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 a 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 ones: the angle a between the X axis and the projection of the radius vector onto the coordinate plane XOY, and the angle b between the coordinate plane XOY 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 Earth satellites). However, for many luminaries observed outside the solar system, these distances are either extremely large compared to the radius of the Earth or are 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.

§ 48. Celestial sphere. Basic points, lines and circles on the celestial sphere

Let us consider the main points and circles of the celestial sphere, the center of which is taken to be the eye of the observer (Fig. 72). Let's draw a plumb line through the center of the celestial sphere. The points of intersection of the plumb line with the sphere are called zenith Z and nadir n.

Rice. 72.

The straight line passing through the center of the celestial sphere parallel to the earth's axis is called the mundi axis. The points of intersection of the axis of the world with the celestial sphere are called poles of the world. One of the poles, corresponding to the poles of the Earth, is called the north celestial pole and is designated Pn, the other is the south celestial pole Ps.

The QQ plane passing through the center of the celestial sphere perpendicular to the axis of the world is called plane of the celestial equator. This plane, intersecting with the celestial sphere, forms a great circle - celestial equator, which divides the celestial sphere into northern and southern parts.

The great circle of the celestial sphere passing through the celestial poles, zenith and nadir, is called observer's meridian PN nPsZ. The mundi axis divides the observer's meridian into the midday PN ZPs and midnight PN nPs parts.

The observer's meridian intersects with the true horizon at two points: the north point N and the south point S. The straight line connecting the points of north and south is called midday line.

If you look from the center of the sphere to point N, then on the right there will be a point of east O st, and on the left - a point of west W. Small circles of the celestial sphere aa", parallel to the plane of the true horizon, are called almucantarates; small bb" parallel to the plane of the celestial equator, - heavenly parallels.

The circles of the celestial sphere Zon passing through the zenith and nadir points are called verticals. The vertical line passing through the points of east and west is called the first vertical.

The circles of the celestial sphere of PNoPs passing through the poles of the world are called declination circles.

The observer's meridian is both a vertical and a circle of declination. It divides the celestial sphere into two parts - eastern and western.

The celestial pole located above the horizon (below the horizon) is called the elevated (lowered) celestial pole. The name of the elevated celestial pole is always the same as the name of the latitude of the place.

The axis of the world makes an angle with the plane of the true horizon equal to geographical latitude of the place.

The position of luminaries on the celestial sphere is determined using spherical coordinate systems. In nautical astronomy, horizontal and equatorial coordinate systems are used.

Determined by their coordinates on the celestial sphere. The equivalents of latitude and longitude on the celestial sphere (in the second equatorial coordinate system) are called declination (measured in degrees from +90? to -90?) and direct elevation (measured in hours from 0 to 24). The celestial poles lie above the Earth's poles, and the celestial equator lies above the Earth's equator. To an observer on earth, it appears as if the celestial sphere is revolving around the Earth. In fact, the imaginary movement of the celestial sphere is caused by the rotation of the Earth around its axis.

1. History of the concept

The idea of ​​the celestial sphere arose in ancient times; it was based on the impression of the existence of a domed sky. 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 bounds the entire world and carries on its surface the stars, the Moon and the Sun. Thus, in their view, the celestial sphere was the most important element of the Universe. With the development of scientific knowledge, this view of the celestial sphere disappeared. However, the geometry of the celestial sphere, laid down in ancient times, as a result of development and improvement, received a modern form, in which it is used in astrometry.

• at the place on the Earth's surface where the observer is located (the celestial sphere is topocentric),
• at the center of the Earth (geocentric celestial sphere),
• in the center of a particular planet (planetocentric celestial sphere),
• at the center of the Sun (heliocentric celestial sphere)
• at any other point in space where the observer (real or hypothetical) is located.

Each luminary on the celestial sphere corresponds to a point at which it is intersected by a straight line connecting the center of the celestial sphere with the luminary (or with the center of the luminary, if it is large and not a point). To study the relative position and visible movements of luminaries on the celestial sphere, choose one or another system of celestial coordinates, which is determined by the main points and lines. The latter are usually large circles of the celestial sphere. Each great circle of a sphere has two poles, which are defined on it by the ends of a diameter perpendicular to the plane of this circle.

