An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation (13)
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.
From ancient times until the 16th and 17th centuries, the motions of the planets were believed to follow perfectly circular geocentric paths as taught by the ancient Greek philosophers Aristotle and Ptolemy. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path (see epicycle). As measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a heliocentric model of the Solar System, although he still believed that the planets traveled in perfectly circular paths centered on the Sun.
Development of the laws
In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe. Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion. The first law states:
More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, which, along with ellipses, belong to a group of curves known as conic sections. Mathematically, the distance between a central body and an orbiting body can be expressed as:
is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis).
Alternately, the equation can be expressed as:
Where is called the semi-latus rectum of the curve. This form of the equation is particularly useful when dealing with parabolic trajectories, for which the semi-major axis is infinite.
Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions.
Between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion and his law of universal gravitation. His second of his three laws of motion states:
The acceleration of a body is parallel and directly proportional to the net force acting on the body, is in the direction of the net force, and is inversely proportional to the mass of the body:
is the force vector
is the mass of the body on which the force is acting
is the acceleration vector, the second time derivative of the position vector
Strictly speaking, this form of the equation only applies to an object of constant mass, which holds true based on the simplifying assumptions made below.
The mechanisms of Newton's law of universal gravitation; a point mass m1 attracts another point mass m2 by a force F2 which is proportional to the product of the two masses and inversely proportional to the square of the distance (r) between them. Regardless of masses or distance, the magnitudes of |F1| and |F2| will always be equal. G is the gravitational constant.
Newton's law of gravitation states:
Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses:
is the magnitude of the gravitational force between the two point masses
From the laws of motion and the law of universal gravitation, Newton was able to derive Kepler's laws, which are specific to orbital motion in astronomy. Since Kepler's laws were well-supported by observation data, this consistency provided strong support of the validity of Newton's generalized theory, and unified celestial and ordinary mechanics. These laws of motion formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics.
To solve for the motion of an object in a two body system, two simplifying assumptions can be made:
The bodies are spherically symmetric and can be treated as point masses.
There are no external or internal forces acting upon the bodies other than their mutual gravitation.
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 its centre. The shell theorem (also proven by Isaac Newton) states 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 asteroids 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. This effect is quite noticeable for artificial Earth satellites, especially those in low orbits.
Planets rotate at varying rates and thus may take a slightly oblate shape because of the centrifugal force. With such an oblate shape, the gravitational attraction will deviate somewhat from that of a homogeneous sphere. At larger distances the effect of this oblateness becomes negligible. Planetary motions in the Solar System can be computed with sufficient precision if they are treated as point masses.
Two point mass objects with masses and and position vectors and relative to some inertial reference frame experience gravitational forces:
where is the relative position vector of mass 1 with respect to mass 2, expressed as:
Dividing by their respective masses and subtracting the second equation from the first yields the equation of motion for the acceleration of the first object with respect to the second:
where is the gravitational parameter and is equal to
In many applications, a third simplifying assumption can be made:
When compared to the central body, the mass of the orbiting body is insignificant. Mathematically, m1 >> m2, so α = G (m1 + m2) ≈ Gm1. Such standard gravitational parameters, often denoted as , are widely available for Sun, major planets and Moon, which have much larger masses than their orbiting satellites.
This assumption is not necessary to solve the simplified two body problem, but it simplifies calculations, particularly with Earth-orbiting satellites and planets orbiting the Sun. Even Jupiter's mass is less than the Sun's by a factor of 1047, which would constitute an error of 0.096% in the value of α. Notable exceptions include the Earth-Moon system (mass ratio of 81.3), the Pluto-Charon system (mass ratio of 8.9) and binary star systems.
Under these assumptions the 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. The small deviations are due to the much weaker gravitational attractions between the planets, and in the case of Mercury, due to general relativity. The orbits of the artificial satellites around the Earth are, with a fair approximation, Kepler orbits with small perturbations due to the gravitational attraction of the Sun, the Moon and the oblateness of the Earth. 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 drag) being taken into account, the Kepler orbit concepts are of paramount importance and heavily used.
Any Keplerian trajectory can be defined by six parameters. The motion of an object moving in three-dimensional space is characterized by a position vector and a velocity vector. Each vector has three components, so the total number of values needed to define a trajectory through space is six. An orbit is generally defined by six elements (known as Keplerian elements) that can be computed from position and velocity, three of which have already been discussed. These elements are convenient in that of the six, five are unchanging for an unperturbed orbit (a stark contrast to two constantly changing vectors). The future location of an object within its orbit can be predicted and its new position and velocity can be easily obtained from the orbital elements.
