Cassini's laws

Cassini's laws describe equilibrium orientations of a synchronously rotating body's spin axis relative to its orbit and the Laplace plane, when the orbit precesses steadily.

Cassini state: An equilibrium configuration in which a body's spin axis, orbital normal, and Laplace-plane normal remain coplanar as the orbit precesses. Each law identifies a possible Cassini state for a synchronously rotating body.
Cassini state 1: The low-obliquity Cassini equilibrium, with the spin axis near the orbit normal and on the same side of the Laplace-plane normal. It is the usual near-aligned state for a synchronously rotating satellite.
Obliquity: The angle between a body's spin axis and the normal to its orbital plane. Cassini-state orientations are distinguished largely by their spin-axis obliquities.
Giovanni Domenico Cassini: An Italian-French astronomer who studied Saturn, its moons, and the motions of celestial bodies in the seventeenth century. The laws bear Cassini's name, although the modern equilibrium framework developed later.
Spin-orbit resonance: A commensurability between a body's rotation rate and its orbital motion, often produced by gravitational torques. Synchronous rotation is the 1:1 spin-orbit resonance assumed in the laws.
Cassini state 2: A high-obliquity Cassini equilibrium in which the spin axis lies on the opposite side of the orbit normal from the Laplace-plane normal. Its large obliquity is a distinct equilibrium allowed by the laws.
Orbital inclination: The angle between an orbit's plane and a chosen reference plane. Inclination measures how the satellite orbit tilts relative to the Laplace plane.
Pierre-Simon Laplace: A French mathematician and astronomer whose work on celestial mechanics shaped the mathematical study of planetary and satellite motion. The Laplace plane in the laws is named for his analysis of satellite orbital dynamics.
Synchronous rotation: Rotation in which a body's spin period equals its orbital period, keeping the same hemisphere directed toward its primary. This rotational state supplies the premise for the equilibria described by the laws.
Cassini state 3: A Cassini equilibrium with the spin axis near the direction opposite the orbit normal, generally on the same side of the Laplace-plane normal. It represents a near-retrograde orientation among the allowed equilibria.

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LemniscateRelated: The lemniscate can be described as a special Cassini oval.