Elements of spherical astronomy

Consider astronomical units of time counting, which have become prototypes of calendar units. Then how in the distant past people, even without knowing the true duration of the year, could measure equal time intervals, observing the sunrise( or sunset) of the Sun at the same point of the horizon, orienting in this direction their temples and pyramids.

The main points and lines of the celestial sphere. First of all, we recall that when studying the view of the starry sky, the concept of a celestial sphere is used - an imaginary sphere of an arbitrary radius, to which the stars are "suspended" to the inner surface. At the center of this sphere( at the point O) is the observer( Fig.).The point of the celestial sphere, located directly above the observer's head, is called the zenith, opposite to it - the nadir. The points of intersection of the imaginary axis of the Earth's rotation( the "axis of the world") with the celestial sphere are called the poles of the world.

We draw through the center of the celestial sphere three imaginary planes: the first perpendicular to the plumb line, the second perpendicular to the axis of the world and the third through the vertical line( through the center of the sphere and the zenith) and the axis of the world( through the world's pole).As a result, on the celestial sphere we obtain three large circles( the centers of which coincide with the center of the celestial sphere): the horizon, the celestial equator and the celestial meridian( Fig.).The heavenly meridian intersects with the horizon at two points: the north point( N) and the south point( S), the celestial equator - at the east point( E) and the west point( W).The SN line defining the direction "north-south" is called the midday line.

Fig. The main points and lines of the celestial sphere;the arrow indicates the direction of its rotation

The apparent annual movement of the center of the disk of the Sun among the stars occurs along the ecliptic - a large circle whose plane makes with the plane of the celestial equator an angle ε = 23 ° 27.With the celestial equator, the ecliptic intersects at two points( Fig.): At the point of the vernal equinox ¡ and at the point of the autumn equinox W.

Fig. The position of the ecliptic on the celestial sphere;the arrow indicates the direction of the apparent annual movement of the Sun

. Recall that the Sun moves along the ecliptic towards a seemingly diurnal rotation of the celestial sphere( ie, from west to east) at a rate of almost 1 ° per day, which is its apparent angular diameter. Through the point of the vernal equinox the Sun passes on March 20( or 21), moving from the southern hemisphere to the northern hemisphere. Six months later on September 22( or 23) it passes through the point of the autumn equinox from the northern hemisphere to the southern hemisphere.

Celestial coordinates. As on the globe-reduced model of the Earth, on the celestial sphere( but from within it!) It is possible to construct a coordinate grid that allows to determine the coordinates of any luminous sphere. The role of terrestrial meridians on the celestial sphere is played by circles of declination that pass from the north pole of the world to the south, instead of terrestrial parallels in the celestial sphere, daily parallels are held. For each luminary it is possible to find( Fig.):

Fig. The direction of reference of the equatorial coordinates a and S, as well as the hour angle t of the star

1. The angular distance α of its declination circle from the vernal equinox point, measured along the celestial equator against the daily motion of the celestial sphere( similar to the way along the Earth's equator we measure the geographic longitude λ- the angular distance of the meridian of the observer from the zero Greenwich meridian).This coordinate is called the direct ascent of the star.

2. The angular distance of the star δ from the celestial equator is the declination of the light measured along the declination circle passing through this star( corresponding to the geographical latitude φ).

The direct ascension of a star α is measured in hours - in hours( h or h), minutes( m or m) and seconds( s or s) from 0h to 24h, δ declines in degrees, with a plus sign( from 0 °to + 90 °) from the celestial equator to the north pole of the world and with a minus sign( from 0 ° to -90 °) to the south pole of the world. In the course of the daily rotation of the celestial sphere, these coordinates for each luminary remain unchanged.

The position of each luminary on the celestial sphere at a given time can also be described by two other coordinates: its azimuth and angular altitude above the horizon. To do this, from the zenith through the star to the horizon we hold a mentally large circle - the vertical. The azimuth of the light A is measured from the point south of S to the west to the point of intersection of the vertical of the star with the horizon. If the azimuth is read from the south point in a counter-clockwise direction, then a minus sign is assigned to it. The height of the light h is measured along the vertical from the horizon to the luminary( Fig.).Turning to Fig., It is clear that the height of the world's pole above the horizon is equal to the geographic latitude of the observer.

Fig. The direction of reference of the azimuth A and the altitude h of the light

The culmination of the luminaries. In the course of the daily rotation of the Earth, each point of the celestial sphere passes twice through the celestial meridian of the observer. The passage of this or that luminary through that part of the arc of the celestial meridian, on which the zenith of the observer is located, is called the upper culmination of the luminary. At the same time, the height of the light above the horizon reaches its highest value. At the moment of the lower culmination of the luminary, the opposite part of the arc of the meridian, on which the nadir is located, passes. The time elapsed after the upper culmination of the luminary, the hour angle of the light t is measured.

If the luminary in the upper climax passes through the celestial meridian to the south of the zenith, then its height above the horizon at this moment is equal to