Osculating orbit

# Osculating orbit

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Encyclopedia
In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, and in particular in astrodynamics
Astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It...

, the osculating orbit of an object in space (at a given moment of time) is the gravitational Kepler orbit
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

(i.e. ellipse or other conic) that it would have about its central body (corresponding to its actual position and velocity for that given moment of time) if perturbations were not present.

An osculating orbit and the object's position upon it can be fully described by the six standard Keplerian orbital elements
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...

(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 perturbation
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

s. However, 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
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...

has been devised to enable representation of the most important aspects of their orbits.

The word "osculate" derives from a Latin word meaning "to kiss". Its use in this context derives from the fact that, at any point in time, an object's osculating orbit is precisely tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to its actual orbit, with the tangent point being the object's location – and has the same curvature as the orbit would have in the absence of perturbing forces.

Perturbation
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

s 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
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

's gravity on objects orbiting Earth.
• A non-gravitational force acting on the body, for example force arising from:
• Thrust from a rocket
Rocket engine
A rocket engine, or simply "rocket", is a jet engineRocket Propulsion Elements; 7th edition- chapter 1 that uses only propellant mass for forming its high speed propulsive jet. Rocket engines are reaction engines and obtain thrust in accordance with Newton's third law...

or ion engine
• Releasing, leaking, venting or ablation
Ablation
Ablation is removal of material from the surface of an object by vaporization, chipping, or other erosive processes. This occurs in spaceflight during ascent and atmospheric reentry, glaciology, medicine, and passive fire protection.-Spaceflight:...

of a material
• Collisions with other objects
• Atmospheric drag
Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the power flux density divided by the speed of light...

• Solar wind
Solar wind
The solar wind is a stream of charged particles ejected from the upper atmosphere of the Sun. It mostly consists of electrons and protons with energies usually between 1.5 and 10 keV. The stream of particles varies in temperature and speed over time...

pressure
• Switch to a non-inertial reference frame (e.g. when a satellite's orbit is described in a reference frame associated with the precessing equator of the planet).

The form of expression of an object's orbital parameters is different in general if it is given with respect to a non-inertial frame of reference (for example, to 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
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...

. Typically (though not necessarily), these curves are chosen as Keplerian conics sharing one of their foci. 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.

• Kepler orbit
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

• Eccentricity vector
Eccentricity vector
In astrodynamics, the eccentricity vector of a Kepler orbit is the vector pointing towards the periapsis having a magnitude equal to the orbit's scalar eccentricity. The magnitude is unitless. For Kepler orbits the eccentricity vector is a constant of motion...

• Orbital elements
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...

• Contact (mathematics)
Contact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...

• Osculating circle
Osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...

• List of orbits

• Diagram of a sequence of osculating orbits for the escape from Earth orbit by the ion-driven SMART-1
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" stands for Small Missions for Advanced Research in Technology...

spacecraft: http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=35722
• A sequence of osculating orbits for the approach to the Moon by the SMART-1
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" stands for Small Missions for Advanced Research in Technology...

spacecraft: http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=36359
• Osculating orbits in a restricted 3-Body problem (video, youtube)
• Osculating orbits in a 3-Body Lagrange problem (video, youtube)
• Osculating orbits in a 4-Body Lagrange problem (video, youtube)
• Osculating orbits in the Pythagorean 3-Body problem (video, youtube)