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Mean Anomaly

Mean anomaly

In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. It is the fraction of the orbital period that has elapsed since the last passage at periapsis z, expressed as an angle. In the diagram below, it is M (the angle z-c-y). Image:Kepler's-equation-scheme.png The point y is defined such that the circular sector area z-c-y is equal to the elliptic sector area z-s-p, scaled up by the ratio of the major to minor axes of the ellipse. Or, in other words, the circular sector area z-c-y is equal to the area x-s-z.

Calculation

In astrodynamics mean anomaly

M\,\!

can be calculated as follows:

M - M_0=n(t-t_0)\,\!

where:
-

M_0\,\!

is the mean anomaly at time

t_0\,\!

,
-

t_0\,\!

is the start time,
-

t\,\!

is the time of interest, and
-

n\,\!

is the mean motion.
Alternatively:

M=E - e \cdot \sin E\,\!

where:
-

E\,\!

is orbit's eccentric anomaly,
-

e\,\!

is orbit's eccentricity.

Orbit

.]] :For other meanings of the term "orbit", see orbit (disambiguation) In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity.

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about 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.

Planetary orbits

Within a planetary system, planets, asteroids, comets and space debris orbit the central star in elliptical orbits. Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet. Due to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Pluto and Mercury have the most eccentric orbits. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune. As an object orbits another, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. In the elliptical orbit, the centre of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity. See also: Kepler's laws of planetary motion

Understanding orbits

There are a few common ways of understanding orbits.
- As the object moves sideways, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall away beneath it.
- A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
- As the object falls, it moves sideways fast enough (has enough tangential velocity) to miss the orbited object. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center. As an illustration of the orbit around a planet (eg Earth), the much-used cannon model may prove useful (see image below). Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and we can ignore the effects of air friction on the cannon ball. 300px If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground (A). As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above). If the cannonball is fired with sufficient velocity, the ground will curve away from the ball at the same rate as the ball falls - it is now in orbit (D). The orbit may be circular like (D) or if the firing velocity is increased even more, the orbit may become more (E) and more (F) elliptical. At a certain even faster velocity (called the escape velocity) the motion changes from an elliptical orbit to a parabola.

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron # As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms. One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :

\frac - r\left( \frac \right)^2

and :

\frac\frac\left( r^2\frac \right)

. Since the force is always radial, the transverse acceleration is zero, and it follows that: :

\frac = hu^2

, where h is a constant of integration and we have introduced the auxiliary variable u defined as 1/r. If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields: :

\frac + u = \frac

. In the case of an inverse square force law the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The equation of the orbit described by the particle is thus: :

r = \frac = \frac

, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

See: Orbital elements For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are: :Semimajor axis = (periapsis + apoapsis)/2 = mean of the extreme radii :Periapsis = semimajor axis × (1 - eccentricity) = least distance :Apoapsis = semimajor axis × (1 + eccentricity) = largest distance Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (mean anomaly), or over the orbital angle as observed by the primary (true anomaly), then you get a different result. See here for details.

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Earth orbits

See Earth orbit for more details.
- Low Earth orbit
- High Earth Orbit
- Intermediate circular orbit
- Geostationary orbit
- Geosynchronous orbit
- Geostationary transfer orbit
- Molniya orbit
- Polar orbit
- Polar Sun Synchronous Orbit (this is not a complete list).

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10⁻¹¹ N·m²/kg²
- 6.6742 × 10⁻¹¹ m³/(kg·s²)
- 6.6742 × 10⁻¹¹(kg/m³)^-1s^-2. Thus the constant has dimension density^-1 time^-2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :

GT^2 \sigma = 3\pi \left( \frac \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.

External links

- An on-line orbit plotter: http://www.bridgewater.edu/departments/physics/ISAW/PlanetOrbit.html
- [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) Category:Celestial mechanics Category:Solar System als:Umlaufbahn ja:軌道 (力学) simple:Orbit th:วงโคจร

Radians

See Radian (band) for the Austrian trio. ---- The radian (symbol: rad) is the SI unit of plane angle.

