Tisseran's Criteria and Gravity Assist Orbit Determination

Below is a graph of Tisserand's Parameter for elliptical orbits (the eccentricity is limited to values from 0 to 0.99). Other limits that restrict the orbits are (1) having Perihelia less than the planet's orbital radius and (2) aphelia greater than the planet's orbital radius. These limits require solutions below the dashed black line on the graph.

Graph of Tisserand's Parameter

Procedure for the example of a gravity assist to Neptune.

STEPS to GET THE ORBIT TO NEPTUNE from EARTH VIA JUPITER GRAVITY ASSIST.

Timing of the Spacecraft Launch for Interaction with Jupiter
and Flyby of Neptune.

The object of launching a spacecraft to Neptune's vicinity is to fly past it. Neptune must be at the orbit crossing location when the spacecraft gets there. The same is true for the interaction with Jupiter to get the gravity assist. The timing of the launch must be at a very special time when Jupiter will be the orbit cross of its orbit when the spacecraft gets there. To match of both the Neptune and Jupiter flybys would be a very difficult task to accomplish. But let's calculate the timing of the events.