Bertrande

Bertrande traces two orbits at once: a Kepler ellipse in one hand, an oscillator ellipse in the other. Every other force law frays into a rosette behind her.

The Physics

Among all central forces, only two make every bounded orbit close on itself, for any starting conditions: the inverse-square law and the linear (Hooke's-law) force. For inverse-square attraction the centre of force sits at a focus of the ellipse; for the isotropic oscillator it sits at the centre. Any other law lets the orbits precess into rosettes. A quick test uses near-circular orbits: β, the ratio of radial to orbital angular frequency, must come out as 1 or 2.

F = −k/r² or F = −k r; β² = 3 + r f′(r)/f(r) = 1 or 4

• F: radial force on the body (N), minus = attractive
• r: distance from the force centre (m)
• k: force constant (Kepler N m², oscillator N/m)
• f(r): magnitude of the attractive central force (N)
• β: radial-to-orbital frequency ratio for near-circular orbits (dimensionless)

Joseph Bertrand proved it in the Comptes Rendus in 1873 (vol. 77, p. 849). For a power law f ∝ rⁿ the test gives β² = n + 3, so n = −2 or n = 1.

Honest caveat: The theorem is about ALL bounded orbits. Other force laws do allow some closed orbits; only these two force every orbit to close.

On the shirt

Front: Bertrande, the pixel-art character, with the title. Back: the two closed-orbit laws, the β test, and three orbit panels (focus, centre, open rosette). Printed on a unisex tee.