The position of the zenith point on the celestial sphere. Lessons
An imaginary auxiliary sphere of arbitrary radius onto which the celestial bodies are projected; serves to solve various astrometric problems. The idea of N. s. arose in ancient times; it was based on the visual impression of the existence of a domed vault of heaven. This impression is due to the fact that, as a result of the enormous distance of the celestial bodies, the human eye is not able to appreciate the differences in the distances to them, and they appear equally distant. Among ancient peoples, this was associated with the presence of a real sphere that bounded the entire world and carried numerous stars on its surface. Thus, in their view N. s. was the most important element Universe. With the development of scientific knowledge, this view of N. s. disappeared. However, the geometry of the N. s. laid down in ancient times. as a result of development and improvement received modern look, in which it is used in astrometry.
Radius N.s. can be taken in any way: in order to simplify the geometric relationships, it is assumed equal to one. Depending on the problem being solved, the center of N. s. can be placed in the place where the observer is located (topocentric n.s.), in the center of the Earth (geocentric n.s.), in the center of a particular planet (planetocentric n.s.), in the center of the Sun (heliocentric n.s.). s.) or to any other point in space. To each luminary on N.S. corresponds to the point at which it is intersected by a straight line connecting the center of the N.S. with the luminary (with its center). When studying the relative positions and visible movements of the luminaries on the N.S. choose one or the other coordinate system, defined by main points and lines. The latter are usually large circles of N. s. Each great circle of a sphere has two poles, defined on it by the ends of a diameter perpendicular to the plane of the given circle.
Figure 1 - Celestial sphere: Z - zenith; Z" - nadir; NESW - mathematical horizon; N, E, S, W - points of north, east, south and west; P and P" - North and South poles of the world; AWA"E - celestial equator; ? - geographic latitude
On rice. 1 a vertical line drawn through the center of this sphere intersects the natural system at points Z and Z", called zenith and respectively. nadir. A plane passing through the center of the vertical line perpendicular to the plumb line intersects the sphere in a great circle. N.E.S.W. called the mathematical (or true) horizon. The mathematical horizon divides the N. s. on the visible and invisible hemispheres; the first is the zenith, the second is the nadir. A straight line passing through the center of N. s. parallel to the axis of rotation of the Earth, called the axis of the world, and the points of its intersection with the N.S. - Northern R and South P" poles of the world. The plane passing through the center of the N. s. perpendicular to the axis of the world, intersects the sphere in a great circle AWA"E, called celestial equator. From the construction it follows that the angle between the axis of the world and the plane of the mathematical horizon, as well as the angle between the plumb line and the plane of the celestial equator, are equal geographical latitude(places of observation. The great circle of the N.S., passing through the poles of the world, zenith and nadir, is called the celestial meridian. Of the two points at which the celestial meridian intersects the mathematical horizon, the one closest to the North Pole of the world N is called the north point, and the diametrically opposite S- the point of the south. Straight N.S. passing through these points is the noon line. Horizon points 90° from points N And S, are called east points E and west W. Points N, E. S, W are called the main points of the horizon. By diameter E.W. The planes of the mathematical horizon and the celestial equator intersect.
The great circle of the solar system, along which the visible annual movement of the center of the Sun occurs, is called the ecliptic ( rice. 2 ).
