In astronomy, the main sequence is a continuous and distinctive band of stars that appears on plots of stellar color versus brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams after their co-developers, Ejnar Hertzsprung and Henry Norris Russell. Stars on this band are known as main-sequence stars or dwarf stars; these are the most numerous true stars in the universe, include the Earth's Sun. After condensation and ignition of a star, it generates thermal energy in its dense core region through nuclear fusion of hydrogen into helium. During this stage of the star's lifetime, it is located on the main sequence at a position determined by its mass, but based upon its chemical composition and age; the cores of main-sequence stars are in hydrostatic equilibrium, where outward thermal pressure from the hot core is balanced by the inward pressure of gravitational collapse from the overlying layers. The strong dependence of the rate of energy generation on temperature and pressure helps to sustain this balance.
Energy generated at the core is radiated away at the photosphere. The energy is carried by either radiation or convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both; the main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. Stars below about 1.5 times the mass of the Sun fuse hydrogen atoms together in a series of stages to form helium, a sequence called the proton–proton chain. Above this mass, in the upper main sequence, the nuclear fusion process uses atoms of carbon and oxygen as intermediaries in the CNO cycle that produces helium from hydrogen atoms. Main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. Below this mass, stars have cores that are radiative with convective zones near the surface. With decreasing stellar mass, the proportion of the star forming a convective envelope increases.
Main-sequence stars below 0.4 M☉ undergo convection throughout their mass. When core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen. In general, the more massive a star is, the shorter its lifespan on the main sequence. After the hydrogen fuel at the core has been consumed, the star evolves away from the main sequence on the HR diagram, into a supergiant, red giant, or directly to a white dwarf. In the early part of the 20th century, information about the types and distances of stars became more available; the spectra of stars were shown to have distinctive features. Annie Jump Cannon and Edward C. Pickering at Harvard College Observatory developed a method of categorization that became known as the Harvard Classification Scheme, published in the Harvard Annals in 1901. In Potsdam in 1906, the Danish astronomer Ejnar Hertzsprung noticed that the reddest stars—classified as K and M in the Harvard scheme—could be divided into two distinct groups; these stars are either much brighter than the Sun, or much fainter.
To distinguish these groups, he called them. The following year he began studying star clusters, he published the first plots of color versus luminosity for these stars. These plots showed a continuous sequence of stars, which he named the Main Sequence. At Princeton University, Henry Norris Russell was following a similar course of research, he was studying the relationship between the spectral classification of stars and their actual brightness as corrected for distance—their absolute magnitude. For this purpose he used a set of stars that had reliable parallaxes and many of, categorized at Harvard; when he plotted the spectral types of these stars against their absolute magnitude, he found that dwarf stars followed a distinct relationship. This allowed the real brightness of a dwarf star to be predicted with reasonable accuracy. Of the red stars observed by Hertzsprung, the dwarf stars followed the spectra-luminosity relationship discovered by Russell. However, the giant stars are much brighter than so do not follow the same relationship.
Russell proposed that the "giant stars must have low density or great surface-brightness, the reverse is true of dwarf stars". The same curve showed that there were few faint white stars. In 1933, Bengt Strömgren introduced the term Hertzsprung–Russell diagram to denote a luminosity-spectral class diagram; this name reflected the parallel development of this technique by both Hertzsprung and Russell earlier in the century. As evolutionary models of stars were developed during the 1930s, it was shown that, for stars of a uniform chemical composition, a relationship exists between a star's mass and its luminosity and radius; that is, for a given mass and composition, there is a unique solution for determining the star's radius and luminosity. This became known as the Vogt–Russell theorem. By this theorem, when a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius. A refined scheme for stellar classification was published in 1943 by William Wilson Morgan and Philip Childs Keenan.
The MK classification assigned each star a spectral type—based on the Harvard classification—and a luminosity class. The Harvard classification had been developed by assigning a different lett
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of the Solar System and our galaxy, the Milky Way; the history of astrometry is linked to the history of star catalogues, which gave astronomers reference points for objects in the sky so they could track their movements. This can be dated back to Hipparchus, who around 190 BC used the catalogue of his predecessors Timocharis and Aristillus to discover Earth's precession. In doing so, he developed the brightness scale still in use today. Hipparchus compiled a catalogue with their positions. Hipparchus's successor, included a catalogue of 1,022 stars in his work the Almagest, giving their location and brightness. In the 10th century, Abd al-Rahman al-Sufi carried out observations on the stars and described their positions and star color. Ibn Yunus observed more than 10,000 entries for the Sun's position for many years using a large astrolabe with a diameter of nearly 1.4 metres.
