Spherical Astronomy

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Spherical Astronomy

the branch of astrometry that deals with mathematical methods for solving problems connected with the study of the apparent positions and motions, on the celestial sphere, of celestial bodies, such as stars, the sun, the moon, planets, and artificial celestial bodies. Spherical astronomy is made use of in various areas of astronomy. It arose in antiquity and constituted the first step in the study of astronomical phenomena.

The basic concept of spherical astronomy is the celestial sphere. Each direction to a heavenly body in space is represented on the sphere by a point, and planes are represented by great circles. The use of the celestial sphere permits a considerable simplification of the mathematical relations between directions to celestial bodies, since complex spatial representations are reduced to simpler figures on the surface of the sphere; hence the term “spherical astronomy.”

In order to study the relative positions and motions of points on the celestial sphere, coordinate systems are established on it. Spherical astronomy makes use of the horizon coordinate system, two equatorial systems, and the ecliptic coordinate system (seeCELESTIAL COORDINATES). The relationships between the different coordinate systems are determined by means of the formulas of spherical trigonometry. Since spherical astronomy studies phenomena associated with the apparent diurnal rotation of the celestial dome, that is, the apparent motions of bodies due to the rotation of the earth, the celestial sphere is regarded as rotating from east to west about the extended axis of the earth at an angular speed equal to that at which the earth rotates. This kinematic model almost exactly reproduces the appearance of the sky to an observer on the rotating earth. The general relations between the horizon and equatorial coordinate systems make it possible to determine, for example, the times and azimuths at which a celestial body rises and sets, the time of transit of a celestial body, the elongation of a celestial body, and the position of a celestial body at a given time. One of the tasks of spherical astronomy is the determination of the conditions under which two suitably chosen stars are at the same altitude. Knowledge of these conditions is important in determining from astronomical observations the geographic coordinates of points on the earth’s surface.

Measurement of time. An important problem in spherical astronomy is the establishment of the theoretical foundations of the astronomical system of reckoning time. Spherical astronomy thus studies units of time and the relationships between them. The measurement of time is based on the natural periodic phenomena of the rotation of the earth about its axis and the revolution of the earth about the sun.

The duration of one rotation of the earth is, depending on whether the vernal equinox or the sun is used as the reference point on the celestial sphere, one sidereal or solar day. In reckoning sidereal days, it is taken into account that the vernal equinox, owing to precession and nutation, does not remain at the same position on the celestial sphere but moves translationally and, at the same time, executes oscillations about its mean position. For the reckoning of solar days, the concept of the mean sun is introduced. The mean sun is a fictitious point that moves uniformly along the equator in coordination with the complicated apparent motion of the true sun along the ecliptic.

The period of one revolution of the earth about the sun is one tropical year. The calendar is based on the tropical year, which corresponds to the time required for one cycle of the four seasons to be completed. Since a tropical year does not contain an integral number of mean days, the duration of a calendar year is set at 365 days in some years and 366 days in other years in order that the average duration of the calendar year over a long interval of time be equal to the length of one tropical year. In astronomy, time is reckoned in tropical years, in calendar years with a mean duration of 365.25 days, or in Julian days.

Observed positions of celestial bodies. The coordinates of celestial bodies obtained directly from observation are distorted by a number of factors. First of all, the coordinate axes associated with the earth’s axis of rotation and with the vernal equinox do not maintain a constant direction but move as a result of precession and nutation. Because of aberration, the apparent positions of celestial bodies on the celestial sphere are somewhat displaced from the positions the bodies would have if the earth were stationary. Observations are also distorted by the refraction of light in the earth’s atmosphere. In addition, parallax effects must be taken into account in processing observational data.

Corrections must be applied to the coordinates of celestial bodies in order to eliminate the enumerated distortions from the observed positions of the celestial bodies and in order to determine positions in the same coordinate system for all observations. The coordinate system used is associated with the position of the earth’s axis of rotation and the vernal equinox at some epoch, such as 1900.0 or 1950.0 (seeMEAN POSITION). The corrections applied take into account the effects of precession, nutation, aberration, parallax, and refraction. Astronomical yearbooks give the values of special reduction quantities that are used in allowing for the effects of precession, nutation, and aberration. The yearbooks also give the values of other quantities necessary for processing astronomical observations.