Figure 2 - Celestial sphere: ЎA A" - celestial equator; ЎE = E" - ecliptic; Ў and - points of the spring and autumn equinox; E and E" - points of summer and winter solstice; P and P" - North and South poles of the world; P and P" - North and South poles of the ecliptic
Does the plane of the ecliptic form an angle with the plane of the celestial equator? = 23°27". The ecliptic intersects the equator at two points, one of which is the vernal equinox (at which the Sun, during its apparent annual movement, passes from Southern Hemisphere N. s. in Northern), and the other, diametrically opposite to it, is the point of the autumnal equinox. The points of the ecliptic, spaced 90° from the points of the spring and autumn equinoxes, are called the points of the summer and winter solstices (the first is in the Northern Hemisphere of the N.S., the second is in the Southern). The great circle of the Earth, passing through the poles of the world and the equinoxes, is called the color of the equinoxes; the great circle of the N. s., passing through the poles of the world and the solstice points, is called the solstices color scheme. Drawn on a star map, these circles cut off the tails of ancient images of constellations Ursa Major(color of the equinoxes) and Ursa Minor (color of the solstices), which is where their name comes from (Greek kуluroi, literally - with a chopped off tail, from kуlos - chopped off, cut off and yoke - tail). The apparent daily movement of the stars, which is a reflection of the actual rotation of the Earth around its axis, corresponds to the rotation of the solar system. around the axis of the world with a period equal to one sidereal day. Due to the rotation of the N. s. All images of luminaries describe circles parallel to the equator in space, called daily parallels of luminaries. Depending on the location of the daily parallels relative to the horizon, luminaries are divided into non-setting (the daily parallels are located entirely above the horizon), non-ascending (the daily parallels are entirely below the horizon), ascending and setting (the daily parallels are intersected by the horizon). The boundaries of these groups of luminaries are parallels KN And SM", touching the horizon at points N and S( rice. 1 ). Since the visibility of celestial bodies is determined by the position of the horizon, the plane of which is perpendicular to the plumb line, the conditions for the visibility of celestial bodies are different for places on the Earth’s surface with different geographic latitudes. This phenomenon, known already in ancient times, served as one of the proofs of the sphericity of the Earth. At the equator (? = 0°) the axis of the world PP" located in the horizon plane and coincides with the noon line NS. Daily parallels ( KK",MM") all luminaries intersect the horizon plane at right angles. Here all the luminaries are rising and setting ( rice. 3 ).
As the observer moves along the earth's surface from the equator to the pole, the inclination of the world axis to the horizon increases. An increasing number of luminaries are becoming non-setting and non-rising. At the pole (? = 90°), the axis of the world coincides with the plumb line, and the equatorial plane coincides with the horizon plane. Here all the luminaries are divided only into non-setting and non-rising, so what are the daily parallels ( KK",MM") are placed in planes parallel to the horizon ( rice. 4 ).
1 Basic provisions celestial sphere
To determine the visible position celestial bodies and the study of their movement in astronomy, the concept is introduced celestial sphere. The sphere has arbitrary dimensions and an arbitrary center. To its center at a point ABOUT an observer is placed, and the rotation of the sphere repeats the rotation of the firmament. Straight ZOZ′ stands for plumb line for the observer, no matter where he is. The highest point above the observer's head Z called Zenith, and its opposite point Z′- called Nadir. Big circle SWNE perpendicular plumb line called true horizon or mathematical horizon. Mathematical horizon divides the sphere into two halves , visible And invisible for the observer. Line RR′- called axis mundi, rotation occurs around this axis celestial sphere. Plane ЕQWQ′ perpendicular to axis mundi called celestial equator. He divides celestial sphere into two hemispheres - northern And southern. Great circle of the celestial sphere PZQSP′Z′Q′N called celestial meridian. The celestial meridian divides the celestial sphere into Eastern And Western hemisphere. Line NOS called midday line.
The position of the main elements of the celestial sphere relative to each other depends on geographic latitude observer positions. At an angle The axis of the world is located towards the plane of the mathematical horizonRR′. The positions of the luminaries in the sky are determined in relation to the main planes and the lines and points associated with them celestial sphere and is expressed quantitatively in two quantities ( central angles or arcs of great circles) which are called celestial coordinates.
2 Horizontal coordinate system
Main plane horizontal system coordinates is mathematical horizonN.W.S.E., and the report is carried out from Z zenith and from one of the points of the mathematical horizon. One coordinate is zenith distancez ( Zenith distance to south zв = φ - δ; To north zн = 180 - φ - δ) or height of the luminary above the horizon h. Height h luminaries M called the height of a vertical circle mM from mathematical horizon before luminaries or central angle mOM between plane mathematical horizon and direction to luminary M. Heights are counted from 0 to 90 k zenith and from 0 to -90 to the nadir. The zenith distance of a luminary is called the arc of a vertical circle ZM from the luminary to zenith. z + h = 90 (1). The position of the vertical circle itself is determined by the coordinate arc - azimuth A. Azimuth A called arc mathematical horizon Sm from point southS to a vertical circle passing through the luminary. Azimuths counted in the direction of rotation celestial sphere, i.e. west of the south point, ranging from 0 to 360. The coordinate system is used to direct definitions visible positions of the luminaries using goniometric instruments.
