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, in the form of an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also tend toward 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 parameterized 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. 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:
The orbit of every planet is an ellipse with the sun at a focus. 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:
r ( θ ) = a ( 1 − e 2 ) 1 + e cos ( θ ) {\displaystyle r(\theta )={\frac {a(1-e^{2})}{1+e\cos(\theta )}}}
where:
r {\displaystyle r} is the distance
a {\displaystyle a} is the semi-major axis, which defines the size of the orbit
e {\displaystyle e} is the eccentricity, which defines the shape of the orbit
θ {\displaystyle \theta } 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:
r ( θ ) = p 1 + e cos ( θ ) {\displaystyle r(\theta )={\frac {p}{1+e\cos(\theta )}}}
Where p {\displaystyle p} 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 produced the first such theory based around the concept of gravity. Albert Einstein's general relativity is the current description of gravitation in modern physics. The two-body problem in general relativity has no closed-form solutions.
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:
F = m a = m d 2 r d t 2 {\displaystyle \mathbf {F} =m\mathbf {a} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}}
Where:
F {\displaystyle \mathbf {F} } is the force vector
m {\displaystyle m} is the mass of the body on which the force is acting
a {\displaystyle \mathbf {a} } is the acceleration vector, the second time derivative of the position vector r {\displaystyle \mathbf {r} }
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:
… excerpt ends here. Continue reading the full article.