PRECESSION AND NUTATION. As a result of precession, the earth’s axis slowly changes its direction, with a period of about 26,000 years, so as to describe a conical surface. Nutational oscillations are superimposed on this motion of the earth’s axis (seeNUTATION). The position in space of the plane of the ecliptic also changes very slowly; associated with this change is a motion of the vernal equinox. Since the vernal equinox is used as a reference point in the equatorial and ecliptic coordinate systems, the coordinates of celestial bodies in these systems change.

ABERRATION. Aberration is the apparent displacement of the position of a celestial body on the celestial sphere from the true position as a result of the observer and the celestial body being in motion relative to each other. In observations of stars, annual and diurnal aberration are taken into account. The former is the aberration due to the motion of the earth about the sun; the latter is the aberration produced by the earth’s rotation about its axis. In observations of artificial earth satellites, the aberration due to the motion of the satellite about the earth is also calculated.

PARALLAX. Because the observer moves in space as a result of the earth’s rotation and the revolution of the earth about the sun, the directions to celestial bodies also change. To obtain comparable quantities, observation results are referred in the first case (when bodies in the solar system are observed) to the center of the earth and in the second case (when stars are observed) to the center of the solar system—the sun. The magnitude of the parallactic displacement depends on the distance to the celestial body.

REFRACTION. Because the light from celestial bodies is refracted in the earth’s atmosphere, the celestial bodies appear displaced in the direction of the zenith. The magnitude of the displacement depends on the refractive index of the air—that is, on such factors as temperature and pressure—and on the zenith distance of the celestial body. In the observation of celestial bodies near the earth, particularly artificial earth satellites, displacements due to refraction parallax are also taken into account. These displacements result from the different effects of refraction on celestial bodies that are located in the same direction from the terrestrial observer but at different distances from him.

Other concerns of spherical astronomy. The effects of the distorting factors listed above must be eliminated before data from the observation of celestial bodies can be used for theoretical studies or for such practical purposes as the determination of geographic coordinates, of azimuths, or of time. To calculate the appropriate corrections, the astronomical constants are used; these constants are numerical quantities characterizing the described phenomena. The determination of the astronomical constants from the data of astronomical observations is a problem that links spherical astronomy with such fields as fundamental astrometry, celestial mechanics, and the study of the structure of the earth.

Practical astronomy makes extensive use of spherical astronomy. Among the matters dealt with in spherical astronomy are problems associated with the determination of coordinates on the surfaces of bodies in the solar system, especially on the surface of the moon, where librations must be taken into account. With the beginning of the age of interplanetary flight and the landing of astronauts on the moon, the determination of coordinates on the moon has taken on particular importance. Spherical astronomy also studies methods of calculating solar and lunar eclipses and similar phenomena, such as occultations of stars by the moon and transits of planets across the solar disk.

REFERENCES

Blazhko, S. N. Kurs sfericheskoi astronomii, 2nd ed. Moscow, 1954.
“Reduktsionnye vychisleniia v astronomii.” In Astronomicheskii ezhegodnik SSSR na 1941 g. Moscow-Leningrad, 1940. (Appendix, pp. 379–432.)
Kazakov, S. A. Kurs sfericheskoi astronomii, 2nd ed. Moscow-Leningrad, 1940.
Kulikov, K. A. Kurs sfericheskoi astronomii. Moscow, 1969.
Zagrebin, D. V. Vvedenie v astrometriiu. Moscow-Leningrad, 1966.
Newcomb, S. A Compendium of Spherical Astronomy. New York-London, 1906.
Chauvenet, W. A Manual of Spherical and Practical Astronomy, 5th ed., vol. 1. Philadelphia, 1891.
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