The Tisserand parameter is a constant in the frame of the restricted circular three-body problem (RC3BP) given by , where (the ratio of the semimajor axes of the body and planet), is the eccentricity, and is the inclination of the body's orbit with respect to the planet's orbit.
This Demonstration plots surfaces of constant Tisserand parameters in the space , where . A body encountering the planet must have , (which means ), and in units of the RC3BP. As the body encounters the planet, its orbital elements change but conserve . The plot shows all possible orbital evolutions for the body for a given value of . The boundary between asteroids and comets in the solar system occurs for .
It is possible to rearrange in terms of , , and . Given the body's orbital elements, calculate and choose that value in the control; the resulting surface is the possible region in the space which the body can reach after successive encounters with the planet. We assume that the body reaches the planet's orbit and that imposes , , and in units of the RC3BP. See details in Solar System Dynamics.