Parallactic Triangle

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Parallactic Triangle

 

in astronomy, the spherical triangle on the celestial sphere with vertices at the celestial pole P, the zenith Z of the observation site, and a given point cr on the celestial sphere; in most cases, σ is the center of a celestial

Figure 1

body (see Figure 1). The sides of the triangle are equal to z, 90° — δ, and 90° — Φ, where z and δ are, respectively, the zenith distance and declination of σ and Φ is the latitude of the observation site. Two of the angles are equal to t and 360° — A, where t and A are, respectively, the hour angle and the azimuth, measured from the north point, of σ; the third angle is called the parallactic angle and is denoted by q. By applying the formulas of spherical trigonometry to the parallactic triangle, we can find the equatorial coordinates t and δ of the point σ from the point’s horizontal coordinates A and z, and vice versa:

cos z = sin Φ sin δ + cos Φ cos δ cos t

sin z cos A = -cos Φ sin δ + sin Φ cos δ cos t

sin z sin A = cos δ sin t

The parallactic triangle can also be used to determine the times and azimuths of the risings and settings of celestial bodies (in this case z = 90°) and to compute the time of onset of twilight.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.