Tisserand Parameter Calculator

Use orbital elements to estimate Tisserand parameter (commonly with respect to Jupiter) for dynamical classification.

Worked example — Tisserand Parameter Calculator

First term	1.5770
Orbital factor	0.6896
Second term	1.3491
Tisserand T	2.9261

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

Loading interactive simulation...

Why 3 is a wall, not a convention 🖖

The four inputs here — a, e, i and the planet's a_P — collapse into one number that is a disguised speed. For a planet on a circular orbit, the small body's encounter velocity U, measured in units of the planet's own orbital speed, satisfies U² = 3 − T_P. That makes the familiar dividing line physical rather than clerical: push the readout above 3 and U² turns negative, so no encounter with that planet is geometrically possible at all. A body with T_P > 3 cannot reach the planet's orbit to be deflected by it, which is why the boundary separates comet-like from asteroid-like orbits as sharply as it does.

A fingerprint that survives a slingshot 🖖

When a comet swings past Jupiter, the planet's gravity can reshape its whole orbit at once — size, stretch, and tilt. Yet one particular blend of those elements, the Tisserand parameter, barely changes. That lets astronomers recognize a comet even after its path has been scrambled. Relative to Jupiter it also sorts small bodies: values near 2–3 flag comet-like orbits, above 3 mark asteroid-like ones.

How space probes plan grand tours 🖖

Mission designers reuse the same invariant to plot gravity-assist tours. A planetary flyby reshapes a spacecraft's orbit around the Sun but leaves its Tisserand parameter with respect to that planet nearly fixed, so a "Tisserand graph" reveals which future flybys are reachable. Trajectories for Galileo, Cassini, and the moon tours of JUICE and Europa Clipper were sketched this way — a comet-sorting formula doubling as an interplanetary road map.

Problem solved in full

Tisserand parameter for an object with respect to Jupiter 5 steps
An object with a = 3.3 AU, e = 0.5, i = 12° has a Tisserand parameter of 2.926 with respect to Jupiter. Work it out, and then find why the value 3 is the number that matters.
1. The parameter combines three orbital elements into one quantity that a Jupiter encounter leaves almost unchanged — which is what makes it useful, since an encounter changes a, e and i individually by a great deal.
  
  $T_{p} = \frac{a _{p}}{a} + 2 cos i \frac{a}{a _{p}} (1 - e^{2})$
2. The first term is the ratio of Jupiter's semi-major axis to the object's, and it is the larger contribution here.
  
  $\frac{a _{p}}{a} = \frac{5.204}{3.3} = 1.5770$
3. The second needs the orbital factor, which folds the eccentricity in through a square root, then the inclination through a cosine.
  
  $2 cos 1 2^{\circ} \times 0.6896 = 1.9563 \times 0.6896 = 1.3491$
4. Together they give 2.926.
  
  $T_{p} = 1.5770 + 1.3491 = 2.9261 < 3$
5. The threshold comes from putting Jupiter's own orbit into the formula: a = a_p, e = 0, i = 0 gives 1 + 2 = 3 exactly. So 3 is not a convention, it is what the expression returns for an object sharing Jupiter's orbit.
Answer

The tool prints 1.5770, 0.6896, 1.3491 and a Tisserand parameter of 2.9261. Below 3 means the orbit can cross Jupiter's and be scattered by it, which is the standard dividing line between Jupiter-family comets and asteroids — and it is a better one than composition or appearance, because it survives the encounters that scramble everything else. That is the whole reason the quantity exists: it is nearly conserved, so it labels an object's dynamical family rather than its current orbit. A dormant comet that has lost its tail still gives itself away here.

$a = a_{p}, e = 0, i = 0 \Rightarrow T_{p} = 3 exactly; T_{p} < 3 is comet-like$

References (3)

The Öpik relation U² = 3 − T that makes 3 a physical boundary: G. B. Valsecchi, A. Milani, G. F. Gronchi and S. R. Chesley, "Resonant returns to close approaches: Analytical theory." Astronomy & Astrophysics 408(3), 1179–1196, 2003.
The parameter used to sort short-period comet orbits: H. F. Levison and M. J. Duncan, "The Long-Term Dynamical Behavior of Short-Period Comets." Icarus 108(1), 18–36, 1994.
The original close-encounter treatment: E. J. Öpik, Interplanetary Encounters: Close-Range Gravitational Interactions. Elsevier, 1976.

By the 42Math team · Updated 2026-07-29

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

All Astronomy & Space lessons →

Example problems

JFC-like - Typical Jupiter-family-comet-like orbit often gives Tⱼ between 2 and 3.
Hilda-like - Hilda-like asteroidal orbits generally lie above the cometary threshold in Tisserand space.
NEO-like - Near-Earth asteroidal cases usually produce larger Tⱼ values than Jupiter-family comets.
Retrograde comet - High-inclination retrograde comet examples can drive Tⱼ well below classical Jupiter-family range.