All points and lines of the celestial sphere. Laboratory work "Basic elements of the celestial sphere

An arbitrary radius onto which celestial bodies are projected: serves to solve various astrometric problems. The eye of the observer is taken as the center of the celestial sphere; in this case, the observer can be both on the surface of the Earth and at other points in space (for example, he can be referred to the center of the Earth). For a terrestrial observer, the rotation of the celestial sphere reproduces the daily movement of the luminaries in the sky.

Each celestial body corresponds to a point on the celestial sphere at which it is crossed by a straight line connecting the center of the sphere with the center of the luminary. When studying the positions and apparent movements of the luminaries on the celestial sphere, one or another system of spherical coordinates is chosen. Calculations of the positions of the bodies on the celestial sphere are made using celestial mechanics and spherical trigonometry and are the subject of spherical astronomy.

Story

The concept of the celestial sphere arose in ancient times; it was based on the visual impression of the existence of a domed firmament. This impression is due to the fact that, as a result of the enormous remoteness of the celestial bodies, the human eye is not able to appreciate the differences in the distances to them, and they appear to be equally distant. Among the ancient peoples, this was associated with the presence of a real sphere that bounds the whole world and carries numerous stars on its surface. Thus, in their view, the celestial sphere was the most important element of the universe. With the development of scientific knowledge, such a view of the celestial sphere fell away. However, the geometry of the celestial sphere laid down in antiquity, as a result of development and improvement, has received a modern form, in which it is used in astrometry.

Elements of the celestial sphere

Plumb line and related concepts

plumb line(or vertical line) - a straight line passing through the center of the celestial sphere and coinciding with the direction of the plumb line at the observation point. The plumb line intersects with the surface of the celestial sphere at two points - zenith over the observer's head and nadir under the feet of the observer.

True (mathematical, or astronomical) horizon- a great circle of the celestial sphere, the plane of which is perpendicular to the plumb line. The true horizon divides the surface of the celestial sphere into two hemispheres: visible hemisphere with the top at the zenith and invisible hemisphere with the top in the nadir. The true horizon does not coincide with the visible horizon due to the elevation of the observation point above the earth's surface, as well as due to the curvature of light rays in the atmosphere.

height circle, or vertical, luminaries - a large semicircle of the celestial sphere, passing through the luminary, zenith and nadir. Almuqantarat(arab. " circle of equal heights") - a small circle of the celestial sphere, the plane of which is parallel to the plane of the mathematical horizon. Altitude circles and almucantarata form a coordinate grid that sets the horizontal coordinates of the luminary.

Daily rotation of the celestial sphere and related concepts

world axis- an imaginary line passing through the center of the world, around which the celestial sphere rotates. The axis of the world intersects with the surface of the celestial sphere at two points - north pole of the world And south pole of the world. The rotation of the celestial sphere occurs counterclockwise around the north pole, when viewed from the inside of the celestial sphere.

Celestial equator- a great circle of the celestial sphere, the plane of which is perpendicular to the axis of the world and passes through the center of the celestial sphere. The celestial equator divides the celestial sphere into two hemispheres: northern And southern.

Luminary declination circle- a large circle of the celestial sphere, passing through the poles of the world and this luminary.

Daily parallel- a small circle of the celestial sphere, the plane of which is parallel to the plane of the celestial equator. The visible daily movements of the luminaries occur along daily parallels. Circles of declination and daily parallels form a coordinate grid on the celestial sphere that sets the equatorial coordinates of the star.

Terms born at the intersection of the concepts "Plumb line" and "Rotation of the celestial sphere"

The celestial equator intersects the mathematical horizon at east point And west point. The point of the east is the one in which the points of the rotating celestial sphere rise from the horizon. The height semicircle passing through the east point is called first vertical.

sky meridian- a large circle of the celestial sphere, the plane of which passes through the plumb line and the axis of the world. The celestial meridian divides the surface of the celestial sphere into two hemispheres: eastern hemisphere And western hemisphere.

noon line- the line of intersection of the plane of the celestial meridian and the plane of the mathematical horizon. The midday line and the celestial meridian cross the mathematical horizon at two points: north point And south point. The north point is the one that is closer to the north pole of the world.

