Locations of Solar System Planetary Mean-Motion Resonances

Mean-motion resonances are denoted here by p+q:p, where p is a positive integer and q is an integer restricted to q > -p. Thus, a massless object orbiting closer to the Sun than and in resonance with a planet would orbit the Sun p+q times for every p times the planet orbits the Sun, and q will be positive. For an object orbiting outside the planet, q will be negative. The calculator below will output the semimajor axis of a massless object in a p+q:p mean-motion resonance with the specified planet, in Astronomical Units.

calculate a mean-motion resonance
 planet p+q p Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

The plots below (push the buttons to see them) contain distributions of the osculating semimajor axes of the 329,654 minor planets from the 2006 March 17 version of the Lowell Observatory database. The distributions are shown as average shifted histograms. There is also one plot of the perihelion distance, 0 < q < 1.6. (This q is different from the resonance integer q!) Overlayed on the plots are colored vertical lines showing the locations of various mean-motion resonances with nearby planets.

The 1.0 < a < 5.5 plot is an overview of the asteroid belt interior to Jupiter. The 0 < q < 1.6 and 0.6 < a < 1.7 plots zoom in on the inner solar system. The 2.1 < a < 3.3 plot shows the main belt. The 35 < a < 50 plot shows the region of the (non-scattered) Trans-Neptunian Objects.

select a mean-motion resonance plot

Distances are AU. Vertical scales are asteroids per histogram bin, with different bin widths for the different plots. The mean-motion resonances, along with the histograms, were calculated and plotted using Maple. In the calculations, values used for the planetary masses and semimajor axes are from JPL.

In all of the plots except the Trans-Neptunian Objects (TNOs) and the perihelia, black is Jupiter, aquamarine is Mars, blue is Earth, and red is Venus. For the perihelia plot (0 < q < 1.6), black is Mercury. For the TNOs (35 < a < 50), aquamarine is Saturn, blue is Uranus, and red is Neptune.

The length of a resonance line corresponds to its order |q|. For example, q=1 is a first-order resonance, etc. The smaller |q| is, the stronger the resonance (i.e., the more it will perturb the resonant object's motion over time if the resonance happens to be stable), and thus the longer the plotted line relative to the other resonant lines from the same planet. Each planet's resonance lines are vertically offset for readability.