His observations on eclipses were still used centuries in Simon Newcomb's investigations on the motion of the Moon, while his other observations of the motions of the planets Jupiter and Saturn inspired Laplace's Obliquity of the Ecliptic and Inequalities of Jupiter and Saturn. In the 15th century, the Timurid astronomer Ulugh Beg compiled the Zij-i-Sultani, in which he catalogued 1,019 stars. Like the earlier catalogs of Hipparchus and Ptolemy, Ulugh Beg's catalogue is estimated to have been precise to within 20 minutes of arc. In the 16th century, Tycho Brahe used improved instruments, including large mural instruments, to measure star positions more than with a precision of 15–35 arcsec. Taqi al-Din measured the right ascension of the stars at the Constantinople Observatory of Taqi ad-Din using the "observational clock" he invented; when telescopes became commonplace, setting circles sped measurements James Bradley first tried to measure stellar parallaxes in 1729. The stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of the Earth's axis.
His cataloguing of 3222 stars was refined in 1807 by Friedrich Bessel, the father of modern astrometry. He made the first measurement of stellar parallax: 0.3 arcsec for the binary star 61 Cygni. Being difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices replaced photographic plates and reduced optical uncertainties to one milliarcsecond; this technology made astrometry less expensive. In 1989, the European Space Agency's Hipparcos satellite took astrometry into orbit, where it could be less affected by mechanical forces of the Earth and optical distortions from its atmosphere. Operated from 1989 to 1993, Hipparcos measured large and small angles on the sky with much greater precision than any previous optical telescopes.
During its 4-year run, the positions and proper motions of 118,218 stars were determined with an unprecedented degree of accuracy. A new "Tycho catalog" drew together a database of 1,058,332 to within 20-30 mas. Additional catalogues were compiled for the 23,882 double/multiple stars and 11,597 variable stars analyzed during the Hipparcos mission. Today, the catalogue most used is USNO-B1.0, an all-sky catalogue that tracks proper motions, positions and other characteristics for over one billion stellar objects. During the past 50 years, 7,435 Schmidt camera plates were used to complete several sky surveys that make the data in USNO-B1.0 accurate to within 0.2 arcsec. Apart from the fundamental function of providing astronomers with a reference frame to report their observations in, astrometry is fundamental for fields like celestial mechanics, stellar dynamics and galactic astronomy. In observational astronomy, astrometric techniques help identify stellar objects by their unique motions, it is instrumental for keeping time, in that UTC is the atomic time synchronized to Earth's rotation by means of exact astronomical observations.
Astrometry is an important step in the cosmic distance ladder because it establishes parallax distance estimates for stars in the Milky Way. Astrometry has been used to support claims of extrasolar planet detection by measuring the displacement the proposed planets cause in their parent star's apparent position on the sky, due to their mutual orbit around the center of mass of the system. Astrometry is more accurate in space missions that are not affected by the distorting effects of the Earth's atmosphere. NASA's planned Space Interferometry Mission was to utilize astrometric techniques to detect terrestrial planets orbiting 200 or so of the nearest solar-type stars; the European Space Agency's Gaia Mission, launched in 2013, applies astrometric techniques in its stellar census. In addition to the detection of exoplanets, it can be used to determine their mass. Astrometric measurements are used by astrophysicists to constrain certain models in celestial mechanics. By measuring the velocities of pulsars, it is possible to put a limit on the asymmetry of supernova explosions.
Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars. The components for proper motion in the equatorial coordinate system are given in the direction of right ascension and of declination, their combined value is computed as the total proper motion. It has dimensions of angle per time arcseconds per year or milliarcseconds per year. Knowledge of the proper motion and radial velocity allows calculations of true stellar motion or velocity in space in respect to the Sun, by coordinate transformation, the motion in respect to the Milky Way. Proper motion is not "proper", because it includes a component due to the motion of the Solar System itself. Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time.
Ursa Major or Crux, for example, looks nearly the same now. However, precise long-term observations show that the constellations change shape, albeit slowly, that each star has an independent motion; this motion is caused by the movement of the stars relative to the Solar System. The Sun travels in a nearly circular orbit about the center of the Milky Way at a speed of about 220 km/s at a radius of 8 kPc from the center, which can be taken as the rate of rotation of the Milky Way itself at this radius; the proper motion is a two-dimensional vector and is thus defined by two quantities: its position angle and its magnitude. The first quantity indicates the direction of the proper motion on the celestial sphere, the second quantity is the motion's magnitude expressed in arcseconds per year or milliarcsecond per year. Proper motion may alternatively be defined by the angular changes per year in the star's right ascension and declination, using a constant epoch in defining these; the components of proper motion by convention are arrived at.