Annual motion of the Sun in the celestial sphere and related concepts

Ecliptic- a large circle of the celestial sphere, along which the apparent annual movement of the Sun occurs. The plane of the ecliptic intersects with the plane of the celestial equator at an angle ε = 23°26".

The two points where the ecliptic intersects the celestial equator are called the equinoxes. IN vernal equinox point The sun in its annual movement passes from the southern hemisphere of the celestial sphere to the northern; V point of the autumnal equinox from the northern hemisphere to the southern. The line passing through these two points is called equinoxes. Two points on the ecliptic that are 90 ° away from the equinoxes and thus as far as possible from the celestial equator are called solstice points. Summer solstice point located in the northern hemisphere winter solstice point- in the southern hemisphere. These four points are denoted by the symbols of the zodiac, corresponding to

The content of the article

CELESTIAL SPHERE. When we observe the sky, all astronomical objects appear to be located on a dome-shaped surface, in the center of which the observer is located. This imaginary dome forms the upper half of an imaginary sphere, which is called the "celestial sphere". It plays a fundamental role in indicating the position of astronomical objects.

Although the Moon, the planets, the Sun, and the stars are located at different distances from us, even the closest of them are so far away that we are not able to estimate their distance by eye. The direction to the star does not change as we move across the surface of the Earth. (True, it changes slightly as the Earth moves along its orbit, but this parallactic shift can be noticed only with the help of the most accurate instruments.)

It seems to us that the celestial sphere rotates, since the luminaries rise in the east and set in the west. The reason for this is the rotation of the Earth from west to east. The apparent rotation of the celestial sphere occurs around an imaginary axis that continues the earth's axis of rotation. This axis intersects the celestial sphere at two points, called the north and south "poles of the world." The north celestial pole lies about a degree from the North Star, and there are no bright stars near the south pole.

The axis of rotation of the Earth is inclined by about 23.5 ° relative to the perpendicular drawn to the plane of the earth's orbit (to the plane of the ecliptic). The intersection of this plane with the celestial sphere gives a circle - the ecliptic, the apparent path of the Sun in a year. The orientation of the earth's axis in space almost does not change. So every year in June, when the northern end of the axis is tilted towards the Sun, it rises high in the sky in the Northern Hemisphere, where the days become long and the nights short. Having moved to the opposite side of the orbit in December, the Earth turns to the Sun with the Southern Hemisphere, and in our north the days become short and the nights long.

However, under the influence of solar and lunar attraction, the orientation of the earth's axis is still gradually changing. The main movement of the axis, caused by the influence of the Sun and Moon on the equatorial bulge of the Earth, is called precession. As a result of precession, the earth's axis slowly rotates around the perpendicular to the orbital plane, describing a cone with a radius of 23.5° in 26 thousand years. For this reason, in a few centuries the pole will no longer be near the North Star. In addition, the Earth's axis makes small fluctuations, called nutation and associated with the ellipticity of the orbits of the Earth and the Moon, as well as the fact that the plane of the lunar orbit is slightly inclined to the plane of the Earth's orbit.

As we already know, the appearance of the celestial sphere changes during the night due to the rotation of the Earth around its axis. But even if you observe the sky at the same time during the year, its appearance will change due to the rotation of the Earth around the Sun. It takes approx. 365 1/4 days - about a degree per day. By the way, a day, or rather a solar day, is the time during which the Earth rotates once around its axis with respect to the Sun. It consists of the time it takes for the Earth to complete one revolution with respect to the stars (“sidereal day”), plus a small amount of time, about four minutes, required for the rotation to compensate for the Earth’s orbital movement by one degree per day. Thus, in a year approx. 365 1/4 solar days and approx. 366 1/4 star.

