Tissue edge parameters

The dimensionless Tisserand parameter (after François Félix Tisserand ) represents an approximation of the Jacobi integral and is approximately a conservation quantity of the circularly simplified three-body problem . It is used in astronomy and space travel . ${\ displaystyle T}$

definition

The tissue edge parameter of a small body (typically an asteroid or comet ) with respect to a planet P is defined by

{\ displaystyle T_ {P} = {\ frac {a_ {P}} {a}} + 2 \ cdot \ cos i \ cdot {\ sqrt {{\ frac {a} {a_ {P}}} (1- e ^ {2})}}}

With

${\ displaystyle a_ {P}}$ the semi- major axis of the planet's orbit

{\ displaystyle a}

the semi-major axis of the orbit of the small body

{\ displaystyle i}

the inclination of the orbit of the small body relative to the planetary orbit

{\ displaystyle e}

the eccentricity of the orbit of the small body.

The Tisserand parameter is usually given in relation to Jupiter ( ), since the interaction with Jupiter has the greatest influence on the orbits of the smaller bodies of the solar system . For objects beyond Jupiter's orbit, however, the Tisserand parameter is also calculated in relation to Saturn , Uranus and Neptune . ${\ displaystyle T_ {J}}$

Circularly simplified three-body problem

The prerequisite, the "circularly simplified three-body problem", means in detail:

the mass of the small body is negligible compared to the mass of the planet (and the sun): - the approximation for the Jacobi integral only applies if the mass of the planet is small compared to the sun: ${\ displaystyle m \ ll m_ {P}}$ ${\ displaystyle m_ {P} \ ll m_ {S}}$
the orbit of the planet is circular ("circular")
the orbit of the small body is only influenced by the sun and the planet under consideration, i.e. H. neither other bodies nor non-gravity influences disturb the path.

While the first assumption is entirely justified in practical application, the other two represent strong idealizations .

history

Due to the interaction with Jupiter, the orbital elements of a comet sometimes change very strongly, so that sometimes it can only be decided after extensive iterative orbital calculations whether two comet observations are the same or two different comets.

The French astronomer François Félix Tisserand published in 1896 a simple criterion to compare the orbits of comets together: by the Tisserandkriterium - the Tisserand's parameter for both observations have nearly identical: - you can decide whether it is at all involve the same comet could , and can therefore in many cases dispense with the time-consuming (manual) calculations. ${\ displaystyle T_ {J_ {1}} \ approx T_ {J_ {2}}}$

In the second half of the 20th century, the tissue edge criterion lost much of its importance due to the use of powerful computers.

Today's applications

In astronomy

The current meaning of the Tisserand parameter lies primarily in a simple classification of the bodies of the solar system:

most asteroids have a value greater than 3 ${\ displaystyle T_ {J}}$
for the comets of the Jupiter family is between 2 and 3. ${\ displaystyle T_ {J}}$

There are exceptions to this “rule”, as it is not easy to distinguish them from asteroids due to the lack of activity of the comets in the outer regions of the solar system. For example, some objects that were originally classified as asteroids were later found to be coma , whereupon they are also classified as comets (e.g. (2060) Chiron ) - other asteroids ( damocloids ) move on typical cometary orbits , but show no activity.

In space travel

When planning a gravity assist maneuver, maintaining the tissue edge parameter plays a crucial role. Since the possible paths after the flyby are very limited by the tissue edge parameter, it is used as the basis for choosing a suitable target path. Once this has been found, this in turn leads directly to the speed and distance required for the flyby maneuver.