Definition

The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle is one radian. 1 rad = m·m^–1 = 1

Explanation

The radian is useful to distinguish between quantities of different nature but the same dimension. For example angular velocity can be measured in radians per second (rad/s). Retaining the word radian emphasizes that angular velocity is equal to 2π times the rotational frequency. In practice, the symbol rad is used where appropriate, but the derived unit "1" is generally omitted in combination with a numerical value. Angle measures in radians are often given without any explicit unit. When a unit is given, sometimes the symbol rad is used, sometimes the symbol ^c (for "circular"). Care must be taken with this symbol since it can be mistaken for the ° (degree) symbol. degree.]] There are 2π (approximately 6.28318531) radians in a complete circle, so: :

2\pi\mbox = 360^\circ

1 \mbox = \frac   = \frac   \approx 57.29577951^\circ

or: :

360^\circ=2\pi\mbox

1^\circ=\frac\mbox=\frac\mbox \approx 0.01745329\mbox

More generally, we can say: :

x \mbox = x \frac

If, for example, -1.570796 in radians was given, the corresponding degree value would be: :

-1.570796 \mbox = -1.570796 \cdot \frac   = -90^\circ

In calculus, angles must be represented in radians in trigonometric functions, to make identities and results as simple and natural as possible. For example, the use of radians leads to the simple identity :

\lim_\frac=1

, which is the basis of many other elegant identities in mathematics, including :

\frac \sin x = \cos x

. The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995. For measuring solid angles, see steradian.

Dimensional analysis

Although the radian is a unit of measure, anything measured in radians is dimensionless. This can be seen easily in that the ratio of an arc's length to its radius is the angle of the arc, measured in radians; yet the quotient of two distances is dimensionless. Another way to see the dimensionlessness of the radian is in the Taylor series for the trigonometric function

\sin(x)

: :

\sin(x) = x - \frac + \cdots

x

had units, then the sum would be meaningless; the linear term

x

can not be added to the cubic term

x^3/3!

, etc. Therefore,

x

must be dimensionless.

SI multiples

Periapsis

: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms). In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. video game publisher In astronomy, an apsis (plural apsides "ap-si-deez") is the point of greatest or least distance of the elliptical orbit of a celestial body from its center of attraction (the center of mass of the system). The point of closest approach is called the periapsis or pericentre and the point of farthest approach is the apoapsis (Greek απο, from), apocentre or apapsis (the latter term, although etymologically more correct, is much less used). A straight line drawn through the periapsis and apoapsis is the line of apsides. This is the major axis of the ellipse, the line through the longest part of the ellipse. Related terms are used to identify the body being orbited. The most common are perigee and apogee, referring to Earth orbits, and perihelion and aphelion, referring to orbits around the Sun (Greek ἡλιος).

Formulae

We have:
- Periapsis: maximum speed

v_\mathrm = \sqrt \,

at minimum distance

r_\mathrm=(1-e)a\!\,

(periapsis distance)
- Apoapsis: minimum speed

v_\mathrm = \sqrt \,

at maximum distance

r_\mathrm=(1+e)a\!\,

(apoapsis distance) where one easily verifies :

h = \sqrt

\epsilon=-\frac

(each the same for both points, like they are for the whole orbit, in accordance with Kepler's laws of planetary motion (conservation of angular momentum) and the conservation of energy) where:
-

a\!\,

is the semi-major axis
-

e\!\,

is the eccentricity
-

h\!\,

is the specific relative angular momentum
-

\epsilon\!\,

is the specific orbital energy
-

\mu\!\,

is the standard gravitational parameter Properties: :

e=\frac=1-\frac=\frac-1

Note that for conversion from heights above the surface to distances, the radius of the central body has to be added, and conversely. The arithmetic mean of the two distances is the semi-major axis

a\!\,

. The geometric mean of the two distances is the semi-minor axis

b\!\,

. The geometric mean of the two speeds is

\sqrt

, the speed corresponding to a kinetic energy which, at any position of the orbit, added to the existing kinetic energy, would allow the orbiting body to escape (the square root of the sum of the squares of the two speeds is the local escape velocity).