When viewed from a certain point on the Earth, stars located near the poles are either always above the horizon or never rise above it. All other stars rise and set, and every day each star rises and sets 4 minutes earlier than the previous day. Some stars and constellations rise in the sky at night during winter - we call them "winter" and others - "summer".

Thus, the view of the celestial sphere is determined by three times: the time of day associated with the rotation of the Earth; time of year associated with circulation around the sun; an epoch associated with precession (although the latter effect is hardly noticeable “by eye” even in 100 years).

Coordinate systems.

There are various ways to indicate the position of objects on the celestial sphere. Each of them is suitable for tasks of a certain type.

Alt-azimuth system.

To indicate the position of an object in the sky in relation to the earthly objects surrounding the observer, an "alt-azimuth", or "horizontal" coordinate system is used. It indicates the angular distance of the object above the horizon, called "altitude", as well as its "azimuth" - the angular distance along the horizon from a conditional point to a point directly below the object. In astronomy, the azimuth is measured from a point south to west, and in geodesy and navigation, from a point north to east. Therefore, before using the azimuth, you need to find out in which system it is indicated. The point in the sky directly above the head has a height of 90 ° and is called the "zenith", and the point diametrically opposite to it (under the feet) is called the "nadir". For many tasks, a large circle of the celestial sphere, called the "celestial meridian" is important; it passes through the zenith, nadir and celestial poles, and crosses the horizon at points north and south.

equatorial system.

Due to the rotation of the Earth, the stars are constantly moving relative to the horizon and cardinal points, and their coordinates in the horizontal system change. But for some tasks of astronomy, the coordinate system must be independent of the position of the observer and the time of day. Such a system is called "equatorial"; its coordinates resemble geographic latitudes and longitudes. In it, the plane of the earth's equator, extended to the intersection with the celestial sphere, sets the main circle - the "celestial equator". The "declination" of a star resembles latitude and is measured by its angular distance north or south of the celestial equator. If the star is visible exactly at the zenith, then the latitude of the place of observation is equal to the declination of the star. Geographic longitude corresponds to the "right ascension" of the star. It is measured east of the intersection point of the ecliptic with the celestial equator, which the Sun passes in March, on the day of the beginning of spring in the Northern Hemisphere and autumn in the Southern. This point, important for astronomy, is called the "first point of Aries", or the "point of the vernal equinox", and is denoted by the sign. Right ascension values ​​are usually given in hours and minutes, considering 24 hours as 360°.

The equatorial system is used when observing with telescopes. The telescope is installed so that it can rotate from east to west around the axis directed to the celestial pole, thereby compensating for the rotation of the Earth.

other systems.

For some purposes, other coordinate systems on the celestial sphere are also used. For example, when studying the motion of bodies in the solar system, they use a coordinate system whose main plane is the plane of the earth's orbit. The structure of the Galaxy is studied in a coordinate system, the main plane of which is the equatorial plane of the Galaxy, represented in the sky by a circle passing along the Milky Way.

Comparison of coordinate systems.

The most important details of the horizontal and equatorial systems are shown in the figures. In the table, these systems are compared with the geographic coordinate system.

Transition from one system to another.

Often there is a need to calculate its equatorial coordinates from the alt-azimuth coordinates of a star, and vice versa. To do this, it is necessary to know the moment of observation and the position of the observer on Earth. Mathematically, the problem is solved using a spherical triangle with vertices at the zenith, the north celestial pole and the star X; it is called the "astronomical triangle".

The angle with a vertex at the north pole of the world between the meridian of the observer and the direction to any point of the celestial sphere is called the "hour angle" of this point; it is measured west of the meridian. The hour angle of the vernal equinox, expressed in hours, minutes and seconds, is called "sidereal time" (Si. T. - sidereal time) at the point of observation. And since the right ascension of a star is also the polar angle between the direction to it and to the vernal equinox, then sidereal time is equal to the right ascension of all points lying on the meridian of the observer.

