Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

A real orbit (and its elements) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time.

Keplerian elements

Orbit1
In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For Earth-orbiting satellites, the reference plane is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. The reference plane, together with the vernal point (♈︎), establishes a reference frame.

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion.

When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. When viewed from a non-inertial frame centred on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary.

The main two elements that define the shape and size of the ellipse:

  • Eccentricity (e)—shape of the ellipse, describing how much it is elongated compared to a circle (not marked in diagram).
  • Semimajor axis (a)—the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the semimajor axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass.

Two elements define the orientation of the orbital plane in which the ellipse is embedded:

  • Inclination (i)—vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane, the green angle i in the diagram). Tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. Any three points on an ellipse will define the ellipse orbital plane. The plane and the ellipse are both two-dimensional objects defined in three-dimensional space.
  • Longitude of the ascending node (Ω)—horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane, symbolized by ) with respect to the reference frame's vernal point (symbolized by ♈︎). This is measured in the reference plane, and is shown as the green angle Ω in the diagram.

And finally:

  • Argument of periapsis (ω) defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis (the closest point the satellite object comes to the primary object around which it orbits, the blue angle ω in the diagram).
  • True anomaly (ν, θ, or f) at epoch (M0) defines the position of the orbiting body along the ellipse at a specific time (the "epoch").

The mean anomaly is a mathematically convenient "angle" which varies linearly with time, but which does not correspond to a real geometric angle. It can be converted into the true anomaly ν, which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting object at any given time. Thus, the true anomaly is shown as the red angle ν in the diagram, and the mean anomaly is not shown.

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

Note that non-elliptic trajectories also exist, but are not closed, and are thus not orbits. If the eccentricity is greater than one, the trajectory is a hyperbola. If the eccentricity is equal to one and the angular momentum is zero, the trajectory is radial. If the eccentricity is one and there is angular momentum, the trajectory is a parabola.

Required parameters

Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position (x, y, z in a Cartesian coordinate system), plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead.

Sometimes the epoch is considered a "seventh" orbital parameter, rather than part of the reference frame.

If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.)

Alternative parametrizations

Keplerian elements can be obtained from orbital state vectors (a three-dimensional vector for the position and another for the velocity) by manual transformations or with computer software.[1]

Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis, and periapsis. (When orbiting the Earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, GM, is given for the central body.

Instead of the mean anomaly at epoch, the mean anomaly M, mean longitude, true anomaly ν0, or (rarely) the eccentric anomaly might be used.

Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified.

Different sets of elements are used for various astronomical bodies. The eccentricity, e, and either the semi-major axis, a, or the distance of periapsis, q, are used to specify the shape and size of an orbit. The angle of the ascending node, Ω, the inclination, i, and the argument of periapsis, ω, or the longitude of periapsis, ϖ, specify the orientation of the orbit in its plane. Either the longitude at epoch, L0, the mean anomaly at epoch, M0, or the time of perihelion passage, T0, are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or the node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known.[2][3]

It is also quite common to see either the mean anomaly (M) or the mean longitude (L) expressed directly, without either M0 or L0 as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate the mean motion (n) into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element.

Mean motion can also be obscured behind citations of the orbital period P.

Sets of orbital elements
Object Elements used
Major planet e, a, i, Ω, ϖ, L0
Comet e, q, i, Ω, ω, T0
Asteroid e, a, i, Ω, ω, M0
Two-line elements e, i, Ω, ω, n, M0

Euler angle transformations

The angles Ω, i, ω are the Euler angles (α, β, γ with the notations of that article) characterizing the orientation of the coordinate system

, ŷ, from the inertial coordinate frame Î, Ĵ,

where:

  • Î, Ĵ is in the equatorial plane of the central body. Î is in the direction of the vernal equinox. Ĵ is perpendicular to Î and with Î defines the reference plane. is perpendicular to the reference plane. Orbital elements of bodies (planets, comets, asteroids,...) in the solar system usually use the ecliptic as that plane.
  • , ŷ are in the orbital plane and with in the direction to the pericenter (periapsis). is perpendicular to the plane of the orbit. ŷ is mutually perpendicular to and .

