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By Jack Wisdom.

Verified locator: Icarus 56 (1983), 51–74. Abstract and sections 1–2 compare the mapping and direct integration and describe the eccentricity jumps and chaotic gap boundary.

Jack Wisdom is a planetary scientist whose work applies Hamiltonian mechanics, chaotic dynamics and numerical methods to the motion of bodies in the Solar system. His MIT profile documents contributions to long-duration integrations and mapping methods for planetary motion.^[ref]

His 1983 study of the three-to-one Kirkwood gap investigated how an orbital resonance with Jupiter can produce large changes in an asteroid's orbital eccentricity. It compared a simplified mapping with direct numerical integration, showing that the sudden eccentricity changes were not merely artifacts of the faster model.^[ref]

The three-to-one label means that an asteroid completes approximately three orbits for one orbit of Jupiter. Using Kepler's third law in an ideal solar two-body approximation givesThis locates the resonance approximately; it does not by itself establish that every trajectory there is unstable. The dynamical calculation asks how repeated perturbations alter the orbit over time. That distinction connects a simple period ratio to the harder problem of explaining an observed depletion of asteroids.

A Kirkwood gap is a depletion in the distribution of asteroid semi-major axes near certain mean-motion resonances with Jupiter. It is a deficit of long-lived orbits in orbital-element space, rather than an empty circular lane that every asteroid must avoid. The JPL orbital histogram shows prominent gaps at the $3:1$, $5:2$, $7:3$ and $2:1$ resonances.^[ref]

At the $3:1$ resonance an asteroid makes three revolutions per Jovian revolution. Kepler's third law gives $a\simeq a_J{3}^{-2/3}\simeq 2.50\mathrm{au}$. Repeated perturbations can drive changes in orbital eccentricity, allowing some initially modest-eccentricity orbits to become planet crossing. Thus depletion need not result from an immediate large change in semi-major axis.

Jack Wisdom tested this mechanism by comparing an efficient approximate mapping with direct numerical integration of the planar elliptic restricted three-body problem. Both produced sudden eccentricity increases and consistent chaotic behavior; the computed chaotic boundary matched the observed $3:1$ gap within orbital uncertainties.^[ref] This numerical comparison was a test of the mechanism, not just an association between a resonance label and a histogram dip.