Thus, the hour angle of any point on the celestial sphere is equal to the difference between sidereal time and its right ascension:

Let the observer's latitude be j. Given the equatorial coordinates of a star a And d, then its horizontal coordinates A And can be calculated using the following formulas:

You can also solve the inverse problem: according to the measured values A And h, knowing the time, calculate a And d. declination d is calculated directly from the last formula, then from the penultimate one is calculated H, and from the first, if sidereal time is known, then a.

Representation of the celestial sphere.

For centuries, scientists have searched for the best way to represent the celestial sphere for study or demonstration. Two types of models were proposed: two-dimensional and three-dimensional.

The celestial sphere can be depicted on a plane in the same way as the spherical Earth is depicted on maps. In both cases, a geometric projection system must be selected. The first attempt to represent sections of the celestial sphere on a plane was rock carvings of stellar configurations in the caves of ancient people. Nowadays, there are various star charts published in the form of hand-drawn or photographic star atlases covering the entire sky.

Ancient Chinese and Greek astronomers represented the celestial sphere in a model known as the "armillary sphere". It consists of metal circles or rings connected together so as to show the most important circles of the celestial sphere. Now stellar globes are often used, on which the positions of the stars and the main circles of the celestial sphere are marked. Armillary spheres and globes have a common drawback: the position of the stars and the markings of the circles are marked on their outer, convex side, which we view from the outside, while we look at the sky "from the inside", and the stars seem to us placed on the concave side of the celestial sphere. This sometimes leads to confusion in the directions of movement of stars and constellation figures.

The planetarium gives the most realistic representation of the celestial sphere. The optical projection of stars onto a hemispherical screen from the inside makes it possible to very accurately reproduce the appearance of the sky and all kinds of movements of the luminaries on it.

Laboratory work

« MAIN ELEMENTS OF THE HEAVENLY SPHERE»

Goal of the work: The study of the main elements and the daily rotation of the celestial sphere on its model.

Benefits: a model of the celestial sphere (or a celestial planisphere replacing it); black globe; mobile map of the starry sky.

Brief theoretical information:

The visible positions of the celestial bodies are determined relative to the basic elements of the celestial sphere.

The main elements of the celestial sphere (Fig. 1) include:

Zenith points Z and nadir Z" , true or mathematical horizon NESWN, world axis RR", world poles ( R- northern and R"- southern), celestial equator QWQ" EQ the celestial meridian PZSP "Z" NP and the points of intersection of the celestial meridian and the celestial equator with the true horizon, i.e. points of the south S, north N, east E and west W.

The elements of the celestial sphere can be studied on its model (Fig. 2), which consists of several rings depicting the main circles of the celestial sphere. In ring 1, representing the celestial meridian, the axis is rigidly fixed RR"- the axis of the world around which the celestial sphere rotates. endpoints R And R" this axis lie on the celestial meridian and represent, respectively, the northern ( R) and southern ( R") the poles of the world.

metal circle 8 depicts the true or mathematical horizon, which should always be set horizontally when working with a celestial sphere model. The axis of the world forms an angle with the plane of the true horizon equal to the geographic latitude at the place of observation, and when the model is set to a given geographic latitude, this angle is fixed with a screw 11 , after which the true horizon 8 is brought to a horizontal position by turning the ring 1 (celestial meridian), which is fixed in the stand 9 clamp 10 .

around the axis RR"(axis of the world) two rings fastened together rotate freely 2 And 3 whose planes are mutually perpendicular. These rings depict declination circles - large circles passing through the poles of the world. Although countless circles of declination pass through the celestial poles on the celestial sphere, only four circles of declination (in the form of two full rings) are made on the model of the celestial sphere, along which one can imagine the entire spherical surface. Attention should be paid to the fact that not a complete circle is taken as a circle of declination, but only its half, enclosed between the poles of the world. Thus, the two rings of the model depict four circles of declination of the celestial sphere, spaced from each other by 90°; they make it possible to demonstrate the equatorial coordinates of celestial bodies.