3 First equatorial coordinate system
Start of counting - celestial equator pointQ. One coordinate is declination. Declension called arc mM hour circle PMmP′ from the celestial equator to the luminary. Counted from 0 to +90 to the north pole and from 0 to -90 to the south. p + = 90 . The position of the hour circle is determined hour anglet. Hour angle luminaries M called the arc of heaven equatorQm from the top point Q celestial equator to hour circle PMmP′, passing through the luminary. Hour angles are counted towards the daily rotation of the celestial sphere, west of Q, ranging from 0 to 360 or from 0 to 24 hours. The coordinate system is used in practical astronomy to determine the exact time and daily rotation of the sky. Determines the daily movement of the Sun, Moon and other luminaries.
4 Second equatorial coordinate system
One coordinate is declination, another right ascensionα . Direct ascent α luminaries M called the arc of the celestial equator ♈ m from point spring equinox♈ to the hour circle passing through the luminary. It is counted in the direction opposite to the daily rotation within the range from 0 to 360 or from 0 to 24 hours. The system is used to determine stellar coordinates and compile catalogs. Determines the annual movement of the Sun and other luminaries.
5 The height of the celestial pole above the horizon, the height of the luminary in the meridian
The height of the celestial pole above the horizon is always equal to the astronomical latitude of the observer:
- If the declination of the luminary less latitude, then it culminates south of the zenith at z = φ - δ or at altitude h = 90 - φ + δ
- If the declination of the luminary equal to geographic latitude, then it culminates at the zenith and z = 0 , A h = + 90
- If the declination of the luminary more latitude, then it culminates north of the zenith at z = s - φ or at altitude h = 90 + φ - s
6 Conditions for sunrise and sunset
never setting luminaries.
the culmination of the luminary.
upper climax, if the lower one is lower climax.
For an observer at the poles there will only be never setting luminaries.
The phenomenon of a luminary crossing the celestial meridian is called the culmination of the luminary.
If the luminary crosses the upper part of the meridian, it comes upper climax, if the lower one is lower climax.
Auxiliary celestial sphere
Coordinate systems used in geodetic astronomy
Geographic latitudes and longitudes of points on the earth's surface and directional azimuths are determined from observations of celestial bodies - the Sun and stars. To do this, you need to know the position of the luminaries both relative to the Earth and relative to each other. The positions of the luminaries can be specified in appropriately chosen coordinate systems. As is known from analytical geometry, to determine the position of the star s, you can use a rectangular Cartesian coordinate system XYZ or polar a,b, R (Fig. 1).
In a rectangular coordinate system, the position of the luminary s is determined by three linear coordinates X, Y, Z. In the polar coordinate system, the position of the luminary s is given by one linear coordinate, the radius vector R = Os and two angular coordinates: the angle a between the X axis and the projection of the radius vector onto the coordinate plane XOY, and the angle b between coordinate plane XOY and radius vector R. The relationship between rectangular and polar coordinates is described by the formulas
X = R cos b cos a,
Y = R cos b sin a,
Z = R sin b,
These systems are used in cases where the linear distances R = Os to celestial bodies are known (for example, for the Sun, Moon, planets, artificial satellites Earth). However, for many luminaries observed beyond solar system, these distances are either extremely large compared to the radius of the Earth or unknown. To simplify the solution of astronomical problems and avoid distances to luminaries, it is believed that all luminaries are at an arbitrary, but equal distance from the observer. Usually this distance is taken equal to unity, as a result of which the position of the luminaries in space can be determined not by three, but by two angular coordinates a and b of the polar system. It is known that the locus of points equidistant from a given point “O” is a sphere with a center at this point.