2. Names of the most important points and arcs on the celestial sphere

2.1. Plumb line

A plumb line (or vertical line) is a straight line passing through the center of the celestial sphere and coincides with the direction of the plumb line (vertical) at the observation location. For an observer on the Earth's surface, a plumb line passes through the center of the Earth and the observation point.

2.2. Zenith and nadir

The plumb line intersects the surface of the celestial sphere at two points - zenith, above the observer's head, and nadir, diametrically opposite the point.

2.3. Mathematical horizon

The mathematical horizon is a great circle of the celestial sphere, the plane of which is perpendicular to the plumb line. The mathematical horizon divides the surface of the celestial sphere into two halves: visible to the observer, with the apex at the zenith, and invisible, with the apex at the nadir. The mathematical horizon, generally speaking, does not coincide with the visible horizon due to the unevenness of the Earth's surface and different heights of observation points, as well as the bending of light rays in the atmosphere.

2.4. axis mundi

The mundi axis is the diameter around which the celestial sphere rotates.

2.5. Poles of the world

The mundi axis intersects with the surface of the celestial sphere at two points - the north celestial pole and the south celestial pole. The north pole is the one from which the celestial sphere rotates clockwise when looking at the sphere from the outside. If you look at the celestial sphere from the inside (which is what we usually do when observing the starry sky), then in the vicinity of the north pole of the world its rotation occurs counterclockwise, and in the vicinity of the south pole of the world it rotates clockwise.

2.6. Celestial equator

The celestial equator is a great circle of the celestial sphere, the plane of which is perpendicular to the axis of the world. It is a projection of the earth's equator onto the celestial sphere. The celestial equator divides the surface of the celestial sphere into two hemispheres: the northern hemisphere, with its apex at the north celestial pole, and the southern hemisphere, with its apex at the south celestial pole.

2.7. Sunrise and sunset points

The celestial equator intersects with the mathematical horizon at two points: the east point and the west point. The vanishing point is the one from which the point of the celestial sphere, due to its rotation, crosses the mathematical horizon, passing from the invisible hemisphere to the visible one.

2.8. Celestial meridian

The celestial meridian is 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 - the eastern hemisphere, with its apex at the point of the east, and the western hemisphere, with its apex at the point of the west.

2.9. Noon Line

The noon line is the line of intersection of the plane of the celestial meridian and the plane of the mathematical horizon.

2.10. North and south points

The celestial meridian intersects the mathematical horizon at two points: the north point and the south point. The north point is the one that is closer to the north pole of the world.

2.11. Ecliptic

The ecliptic is the great circle of the celestial sphere, the intersection of the celestial sphere and the plane of the earth's orbit. The ecliptic carries out the visible annual movement of the Sun across the celestial sphere. The plane of the ecliptic intersects with the plane of the celestial equator at an angle ε = 23? 26".

2.12. Equinox points

The ecliptic intersects with the celestial equator at two points - the vernal equinox and the autumn equinox. The vernal equinox point is the point at which the Sun, in its annual movement, passes from the southern hemisphere of the celestial sphere to the northern. At the point of the autumn equinox, the Sun moves from the northern hemisphere of the celestial sphere to the southern.

2.13. Solstice points

Points of the ecliptic separated from the equinox points by 90? are called the summer solstice point (in the northern hemisphere) and the winter solstice point (in the southern hemisphere).

2.14. Ecliptic axis

The ecliptic axis is the diameter of the celestial sphere perpendicular to the ecliptic plane.

2.15. Poles of the ecliptic

The ecliptic axis intersects with the surface of the celestial sphere at two points - the north pole of the ecliptic, which lies in the northern hemisphere, and the south pole of the ecliptic, which lies in the southern hemisphere.