Suppose an object moves from coordinates to coordinates in a time Δt. The proper motions are given by: μ α = α 2 − α 1 Δ t, μ δ = δ 2 − δ 1 Δ t; the magnitude of the proper motion μ is given by the Pythagorean theorem: μ 2 = μ δ 2 + μ α 2 ⋅ cos 2 δ, μ 2 = μ δ 2 + μ α ∗ 2, where δ is the declination. The factor in cos2δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cosδ, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in α is smaller the further north the object's location; the change μα, which must be multiplied by cosδ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", μδ the "proper motion in declination". If the proper motion in right ascension has been converted by cosδ, the result is designated μα*. For example, the proper motion results in right ascension in the Hipparcos Catalogue have been converted. Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.
The position angle θ is related to these components by: μ sin θ = μ α cos δ = μ α ∗, μ cos θ = μ δ. Motions in equatorial coordinates can be converted to motions in galactic coordinates. For the majority of stars seen in the sky, the observed proper motions are small and unremarkable; such stars are either faint or are distant, have changes of below 10 milliarcseconds per year, do not appear to move appreciably over many millennia. A few do have significant motions, are called high-proper motion stars. Motions can be in seemingly random directions. Two or more stars, double stars or open star clusters, which are moving in similar directions, exhibit so-called shared or common proper motion, suggesting they may be gravitationally attached or share similar motion in space. Barnard's Star has the largest proper motion of all stars, moving at 10.3 seconds of arc per year. L
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is used as an estimate of a body's surface temperature when the body's emissivity curve is not known; when the star's or planet's net emissivity in the relevant wavelength band is less than unity, the actual temperature of the body will be higher than the effective temperature. The net emissivity may be low due to surface or atmospheric properties, including greenhouse effect; the effective temperature of a star is the temperature of a black body with the same luminosity per surface area as the star and is defined according to the Stefan–Boltzmann law FBol = σTeff4. Notice that the total luminosity of a star is L = 4πR2σTeff4, where R is the stellar radius; the definition of the stellar radius is not straightforward. More rigorously the effective temperature corresponds to the temperature at the radius, defined by a certain value of the Rosseland optical depth within the stellar atmosphere.
The effective temperature and the bolometric luminosity are the two fundamental physical parameters needed to place a star on the Hertzsprung–Russell diagram. Both effective temperature and bolometric luminosity depend on the chemical composition of a star; the effective temperature of our Sun is around 5780 kelvins. Stars have a decreasing temperature gradient; the "core temperature" of the Sun—the temperature at the centre of the Sun where nuclear reactions take place—is estimated to be 15,000,000 K. The color index of a star indicates its temperature from the cool—by stellar standards—red M stars that radiate in the infrared to the hot blue O stars that radiate in the ultraviolet; the effective temperature of a star indicates the amount of heat that the star radiates per unit of surface area. From the warmest surfaces to the coolest is the sequence of stellar classifications known as O, B, A, F, G, K, M. A red star could be a tiny red dwarf, a star of feeble energy production and a small surface or a bloated giant or supergiant star such as Antares or Betelgeuse, either of which generates far greater energy but passes it through a surface so large that the star radiates little per unit of surface area.
A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel. To find the effective temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature T. Take the case of a planet at a distance D from the star, of luminosity L. Assuming the star radiates isotropically and that the planet is a long way from the star, the power absorbed by the planet is given by treating the planet as a disc of radius r, which intercepts some of the power, spread over the surface of a sphere of radius D; the calculation assumes the planet reflects some of the incoming radiation by incorporating a parameter called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed; the expression for absorbed power is then: P a b s = L r 2 4 D 2 The next assumption we can make is that the entire planet is at the same temperature T, that the planet radiates as a blackbody.
