Kepler orbit
In celestial mechanics, a Kepler orbit 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. 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, 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 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, 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 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. As measurements of the planets became 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 circular paths centered on the Sun. History of Kepler and the telescope Kepler started working with Tycho Brahe. After inheriting Tycho's work in 1601, Kepler discovered; this let him to publish Astronomia Nova. He suggested. Kepler’s book Astronomia Nova contained information about treatises on the nature of light and a “new star.” Kepler first noticed the star in October 1604. This sighting of the new star was called a supernova. New questions came to Kepler's mind, he questioned how light from the region of the heavenly bodies could reflect into the atmosphere of Earth and be seen with the human eye. Kepler questioned how the light could be pinpoint against a single point on the retina and how images were conveyed upside down.
Galileo sent Kepler his account of the skies in Siderius Nunicus. Kepler responded to Galileo with three treatises. Conversation with the Sidereal Messenger which speculates the distance of the newly discovered Jupiter’s moons; the theoretical work of the new telescope using two convex lenses instead of two. This allowed the image to be seen right side up; that the observations of Jupiter in Narration Concerning the Jovian Satellites and how the work had strong support of Galileo’s discoveries. In return of the treatises, Galileo wrote to Kepler saying, “I thank you because you were the first one, the only one, to have complete faith in my assertions.” Kepler’s Laws of Planetary Motion When Kepler started working with Tycho Brahe around 1600, Tycho gave him the task to review all the information he had on Mars. Kepler noted that the position of Mars created problems for a lot of models; this led Kepler to configure 3 Laws of Planetary Motion. First Law: Planets move in ellipses with the Sun at one focus The law would change an eccentricity of 0.0. and focus more of an eccentricity of 0.8.
Which show that Circular and Elliptical orbits have the same period and focus, but different sweeps of area defined by the Sun. This leads to the Second Law: The radius vector describes equal areas in equal times; these two laws were published in Kepler’s book Astronomia Nova in 1609. For a circles motion is uniform, however for the elliptical to sweep the area in a uniform rate, the object moves when the radius vector is short and moves slower when the radius vector is long. Kepler published his Third Law of Planetary Motion in his book Harmonices Mundi. With this law, Newton used it to define his laws of gravitation; the Third Law: The squares of the periodic times are to each other as the cubes of the mean distances. Along with The Third Law, that talks about gravitation, it led to Johannes Kepler working on his celestial mechanics, he became known as the father of calculus. 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 the path of an object undergoing Keplerian motion may follow a parabola or a hyperbola, 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 = a 1 + e cos where: r is the distance a is the semi-major axis, which defines the size of the orbit e is the eccentricity, which defines the shape of the orbit θ is the true anomaly, the angle between th
N-body problem
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem; the n-body problem in general relativity is more difficult to solve. The classical physical problem can be informally stated as the following: Given the quasi-steady orbital properties of a group of celestial bodies, predict their interactive forces. For this reason, the two-body problem has been solved and is discussed below, as well as the famous restricted three-body problem. Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton from astronomer John Flamsteed – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion.
Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits well or correctly. Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits; the above discovery goes right to the heart of the matter as to what the n-body problem is physically: as Newton realized, it is not sufficient to just specify the initial position and velocity, or three orbital positions either, to determine a planet's true orbit: the gravitational interactive forces have to be known too. Thus came the awareness and rise of the n-body "problem" in the early 17th century; these gravitational attractive forces do conform to Newton's laws of motion and to his law of universal gravitation, but the many multiple interactions have made any exact solution intractable. This conformity led to the wrong approach. After Newton's time the n-body problem was not stated because it did not include a reference to those gravitational interactive forces.
Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of those gravitational interactive forces. Newton said in his Principia, paragraph 21: And hence it is that the attractive force is found in both bodies; the Sun attracts Jupiter and the other planets, Jupiter attracts its satellites and the satellites act on one another. And although the actions of each of a pair of planets on the other can be distinguished from each other and can be considered as two actions by which each attracts the other, yet inasmuch as they are between the same, two bodies they are not two but a simple operation between two termini. Two bodies can be drawn to each other by the contraction of rope between them; the cause of the action is twofold, namely the disposition of each of the two bodies. This last statement, which implies the existence of gravitational interactive forces, is key; as shown below, the problem conforms to Jean Le Rond D'Alembert's non-Newtonian first and second Principles and to the nonlinear n-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces.