True anomaly () defines the position of the orbiting body along the trajectory, measured from periapsis. Several alternate values can be used instead of true anomaly, the most common being the mean anomaly and , the time since periapsis.
Because , and are simply angular measurements defining the orientation of the trajectory in the reference frame, they are not strictly necessary when discussing the motion of the object within the orbital plane. They have been mentioned here for completeness, but are not required for the proofs below.
Mathematical solution of the differential equation (1) above
Since the cross product of the position vector and its velocity stays constant, they must lie in the same plane, orthogonal to . This implies the vector function is a plane curve.
Because the equation has symmetry around its origin, it is easier to solve in polar coordinates. However, it is important to note that equation (1) refers to linear acceleration as opposed to angular or radial acceleration. Therefore, one must be cautious when transforming the equation.
Introducing a cartesian coordinate system and polar unit vectors in the plane orthogonal to :
We can now rewrite the vector function and its derivatives as:
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
Another way to solve this equation without the use of polar differential equations is as follows:
Define a unit vector , , such that and . It follows that
Substituting these values into the previous equation gives:
Integrating both sides:
where c is a constant vector. Dotting this with r yields an interesting result:
where is the angle between and . Solving for r :
Notice that are effectively the polar coordinates of the vector function. Making the substitutions and , we again arrive at the equation
This is the equation in polar coordinates for a conic section with origin in a focal point. The argument is called "true anomaly".
Notice also that, since is the angle between the position vector and the integration constant , the vector must be pointing in the direction of the periapsis of the orbit. We can then define the eccentricity vector associated with the orbit as:
where is the constant angular momentum vector of the orbit, and is the velocity vector associated with the position vector .
Obviously, the eccentricity vector, having the same direction as the integration constant , also points to the direction of the periapsis of the orbit, and it has the magnitude of orbital eccentricity. This makes it very useful in orbit determination (OD) for the orbital elements of an orbit when a state vector  or  is known.
The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue)
A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory (e > 1). Green is a parabolic trajectory (e = 1). Red is an elliptical orbit (0 < e < 1). Grey is a circular orbit (e = 0).
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 parabola the range is
Using the chain rule for differentiation (5), the equation (2) and the definition of p as one gets that the radial velocity component is
and that the tangential component (velocity component perpendicular to ) is
The connection between the polar argument and time t is slightly different for elliptic and hyperbolic orbits.
To find what time t that corresponds to a certain true anomaly one computes corresponding parameter E connected to time with relation (27) for an elliptic and with relation (34) for a hyperbolic orbit.
Note that the relations (27) and (34) define a mapping between the ranges
Some additional formulae
For an elliptic orbit one gets from (20) and (21) that
From the geometrical construction defining the eccentric anomaly 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
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) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN can be used.
Note that this is a mapping between the ranges
For a hyperbolic orbit one gets from (28) and (29) that
and therefore that
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 (34). Note that this is a mapping between the ranges
and that can be computed using the relation
From relation (27) follows that the orbital period P for an elliptic orbit is
As the potential energy corresponding to the force field of relation (1) is
it follows from (13), (14), (18) and (19) that the sum of the kinetic and the potential energy
for an elliptic orbit is
and from (13), (16), (18) and (19) 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 (18) and (19) that the velocity components are
one gets a Kepler orbit that for true anomaly has the same r, and values as those defined by (50) and (51).
If this Kepler orbit then also has the same vectors for this true anomaly as the ones defined by (50) and (51) 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 (53) and (54) has a singularity when and
which is the case that it is a circular orbit that is fitting the initial state
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 (56) and (57).
If now the equation of motion is
is a function other than
the resulting parameters
defined by will all vary with time as opposed to the case of a Kepler orbit for which only the parameter
The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" (59) at time t is said to be "osculating" at this time.
This concept is for example useful in case
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 case the thrust is switched off.
For a "close to circular" orbit the concept "eccentricity vector" defined as is useful. From (53), (54) and (56) follows that
i.e. is a smooth differentiable function of the state vector also if this state corresponds to a circular orbit.
Copernicus, Nicolaus (1952), "Book I, Chapter 4, The Movement of the Celestial Bodies Is Regular, Circular, and Everlasting-Or Else Compounded of Circular Movements", On the Revolutions of the Heavenly Spheres, Great Books of the Western World, vol. 16, translated by Charles Glenn Wallis, Chicago: William Benton, pp. 497–838