Asteroid belt

The main asteroid belt is a population of asteroids and other Small Solar System body objects orbiting chiefly between Mars and Jupiter. Its members occupy many separate orbits, rather than forming a solid ring. Their orbital distribution records both the history of planet formation and continuing gravitational perturbations.

A particularly clear observation is the series of Kirkwood gaps, deficits in asteroid counts at certain semi-major axes. The JPL Solar System Dynamics histogram, constructed in June 2007 from 156,929 numbered asteroids with well-determined orbits, shows gaps associated with mean-motion resonances with Jupiter.^[ref]

For the $3:1$ resonance, an asteroid makes three revolutions during one Jovian revolution, so $T_a=T_J/3$. Using Kepler's third law and Jupiter's semimajor axis of about $5.20$ astronomical units givesRepeated gravitational forcing can change orbital elements and remove bodies from parts of this region. A resonance label specifies the ratio of orbital frequencies here, so reversing it into a period ratio would give the wrong location. The histogram shows the observed distribution under a stated selection, rather than a complete census of every small asteroid.

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.