Tisserand's parameter (or Tisserand's invariant) is a number calculated from several

orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same o ...

(

semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...

orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values be ...

, and

inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Eart ...

) of a relatively small object and a larger " perturbing body". It is used to distinguish different kinds of orbits. The term is named after French astronomer Félix Tisserand who derived it and applies to restricted

three-body problem In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...

s in which the three objects all differ greatly in mass.

Definition

For a small body with semi-major axis

a\,\!

, orbital eccentricity

e\,\!

, and orbital inclination

i\,\!

, relative to the orbit of a perturbing larger body with semimajor axis

a_P

, the parameter is defined as follows: :

T_P\ = \frac + 2\cos i\sqrt

Tisserand invariant conservation

In the three-body problem, the quasi-conservation of Tisserand's invariant is derived as the limit of the

Jacobi integral In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem. Numerical simulations show that the Tisserand invariant of orbit-crossing bodies is conserved in the three-body problem on Gigayear timescales.

Applications

The Tisserand parameter's conservation was originally used by Tisserand to determine whether or not an observed orbiting body is the same as one previously observed. This is usually known as the

Tisserand's criterion Tisserand's criterion is used to determine whether or not an observed orbiting body, such as a comet or an asteroid, is the same as a previously observed orbiting body. While all the orbital parameters of an object orbiting the Sun during the close ...

Orbit classification

The value of the Tisserand parameter with respect to the planet that most perturbs a small body in the

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

can be used to delineate groups of objects that may have similar origins. * T_J, Tisserand's parameter with respect to

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

as perturbing body, is frequently used to distinguish

asteroid An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...

s (typically

T_J > 3

) from Jupiter-family comets (typically

2< T_J < 3

). * The minor planet group of damocloids are defined by a Jupiter Tisserand's parameter of 2 or less (). * T_N, Tisserand's parameter with respect to

Neptune Neptune is the eighth and farthest known planet from the Sun. It is the List of Solar System objects by size, fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 t ...

, has been suggested to distinguish near-

scattered Scattered may refer to: Music * ''Scattered'' (album), a 2010 album by The Handsome Family * "Scattered" (The Kinks song), 1993 * "Scattered", a song by Ace Young * "Scattered", a song by Lauren Jauregui * "Scattered", a song by Green Day from ...

(affected by Neptune) from extended-scattered

trans-Neptunian object A trans-Neptunian object (TNO), also written transneptunian object, is any minor planet in the Solar System that orbits the Sun at a greater average distance than Neptune, which has an orbital semi-major axis of 30.1 astronomical units (AU). ...

s (not affected by Neptune; e.g.

90377 Sedna Sedna ( minor-planet designation: 90377 Sedna) is a dwarf planet in the outermost reaches of the Solar System, orbiting the Sun far beyond the orbit of Neptune. Discovered in 2003, the frigid planetoid is one of the reddest known among Solar ...

). * T_N, Tisserand's parameter with respect to

may also be used to distinguish Neptune-crossing trans-neptunian objects that may be injected onto retrograde and polar Centaur orbits () and those that may be injected onto prograde Centaur orbits ().

Other uses

*The quasi-conservation of Tisserand's parameter constrains the orbits attainable using

gravity assist A gravity assist, gravity assist maneuver, swing-by, or generally a gravitational slingshot in orbital mechanics, is a type of spaceflight flyby (spaceflight), flyby which makes use of the relative movement (e.g. orbit around the Sun) and gra ...

for

outer Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sol ...

exploration. * Tisserand's parameter could be used to infer the presence of an

intermediate-mass black hole An intermediate-mass black hole (IMBH) is a class of black hole with mass in the range of one hundred to one hundred thousand (102–105) solar masses: significantly higher than stellar black holes but lower than the hundred thousand to more than ...

at the center of the

Milky Way The Milky Way or Milky Way Galaxy is the galaxy that includes the Solar System, with the name describing the #Appearance, galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars in other arms of the galax ...

using the motions of orbiting stars.

Related notions

The parameter is derived from one of the so-called Delaunay standard variables, used to study the perturbed

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

in a three-body system. Ignoring higher-order perturbation terms, the following value is conserved: :

\sqrt \cos i

Consequently, perturbations may lead to the

resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...

between the orbital inclination and eccentricity, known as

Kozai resonance In celestial mechanics, the Kozai mechanism is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions. The mechanism is also named von Zeipel-Kozai-Lidov, Lidov–Kozai, Kozai–Lid ...

. Near-circular, highly inclined orbits can thus become very eccentric in exchange for lower inclination. For example, such a mechanism can produce sungrazing comets, because a large eccentricity with a constant semimajor axis results in a small

perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...

References

{{Reflist , refs= {{cite web , title = The Damocloids , date = August 2013 , publisher = UCLA – Department of Earth and Space Sciences , author = Jewitt, David C. , url = http://www2.ess.ucla.edu/~jewitt/damocloid.html , access-date = 15 February 2017, author-link = David C. Jewitt

External links

* David Jewitt's page o
Tisserand's parameter

Orbits Equations of astronomy