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61-498: It was placed in a highly elliptical orbit with an apogee altitude of 21,400 km and a perigee altitude of 560 km, with an orbital period of approximately 6.3 hours. This orbit allowed imaging of celestial radio sources by the satellite in conjunction with an array of ground-based radio telescopes, such that both good (u,v) plane coverage and very high resolution were obtained. Although designed to observe in three frequency bands: 1.6 GHz, 5.0 GHz, and 22 GHz, it

122-479: A 3 . {\displaystyle n={\sqrt {\frac {\mu }{a^{3}}}}.} where μ is the standard gravitational parameter . Hence if at any instant t 0 the orbital parameters are ( e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 ) , then the elements at time t = t 0 + δt is given by ( e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 + n δt ) . Unperturbed, two-body , Newtonian orbits are always conic sections , so

183-590: 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

244-440: A non-inertial frame centered on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body (usually the most massive) is called the primary , the other body is called the secondary . The primary does not necessarily possess more mass than

305-400: A real geometric angle, but rather varies linearly with time, one whole orbital period being represented by an "angle" of 2 π radians . It can be converted into the true anomaly ν , which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting body at any given time. Thus, the true anomaly

366-423: A specific orbit . In celestial mechanics these elements are considered in two-body systems using a Kepler orbit . There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics . A real orbit and its elements change over time due to gravitational perturbations by other objects and

427-525: 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

488-533: A two-line element: The Delaunay orbital elements were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon . Commonly called Delaunay variables , they are a set of canonical variables , which are action-angle coordinates . The angles are simple sums of some of the Keplerian angles: along with their respective conjugate momenta , L , G , and H . The momenta L , G , and H are

549-468: 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,

610-435: 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

671-401: Is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified. Different sets of elements are used for various astronomical bodies. The eccentricity, e , and either the semi-major axis, a , or the distance of periapsis, q , are used to specify the shape and size of an orbit. The longitude of

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732-412: Is because the problem contains six degrees of freedom . These correspond to the three spatial dimensions which define position ( x , y , z in a Cartesian coordinate system ), plus the velocity in each of these dimensions. These can be described as orbital state vectors , but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead. Sometimes

793-473: 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

854-436: Is greater than one, the trajectory is a hyperbola . If the eccentricity is equal to one, the trajectory is a parabola . Regardless of eccentricity, the orbit degenerates to a radial trajectory if the angular momentum equals zero. Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit. This

915-432: 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

976-461: Is shown as the red angle ν in the diagram, and the mean anomaly is not shown. The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system. Note that non-elliptic trajectories also exist, but are not closed, and are thus not orbits. If the eccentricity

1037-565: Is the NASA / NORAD "two-line elements" (TLE) format, originally designed for use with 80 column punched cards, but still in use because it is the most common format, and 80-character ASCII records can be handled efficiently by modern databases. Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through simplified perturbation models ( SGP4 / SDP4 / SGP8 / SDP8). Example of

1098-576: The action variables and are more elaborate combinations of the Keplerian elements a , e , and i . Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems. The advantage of the Delaunay variables is that they remain well defined and non-singular (except for h , which can be tolerated) when e and / or i are very small: When

1159-483: 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

1220-622: 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

1281-463: 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

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1342-428: 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

1403-3678: The 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of matrix algebra , the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles. That is, [ i 1 i 2 i 3 j 1 j 2 j 3 k 1 k 2 k 3 ] = [ cos ⁡ Ω − sin ⁡ Ω 0 sin ⁡ Ω cos ⁡ Ω 0 0 0 1 ] [ 1 0 0 0 cos ⁡ i − sin ⁡ i 0 sin ⁡ i cos ⁡ i ] [ cos ⁡ ω − sin ⁡ ω 0 sin ⁡ ω cos ⁡ ω 0 0 0 1 ] ; {\displaystyle {\begin{bmatrix}i_{1}&i_{2}&i_{3}\\j_{1}&j_{2}&j_{3}\\k_{1}&k_{2}&k_{3}\end{bmatrix}}={\begin{bmatrix}\cos \Omega &-\sin \Omega &0\\\sin \Omega &\cos \Omega &0\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}1&0&0\\0&\cos i&-\sin i\\0&\sin i&\cos i\end{bmatrix}}\,{\begin{bmatrix}\cos \omega &-\sin \omega &0\\\sin \omega &\cos \omega &0\\0&0&1\end{bmatrix}}\,;} where I ^ = i 1 x ^ + i 2 y ^ + i 3 z ^   ; J ^ = j 1 x ^ + j 2 y ^ + j 3 z ^   ; K ^ = k 1 x ^ + k 2 y ^ + k 3 z ^   . {\displaystyle {\begin{aligned}\mathbf {\hat {I}} &=i_{1}\mathbf {\hat {x}} +i_{2}\mathbf {\hat {y}} +i_{3}\mathbf {\hat {z}} ~;\\\mathbf {\hat {J}} &=j_{1}\mathbf {\hat {x}} +j_{2}\mathbf {\hat {y}} +j_{3}\mathbf {\hat {z}} ~;\\\mathbf {\hat {K}} &=k_{1}\mathbf {\hat {x}} +k_{2}\mathbf {\hat {y}} +k_{3}\mathbf {\hat {z}} ~.\\\end{aligned}}} The transformation from x̂ , ŷ , ẑ to Euler angles Ω , i , ω is: Ω = arg ⁡ ( − z 2 , z 1 ) i = arg ⁡ ( z 3 , z 1 2 + z 2 2 ) ω = arg ⁡ ( y 3 , x 3 ) {\displaystyle {\begin{aligned}\Omega &=\operatorname {arg} \left(-z_{2},z_{1}\right)\\i&=\operatorname {arg} \left(z_{3},{\sqrt {{z_{1}}^{2}+{z_{2}}^{2}}}\right)\\\omega &=\operatorname {arg} \left(y_{3},x_{3}\right)\\\end{aligned}}} where arg( x , y ) signifies

