The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to an observer at the Earth's North Pole and South Pole, respectively; as the Earth spins on its axis, the two celestial poles remain fixed in the sky, all other points appear to rotate around them, completing one circuit per day. The celestial poles are the poles of the celestial equatorial coordinate system, meaning they have declinations of +90 degrees and −90 degrees; the celestial poles do not remain permanently fixed against the background of the stars. Because of a phenomenon known as the precession of the equinoxes, the poles trace out circles on the celestial sphere, with a period of about 25,700 years; the Earth's axis is subject to other complex motions which cause the celestial poles to shift over cycles of varying lengths. Over long periods the positions of the stars themselves change, because of the stars' proper motions.
An analogous concept applies to other planets: a planet's celestial poles are the points in the sky where the projection of the planet's axis of rotation intersects the celestial sphere. These points vary. Celestial bodies other than Earth have defined celestial poles; the north celestial pole is within a degree of the bright star Polaris. This makes Polaris useful for navigation in the northern hemisphere: not only is it always above the north point of the horizon, but its altitude angle is always equal to the observer's geographic latitude. Polaris can, of course, only be seen from locations in the northern hemisphere. Polaris is near the celestial pole for only a small fraction of the 25,700-year precession cycle, it will remain a good approximation for about 1,000 years, by which time the pole will have moved to be closer to Alrai. In about 5,500 years, the pole will have moved near the position of the star Alderamin, in 12,000 years, Vega will become our north star, but it will be about six degrees from the true north celestial pole.
To find Polaris, face north and locate the Big Dipper and Little Dipper asterisms. Looking at the "cup" part of the Big Dipper, imagine that the two stars at the outside edge of the cup form a line pointing upward out of the cup; this line points directly at the star at the tip of the Little Dipper's handle. That star is the North Star; the south celestial pole is visible only from the Southern Hemisphere. It lies in the Octant. Sigma Octantis is identified as the south pole star, over a degree away from the pole, but with a magnitude of 5.5 it is visible on a clear night. The south celestial pole can be located from the Southern Cross and its two "pointer" stars α Centauri and β Centauri. Draw an imaginary line from γ Crucis to α Crucis—the two stars at the extreme ends of the long axis of the cross—and follow this line through the sky. Either go four-and-a-half times the distance of the long axis in the direction the narrow end of the cross points, or join the two pointer stars with a line, divide this line in half at right angles draw another imaginary line through the sky until it meets the line from the Southern Cross.
This point is 6 degrees from the south celestial pole. Few bright stars of importance lie between Crux and the pole itself, although the constellation Musca is easily recognised beneath Crux; the second method uses Achernar. Make a large equilateral triangle using these stars for two of the corners; the third imaginary corner will be the south celestial pole. If Canopus has not yet risen, the second-magnitude Alpha Pavonis can be used to form the triangle with Achernar and the pole; the third method is best for moonless and clear nights, as it uses two faint "clouds" in the Southern Sky. These are marked in astronomy books as Small Magellanic Clouds; these "clouds" are dwarf galaxies near the Milky Way. Make an equilateral triangle, the third point of, the south celestial pole. A line from Sirius, the brightest star in the sky, through Canopus, the second-brightest, continued for the same distance lands within a couple of degrees of the pole. In other words, Canopus is halfway between the pole. Celestial sphere Celestial equator Circumpolar star Orbital pole Polaris Pole star visual representation of finding Polaris using the Big Dipper
A compass is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions. A diagram called a compass rose shows the directions north, south and west on the compass face as abbreviated initials; when the compass is used, the rose. Compasses display markings for angles in degrees in addition to the rose. North corresponds to 0°, the angles increase clockwise, so east is 90° degrees, south is 180°, west is 270°; these numbers allow the compass to show magnetic North azimuths or true North azimuths or bearings, which are stated in this notation. If magnetic declination between the magnetic North and true North at latitude angle and longitude angle is known direction of magnetic North gives direction of true North. Among the Four Great Inventions, the magnetic compass was first invented as a device for divination as early as the Chinese Han Dynasty, adopted for navigation by the Song Dynasty Chinese during the 11th century; the first usage of a compass recorded in Western Europe and the Islamic world occurred around 1190.
