The mass of the Sun is much greater than that of Jupiter. It follows that the gravitational effect of Jupiter on the cometary orbit is negligible unless the comet makes a very close approach to Jupiter. Hence, as described in Chapter 4, before and after such an approach, the comet executes a Keplerian elliptical orbit about the Sun with fixed orbital parameters; that is, fixed major radius, eccentricity, and inclination to the ecliptic plane. However, in general, the orbital parameters before and after the close approach will not be the same as one another. Let us investigate further.
Because , we have , and . Hence, according to Equations (4.34) and (4.44), the (approximately) conserved energy (per unit mass) of the comet before and after its close approach to Jupiter is
Let , , and be the major radius, eccentricity, and inclination angle of the cometary orbit before the close encounter with Jupiter, and let , , and be the corresponding parameters after the encounter. It follows from Equations (9.15), (9.16), and (9.18), and the fact that is conserved during the encounter, whereas and are not, that
The Tisserand criterion is extremely useful. For instance, whenever a new comet is discovered, astronomers immediately calculate its Tisserand parameter,
Figure 9.2 shows the changing orbit of a hypothetical comet that has a close approach to Jupiter. The initial orbit is such that (in units in which the major radius of the Jovian orbit is unity), , and , whereas the final orbit is such that , , and . Figure 9.3 shows the comet's major radius, , eccentricity, , and Tisserand parameter, , as functions of time, before, during, and after, the encounter with Jupiter. It can be seen that the major radius and the eccentricity are both modified by the encounter (which occurs when ), while the Tisserand parameter remains constant in time. This remains true even when the small eccentricity of the Jovian orbit is taken into account in the calculation.
The Tisserand parameter is often employed to distinguish between comets and asteroids in the solar system. This idea is illustrated in Figure 9.4, which shows the Jovian Tisserand parameter, , plotted against the major radius, , of the principal asteroids and comets in the solar system. The Tisserand parameter of Jupiter (which is almost exactly three) is also shown. It can be seen that, roughly speaking, asteroids have higher Tisserand parameters than Jupiter, whereas comets have lower Tisserand parameters. The only major exception to this rule is the so-called Trojan asteroids (see Section 9.8), which all have very similar major radii to Jupiter (because, by definition, they must have the same orbital period as Jupiter), and consequently have lower Tisserand parameters (because they generally have higher eccentricities and inclinations than Jupiter). The lower Tisserand parameters of comets with respect to Jupiter, and of Jupiter with respect to regular asteroids, is indicative of the fact that comets generally originated beyond the Jovian orbit, whereas regular asteroids generally originated within the Jovian orbit.
The Tisserand criterion is also applicable to so-called gravity assists, in which a spacecraft gains energy due to a close encounter with a moving planet. Such assists are often employed in missions to the outer planets to reduce the amount of fuel which the spacecraft must carry in order to reach its destination. In fact, it is clear, from Equations (9.16) and (9.19), that a spacecraft can make use of a close encounter with a moving planet to increase (or decrease) its orbital major radius , and, hence, to increase (or decrease) its total orbital energy.