Laboratory work “Basic elements of the celestial sphere. Celestial sphere

Topic 4. HEAVENLY SPHERE. ASTRONOMICAL COORDINATE SYSTEMS

4.1. CELESTIAL SPHERE

Celestial sphere - an imaginary sphere of arbitrary radius onto which the celestial bodies are projected. Serves to solve various astrometric problems. The eye of the observer is usually taken to be the center of the celestial sphere. For an observer on the Earth's surface, the rotation of the celestial sphere reproduces the daily movement of the luminaries in the sky.

The idea of ​​the Celestial Sphere 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, 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.

The radius of the celestial sphere can be taken in any way: in order to simplify geometric relationships, it is assumed to be equal to unity. Depending on the problem being solved, the center of the celestial sphere can be placed in the place:

where is the observer located (topocentric celestial sphere),

to the center of the Earth (geocentric celestial sphere),

to the center of a particular planet (planetocentric celestial sphere),

to the center of the Sun (heliocentric celestial sphere) or to any other point in space.

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 (with its center). When studying the relative positions and visible movements of luminaries on the celestial sphere, one or another coordinate system is chosen, 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, defined on it by the ends of a diameter perpendicular to the plane of the given circle.

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

Plumb line (or vertical line) - a straight line passing through the centers of the Earth and the celestial sphere. A plumb line intersects the surface of the celestial sphere at two points - zenith , above the observer's head, and nadir – diametrically opposite point.

Mathematical horizon - a great circle of the celestial sphere, the plane of which is perpendicular to the plumb line. The plane of the mathematical horizon passes through the center of the celestial sphere and divides its surface into two halves: visible for the observer, with the vertex at the zenith, and invisible, with the top at nadir. The mathematical horizon may 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.

Rice. 4.1. Celestial sphere

axis mundi – the axis of apparent rotation of the celestial sphere, parallel to the Earth’s axis.

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 .

Celestial pole - a point on the celestial sphere around which the visible daily movement of stars occurs due to the rotation of the Earth around its axis. The North Pole of the world is located in the constellation Ursa Minor, southern in the constellation Octant. As a result precession The world's poles shift about 20" per year.

The height of the celestial pole is equal to the latitude of the observer. The celestial pole located in the above-horizon part of the sphere is called elevated, while the other celestial pole located in the subhorizon part of the sphere is called low.

Celestial equator - a great circle of the celestial sphere, the plane of which is perpendicular to the axis of the world. The celestial equator divides the surface of the celestial sphere into two hemispheres: northern hemisphere , with its summit at the north celestial pole, and Southern Hemisphere , with its peak at the south celestial pole.

The celestial equator intersects the mathematical horizon at two points: point east And point west . The eastern point is the one at which the points of the rotating celestial sphere intersect the mathematical horizon, passing from the invisible hemisphere to the visible one.

Celestial meridian - a great 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 , with its apex at the east point, and western hemisphere , with its apex at the point west.

Noon Line – the line of intersection of the plane of the celestial meridian and the plane of the mathematical horizon.

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

Ecliptic – the trajectory of the apparent 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".

The ecliptic intersects the celestial equator at two points - spring And autumn equinox . At the point of the vernal equinox, the Sun moves from the southern hemisphere of the celestial sphere to the northern, at the point of the autumn equinox - from the northern hemisphere of the celestial sphere to the southern.

Points of the ecliptic that are 90° from the equinoxes are called dot summer solstice (in the northern hemisphere) and dot winter solstice (in the southern hemisphere).

Axis ecliptic – diameter of the celestial sphere perpendicular to the ecliptic plane.

4.2. Main lines and planes of the celestial sphere

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

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

Height circle or vertical circle or vertical luminaries - a large semicircle of the celestial sphere passing through the zenith, luminary and nadir.

Daily parallel luminary - 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.

Circle declination luminaries - a large semicircle of the celestial sphere, passing through the poles of the world and the luminary.

Circle ecliptic latitude , or simply the circle of latitude of the luminary - a large semicircle of the celestial sphere, passing through the poles of the ecliptic and the luminary.

