Spherical astronomy

While astrometry in practice implements measurements of the positions and relative movements of astronomical objects, spherical astronomy, as a more theoretical discipline closely related to mathematics , deals with the establishment of celestial coordinate systems and time frames, as well as transitions between them. In fact, the main task of spherical astronomy is the reduction of observations, that is, the calculation of the coordinates and velocities of celestial bodies in a particular coordinate system at a given point in time, based on their observations ^[1] .

The basic concept of spherical astronomy is the celestial sphere , that is, an imaginary sphere of arbitrary radius with a center in the observer onto which visible positions of astronomical objects are projected and on which celestial coordinate systems are introduced, the most used of which are horizontal, two equatorial, ecliptic and galactic. Transitions between them are carried out according to the formulas of spherical trigonometry ^[1] .

When observing, the visible coordinates of celestial bodies on the celestial sphere, in addition to the body's own motion in space , are subject to the influence of several factors: precession , nutation , refraction , aberration, and parallax displacement . The first two reasons lead to global displacements of the celestial coordinate systems, and the last three reasons, also known in classical physics , as well as the deviation of light by the gravitational field , predicted by relativistic physics (and the refraction of solar wind plasma by the plasma), lead to small quasi-periodic changes in visible coordinates over time, the removal (reduction) which leads to the coordinates of bodies topocentric coordinate system associated with the observer at the time of observation and the direction of the axes to Torah observer specifies the position on the Earth's surface ^[1] .

The next step is the reduction to the Earth's coordinate system, associated with the Earth as a whole, and from it, through consideration of precession and nutation, to the inertial coordinate system, for which it is necessary to know the parameters of the shape and rotation of the Earth. In this, spherical astronomy merges with geodesy , cartography, and gravimetry . In addition, the reduction of the observation time to the inertial system is also performed, which requires knowledge of the parameters of the Earth’s motion in the Solar System and accounting for corrections of the general theory of relativity ^[1] .

The main elements of spherical astronomy are coordinate systems and time. To indicate the position of celestial bodies, an equatorial coordinate system is used , based on the projection of the Earth's equator on the celestial sphere. The position of the object is determined by its right ascension (α) and declination (δ). Based on these data, latitude and local time, you can determine the position of the object in the horizontal coordinate system , namely, its height and azimuth ^[1] .

The coordinates of objects in the starry sky, such as stars and galaxies, are recorded in catalogs , in which the position of the object at a specific time is given, usually the year called the catalog epoch . It is the reference catalogs, together with the procedures for measuring and reducing observations, that implement the celestial coordinate systems in practice. However, the combined effect of precession , nutation, and proper motions of celestial bodies leads to the fact that their coordinates change somewhat with time. The impact of such changes in the motion of the Earth is compensated by the periodical publication of new versions of the catalogs ^[1] .

To determine the positions of the Sun and planets, astronomical ephemeris is used (a table of values ​​that allows determining the position of celestial bodies at a certain time, calculated by the methods of celestial mechanics ) ^[1] .

Spherical astronomy is the basis for astronomy as a whole and has numerous applications. In fundamental astronomy, as a result of processing the reduced astrometric observations, the parameters of the celestial coordinate systems and time scales are determined, and the reduction parameters are refined, and the systems of astronomical constants are compiled. In applied astronomy, spherical astronomy is routinely used in the process of solving problems of navigation , that is, determining the coordinates of an observer, both on Earth and in space ^[1] .

Astronomy originated from the need to identify moments of certain events, both economic and religious significance. The establishment of the calendar was required for agriculture, and therefore the ancient inhabitants of Mesopotamia and Egypt rather accurately determined the duration of the year , and also by long chains of solar and lunar eclipses learned how to predict them. The hexadecimal number system of the ancient Babylonians is still used in the calculation of time ^[2] .

Further progress is associated with the flourishing of philosophy and mathematics in ancient Greece. The first ancient Greek astronomer Thales of Miletus (the end of VII — the first half of the VI century BC) —one of the “ seven wise men ”, according to legends, set the time of equinoxes and solstices, determined the duration of the year to 365 days and realized that the Moon does not shine herself, and so on. At the same time, he considered the Earth to be a flat disk, and did not understand the causes of eclipses ^[2] .