Terminology

Various related terms are used for other celestial objects. The '-gee', '-helion' and '-astron' and '-galacticon' forms are frequently used in the astronomical literature, while the other listed forms are occasionally used, although '-saturnium' has very rarely been used in the last 50 years. The '-gee' form is commonly (although incorrectly) used as a generic 'closest approach to planet' term instead of specifically applying to the Earth. The term peri/apomelasma was used by Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002. ⁽¹⁾ Properly pronounced 'affelion', although 'ap-helion' is commonly heard. Since "peri" and "apo" are Greek, it is considered by purists more correct to use the Greek form for the body, giving forms such as '-zene' for Jupiter and '-krone' for Saturn. For Venus, the alternate form '-krition' (from Kritias, an older name for Aphrodite) has also been suggested. In the Moon's case, in practice all three forms are used, albeit very infrequently. The '-cynthion' form is, according to some, reserved for artificial bodies, whilst others reserve '-lune' for an object launched from the Moon and '-cynthion' for an object launched from elsewhere. For Jupiter, the '-jove' form is occasionally used by astronomers whilst the '-zene' form is never used, like the other pure Greek forms ('-cytherion' (Venus), '-areion' (Mars), '-hermion' (Mercury), '-krone' (Saturn), '-uranion' (Uranus), '-poseidion' (Neptune) and '-hadion' (Pluto)). The daunting prospect of having to maintain a different word for every orbitable body in the solar system (and beyond) is the main reason why the generic '-apsis' has become the almost universal norm.

Orbital period

The orbital period is the time it takes a planet (or another object) to make one full orbit. There are several kinds of orbital periods for objects around the Sun:
- The sidereal period is the time that it takes the object to make one full orbit around the Sun, relative to the stars. This is considered to be an object's true orbital period.
- The synodic period is the time that it takes for the object to reappear at the same spot in the sky, relative to the Sun, as observed from Earth. This is the time that elapses between two successive conjunctions with the Sun and is the object's Earth-apparent orbital period. The synodic period differs from the sidereal period since Earth itself revolves around the Sun.
- The draconitic period is the time that elapses between two passages of the object at its ascending node, the point of its orbit where it crosses the ecliptic from the southern to the northern hemisphere. It differs from the sidereal period because the object's line of nodes typically precesses or recesses slowly.
- The anomalistic period is the time that elapses between two passages of the object at its perihelion, the point of its closest approach to the Sun. It differs from the sidereal period because the object's semimajor axis typically precesses or recesses slowly.
- The tropical period, finally, is the time that elapses between two passages of the object at right ascension zero. It is slightly shorter than the sidereal period because the vernal point precesses.

Relation between sidereal and synodic period

Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period. Using the abbreviations :E = the sidereal period of Earth (a sidereal year, not the same as a tropical year) :P = the sidereal period of the other planet :S = the synodic period of the other planet (wrt Earth) During the time S, the Earth moves over an angle of (360°/E)S (assuming a circular orbit) and the planet moves (360°/P)S. Let us consider the case of an inferior planet, i.e. a planet that will complete one orbit more than Earth before the two return to the same position relative to the Sun. :

\frac 360^\circ = \frac 360^\circ + 360^\circ

and using algebra we obtain :

P = \frac1

For a superior planet one derives likewise: :

P = \frac1

The above formulæ are easily understood by considering the angular velocities of the Earth and the object: the object's apparent angular velocity is its true (sidereal) angular velocity minus the Earth's, and the synodic period is then simply a full circle divided by that apparent angular velocity. Table of synodic periods in the Solar System, relative to Earth: In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to run its phases, coming back to the same solar aspect angle for an observer on the planet's surface —the Earth's motion does not affect this value, because an Earth observer is not involved. For example, Deimos' synodic period is 1.2648 days, 0.18% longer than Deimos' sidereal period of 1.2624 d.

Calculation

Small body orbiting a central body

In astrodynamics the orbital period

T\,

of a small body orbiting a central body in a circular or elliptical orbit is: :

T = 2\pi\sqrt

and :

\mu = GM \,

(standard gravitational parameter) where:
-

a\,

is length of orbit's semi-major axis,
-

G \,

is the gravitational constant,
-

M \,

the mass of the central body. Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. For the Earth (and any other spherically symmetric body with the same average density) as central body we get :

T = 1.4 \sqrt

and for a body of water :

T = 3.3 \sqrt

T in hours, with R the radius of the body. Thus, as an alternative for using a very small number like G, the strength of universal gravity can be described using some reference material, like water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time. For the Sun as central body we simply get :

T = \sqrt

T in years, with a in astronomical units. This is the same as Kepler's Third Law

Two bodies orbiting each other

In celestial mechanics when both orbiting bodies' masses have to be taken into account the orbital period

