## Restrictions on Gravity Assist Orbits

When using gravity assist from a planet to change a spacecraft's orbit, it is not possible
to put the spacecraft into any desired final orbit.

The spacecraft interacts with Jupiter in a manner similar to a small projectile hitting
a larger one. The interaction is governed by the conservation of angular momentum and
energy and as a result only certain paths are possible. Do the orbits the spacecraft in this
example satisfy those restrictions?

As the spacecraft is lifted into a larger orbit, it gains energy and momentum. Those gains
should equal to the loses of energy and momentum by Jupiter. Is that in the example here?

### Summary of Gravity Assist Orbits

| Semi-major axis AU | Eccentricity | Energy per mass (km/s)^{2} = MJ/kg | Angular Momentum per mass AU^{2}/yr |

Spacecraft Orbit to Jupiter | 3.50 | 0.67 | -126.7 | 8.727 |

Spacecraft Orbit past Jupiter | 11.16 | 0.714 | -39.8 | 14.274 |

Jupiter's Orbit | 5.20 | 0.0 | -85.3 | 14.328 |

### Tisserand's Criteria

In the 19th century, ....Tisserand, showed that in a three body interaction, where a small body orbiting the Sun interacts with a planet, a certain parameter defined by the orbital elements of the small objects orbit, is unchanged by the interaction with the planet. The parameter is given by
#### C = a_{P}/a + 2 cos(i) sqrt[ a/a_{P} (1-e^{2})]

**a** = semi-major axis of the object's orbit
**e** = eccentricity of the objects's orbit
**a**_{P} = the semi-major axis of the planet's orbit (here assumed to be a circular orbit)
**i** = inclination of the orbit to the planetary orbit

If we calculate the value of Tisserand's Parameter for the orbits before and after the spacecraft's closest approach to Jupiter we can determine if our example satisfies Tisserand's Criteria.

- | a/a_{P} | (1-e^{2}) | Tisserand Parameter |

Before | 0.673 | 0.496 | 2.641 |

After | 2.153 | 0.551 | 2.643 |

We see that our example does satisfy the criteria within the accuracy of our calculations. In the example here, we arbitrarily selected a path of closest approach and took whatever resulting solar orbit it produced, since I was only wanted to show the principles of gravity assist. Tisserand's
criteria is more useful than testing example orbits; it can be used to predict what approach to
the planet is needed to get into a desired final orbit.

For more on the Tisserand Criteria see the Calculation of the Gravity Assist to Neptune

- Who Was Tisserand?

Online Tisserand Biography

In Print:

"The Shadow Effect and the Case of Flex Tisserand"

by William McLaughlin and Slyvia Miller, *American Scientist* Vol 92, No.3, p262

- How Does the Parameter come about from Conservation of Energy and Momentum?

Derivation of Tisserand's Criteria

- What Use is Tisserand's Criteria?

Orbit Determination using Tisserand's Criteria.

*L.Bogan - May-June 2005*