Kepler orbit
Overview
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, a Kepler orbit describes the motion of an orbiting body as an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, or hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
, which forms a twodimensional orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semiflat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
in threedimensional space. (A Kepler orbit can also form a straight line.) It considers only the gravitational attraction of two bodies, neglecting perturbations
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....
due to gravitational interactions with other objects, atmospheric drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
, solar radiation pressure, a nonspherical central body, and so on.
Unanswered Questions
Encyclopedia
In celestial mechanics
, a Kepler orbit describes the motion of an orbiting body as an ellipse
, parabola
, or hyperbola
, which forms a twodimensional orbital plane
in threedimensional space. (A Kepler orbit can also form a straight line.) It considers only the gravitational attraction of two bodies, neglecting perturbations
due to gravitational interactions with other objects, atmospheric drag
, solar radiation pressure, a nonspherical central body, and so on. It is thus said to be a solution of a special case of the twobody problem
, known as the Kepler problem. As a theory in classical mechanics
, it also does not take into account the effects of general relativity
. Keplerian orbits can be parametrized
into six orbital elements
in various ways.
In most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of similar mass can be described as Kepler orbits around their common center of mass, their barycenter
.
developed his laws of planetary motion
around 1605, from astronomical tables detailing the movements of the visible planet
s. Kepler's First Law is:
The mathematics of ellipses are thus the mathematics of Kepler orbits, later expanded to include parabola
s and hyperbola
s.
Isaac Newton
published his law of universal gravitation
in 1687, which says:
The shapes of large celestial bodies are close to spheres. By symmetry, the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards the centre of the sphere. The shell theorem
(also proven by Isaac Newton) says that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere, even if the density of the sphere varies with depth (as it does for most celestial bodies). From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to its center.
Smaller objects, like asteroid
s or spacecraft
often have a shape strongly deviating from a sphere. But the gravitational forces produced by these irregularities are generally small compared to the gravity of the central body. The difference between an irregular shape and a perfect sphere also diminishes with distances, and most orbital distances are very large when compared with the diameter of a small orbiting body. Thus for some applications, shape irregularity can be neglected without significant impact on accuracy.
Planets that rotate (such as the Earth) take a slightly oblate shape because of the centrifugal force and with such an oblate shape the gravitational attraction will deviate somewhat from that of a homogeneous sphere. This phenomenon is quite noticeable for artificial Earth satellites, especially those in low orbits. But at a large distance the effect of this oblateness is very small and the planetary motions in the Solar System can be computed with sufficient precision using the point mass assumption, using the equation given above.
The twobody problem
is the case that there are only two point masses (or homogeneous spheres).
If the two mass points (or homogenous spheres) have the masses and and the position vectors
and relative a point fixed with respect to inertial space (for example relative their common centre of mass
) the equations of motion for the two mass points are
where
is the distance between the bodies
and
is the unit vector pointing from body 2 to body 1.
Dividing with the factors and and subtracting the resulting equations one gets the differential equation
for the vector from body 2 to body 1
where
This differential equation for the two body case can be completely solved mathematically and the resulting orbit which follows Kepler's laws of planetary motion is called a "Kepler orbit". The orbits of all planets are to high accuracy Kepler orbits around the Sun (as observed by Johannes Kepler!), the small deviations being due to the much weaker gravitational attractions between the planets, or in the case of Mercury
, due to general relativity
. Also the orbits around the Earth of the Moon and of the artificial satellites are with a fair approximation Kepler orbits. In fact, the gravitational acceleration towards the Sun is about the same for the Earth and the satellite and therefore in a first approximation "cancels out". Also in high accuracy applications for which the equation of motion must be integrated numerically with all gravitational and nongravitational forces (such as solar radiation pressure and atmospheric drag
) being taken into account, the Kepler orbit concepts are of paramount importance and heavily used.