Circle galactic latitude luminaries - a large semicircle of the celestial sphere passing through the galactic poles and luminaries.

2. ASTRONOMICAL COORDINATE SYSTEMS

The celestial coordinate system is used in astronomy to describe the position of luminaries in the sky or points on an imaginary celestial sphere. The coordinates of luminaries or points are specified by two angular values ​​(or arcs), which uniquely determine the position of objects on the celestial sphere. Thus, the celestial coordinate system is a spherical coordinate system in which the third coordinate - distance - is often unknown and does not play a role.

Celestial coordinate systems differ from each other in the choice of the main plane. Depending on the task at hand, it may be more convenient to use one or another system. The most commonly used are horizontal and equatorial coordinate systems. Less often - ecliptic, galactic and others.

Horizontal coordinate system

The horizontal coordinate system (horizontal) is a system of celestial coordinates in which the main plane is the plane of the mathematical horizon, and the poles are zenith and nadir. It is used when observing stars and the movement of celestial bodies of the Solar System on the ground with the naked eye, through binoculars or a telescope. The horizontal coordinates of the planets, the Sun and stars continuously change during the day due to the daily rotation of the celestial sphere.

Lines and planes

The horizontal coordinate system is always topocentric. The observer is always located at a fixed point on the surface of the earth (marked with the letter O in the figure). We will assume that the observer is located in the Northern Hemisphere of the Earth at latitude φ. Using a plumb line, the direction to the zenith (Z) is determined as the top point to which the plumb line is directed, and the nadir (Z") is determined as the bottom (under the Earth). Therefore, the line (ZZ") connecting the zenith and nadir is called a plumb line.

4.3. Horizontal coordinate system

The plane perpendicular to the plumb line at point O is called the plane of the mathematical horizon. On this plane, the direction to the south (geographic) and north is determined, for example, in the direction of the shortest shadow of the gnomon during the day. It will be shortest at true noon, and the line (NS) connecting south to north is called the noon line. The points of east (E) and west (W) are taken to be 90 degrees from the point of south, respectively, counterclockwise and clockwise, as viewed from the zenith. Thus, NESW is the plane of the mathematical horizon

The plane passing through the noon and plumb lines (ZNZ"S) is called plane of the celestial meridian , and the plane passing through the celestial body is vertical plane of a given celestial body . The great circle in which it crosses the celestial sphere, called the vertical of the celestial body .

In a horizontal coordinate system, one coordinate is either height of the luminary h, or his zenith distance z. The other coordinate is azimuth A.

Height h of the luminary is called the arc of the vertical of the luminary from the plane of the mathematical horizon to the direction towards the luminary. Heights are measured from 0° to +90° to the zenith and from 0° to −90° to the nadir.

Zenith distance z of the luminary is called the arc of the vertical of the luminary from the zenith to the luminary. Zenith distances are measured from 0° to 180° from zenith to nadir.

Azimuth A of the luminary is called the arc of the mathematical horizon from the point south to the vertical of the luminary. Azimuths are measured in the direction of the daily rotation of the celestial sphere, that is, to the west of the south point, ranging from 0° to 360°. Sometimes azimuths are measured from 0° to +180° west and from 0° to −180° east (in geodesy, azimuths are measured from the north point).

Features of changes in the coordinates of celestial bodies

During the day, the star describes a circle perpendicular to the axis of the world (PP"), which at latitude φ is inclined to the mathematical horizon at an angle φ. Therefore, it will move parallel to the mathematical horizon only at φ equal to 90 degrees, that is, at the North Pole. Therefore, all stars, visible there will be non-setting (including the Sun for six months, see the length of the day) and their height h will be constant.At other latitudes, the stars available for observation at a given time of year are divided into:

descending and ascending (h passes through 0 during the day)

non-coming (h is always greater than 0)

non-ascending (h is always less than 0)

The maximum height h of the star will be observed once a day during one of its two passages through the celestial meridian - the upper culmination, and the minimum - during the second of them - the lower culmination. From the lower to the upper culmination, the height h of the star increases, from the upper to the lower it decreases.

