Below is an example Tisserand graph showing interplanetary trajectories (in bold black).

The first one represents a trajectory from Earth till the Mercury system. In the second one, the spacecraft instead switches to the a path that increases the energy of the spacecraft upon its return to Earth from a Venus flyby (so that it can go on to Jupiter, for example).

This graph is from "Lunar and Interplanetary Trajectories" by Robin Biesbroek (Springer, 2015).

Each line represents a paths of constant v-infinity computed via Tisserand's criterion. Along a path, perihelion and period vary, but the v-inf remains constant. The graph assumes that the orbits are coplanar.

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I am a bit confused regarding how this actually works out for the spacecraft:

  1. What does it mean to move along a path with respect to one of the planets? I reckon that the spacecraft would naturally follow a path because at one edge of the line it is approaching the planet, and at the other end it is coming out of a flyby about that planet. If no flybys were to occur, the spacecraft would remain on a single point on the graph. Is this correct?
  2. How does the spacecraft jump between lines that intersect? I understand that these lines, albeit representing different v-inf with respect to different planets, do share the same heliocentric orbital elements at the intersection point. I do not quite understand how the spacecraft changes line and goes on moving along a different line? What does the maneuver look like, qualitatively?

Thank you for posting this interesting question. I learned a little more about navigation than I new before. This Paper from Purdue talks about the method in a little more detail. Although one can graphically follow a set of curves to identify that there is a path at a certain energy level (V-infinity) that takes one from one gravity assist to another, it is still necessary to target your departure time and direction so that the spacecraft will travel from one body to the next.

Once you identify graphically that there is a path of interest, say one that goes from Venus to Earth then to Mars at 3 km/s, this data is input to a path finding program like STOUR (a satellite tour design program originally from JPL and modified by Purdue). This tool determines the time of Earth departure and the departure vector that fits the V infinity and rendezvous constraints so solve the problem.

These programs typically work with some sort of guess and propagate technique. They try an approximate solution, see where it ends up, and then they vary the input slightly and see where that ends up, then repeat until they home in on the initial conditions that set up the desired parameters.

  • $\begingroup$ These feasible paths are indeed only from an energy perspective. Before running optimizers like STOUR, the information in the graph needs to be translated into nodes with epoch and state that will be the first guess inputted into the software. Therefore, my question remains: how is the navigating along a path or jumping between path reflected in that input? $\endgroup$ – Paek Se Jun 12 '20 at 3:52

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