Ring 4 , whose plane is perpendicular to the axis of the world, depicts the celestial equator. at an angle to him 23°.5 attached ring 5 representing the ecliptic.

Rings depicting the celestial meridian 1 , celestial equator 4 , the ecliptic 5 , declination circles 2 And 3 and true horizon 8 , are great circles of the celestial sphere - their planes pass through the center O model in which the observer is conceived.

Perpendicular to the plane of the true horizon, raised from the center O models of the celestial sphere, crosses the celestial meridian at points called the zenith Z(above the observer's head) and nadir Z" (the nadir is under the observer's feet and is hidden from him by the earth's surface).

At the zenith, on the celestial meridian, a moving rider is being strengthened 12 , with an arc freely rotating on it 13 , whose plane also passes through the center of the celestial sphere model. Arc 13 depicts a circle of height (vertical) and allows you to demonstrate the horizontal coordinates of celestial bodies.

In addition to the large circles, two small circles are shown on the model of the celestial sphere. 6 And 7 -two celestial parallels, separated from the celestial equator by 23°.5. Other celestial parallels are not shown on the model. The planes of celestial parallels do not pass through the center of the celestial sphere, they are parallel to the plane of the celestial equator and are perpendicular to the axis of the world.

Two nozzles are attached to the model of the celestial sphere, one is in the form of a circle, the other is in the form of an asterisk. These attachments are used to depict celestial bodies and can be mounted on any circle of the celestial sphere model.

In the future, all elements of the model of the celestial sphere are referred to by the same terms that are accepted for the corresponding elements of the celestial sphere.

Due to the uniform rotation of the Earth around its axis in the direction from west to east (or counterclockwise), it seems to the observer that the celestial sphere rotates uniformly around the axis of the world RR" in the opposite direction, i.e. clockwise, if you look at it from the outside from the north celestial pole (or if the observer in the center of the sphere has his back to the north celestial pole, and his face to the south). The celestial sphere makes one revolution per day; this apparent rotation is called diurnal. The direction of the daily rotation of the celestial sphere is shown in fig. 1 arrow.

On the model of the celestial sphere, one can clearly understand that although the celestial sphere rotates as a whole, most of its main elements do not participate in the daily rotation of the sphere, remaining motionless relative to the observer. The celestial equator rotates in its plane along with the celestial sphere, sliding in the fixed points of east E and west W. In the process of daily rotation, all points of the celestial sphere (except for fixed points) cross the celestial meridian twice a day, once its southern half (south of the north celestial pole, arc RZSR"), another time - its northern half (north of the north pole of the world, arc RNZ" P" ). These passages of points through the celestial meridian are called, respectively, the upper and lower climaxes. Through the zenith Z and nadir Z" not all pass, but only certain points of the celestial sphere, the declination δ of which (as will be seen later) is equal to the geographical latitude φ of the observer's place (δ = φ). Points of the celestial sphere above the true horizon are visible to the observer; the hemisphere under the true horizon is inaccessible to observations (in Fig. 1 it is indicated by vertical shading).

Arc NES the true horizon, above which the points of the celestial sphere rise, is called its eastern half and extends 180º from the north point N, through the east point E, to the point south S. Opposite, western half SWN the true horizon, beyond which the points of the celestial sphere go, also contains 180º and is also limited by the points of the south S and north N, but passes through the west point W. The eastern and western halves of the true horizon should not be confused with its sides, which are determined by its main points - the points of the east, south, west and north.

Particular attention should be paid to the fact that the celestial sphere is divided into northern and southern hemispheres by the celestial equator, and not by the true horizon, above which there are always regions of both hemispheres, both northern and southern. The size of these areas depends on the geographical latitude at the place of observation: the closer to the north pole of the Earth is the place of observation (the greater its φ), the smaller the area of ​​the southern celestial hemisphere is available for observations, and the larger the area of ​​the northern celestial hemisphere is simultaneously visible above the true horizon (and the southern hemisphere of the Earth - on the contrary).