First equatorial coordinate system

In this system, the main plane is the plane of the celestial equator. One coordinate in this case is the declination δ (more rarely, the polar distance p). Another coordinate is the hour angle t.

The declination δ of a luminary is the arc of the circle of declination from the celestial equator to the luminary, or the angle between the plane of the celestial equator and the direction to the luminary. Declinations are measured from 0° to +90° to the north celestial pole and from 0° to −90° to the south celestial pole.

4.4. Equatorial coordinate system

The polar distance p of a luminary is the arc of the circle of declination from the north celestial pole to the luminary, or the angle between the axis of the world and the direction to the luminary. Polar distances are measured from 0° to 180° from the north celestial pole to the south.

The hour angle t of a luminary is the arc of the celestial equator from the highest point of the celestial equator (that is, the point of intersection of the celestial equator with the celestial meridian) to the circle of declination of the luminary, or the dihedral angle between the planes of the celestial meridian and the circle of declination of the luminary. Hour angles are counted in the direction of the daily rotation of the celestial sphere, that is, to the west of the highest point of the celestial equator, ranging from 0° to 360° (in degree measure) or from 0h to 24h (in hourly measure). Sometimes hour angles are measured from 0° to +180° (0h to +12h) to the west and from 0° to −180° (0h to −12h) to the east.

Second equatorial coordinate system

In this system, as in the first equatorial system, the main plane is the plane of the celestial equator, and one coordinate is the declination δ (less often, the polar distance p). The other coordinate is right ascension α. The right ascension (RA, α) of a luminary is the arc of the celestial equator from the point of the vernal equinox to the circle of declination of the luminary, or the angle between the direction to the point of the vernal equinox and the plane of the circle of declination of the luminary. Right ascensions are counted in the direction opposite to the daily rotation of the celestial sphere, ranging from 0° to 360° (in degree measure) or from 0h to 24h (in hourly measure).

RA is the astronomical equivalent of Earth's longitude. Both RA and longitude measure the east-west angle along the equator; both measures are based on the zero point at the equator. For longitude, the zero point is the prime meridian; for RA, the zero mark is the place in the sky where the Sun crosses the celestial equator at the spring equinox.

Declination (δ) in astronomy is one of two coordinates of the equatorial coordinate system. Equal to the angular distance on the celestial sphere from the plane of the celestial equator to the luminary and is usually expressed in degrees, minutes and seconds of arc. The declination is positive north of the celestial equator and negative south. The declination always has a sign, even if the declination is positive.

The declination of a celestial object passing through the zenith is equal to the latitude of the observer (if we consider northern latitude with a + sign, and southern latitude as negative). In the northern hemisphere of the Earth, for a given latitude φ, celestial objects with declination

δ > +90° − φ do not go beyond the horizon, therefore they are called non-setting. If the object’s declination is δ

Ecliptic coordinate system

In this system, the main plane is the ecliptic plane. One coordinate in this case is the ecliptic latitude β, and the other is the ecliptic longitude λ.

4.5. Relationship between the ecliptic and second equatorial coordinate systems

The ecliptic latitude of a β luminary is the arc of the circle of latitude from the ecliptic to the luminary, or the angle between the plane of the ecliptic and the direction towards the luminary. Ecliptic latitudes are measured from 0° to +90° to the north pole of the ecliptic and from 0° to −90° to the south pole of the ecliptic.

The ecliptic longitude λ of a luminary is the arc of the ecliptic from the point of the vernal equinox to the circle of latitude of the luminary, or the angle between the direction to the point of the vernal equinox and the plane of the circle of latitude of the luminary. Ecliptic longitudes are measured in the direction of the apparent annual movement of the Sun along the ecliptic, that is, east of the vernal equinox in the range from 0° to 360°.

Galactic coordinate system

In this system, the main plane is the plane of our Galaxy. One coordinate in this case is the galactic latitude b, and the other is the galactic longitude l.

