The ecliptic is the mean plane of the apparent path in the Earth's sky that the Sun follows over the course of one year. This plane of reference is coplanar with Earth's orbit around the Sun; the ecliptic is not noticeable from Earth's surface because the planet's rotation carries the observer through the daily cycles of sunrise and sunset, which obscure the Sun's apparent motion against the background of stars during the year. The motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, the apparent path of the Sun wobbles with a period of about one month. Due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles around a mean position in a complex fashion; the ecliptic is the apparent path of the Sun throughout the course of a year. Because Earth takes one year to orbit the Sun, the apparent position of the Sun takes one year to make a complete circuit of the ecliptic. With more than 365 days in one year, the Sun moves a little less than 1° eastward every day.
This small difference in the Sun's position against the stars causes any particular spot on Earth's surface to catch up with the Sun about four minutes each day than it would if Earth would not orbit. Again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun; the actual speed with which Earth orbits the Sun varies during the year, so the speed with which the Sun seems to move along the ecliptic varies. For example, the Sun is north of the celestial equator for about 185 days of each year, south of it for about 180 days; the variation of orbital speed accounts for part of the equation of time. Because Earth's rotational axis is not perpendicular to its orbital plane, Earth's equatorial plane is not coplanar with the ecliptic plane, but is inclined to it by an angle of about 23.4°, known as the obliquity of the ecliptic. If the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes.
The Sun, in its apparent motion along the ecliptic, crosses the celestial equator at these points, one from south to north, the other from north to south. The crossing from south to north is known as the vernal equinox known as the first point of Aries and the ascending node of the ecliptic on the celestial equator; the crossing from north to south is descending node. The orientation of Earth's axis and equator are not fixed in space, but rotate about the poles of the ecliptic with a period of about 26,000 years, a process known as lunisolar precession, as it is due to the gravitational effect of the Moon and Sun on Earth's equatorial bulge; the ecliptic itself is not fixed. The gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earth's orbit, hence of the ecliptic, known as planetary precession; the combined action of these two motions is called general precession, changes the position of the equinoxes by about 50 arc seconds per year.
Once again, this is a simplification. Periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earth's axis, hence the celestial equator, known as nutation; this adds a periodic component to the position of the equinoxes. Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic, it is about 23.4° and is decreasing 0.013 degrees per hundred years due to planetary perturbations. The angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, from these ephemerides various astronomical values, including the obliquity, are derived; until 1983 the obliquity for any date was calculated from work of Newcomb, who analyzed positions of the planets until about 1895: ε = 23° 27′ 08″.26 − 46″.845 T − 0″.0059 T2 + 0″.00181 T3 where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question.
From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated: ε = 23° 26′ 21″.45 − 46″.815 T − 0″.0006 T2 + 0″.00181 T3 where hereafter T is Julian centuries from J2000.0. JPL's fundamental ephemerides have been continually updated; the Astronomical Almanac for 2010 specifies:ε = 23° 26′ 21″.406 − 46″.836769 T − 0″.0001831 T2 + 0″.00200340 T3 − 0″.576×10−6 T4 − 4″.34×10−8 T5 These expressions for the obliquity are intended for high precision over a short time span ± several centuries. J. Laskar computed an expression to order T10 good to 0″.04/1000 years over 10,000 years. All of these expressions are for the mean obliquity, that is, without the nutation of the equator included; the true or instantaneous obliquity includes the nutation. Most of the major bodies of the Solar System o
An asteroid family is a population of asteroids that share similar proper orbital elements, such as semimajor axis and orbital inclination. The members of the families are thought to be fragments of past asteroid collisions. An asteroid family is a more specific term than asteroid group whose members, while sharing some broad orbital characteristics, may be otherwise unrelated to each other. Large prominent families contain several hundred recognized asteroids. Small, compact families may have only about ten identified members. About 33% to 35% of asteroids in the main belt are family members. There are about 20 to 30 reliably recognized families, with several tens of less certain groupings. Most asteroid families are found in the main asteroid belt, although several family-like groups such as the Pallas family, Hungaria family, the Phocaea family lie at smaller semi-major axis or larger inclination than the main belt. One family has been identified associated with the dwarf planet Haumea; some studies have tried to find evidence of collisional families among the trojan asteroids, but at present the evidence is inconclusive.
