale of the cosmological acceleration, which suggest new-physics below 1 mm [1Jump To The Next Citation Point].

Modern tests of Newton’s inverse-square law of gravity often search for an additional Yukawa contribution to the gravitational potential:

m1m2--( −r∕λ ) V (r) = − G r 1 + αe , (12 )
where α is the dimensionless strength and λ is the length scale. Such a potential would generate a precession of the Moon’s perigee with frequency δω [1]:
( ) δω- α- a- 2 −a∕λ ω = 2 λ e , (13 )
where a is the mean radius of the Moon’s orbit. The agreement of geodetic precession with GR described below leads to a limit on an anomalous precession of δω ∕ω < 1.6 × 10− 11. This translates into a limit on the strength of a new Yukawa potential of α < 5.9 × 10− 11 at λ = a ∕2 where the lunar test is most sensitive.

Recent analysis of LLR data includes specifically fitting for Yukawa perturbation terms in the equations of motion leading to a measurement of α = (3 ± 2) × 10−11 at λ = 4 × 105 km. While intriguing, this possible non-null result has yet to be thoroughly investigated [38Jump To The Next Citation Point].


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