The problem of finding the general solution of the n-body problem was considered important and challenging. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem; the announcement was quite specific: Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points collide, try to find a representation of the coordinates of each point as a series in a variable, some known function of time and for all of whose values the series converges uniformly. In case the problem could not be solved, any other important contribution to classical mechanics would be considered to be prizeworthy; the prize was awarded to Poincaré though he did not solve the original problem.. The version printed contained many important ideas which led to the development of chaos theory; the problem as stated was solved by Karl Fritiof Sundman for n = 3.
The n-body problem considers n point masses mi, i = 1, 2, …, n in an inertial reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi. Newton's second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass. Newton's law of gravity says that the gravitational force felt on mass mi by a single mass mj is given by F i j = G m i m j ‖ q j − q i ‖ 3, {\displaystyle \mathbf _={\frac {\left\|\mathbf _-\m
Aerospace engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: astronautical engineering. Avionics engineering deals with the electronics side of aerospace engineering. Aeronautical engineering was the original term for the field; as flight technology advanced to include craft operating in outer space, the broader term "aerospace engineering" has come into common use. Aerospace engineering the astronautics branch is colloquially referred to as "rocket science". Flight vehicles are subjected to demanding conditions such as those caused by changes in atmospheric pressure and temperature, with structural loads applied upon vehicle components, they are the products of various technological and engineering disciplines including aerodynamics, avionics, materials science, structural analysis and manufacturing. The interaction between these technologies is known as aerospace engineering; because of the complexity and number of disciplines involved, aerospace engineering is carried out by teams of engineers, each having their own specialized area of expertise.
The origin of aerospace engineering can be traced back to the aviation pioneers around the late 19th to early 20th centuries, although the work of Sir George Cayley dates from the last decade of the 18th to mid-19th century. One of the most important people in the history of aeronautics, Cayley was a pioneer in aeronautical engineering and is credited as the first person to separate the forces of lift and drag, which are in effect on any flight vehicle. Early knowledge of aeronautical engineering was empirical with some concepts and skills imported from other branches of engineering. Scientists understood some key elements of aerospace engineering, like fluid dynamics, in the 18th century. Many years after the successful flights by the Wright brothers, the 1910s saw the development of aeronautical engineering through the design of World War I military aircraft. Between World Wars I and II, great leaps were made in Aeronautical Engineering; the advent of mainstream civil aviation accelerated this process.
Notable airplanes of this era include the Curtiss JN 4, the Farman F.60 Goliath, Fokker trimotor. Notable military airplanes of this period include the Mitsubishi A6M Zero, the Supermarine Spitfire and the Messerschmitt Bf 109 from Japan, Great Britain, Germany respectively. A significant development in Aerospace engineering came with the first Jet engine-powered airplane, the Messerschmitt Me 262 which entered service in 1944 towards the end of the second World War; the first definition of aerospace engineering appeared in February 1958. The definition considered the Earth's atmosphere and the outer space as a single realm, thereby encompassing both aircraft and spacecraft under a newly coined word aerospace. In response to the USSR launching the first satellite, Sputnik into space on October 4, 1957, U. S. aerospace engineers launched the first American satellite on January 31, 1958. The National Aeronautics and Space Administration was founded in 1958 as a response to the Cold War. In 1969, Apollo 11, the first manned space mission to the moon took place.
It saw three astronauts enter orbit around the Moon, with two, Neil Armstrong and Buzz Aldrin, visiting the lunar surface. The third astronaut, Michael Collins, stayed in orbit to rendezvous with Armstrong and Aldrin after their visit to the lunar surface; some of the elements of aerospace engineering are: Radar cross-section – the study of vehicle signature apparent to Radar remote sensing. Fluid mechanics – the study of fluid flow around objects. Aerodynamics concerning the flow of air over bodies such as wings or through objects such as wind tunnels. Astrodynamics – the study of orbital mechanics including prediction of orbital elements when given a select few variables. While few schools in the United States teach this at the undergraduate level, several have graduate programs covering this topic. Statics and Dynamics – the study of movement, moments in mechanical systems. Mathematics – in particular, differential equations, linear algebra. Electrotechnology – the study of electronics within engineering.
