Kepler orbit
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, 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 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. Introduction. 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: where: Alternately, the equation can be expressed as: Where formula_7 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. Isaac Newton. 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: Where: 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. 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: where: 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. Simplified two body problem. To solve for the motion of an object in a two body system, two simplifying assumptions can be made: 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 formula_16 and formula_17 and position vectors formula_21 and formula_22 relative to some inertial reference frame experience gravitational forces: where formula_12 is the relative position vector of mass 1 with respect to mass 2, expressed as: and formula_27 is the unit vector in that direction and formula_2 is the length of that vector. 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 formula_29 is the gravitational parameter and is equal to In many applications, a third  simplifying assumption can be made: 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. Keplerian elements. 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. Two define the size and shape of the trajectory: Three define the orientation of the orbital plane: And finally: Because formula_33, formula_34 and formula_35 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 () above. For movement under any central force, i.e. a force parallel to r, the specific relative angular momentum formula_42 stays constant: Since the cross product of the position vector and its velocity stays constant, they must lie in the same plane, orthogonal to formula_44. 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 () refers to linear acceleration formula_45 as opposed to angular formula_46 or radial formula_47 acceleration. Therefore, one must be cautious when transforming the equation. Introducing a cartesian coordinate system formula_48 and polar unit vectors formula_49 in the plane orthogonal to formula_44: We can now rewrite the vector function formula_12 and its derivatives as: (see "Vector calculus"). Substituting these into (), we find: This gives the non-ordinary polar differential equation: In order to solve this equation, all time derivatives must be eliminated. This brings: Taking the time derivative of () gets Equations  () and () allow us to eliminate the time derivatives of formula_5. In order to eliminate the time derivatives of formula_2, the chain rule is used to find appropriate substitutions: Using these four substitutions, all time derivatives in () can be eliminated, yielding an ordinary differential equation for formula_2 as function of formula_62 The differential equation () can be solved analytically by the variable substitution Using the chain rule for differentiation gets: Using the expressions () and () for formula_66 and formula_67 gets with the general solution where "e" and formula_68 are constants of integration depending on the initial values for "s" and formula_69 Instead of  using the constant of integration formula_68 explicitly one introduces the convention that the unit vectors formula_71 defining the coordinate system in the orbital plane are selected such that formula_68 takes the value zero and "e" is positive. This then means that formula_5 is zero at the point where formula_74 is maximal and therefore formula_75 is minimal. Defining the parameter "p" as formula_76 one has that Alternate derivation. Another way to solve this equation without the use of polar differential equations is as follows: Define a unit vector formula_78 such that formula_79 and formula_80. It follows that Now consider (see Vector  triple product). Notice 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 formula_5 is the angle between formula_12 and formula_90. Solving for "r": and for a parabola the range is Using the chain rule for differentiation (), the equation () and the definition of "p" as formula_97 one gets that the radial velocity component is and that the tangential component (velocity component perpendicular to formula_98) is The connection between the polar argument formula_5 and time "t" is slightly different for elliptic and hyperbolic orbits . For an elliptic orbit one switches to the "eccentric anomaly" "E" for which and consequently and the angular momentum "H" is Integrating with respect to time "t" gives under the assumption that time formula_100 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" gets i.e. To find what time t that corresponds to a certain true anomaly formula_101 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. For an "elliptic orbit" one gets from () and () that and therefore that From () then follows that From the geometrical construction defining the eccentric anomaly it is clear that the vectors formula_104 and formula_105 are on the same side of the "x"-axis. From this then follows that the vectors formula_106 and formula_107 are in the same quadrant. One therefore has that and that where "formula_108" is the polar argument of  the vector formula_109 and "n" is selected such that formula_110 For the numerical computation of formula_111 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 () and () that and therefore that As and as formula_114 and formula_115 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 formula_117 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 "See also Equation of the center – Analytical expansions" The Equation of the center relates mean anomaly to true anomaly for elliptical orbits, for small numerical eccentricity. Determination of the Kepler orbit that corresponds to a given initial state. This is the "initial value problem" for the differential equation () which is  a first order equation for the 6-dimensional "state vector" formula_121 when written as For any values for the initial "state vector" formula_122 the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm: Define the orthogonal unit vectors formula_123 through with formula_124 and formula_125 From (), () and () follows that by setting one gets a Kepler orbit that for true anomaly formula_5 has the same "r", formula_98 and formula_128 values as those defined by () and (). If this Kepler orbit then also has the same formula_129 vectors for this true anomaly formula_5 as the  ones defined by () and () the state vector formula_131 of the Kepler orbit takes the desired values formula_132 for true anomaly formula_5. The standard inertially fixed coordinate system formula_134 in the orbital plane (with formula_135 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 formula_136 and i.e. which is the case that it is a circular orbit that is fitting the initial state formula_138 The osculating Kepler orbit. For any state vector  formula_131 the Kepler orbit corresponding to this state can be computed with the algorithm defined above. First the parameters formula_140 are determined from formula_141 and then the orthogonal unit vectors in the orbital plane formula_71 using the relations () and (). If now the equation of motion is where is a function other than the resulting parameters formula_145 defined by formula_146 will all vary with time as opposed to the case of a Kepler orbit for which only the parameter formula_5 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 case  the thrust is switched off. For a "close to circular" orbit the concept "eccentricity vector" defined as formula_150 is useful. From (), () and () follows that i.e. formula_151 is a smooth differentiable function of the state vector formula_121 also if this state corresponds to a circular orbit.