Then, the transformation from the Î, Ĵ, coordinate frame to the , ŷ, frame with the Euler angles Ω, i, ω is:

where

The inverse transformation, which computes the 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of matrix algebra, the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles.

The transformation from , ŷ, to Euler angles Ω, i, ω is:

where arg(x,y) signifies the polar argument that can be computed with the standard function atan2(y,x) available in many programming languages.

Orbit prediction

Under ideal conditions of a perfectly spherical central body and zero perturbations, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion,[2]

Hence if at any instant t0 the orbital parameters are [e0, a0, i0, Ω0, ω0, M0], then the elements at time t0 + δt is given by [e0, a0, i0, Ω0, ω0, M0 + n δt]

Perturbations and elemental variance

Unperturbed, two-body, Newtonian orbits are always conic sections, so the Keplerian elements define an ellipse, parabola, or hyperbola. Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, and so on.

Keplerian elements can often be used to produce useful predictions at times near the epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations, differential equations which come in different forms developed by Lagrange, Gauss, Delaunay, Poincaré, or Hill.

Two-line elements

Keplerian elements parameters can be encoded as text in a number of formats. The most common of them is the NASA/NORAD "two-line elements" (TLE) format,[4] originally designed for use with 80-column punched cards, but still in use because it is the most common format, and can be handled easily by all modern data storages as well.

Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through the SGP/SGP4/SDP4/SGP8/SDP8 algorithms.[5]

Example of a two-line element:[6]

 1 27651U 03004A   07083.49636287  .00000119  00000-0  30706-4 0  2692
 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

Delaunay variables

The Delaunay orbital elements, commonly referred to as Delaunay variables, are action-angle coordinates consisting of the argument of periapsis, the mean anomaly and the longitude of the ascending node, along with their conjugate momenta.[7] They are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems.[7] They were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon.[8]

See also

References

  1. ^ For example, with VEC2TLE
  2. ^ a b Green, Robin M. (1985). Spherical Astronomy. Cambridge University Press. ISBN 978-0-521-23988-2.
  3. ^ Danby, J. M. A. (1962). Fundamentals of Celestial Mechanics. Willmann-Bell. ISBN 978-0-943396-20-0.
  4. ^ Kelso, T. S. "CelesTrak: "FAQs: Two-Line Element Set Format"". celestrak.com. Archived from the original on 26 March 2016. Retrieved 15 June 2016.
  5. ^ Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed., University Science Books, Mill Valley, California.
  6. ^ SORCE Archived 2007-09-27 at the Wayback Machine. – orbit data at Heavens-Above.com
  7. ^ a b Shevchenko, Ivan (2017). The Lidov–Kozai effect: applications in exoplanet research and dynamical astronomy. Cham: Springer. ISBN 978-3-319-43522-0.
  8. ^ Aubin, David (2014). "Delaunay, Charles-Eugène". Biographical Encyclopedia of Astronomers. New York, NY: Springer New York. pp. 548–549. doi:10.1007/978-1-4419-9917-7_347. ISBN 978-1-4419-9916-0.

External links

2011 FW62

2011 FW62 also designated 2015 AJ281 is a trans-Neptunian object that was discovered in 2011. With an absolute magnitude of 5.0, it is likely a dwarf planet. Its orbital elements were very uncertain and it was lost. It was recovered on 6 January 2015 as 2015 AJ281.