P\,

can be calculated as follows: :

P = 2\pi\sqrt

where:
-

a\,

is the sum of the semi-major axes of the ellipses in which the centers of the bodies move, or equivalently, the semi-major axis of the ellipse in which one body moves, in the frame of reference with the other body at the origin (which is equal to their constant separation for circular orbits),
-

M_1\,

and

M_2\,

are the masses of the bodies,
-

G\,

is the gravitational constant. Note that the orbital period is independent of size: for a scale model it would be the same, when densities are the same (see also Orbit#Scaling in gravity). In a parabolic or hyperbolic trajectory the motion is not periodic, and the duration of the full trajectory is infinite.

Major axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis

b\,\!

through the eccentricity

e\,\!

and the semi-latus rectum

\ell\,\!

, as follows: :

b = a \sqrt\,\!

\ell=a(1-e^2)\,\!

. :

a\ell=b^2\,\!

. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping

\ell\,\!

fixed. Thus

a\,\!

and

b\,\!

tend to infinity,

a\,\!

faster than

b\,\!

. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, :

r(1-e\cos\theta)=l\,\!

The mean value of

r=\,\!

and

r=\,\!

, is

a=\,\!

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is:

\frac - \frac = 1

In terms of the semi-latus rectum and the eccentricity we have

a=

Astronomy

Orbital period

In astrodynamics the orbital period

T\,

of a small body orbiting a central body in a circular or elliptical orbit is: :

T = 2\pi\sqrt

where: :

a\,

is the length of the orbit's semi-major axis :

\mu

is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :

P^2=a^3\,

where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :

P^2= \fraca^3\,

where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis

b = a \sqrt\,\!

.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman):

a (1 + \frac)\,\!

. The time-average of the inverse of the radius,

r^\,\!

, is

a^\,\!

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis

a \,

can be calculated from orbital state vectors:

a = \,

for an elliptical orbit and

a = \,

for a hyperbolic trajectory and

\epsilon =  -

(specific orbital energy) and

\mu = GM \,

(standard gravitational parameter), where:
-

v\,

is orbital velocity from velocity vector of an orbiting object,
-

\mathbf\,

is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
-

G \,

is the gravitational constant,
-

M \,

the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References

- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix

A = PDP^T

, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :

\frac + \frac = 1

The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted

l\,\!

(lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to

a\,\!

and

b\,\!

(the ellipse's semi-axes) by the formula

al=b^2\,\!

or, if using the eccentricity,

l=a(1-e^2)\,\!

. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation :

r (1 + e \cos \theta) = l \,\!

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

Mean motion

Mean Motion,

n\,\!

, is a meansure of how far a satellite has progressed around its orbit, from perigee. Unless the orbit is circular, the Mean Motion is only an average value, and does not represent the instantaneous angular rate. In databases of satellite orbital parameters the Mean Motion is typically specified in revolutions per day.

Calculation

n =  \sqrt\,\!

will return units of radians/second. where:
-

G\,\!

is the gravitational constant in m³ kg^-1 s^{-2,

- $M\,\!$ is the mass of the planet, in kg,
(note that $G\,\!$ and $M\,\!$ are typically combined to give the Standard gravitational parameter, $\mu\,\!$ )

- $a\,\!$ is semi-major axis in meters.

Category:Astronomy

Orbital eccentricity__NOTOC__
:This page refers to eccentricity in astrodynamics. For other uses, see the disambiguation page eccentricity.

eccentricity
In astrodynamics, under standard assumptions any orbit must be of conic section shape. The eccentricity of this conic section, the orbit's eccentricity, is an important parameter of the orbit that defines its absolute shape. Eccentricity may be interpreted as a measure of how much this shape deviates from a circle.

Under standard assumptions eccentricity ( $e\,\!$ ) is strictly defined for all circular, elliptic, parabolic and hyperbolic orbits and may take following values:

- for circular orbits: $e=0\,\!$ ,

- for elliptic orbits: $0,$
- for parabolic trajectories: $e=1\,\!$ ,

- for hyperbolic trajectories: $e>1\,\!$ .