4.6. Galactic and second equatorial coordinate systems.

The galactic latitude b of a luminary is the arc of the circle of galactic latitude from the ecliptic to the luminary, or the angle between the plane of the galactic equator and the direction towards the luminary.

Galactic latitudes range from 0° to +90° to the north galactic pole and from 0° to −90° to the south galactic pole.

The galactic longitude l of a luminary is the arc of the galactic equator from the reference point C to the circle of the galactic latitude of the luminary, or the angle between the direction to the reference point C and the plane of the circle of the galactic latitude of the luminary. Galactic longitudes are measured counterclockwise when viewed from the north galactic pole, that is, east of datum C, ranging from 0° to 360°.

The reference point C is located close to the direction of the galactic center, but does not coincide with it, since the latter, due to the slight elevation of the Solar system above the plane of the galactic disk, lies approximately 1° south of the galactic equator. The starting point C is chosen so that the intersection point of the galactic and celestial equators with a right ascension of 280° has a galactic longitude of 32.93192° (for the epoch 2000).

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 with 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 a point on 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.

An arbitrary radius onto which celestial bodies are projected: used 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 located 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 intersected by a straight line connecting the center of the sphere with the center of the body. When studying the positions and apparent movements of luminaries on the celestial sphere, one or another system of spherical coordinates is chosen. Calculations of the positions of luminaries on the celestial sphere are made using celestial mechanics and spherical trigonometry and form the subject of spherical astronomy.

Story

The idea of ​​the celestial sphere 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, 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.

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 location. A plumb line intersects the surface of the celestial sphere at two points - zenith above the observer's head and nadir under the observer's feet.

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 at 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 bending of light rays in the atmosphere.

height circle, or vertical, luminary - a large semicircle of the celestial sphere passing through the luminary, zenith and nadir. Almucantarat(Arabic “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 almucantarates form a coordinate grid that specifies the horizontal coordinates of the luminary.

Daily rotation of the celestial sphere and related concepts

axis mundi- 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 looking at the celestial sphere from the inside.

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.

Declination circle of the luminary- a large circle of the celestial sphere passing through the poles of the world and a given 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. Declination circles and daily parallels form a coordinate grid on the celestial sphere that specifies 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 point of the east And point west. The eastern point is the one at which the points of the rotating celestial sphere rise from the horizon. The semicircle of altitude passing through the east point is called first vertical.

Celestial meridian- a great 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 noon line and the celestial meridian intersect the mathematical horizon at two points: north point And point south. The north point is the one that is closer to the north pole of the world.

The annual movement of the Sun across 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 equinoxes. IN vernal equinox The Sun in its annual movement moves from the southern hemisphere of the celestial sphere to the northern; V autumnal equinox- from the northern hemisphere to the southern. The straight line passing through these two points is called line of equinoxes. Two points of the ecliptic, spaced 90° from the equinoxes and thereby furthest from the celestial equator, are called solstice points. Summer solstice point is located in the northern hemisphere, winter solstice point- in the southern hemisphere. These four points are indicated by the zodiac symbols corresponding to

Basic elements of the celestial sphere

The sky appears to the observer as a spherical dome surrounding him on all sides. In this regard, even in ancient times, the concept of the celestial sphere (vault of heaven) arose and its main elements were defined.

Celestial sphere called an imaginary sphere of arbitrary radius, on the inner surface of which, as it seems to the observer, the celestial bodies are located. It always seems to the observer that he is in the center of the celestial sphere (i.e. in Fig. 1.1).

Rice. 1.1. Basic elements of the celestial sphere

Let the observer hold a plumb line in his hands - a small massive weight on a thread. The direction of this thread is called plumb line. Let's draw a plumb line through the center of the celestial sphere. It will intersect this sphere at two diametrically opposite points called zenith And nadir. The zenith is located exactly above the observer's head, and the nadir is hidden by the earth's surface.