The families are thought to form as a result of collisions between asteroids. In many or most cases the parent body was shattered, but there are several families which resulted from a large cratering event which did not disrupt the parent body; such cratering families consist of a single large body and a swarm of asteroids that are much smaller. Some families have complex internal structures which are not satisfactorily explained at the moment, but may be due to several collisions in the same region at different times. Due to the method of origin, all the members have matching compositions for most families. Notable exceptions are those families. Asteroid families are thought to have lifetimes of the order of a billion years, depending on various factors; this is shorter than the Solar System's age, so few if any are relics of the early Solar System. Decay of families occurs both because of slow dissipation of the orbits due to perturbations from Jupiter or other large bodies, because of collisions between asteroids which grind them down to small bodies.
Such small asteroids become subject to perturbations such as the Yarkovsky effect that can push them towards orbital resonances with Jupiter over time. Once there, they are rapidly ejected from the asteroid belt. Tentative age estimates have been obtained for some families, ranging from hundreds of millions of years to less than several million years as for the compact Karin family. Old families are thought to contain few small members, this is the basis of the age determinations, it is supposed that many old families have lost all the smaller and medium-sized members, leaving only a few of the largest intact. A suggested example of such old family remains are 113 Amalthea pair. Further evidence for a large number of past families comes from analysis of chemical ratios in iron meteorites; these show that there must have once been at least 50 to 100 parent bodies large enough to be differentiated, that have since been shattered to expose their cores and produce the actual meteorites. When the orbital elements of main belt asteroids are plotted, a number of distinct concentrations are seen against the rather uniform distribution of non-family background asteroids.
These concentrations are the asteroid families. Interlopers are asteroids classified as family members based on their so-called proper orbital elements but having spectroscopic properties distinct from the bulk of the family, suggesting that they, contrary to the true family members, did not originate from the same parent body that once fragmented upon a collisional impact. Speaking and their membership are identified by analysing the proper orbital elements rather than the current osculating orbital elements, which fluctuate on timescales of tens of thousands of years; the proper elements are related constants of motion that remain constant for times of at least tens of millions of years, longer. The Japanese astronomer Kiyotsugu Hirayama pioneered the estimation of proper elements for asteroids, first identified several of the most prominent families in 1918. In his honor, asteroid families are sometimes called Hirayama families; this applies to the five prominent groupings discovered by him.
Present day computer-assisted searches have identified more than a hundred asteroid families. The most prominent algorithms have been the hierarchical clustering method, which looks for groupings with small nearest-neighbour distances in orbital element space, wavelet analysis, which builds a density-of-asteroids map in orbital element space, looks for density peaks; the boundaries of the families are somewhat vague because at the edges they blend into the background density of asteroids in the main belt. For this reason the number of members among discovered asteroids is only known and membership is uncertain for asteroids near the edges. Additionally, some interlopers from the heterogeneous background asteroid population are expected in the central regions of a family. Since the true family members caused by the collision are expected to have similar compositions, most such interlopers can in principle be recognised by spectral properties which do not matc
Konkoly Observatory is an astronomical observatory owned and operated by the Hungarian Academy of Sciences, located in Budapest, Hungary. It was founded in 1871 by Hungarian astronomer Miklós Konkoly-Thege as a private observatory, was donated to the state in 1899. Konkoly Observatory known as MTA CSFK Konkoly Thege Miklós Csillagászati Intézete in Hungarian, is the largest astronomical research institute in Hungary, hosts the largest telescopes in the country; as well as performing astronomical research, the observatory publishes the Information Bulletin on Variable Stars on behalf of the International Astronomical Union. Hungarian nobleman Miklós Konkoly-Thege founded his private observatory in 1871 at Hurbanovo, in Northern Hungary, known as Ógyalla in Hungarian at the time; the instrumentation included a 6" refractor, a 10.5" Browning reflector, a meridian circle, a spectrograph, as well as various meteorological and geomagnetic instruments. Konkoly-Thege, who had no offspring, became afraid over time that his legacy will perish with his death.