2.16. Galactic poles and galactic equator

A point on the celestial sphere with equatorial coordinates α = 192.85948? β = 27.12825 ? is called the north galactic pole, and the point diametrically opposite to it is called the south galactic pole. The great circle of the celestial sphere, the plane of which is perpendicular to the line connecting the galactic poles, is called the galactic equator.

3. The names of arcs on the celestial sphere associated with the position of the luminaries

3.1. Almucantarat

Almucantarat - Arabic. circle of equal heights. Almucantarat of a luminary is a small circle of the celestial sphere passing through the luminary, the plane of which is parallel to the plane of the mathematical horizon.

3.2. Vertical circle

The circle of altitude or vertical circle or vertical of the luminary is a large semicircle of the celestial sphere, passing through the zenith, luminary and nadir.

3.3. Daily parallel

The daily parallel of a luminary is a small circle of the celestial sphere passing through the luminary, the plane of which is parallel to the plane of the celestial equator. The visible daily movements of the luminaries occur along daily parallels.

3.4. Tilt circle

The circle of inclination of the luminary is a large semicircle of the celestial sphere, passing through the poles of the world and the luminary.

3.5. Circle Ecliptic latitudes

The circle of Ecliptic latitudes, or simply the circle of latitude of a luminary, is a large semicircle of the celestial sphere, passing through the poles of the ecliptic and the luminary.

3.6. Circle of galactic latitude

The circle of the galactic latitude of a luminary is a large semicircle of the celestial sphere, passing through the galactic poles and the luminary.

TEST . Celestial sphere (Gomulina N.N.)

1. The celestial sphere is:
A) an imaginary sphere of infinitely large radius, described around the center of the Galaxy;
B) a crystal sphere on which, according to the ancient Greeks, luminaries are attached;
C) an imaginary sphere of arbitrary radius, the center of which is the observer’s eye.
D) an imaginary sphere - the conditional border of our Galaxy.

2. Celestial sphere:
A) motionless, the Sun, Earth, other planets and their satellites move on its inner surface;
B) rotates around an axis passing through the center of the Sun, the period of rotation of the celestial sphere is equal to the period of revolution of the Earth around the Sun, i.e. one year;
B) rotates around the earth's axis with a period equal to the period of the earth's rotation around its axis, i.e. one day;
D) rotates around the center of the Galaxy, the period of rotation of the celestial sphere is equal to the period of rotation of the Sun around the center of the Galaxy.

3. The reason for the daily rotation of the celestial sphere is:
A) Proper motion of stars;
B) Rotation of the Earth around its axis;
B) The movement of the Earth around the Sun;
D) The movement of the Sun around the center of the Galaxy.

4. Center of the celestial sphere:
A) coincides with the eye of the observer;
B) coincides with the center of the Solar system;
B) coincides with the center of the Earth;
D) coincides with the center of the Galaxy.

5. The North Pole of the world at present:
A) coincides with the North Star;
B) is located 1°.5 from a Ursa Minor;
C) is located near the brightest star in the entire sky - Sirius;
D) is located in the constellation Lyra near the star Vega.

6. The constellation Ursa Major makes a full revolution around the North Star in a time equal to
A) one night;
B) one day;
B) one month;
D) one year.

7. The axis of the world is:
A) a line passing through the zenith Z and nadir Z" and passing through the eye of the observer;
B) a line connecting the points south S and north N and passing through the observer’s eye;
B) a line connecting points east E and west W and passing through the observer's eye;
D) A line connecting the poles of the world P and P" and passing through the eye of the observer.

8. The poles of the world are the points:
A) points north N and south S.
B) points of east E and west W.
C) the points of intersection of the axis of the world with the celestial sphere P and P";
D) the north and south poles of the Earth.

9. The zenith point is called:

10. The nadir point is called:
A) the point of intersection of the celestial sphere with a plumb line located above the horizon;
B) the point of intersection of the celestial sphere with a plumb line, located below the horizon;
C) the point of intersection of the celestial sphere with the axis of the world, located in the northern hemisphere;
D) the point of intersection of the celestial sphere with the axis of the world, located in the southern hemisphere.