Propulsion – the energy to move a vehicle through the air is provided by internal combustion engines, jet engines and turbomachinery, or rockets. A more recent addition to this module is ion propulsion. Control engineering – the study of mathematical modeling of the dynamic behavior of systems and designing them using feedback signals, so that their dynamic behavior is desirable; this applies to the dynamic behavior of aircraft, propulsion systems, subsystems that exist on aerospace vehicles. Aircraft structures – design of the physical configuration of the craft to withstand the forces encountered during flight. Aerospace engineering aims to keep structures lightweight and low-cost while maintaining structural integrity. Materials science – related to structures, aerospace engineering studies the materials of which the aerospace structures are to be built. New materials with specific properties are invented, or existing ones are modified to improve their performance. Solid mechanics – Closely related to material science is solid mechanics which deals with stress and strain analysis of the components of the vehicle.
Nowadays there are several Finite Element programs such as MSC
Circular orbit
A circular orbit is the orbit with a fixed distance around the barycenter, that is, in the shape of a circle. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. In this case, not only the distance, but the speed, angular speed and kinetic energy are constant. There is no apoapsis; this orbit has no radial version. Transverse acceleration causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have a = v 2 r = ω 2 r where: v is orbital velocity of orbiting body, r is radius of the circle ω is angular speed, measured in radians per unit time; the formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of a is measured in meters per second per second the numerical values for v will be in meters per second, r in meters, ω in radians per second.
The relative velocity is constant: v = G M r = μ r where: G, is the gravitational constant M, is the mass of both orbiting bodies, although in common practice, if the greater mass is larger, the lesser mass is neglected, with minimal change in the result. Μ = G M, is the standard gravitational parameter. The orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to: r = h 2 μ where: h = r v is specific angular momentum of the orbiting body; this is because μ = r v 2 ω 2 r 3 = μ Hence the orbital period can be computed as: T = 2 π r 3 μ Compare two proportional quantities, the free-fall time T f f = π 2 2 r 3 μ and the time to fall to a point mass in a radial parabolic orbit T p a r = 2 3 r 3 μ The fact that the formulas only differ by a constant factor is a priori clear from dimensional analysis. The specific orbital energy is negative, ϵ = − v 2 2 ϵ = − μ 2 r Thus the virial theorem applies without taking a time-average: the kinetic energy of the system is equal to the absolute value of the total energy the potential energy of the system is equal to twice the total energyThe escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.
Maneuvering into a large circular orbit, e.g. a geostationary orbit, requires a larger delta-v than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed of the circular orbit. It is a matter of maneuvering into the orbit. See Hohmann transfer orbit. In Schwarzschild metric, the orbital velocity for a circular orbit with radius r is given by the following formula: v = G M r − r S where r S = 2 G M c 2 is the Schwarzschild radius of the central body. For the sake of convenience, the derivation will be written in units in which c = G = 1; the four-velocity of a body on a circular orbit is given by: u μ
Azimuth
An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer to a point of interest is projected perpendicularly onto a reference plane; when used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area around an observer on Earth's surface, the reference vector points to true north; the azimuth is the star's vector on the horizontal plane. Azimuth is measured in degrees; the concept is used in navigation, engineering, mapping and ballistics. In land navigation, azimuth is denoted alpha, α, defined as a horizontal angle measured clockwise from a north base line or meridian. Azimuth has been more defined as a horizontal angle measured clockwise from any fixed reference plane or established base direction line. Today, the reference plane for an azimuth is true north, measured as a 0° azimuth, though other angular units can be used.
Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is defined. Quite azimuths or compass bearings are stated in a system in which either north or south can be the zero, the angle may be measured clockwise or anticlockwise from the zero. For example, a bearing might be described as " south, thirty degrees east", abbreviated "S30°E", the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, the turning direction, stated last, is east or west; the directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be in the direction of one of the cardinal points, a different notation, e.g. "due east", is used instead. The cartographical azimuth can be calculated when the coordinates of 2 points are known in a flat plane: α = 180 π atan2 Remark that the reference axes are swapped relative to the mathematical polar coordinate system and that the azimuth is clockwise relative to the north.
This is the reason why the Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°; the formula in radians would be easier: α = atan2 Caveat: Most computer libraries reverse the order of the atan2 parameters. When the coordinates of one point, the distance L, the azimuth α to another point are known, one can calculate its coordinates: X 2 = X 1 + L sin α Y 2 = Y 1 + L cos α This is used in triangulation. We are standing at latitude φ 1, longitude zero. We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth α is given by tan α = sin L cos φ 1 tan φ 2 − sin φ 1 cos L A better approximation assumes the Earth is a slightly-squashed sphere. Normal-section azimuth is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; the difference is immeasurably small. Various websites will calculate geodetic azimuth. Formulas for calculating geodetic azimuth are linked in the distance
Orbit
In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Orbit refers to a repeating trajectory, although it may refer to a non-repeating trajectory. To a close approximation and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion; the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached.
It assumed the heavens were fixed apart from the motion of the spheres, was developed without any understanding of gravity. After the planets' motions were more measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably predicting the planets' positions in the sky and more epicycles were required as the measurements became more accurate, hence the model became unwieldy. Geocentric it was modified by Copernicus to place the Sun at the centre to help simplify the model; the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular, as had been believed, that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had been thought, but rather that the speed depends on the planet's distance from the Sun.
Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are about 5.2 and 0.723 AU distant from the Sun, their orbital periods about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.23/11.862, is equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, that those bodies orbit their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Advances in Newtonian mechanics were used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously; this led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are approximated well by the Newtonian predictions but the differences are measurable.
All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is easier to use and sufficiently accurate. Within a planetary system, dwarf planets and other minor planets and space debris orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time.
Mercury, the smallest planet in the Solar System, has the most eccentric orbit
Orbital eccentricity
The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, greater than 1 is a hyperbola; the term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy. In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit; the eccentricity of this Kepler orbit is a non-negative number. The eccentricity may take the following values: circular orbit: e = 0 elliptic orbit: 0 < e < 1 parabolic trajectory: e = 1 hyperbolic trajectory: e > 1 The eccentricity e is given by e = 1 + 2 E L 2 m red α 2 where E is the total orbital energy, L is the angular momentum, mred is the reduced mass, α the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics: F = α r 2 or in the case of a gravitational force: e = 1 + 2 ε h 2 μ 2 where ε is the specific orbital energy, μ the standard gravitational parameter based on the total mass, h the specific relative angular momentum.
For values of e from 0 to 1 the orbit's shape is an elongated ellipse. The limit case between an ellipse and a hyperbola, when e equals 1, is parabola. Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1. For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury, one must calculate the inverse sine to find the projection angle of 11.86 degrees. Next, tilt any circular object by that angle and the apparent ellipse projected to your eye will be of that same eccentricity; the word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + κέντρον kentron "center".
"Eccentric" first appeared in English in 1551, with the definition "a circle in which the earth, sun. Etc. deviates from its center". By five years in 1556, an adjectival form of the word had developed; the eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector: e = | e | where: e is the eccentricity vector. For elliptical orbits it can be calculated from the periapsis and apoapsis since rp = a and ra = a, where a is the semimajor axis. E = r a − r p r a + r p = 1 − 2 r a r p + 1 where: ra is the radius at apoapsis. Rp is the radius at periapsis; the eccentricity of an elliptical orbit can be used to obtain the ratio of the periapsis to the apoapsis: r p r a = 1 − e 1 + e For Earth, orbital eccentricity ≈ 0.0167, apoapsis= aphelion and periapsis= perihelion relative to sun. For Earth's annual orbit path, ra/rp ratio = longest_radius / shortest_radius ≈ 1.034 relative to center point of path. The eccentricity of the Earth's orbit is about 0.0167.
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