The magnetic compass is the most familiar compass type. It functions as a pointer to "magnetic north", the local magnetic meridian, because the magnetized needle at its heart aligns itself with the horizontal component of the Earth's magnetic field; the magnetic field exerts a torque on the needle, pulling the North end or pole of the needle toward the Earth's North magnetic pole, pulling the other toward the Earth's South magnetic pole. The needle is mounted on a low-friction pivot point, in better compasses a jewel bearing, so it can turn easily; when the compass is held level, the needle turns until, after a few seconds to allow oscillations to die out, it settles into its equilibrium orientation. In navigation, directions on maps are expressed with reference to geographical or true north, the direction toward the Geographical North Pole, the rotation axis of the Earth. Depending on where the compass is located on the surface of the Earth the angle between true north and magnetic north, called magnetic declination can vary with geographic location.
The local magnetic declination is given on most maps, to allow the map to be oriented with a compass parallel to true north. The location of the Earth's magnetic poles change with time, referred to as geomagnetic secular variation; the effect of this means. Some magnetic compasses include means to manually compensate for the magnetic declination, so that the compass shows true directions. There are other ways to find north than the use of magnetism, from a navigational point of view a total of seven possible ways exist. Two sensors that utilize two of the remaining six principles are also called compasses, i.e. gyrocompass and GPS-compass. A gyrocompass is similar to a gyroscope, it is a non-magnetic compass that finds true north by using an fast-spinning wheel and friction forces in order to exploit the rotation of the Earth. Gyrocompasses are used on ships, they have two main advantages over magnetic compasses: they find true north, i.e. the direction of Earth's rotational axis, as opposed to magnetic north, they are not affected by ferromagnetic metal in a ship's hull.
Large ships rely on a gyrocompass, using the magnetic compass only as a backup. Electronic fluxgate compasses are used on smaller vessels. However, magnetic compasses are still in use as they can be small, use simple reliable technology, are comparatively cheap, are easier to use than GPS, require no energy supply, unlike GPS, are not affected by objects, e.g. trees, that can block the reception of electronic signals. GPS receivers using two or more antennae mounted separately and blending the data with an inertial motion unit can now achieve 0.02° in heading accuracy and have startup times in seconds rather than hours for gyrocompass systems. The devices determine the positions of the antennae on the Earth, from which the cardinal directions can be calculated. Manufactured for maritime and aviation applications, they can detect pitch and roll of ships. Small, portable GPS receivers with only a single antenna can determine directions if they are being moved if only at walking pace. By determining its position on the Earth at times a few seconds apart, the device can calculate its speed and the true bearing of its direction of motion.
It is preferable to measure the direction in which a vehicle is moving, rather than its heading, i.e. the direction in which its nose is pointing. These directions may be different if there is tidal current. GPS compasses share the main advantages of gyrocompasses, they determine true North, as opposed to magnetic North, they are unaffected by perturbations of the Earth's magnetic field. Additionally, compared with gyrocompasses, they are much cheaper, they work better in polar regions, they are less prone to be affected by mechanical vibration, they can be initialized far more quickly. However, they depend on the functioning of, communication with, the GPS satellites, which might be disrupted by an electronic attack or by the effects of
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
Abramowitz and Stegun
Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards, now the National Institute of Standards and Technology. Its full title is Handbook of Mathematical Functions with Formulas and Mathematical Tables. A digital successor to the Handbook was released as the "Digital Library of Mathematical Functions" on May 11, 2010, along with a printed version, the NIST Handbook of Mathematical Functions, published by Cambridge University Press. Since it was first published in 1964, the 1046 page Handbook has been one of the most comprehensive sources of information on special functions, containing definitions, approximations and tables of values of numerous functions used in all fields of applied mathematics; the notation used in the Handbook is the de facto standard for much of applied mathematics today. At the time of its publication, the Handbook was an essential resource for practitioners.
