Tisserand's parameter (or Tisserand's invariant) is a value 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 ...
(
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 lo ...
,
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 b ...
and
inclination
Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
) 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, and applies to restricted
three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
s in which the three objects all differ greatly in mass.
Definition
For a small body with semi-major axis
, orbital eccentricity
, and orbital inclination
, relative to the orbit of a perturbing larger body with
semimajor axis
, the parameter is defined as follows:
:
The quasi-conservation of Tisserand's parameter is a consequence of
Tisserand's relation.
Applications
* T
J, Tisserand's parameter with respect to
Jupiter
Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousandt ...
as perturbing body, is frequently used to distinguish
asteroid
An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic o ...
s (typically
) from
Jupiter-family comets
A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
(typically
).
* The minor planet group of
damocloid
Damocloids are a class of minor planets such as 5335 Damocles and 1996 PW that have Halley-type or long-period highly eccentric orbits typical of periodic comets such as Halley's Comet, but without showing a cometary coma or tail. ...
s are defined by a Jupiter Tisserand's parameter of 2 or less ().
* The roughly constant value of the parameter before and after the interaction (encounter) is used to determine whether or not an observed orbiting body is the same as one previously observed in Tisserand's criterion.
*The quasi-conservation of Tisserand's parameter constrains the orbits attainable using
gravity assist
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the ...
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 "Solar S ...
exploration.
* T
N, Tisserand's parameter with respect to
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 time ...
, 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 a semi-major axis of 30.1 astronomical units (au).
Typicall ...
s (not affected by Neptune; e.g.
90377 Sedna
Sedna ( minor-planet designation 90377 Sedna) is a dwarf planet in the outer reaches of the Solar System that is in the innermost part of its orbit; it is 84 astronomical units (AU), or 1.26×1010 km, from the Sun, almost three times farth ...
).
* 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 102–105 solar masses: significantly more than stellar black holes but less than the 105–109 solar mass supermassive black holes. Several IMBH candidate obje ...
at the center of the
Milky Way
The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked ey ...
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 in a
three-body system. Ignoring higher-order perturbation terms, the following value is
conserved:
:
Consequently, perturbations may lead to the
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
between the orbital inclination and eccentricity, known as
Kozai resonance. Near-circular, highly inclined orbits can thus become very eccentric in exchange for lower inclination. For example, such a mechanism can produce
sungrazing comet
A sungrazing comet is a comet that passes extremely close to the Sun at perihelion – sometimes within a few thousand kilometres of the Sun's surface. Although small sungrazers can completely evaporate during such a close approach to the Sun, la ...
s, 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. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any e ...
.
See also
*
Tisserand's relation for the derivation and the detailed assumptions
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