As a member of the Hungarian Parliament, he was able to convince the government to take the observatory into state property. In 1899, the Hurbanovo site was renamed to Royal Konkoly-founded Astrophysical Observatory, operated under the direction of Konkoly-Thege and Radó Kövesligethy, a renowned geophysicist of the time; the main scientific profile of the institute was the photometry of stars, the observation of the Sun, meteor counting, providing time service for the government. In 1913, an order for a 60 cm telescope was finalized with the German Heyde company, but with the start of World War I, the telescope was never built. After the end of World War I, Ógyalla became part of the newly formed Czechoslovak Republic; the government transferred the scientific equipment to Budapest before the newly formed border was shut down, in 1921 allocated a new property in the Buda hills just west to Budapest to build a new astronomical observatory. The new building, known as the Svábhegy observatory after a nearby hill, was finished by 1924, the 6" refractor and the re-ordered 60 cm Heyde telescope were set up in new domes by 1928.
Scientific work in the observatory focussed on the light variations of pulsating stars, the orbits of binary stars, searching for asteroids. Measurements were made with photographic plates and wedge photometers; the observatory survived World War II with minor damages. In late 1944, observations were put on hold, the optical elements of the telescopes were removed to protect them from aerial bombings. Soviet troops occupied the buildings from December 1944 until February 1945, but the library and the photographic laboratories were spared. Observations were resumed by July 1945. After the communist takeover of Hungary in 1948, a new network of research institutes, independent from universities, was set up under the Hungarian Academy of Sciences; the observatory was renamed to the Astronomical Institute of the HAS, but retained the traditional name, Konkoly Observatory, in English correspondences. With the communist isolation of the country, international relations shifted from predominantly German and American to Soviet and Eastern-bloc partnerships.
However, the international recognition of director László Detre kept some connections to the West alive. During one of his visits to the Western bloc, Detre received an RCA 1P21 photomultiplier tube from American astronomer Harlow Shapley to start photometric measurements at the observatory in 1948, he subsequently smuggled in to the country despite the strict trade restrictions at the time. In 1957, after the launch of Sputnik, multiple independent satellite observing and tracking stations were set up in the country at the suggestion of the Soviet Union; the institute provided coordination for these stations, in 1966, the Baja station merged into the institute. Relations with the West soon eased and at the 1961 General Assembly of the International Astronomical Union at Berkeley, the institute was tasked with setting up and circulating the Information Bulletin on Variable Stars; the Bulletin was envisaged to be a rapid communications platform between variable star observers, but it expanded into a peer-reviewed journal for short papers and notes about variable stars.
The institute expanded in the 1960-70s with the foundation and subsequent independence of the Debrecen Heliophysical Observatory and the installation of new telescopes at Piszkéstető Mountain Station. In 1982, the observatories in Debrecen and Budapest were merged back together to form the Research Institute for Astronomy of the HAS. In 1992, the Baja station was separated from the institute and handed over to the county council of Bács-Kiskun. In 2012, the Hungarian Academy of Sciences reorganized the structure of its institute network: the Astronomical Institute was merged with three other entities to form the Research Center for Astronomy and Earth Sciences. In 1946, a new Solar physics department was initiated under the leadership of Lóránt Dezső. In 1958, the department moved to Debrecen to form a new solar observatory independent of the Astronomical Institute. Observations of the Sun started with two photoheliograph telescopes, one of, relocated to the top of the water tower at Gyula in 1972.
In 1973, a large coronagraph telescope with a 53 cm aperture was set up, but observations were hindered by the cumbersome mechanical setup. Given the low elevation of the observatory of just 124 m above sea level, the coronagraph was only capable to observe the chromosphere of the Sun, not the corona itself; the main scientific output of the observatory has
The astronomical unit is a unit of length the distance from Earth to the Sun. However, that distance varies as Earth orbits the Sun, from a maximum to a minimum and back again once a year. Conceived as the average of Earth's aphelion and perihelion, since 2012 it has been defined as 149597870700 metres or about 150 million kilometres; the astronomical unit is used for measuring distances within the Solar System or around other stars. It is a fundamental component in the definition of another unit of astronomical length, the parsec. A variety of unit symbols and abbreviations have been in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union used the symbol A to denote a length equal to the astronomical unit. In the astronomical literature, the symbol AU was common. In 2006, the International Bureau of Weights and Measures recommended ua as the symbol for the unit. In the non-normative Annex C to ISO 80000-3, the symbol of the astronomical unit is "ua". In 2012, the IAU, noting "that various symbols are presently in use for the astronomical unit", recommended the use of the symbol "au".
