Sphere of influence (astrodynamics)
A sphere of influence in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by; the general equation describing the radius of the sphere r S O I of a planet: r S O I ≈ a 2 / 5 where a is the semimajor axis of the smaller object's orbit around the larger body. M and M are the masses of the larger object, respectively. In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun.
Because the definition of rSOI relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem; the table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun.: The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance θ from the massive body. A more accurate formula is given by r S O I ≈ a 2 / 5 1 1 + 3 cos 2 10 Averaging over all possible directions we get r S O I ¯ = 0.9431 a 2 / 5 Consider two point masses A and B at locations r A and r B, with mass m A and m B respectively. The distance R = | r B − r A | separates the two objects. Given a massless third point C at location r C, one can ask whether to use a frame centered on A or on B to analyse the dynamics of C. Let's consider a frame centered on A.
The gravity of B is denoted as g B and will be treated as a perturbation to the dynamics of C due to the gravity g A of body A. Due their gravitational interactions, point A is attracted to point B with acceleration a A = G m B R 3, this frame is therefore non-inertial. To quantify the effects of the perturbations in this frame, one should consider the ratio of the perturbations to the main body gravity i.e. χ A = | g B − a A | | g A |. The perturbation g B − a A is known as the tidal forces due to body B, it is possible to construct the perturbation ratio χ B for the frame centered on B by interchanging A ↔ B. As C gets close to A, χ A →
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
Potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, the electric potential energy of an electric charge in an electric field; the unit for energy in the International System of Units is the joule, which has the symbol J. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space; these forces, that are called conservative forces, can be represented at every point in space by vectors expressed as gradients of a certain scalar function called potential.
Since the work of potential forces acting on a body that moves from a start to an end position is determined only by these two positions, does not depend on the trajectory of the body, there is a function known as potential that can be evaluated at the two positions to determine this work. There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy has two components: the kinetic energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are called conservative forces; the work done by a conservative force is W = − Δ U where Δ U is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy.
Common notations for potential energy are PE, U, V, Ep. Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity; the action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. Consider a ball whose mass is m and whose height is h; the acceleration g of free fall is constant, so the weight force of the ball mg is constant. Force × displacement gives the work done, equal to the gravitational potential energy, thus U g = m g h The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.
Potential energy is linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for an applied force is independent of the path the work done by the force is evaluated at the start and end of the trajectory of the point of application; this means that there is a function U, called a "potential," that can be evaluated at the two points xA and xB to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, W = ∫ C F ⋅ d x = U − U where C is the trajectory taken from A to B; because the work done is independent of the path taken this expression is true for any trajectory, C, from A to B.
The function U is called the potential energy associated with the applied force. Examples of forces that have potential energies are spring forces. In this section the relationship between work and potential energy is presented in more detail; the line integral that defines work along curve C takes a special form if the force F is related to a scalar field φ so that F = ∇ φ = ( ∂ φ ∂ x, ∂
Kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes; the same amount of work is done by the body when decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1 2 m v 2. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light; the standard unit of kinetic energy is the joule. The imperial unit of kinetic energy is the foot-pound; the adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality; the principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva.
Willem's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51. Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy; these can be categorized in two main classes: kinetic energy. Kinetic energy is the movement energy of an object.
Kinetic energy can be transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction; the chemical energy has been converted into kinetic energy, the energy of motion, but the process is not efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top; the kinetic energy has now been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling.
The energy is not destroyed. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent; the bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity, a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an circular orbit, this kinetic energy remains constant because there is no friction in near-earth space. However, it becomes apparent at re-entry. If the orbit is elliptical or hyperbolic throughout the orbit kinetic and potential energy are exchanged.
Without loss or gain, the sum of the kinetic and potential energy remains constant. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, binding energy. Flywheels have been developed as a method of energy storage; this illustrates that kinetic energy is stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian mechanics is suitable. However, if the speed of the object is comparabl
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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Apsis
The term apsis refers to an extreme point in the orbit of an object. It denotes either the respective distance of the bodies; the word comes via Latin from Greek, there denoting a whole orbit, is cognate with apse. Except for the theoretical possibility of one common circular orbit for two bodies of equal mass at diametral positions, there are two apsides for any elliptic orbit, named with the prefixes peri- and ap-/apo-, added in reference to the body being orbited. All periodic orbits are, according to Newton's Laws of motion, ellipses: either the two individual ellipses of both bodies, with the center of mass of this two-body system at the one common focus of the ellipses, or the orbital ellipses, with one body taken as fixed at one focus, the other body orbiting this focus. All these ellipses share a straight line, the line of apsides, that contains their major axes, the foci, the vertices, thus the periapsis and the apoapsis; the major axis of the orbital ellipse is the distance of the apsides, when taken as points on the orbit, or their sum, when taken as distances.
The major axes of the individual ellipses around the barycenter the contributions to the major axis of the orbital ellipses are inverse proportional to the masses of the bodies, i.e. a bigger mass implies a smaller axis/contribution. Only when one mass is sufficiently larger than the other, the individual ellipse of the smaller body around the barycenter comprises the individual ellipse of the larger body as shown in the second figure. For remarkable asymmetry, the barycenter of the two bodies may lie well within the bigger body, e.g. the Earth–Moon barycenter is about 75% of the way from Earth's center to its surface. If the smaller mass is negligible compared to the larger the orbital parameters are independent of the smaller mass. For general orbits, the terms periapsis and apoapsis are used. Pericenter and apocenter are equivalent alternatives, referring explicitly to the respective points on the orbits, whereas periapsis and apoapsis may refer to the smallest and largest distances of the orbiter and its host.
For a body orbiting the Sun, the point of least distance is the perihelion, the point of greatest distance is the aphelion. The terms become apastron when discussing orbits around other stars. For any satellite of Earth, including the Moon, the point of least distance is the perigee and greatest distance the apogee, from Ancient Greek Γῆ, "land" or "earth". For objects in lunar orbit, the point of least distance is sometimes called the pericynthion and the greatest distance the apocynthion. Perilune and apolune are used. In orbital mechanics, the apsides technically refer to the distance measured between the barycenters of the central body and orbiting body. However, in the case of a spacecraft, the terms are used to refer to the orbital altitude of the spacecraft above the surface of the central body; these formulae characterize the pericenter and apocenter of an orbit: Pericenter Maximum speed, v per = μ a, at minimum distance, r per = a. Apocenter Minimum speed, v ap = μ a, at maximum distance, r ap = a.
While, in accordance with Kepler's laws of planetary motion and the conservation of energy, these two quantities are constant for a given orbit: Specific relative angular momentum h = μ a Specific orbital energy ε = − μ 2 a where: a is the semi-major axis: a = r per + r ap 2 μ is the standard gravitational parameter e is the eccentricity, defined as e = r ap − r per r ap + r per = 1 − 2 r ap r per + 1 Note t
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