289P/Blanpain

289P/Blanpain, formerly D/1819 W1 (Blanpain) is a short-period comet that was discovered by Jean-Jacques Blanpain on November 28, 1819. Blanpain described the comet as having a "very small and confused nucleus". Another independent discovery was made on December 5 of that year by J. L. Pons. Following this the comet was lost, and was given the designation 'D' (Disappeared or Dead). However, in 2003, the orbital elements of newly discovered asteroid 2003 WY25 were calculated by Marco Micheli and others to be a probable match for the lost comet. Further observations of the asteroid in 2005 by David Jewitt using the University of Hawaii 2.2 m telescope on Mauna Kea, appeared to reveal a faint coma, which supports the theory that 2003 WY25 is the lost comet, or a part of it. The comet was officially established as periodic comet 289P in July 2013, after being rediscovered by the Pan-STARRS survey during an outburst event.

Argument of periapsis

The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the body's ascending node to its periapsis, measured in the direction of motion.

For specific types of orbits, words such as perihelion (for heliocentric orbits), perigee (for geocentric orbits), periastron (for orbits around stars), and so on may replace the word periapsis. (See apsis for more information.)

An argument of periapsis of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North. An argument of periapsis of 90° means that the orbiting body will reach periapsis at its northmost distance from the plane of reference.

Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis. However, especially in discussions of binary stars and exoplanets, the terms "longitude of periapsis" or "longitude of periastron" are often used synonymously with "argument of periapsis".

C/1999 F1

C/1999 F1 (Catalina) is one of the longest known long-period comets. It was discovered on March 23, 1999, by the Catalina Sky Survey.The comet has an observation arc of 2,360 days allowing a good estimate of the orbit. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. C/1999 F1 made its closest approach to Neptune in August 2017. Using JPL Horizons, the barycentric orbital elements for epoch 2035-Jan-01 generate a semi-major axis of 33,300 AU, an apoapsis distance of 66,600 AU, and a period of approximately 6 million years. Comet West has a similar period.

The generic JPL Small-Body Database browser uses a near-perihelion epoch of 2001-May-19 which is before the comet left the planetary region and makes the highly eccentric aphelion point inaccurate since it does not account for any planetary perturbations. The heliocentric JPL Small-Body Database solution also does not account for the mass of Jupiter.

C/1999 S4

C/1999 S4 (LINEAR) is a long-period comet discovered on September 27, 1999, by LINEAR.The comet made its closest approach to the Earth on July 22, 2000, at a distance of 0.3724 AU (55,710,000 km; 34,620,000 mi). It came to perihelion on July 26, 2000, at a distance of 0.765 AU from the Sun.The comet nucleus was estimated to be about 0.9 km in diameter. Before the comet broke up, the (dust and water) nucleus erosion rate was about 1 cm per day. The comet brightened near July 5, 2000, and underwent a minor fragmentation event.

The comet brightened again around July 20, 2000, and then disintegrated. The published optical and most radio data support that the main nuclear decay started July 23, 2000. The dust cloud expanded at about 20 meters per second (45 miles per hour) while the fragments expanded at around 7 m/s (16 mph). Other comets are known to have disappeared, but Comet LINEAR is the first one to have been caught in the act.The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2010-Jan-01 generate a semi-major axis of 700 AU, an aphelion distance of 1400 AU, and a period of approximately 18,700 years.

C/2000 W1

C/2000 W1 (Utsunomiya-Jones) is a long-period comet discovered on November 18, 2000, by Syogo Utsunomiya and Albert F. A. L. Jones.The comet has an observation arc of 58 days allowing a reasonable estimate of the orbit. But the near-parabolic trajectory with an osculating perihelion eccentricity of 0.9999996 generates an extreme unperturbed aphelion distance of 2,034,048 AU (32 light-years). The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2020-Jan-01 generate a semi-major axis of 832 AU, an aphelion distance of 1660 AU, and a period of approximately 24,000 years.C/2000 W1 disappeared in February 2001.