The duration of stay of the points of the celestial sphere during the day above the true horizon (and below it) depends on the ratio of the declination δ of these points with the geographical latitude φ of the place of observation, and for a certain φ, only on their declination δ. Since the celestial equator and the true horizon intersect at diametrically opposite points, then any point of the celestial equator (δ = 0°) is always half a day above the true horizon and half a day below it, regardless of the geographical latitude at the place of observation (except for the geographic poles of the Earth, φ = ± 90°).

To study the main elements of the celestial sphere, in the absence of a model, you can use the celestial planisphere (tablet 10), which, of course, is not as clear as a spatial model, but still can give a correct idea of ​​the main elements and the daily rotation of the celestial sphere. The planisphere is an orthogonal (rectangular) projection of the celestial sphere onto the plane of the celestial meridian and consists of a circle SZNZ" , depicting the celestial meridian, through the center ABOUT which a plumb line is drawn ZZ" and the trace of the true horizon plane NS. points of the east E and west W are projected to the center of the planisphere. Degree divisions on the celestial meridian give height h almucantarats (small circles parallel to the true horizon), which above the true horizon is considered positive (h > 0°), and below it - negative (h< 0°).

world axis RR", celestial equator QQ" and the celestial parallels are shown in the same projection on a tracing paper, on which the two positions of the ecliptic are also shown in dotted lines, corresponding to its highest ξξ") and lowest (ξоξо") position above the true horizon. Degree digitization on the tracing paper gives the angular distance of the celestial parallels from the celestial equator, i.e. their declination δ, considered positive in the northern celestial hemisphere (δ > 0°), and negative in the southern celestial hemisphere (δ< 0°).

Putting a tracing paper symmetrically on the circle of the celestial meridian and turning it around a common center ABOUT at a certain angle of 90 ° - φ, we will get a view of the celestial sphere (in projection onto the plane of the celestial meridian) at the geographical latitude φ. Then the location of the elements of the celestial sphere relative to the true horizon will immediately become clear. NS and with respect to the observer at the center ABOUT celestial sphere. The direction of the daily rotation of the celestial sphere around the axis of the world has to be depicted by arrows along the celestial equator and celestial parallels.

It is very useful to imagine the correspondence of the elements of the celestial sphere to the points and circles of the earth's surface. To illustrate this correspondence, it is best to represent the radius of the celestial sphere as large as desired, but not infinite, since in the case of an infinitely large radius, parts of the sphere degenerate into a plane. For an arbitrarily large radius of the celestial sphere, the observer ABOUT, located at some point on the earth's surface, sees the celestial sphere in the same way as from the center of the Earth WITH(Fig. 3), but with the same direction to the zenith Z. Then it becomes clear that the plumb line oz is an extension of the earth's radius SO in the place of observation (the Earth is taken as a ball), the axis of the world RR" identical to the earth's axis of rotation rr", poles of the world R And R" correspond to the geographic poles of the Earth R And R", celestial equator QQ" formed on the celestial sphere by the plane of the earth's equator qq" , and the celestial meridian RZR"Z"R formed on the celestial sphere by the plane of the earth's meridian roqR"q" p on which the observer is located ABOUT. The plane of the true horizon is tangent to the surface of the Earth at the point of observation ABOUT. This explains the immobility of the celestial meridian, zenith, nadir and the true horizon relative to the observer, which rotate with him around the earth's axis. Poles of the world R And R" are also motionless relative to the observer, since they lie on the earth's axis, which does not participate in the daily rotation of the earth. Any terrestrial parallel kO with geographical latitude a corresponds to the celestial parallel TOZ. with declination and δ = φ. Therefore, the points of this celestial parallel pass through the zenith of the observation site ABOUT.

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Name

Position relative to the observer

Location relative to true horizon

3. On the globe can be depicted:

4. The movable map shows:

7. Matching dots and circles:

The drawing is attached.

