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Series — page 10

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.

The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed.

Tisserand parameter

1 The number that survives the encounter
2 A map of the transfers that are free
3 An invariant that is only almost one

3 essays · gravitation