In
celestial mechanicsCelestial 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
ellipseIn 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...
,
parabolaIn 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
hyperbolaIn 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 two-dimensional
orbital planeAll 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 semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
in three-dimensional space. (A Kepler orbit can also form a straight line.) It considers only the gravitational attraction of two bodies, neglecting
perturbationsPerturbation 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 dragIn 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 non-spherical central body, and so on. It is thus said to be a solution of a special case of the
two-body problemIn classical mechanics, the two-body 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 mechanicsIn physics, classical mechanics is one of the two major sub-fields 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 relativityGeneral 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
parametrizedParametrization 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 elementsOrbital 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...
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
barycenterIn astronomy, barycentric coordinates are non-rotating 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 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 motionIn 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
planetA 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 orbit
In 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 ellipseIn 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 focusIn 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
parabolaIn 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
hyperbolaIn 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 NewtonSir 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 gravitationNewton'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 forceIn 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 lineThe 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 proportionalIn 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 productIn 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 massMass 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,
- m1 is the mass of the first point mass,
- m2 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 theoremIn 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
asteroidAsteroids 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
spacecraftA 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
two-body problemIn classical mechanics, the two-body 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 massIn 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
MercuryMercury 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 relativityGeneral 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 non-gravitational forces (such as solar radiation pressure and
atmospheric dragIn 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 elementsOrbital 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...
:
- Semi-major axis
The 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...
- eccentricity
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,...
- Inclination
Inclination 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 node
The 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 periapsis
The argument of periapsis , symbolized as ω, is one of the orbital elements of an orbiting body...
- True anomaly
In 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 momentumThe 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 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 ellipseIn 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 parabolaIn 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 hyperbolaIn 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 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 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 x-axis. 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)In trigonometry, the two-argument 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 precisionIn computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32-bit storage locations...
DATAN2(y,x)) available in for example the programming language FORTRANFortran is a general-purpose, 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 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 6-dimensional "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 switched-off.
For a "close to circular" orbit the concept "eccentricity 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 planeAll 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 semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
:
- Inclination
Inclination 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 node
The 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 periapsis
The 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 anomaly
In 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 epochIn astronomy, an epoch is a moment in time used as a reference point for some time-varying 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 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 orbit
In 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 trajectory
In 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 trajectory
In 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