Eclipses were able to correctly explain Anaxagoras from Clazomen (about 500 — about 428 BC), and the hypothesis about the Earth's sphericity was formulated by the Pythagoreans , they also own the model of the celestial spheres, from which this concept remained in modern astronomy. In the second half of the 5th century BC. er By observing the equinoxes and solstices, the Athenian astronomers Meton and Euktemon established, with an accuracy of up to half an hour, the length of the tropical year and found the inequality of the seasons, that is, the uneven movement of the Sun along the ecliptic ^[2] .

The development of the first rigorous mathematical theories of astronomy belongs to Eudoxus of Knidd (c. 400–355 BCE). Proceeding from the sphere and the circle as ideal figures, he invented a system for decomposing the visible motion of the Sun and the planets into uniform rotations of the spheres, dragging other spheres, the celestial body attached to the last one at the equator. In his model there were 27 such spheres, in Calypp - 34, and Aristotle (384–322 BC), thanks to whose authority this model became dominant, already considered 56 spheres ^[2] .

Heraclides of Pontus suggested that the apparent rotation of the outermost sphere of fixed stars is actually caused by the rotation of the Earth, and changes in the brightness of Mercury and Venus, which were a problem of Eudox's scheme, are caused by their rotation around the Sun, and not the Earth as the center. Aristarkh Samos (310–230 BC) showed, relying on observations, that the Sun is far beyond the Moon, and on this basis developed the first heliocentric model, explaining also the absence of visible parallax of stars by their very large distances from the Earth ^[2] .

Observer astronomers Aristill and Timokharis (3rd century BC) pioneered the definitions of the positions of the stars and made the first star catalog in the equatorial system, finding the right ascents and declinations of stars. Eratosthenes from Cyrene (276-194 BC), with an accuracy of up to 50 km, determined the radius of the Earth and, with an accuracy of up to 8 seconds of arc, the slope of the ecliptic to the equator ^[2] .

Hipparchus (circa 180-125 BC) systematized and generalized all predecessors. After conducting his own measurements of the positions of the stars and compiling a catalog, he found changes in longitudes relative to the data of Aristill and Timorhovis and came to the conclusion that there was a precession , that is, movement of equinox points along the ecliptic, which allowed him to specify the duration of the year. In addition, to describe the motion of the Sun along the ecliptic, he introduced a system of epicycles and eccentres and derived the “first inequality”, the difference in the center position of the true and middle Sun, which is now called the “ equation of time ” ^[2] .

Further, in the development of astronomy followed by a pause, which ended at the end of the I century AD. er works on spherical trigonometry of the Greek astronomer Menelaus of Alexandria , the results of which were then used by Ptolemy (about 100 - about 165 years), 13 books of the Almagest of which became the main source of astronomical knowledge for the next fifteen hundred years across Eurasia. Ptolemy's star catalog was then repeatedly updated: al-Battani (880), al-Sufi (964), Alphonse Tables (1252), Ulugbek (1437), which made it possible to specify the constant precession and ecliptic slope to a few minutes arc ^[2] .

Copernicus' heliocentric theory, published in 1543, was the next big step, the meaning of which was understood only later, after the work of Tycho Brahe (1546-1601), which achieved the best known accuracy of observations of stars and planets with the naked eye and compiled a new catalog of 777 stars with accuracy positions in half a minute arc. His observations of Mars allowed Kepler to deduce the laws of planetary motion , which finally confirmed the priority of the heliocentric system ^[2] .

John Napier (1550–1617), the inventor of logarithms , also developed problems for solving spherical triangles, finding Napier’s analogies . The rapid development of navigation made it imperative to accurately determine the time, for which Huygens first invented a pendulum (1656) and then a spring clock (1675). In the observatories for storing time, such clocks could be used, but the definition of longitude in the open sea was still a difficult problem - the accuracy of the clock in the conditions of ship rolling and temperature drops was completely inadequate. The calculated tables of the Moon's motion and star catalogs served as a palliative, on the basis of which it was possible to determine the longitude, for example, the Euler tables gave an accuracy of about a degree. The relatively stable spring clock — the chronometer — was invented by John Harrison in 1735, but only in 1761 did his son William improve them so that, when traveling to Jamaica, he achieved a measurement accuracy of 1/3 degree longitude ^[2] .

By the end of the 18th century, mechanical watches had already been produced by tens of thousands of pieces, their mechanisms were quickly improved, and accuracy increased. The globalization of trade and movement of people demanded the introduction of a single time, and in 1884 the international time in Washington was taken as the standard time , the starting point of which was Greenwich time - the average solar time at the selected zero meridian, the Greenwich meridian. In the same place determined a line of change of dates ^[2] .