For example, the orbital elements
:
are only defined for a Kepler orbit. The use of orbital elements therefore always implies that the orbit is approximated with the Kepler orbit having these orbital elements.
stays constant:
Introducing a coordinate system in the plane orthogonal to and polar coordinates
the differential equation takes the form (see "Polar coordinates#Vector calculus")
Taking the time derivative of one gets
Using the chain rule for differentiation one gets
Using the expressions for of equations , , and all time derivatives in can be replaced by derivatives of r as function of . After some simplification one gets
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, a Kepler orbit describes the motion of an orbiting body as an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, or hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
, which forms a twodimensional orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semiflat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
in threedimensional space. (A Kepler orbit can also form a straight line.) It considers only the gravitational attraction of two bodies, neglecting perturbations
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....
due to gravitational interactions with other objects, atmospheric drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
, solar radiation pressure, a nonspherical central body, and so on. It is thus said to be a solution of a special case of the twobody problem
Twobody problem
In classical mechanics, the twobody problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...
, known as the Kepler problem. As a theory in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, it also does not take into account the effects of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. Keplerian orbits can be parametrized
Parametrization
Parametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....
into six 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 twobody systems, where a Kepler orbit is used...
in various ways.
In most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of similar mass can be described as Kepler orbits around their common center of mass, their barycenter
Barycentric coordinates (astronomy)
In astronomy, barycentric coordinates are nonrotating coordinates with origin at the center of mass of two or more bodies.The barycenter is the point between two objects where they balance each other. For example, it is the center of mass where two or more celestial bodies orbit each other...
.
History and relationship with Newton's Laws
Johannes KeplerJohannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
developed his laws of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
around 1605, from astronomical tables detailing the movements of the visible planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s. Kepler's First Law is:
 "The orbitOrbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
of every planet is an ellipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
with the sun at a focusFocus (geometry)In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
."
The mathematics of ellipses are thus the mathematics of Kepler orbits, later expanded to include parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
s and hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
s.
Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
published his law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
in 1687, which says:
Every point mass attracts every other point mass by a forceForceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
pointing along the lineLine (mathematics)The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
intersecting both points. The force is proportionalProportionality (mathematics)In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
to the productProduct (mathematics)In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of the two massMassMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es and inversely proportional to the square of the distance between the point masses:
where:
 F is the magnitude of the gravitational force between the two point masses,
 G is the gravitational constant,
 m_{1} is the mass of the first point mass,
 m_{2} is the mass of the second point mass,
 r is the distance between the two point masses.
The shapes of large celestial bodies are close to spheres. By symmetry, the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards the centre of the sphere. The shell theorem
Shell theorem
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body...
(also proven by Isaac Newton) says that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere, even if the density of the sphere varies with depth (as it does for most celestial bodies). From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to its center.
Smaller objects, like asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s or spacecraft
Spacecraft
A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....
often have a shape strongly deviating from a sphere. But the gravitational forces produced by these irregularities are generally small compared to the gravity of the central body. The difference between an irregular shape and a perfect sphere also diminishes with distances, and most orbital distances are very large when compared with the diameter of a small orbiting body. Thus for some applications, shape irregularity can be neglected without significant impact on accuracy.
Planets that rotate (such as the Earth) take a slightly oblate shape because of the centrifugal force and with such an oblate shape the gravitational attraction will deviate somewhat from that of a homogeneous sphere. This phenomenon is quite noticeable for artificial Earth satellites, especially those in low orbits. But at a large distance the effect of this oblateness is very small and the planetary motions in the Solar System can be computed with sufficient precision using the point mass assumption, using the equation given above.
The twobody problem
Twobody problem
In classical mechanics, the twobody problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...
is the case that there are only two point masses (or homogeneous spheres).
If the two mass points (or homogenous spheres) have the masses and and the position vectors
and relative a point fixed with respect to inertial space (for example relative their common centre of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
) the equations of motion for the two mass points are
where
is the distance between the bodies
and
is the unit vector pointing from body 2 to body 1.
Dividing with the factors and and subtracting the resulting equations one gets the differential equation
for the vector from body 2 to body 1
where
This differential equation for the two body case can be completely solved mathematically and the resulting orbit which follows Kepler's laws of planetary motion is called a "Kepler orbit". The orbits of all planets are to high accuracy Kepler orbits around the Sun (as observed by Johannes Kepler!), the small deviations being due to the much weaker gravitational attractions between the planets, or in the case of Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
, due to general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. Also the orbits around the Earth of the Moon and of the artificial satellites are with a fair approximation Kepler orbits. In fact, the gravitational acceleration towards the Sun is about the same for the Earth and the satellite and therefore in a first approximation "cancels out". Also in high accuracy applications for which the equation of motion must be integrated numerically with all gravitational and nongravitational forces (such as solar radiation pressure and atmospheric drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
) being taken into account, the Kepler orbit concepts are of paramount importance and heavily used.