1464-526: 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

1525-434: 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

1586-584: 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

1647-525: The Keplerian elements define an ellipse , parabola , or hyperbola . Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag , relativistic effects , radiation pressure , electromagnetic forces , and so on. Keplerian elements can often be used to produce useful predictions at times near

1708-472: The Russian Spektr-R mission as the only then operational space VLBI facility. Spektr-R stopped operating in 2019. The large 8 meter antenna was designed to unfold in space as the unfolded configuration did not fit inside the rocket fairing. The antenna was a metal mesh of 6000 cables. To form an ideal shape the length of the cables were adjusted on the backside of the antenna. One concern was that

1769-404: 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

1830-452: 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

1891-400: The ascending node, Ω , the inclination, i , and the argument of periapsis, ω , or the longitude of periapsis, ϖ , specify the orientation of the orbit in its plane. Either the longitude at epoch, L 0 , the mean anomaly at epoch, M 0 , or the time of perihelion passage, T 0 , are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or

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1952-427: The background stars. Due to statistics of small numbers, trans-Neptunian objects such as 2015 TH 367 when it had only 8 observations over an observation arc of 1 year that have not or will not come to perihelion for roughly 100 years can have a 1-sigma uncertainty of 77.3 years (28,220 days) in the perihelion date. Orbital elements Orbital elements are the parameters required to uniquely identify

2013-399: The cables could entangle. The deployment of the main reflector started on February 27, 1997. The deployment was done over three hours on the first day and was completed in 20 minutes during the next day. Apogee An apsis (from Ancient Greek ἁψίς ( hapsís )  'arch, vault'; pl.   apsides / ˈ æ p s ɪ ˌ d iː z / AP -sih-deez ) is

2074-422: 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

2135-409: 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 ,

2196-444: The effects of general relativity . A Kepler orbit is an idealized, mathematical approximation of the orbit at a particular time. The traditional orbital elements are the six Keplerian elements , after Johannes Kepler and his laws of planetary motion . When viewed from an inertial frame , two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass . When viewed from

2257-524: The epoch is considered a "seventh" orbital parameter, rather than part of the reference frame. If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.) Keplerian elements can be obtained from orbital state vectors (a three-dimensional vector for

2318-439: The epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations , differential equations which come in different forms developed by Lagrange , Gauss , Delaunay , Poincaré , or Hill . Keplerian elements parameters can be encoded as text in a number of formats. The most common of them

2379-405: 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

2440-449: 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

2501-401: 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

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2562-465: 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,

2623-522: The loss of attitude control. All operations were officially ended in November 2005. A follow-up mission ASTRO-G (VSOP-2) was planned, with a proposed launch date of 2012, but the project was eventually cancelled in 2011 due to increasing costs and the difficulties of achieving its science goals. It was expected to achieve resolutions up to ten times higher and up to ten times greater sensitivity than its predecessor HALCA. The cancellation of ASTRO-G left

2684-421: The mean motion ( n ) into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element. Mean motion can also be obscured behind citations of the orbital period P . The angles Ω , i , ω are the Euler angles (corresponding to α , β , γ in the notation used in that article) characterizing

2745-413: 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 ,

2806-399: The node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known. It is also quite common to see either the mean anomaly ( M ) or the mean longitude ( L ) expressed directly, without either M 0 or L 0 as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate

2867-503: 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

2928-5056: The orientation of the coordinate system where: Then, the transformation from the Î , Ĵ , K̂ coordinate frame to the x̂ , ŷ , ẑ frame with the Euler angles Ω , i , ω is: x 1 = cos ⁡ Ω ⋅ cos ⁡ ω − sin ⁡ Ω ⋅ cos ⁡ i ⋅ sin ⁡ ω   ; x 2 = sin ⁡ Ω ⋅ cos ⁡ ω + cos ⁡ Ω ⋅ cos ⁡ i ⋅ sin ⁡ ω   ; x 3 = sin ⁡ i ⋅ sin ⁡ ω ; y 1 = − cos ⁡ Ω ⋅ sin ⁡ ω − sin ⁡ Ω ⋅ cos ⁡ i ⋅ cos ⁡ ω   ; y 2 = − sin ⁡ Ω ⋅ sin ⁡ ω + cos ⁡ Ω ⋅ cos ⁡ i ⋅ cos ⁡ ω   ; y 3 = sin ⁡ i ⋅ cos ⁡ ω   ; z 1 = sin ⁡ i ⋅ sin ⁡ Ω   ; z 2 = − sin ⁡ i ⋅ cos ⁡ Ω   ; z 3 = cos ⁡ i   ; {\displaystyle {\begin{aligned}x_{1}&=\cos \Omega \cdot \cos \omega -\sin \Omega \cdot \cos i\cdot \sin \omega \ ;\\x_{2}&=\sin \Omega \cdot \cos \omega +\cos \Omega \cdot \cos i\cdot \sin \omega \ ;\\x_{3}&=\sin i\cdot \sin \omega ;\\\,\\y_{1}&=-\cos \Omega \cdot \sin \omega -\sin \Omega \cdot \cos i\cdot \cos \omega \ ;\\y_{2}&=-\sin \Omega \cdot \sin \omega +\cos \Omega \cdot \cos i\cdot \cos \omega \ ;\\y_{3}&=\sin i\cdot \cos \omega \ ;\\\,\\z_{1}&=\sin i\cdot \sin \Omega \ ;\\z_{2}&=-\sin i\cdot \cos \Omega \ ;\\z_{3}&=\cos i\ ;\\\end{aligned}}} [ x 1 x 2 x 3 y 1 y 2 y 3 z 1 z 2 z 3 ] = [ cos ⁡ ω sin ⁡ ω 0 − sin ⁡ ω cos ⁡ ω 0 0 0 1 ] [ 1 0 0 0 cos ⁡ i sin ⁡ i 0 − sin ⁡ i cos ⁡ i ] [ cos ⁡ Ω sin ⁡ Ω 0 − sin ⁡ Ω cos ⁡ Ω 0 0 0 1 ] ; {\displaystyle {\begin{bmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{bmatrix}}={\begin{bmatrix}\cos \omega &\sin \omega &0\\-\sin \omega &\cos \omega &0\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}1&0&0\\0&\cos i&\sin i\\0&-\sin i&\cos i\end{bmatrix}}\,{\begin{bmatrix}\cos \Omega &\sin \Omega &0\\-\sin \Omega &\cos \Omega &0\\0&0&1\end{bmatrix}}\,;} where x ^ = x 1 I ^ + x 2 J ^ + x 3 K ^   ; y ^ = y 1 I ^ + y 2 J ^ + y 3 K ^   ; z ^ = z 1 I ^ + z 2 J ^ + z 3 K ^   . {\displaystyle {\begin{aligned}\mathbf {\hat {x}} &=x_{1}\mathbf {\hat {I}} +x_{2}\mathbf {\hat {J}} +x_{3}\mathbf {\hat {K}} ~;\\\mathbf {\hat {y}} &=y_{1}\mathbf {\hat {I}} +y_{2}\mathbf {\hat {J}} +y_{3}\mathbf {\hat {K}} ~;\\\mathbf {\hat {z}} &=z_{1}\mathbf {\hat {I}} +z_{2}\mathbf {\hat {J}} +z_{3}\mathbf {\hat {K}} ~.\\\end{aligned}}} The inverse transformation, which computes

2989-416: The other provided the standard gravitational parameter , GM , is given for the central body. Instead of the mean anomaly at epoch , the mean anomaly M , mean longitude , true anomaly ν 0 , or (rarely) the eccentric anomaly might be used. Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it

3050-588: 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

3111-409: 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

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3172-415: The polar argument that can be computed with the standard function atan2(y,x) available in many programming languages. Under ideal conditions of a perfectly spherical central body, zero perturbations and negligible relativistic effects, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion , n = μ

3233-423: The position and another for the velocity) by manual transformations or with computer software. Other orbital parameters can be computed from the Keplerian elements such as the period , apoapsis, and periapsis . (When orbiting the Earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis a in Keplerian element sets, as each can be computed from

3294-504: 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

3355-417: 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

3416-417: The secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary. Two elements define the shape and size of the ellipse: Two elements define the orientation of the orbital plane in which the ellipse is embedded: The remaining two elements are as follows: The mean anomaly M is a mathematically convenient fictitious "angle" which does not correspond to

3477-401: 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

3538-415: 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

3599-412: 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

3660-491: 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

3721-531: Was found that the sensitivity of the 22 GHz band had severely degraded after orbital deployment, probably caused by vibrational deformation of the dish shape at launch, thus limiting observations to the 1.6 GHz and 5.0 GHz bands. HALCA was launched in February 1997 from Kagoshima Space Center , and made its final VSOP observations in October 2003, far exceeding its 3-year predicted lifespan, before

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