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57-605: The Hayford ellipsoid is defined by its semi-major axis a = 6 378 388 .000 m and its flattening f = 1:297.00. Unlike some of its predecessors, such as the Bessel ellipsoid ( a = 6 377 397  m , f = 1:299.15), which was a European ellipsoid, the Hayford ellipsoid also included measurements from North America , as well as other continents (to a lesser extent). It also included isostatic measurements to reduce plumbline divergences. Hayfords ellipsoid did not reach

114-485: A and b tend to infinity, a faster than b . The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola. The equation of an ellipse is where ( h ,  k ) is the center of the ellipse in Cartesian coordinates , in which an arbitrary point is given by ( x ,  y ). The semi-major axis

171-437: A hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola. 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 ℓ {\displaystyle \ell } fixed. Thus

228-427: A hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is: In terms of the semi-latus rectum and the eccentricity, we have The transverse axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate axis or minor axis of length 2 b {\displaystyle 2b} , corresponding to

285-399: A line segment that runs through the center and both foci , with ends at the two most widely separated points of the perimeter . The semi-major axis ( major semiaxis ) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus , and to the perimeter. The semi-minor axis ( minor semiaxis ) of an ellipse or hyperbola is a line segment that

342-473: A central body in a circular or elliptical orbit is: where: Note that for all ellipses with a given semi-major axis, the orbital period is the same, disregarding their eccentricity. The specific angular momentum h of a small body orbiting a central body in a circular or elliptical orbit is where: 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,

399-590: A generic two-body model ) of the actual minimum distance to the Sun using the full dynamical model . Precise predictions of perihelion passage require numerical integration . The two images below show the orbits, orbital nodes , and positions of perihelion (q) and aphelion (Q) for the planets of the Solar System as seen from above the northern pole of Earth's ecliptic plane , which is coplanar with Earth's orbital plane . The planets travel counterclockwise around

456-525: A story published in 1998, thus appearing before perinigricon and aponigricon (from Latin) in the scientific literature in 2002. The suffixes shown below may be added to prefixes peri- or apo- to form unique names of apsides for the orbiting bodies of the indicated host/ (primary) system. However, only for the Earth, Moon and Sun systems are the unique suffixes commonly used. Exoplanet studies commonly use -astron , but typically, for other host systems

513-399: Is In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci ) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is

570-468: Is -gee , so the apsides' names are apogee and perigee . For the Sun, the suffix is -helion , so the names are aphelion and perihelion . According to Newton's laws of motion , all periodic orbits are ellipses. 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, compared to the larger mass,

627-430: Is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives a geocentric lunar average orbital speed of 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value. It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body. This is not quite accurate, because it depends on what

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684-435: Is 236 years early, less accurately shows Eris coming to perihelion in 2260. 4 Vesta came to perihelion on 26 December 2021, but using a two-body solution at an epoch of July 2021 less accurately shows Vesta came to perihelion on 25 December 2021. Trans-Neptunian objects discovered when 80+ AU from the Sun need dozens of observations over multiple years to well constrain their orbits because they move very slowly against

741-597: Is also based on the eccentricity and is computed as r a r p = 1 + e 1 − e {\displaystyle {\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {1+e}{1-e}}} . Due to the large difference between aphelion and perihelion, Kepler's second law is easily visualized. 1 AU (astronomical unit) equals 149.6 million km. Apsis An apsis (from Ancient Greek ἁψίς ( hapsís )  'arch, vault'; pl.   apsides / ˈ æ p s ɪ ˌ d iː z / AP -sih-deez )

798-464: Is at right angles with the semi-major axis and has one end at the center of the conic section . For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum ℓ {\displaystyle \ell } , as follows: The semi-major axis of

855-441: Is based on the eccentricity and is computed as a b = 1 1 − e 2 {\displaystyle {\frac {a}{b}}={\frac {1}{\sqrt {1-e^{2}}}}} , which for typical planet eccentricities yields very small results. The reason for the assumption of prominent elliptical orbits lies probably in the much larger difference between aphelion and perihelion. That difference (or ratio)

912-473: Is currently about 1.016 71  AU or 152,097,700 km (94,509,100 mi). The dates of perihelion and aphelion change over time due to precession and other orbital factors, which follow cyclical patterns known as Milankovitch cycles . In the short term, such dates can vary up to 2 days from one year to another. This significant variation is due to the presence of the Moon: while the Earth–Moon barycenter

