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.[1] That is, it is the orbit that coincides with the current orbital state vectors (position and velocity).

Enckes method-vector
Osculating orbit (inner, black) and perturbed orbit (red)

Etymology

The word osculate is Latin for “kiss”. In mathematics, two curves osculate when they just touch, without (necessarily) crossing, at a point, where both have the same position and slope, i.e. the two curves “kiss”.

Kepler elements

An osculating orbit and the object's position upon it can be fully described by the six standard Kepler orbital elements (osculating elements), which are easy to calculate as long as one knows the object's position and velocity relative to the central body. The osculating elements would remain constant in the absence of perturbations. Real astronomical orbits experience perturbations that cause the osculating elements to evolve, sometimes very quickly. In cases where general celestial mechanical analyses of the motion have been carried out (as they have been for the major planets, the Moon, and other planetary satellites), the orbit can be described by a set of mean elements with secular and periodic terms. In the case of minor planets, a system of proper orbital elements has been devised to enable representation of the most important aspects of their orbits.

Perturbations

Perturbations that cause an object's osculating orbit to change can arise from:

  • A non-spherical component to the central body (when the central body can be modeled neither with a point mass nor with a spherically symmetrical mass distribution, e.g. when it is an oblate spheroid).
  • A third body or multiple other bodies whose gravity perturbs the object's orbit, for example the effect of the Moon's gravity on objects orbiting Earth.
  • A relativistic correction.
  • A non-gravitational force acting on the body, for example force arising from:

Parameters

An object's orbital parameters will be different if they are expressed with respect to a non-inertial reference frame (for example, a frame co-precessing with the primary's equator), than if it is expressed with respect to a (non-rotating) inertial reference frame.

Put in more general terms, a perturbed trajectory can be analysed as if assembled of points, each of which is contributed by a curve out of a sequence of curves. Variables parameterising the curves within this family can be called orbital elements. Typically (though not necessarily), these curves are chosen as Keplerian conics, all of which share one focus. In most situations, it is convenient to set each of these curves tangent to the trajectory at the point of intersection. Curves that obey this condition (and also the further condition that they have the same curvature at the point of tangency as would be produced by the object's gravity towards the central body in the absence of perturbing forces) are called osculating, while the variables parameterising these curves are called osculating elements. In some situations, description of orbital motion can be simplified and approximated by choosing orbital elements that are not osculating. Also, in some situations, the standard (Lagrange-type or Delaunay-type) equations furnish orbital elements that turn out to be non-osculating.[2]

See also

References

  1. ^ Moulton, Forest R. (1970) [1902]. Introduction to Celestial Mechanics (2nd revised ed.). Mineola, New York: Dover. pp. 322–23. ISBN 0486646874.
  2. ^ For details see: Efroimsky, M. (2005). "Gauge Freedom in Orbital Mechanics". Annals of the New York Academy of Sciences. 1065: 346–74. arXiv:astro-ph/0603092. Bibcode:2005NYASA1065..346E. doi:10.1196/annals.1370.016. PMID 16510420.; Efroimsky, Michael; Goldreich, Peter (2003). "Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach". Journal of Mathematical Physics. 44 (12): 5958–5977. arXiv:astro-ph/0305344. Bibcode:2003JMP....44.5958E. doi:10.1063/1.1622447.

External links

Videos
(308933) 2006 SQ372

(308933) 2006 SQ372 is a trans-Neptunian object and highly eccentric centaur on a cometary-like orbit in the outer region of the Solar System, approximately 123 kilometers (76 miles) in diameter. It was discovered through the Sloan Digital Sky Survey by astronomers Andrew Becker, Andrew Puckett and Jeremy Kubica on images first taken on 27 September 2006 (with precovery images dated to 13 September 2005).

1915 Quetzálcoatl

1915 Quetzálcoatl, provisional designation 1953 EA, is a very eccentric, stony asteroid classified as near-Earth object, about half a kilometer in diameter. It was discovered on 9 March 1953, by American astronomer Albert George Wilson at Palomar Observatory, California. It was named for Quetzalcoatl from Aztec mythology.