11. The celestial meridian is called:
A) a plane passing through the noon line NS;
B) a plane perpendicular to the world axis P and P";
B) a plane perpendicular to the plumb line passing through the zenith Z and nadir Z";
D) a plane passing through the north point N, the world poles P and P, the zenith Z, the south point S.

12. The noon line is called:
A) a line connecting points east E and west W;
B) a line connecting points south S and north N;
B) a line connecting the points of the celestial pole P and the celestial poles P";
D) a line connecting the points of zenith Z and nadir Z".

13. The visible paths of stars when moving across the sky are parallel
A) the celestial equator;
B) celestial meridian;
B) ecliptic;
D) horizon.

14. The upper climax is:
A) the position of the luminary in which the height above the horizon is minimal;
B) the passage of the luminary through the zenith point Z;
C) the passage of the luminary through the celestial meridian and reaching its greatest height above the horizon;
D) the passage of a star at an altitude equal to the geographic latitude of the observation site.

15. In the equatorial coordinate system, the main plane and the main point are:
A) the plane of the celestial equator and the vernal equinox point g;
B) horizon plane and south point S;
B) meridian plane and south point S;
D) the plane of the ecliptic and the point of intersection of the ecliptic and the celestial equator.

16. Equatorial coordinates are:
A) declination and right ascension;
B) zenith distance and azimuth;
B) altitude and azimuth;
D) zenith distance and right ascension.

17. The angle between the axis of the world and the earth’s axis is equal to: A) 66°.5; B) 0°; B) 90°; D) 23°.5.

18. The angle between the plane of the celestial equator and the axis of the world is equal to: A) 66°.5; B) 0°; B) 90°; D) 23°.5.

19. The angle of inclination of the earth’s axis to the plane of the earth’s orbit is equal to: A) 66°.5; B) 0°; B) 90°; D) 23°.5.

20. In what place on Earth does the daily movement of stars occur parallel to the horizon plane?
A) at the equator;
B) at mid-latitudes of the Earth’s northern hemisphere;
B) at the poles;
D) at mid-latitudes of the Earth's southern hemisphere.

21. Where would you look for the North Star if you were at the equator?
A) at the zenith point;

B) on the horizon;

22. Where would you look for the North Star if you were at the north pole?
A) at the zenith point;
B) at a height of 45° above the horizon;
B) on the horizon;
D) at an altitude equal to the geographic latitude of the observation site.

23. A constellation is called:
A) a certain figure of stars into which the stars are conventionally united;
B) a section of sky with established boundaries;
C) the volume of a cone (with a complex surface) extending to infinity, the apex of which coincides with the observer’s eye;
D) lines connecting stars.

24. If the stars in our Galaxy move in different directions, and the relative speed of the stars reaches hundreds of kilometers per second, then we should expect that the outlines of the constellations change noticeably:
A) within one year;
B) for a time equal to the average duration of human life;
B) for centuries;
D) for thousands of years.

25. There are a total of constellations in the sky: A) 150; B)88; B)380; D)118.

Celestial sphere is an imaginary sphere of arbitrary radius with a center at an arbitrary point, on the surface of which the positions of the luminaries are plotted as they are visible in the sky at some point in time from a given point.

The celestial sphere rotates. It is not difficult to verify this simply by observing the change in the position of the celestial bodies relative to the observer or the horizon. If you point the camera at the Ursa Minor star and open the lens for several hours, the images of the stars on the photographic plate will describe arcs, the central angles of which are the same (Fig. 17). Material from the site

Due to the rotation of the celestial sphere, each luminary moves in a small circle, the plane of which is parallel to the plane of the equator - daily parallel. As can be seen from Figure 18, the daily parallel may intersect the mathematical horizon, but may not intersect it. The intersection of the horizon by a luminary is called sunrise, if it passes into the upper part of the celestial sphere, and sets when the luminary passes into the lower part of the celestial sphere. In the event that the daily parallel along which the luminary moves does not cross the horizon, the luminary is called non-ascending or non-visitors depending on where it is located: always in the upper or always in the lower part of the celestial sphere.