969-432: Is moving on a stable orbit around the Sun, the position of the Earth's center which is on average about 4,700 kilometres (2,900 mi) from the barycenter, could be shifted in any direction from it—and this affects the timing of the actual closest approach between the Sun's and the Earth's centers (which in turn defines the timing of perihelion in a given year). Because of the increased distance at aphelion, only 93.55% of

1026-701: Is the geometric mean of these distances: The eccentricity of an ellipse is defined as so Now consider the equation in polar coordinates , with one focus at the origin and the other on the θ = π {\displaystyle \theta =\pi } direction: The mean value of r = ℓ / ( 1 − e ) {\displaystyle r=\ell /(1-e)} and r = ℓ / ( 1 + e ) {\displaystyle r=\ell /(1+e)} , for θ = π {\displaystyle \theta =\pi } and θ = 0 {\displaystyle \theta =0}

1083-452: Is the farthest or nearest point in the orbit of a planetary body about its primary body . The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values . Apsides pertaining to orbits around the Sun have distinct names to differentiate themselves from other apsides; these names are aphelion for the farthest and perihelion for

1140-405: Is the mean value of the maximum and minimum distances r max {\displaystyle r_{\text{max}}} and r min {\displaystyle r_{\text{min}}} of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis In astronomy these extreme points are called apsides . The semi-minor axis of an ellipse

1197-483: The First Point of Aries not in terms of days and hours, but rather as an angle of orbital displacement, the so-called longitude of the periapsis (also called longitude of the pericenter). For the orbit of the Earth, this is called the longitude of perihelion , and in 2000 it was about 282.895°; by 2010, this had advanced by a small fraction of a degree to about 283.067°, i.e. a mean increase of 62" per year. For

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1254-622: The Galactic Center respectively. The suffix -jove is occasionally used for Jupiter, but -saturnium has very rarely been used in the last 50 years for Saturn. The -gee form is also used as a generic closest-approach-to "any planet" term—instead of applying it only to Earth. During the Apollo program , the terms pericynthion and apocynthion were used when referring to orbiting the Moon ; they reference Cynthia, an alternative name for

1311-463: The comets , and the asteroids of the Solar System . There are two apsides in any elliptic orbit . The name for each apsis is created from the prefixes ap- , apo- (from ἀπ(ό) , (ap(o)-)  'away from') for the farthest or peri- (from περί (peri-)  'near') for the closest point to the primary body , with a suffix that describes the primary body. The suffix for Earth

1368-406: The impact parameter , this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus. The semi-minor axis and the semi-major axis are related through the eccentricity, as follows: Note that in a hyperbola b can be larger than a . In astrodynamics the orbital period T of a small body orbiting

1425-428: The precession of the axes .) The dates and times of the perihelions and aphelions for several past and future years are listed in the following table: The following table shows the distances of the planets and dwarf planets from the Sun at their perihelion and aphelion. These formulae characterize the pericenter and apocenter of an orbit: While, in accordance with Kepler's laws of planetary motion (based on

1482-526: The Earth reaches perihelion in early January, approximately 14 days after the December solstice . At perihelion, the Earth's center is about 0.983 29 astronomical units (AU) or 147,098,070 km (91,402,500 mi) from the Sun's center. In contrast, the Earth reaches aphelion currently in early July, approximately 14 days after the June solstice . The aphelion distance between the Earth's and Sun's centers

1539-434: The Earth's distance from the Sun. In the northern hemisphere, summer occurs at the same time as aphelion, when solar radiation is lowest. Despite this, summers in the northern hemisphere are on average 2.3 °C (4 °F) warmer than in the southern hemisphere, because the northern hemisphere contains larger land masses, which are easier to heat than the seas. Perihelion and aphelion do however have an indirect effect on

1596-584: The Greek Moon goddess Artemis . More recently, during the Artemis program , the terms perilune and apolune have been used. Regarding black holes, the term peribothron was first used in a 1976 paper by J. Frank and M. J. Rees, who credit W. R. Stoeger for suggesting creating a term using the greek word for pit: "bothron". The terms perimelasma and apomelasma (from a Greek root) were used by physicist and science-fiction author Geoffrey A. Landis in

1653-404: The Sun and for each planet, the blue part of their orbit travels north of the ecliptic plane, the pink part travels south, and dots mark perihelion (green) and aphelion (orange). The first image (below-left) features the inner planets, situated outward from the Sun as Mercury, Venus, Earth, and Mars. The reference Earth-orbit is colored yellow and represents the orbital plane of reference . At