C/2006 M4 (SWAN)

C/2006 M4 is one of several SWAN comets; the others are C/2002 O6, C/2004 H6, C/2004 V13, C/2005 P3, P/2005 T4, C/2009 F6, C/2011 Q4 and C/2012 E2.Comet C/2006 M4 (SWAN) is a non-periodic comet discovered in late June 2006 by Robert D. Matson of Irvine, California and Michael Mattiazzo of Adelaide, South Australia in publicly available images of the Solar and Heliospheric Observatory (SOHO). These images were captured by the Solar Wind ANisotropies (SWAN) Lyman-alpha all-sky camera on board the SOHO. The comet was officially announced after a ground-based confirmation by Robert McNaught (Siding Spring Survey) on July 12.Although perihelion was Sept 28, 2006, the comet flared dramatically from seventh magnitude to fourth magnitude on October 24, 2006, becoming visible with the naked eye.Comet C/2006 M4 is in a hyperbolic trajectory (with an osculating eccentricity larger than 1) during its passage through the inner solar system. After leaving the influence of the planets, the eccentricity will drop below 1 and it will remain bound to the Solar System as an Oort cloud comet.

Given the extreme orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance (maximum distance) of this object. For objects at such high eccentricity, the Suns barycentric coordinates are more stable than heliocentric coordinates. Using JPL Horizons, the barycentric orbital elements for epoch 2013-May-14 generate a semi-major axis of about 1300 AU and a period of about 47,000 years.

C/2007 Q3

C/2007 Q3 (Siding Spring), is an Oort cloud comet that was discovered by Donna Burton in 2007 at Siding Spring Observatory in New South Wales, Australia. Siding Spring came within 1.2 astronomical units of Earth and 2.25 AU of the Sun on October 7, 2009. The comet was visible with binoculars until January 2010.Images of the comet taken in March 2010 by N.Howes using the Faulkes telescope, showed that the nucleus had fragmented.The comet has an observation arc of 1,327 days and is still been observed as of April 2011. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2030-Jan-01 generate a semi-major axis of 7,500 AU, an apoapsis distance of 15,000 AU, and a period of approximately 650,000 years.Before entering the planetary region (epoch 1950), C/2007 Q3 had a calculated barycentric orbital period of ~6.4 million years with an apoapsis (aphelion) distance of about 69,000 AU (1.09 light-years). The comet was probably in the outer Oort cloud for millions or billions of years with a loosely bound chaotic orbit until it was perturbed inward.

C/2007 W1 (Boattini)

C/2007 W1 (Boattini) is a long-period comet discovered on 20 November 2007, by Andrea Boattini at the Mt. Lemmon Survey. At the peak the comet had an apparent magnitude around 5.On 3 April 2008, when C/2007 W1 was 0.66AU from the Earth and 1.7AU from the Sun, the coma (expanding tenuous dust atmosphere) of the comet was estimated to be as large as 10 arcminutes. This made the coma roughly 290,000 km in diameter.On 12 June 2008, the comet passed within about 0.21005 AU (31,423,000 km; 19,525,000 mi) of the Earth. The comet came to perihelion (closest approach to the Sun) on 24 June 2008 at a distance of 0.8497 AU.The comet has an observation arc of 285 days allowing a good estimate of the orbit. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2020-Jan-01 generate a semi-major axis of 1,582 AU, an apoapsis distance of 3,163 AU, and a period of approximately 63,000 years.Before entering the planetary region, C/2007 W1 had a hyperbolic trajectory. The comet was probably in the outer Oort cloud with a loosely bound chaotic orbit that was easily perturbed by passing stars.

C/2015 ER61 (PANSTARRS)

C/2015 ER61 (PANSTARRS) is a comet, inner Oort cloud object, Amor near-Earth asteroid, and possibly a damocloid. When classified as a minor planet, it had the fourth-largest aphelion of any known minor planet in the Solar System, after 2005 VX3, 2012 DR30, and 2013 BL76. It additionally had the most eccentric orbit of any known minor planet, with its distance from the Sun varying by about 99.9% during the course of its orbit, followed by 2005 VX3 with an eccentricity of 0.9973. On January 30, 2016, it was classified as a comet when it was 5.7 AU from the Sun. It comes close to Jupiter, and a close approach in the past threw it on the distant orbit it is on now.

