For ground-based observatories, the Earth atmosphere acts like a prism which disperses light of different wavelengths such that a star generates a rainbow along the direction that points to the zenith.
With an equatorial mount, the cardinal points of the celestial object's disk are aligned with the vertical and horizontal direction of the view in the telescope.
The orientation of the disk of the Moon, as related to the horizon, changes throughout its diurnal motion and the parallactic angle changes equivalently.
In the equatorial system of right ascension, α, and declination, δ, the star is at The North Celestial Pole is at In this same coordinate system the zenith is found by inserting altitude, a=π/2, cos a=0, into the transformation formulas to get where φ is the observer's geographic latitude, and l the local sidereal time.
The plane tangential to the celestial sphere at the star is spanned by the unit vectors to the north, and to the east These are orthogonal: The parallactic angle q is the angle of the initial section of the great circle at s, east of north,[6] (The previous formula is the sine formula of spherical trigonometry.
[7]) The values of sin z and of cos φ are positive, so using atan2 functions one may divide both expressions through these without losing signs; eventually yields the angle in the full range -π ≤ q ≤ π.