C/1992 J1

C/1992 J1 is a comet that was discovered 1 May 1992 by David Rabinowitz of the Spacewatch Project. This was the first comet to be discovered using an automated system.Using a generic heliocentric (two-body) solution calculated near the time of perihelion (closest approach to the Sun), it is estimated to have an aphelion (Q) (furthest distance from the Sun) of 154,202 AU (more than 2 Light-years). But 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. After leaving the planetary region of the Solar System, the post-perihelion orbital period is estimated to be about 78,000 years with aphelion around 3,650 AU. In 2007 it became more than 30 AU from the Sun.

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/2002 V1 (NEAT)

Comet C/2002 V1 (NEAT) is a non-periodic comet that appeared in November 2002. The comet peaked with an apparent magnitude of approximately –0.5, making it the eighth-brightest comet seen since 1935. It was seen by SOHO in February 2003. At perihelion the comet was only 0.099258 astronomical units (14,848,800 kilometres; 9,226,600 miles) from the Sun. (Slight controversy arose when the comet failed to break up when it approached the Sun, as expected by some scientists if it were a small comet.)The comet was hit by a coronal mass ejection during its pass near the Sun; some rumoured it had "disturbed" the Sun, but scientists dismissed this notion. The scientific consensus is that there is no link between comets and CMEs that can not be explained through simple coincidence, and there were 56 CMEs in February 2003. On February 18, 2003, comet C/2002 V1 (NEAT) passed 5.7 degrees from the Sun. C/2002 V1 (NEAT) appeared impressive as viewed by the Solar and Heliospheric Observatory (SOHO) as a result of the forward scattering of light off of the dust in the coma and tail. After the comet left LASCO's field of view, on February 20, 2003, an object was seen at the bottom of a single frame. Although technicians dismissed this as a software bug, rumours persisted that the object had been expelled from the Sun.

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 1,100 AU, an apoapsis distance of 2,230 AU, and a period of approximately 37,000 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.

Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data.

Ephemeris

In astronomy and celestial navigation, an ephemeris (plural: ephemerides) gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly velocity) over time. The etymology is from Latin ephemeris, meaning 'diary' and from Greek, Modern εφημερίς (ephemeris), meaning 'diary, journal'. Historically, positions were given as printed tables of values, given at regular intervals of date and time. The calculation of these tables was one of the first applications of mechanical computers. Modern ephemerides are often computed electronically, from mathematical models of the motion of astronomical objects and the Earth. However, printed ephemerides are still produced, as they are useful when computational devices are not available.

The astronomical position calculated from an ephemeris is given in the spherical polar coordinate system of right ascension and declination. Some of the astronomical phenomena of interest to astronomers are eclipses, apparent retrograde motion/planetary stations, planetary ingresses, sidereal time, positions for the mean and true nodes of the moon, the phases of the Moon, and the positions of minor celestial bodies such as Chiron.

Ephemerides are used in celestial navigation and astronomy. They are also used by some astrologers.

Index of aerospace engineering articles

This is an alphabetical list of articles pertaining specifically to aerospace engineering. For a broad overview of engineering, see List of engineering topics. For biographies, see List of engineers.

Index of physics articles (O)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Orbit modeling

Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.

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.

Osculate

In mathematics, osculate, meaning to touch (from the Latin osculum meaning kiss), may refer to:

osculant, an invariant of hypersurfaces.

osculating circle

osculating curve

osculating plane

osculating orbitThe obsolete Quinarian system of biological classification attempted to group creatures into circles which could touch or overlap with adjacent circles, a phenomenon called 'osculation'.

Osculating curve

In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve.

That is, if F is a family of smooth curves, C is a smooth curve (not in general belonging to F), and p is a point on C, then an osculating curve from F at p is a curve from F that passes through p and has as many of its derivatives at p equal to the derivatives of C as possible.The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency.

Perturbation (astronomy)

In astronomy, perturbation is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.

SMART-1

SMART-1 was a Swedish-designed European Space Agency satellite that orbited around the Moon. It was launched on September 27, 2003 at 23:14 UTC from the Guiana Space Centre in Kourou, French Guiana. "SMART-1" stands for Small Missions for Advanced Research in Technology-1. On September 3, 2006 (05:42 UTC), SMART-1 was deliberately crashed into the Moon's surface, ending its mission.

Gravitational orbits
Types
Parameters
Maneuvers
Orbital mechanics

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