The Tisserand Parameter

The Tisserand parameter is a dynamical quantity that is approximately conserved during an encounter between a planet and an interplanetary body. It therefore provides a way to connect the post-encounter dynamical properties with the pre-encounter properties. The Tisserand parameter also provides a measure of the relative speed of an object when it crosses the orbit of a planet. Therefore, different Tisserand parameters exist for different planets, for a given interplanetary body.

Jupiter is the most massive planet in the solar system and exerts the greatest dynamical influence on the comets and asteroids. For this reason, the most commonly seen Tisserand parameter is the one computed with respect to Jupiter and denoted TJ. Here is the definition of Tisserand parameter relative to Jupiter under the "restricted circular three-body" problem:

latex equation

Here, "three-body" refers to Sun, planet and object and the assumption is made that no other bodies influence the dynamics, that the planet orbit is circular (radius 5.204 AU), and that the mass of the object is negligible compared to the masses of the Sun and planet. Only the last assumption is likely to be really solid.

Of What Use Is It?
The rough constancy of TJ during the interaction between a small body and a planet is useful in understanding the outcome of the interaction. Constancy of TJ allows a qualitative relation between the orbital parameters before and after encounter.

TJ is also of practical use in the classification of planet- crossing, solar system bodies. For example, the Jupiter Family Comets have Tisserand parameter TJ between 2 and 3. Most asteroids have TJ larger than 3. These distinctions are not cast in stone, however, and some comets have TJ larger than 3, while some asteroids have TJ < 3.

The Tisserand parameter is not an iron-clad diagnostic of the nature of an interplanetary body, but it is still useful.

What Did Tisserand Look Like?
His portrait, at the Paris Observatory, shows a colorful jacket, a couple of medals and a rather uninformative facial expression. A short biography.

David Jewitt.

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