Nowadays, computer algebra systems have replaced the function tables, but the Handbook remains an important reference source. The foreword discusses a meeting in 1954 in which it was agreed that "the advent of high-speed computing equipment changed the task of table making but did not remove the need for tables". More than 1,000 pages long, the Handbook of Mathematical Functions was first published in 1964 and reprinted many times, with yet another reprint in 1999, its influence on science and engineering is evidenced by its popularity. In fact, when New Scientist magazine asked some of the world’s leading scientists what single book they would want if stranded on a desert island, one distinguished British physicist said he would take the Handbook; the Handbook is the most distributed and most cited NIST technical publication of all time. Government sales exceed 150,000 copies, an estimated three times as many have been reprinted and sold by commercial publishers since 1965. During the mid-1990s, the book was cited every 1.5 hours of each working day.
And its influence will persist as it is being updated in digital format by NIST. Because the Handbook is the work of U. S. federal government employees acting in their official capacity, it is not protected by copyright in the United States. While it could be ordered from the Government Printing Office, it has been reprinted by commercial publishers, most notably Dover Publications, can be viewed on and downloaded from the web. While there was only one edition of the work, it went through many print runs including a growing number of corrections. Original NBS edition: 1st printing: June 1964. 9th printing with additional corrections Michael Danos and Johann Rafelski edited the Pocketbook of Mathematical Functions, published by Verlag Harri Deutsch in 1984. The book is an abridged version of Abramowitz's and Stegun's Handbook, retaining most of the formulas, but reducing the numerical tables to a minimum, which, by this time, could be calculated with scientific pocket calculators; the references were removed as well.
Most known errata were incorporated, the physical constants updated and the now-first chapter saw some slight enlargement compared to the former second chapter. The numbering of formulas was kept for easier cross-reference. A digital successor to the Handbook, long under development at NIST, was released as the “Digital Library of Mathematical Functions” on May 11, 2010, along with a printed version, the NIST Handbook of Mathematical Functions, published by Cambridge University Press. Mathematical Tables Project, a 1938–48 Works Progress Administration project to calculate mathematical tables, including those used in Abramowitz and Stegun's Handbook of Mathematical Functions Numerical analysis Philip J. Davis, author of the Gamma function section and other sections of the book Digital Library of Mathematical Functions, from the National Institute of Standards and Technology, is intended to be a replacement for Abramowitz and Stegun's Handbook of Mathematical Functions Boole's rule, a mathematical rule of integration sometimes known as Bode's rule, due to a typo in Abramowitz and Stegun, subsequently propagated Abramowitz, Milton.
Handbook of Mathematical Functions with Formulas and Mathematical Tables. Applied Mathematics Series. 55. Washington D. C.. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. Boisvert, Ronald F.. "Handbook of Mathematical Functions". In Lide, David R. A Century of Excellence in Measurements Standards and Technology - A Chronicle of Selected NBS/NIST Publications 1901-2000. Washington, D. C. USA: U. S. Department of Commerce, National Institute of Standards and Technology / CRC Press. Pp. 135–139. ISBN 978-0-8493-1247-2. CODEN NSPUE2. NIST Special Publication 958. 20402–9325. Ret
International Astronomical Union
The International Astronomical Union is an international association of professional astronomers, at the PhD level and beyond, active in professional research and education in astronomy. Among other activities, it acts as the internationally recognized authority for assigning designations and names to celestial bodies and any surface features on them; the IAU is a member of the International Council for Science. Its main objective is to promote and safeguard the science of astronomy in all its aspects through international cooperation; the IAU maintains friendly relations with organizations that include amateur astronomers in their membership. The IAU has its head office on the second floor of the Institut d'Astrophysique de Paris in the 14th arrondissement of Paris. Working groups include the Working Group for Planetary System Nomenclature, which maintains the astronomical naming conventions and planetary nomenclature for planetary bodies, the Working Group on Star Names, which catalogs and standardizes proper names for stars.