8. Three drawings are attached.

Auxiliary celestial sphere

Coordinate systems used in geodetic astronomy

Geographic latitudes and longitudes of points on the earth's surface and azimuths of directions are determined from observations of celestial bodies - the Sun and stars. To do this, it is necessary to know the position of the luminaries both relative to the Earth and relative to each other. The positions of the luminaries can be set in expediently 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 star s is determined by three linear coordinates X, Y, Z. In the polar coordinate system, the position of the star s is given by one linear coordinate, the radius vector R = Оs and two angular ones: the angle a between the X axis and the projection of the radius vector onto the XOY coordinate plane, and the angle b between the XOY coordinate plane and the 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,

where R= .

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 of the Earth). However, for many luminaries observed outside the 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 to do without distances to the luminaries, it is believed that all the luminaries are at an arbitrary, but the same distance from the observer. Usually, this distance is taken equal to one, 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 centered 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 body 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 the celestial sphere - the center is aligned with the center of the Sun;

4) barycentric celestial sphere - the center is located in 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 point of observation. This direction intersects the celestial sphere at two diametrically opposite points - Z and Z. The Z point is above the center and is called zenith, Z" - under the center and is called nadir.

Draw through the center a plane 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. It has four points S, W, N, E, where S is south point,N- north point, W - point of the West, E- point of the East. The straight line 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 of the world. Points P N - north pole of the world; P S - south pole of the world. Around the axis of the World there is a visible daily movement of the celestial sphere.

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 of the celestial sphere, is called celestial (astronomical) equator. Here Q is the highest point of the equator(above the horizon), Q "- the lowest point of the equator(under 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 the equator. It is called the initial coordinate plane.

Draw through ZZ "a vertical plane 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 vertically or around 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 star parallel to the celestial equator, is called daily parallel. The visible daily movement of the luminaries occurs along the daily parallels.

The small circle asa "passing through the luminary parallel to the celestial horizon is called circle of equal heights, or almucantarat.

In the first approximation, the Earth's orbit can be taken as a flat curve - an ellipse, in one of the foci of which is the Sun. The plane of the ellipse taken as the orbit of the Earth , called a plane ecliptic.

In spherical astronomy, it is customary to talk about apparent annual motion of the sun. The great circle ЕgЕ "d, along which the apparent 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 . On fig. 4 shown:

g is the vernal equinox point;

d is the point of the autumnal equinox;

E is the point of the summer solstice; E" - the point of the winter solstice; R N R S - the axis of the ecliptic; R N - the north pole of the ecliptic; R S - the south pole of the ecliptic; e - the inclination of the ecliptic to the equator.

The celestial sphere is an imaginary sphere of arbitrary radius used in astronomy to describe the relative positions of the stars in the sky. For simplicity of calculations, its radius is taken equal to unity; the center of the celestial sphere, depending on the problem being solved, is combined with the pupil of the observer, with the center of the Earth, the Moon, the Sun, or in general with an arbitrary point in space.

The concept of the celestial sphere arose in ancient times. It was based on the visual impression of the existence of a crystal dome of the sky, on which the stars seemed to be fixed. The celestial sphere in the view of the ancient peoples was the most important element of the universe. With the development of astronomy, such a view of the celestial sphere fell away. However, the geometry of the celestial sphere, laid down in antiquity, as a result of development and improvement, has received a modern form, in which, for the convenience of various calculations, it is used in astrometry.

Let us consider the celestial sphere as it appears to the Observer at mid-latitudes from the Earth's surface (Fig. 1).

Two straight lines, the position of which can be established experimentally with the help of physical and astronomical instruments, play an important role in defining concepts related to the celestial sphere. The first of them is a plumb line; is a straight line coinciding at a given point with the direction of gravity. This line, drawn through the center of the celestial sphere, crosses it at two diametrically opposite points: the upper one is called the zenith, the lower one is called the nadir. The plane passing through the center of the celestial sphere perpendicular to the plumb line is called the plane of the mathematical (or true) horizon. The line of intersection of this plane with the celestial sphere is called the horizon.

