Orbital Resonance Checker

Compare measured period ratio against low-integer commensurabilities to detect potential mean-motion resonance structure.

Worked example — Orbital Resonance Checker

Nearest resonance	5:2
Target ratio	2.5
Detuning	0.671564%
Resonance order (p - q)	3
Repeat pattern	5 inner / 2 outer
Measured ratio (Pouter/Pinner)	2.48332291

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Orbital Resonance Checker

Two bodies are in mean-motion resonance when their orbital periods form a ratio of small whole numbers, so they return to the same relative positions again and again. Because the nudges they give each other then repeat in step rather than averaging away, resonance is not a coincidence of numbers — it is a mechanism.

What each symbol means

P_inner: the inner body’s orbital period. The default 4332.59 days is Jupiter’s.
P_outer: the outer body’s period, 10759.22 days — Saturn’s.
p:q: the resonance: p inner orbits for every q outer ones. Here 5:2.
detuning: how far the real ratio sits from the exact one, as a percentage — 0.671564% for Jupiter and Saturn.
order: the order p − q, which is 3 here. Low order means a stronger effect.

Where the formula comes from

Divide the two periods. Jupiter and Saturn give 10759.22 / 4332.59 = 2.48332291.
Now find the closest ratio of small whole numbers to that decimal, allowing denominators up to the limit set above. 2.48332291 sits very near 5/2 = 2.5.
Compare the two: (2.5 − 2.48332291) / 2.5 is about 0.67%. That residue is the detuning, and it is what decides whether the pair is genuinely locked or merely nearby.

Assumes: Circular, coplanar orbits and periods that do not change. Real resonances are held by a balance of perturbations that this ratio arithmetic does not model, so a small detuning is evidence of a resonance rather than proof of one.
Breaks when: The defaults are the classic near-miss. Jupiter and Saturn are famously close to 5:2 but not in it, and the readout puts a figure on the gap: a 0.67% detuning. Note also that with a high enough integer limit some ratio always looks close: a small detuning only means something if the integers are small too, which is what the order column is telling you.

The same resonance can protect an orbit or empty it 🖖

Resonance is often described as stabilising, which is only half true — the identical mechanism does the opposite depending on the geometry. Pluto crosses Neptune’s orbit, which sounds fatal, yet its 3:2 lock (period ratio 1.5045) guarantees the two are never near each other when it happens, and the arrangement is stable over billions of years. Meanwhile the asteroid belt has gaps precisely at its resonances with Jupiter — the Kirkwood gaps at 3:1, 5:2, 7:3 and 2:1 — where repeated tugs arriving in step pumped up eccentricities until those rocks were thrown out entirely. Whether the repeated nudges cancel or accumulate is what decides which case you get, and the period ratio alone does not tell you.

The clockwork behind a period ratio 🖖

A resonance simply means two orbits keep the same beat. When the period ratio sits near a small whole-number ratio like 3:2, the bodies return to the same relative positions over and over. Pluto circles the Sun twice for every three orbits of Neptune, so the pair never make a close approach even though their paths cross. This tool finds the simplest ratio your periods lie near, and the detuning tells you how nearly exact that rhythm is.

The 5:2 near-miss that alarmed astronomers 🖖

Jupiter and Saturn sit just short of an exact 5:2 resonance — their period ratio is about 2.48, not 2.50. That small detuning makes their mutual tugs beat slowly in and out over roughly 900 years, so Jupiter appeared to speed up while Saturn drifted away. Eighteenth-century observers feared the solar system was unravelling, until Laplace proved in 1785 that this 'great inequality' was a harmless periodic swing set by the offset from exact resonance.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Jupiter–Saturn is the famous resonance in this list. Run all seven presets and find which pair is actually the most precisely locked.

Show answer

Titan–Hyperion, at a detuning of 0.079898% — more than eight times tighter than Jupiter–Saturn’s 0.671564%, and tighter than Pluto–Neptune (0.305322%), Io–Europa (0.366094%) or Europa–Ganymede (0.740741%). Jupiter–Saturn is partly famous for being loose: it is a 5:2 near-miss of order 3, and the slow drift of that mismatch is the “great inequality” that spoiled predictions of both planets until Laplace accounted for it in 1785. Read the two columns together rather than one — Titan–Hyperion is 4:3, order 1, while Jupiter–Saturn is order 3, and low order is what lets a resonance hold.
The near 3:2 example preset has a period ratio of exactly 1.52, and the panel calls it 3:2 with 1.315789% detuning. Raise the max resonance integer from 10 to 20, then 26, then 30, and watch two columns move in opposite directions.