For example, the 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 twobody systems, where a Kepler orbit is used...
:
 Semimajor axisSemimajor axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
 eccentricityEccentricity (mathematics)In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
 InclinationInclinationInclination in general is the angle between a reference plane and another plane or axis of direction.Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
 Longitude of the ascending nodeLongitude of the ascending nodeThe longitude of the ascending node 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...
 Argument of periapsisArgument of periapsisThe argument of periapsis , symbolized as ω, is one of the orbital elements of an orbiting body...
 True anomalyTrue anomalyIn celestial mechanics, the true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse .The true anomaly is usually...
are only defined for a Kepler orbit. The use of orbital elements therefore always implies that the orbit is approximated with the Kepler orbit having these orbital elements.
Mathematical solution of the differential equation above
For the movement under any central force, i.e. a force aligned with , the specific relative angular momentumSpecific relative angular momentum
The specific relative angular momentum is also known as the areal momentum .In astrodynamics, the specific relative angular momentum of two orbiting bodies is the vector product of the relative position and the relative velocity. Equivalently, it is the total angular momentum divided by the...
stays constant:
Introducing a coordinate system in the plane orthogonal to and polar coordinates
the differential equation takes the form (see "Polar coordinates#Vector calculus")
Taking the time derivative of one gets
Using the chain rule for differentiation one gets
Using the expressions for of equations , , and all time derivatives in can be replaced by derivatives of r as function of . After some simplification one gets

The differential equation can be solved analytically by the variable substitution
Using the chain rule for differentiation one gets:

Using the expressions and for and
one gets
with the general solution
where e and are constants of integration depending on the initial values for s and .
Instead of using the constant of integration explicitly one introduces the convention that the unit vectors defining the coordinate system in the orbital plane are selected such that takes the value zero and e is positive. This then means that is zero at the point where is maximal and therefore is minimal. Defining the parameter p as one has that
This is the equation in polar coordinates for a conic sectionConic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
with origin in a focal point. The argument is called "true anomaly".
For this is a circle with radius p.
For this is an ellipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
with
For this is a parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
with focal length
For this is a hyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
with
The following image illustrates an ellipse (red), a parabola (green) and a hyperbola (blue)
The point on the horizontal line going out to the right from the focal point is the point with for which the distance to the focus takes the minimal value , the pericentre. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value . For the hyperbola the range for is
and for a parobola the range is
Using the chain rule for differentiation , the equation and the definition of p as one gets that the radial velocity component is
and that the tangential component is
The connection between the polar argument and time t is slightly different for elliptic and hyperbolic orbits.
For an elliptic orbit one switches to the "eccentric anomalyEccentric anomalyIn celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure...
" E for which
and consequently
and the angular momentum H is
Integrating with respect to time t one gets
under the assumption that time is selected such that the integration constant is zero.
As by definition of p one has
this can be written
For a hyperbolic orbit one uses the hyperbolic functions for the parameterisation
for which one has
and the angular momentum H is
Integrating with respect to time t one gets
i.e.
To find what time t that corresponds to a certain true anomaly one computes corresponding parameter E connected to time with relation for an elliptic and with relation for a hyperbolic orbit.
Note that the relations and define a mapping between the ranges
Some additional formulae
See also Equation of the center – Analytical expansions
For an elliptic orbit one gets from and that
and therefore that
From then follows that
From the geometrical construction defining the eccentric anomalyEccentric anomalyIn celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure...
it is clear that the vectors and
are on the same side of the xaxis. From this then follows that the vectors
and are in the same quadrant. One therefore has that
and that
where "" is the polar argument of the vector and n is selected such that
For the numerical computation of the standard function ATAN2(y,x)Atan2In trigonometry, the twoargument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive axis of a plane and the point given by the coordinates on it...
(or in double precisionDouble precisionIn computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A doubleprecision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32bit storage locations...
DATAN2(y,x)) available in for example the programming language FORTRANFortranFortran is a generalpurpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
can be used.