Calculation
Eccentricity of an orbit can be calculated from orbital state vectors as a magnitude of eccentricity vector:
: $e= \left | \mathbf \right |$
where:

- $\mathbf\,\!$ is eccentricity vector.
----
For elliptic orbits it can also be calculated from distance at periapsis and apoapsis:
: $e==1-\frac=\frac-1$
where:

- $d_p\,\!$ is distance at periapsis,

- $d_a\,\!$ is distance at apoapsis.
Examples
For example, the eccentricity of the Earth's orbit today is 0.0167. Through time, the eccentricity of the Earth's orbit slowly changes from nearly 0 to almost 0.05 as a result of gravitational attractions between the planets (see graph [http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html]).

Other values: Pluto 0.2488 (largest value among the planets of the Solar System), Mercury 0.2056, Moon 0.0554. For the values for all planets in one table, see :de:Planet (Tabelle).

See also

- Eccentricity vector

External links

- [http://scienceworld.wolfram.com/physics/Eccentricity.html World of Physics: Eccentricity]

Category:Astrodynamics
Category:Celestial mechanics

Kepler's laws of planetary motion
Johannes Kepler's primary contributions to astronomy/astrophysics were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the observations of the keen-sighted Danish astronomer Tycho Brahe. The article on Johannes Kepler gives a less mathematical description of the laws, as well as a treatment of their historical and intellectual context.

Sir Isaac Newton's later discovery of the laws of motion and universal gravitation depended strongly on Kepler's work. Although from the modern point of view, Kepler's laws can be seen as a consequence of Newton's laws, historically, it was the other way around: Kepler provided a mathematically distilled description of the empirical observations, which Newton then interpreted.

Kepler's first law
Kepler's first law
The orbit of a planet about a star is an ellipse with the star at one focus.

There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near apocentre than near pericentre.

Kepler's second law

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time.

This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day.

As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. The physical meaning of the law is that the planet moves faster when it is closer to the sun. This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out.

Kepler's third law

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis.

: $P^2 \propto a^3$
:P = object's sidereal period
:a = orbit's semimajor axis

Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional.

See the actual figures: attributes of major planets.

This law is also known as the harmonic law.

Accuracy and limitations
As discussed below, Kepler's third law needs to be modified when the orbiting body's mass is not negligible compared to the mass of the body being orbited. However, the correction is fairly small in the case of the planets orbiting the Sun.

A more serious limitation of Kepler's laws is that they assume a two-body system. This is a particularly bad approximation in the case of the Earth-Sun-Moon system, and for calculations of the Moon's orbit, Kepler's laws are far less accurate than the empirical method invented by Ptolemy hundreds of years before.

Newton generalized Kepler's first law with the realization that an object moving at greater than escape velocity (e.g., some comets) would have an open parabolic or hyperbolic orbit rather than a closed elliptical one. Thus, all of the conic sections are possible orbits. The second law is still valid for open orbits (since angular momentum is still conserved), but the third law is inapplicable because the motion is not periodic.

Because electrical forces also obey an inverse square law, Kepler's laws also apply to bodies interacting electrically. If the interaction is repulsive, as in Rutherford scattering, then the orbit is hyperbolic, with the center of repulsion outside the hyperbola. In the planetary model of the atom, the electrons were imagined to orbit the nucleus exactly like the planets orbiting the Sun, and Niels Bohr even went so far as to attempt to assign specific atomic energy levels to specific elliptical orbits. The Keplerian approximation is successful when the quantum-mechanical wavelengths are small compared to the size of the orbit (as in Rutherford scattering, and in some unusual artificially populated electron orbitals with very high values of the principal quantum number).

Kepler's laws do not consider the emission of radiation. The emission of gravitational radiation is negligible in our solar system, but important in some stellar systems containing black holes or neutron stars (any system involving large masses combined with large accelerations). In the planetary model of the atom, the emission of electromagnetic radiation should have led to the collapse of all atoms, and this was a hint that classical physics was in need of modification.

Kepler's laws don't incorporate relativity, so, for example, they don't correctly predict the precession of Mercury's orbit. The successful explanation of this non-Keplerian behavior was one of the most dramatic early successes of general relativity. Kepler's laws fail even more dramatically in the case of an object orbiting a black hole; here, general relativity predicts that many orbits —those that pass through the event horizon— are one-way streets to a dead-end region of spacetime.

Connection to Newton's laws and conservation laws
Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. The second law can also be seen as a statement of conservation of angular momentum, which is a logical consequence of Newton's laws in the special case of a force that acts along the line connecting two objects.