1710-452: The Sun. The words are formed from the prefixes peri- (Greek: περί , near) and apo- (Greek: ἀπό , away from), affixed to the Greek word for the Sun, ( ἥλιος , or hēlíos ). Various related terms are used for other celestial objects . The suffixes -gee , -helion , -astron and -galacticon are frequently used in the astronomical literature when referring to the Earth, Sun, stars, and

1767-417: The accuracy of Helmert's ellipsoid published 1906 ( a = 6 378 200  m , f = 1:298.3). It has since been replaced as the "International ellipsoid" by the newer Lucerne ellipsoid (1967) and GRS 80 (1980). This geodesy -related article is a stub . You can help Misplaced Pages by expanding it . Semi-major axis In geometry , the major axis of an ellipse is its longest diameter :

Hayford ellipsoid - Misplaced Pages Continue

1824-411: The average is taken over. The time- and angle-averaged distance of the orbiting body can vary by 50-100% from the orbital semi-major axis, depending on the eccentricity. The time-averaged value of the reciprocal of the radius, r − 1 {\displaystyle r^{-1}} , is a − 1 {\displaystyle a^{-1}} . In astrodynamics ,

1881-422: The conservation of angular momentum ) and the conservation of energy, these two quantities are constant for a given orbit: where: Note that for conversion from heights above the surface to distances between an orbit and its primary, the radius of the central body has to be added, and conversely. The arithmetic mean of the two limiting distances is the length of the semi-major axis a . The geometric mean of

1938-409: The distance of the line that joins the nearest and farthest points across an orbit; it also refers simply to the extreme range of an object orbiting a host body (see top figure; see third figure). In orbital mechanics , the apsides technically refer to the distance measured between the barycenter of the 2-body system and the center of mass of the orbiting body. However, in the case of a spacecraft ,

1995-405: The extreme range—from the closest approach (perihelion) to farthest point (aphelion)—of several orbiting celestial bodies of the Solar System : the planets, the known dwarf planets, including Ceres , and Halley's Comet . The length of the horizontal bars correspond to the extreme range of the orbit of the indicated body around the Sun. These extreme distances (between perihelion and aphelion) are

2052-401: The generic suffix, -apsis , is used instead. The perihelion (q) and aphelion (Q) are the nearest and farthest points respectively of a body's direct orbit around the Sun . Comparing osculating elements at a specific epoch to those at a different epoch will generate differences. The time-of-perihelion-passage as one of six osculating elements is not an exact prediction (other than for

2109-414: The hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b . Denoting the semi-major axis length (distance from the center to a vertex) as a , the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows: The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called

2166-465: The lines of apsides of the orbits of various objects around a host body. Distances of selected bodies of the Solar System from the Sun. The left and right edges of each bar correspond to the perihelion and aphelion of the body, respectively, hence long bars denote high orbital eccentricity . The radius of the Sun is 0.7 million km, and the radius of Jupiter (the largest planet) is 0.07 million km, both too small to resolve on this image. Currently,

2223-399: The longest line segment perpendicular to the major axis that connects two points on the ellipse's edge. The semi-minor axis b is related to the semi-major axis a through the eccentricity e and the semi-latus rectum ℓ {\displaystyle \ell } , as follows: A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as

2280-426: The mass ratio of the primary to the secondary is significantly large ( M ≫ m {\displaystyle M\gg m} ); 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 81.300 59 . The Earth–Moon characteristic distance,

2337-399: The minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints ( 0 , ± b ) {\displaystyle (0,\pm b)} of the minor axis lie at the height of the asymptotes over/under

Hayford ellipsoid - Misplaced Pages Continue

2394-413: The nearest point in the solar orbit. The Moon 's two apsides are the farthest point, apogee , and the nearest point, perigee , of its orbit around the host Earth . Earth's two apsides are the farthest point, aphelion , and the nearest point, perihelion , of its orbit around the host Sun. The terms aphelion and perihelion apply in the same way to the orbits of Jupiter and the other planets ,

2451-503: The orbit of the Earth around the Sun, the time of apsis is often expressed in terms of a time relative to seasons, since this determines the contribution of the elliptical orbit to seasonal variations. The variation of the seasons is primarily controlled by the annual cycle of the elevation angle of the Sun, which is a result of the tilt of the axis of the Earth measured from the plane of the ecliptic . The Earth's eccentricity and other orbital elements are not constant, but vary slowly due to

2508-428: 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. The orbiting body's path around the barycenter and its path relative to its primary are both ellipses. The semi-major axis is sometimes used in astronomy as the primary-to-secondary distance when

