Tisserand Parameter Flyby Invariant Calculator
Celestial Mechanics & Mission Planning: Evaluate the Tisserand invariant ($T_p$) for gravitational encounters to map achievable post-flyby perihelia, aphelia, and inclination limits.
Encounter Planet & Pre-Flyby Orbit
Tisserand Invariant & Orbit Boundaries
Tisserand's Criterion & The Jacobi Invariant in Astrodynamics
Tisserand's criterion governs planetary gravity-assist tours, constraining the set of reachable orbits following a flyby.
1. Mathematical Formulation
T_p = ( a_p / a ) + 2 · √[ (a / a_p) · (1 - e²) ] · cos i
2. Asymptotic Flyby Velocity Relation
U_rel² = 3 - T_p [non-dimensional relative velocity]
Frequently Asked Questions
What is Tisserand's parameter (Tp) and why is it invariant during a flyby?
Tisserand's parameter ($T_p = \frac{a_p}{a} + 2\sqrt{\frac{a}{a_p}(1-e^2)}\cos i$) is an approximation of the circular restricted three-body problem Jacobi integral. During a close gravitational flyby of a planet, the spacecraft's semi-major axis ($a$), eccentricity ($e$), and inclination ($i$) change drastically, but $T_p$ remains nearly unchanged, linking the pre- and post-encounter orbital elements.
How do astronomers use Tisserand's parameter to classify comets vs. asteroids?
With respect to Jupiter ($a_p = 5.2\,\text{AU}$), solar system bodies are categorized by $T_J$: asteroids have $T_J > 3$ (stable, non-Jupiter crossing orbits); Jupiter-Family Comets have $2 < T_J < 3$ (frequent close gravitational interactions with Jupiter); and Halley-type / Oort cloud comets have $T_J < 2$ (highly inclined or retrograde orbits).
How do mission planners exploit the Tisserand criterion in gravity-assist tours?
Because $(3 - T_p) = (U_{rel} / V_{planet})^2$, the encounter relative velocity $U_{rel}$ is fixed. Mission designers plot "Tisserand graphs" ($a$ vs $r_p$ or $i$) to trace valid stepping stones for planetary tours (e.g. Cassini, Voyager, JUICE) without requiring onboard propulsion.