The invention of the telescope in the 17th century by Galileo and the improvement of it by Newton led to a rapid progress in the accuracy of astronomical observations. In 1725, the British royal astronomer James Bradley derived from observation the aberration of light , manifested as a periodic change in the apparent positions of stars due to a change in the direction and magnitude of the velocity of the Earth relative to them. In 1837, Friedrich Bessel for the first time managed to measure also the annual parallax of a star - the relative displacement of the 61 Cygni star relative to its nearest due to a change in the position of the observer along with the Earth in space ^[2] .

The development of the theory of motion of the moon and the solar system, based on the law of the world of Newton, took the whole XVIII and XIX century, this was done by Euler , Klero , D'Alembert , Lagrange and Laplace . Accuracy and power of methods steadily increased, starting with Newton, who qualitatively explained the flattening of the Earth due to centrifugal force and indicated that the gravitational effect of the Moon, the Sun and the planets on the equatorial hump would cause precession. The d'Alembert gave a quantitative theory of this phenomenon in 1749, explaining also with this influence the nutation discovered by Bradley in 1745. This theory was clarified by taking into account the oceans and the atmosphere, as well as the tides of Laplace, he also introduced the concept of potential , which then became fundamental in physics, and put forward an assumption about the movement of the poles and the uneven rotation of the Earth. Klero dealt with the question of the Earth's figure, finding how its compression could be determined from gravimetric measurements ^[2] .

The progress of observation accuracy by the end of the 19th century made it possible to detect the movement of the poles, the fluctuation of which with a period of about 1.2 years was found by Seth Chandler in 1891 and bears his name. By the end of the XIX century, the theory of rotation of absolutely solid Earth was completed and Oppolzer obtained formulas describing precession and nutation. However, Simon Newcomb , who introduced the modern system of precession parameters, in 1892 put forward the idea that the Chandler oscillation is caused by the influence of the elasticity of the Earth on the free Eulerian oscillations of the solid pole of the Earth. Thus, it turned out that the movement of the pole cannot be obtained theoretically without an exact knowledge of the structure of the Earth, which makes it necessary to determine this movement by regular measurements. For this, in 1898, the International Latitude Service was created, whose functions were then transferred to the International Earth Rotation Service ^[2] .

Observations of the Moon and the Sun, including the ancients, when compared with the exact theories of the solar system of the late XIX-early XX century, developed by Newcomb, Brown and de Sitter , led to the discovery of the secular slowing down of the Earth's rotation . The theory of motion of the Sun of Newcomb was so accurate that it became the basis for creating the first dynamic time scale - the scale of ephemeris time , and the definition of an ephemeris second . Only by the middle of the 20th century, the accuracy of clocks - atomic frequency standards - became better than for ephemeris time, and the transition to the atomic scale made it possible to directly measure the unevenness of the Earth's rotation ^[2] .

A new development of observation techniques at the end of the 20th century — radio interferometry with super-long bases , laser ranging and other methods — made it possible to further improve the accuracy of astrometric measurements and the Earth’s figure to millimeter precision, forcing to take into account both the relativistic effects of deflection and lagging of electromagnetic signals in gravitational fields, which was formalized by decisions of the International Astronomical Union in 2000. The use of high-precision equipment made it possible to map the gravitational field of the Earth, measure the effect of the gravitational field on the speed of the clock and introduce global GPS navigation systems (Global Positioning System) and GLONASS (GLOBAL Navigation Satellite System) into practice. The new reference catalogs, for which the celestial coordinates are determined, have reached an accuracy of 0.1 milliseconds of arc in the radio and few milliseconds in the optical range ^{[3] [2]} .

• Robin M. Green. Spherical Astronomy. - Cambridge University Press, 1985 . - ISBN 0-521-31779-7 .
• William M. Smart, edited by Robin M. Green. Textbook on Spherical Astronomy. - Cambridge University Press, 1977 . - ISBN 0-521-29180-1 .
• Zharov V. Ye. Spherical astronomy . - M. , 2006. - 480 p. - (Monographs and textbooks). - 500 copies - ISBN 5-85099-168-9 .

• Spherical astronomy // Great Soviet Encyclopedia : [in 30 t.] / Ch. ed. A. M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia, 1969-1978.