Note that this is a mapping between the ranges
For an hyperbolic orbit one gets from and that
and therefore that
As
and as and have the same sign it follows that
This relation is convenient for passing between "true anomaly" and the parameter
E, the latter being connected to time through relation . Note that this is a mapping between the ranges
and that can be computed using the relation
From relation follows that the orbital period P for an elliptic orbit is
As the potential energy corresponding to the force field of relation is
it follows from , , and that the sum of the kinetic and the potential energy
for an elliptic orbit is
and from , , and that the sum of the kinetic and the potential energy for a hyperbolic orbit is
Relative the inertial coordinate system
in the orbital plane with towards pericentre one gets from and that the velocity components are
Determination of the Kepler orbit that corresponds to a given initial state
This is the "initial value problemInitial value problemIn mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
" for the differential equation which is a first order equation
for the 6dimensional "state vector" when written as
For any values for the initial "state vector" the Kepler orbit corresponding
to the solution of this initial value problem can be found with the following algorithm:
Define the orthogonal unit vectors through
with and
From , and follows that by setting
and by defining and such that
where
one gets a Kepler orbit that for true anomaly has the same r, and values as those defined by and .
If this Kepler orbit then also has the same vectors for this true anomaly as the ones defined by and the state vector of the Kepler orbit takes the desired values for true anomaly .
The standard inertially fixed coordinate system in the orbital plane (with directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation
Note that the relations and has a singularity when and
i.e.
which is the case that it is a circular orbit that is fitting the initial state
The osculating Kepler orbit
For any state vector the Kepler orbit corresponding to this state can be computed with the algorithm defined above.
First the parameters are determined from and then
the orthogonal unit vectors in the orbital plane using the relations and .
If now the equation of motion is
where
is another function then
the resulting parameters
defined by will all vary with time as opposed to the case of a Kepler orbit for which only the parameter
will vary
The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" at time t is said to be "osculating" at this time.
This concept is for example useful in case
where
is a small "perturbing force" due to for example a faint gravitational pull from other celestial bodies. The parameters of the osculating Kepler orbit will then only slowly change and the osculating Kepler orbit is a good approximation to the real orbit for a considerable time period before and after the time of osculation.
This concept can also be useful for a rocket during powered flight as it then tells which Kepler orbit
the rocket would continue in in case the thrust is switchedoff.
For a "close to circular" orbit the concept "eccentricity vectorEccentricity vectorIn 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...
" defined as is useful. From , and follows that
i.e. is a smooth differentiable function of the state vector also if this state corresponds to a circular orbit.
Orbital elements
A Kepler orbit is specified by six orbital elements, normally the following (assuming an elliptical orbit; parabolas and hyperbolas are also possible if eccentricity >= 1).
Two define the shape and size of the ellipse: Eccentricity ()
 Semimajor axis ()
Two define the orientation of the orbital planeOrbital plane (astronomy)All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semiflat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
: InclinationInclinationInclination in general is the angle between a reference plane and another plane or axis of direction.Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
()  Longitude of the ascending nodeLongitude of the ascending nodeThe longitude of the ascending node 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...
()
And finally: Argument of periapsisArgument of periapsisThe argument of periapsis , symbolized as ω, is one of the orbital elements of an orbiting body...
() defines the orientation of the ellipse in the orbital plane.  Mean anomalyMean anomalyIn celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....
at epochEpoch (astronomy)In astronomy, an epoch is a moment in time used as a reference point for some timevarying astronomical quantity, such as celestial coordinates, or elliptical orbital elements of a celestial body, where these are subject to perturbations and vary with time...
() defines the position of the orbiting body along the ellipse. (True anomalyTrue anomalyIn celestial mechanics, the true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse .The true anomaly is usually...
() is shown on the diagram, since mean anomaly does not represent a real geometric angle.)
See also
 Elliptic orbitElliptic orbitIn astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 . In a...
 Hyperbolic trajectoryHyperbolic trajectoryIn astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to...
 Parabolic trajectoryParabolic trajectoryIn astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...
External links
 JAVA applet animating the orbit of a satellite in an elliptic Kepler orbit around the Earth with any value for semimajor axis and eccentricity.