In the 2014 revision of the SI Brochure, the BIPM used the unit symbol "au". Earth's orbit around the Sun is an ellipse; the semi-major axis of this elliptic orbit is defined to be half of the straight line segment that joins the perihelion and aphelion. The centre of the Sun lies on this straight line segment, but not at its midpoint; because ellipses are well-understood shapes, measuring the points of its extremes defined the exact shape mathematically, made possible calculations for the entire orbit as well as predictions based on observation. In addition, it mapped out the largest straight-line distance that Earth traverses over the course of a year, defining times and places for observing the largest parallax in nearby stars. Knowing Earth's shift and a star's shift enabled the star's distance to be calculated, but all measurements are subject to some degree of error or uncertainty, the uncertainties in the length of the astronomical unit only increased uncertainties in the stellar distances.
Improvements in precision have always been a key to improving astronomical understanding. Throughout the twentieth century, measurements became precise and sophisticated, more dependent on accurate observation of the effects described by Einstein's theory of relativity and upon the mathematical tools it used. Improving measurements were continually checked and cross-checked by means of improved understanding of the laws of celestial mechanics, which govern the motions of objects in space; the expected positions and distances of objects at an established time are calculated from these laws, assembled into a collection of data called an ephemeris. NASA's Jet Propulsion Laboratory HORIZONS System provides one of several ephemeris computation services. In 1976, in order to establish a yet more precise measure for the astronomical unit, the IAU formally adopted a new definition. Although directly based on the then-best available observational measurements, the definition was recast in terms of the then-best mathematical derivations from celestial mechanics and planetary ephemerides.
It stated that "the astronomical unit of length is that length for which the Gaussian gravitational constant takes the value 0.01720209895 when the units of measurement are the astronomical units of length and time". Equivalently, by this definition, one AU is "the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass, moving with an angular frequency of 0.01720209895 radians per day". Subsequent explorations of the Solar System by space probes made it possible to obtain precise measurements of the relative positions of the inner planets and other objects by means of radar and telemetry; as with all radar measurements, these rely on measuring the time taken for photons to be reflected from an object. Because all photons move at the speed of light in vacuum, a fundamental constant of the universe, the distance of an object from the probe is calculated as the product of the speed of light and the measured time. However, for precision the calculations require adjustment for things such as the motions of the probe and object while the photons are transiting.
In addition, the measurement of the time itself must be translated to a standard scale that accounts for relativistic time dilation. Comparison of the ephemeris positions with time measurements expressed in the TDB scale leads to a value for the speed of light in astronomical units per day. By 2009, the IAU had updated its standard measures to reflect improvements, calculated the speed of light at 173.1446326847 AU/d. In 1983, the International Committee for Weights and Measures modified the International System of Units to make the metre defined as the distance travelled in a vacuum by light in 1/299792458 second; this replaced the previous definition, valid between 1960 and 1983, that the metre equalled a certain number of wavelengths of a certain emission line of krypton-86. The speed of light could be expressed as c0 = 299792458 m/s, a standard adopted by the IERS numerical standards. From this definition and the 2009 IAU standard, the time for light to traverse an AU is found to be
In astronomy, a light curve is a graph of light intensity of a celestial object or region, as a function of time. The light is in a particular frequency interval or band. Light curves can be periodic, as in the case of eclipsing binaries, Cepheid variables, other periodic variables, transiting extrasolar planets, or aperiodic, like the light curve of a nova, a cataclysmic variable star, a supernova or a microlensing event or binary as observed during occultation events; the study of the light curve, together with other observations, can yield considerable information about the physical process that produces it or constrain the physical theories about it. Graphs of the apparent magnitude of a variable star over time are used to visualise and analyse their behaviour. Although the categorisation of variable star types is done from their spectral properties, the amplitudes and regularity of their brightness changes are still important factors; some types such as Cepheids have regular light curves with the same period and shape in each cycle.