The IAU is responsible for the system of astronomical telegrams which are produced and distributed on its behalf by the Central Bureau for Astronomical Telegrams. The Minor Planet Center operates under the IAU, is a "clearinghouse" for all non-planetary or non-moon bodies in the Solar System; the Working Group for Meteor Shower Nomenclature and the Meteor Data Center coordinate the nomenclature of meteor showers. The IAU was founded on 28 July 1919, at the Constitutive Assembly of the International Research Council held in Brussels, Belgium. Two subsidiaries of the IAU were created at this assembly: the International Time Commission seated at the International Time Bureau in Paris and the International Central Bureau of Astronomical Telegrams seated in Copenhagen, Denmark; the 7 initial member states were Belgium, France, Great Britain, Greece and the United States, soon to be followed by Italy and Mexico. The first executive committee consisted of Benjamin Baillaud, Alfred Fowler, four vice presidents: William Campbell, Frank Dyson, Georges Lecointe, Annibale Riccò.
Thirty-two Commissions were appointed at the Brussels meeting and focused on topics ranging from relativity to minor planets. The reports of these 32 Commissions formed the main substance of the first General Assembly, which took place in Rome, Italy, 2–10 May 1922. By the end of the first General Assembly, ten additional nations had joined the Union, bringing the total membership to 19 countries. Although the Union was formed eight months after the end of World War I, international collaboration in astronomy had been strong in the pre-war era; the first 50 years of the Union's history are well documented. Subsequent history is recorded in the form of reminiscences of past IAU Presidents and General Secretaries. Twelve of the fourteen past General Secretaries in the period 1964-2006 contributed their recollections of the Union's history in IAU Information Bulletin No. 100. Six past IAU Presidents in the period 1976–2003 contributed their recollections in IAU Information Bulletin No. 104. The IAU includes a total of 12,664 individual members who are professional astronomers from 96 countries worldwide.
83% of all individual members are male, while 17% are female, among them the union's former president, Mexican astronomer Silvia Torres-Peimbert. Membership includes 79 national members, professional astronomical communities representing their country's affiliation with the IAU. National members include the Australian Academy of Science, the Chinese Astronomical Society, the French Academy of Sciences, the Indian National Science Academy, the National Academies, the National Research Foundation of South Africa, the National Scientific and Technical Research Council, KACST, the Council of German Observatories, the Royal Astronomical Society, the Royal Astronomical Society of New Zealand, the Royal Swedish Academy of Sciences, the Russian Academy of Sciences, the Science Council of Japan, among many others; the sovereign body of the IAU is its General Assembly. The Assembly determines IAU policy, approves the Statutes and By-Laws of the Union and elects various committees; the right to vote on matters brought before the Assembly varies according to the type of business under discussion.
The Statutes consider such business to be divided into two categories: issues of a "primarily scientific nature", upon which voting is restricted to individual members, all other matters, upon which voting is restricted to the representatives of national members. On budget matters, votes are weighted according to the relative subscription levels of the national members. A second category vote requires a turnout of at least two-thirds of national members in order to be valid. An absolute majority is sufficient for approval in any vote, except for Statute revision which requires a two-thirds majority. An equality of votes is resolved by the vote of the President of the Union. Since 1922, the IAU General Assembly meets every three years, with the ex
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.
It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".
Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.
Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.
Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published
A great circle known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle. Any diameter of any great circle coincides with a diameter of the sphere, therefore all great circles have the same center and circumference as each other; this special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of one sphere. For most pairs of points on the surface of a sphere, there is a unique great circle through the two points; the exception is a pair of antipodal points. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry; the length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles.
These great circles are the geodesics of the sphere. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1. To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all regular paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow ϕ to take on arbitrary real values; the infinitesimal arc length in these coordinates is d s = r θ ′ 2 + ϕ ′ 2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′ 2 + ϕ ′ 2 sin 2 θ d t. According to the Euler–Lagrange equation, S is minimized if and only if sin 2 θ ϕ ′ θ ′ 2 + ϕ ′ 2 sin 2 θ = C,where C is a t -independent constant, sin θ cos θ ϕ ′ 2 θ ′ 2 + ϕ ′ 2 sin 2 θ = d d t θ ′ θ ′ 2 + ϕ ′ 2 sin 2 θ.
From the first equation of these two, it can be obtained that ϕ ′ = C θ ′ sin θ sin 2 θ − C 2. Integrating both sides and considering the boundary condition, the real solution of C is zero. Thus, ϕ ′ = 0 and θ can be any value between 0 and θ 0, indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is x sin ϕ 0 − y cos ϕ 0 = 0, a