2565-428: The other is allowed to move arbitrarily far away in one direction, keeping ℓ {\displaystyle \ell } fixed. Thus a and b tend to infinity, a faster than b . The length of the semi-minor axis could also be found using the following formula: where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse. The semi-major axis of

2622-588: The perihelion passage. For example, using an epoch of 1996, Comet Hale–Bopp shows perihelion on 1 April 1997. Using an epoch of 2008 shows a less accurate perihelion date of 30 March 1997. Short-period comets can be even more sensitive to the epoch selected. Using an epoch of 2005 shows 101P/Chernykh coming to perihelion on 25 December 2005, but using an epoch of 2012 produces a less accurate unperturbed perihelion date of 20 January 2006. Numerical integration shows dwarf planet Eris will come to perihelion around December 2257. Using an epoch of 2021, which

2679-409: The perturbing effects of the planets and other objects in the solar system (Milankovitch cycles). On a very long time scale, the dates of the perihelion and of the aphelion progress through the seasons, and they make one complete cycle in 22,000 to 26,000 years. There is a corresponding movement of the position of the stars as seen from Earth, called the apsidal precession . (This is closely related to

2736-504: The radiation from the Sun falls on a given area of Earth's surface as does at perihelion, but this does not account for the seasons , which result instead from the tilt of Earth's axis of 23.4° away from perpendicular to the plane of Earth's orbit. Indeed, at both perihelion and aphelion it is summer in one hemisphere while it is winter in the other one. Winter falls on the hemisphere where sunlight strikes least directly, and summer falls where sunlight strikes most directly, regardless of

2793-443: The ratio of the masses. Conversely, for a given total mass and semi-major axis, the total specific orbital energy is always the same. This statement will always be true under any given conditions. Planet orbits are always cited as prime examples of ellipses ( Kepler's first law ). However, the minimal difference between the semi-major and semi-minor axes shows that they are virtually circular in appearance. That difference (or ratio)

2850-417: The seasons: because Earth's orbital speed is minimum at aphelion and maximum at perihelion, the planet takes longer to orbit from June solstice to September equinox than it does from December solstice to March equinox. Therefore, summer in the northern hemisphere lasts slightly longer (93 days) than summer in the southern hemisphere (89 days). Astronomers commonly express the timing of perihelion relative to

2907-404: The semi-major axis a can be calculated from orbital state vectors : for an elliptical orbit and, depending on the convention, the same or for a hyperbolic trajectory , and ( specific orbital energy ) and ( standard gravitational parameter ), where: Note that for a given amount of total mass, the specific energy and the semi-major axis are always the same, regardless of eccentricity or

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2964-403: The semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived): where T is the period, and a is the semi-major axis. This form turns out to be a simplification of the general form for the two-body problem , as determined by Newton : where G is the gravitational constant , M is the mass of the central body, and m is the mass of

3021-421: The semi-major axis of the geocentric lunar orbit, is 384,400 km. (Given the lunar orbit's eccentricity e  = 0.0549, its semi-minor axis is 383,800 km. Thus the Moon's orbit is almost circular.) The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,730 km, the Earth's counter-orbit taking up the difference, 4,670 km. The Moon's average barycentric orbital speed

3078-401: The smaller mass is negligible (e.g., for satellites), then the orbital parameters are independent of the smaller mass. When used as a suffix—that is, -apsis —the term can refer to the two distances from the primary body to the orbiting body when the latter is located: 1) at the periapsis point, or 2) at the apoapsis point (compare both graphics, second figure). The line of apsides denotes

3135-415: The terms are commonly used to refer to the orbital altitude of the spacecraft above the surface of the central body (assuming a constant, standard reference radius). The words "pericenter" and "apocenter" are often seen, although periapsis/apoapsis are preferred in technical usage. The words perihelion and aphelion were coined by Johannes Kepler to describe the orbital motions of the planets around

3192-412: The time of vernal equinox, the Earth is at the bottom of the figure. The second image (below-right) shows the outer planets, being Jupiter, Saturn, Uranus, and Neptune. The orbital nodes are the two end points of the "line of nodes" where a planet's tilted orbit intersects the plane of reference; here they may be 'seen' as the points where the blue section of an orbit meets the pink. The chart shows

3249-491: The two distances is the length of the semi-minor axis b . The geometric mean of the two limiting speeds is which is the speed of a body in a circular orbit whose radius is a {\displaystyle a} . Orbital elements such as the time of perihelion passage are defined at the epoch chosen using an unperturbed two-body solution that does not account for the n-body problem . To get an accurate time of perihelion passage you need to use an epoch close to

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