Virial theorem · n = 2⟨T⟩/⟨U⟩

Kepler's third law reads a period off an orbit. Can a SINGLE orbit read off the EXPONENT of the force law that made it — the −1 of gravity, the +2 of a spring — from nothing but its time-averaged energies?

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Units: dimensionless (n = homogeneity/scaling degree of U(r) ∝ rⁿ: gravity/Kepler −1, linear +1, harmonic +2; 2⟨T⟩/⟨U⟩ is a ratio of energies, independent of mass, coupling, and orbit size)

How the lab tests it

Integrate a from-scratch bound orbit with velocity Verlet (symplectic, energy-conserving) around a fixed central mass, and time-average the kinetic energy ⟨T⟩ and potential energy ⟨U⟩. The virial theorem says that for a homogeneous potential U ∝ rⁿ, the pure number 2⟨T⟩/⟨U⟩ equals the exponent n — never plugged in, only T(t) and U(t) are measured. Do it for three pure power laws: gravity U=−μ/r (Kepler), a linear confining U=k·r, and an isotropic oscillator U=½k·r². Controls: (i) the INSTANTANEOUS ratio 2T/U, (ii) an eccentricity scan, (iii) a MIXED non-power-law potential. ?world=virial.

What it checks

the virial theorem (Clausius 1870): 2⟨T⟩ = ⟨r·∇U⟩ = n⟨U⟩ for U∝rⁿ, so 2⟨T⟩/⟨U⟩ recovers the potential's homogeneity degree — n=−1 for gravity (the Kepler identity 2⟨T⟩=−⟨U⟩, equivalently ⟨T⟩=−E, ⟨U⟩=2E), n=+1 for the linear, n=+2 for the harmonic, each to ≤0.02 with a slope-1 line through (n_true, n_recovered). DECISIVE CONTROL: the mean of the INSTANTANEOUS ratio ⟨2T/U⟩ MISSES n (−0.82 not −1, 3.80 not 2) — only the ratio of separately time-AVERAGED energies is the virial invariant. The recovered n is INDEPENDENT of eccentricity (−1 across e=0…0.85 — it is a property of the force law, not the orbit shape), and a MIXED potential U=−A/r+½Br² gives a non-integer, orbit-DEPENDENT ratio, so the clean integers require a pure power law. The kinetic-theory sibling of Kepler's third law and the bridge from orbits to the virial mass estimates of galaxies and clusters.

Virial theorem, force-law exponent & cluster virial-mass calculator

One bound orbit, two averaged energies, and the exponent of the force that made it. The virial theorem is the cheapest instrument in mechanics: because the quantity G = Σp·r is bounded on a bound orbit, its long-run rate of change is zero, and what is left is 2⟨T⟩ = ⟨r·∇U⟩ — which for a potential that is a pure power rⁿ collapses by Euler's theorem to 2⟨T⟩ = n⟨U⟩. So the pure number 2⟨T⟩/⟨U⟩ IS the power. It is dimensionless, so the mass, the coupling and the size of the orbit all cancel; it is the same for a comet and an electron; and it is independent of the orbit's shape, which is the part that makes it an instrument rather than a curiosity. The ratio is never typed into this page. It is RECOVERED, by a radial quadrature that integrates dr/v_r between the two turning points and never once looks at the exponent — the same discipline the simulation's oracle runs, which is why the page agrees with it to eleven decimals on gravity, on a linear pull and on a spring. Two controls come with it, and both are closed forms that the oracle upstairs could only measure: the naive pointwise average ⟨2T/U⟩, which for a Kepler orbit is exactly e²/2 − 1 and therefore −0.82 rather than −1 at e = 0.6, and for a centred harmonic ellipse 2(1+q)/√q − 2, which is 3.8 rather than 2 at axis ratio 0.4 — the mean of a ratio is not the ratio of the means, and here the gap is 18% and 90%. The second control is a potential with no single degree at all, U = −A/r + ½Br², whose ratio is the ENERGY-WEIGHTED mean of its two exponents and therefore drifts from orbit to orbit: the clean integers demand a pure power law, and the invariance across orbits is the real signature rather than the integer itself. The last box points the same identity at a galaxy cluster, because 2⟨T⟩ = −⟨U⟩ with T from a velocity dispersion and U from a radius is a mass — the measurement Zwicky made of Coma in 1933 and could not reconcile with its light. Three things this page will not do: give you a speed at a radius, a period, or a semi-major axis from an energy (vis-viva questions, and the Kepler-orbit page owns them); measure G (the Cavendish page does that); or claim this lab weighed a cluster. It did not, and the cluster box says so in its own last line.