Though the comet nucleus was probably mildly active, early asteroidal estimates gave an absolute magnitude (H) of 12.3, which would suggest a nucleus as large as 8–20 km in diameter. But it could easily be half that size due to activity brightening the nucleus.

Comet Machholz

Comet Machholz, formally designated C/2004 Q2, is a long-period comet discovered by Donald Machholz on August 27, 2004.

It reached naked eye brightness in January 2005. Unusual for such a relatively bright comet, its perihelion was farther from the Sun than the Earth's orbit.

Comet Pojmański

Comet Pojmański is a non-periodic comet discovered by Grzegorz Pojmański on January 2, 2006 and formally designated C/2006 A1. Pojmański discovered the comet at Warsaw University Astronomic Observatory using the Las Campanas Observatory in Chile as part of the All Sky Automated Survey (ASAS). Kazimieras Cernis at the Institute of Theoretical Physics and Astronomy at Vilnius, Lithuania, located it the same night and before the announcement of Pojmański's discovery, in ultraviolet images taken a few days earlier by the SWAN instrument aboard the SOHO satellite. A pre-discovery picture was later found from December 29, 2005.

At the time of its discovery, the comet was roughly 113 million miles (181 million kilometers) from the Sun. But orbital elements indicated that on February 22, 2006, it would reach perihelion at a distance of 51.6 million miles — almost half the Earth's average distance from the Sun.

The comet moved on a northward path across the night sky, and reached maximum brightness around the beginning of March. Comet Pojmański reached the very fringe of naked-eye visibility at about magnitude 5, and was best visible through binoculars or a telescope. It could be found in the dawn sky within the constellation Capricornus, close to the horizon in the northern hemisphere, during late February, but viewing circumstances became better for the northern hemisphere as the comet departed southern skies and continued north.

By early March, the comet was located in Aquila, the Eagle, and by March 7 was located in the constellation Delphinus, the Dolphin.

Comet Pojmański brightened more than initially estimated, perhaps due to over-cautious estimates by astronomers. It had previously been estimated to reach a maximum brightness of around 6.5 magnitude, but became considerably brighter.

During the comet's appearance, it sported a tail of three to seven degrees (six to fourteen times the apparent lunar diameter) and a coma of up to about 10 arcseconds.

Comet Zhu–Balam

Comet Zhu–Balam (C/1997 L1) is a long-period comet first identified by David D. Balam on June 8, 1997 and originally photographed by Gin Zhu on June 3, 1997. The comet is estimated at 10 kilometres in diameter with a period of approximately 36,800 years.Given the orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance (maximum distance) of this object. For objects at such high eccentricity, the Suns barycentric coordinates are more stable than heliocentric coordinates. Using JPL Horizons the barycentric orbital elements for epoch 2015-Jan-01 generate a semi-major axis of 1100 AU and a period of approximately 36,800 years.

Great Comet of 1819

The Great Comet of 1819, officially designated as C/1819 N1, also known as Comet Tralles, was an easily visible brilliant comet, approaching an apparent magnitude of 1–2, discovered July 1, 1819 by Johann Georg Tralles in Berlin, Germany. It was the first comet analyzed using polarimetry, by François Arago.

Longitude of the ascending node

The longitude of the ascending node (☊ or Ω) is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane. The ascending node is the point where the orbit of the object passes through the plane of reference, as seen in the adjacent image. Commonly used reference planes and origins of longitude include:

For a geocentric orbit, Earth's equatorial plane as the reference plane, and the First Point of Aries as the origin of longitude. In this case, the longitude is also called the right ascension of the ascending node, or RAAN. The angle is measured eastwards (or, as seen from the north, counterclockwise) from the First Point of Aries to the node.

For a heliocentric orbit, the ecliptic as the reference plane, and the First Point of Aries as the origin of longitude. The angle is measured counterclockwise (as seen from north of the ecliptic) from the First Point of Aries to the node.