The second straight line is the axis of the world - a straight line passing through the center of the celestial sphere parallel to the axis of rotation of the Earth; around the axis of the world there is a visible daily rotation of the entire sky. The points of intersection of the axis of the world with the celestial sphere are called the North and South Poles of the world. The most conspicuous of the stars near the North Pole of the world is the North Star. There are no bright stars near the South Pole of the World.

The plane passing through the center of the celestial sphere perpendicular to the axis of the world is called the plane of the celestial equator. The line of intersection of this plane with the celestial sphere is called the celestial equator.

Recall that the circle that is obtained when the celestial sphere intersects with a plane passing through its center is called in mathematics a large circle, and if the plane does not pass through the center, then a small circle is obtained. The horizon and the celestial equator are great circles of the celestial sphere and divide it into two equal hemispheres. The horizon divides the celestial sphere into visible and invisible hemispheres. The celestial equator divides it into the northern and southern hemispheres, respectively.

With the daily rotation of the firmament, the luminaries rotate around the axis of the world, describing small circles on the celestial sphere, called daily parallels; the luminaries, 90 ° removed from the poles of the world, move along the great circle of the celestial sphere - the celestial equator.

Having defined the plumb line and the axis of the world, it is not difficult to define all other planes and circles of the celestial sphere.

The plane passing through the center of the celestial sphere, in which both the plumb line and the axis of the world lie simultaneously, is called the plane of the celestial meridian. The great circle from the intersection of this plane of the celestial sphere is called the celestial meridian. That of the points of intersection of the celestial meridian with the horizon, which is closer to the North Pole of the world, is called the north point; diametrically opposite - the point of the south. The line passing through these points is the noon line.

Points on the horizon that are 90° from north and south are called east and west. These four points are called the principal points of the horizon.

Planes passing through a plumb line cross the celestial sphere in large circles and are called verticals. The celestial meridian is one of the verticals. The vertical perpendicular to the meridian and passing through the points of east and west is called the first vertical.

By definition, the three main planes - the mathematical horizon, the celestial meridian and the first vertical - are mutually perpendicular. The plane of the celestial equator is perpendicular only to the plane of the celestial meridian, forming a dihedral angle with the plane of the horizon. At the geographic poles of the Earth, the plane of the celestial equator coincides with the plane of the horizon, and at the equator of the Earth it becomes perpendicular to it. In the first case, at the geographic poles of the Earth, the axis of the world coincides with a plumb line, and any of the verticals can be taken as the celestial meridian, depending on the conditions of the task at hand. In the second case, at the equator, the axis of the world lies in the plane of the horizon and coincides with the midday line; In this case, the North Pole of the World coincides with the point of the north, and the South Pole of the World coincides with the point of the south (see Fig.).

When using the celestial sphere, the center of which is aligned with the center of the Earth or some other point in space, a number of features also arise, but the principle of introducing the basic concepts - the horizon, the celestial meridian, the first vertical, the celestial equator, etc. - remains the same.

The main planes and circles of the celestial sphere are used in the introduction of horizontal, equatorial and ecliptic celestial coordinates, as well as in describing the features of the visible daily rotation of the stars.

The great circle formed by the intersection of the celestial sphere with a plane passing through its center and parallel to the plane of the earth's orbit is called the ecliptic. The apparent annual movement of the Sun occurs along the ecliptic. The point of intersection of the ecliptic with the celestial equator, at which the Sun passes from the southern hemisphere of the celestial sphere to the northern one, is called the vernal equinox. The opposite point of the celestial sphere is called the autumnal equinox. A straight line passing through the center of the celestial sphere perpendicular to the plane of the ecliptic intersects the sphere at two ecliptic poles: the North Pole in the Northern Hemisphere and the South Pole in the Southern Hemisphere.