Others such as Mira variables have somewhat less regular light curves with large amplitudes of several magnitudes, while the semiregular variables are less regular still and have smaller amplitudes. The shapes of variable star light curves give valuable information about the underlying physical processes producing the brightness changes. For eclipsing variables, the shape of the light curve indicates the degree of totality, the relative sizes of the stars, their relative surface brightnesses, it may show the eccentricity of the orbit and distortions in the shape of the two stars. For pulsating stars, the amplitude or period of the pulsations can be related to the luminosity of the star, the light curve shape can be an indicator of the pulsation mode. Light curves from supernovae can be indicative of the type of supernova. Although supernova types are defined on the basis of their spectra, each has typical light curve shapes. Type I supernovae have light curves with a sharp maximum and decline, while Type II supernovae have less sharp maxima.
Light curves are helpful for classification of faint supernovae and for the determination of sub-types. For example, the type II-P have similar spectra to the type II-L but are distinguished by a light curve where the decline flattens out for several weeks or months before resuming its fade. In planetary science, a light curve can be used to derive the rotation period of a minor planet, moon, or comet nucleus. From the Earth there is no way to resolve a small object in the Solar System in the most powerful of telescopes, since the apparent angular size of the object is smaller than one pixel in the detector. Thus, astronomers measure the amount of light produced by an object as a function of time; the time separation of peaks in the light curve gives an estimate of the rotational period of the object. The difference between the maximum and minimum brightnesses can be due to the shape of the object, or to bright and dark areas on its surface. For example, an asymmetrical asteroid's light curve has more pronounced peaks, while a more spherical object's light curve will be flatter.
This allows astronomers to infer information about the spin of asteroids. The Asteroid Lightcurve Database of the Collaborative Asteroid Lightcurve Link uses a numeric code to assess the quality of a period solution for minor planet light curves, its quality code parameter "U" ranges from 0 to 3: U = 0 → Result proven incorrect U = 1 → Result based on fragmentary light curve, may be wrong. U = 2 → Result based on less than full coverage. Period may be wrong by ambiguous. U = 3 → Secure result within the precision given. No ambiguity. U = n.a. → Not available. Incomplete or inconclusive result. A trailing plus sign or minus sign is used to indicate a better or worse quality than the unsigned value; the occultation light curve is characterised as binary, where the light from the star is terminated instantaneously, remains constant for the duration, is reinstated instantaneously. The duration is equivalent to the length of a chord across the occulting body. Circumstances where the transitions are not instantaneous are.
When the occulted body is large, e.g. a star like Antares the transitions are gradual. When the occulting body has an atmosphere, e.g. the moon TitanThe observations are recorded using video equipment and the disappearance and reappearance timed using a GPS disciplined Video Time Inserter. Occultation light curves are archived at the VizieR service. Light curve inversion is a mathematical technique used to model the surfaces of rotating objects from their brightness variations; this can be used to image starspots or asteroid surface albedos. Microlensing is a process where small and low-mass astronomical objects cause a brief small increase in the brightness of a more distant object; this is caused by the small relativistic effect as larger gravitational lenses, but allows the detection and analysis of otherwise-invisible stellar and planetary mass objects. The properties of these objects can be inferred from the shape of the lensing light curve. For example, PA-99-N2 is a microlensing event that may have been due to a star in the Andromeda galaxy that has an exoplanet.
The AAVSO online light curve generator can plot light curves for thousands of variable stars The Open Astronomy
C-type asteroids are the most common variety, forming around 75% of known asteroids. They are distinguished by a low albedo because their composition includes a large amount of carbon, in addition to rocks and minerals, they occur most at the outer edge of the asteroid belt, 3.5 astronomical units from the Sun, where 80% of the asteroids are of this type, whereas only 40% of asteroids at 2 AU from the Sun are C-type. The proportion of C-types may be greater than this, because C-types are much darker than most other asteroid types except for D-types and others that are at the extreme outer edge of the asteroid belt. Asteroids of this class have spectra similar to those of carbonaceous chondrite meteorites; the latter are close in chemical composition to the Sun and the primitive solar nebula, except for the absence of hydrogen and other volatiles. Hydrated minerals are present. C-type asteroids are dark, with albedos in the 0.03 to 0.10 range. Whereas a number of S-type asteroids can be viewed with binoculars at opposition the largest C-type asteroids require a small telescope.