For an orbit outside the Solar System, the plane tangent to the celestial sphere at the point of interest (called the plane of the sky) as the reference plane, and north, i.e. the perpendicular projection of the direction from the observer to the North Celestial Pole onto the plane of the sky, as the origin of longitude. The angle is measured eastwards (or, as seen by the observer, counterclockwise) from north to the node., pp. 40, 72, 137; , chap. 17.In the case of a binary star known only from visual observations, it is not possible to tell which node is ascending and which is descending. In this case the orbital parameter which is recorded is the longitude of the node, Ω, which is the longitude of whichever node has a longitude between 0 and 180 degrees., chap. 17;, p. 72.

Lysithea (moon)

Lysithea ( ly-SITH-ee-ə, li-SITH-ee-ə; Greek: Λυσιθέα) is a prograde irregular satellite of Jupiter. It was discovered by Seth Barnes Nicholson in 1938 at Mount Wilson Observatory and is named after the mythological Lysithea, daughter of Oceanus and one of Zeus' lovers.Lysithea did not receive its present name until 1975; before then, it was simply known as Jupiter X. It was sometimes called "Demeter" from 1955 to 1975.

It belongs to the Himalia group, five moons orbiting between 11 and 13 Gm from Jupiter at an inclination of about 28.3°. Its orbital elements are as of January 2000. They are continuously changing due to solar and planetary perturbations.

Osculating orbit

In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent. That is, it is the orbit that coincides with the current orbital state vectors (position and velocity).

Proper orbital elements

The proper orbital elements of an orbit are constants of motion of an object in space that remain practically unchanged over an astronomically long timescale. The term is usually used to describe the three quantities:

proper semimajor axis (ap),

proper eccentricity (ep), and

proper inclination (ip).The proper elements can be contrasted with the osculating Keplerian orbital elements observed at a particular time or epoch, such as the semi-major axis, eccentricity, and inclination. Those osculating elements change in a quasi-periodic and (in principle) predictable manner due to such effects as perturbations from planets or other bodies, and precession (e.g. perihelion precession). In the Solar System, such changes usually occur on timescales of thousands of years, while proper elements are meant to be practically constant over at least tens of millions of years.

For most bodies, the osculating elements are relatively close to the proper elements because precession and perturbation effects are relatively small (see diagram). For over 99% of asteroids in the asteroid belt, the differences are less than 0.02 AU (for semi-major axis a), 0.1 (for eccentricity e), and 2° (for inclination i).

Nevertheless, this difference is non-negligible for any purposes where precision is of importance. As an example, the asteroid Ceres has osculating orbital elements (at epoch November 26, 2005)

while its proper orbital elements (independent of epoch) are

A notable exception to this small-difference rule are asteroids lying in the Kirkwood gaps, which are in strong orbital resonance with Jupiter.

To obtain proper elements for an object, one usually conducts a detailed simulation of its motion over timespans of several millions of years. Such a simulation must take into account many details of celestial mechanics including perturbations by the planets. Subsequently, one extracts quantities from the simulation which remain unchanged over this long timespan; for example, the mean inclination, eccentricity, and semi-major axis. These are the proper orbital elements.

Historically, various approximate analytic calculations were made, starting with those of Kiyotsugu Hirayama in the early 20th century. Later analytic methods often included thousands of perturbing corrections for each particular object. Presently, the method of choice is to use a computer to numerically integrate the equations of celestial dynamics, and extract constants of motion directly from a numerical analysis of the predicted positions.

At present the most prominent use of proper orbital elements is in the study of asteroid families, following in the footsteps of the pioneering work of Hirayama. A Mars-crosser asteroid 132 Aethra is the lowest numbered asteroid to not have any proper orbital elements.

Tisserand's parameter

Tisserand's parameter (or Tisserand's invariant) is a value calculated from several orbital elements (semi-major axis, orbital eccentricity and inclination) 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 problems in which the three objects all differ greatly in mass.

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