Let us draw a plane through the center of the celestial sphere perpendicular to the plumb line. It will cross the sphere in a great circle called mathematical or true horizon. (Recall that a circle formed by a section of a sphere by a plane passing through the center is called big; if the plane cuts the sphere without passing through its center, then the section forms small circle). The mathematical horizon is parallel to the observer's apparent horizon, but does not coincide with it.

Through the center of the celestial sphere we draw an axis parallel to the axis of rotation of the Earth and call it axis mundi(in Latin - Axis Mundi). The axis of the world intersects the celestial sphere at two diametrically opposite points called poles of the world. There are two poles of the world - northern And southern. The north celestial pole is taken to be the one in relation to which the daily rotation of the celestial sphere, arising as a result of the rotation of the Earth around its axis, occurs counterclockwise when looking at the sky from inside the celestial sphere (as we look at it). Near the north pole of the world is the North Star - Ursa Minor - the brightest star in this constellation.

Contrary to popular belief, Polaris is not the brightest star in the starry sky. It has a second magnitude and is not one of the brightest stars. An inexperienced observer is unlikely to quickly find it in the sky. It is not easy to search for the Polaris by the characteristic shape of the Ursa Minor bucket - the other stars of this constellation are even fainter than Polaris and cannot be reliable reference points. The easiest way for a novice observer to find the North Star in the sky is to navigate by the stars of the nearby bright constellation Ursa Major (Fig. 1.2). If you mentally connect the two outermost stars of the Ursa Major bucket, and , and continue the straight line until it intersects with the first more or less noticeable star, then this will be the North Star. The distance in the sky from the star Ursa Major to Polaris is approximately five times greater than the distance between the stars and Ursa Major.

Rice. 1.2. Circumpolar constellations Ursa Major
and Ursa Minor

The south celestial pole is marked in the sky by the barely visible star Sigma Octanta.

The point on the mathematical horizon closest to the north celestial pole is called north point. The farthest point of the true horizon from the north pole of the world is south point. It is also located closest to the south pole of the world. A line in the plane of the mathematical horizon passing through the center of the celestial sphere and the points of north and south is called noon line.

Let us draw a plane through the center of the celestial sphere perpendicular to the axis of the world. It will cross the sphere in a great circle called celestial equator. The celestial equator intersects with the true horizon at two diametrically opposite points east And west. The celestial equator divides the celestial sphere into two halves - North hemisphere with its peak at the north celestial pole and Southern Hemisphere with its top at the south celestial pole. The plane of the celestial equator is parallel to the plane of the earth's equator.

The points north, south, west and east are called sides of the horizon.

The great circle of the celestial sphere passing through the celestial poles and, zenith and nadir Na, called celestial meridian. The plane of the celestial meridian coincides with the plane of the observer's earthly meridian and is perpendicular to the planes of the mathematical horizon and the celestial equator. The celestial meridian divides the celestial sphere into two hemispheres - eastern, with apex at the east point , And western, with apex at the west point . The celestial meridian intersects the mathematical horizon at the points north and south. This is the basis for the method of orientation by stars on the earth's surface. If you mentally connect the zenith point, lying above the observer’s head, with the North Star and continue this line further, then the point of its intersection with the horizon will be the north point. The celestial meridian crosses the mathematical horizon along the noon line.

A small circle parallel to the true horizon is called almucantarate(in Arabic - a circle of equal heights). You can perform as many almucantarats as you like on the celestial sphere.

Small circles parallel to the celestial equator are called celestial parallels, they can also be carried out infinitely many. The daily movement of stars occurs along celestial parallels.

The great circles of the celestial sphere passing through the zenith and nadir are called height circles or vertical circles (verticals). Vertical circle passing through the points of east and west W, called first vertical. The vertical planes are perpendicular to the mathematical horizon and almucantarates.

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 called the "celestial sphere." It plays a fundamental role in indicating the position of astronomical objects.

Although the Moon, planets, Sun and 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 towards a star does not change as we move across the Earth's surface. (True, it changes slightly as the Earth moves along its orbit, but this parallax shift can only be noticed with the help of the most precise 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 “celestial poles.” The celestial north pole lies about a degree from the North Star, and there are no bright stars near the south pole.