The Stefan–Boltzmann law gives an expression for the power radiated by the planet: P r a d = 4 π r 2 σ T 4 Equating these two expressions and rearranging gives an expression for the effective temperature: T = L 16 π σ D 2 4 Note that the planet's radius has cancelled out of the final expression. The effective temperature for Jupiter from this calculation is 88 K and 51 Pegasi b is 1,258 K. A better estimate of effective temperature for some planets, such as Jupiter, would need to include the internal heating as a power input; the actual temperature depends on atmosphere effects. The actual temperature from spectroscopic analysis for HD 209458 b is 1,130 K, but the effective temperature is 1,359 K; the internal heating within Jupiter raises the effective temperature to about 152 K. The surface temperature of a planet can be estimated by modifying the effective-temperature calculation to account for emissivity and temperature variation; the area of the planet that absorbs the power from the star is Aabs, some fraction of the total surface area Atotal = 4πr2, where r is the radius of the planet.
This area intercepts some of the power, spread over the surface of a sphere of radius D. We allow the planet to reflect some of the incoming radiation by incorporating a parameter a called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo
The parsec is a unit of length used to measure large distances to astronomical objects outside the Solar System. A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond, which corresponds to 648000/π astronomical units. One parsec is equal to 31 trillion kilometres or 19 trillion miles; the nearest star, Proxima Centauri, is about 1.3 parsecs from the Sun. Most of the stars visible to the unaided eye in the night sky are within 500 parsecs of the Sun; the parsec unit was first suggested in 1913 by the British astronomer Herbert Hall Turner. Named as a portmanteau of the parallax of one arcsecond, it was defined to make calculations of astronomical distances from only their raw observational data quick and easy for astronomers. For this reason, it is the unit preferred in astronomy and astrophysics, though the light-year remains prominent in popular science texts and common usage. Although parsecs are used for the shorter distances within the Milky Way, multiples of parsecs are required for the larger scales in the universe, including kiloparsecs for the more distant objects within and around the Milky Way, megaparsecs for mid-distance galaxies, gigaparsecs for many quasars and the most distant galaxies.
In August 2015, the IAU passed Resolution B2, which, as part of the definition of a standardized absolute and apparent bolometric magnitude scale, mentioned an existing explicit definition of the parsec as 648000/π astronomical units, or 3.08567758149137×1016 metres. This corresponds to the small-angle definition of the parsec found in many contemporary astronomical references; the parsec is defined as being equal to the length of the longer leg of an elongated imaginary right triangle in space. The two dimensions on which this triangle is based are its shorter leg, of length one astronomical unit, the subtended angle of the vertex opposite that leg, measuring one arc second. Applying the rules of trigonometry to these two values, the unit length of the other leg of the triangle can be derived. One of the oldest methods used by astronomers to calculate the distance to a star is to record the difference in angle between two measurements of the position of the star in the sky; the first measurement is taken from the Earth on one side of the Sun, the second is taken half a year when the Earth is on the opposite side of the Sun.
The distance between the two positions of the Earth when the two measurements were taken is twice the distance between the Earth and the Sun. The difference in angle between the two measurements is twice the parallax angle, formed by lines from the Sun and Earth to the star at the distant vertex; the distance to the star could be calculated using trigonometry. The first successful published direct measurements of an object at interstellar distances were undertaken by German astronomer Friedrich Wilhelm Bessel in 1838, who used this approach to calculate the 3.5-parsec distance of 61 Cygni. The parallax of a star is defined as half of the angular distance that a star appears to move relative to the celestial sphere as Earth orbits the Sun. Equivalently, it is the subtended angle, from that star's perspective, of the semimajor axis of the Earth's orbit; the star, the Sun and the Earth form the corners of an imaginary right triangle in space: the right angle is the corner at the Sun, the corner at the star is the parallax angle.
The length of the opposite side to the parallax angle is the distance from the Earth to the Sun (defined as one astronomical unit, the length of the adjacent side gives the distance from the sun to the star. Therefore, given a measurement of the parallax angle, along with the rules of trigonometry, the distance from the Sun to the star can be found. A parsec is defined as the length of the side adjacent to the vertex occupied by a star whose parallax angle is one arcsecond; the use of the parsec as a unit of distance follows from Bessel's method, because the distance in parsecs can be computed as the reciprocal of the parallax angle in arcseconds. No trigonometric functions are required in this relationship because the small angles involved mean that the approximate solution of the skinny triangle can be applied. Though it may have been used before, the term parsec was first mentioned in an astronomical publication in 1913. Astronomer Royal Frank Watson Dyson expressed his concern for the need of a name for that unit of distance.