Auxiliary celestial sphere – an imaginary sphere of arbitrary or unit radius onto which images of celestial bodies are projected (Fig. 2). The position of any luminary s on the celestial sphere is determined using two spherical coordinates, a and b:
x = cos b cos a,
y = cos b sin a,
z = sin b.
Depending on where the center of the celestial sphere O is located, there are:
1)topocentric celestial sphere - the center is on the surface of the Earth;
2)geocentric celestial sphere - the center coincides with the center of mass of the Earth;
3)heliocentric celestial sphere - the center is aligned with the center of the Sun;
4) barycentric celestial sphere - the center is located at the center of gravity of the solar system.
The main circles, points and lines of the celestial sphere are shown in Fig. 3.
One of the main directions relative to the Earth's surface is the direction plumb line, or gravity at the observation point. This direction intersects the celestial sphere at two diametrically opposite points - Z and Z". Point Z is located above the center and is called zenith, Z" – under the center and is called nadir.
Let us draw a plane through the center perpendicular to the plumb line ZZ". The great circle NESW formed by this plane is called celestial (true) or astronomical horizon. This is the main plane of the topocentric coordinate system. There are four points on it S, W, N, E, where S is point of the South, N- North point,W- West point,E- point of the East. Direct NS is called noon line.
The straight line P N P S drawn through the center of the celestial sphere parallel to the axis of rotation of the Earth is called axis mundi. Points P N - north celestial pole; P S - south celestial pole. The visible daily movement of the celestial sphere occurs around the axis of the World.
Let us draw a plane through the center perpendicular to the axis of the world P N P S . The great circle QWQ"E formed as a result of the intersection of this plane with the celestial sphere is called celestial (astronomical) equator. Here Q is highest point of the equator(above the horizon), Q"- lowest point of the equator(below the horizon). The celestial equator and celestial horizon intersect at points W and E.
The plane P N ZQSP S Z"Q"N, containing a plumb line and the axis of the World, is called true (celestial) or astronomical meridian. This plane is parallel to the plane of the earth's meridian and perpendicular to the plane of the horizon and equator. It is called the initial coordinate plane.
Let us draw a vertical plane through ZZ" perpendicular to the celestial meridian. The resulting circle ZWZ"E is called first vertical.
The great circle ZsZ", along which the vertical plane passing through the luminary s intersects the celestial sphere, is called vertical or circle of the heights of the luminary.
The great circle P N sP S passing through the star perpendicular to the celestial equator is called around the declination of the luminary.
The small circle nsn" passing through the luminary parallel to the celestial equator is called daily parallel. The apparent daily movement of the luminaries occurs along diurnal parallels.
The small circle "asa", passing through the luminary parallel to the celestial horizon, is called circle of equal heights, or almucantarate.
To a first approximation, the Earth's orbit can be taken as a flat curve - an ellipse, at one of the foci of which the Sun is located. The plane of the ellipse taken as the Earth's orbit , called a plane ecliptic.
In spherical astronomy it is customary to talk about apparent annual movement of the Sun. The great circle EgE"d, along which the visible movement of the Sun occurs during the year, is called ecliptic. The plane of the ecliptic is inclined to the plane of the celestial equator at an angle approximately equal to 23.5 0. In Fig. 4 shown:
g – vernal equinox point;
d – autumnal equinox point;
E – summer solstice point; E" – winter solstice point; R N R S – ecliptic axis; R N – north pole of the ecliptic; R S – south pole of the ecliptic; e – inclination of the ecliptic to the equator.
Celestial sphere(Fig. 8.1) is an imaginary sphere of arbitrary radius, the center of which is the observer (O).
Zenith(Z) is a point on the celestial sphere located vertically above the observer’s head.
Nadir(Z") is a point on the celestial sphere opposite to the zenith.
True horizon(NESW circle) is a large circle on the celestial sphere, the plane of which is perpendicular to the vertical line (ZZ").