Kepler's first law

Newton proposed that "every object in the universe attracts every other object along a line of the centres of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects."

The following assumes that acceleration is of the form
: $\frac = f(r)\hat$ .

Recall that in polar coordinates:

: $\frac = \dot r\hat + r\dot\theta\hat$

: $\frac = (\ddot r - r\dot\theta^2)\hat + (r\ddot\theta + 2\dot r \dot\theta)\hat$

In component form, we have:

: $\ddot r - r\dot\theta^2 = f(r)$

: $r\ddot\theta + 2\dot r\dot\theta = 0$

Since $\dot r = dr/dt$ and $\ddot\theta=/$ , the latter equation is equivalent to
: $\frac = -2\frac$ .
When integrated, this yields
: $\log\dot\theta = -2\log r + \log\ell$ ,
: $\ell = r^2\dot\theta$ ,
for some constant $\ell$ , which can be shown to be the specific angular momentum. Now we substitute. Let:

: $r = \frac$

: $\dot r = -\frac\dot u = -\frac\frac\frac= -\ell\frac$

: $\ddot r = -\ell\frac\frac = -\ell\dot\theta\frac= -\ell^2u^2\frac$

The equation of motion in the $\hat$ direction becomes:

: $\frac + u = - \fracf\left(\frac\right)$

If $f(r)=-k/r^2$ , as Newton's law of gravitation claims, then:

: $\frac + u = \frac$

where -k is our proportionality constant.

This differential equation has the general solution:

: $u = A\cos(\theta-\theta_0) + \frac.$

Replacing u with r and letting θ₀=0:

: $r = \frac$ .

This is indeed the equation of a conic section with the origin at one focus. Thus, Kepler's first law is a direct result of Newton's law of gravitation and Newton's second law of motion.

Kepler's second law

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum $\mathbf$ of a point mass with mass $m$ and velocity $\mathbf$ is :

: $\mathbf \equiv m \mathbf \times \mathbf$ .

where $\mathbf$ is the position vector of the particle.

Since $\mathbf = \frac$ , we have:

: $\mathbf = \mathbf \times m\frac$

taking the time derivative of both sides:

: $\frac = \mathbf \times \mathbf = 0$

since the cross product of parallel vectors is 0. We can now say that $|\mathbf|$ is constant.

The area swept out by the line joining the planet and the sun is half the area of the parallelogram formed by $\mathbf$ and $d\mathbf$ .

: $dA = \begin\frac\end |\mathbf \times d\mathbf| = \begin\frac\end \left|\mathbf \times \fracdt\right| = \fracdt$

Since $|\mathbf|$ is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law
Newton used the third law as one of the pieces of evidence used to build the conceptual and mathematical framework supporting his law of gravity. If we take Newton's laws of motion as given, and consider a hypothetical planet that happens to be in a perfectly circular orbit of radius r, then we have $F=mv^2/r$ for the sun's force on the planet. The velocity is proportional to r/T, which by Kepler's third law varies as one over the square root of r. Substituting this into the equation for the force, we find that the gravitational force is proportional to one over r squared. (Newton's actual historical chain of reasoning is not known with certainty, because in his writing he tended to erase any traces of how he had reached his conclusions.) Reversing the direction of reasoning, we can consider this as a proof of Kepler's third law based on Newton's of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have:

: $T^2 = \frac \cdot r^3$

where:

- T = planet's sidereal period

- r = radius of the planet's circular orbit

- G = the gravitational constant

- M = mass of the sun

The same arguments can be applied to any object orbiting any other object. This discussion implicitly assumed that the planet orbits around the stationary sun, although in reality both the planet and the sun revolve around their common center of mass. Newton recognized this, and modified this third law, noting that the period is also affected by the orbiting body's mass. However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius. The most general result is:

: $T^2 = \frac \cdot a^3$

where:

- T = object's sidereal period

- a = object's semimajor axis

- G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant

- M = mass of one object

- m = mass of the other object

For objects orbiting the sun, it can be convenient to use units of years, AU, and solar masses, so that G, 4π² and the various conversion factors cancel out. Also with m<<M we can set m+M = M, so we have simply $T^2=a^3$ .