He proposed the name astron, but mentioned that Carl Charlier had suggested siriometer and Herbert Hall Turner had proposed parsec. It was Turner's proposal. In the diagram above, S represents the Sun, E the Earth at one point in its orbit, thus the distance ES is one astronomical unit. The angle SDE is one arcsecond so by definition D is a point in space at a distance of one parsec from the Sun. Through trigonometry, the distance SD is calculated as follows: S D = E S tan 1 ″ S D ≈ E S 1 ″ = 1 au 1 60 × 60 × π
Minute and second of arc
A minute of arc, arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn – it is for this reason that the Earth's circumference is exactly 21,600 nautical miles. A minute of arc is π/10800 of a radian. A second of arc, arcsecond, or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, π/648000 of a radian; these units originated in Babylonian astronomy as sexagesimal subdivisions of the degree. To express smaller angles, standard SI prefixes can be employed; the number of square arcminutes in a complete sphere is 4 π 2 = 466 560 000 π ≈ 148510660 square arcminutes. The names "minute" and "second" have nothing to do with the identically named units of time "minute" or "second"; the identical names reflect the ancient Babylonian number system, based on the number 60. The standard symbol for marking the arcminute is the prime, though a single quote is used where only ASCII characters are permitted.
One arcminute is thus written 1′. It is abbreviated as arcmin or amin or, less the prime with a circumflex over it; the standard symbol for the arcsecond is the double prime, though a double quote is used where only ASCII characters are permitted. One arcsecond is thus written 1″, it is abbreviated as arcsec or asec. In celestial navigation, seconds of arc are used in calculations, the preference being for degrees and decimals of a minute, for example, written as 42° 25.32′ or 42° 25.322′. This notation has been carried over into marine GPS receivers, which display latitude and longitude in the latter format by default; the full moon's average apparent size is about 31 arcminutes. An arcminute is the resolution of the human eye. An arcsecond is the angle subtended by a U. S. dime coin at a distance of 4 kilometres. An arcsecond is the angle subtended by an object of diameter 725.27 km at a distance of one astronomical unit, an object of diameter 45866916 km at one light-year, an object of diameter one astronomical unit at a distance of one parsec, by definition.
A milliarcsecond is about the size of a dime atop the Eiffel Tower. A microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth. A nanoarcsecond is about the size of a penny on Neptune's moon Triton as observed from Earth. Notable examples of size in arcseconds are: Hubble Space Telescope has calculational resolution of 0.05 arcseconds and actual resolution of 0.1 arcseconds, close to the diffraction limit. Crescent Venus measures between 66 seconds of arc. Since antiquity the arcminute and arcsecond have been used in astronomy. In the ecliptic coordinate system and longitude; the principal exception is right ascension in equatorial coordinates, measured in time units of hours and seconds. The arcsecond is often used to describe small astronomical angles such as the angular diameters of planets, the proper motion of stars, the separation of components of binary star systems, parallax, the small change of position of a star in the course of a year or of a solar system body as the Earth rotates.
These small angles may be written in milliarcseconds, or thousandths of an arcsecond. The unit of distance, the parsec, named from the parallax of one arc second, was developed for such parallax measurements, it is the distance at which the mean radius of the Earth's orbit would subtend an angle of one arcsecond. The ESA astrometric space probe Gaia, launched in 2013, can approximate star positions to 7 microarcseconds. Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red giant with a diameter of 0.05 arcsecond. Because of the effects of atmospheric seeing, ground-based telescopes will smear the image of a star to an angular diameter of about 0.5 arcsecond. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond. Space telescopes are diffraction limited. For example, the Hubble Space Telescope can reach an angular size of stars down to about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics, for example, can produce images around 0.05 arcsecond on a 10 m class telescope.
Minutes and seconds of arc are used in cartography and navigation. At sea level one minute of arc
Washington Double Star Catalog
The Washington Double Star Catalog, or WDS, is a catalog of double stars, maintained at the United States Naval Observatory. The catalog contains positions, proper motions and spectral types and has entries for 141,743 pairs of double stars; the catalog includes multiple stars. In general, a multiple star with n components will be represented by entries in the catalog for n-1 pairs of stars; the database used to construct the WDS originated at Lick Observatory, where it was used to construct the Index Catalog of Visual Double Stars, published in 1963. In 1965, under the initiative of Charles Worley, it was transferred to the Naval Observatory; the catalog has since been augmented by a large number of measurements from the Hipparcos and Tycho catalogues and results from speckle interferometry, as well as other sources. Aitken Double Star Catalogue Burnham Double Star Catalogue The WDS at the US Naval Observatory The Washington Double Star Catalog at VizieR StelleDoppie interface to the WDS