The Earth's rotation axis is tilted approximately 23.5° relative to the perpendicular to the plane of the Earth's orbit (to the ecliptic plane). The intersection of this plane with the celestial sphere gives a circle - the ecliptic, the apparent path of the Sun over a year. The orientation of the earth's axis in space remains almost unchanged. Therefore, 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 out to be turned towards the Sun by the Southern Hemisphere, and in our north the days become short and the nights long.

However, under the influence of solar and lunar gravity, the orientation of the earth's axis gradually changes. 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 a perpendicular to the orbital plane, describing a cone with a radius of 23.5° over 26 thousand years. For this reason, after a few centuries the pole will no longer be near the North Star. In addition, the Earth's axis undergoes small oscillations called nutation, which are associated with the ellipticity of the orbits of the Earth and the Moon, as well as with the fact that the plane of the Moon's 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 throughout the year, its appearance will change due to the Earth's revolution around the Sun. For a complete 360° orbit, the Earth requires approx. 365 1/4 days – approximately one degree per day. By the way, a day, or more precisely a solar day, is the time during which the Earth rotates once around its axis in relation to the Sun. It consists of the time it takes for the Earth to rotate relative to the stars (“sidereal day”), plus a short 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 stars.

When observed 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 each day the rising and setting of each star occurs 4 minutes earlier than the previous day. Some stars and constellations rise in the sky at night in winter - we call them “winter”, while others are “summer”.

Thus, the appearance of the celestial sphere is determined by three times: the time of day associated with the rotation of the Earth; the time of year associated with revolution 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 a specific type of task.

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 an object above the horizon, called “height,” as well as its “azimuth” - the angular distance along the horizon from a conventional point to a point lying directly below the object. In astronomy, azimuth is measured from the point south to the west, and in geodesy and navigation - from the point north to the east. Therefore, before using azimuth, you need to find out in which system it is indicated. The point in the sky directly above your head has a height of 90° and is called “zenith,” and the point diametrically opposite to it (under your feet) is called “nadir.” For many problems, the great circle of the celestial sphere, called the “celestial meridian”, is important; it passes through the zenith, nadir and poles of the world, and crosses the horizon at the points of north and south.

Equatorial system.

Due to the rotation of the Earth, stars constantly move relative to the horizon and cardinal points, and their coordinates in the horizontal system change. But for some astronomy problems, the coordinate system must be independent of the observer’s position and 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, defines 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 observation location is equal to the declination of the star. Geographic longitude corresponds to the “right ascension” of the star. It is measured east of the point of intersection 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 “vernal equinox point”, and is designated by the sign. Right ascension values ​​are usually given in hours and minutes, considering 24 hours to be equal to 360°.

The equatorial system is used when observing with telescopes. The telescope is installed so that it can rotate from east to west around an axis directed towards 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 movement 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-azimuthal 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 the vertex at the north celestial pole between the observer’s meridian and the direction to some point on 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 observation point. And since the right ascension of a star is also the polar angle between the direction towards it and the point of the vernal equinox, sidereal time is equal to the right ascension of all points lying on the observer’s meridian.

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. If the equatorial coordinates of the star are given a And d, then its horizontal coordinates A And can be calculated using the following formulas:

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

Representation of the celestial sphere.

For many centuries, scientists have searched for the best ways 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, it is necessary to select a geometric projection system. The first attempt to represent parts of the celestial sphere on a plane were rock paintings of star configurations in the caves of ancient people. Nowadays, there are various star maps, published in the form of hand-drawn or photographic star atlases covering the entire sky.

Ancient Chinese and Greek astronomers conceptualized 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. Nowadays, star 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 positions 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 to be 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 most realistic representation of the celestial sphere is provided by a planetarium. The optical projection of stars onto a hemispherical screen from the inside allows you to very accurately reproduce the appearance of the sky and all kinds of movements of the luminaries on it.