The brightest C-type asteroid is 324 Bamberga, but that object's high eccentricity means it reaches its maximum magnitude. Their spectra contain moderately strong ultraviolet absorption at wavelengths below about 0.4 μm to 0.5 μm, while at longer wavelengths they are featureless but reddish. The so-called "water" absorption feature of around 3 μm, which can be an indication of water content in minerals, is present; the largest unequivocally C-type asteroid is 10 Hygiea, although the SMASS classification places the largest asteroid, 1 Ceres, here as well, because that scheme lacks a G-type. In the Tholen classification, the C-type is grouped along with three less numerous types into a wider C-group of carbonaceous asteroids which contains: B-type C-type F-type G-type In the SMASS classification, the wider C-group contains the types: B-type corresponding to the Tholen B and F-types a core C-type for asteroids having the most "typical" spectra in the group Cg and Cgh types corresponding to the Tholen G-type Ch type with an absorption feature around 0.7μm Cb type corresponding to transition objects between the SMASS C and B types Asteroid spectral types
The term apsis refers to an extreme point in the orbit of an object. It denotes either the respective distance of the bodies; the word comes via Latin from Greek, there denoting a whole orbit, is cognate with apse. Except for the theoretical possibility of one common circular orbit for two bodies of equal mass at diametral positions, there are two apsides for any elliptic orbit, named with the prefixes peri- and ap-/apo-, added in reference to the body being orbited. All periodic orbits are, according to Newton's Laws of motion, ellipses: either the two individual ellipses of both bodies, with the center of mass of this two-body system at the one common focus of the ellipses, or the orbital ellipses, with one body taken as fixed at one focus, the other body orbiting this focus. All these ellipses share a straight line, the line of apsides, that contains their major axes, the foci, the vertices, thus the periapsis and the apoapsis; the major axis of the orbital ellipse is the distance of the apsides, when taken as points on the orbit, or their sum, when taken as distances.
The major axes of the individual ellipses around the barycenter the contributions to the major axis of the orbital ellipses are inverse proportional to the masses of the bodies, i.e. a bigger mass implies a smaller axis/contribution. Only when one mass is sufficiently larger than the other, the individual ellipse of the smaller body around the barycenter comprises the individual ellipse of the larger body as shown in the second figure. For remarkable asymmetry, the barycenter of the two bodies may lie well within the bigger body, e.g. the Earth–Moon barycenter is about 75% of the way from Earth's center to its surface. If the smaller mass is negligible compared to the larger the orbital parameters are independent of the smaller mass. For general orbits, the terms periapsis and apoapsis are used. Pericenter and apocenter are equivalent alternatives, referring explicitly to the respective points on the orbits, whereas periapsis and apoapsis may refer to the smallest and largest distances of the orbiter and its host.
For a body orbiting the Sun, the point of least distance is the perihelion, the point of greatest distance is the aphelion. The terms become apastron when discussing orbits around other stars. For any satellite of Earth, including the Moon, the point of least distance is the perigee and greatest distance the apogee, from Ancient Greek Γῆ, "land" or "earth". For objects in lunar orbit, the point of least distance is sometimes called the pericynthion and the greatest distance the apocynthion. Perilune and apolune are used. In orbital mechanics, the apsides technically refer to the distance measured between the barycenters of the central body and orbiting body. However, in the case of a spacecraft, the terms are used to refer to the orbital altitude of the spacecraft above the surface of the central body; these formulae characterize the pericenter and apocenter of an orbit: Pericenter Maximum speed, v per = μ a, at minimum distance, r per = a. Apocenter Minimum speed, v ap = μ a, at maximum distance, r ap = a.
While, in accordance with Kepler's laws of planetary motion and the conservation of energy, these two quantities are constant for a given orbit: Specific relative angular momentum h = μ a Specific orbital energy ε = − μ 2 a where: a is the semi-major axis: a = r per + r ap 2 μ is the standard gravitational parameter e is the eccentricity, defined as e = r ap − r per r ap + r per = 1 − 2 r ap r per + 1 Note t