Proving Kepler's third law
Define point A to be the periapsis, and point B as the apoapsis of the planet when orbiting the sun.

Kepler's second law states that the orbiting body will sweep out equal areas in equal quantities of time. If we now look at a very small periods of time at the moments when the planet is at points A and B, then we can approximate the area swept out as a triangle with an altitude equal to the distance between the planet and the sun, and the base equal to the time times the speed of the planet.

: $\frac\cdot(1-\epsilon)a\cdot V_A\,dt=\frac\cdot(1+\epsilon)a\cdot V_B\,dt$

: $(1-\epsilon)\cdot V_A=(1+\epsilon)\cdot V_B$

: $V_A=V_B\cdot\frac$

Using the law of conservation of energy for the total energy of the planet at points A and B,

: $\frac-\frac=\frac-\frac$

: $\frac-\frac=\frac-\frac$

: $\frac=\frac\cdot \left ( \frac-\frac \right )$

: $\frac=\frac\cdot \left ( \frac \right )$

: $V_B^2 \cdot \left ( \frac\right ) ^2-V_B^2=\frac\cdot \left ( \frac \right )$

: $V_B^2 \cdot \left ( \frac\right )=\frac$

: $V_B^2 \cdot \left ( \frac\right )=\frac$

: $V_B^2 \cdot 4\epsilon=\frac$

: $V_B=\sqrt.$

Now that we have $V_B$ , we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B.

: $\frac=\frac=\frac\cdot(1+\epsilon)a\cdot V_B=\frac\cdot(1+\epsilon)a\cdot\sqrt=
\frac\cdot\sqrt$

However, the total area of the ellipse is equal to $\pi a \sqrta$ . (That's the same as $\pi a b$ , because $b=\sqrta$ ). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so,

: $T\cdot \frac=\pi a \sqrta$

: $T\cdot \frac\cdot\sqrt=\pi \sqrta^2$

: $T=\frac=\frac=
\frac\sqrt$

: $T^2=\fraca^3.$

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass $M+m$ (see reduced_mass). To integrate that in the above formula, M must be replaced with $M+m$ , to give $T^2=\fraca^3$ . Q.E.D.

Solution for the motion as a function of time

The Keplerian problem assumes an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:

- c center of auxiliary circle and ellipse

- s sun (at one focus of ellipse); $\mboxcs=a\varepsilon$

- p the planet

- z perihelion

- x is the projection of the planet to the auxiliary circle; then $\mboxsxz=\frac ba\mboxspz$

- y is a point on the circle such that $\mboxcyz=\mboxsxz=\frac ba\mboxspz$
and three angles measured from perihelion:

- true anomaly $T=\angle zsp$ , the planet as seen from the sun

- eccentric anomaly $E=\angle zcx$ , x as seen from the centre

- mean anomaly $M=\angle zcy$ , y as seen from the centre

image:kepler's-equation-scheme.png

Then

: $\mboxcxz=\mboxcxs+\mboxsxz$ $=\mboxcxs+\mboxcyz$

: $\frac2E=a\varepsilon\frac a2\sin E+\frac2M$
giving Kepler's equation

: $M=E-\varepsilon\sin E$ .

To connect E and T, assume $r=\mboxsp$ then

: $a\varepsilon+r\cos T=a\cos E$ and $r\sin T=b\sin E$

: $r=\frac=\frac$

: $\tan T=\frac=\frac ba\frac=\frac$

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities:

: $\tan\frac T2=\sqrt\frac\tan\frac E2$

(So far only laws of geometry have been used.)

Note that $\mboxspz$ is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined $\mboxspz=\frac ab\mboxcyz=\frac ab\frac2M$ and so M is also proportional to time since perihelion—this is why it was introduced.

We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as
: $E\approx M+\left(\varepsilon-\frac18\varepsilon^3\right)\sin M+\frac12\varepsilon^2\sin 2M+\frac38\varepsilon^3\sin 3M$
via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.[http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html]

See also

- Circular motion

- Gravity

- Two-body problem

- Free-fall time

External links

- Crowell, Benjamin, Conservation Laws, [http://www.lightandmatter.com/area1book2.html http://www.lightandmatter.com/area1book2.html], an online book that gives a proof of the first law without the use of calculus.