Vertical of the luminary(ZCZ") is a great circle of the celestial sphere passing through the zenith of the observer and the given luminary. It is perpendicular to the plane of the true horizon. The vertical passing through points E and W is called the first vertical.
Rice. 8.1. Basic points and circles on the celestial sphere
Almucantarat(DCD 1) - a small circle on the celestial sphere, parallel to the plane of the true horizon.
axis mundi(PP") is a straight line parallel to the axis of rotation of the earth. The points of its intersection with the celestial sphere P and P" are called the celestial poles, respectively - north and south.
Celestial equator(QWQ"E) is a great circle on the celestial sphere, the plane of which is perpendicular to the axis of the world.
Declension circle(hour circle) luminaries (PCP") - a large circle passing through the poles of the world and the luminary.
Celestial meridian(ZPQZ"P"Q") - a great circle on the celestial sphere, passing through the pole and zenith of the observer. Its intersection with the true horizon at point N is called north point, at point S – point south.
The intersection of the celestial equator with the true horizon at point E is called east point, at point W – west point.
Noon Line– a straight line connecting points N and S.
Daily parallel of the luminary(KCK 1) - a small circle on the celestial sphere drawn through the luminary parallel to the celestial equator
8.3. Celestial coordinate systems
Horizontal coordinate system (HCS). In this system (Fig. 8.2), the main circles relative to which the location of the luminary is determined are the true horizon and the celestial meridian; the coordinates are the height of the luminary ( 
) and its azimuth ( 
).
Height of the luminary
(
) – the angle between the plane of the true horizon and the direction towards the luminary. Counted from 0° to ±90° (positive value towards the zenith from the horizon, negative towards the nadir).
Zenith distance
(
) – the angle in the vertical plane from the plumb line to the direction towards the luminary. Measures from 0° to 180° and is the complement of height up to 90°
. (8.1)
Rice. 8.2. Horizontal coordinate system
Azimuth of the star
(
) – the angle in the plane of the true horizon between the northern direction of the noon line and the vertical plane of the luminary. Measured from 0° to 360° in the east direction.
Equatorial coordinate system (ECS). In this system (Fig. 8.3), the main circles relative to which the location of the luminary is determined are the celestial equator and the celestial meridian. The coordinates are: the declination of the luminary ( 
), its hour angle ( 
) and right ascension ( 
).
Rice. 8.3. Equatorial coordinate system
Declination of the luminary
(
) – the angle between the plane of the celestial equator and the direction of the luminary. Measured from 0° to ±90° (positive value is north of the equator, negative value is south).
Hour angle of the luminary
(
) – angle between southern part the plane of the celestial meridian and the plane of the circle of declination of the luminary. Measured from 0° to 180° in the western and eastern directions. In the Aviation Astronomical Yearbook (AAE), the hour angle is given as western, ranging from 0° to 360°.
Right ascension of the luminary
(
) – the angle between the plane of the declination circle of the vernal equinox and the plane of the declination circle of the luminary. Measured from 0° to 360° against the daily rotation of the sky.
2.1.1. Basic planes, lines and points of the celestial sphere
A celestial sphere is an imaginary sphere of arbitrary radius with a center at a selected observation point, on the surface of which the luminaries are located as they are visible in the sky at some point in time from a given point in space. To correctly imagine an astronomical phenomenon, it is necessary to consider the radius of the celestial sphere to be much greater than the radius of the Earth (R sf >> R Earth), i.e., to assume that the observer is in the center of the celestial sphere, and the same point of the celestial sphere (the same the same star) is visible from different places on the earth's surface in parallel directions.
The celestial vault or sky is usually understood as the inner surface of the celestial sphere onto which celestial bodies (luminaries) are projected. For an observer on Earth, the Sun, sometimes the Moon, and even less often Venus are visible in the sky during the day. On a cloudless night, stars, the Moon, planets, sometimes comets and other bodies are visible. There are about 6000 stars visible to the naked eye. Mutual arrangement stars almost does not change due to the large distances to them. Celestial bodies belonging to the Solar system change their position relative to the stars and each other, which is determined by their noticeable angular and linear daily and annual displacement.