Show answer

Detuning falls — 1.214575% at 20:13, then 0.619195% at 26:17, then 0.415512% at 29:19 — while the order climbs 1 → 7 → 9 → 10. Allow bigger integers and you can always get closer, because every ratio is approximated arbitrarily well by fractions, so a small detuning on its own proves nothing at all. The order column is the guard against that: a 10th-order resonance is far too weak to hold two bodies in step, which makes 3:2 at 1.3% the more meaningful reading than 29:19 at 0.4%. The question is never “is there a fraction?” but “is there a fraction with small integers?” The limit stops at 30, so the tool will not chase this past 29:19.

Problem solved in full

The 5:2 period ratio near-miss of Jupiter and Saturn 5 steps
Jupiter takes 4332.59 days and Saturn 10 759.22. Their period ratio is 2.4833 — very nearly 5:2, and not quite. Work out how near, and then what that near-miss actually does.
1. The ratio is the observation, and it is close enough to a simple fraction that the closeness cannot be coincidence.
  
  $\frac{P _{out}}{P _{in}} = \frac{10 759.22}{4332.59} = 2.48332291$
2. The nearest simple ratio is 5:2 — five Jupiter orbits to two of Saturn's. The order of a resonance is the difference of those integers, and 3 is high enough that the effect is weak per encounter.
  
  $nearest simple ratio = \frac{5}{2} = 2.5, order = 5 - 2 = 3$
3. The detuning is under 0.7%. That is the number the panel reports, and on its own it invites the wrong conclusion: that the planets are almost locked and the mismatch is a small error.
  
  $detuning = \frac{∣2.48332291 - 2.5∣}{2.48332291} = 0.671564%$
4. It is not an error, it is a frequency. Combine the two orbital rates in the 5:2 proportion and what is left over is a slow beat — the rate at which the conjunction point drifts around the sky.
  
  $\frac{2}{P _{J}} - \frac{5}{P _{S}} = \frac{2}{4332.59} - \frac{5}{10 759.22} = 3.10 \times 1 0^{- 6} d^{- 1}$
5. That beat has a period of about 883 years. Jupiter and Saturn do not stay misaligned; they cycle through the whole pattern and come back.
  
  $T_{beat} = \frac{2}{P _{J}} - \frac{5}{P _{S}}^{- 1} = 3.23 \times 1 0^{5} d = 883 yr$
Answer

The tool prints a measured ratio of 2.48332291, a nearest resonance of 5:2 and a detuning of 0.671564%. The consequence is the Great Inequality: because the mismatch is a beat and not a drift, Jupiter and Saturn swap a little orbital energy back and forth over roughly nine centuries, and their positions wander from any fixed prediction by amounts big enough to be measured with pre-telescopic instruments. It defeated everyone until Laplace showed the wandering was periodic rather than cumulative — which is to say, that the solar system was stable after all. A 0.67% mismatch is not a rounding error; it is the clock rate of the whole phenomenon.

$0.67% from 5 : 2 ⟹ T_{beat} = 883 yr, not a drift$

References (2)

The great inequality of Jupiter and Saturn, and the century it took to explain: C. Wilson, "The great inequality of Jupiter and Saturn: from Kepler to Laplace." Archive for History of Exact Sciences 33(1–3), 15–290, 1985.
Why the order of a resonance decides its strength — the point the order column is making: T. Gallardo, "Strength, stability and three dimensional structure of mean motion resonances in the solar system." Icarus 317, 121–134, 2019.

By the 42Math team · Updated 2026-07-10

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Jupiter-Saturn - Jupiter-Saturn lies near a 5:2 structure often discussed in long-term secular dynamics.
Pluto-Neptune - Pluto-Neptune demonstrates a classic near-3:2 resonance that protects against close encounters.
Io-Europa - Io-Europa sits near 2:1, part of the Laplace resonance chain in the Galilean moons.
Europa-Ganymede - Europa and Ganymede sit 0.74% off an exact 2:1, and the runner-up in the candidate list, 7:4, misses by 13%. That gap is the diagnostic. When the top two candidates land within a factor of two of each other the ratio has told you nothing; here the winner is alone by an order of magnitude. Io-Europa gives the neighbouring link of the same Laplace chain.
Titan-Hyperion - Hyperion sits 0.0799% from an exact 4:3 with Titan — eight parts in ten thousand, the tightest lock among these presets. Hyperion is also the moon whose rotation is genuinely chaotic, tumbling with no predictable spin axis from one orbit to the next. The resonance is why: it keeps Hyperion’s orbit eccentric, and an eccentric orbit around a lumpy moon is what makes the tumbling unpredictable.
Near 3:2 example - Custom near-3:2 case highlights practical detuning from exact commensurability.
1:1 co-orbital - Two identical periods, 365.25 days each, and the checker returns a perfect score it cannot interpret: 0% detuning, order 0. Order is p − q, the thing that ranks how strong a resonance is, and at 1:1 it collapses. Earth’s Trojan asteroid, a horseshoe orbit and two rocks about to collide all produce exactly this readout.