= Kirkwood gap
{c}
{title2=$a\simeq a_J(m/n)^{-2/3}$}
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.https://ssd.jpl.nasa.gov/diagrams/mb_hist.html[Main-Belt Distribution; June 2007 sample of numbered asteroids]{ref}
At the $3:1$ resonance an asteroid makes three revolutions per Jovian revolution. <Kepler's third law> gives $a\simeq a_J3^{-2/3}\simeq2.50\,\mathrm{au}$. 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.<Chaotic Behavior and the Origin of the 3 1 Kirkwood Gap by Jack Wisdom>[1983, abstract and sections 1–2]{ref} This numerical comparison was a test of the mechanism, not just an association between a resonance label and a histogram dip.
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