The vault of heaven rotates as a single whole with all the luminaries located on it about an imaginary axis. This rotation is daily. If you observe the daily rotation of stars in the northern hemisphere of the Earth and face the north pole, then the rotation of the sky will occur counterclockwise.
Center O of the celestial sphere is the observation point. The straight line ZOZ" coinciding with the direction of the plumb line at the observation location is called a plumb or vertical line. The plumb line intersects with the surface of the celestial sphere at two points: at the zenith Z, above the observer's head, and at the diametrically opposite point Z" - the nadir. The great circle of the celestial sphere (SWNE), the plane of which is perpendicular to the plumb line, is called the mathematical or true horizon. The mathematical horizon is a plane tangent to the surface of the Earth at the observation point. The small circle of the celestial sphere (aMa"), passing through the luminary M, and the plane of which is parallel to the plane of the mathematical horizon, is called the almucantarate of the luminary. The large semicircle of the celestial sphere ZMZ" is called the circle of height, vertical circle, or simply the vertical of the luminary.The diameter PP" around which the celestial sphere rotates is called the mundi axis. The mundi axis intersects with the surface of the celestial sphere at two points: at the north celestial pole P, from which the celestial sphere rotates clockwise when looking at the sphere from the outside, and at the south pole of the world R". The world axis is inclined to the plane of the mathematical horizon at an angle equal to the geographic latitude of the observation point φ. The great circle of the celestial sphere QWQ"E, the plane of which is perpendicular to the axis of the world, is called the celestial equator. The small circle of the celestial sphere (bМb"), the plane of which is parallel to the plane of the celestial equator, is called the celestial or daily parallel of the luminary M. The great semicircle of the celestial sphere RMR* is called hour circle or circle of declination of the luminary.
The celestial equator intersects with the mathematical horizon at two points: at the east point E and at the west point W. The circles of heights passing through the points of east and west are called the first verticals - east and west.
The great circle of the celestial sphere PZQSP"Z"Q"N, the plane of which passes through the plumb line and the axis of the world, is called the celestial meridian. The plane of the celestial meridian and the plane of the mathematical horizon intersect along a straight line NOS, which is called the noon line. The celestial meridian intersects with the mathematical horizon at the north point N and at the south point S. The celestial meridian also intersects with the celestial equator at two points: at the upper point of the equator Q, which is closer to the zenith, and at the lower point of the equator Q", which is closer to the nadir.
2.1.2. Luminaries, their classification, visible movements.
Stars, Sun and Moon, planets
In order to navigate the sky, bright stars are grouped into constellations. There are 88 constellations in the sky, of which 56 are visible to an observer located in the middle latitudes of the Earth’s northern hemisphere. All constellations have proper names, associated with the names of animals (Ursa Major, Lion, Dragon), names of heroes Greek mythology(Cassiopeia, Andromeda, Perseus) or the names of objects whose outlines resemble (Northern Crown, Triangle, Libra). Individual stars in constellations are designated by letters Greek alphabet, and the brightest of them (about 200) received “proper” names. For example, α Canis Major– “Sirius”, α Orion – “Betelgeuse”, β Perseus – “Algol”, α Ursa Minor – “ polar Star", near which the point of the north pole of the world is located. The paths of the Sun and Moon against the background of the stars almost coincide and come through twelve constellations, which are called zodiac constellations, since most of them are named after animals (from the Greek “zoon” - animal). These include the constellations of Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces. The trajectory of Mars across the celestial sphere in 2003 |
Even in ancient times, 5 luminaries were noticed, similar to stars, but “wandering” through the constellations. They were called planets - “wandering luminaries”. Later, 2 more planets and a large number of smaller celestial bodies (dwarf planets, asteroids) were discovered.
The planets move most of the time across the zodiacal constellations from west to east (direct motion), but part of the time from east to west (retrograde motion).
| Your browser does not support the video tag. The movement of stars in the celestial sphere |
