Now, 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 5, before and after such an approach, the comet executes a
standard elliptical orbit about the Sun with fixed orbital parameters: *i.e.*, 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.

Now, since , we have
, and
.
Hence, according to Equations (260) and (269), the (approximately) conserved
energy (per unit mass) of the comet before and after its close approach to Jupiter is

(1046) |

since in our adopted system of units. Here, is the angle of inclination of the normal to the comet's orbital plane to that of Jupiter's orbital plane.

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 (1044), (1045), and
(1047), and the fact that is conserved during the encounter, whereas
and are not, that

The Tisserand criterion is very useful. For instance, whenever a new comet is discovered, astronomers
immediately calculate its *Tisserand parameter*,

(1049) |

The Tisserand criterion is also applicable to so-called *gravity assists*, in which a
space-craft 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 space-craft must carry in order to reach its destination. In fact, it is clear,
from Equations (1045) and (1048), that a space-craft 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.