Category:Celestial mechanics
Category:Equations
Category:Eponymous laws

ko:케플러 법칙

ja:ケプラーの法則

Eccentric anomaly
The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. In the diagram below, it is E (the angle zcx).

ellipse

Calculation
In astrodynamics eccentric anomaly E can be calculated as follows:

: $E=\arccos$

where:

- $\mathbf\,\!$ is the orbiting body's position vector (segment sp),

- $a\,\!$ is the orbit's semi-major axis (segment cz), and

- $e\,\!$ is the orbit's eccentricity.

The relation between E and M, the mean anomaly, is:

: $M = E - e \cdot \sin.\,\!$

For small values of $e$ ( $e < 0.6627434$ ) this equation can be solved iteratively, starting from $E_0 = M$ and using the relation $E_ = M + e\,\sin E_i$ . The first few terms of the expansion in powers of $e$ are:

- $E_1 = M + e\,\sin M$

- $E_2 = M + e\,\sin M + \frac e^2 \sin 2M$

- $E_3 = M + e\,\sin M + \frac e^2 \sin 2M
+ \frac e^3 (3\sin 3M - \sin M)$ .
For references on details of this derivation, as well as other more efficient methods of solution, see Murray and Dermott (1999, p.35). For a derivation of the limiting value of $e$ see Plummer (1960, section 46).

The relation between E and T, the true anomaly, is:

: $\cos =$

or equivalently

: $\tan = \sqrt \tan.\,$

The relations between the radius (position vector magnitude) and the anomalies are:

: $r = a \left ( 1 - e \cdot \cos \right )\,\!$

and

: $r = a.\,\!$

See also

- Kepler's laws of planetary motion

- Mean anomaly

- True anomaly

References

- Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.

- Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)

Category:Astrodynamics
Category:Celestial mechanics

Category:Astrodynamics
Astrodynamics is a term application of physics, particularly Newtonian mechanics, to space objects such as stars, planets and satellites.

Category: Celestial mechanics
Category: Astronomy

Marshall, Minnesota
Marshall is a city located in Lyon County, Minnesota. As of the 2000 census, the city had a total population of 12,735. It is the county seat of Lyon County⁶. Marshall is home to Southwest Minnesota State University.

Geography
Southwest Minnesota State University
According to the United States Census Bureau, the city has a total area of 21.5 km² (8.3 mi²). 21.5 km² (8.3 mi²) of it is land and none of it is covered by water.

Demographics
As of the census² of 2000, there are 12,735 people, 4,914 households, and 2,914 families residing in the city. The population density is 593.1/km² (1,537.0/mi²). There are 5,182 housing units at an average density of 241.3/km² (625.4/mi²). The racial makeup of the city is 91.35% White, 2.79% African American, 0.35% Native American, 1.52% Asian, 0.03% Pacific Islander, 2.61% from other races, and 1.34% from two or more races. 5.93% of the population are Hispanic or Latino of any race.

There are 4,914 households out of which 30.5% have children under the age of 18 living with them, 48.0% are married couples living together, 8.6% have a female householder with no husband present, and 40.7% are non-families. 30.4% of all households are made up of individuals and 12.1% have someone living alone who is 65 years of age or older. The average household size is 2.39 and the average family size is 3.04.

In the city the population is spread out with 23.9% under the age of 18, 19.1% from 18 to 24, 26.8% from 25 to 44, 17.7% from 45 to 64, and 12.4% who are 65 years of age or older. The median age is 30 years. For every 100 females there are 91.7 males. For every 100 females age 18 and over, there are 90.5 males.

The median income for a household in the city is $37,950, and the median income for a family is $52,284. Males have a median income of $35,478 versus $21,640 for females. The per capita income for the city is $18,588. 12.4% of the population and 7.8% of families are below the poverty line. Out of the total population, 10.3% of those under the age of 18 and 16.7% of those 65 and older are living below the poverty line.

Key Business(s)

The Schwan Food Company [http://www.theschwanfoodcompany.com], headquartered in Marshall, Minn., is one of the largest, branded frozen-food companies in the United States and the second-largest privately-held corporation in Minnesota.

With approximately 22,000 employees worldwide, The Schwan Food Company has grown to become one of the largest producers of frozen pizza and egg rolls. The company is also a national leader in frozen dessert and premium ice cream manufacturing and distribution.

Category:Cities in Minnesota
Category:Lyon County, Minnesota

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