Astronomy & Celestial Mechanics
the sky, computed
Concept

Torsion balance — where it appears

An instrument in which a small force twists a thin fibre carrying masses, the twist being read optically. Cavendish used it to measure gravity between lead spheres, and Eötvös to compare how materials fall. Modern versions test the inverse-square law below a tenth of a millimetre.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

G: 14 determinations in two families. Published determinations of G, each with its quoted one-sigma interval, sorted into two families — torsion balance, in one form or another, against beam balance, pendulum, atom interferometry. The shaded band behind each family is that family's inverse-variance weighted mean: 6.67435 ± 0.00004 across 11 of them, against 6.67343 ± 0.00009 across 3. The difference is 0.00092 ± 0.00010 10⁻¹¹ m³ kg⁻¹ s⁻², which is 9.3 standard deviations, computed here from the quoted errors alone. The arithmetic is the same one the Hubble figure uses and here it should be distrusted, because the scatter inside each family already exceeds what the intervals allow: eleven torsion-balance determinations spread over 500 parts per million with quoted intervals of 12 to 130 cannot all be right, whatever the difference between the families comes to. That is why the recommended value's uncertainty is expanded far beyond any single experiment's rather than being the weighted combination drawn here — the disagreement is between laboratories using the same method, not between methods.

Nothing in the sky is weighed in kilograms

The Sun's gravitational parameter is known to eleven significant figures. The Sun's mass is known to five. The two statements are about the same object and the difference between them is a constant measured in basements, which is the worst-determined fundamental constant in physics.

gravitation · Gravitational constant A valley of limits, and how much of it each experiment holds. The strength α a Yukawa term could have, against its range, with both axes logarithmic and everything above the curve excluded at about 95% confidence. The curve is a rounded envelope of published limits: laboratory torsion balances below a metre, tower and mine gravimetry to a few kilometres, the orbits of geodetic satellites near 10⁷ metres, lunar laser ranging near 4×10⁸ metres at a few parts in 10¹¹, and planetary ranging beyond. It does not rule out a term at ranges under about ten micrometres or above about 10¹³ metres; the limits are weakest where no pair of masses of the right size can be arranged.

A force that can hide at any one distance

The inverse-square law is tested by comparing the strength of gravity at two distances, and every instrument that does so is sensitive over about one decade of range. A Yukawa force with a range outside that decade is invisible to it: much shorter and it never reaches, much longer and it only rescales the constant. So testing the law is not one experiment made more precise but a chain of unrelated ones, from torsion pendulums to the orbits of planets, and the chain has weak links exactly where no pair of masses of the right size can be arranged.

gravitation · Gravitational constant

Named alongside it

The objects these essays reach for when they reach for this one.

Astronomical unit (AU)Dimensional analysisEquivalence principleFifth forceGravitational constantInverse square lawKepler's third lawLunar laser rangingMass ratioRadar rangingSolar-massStandard gravitational parameter

All concepts