Alexander Thom's Declination equation (I understand that Thom did not create this formula, but his book is where I found it.):
inv sin D = (sin L * sin H) + (cos L * cos H * cos Az)
D = declination
L = latitude
H = horizon height (degrees)
AZ = Azimuth can be more practically expressed as;
D = asin ((sin L * sin H) + (cos L * cos H * cos AZ))
This finds the declination for a given azimuth and elevation in degrees Today, the Sun's declination at equinox = 0 deg. At solstice = 23.45 degs.
from Thom, Alexander, 1967. Megalithic Sites in Britain. London: Oxford University Press. p. 17.
Chuck Pettis' Azimuth equation:
AZ = acos ((sin D - (sin L * sin H) / (cos L * cos H))
Byron Dix's equation to calculate sun's declination for any day of the year:
D = 23.45 sin (360/365.25 * t)
where t = number of days from vernal equinox (NB vernal equinox date is variable!) WITH EQUINOX AS DAY 1 or we can reverse that to show the days past equinox for a given declination between +/- 23.45 (sun's in this case) : t = (inv sin (D/23.45))/(360/365.25) or t = (asin(D/23.45))/(360/365.25).
Grahame Gardner writes:
radians = 360/Pi 2 x deg. degrees = 2 Pi R/360
To find solar declination (D) for a particular day (t):
D/23.45 = sin (0.985 * t)
(the 0.985 is 360/365.25 from Bryon's equations)
asin (D/23.45) = 0.985 * t
therefore: (asin (D/23.45)/0.985) = t in other words, t = asin(D/23.45)/0.985 so, do:
find the asin (inv. sin) of that result, divide it by 0.985 and that should be our desired result.
Refraction and Parallax
We do not account for refraction – the degree to which the apparent position of a celestial body (in this case the Sun) is distorted by the redirection of its light as it passes through the Earth‚s atmosphere - since it has little effect on the azimuth. Since we're talking about sunrise and sunset times, the refraction is going to be pretty much constant since we are theoretically always looking through the same density of atmosphere. The horizontal refraction of the sun at 90 degs. of zenith (i.e. rising or setting) is 35 minutes of arc, which is just over half a degree of displacement; therefore sunrise will occur slightly earlier and sunset slightly later. For example, at the equator on the equinoxes, the sun will appear to rise about four and a half minutes earlier, and set four and a half minutes later. This refraction will be increased slightly if the observer is at altitude and the horizon is a great distance away. Refraction also accounts for why the sun appears slightly oval in shape when rising or setting.
Parallax is the apparent vertical displacement of a heavenly body due to the latitude of the observer. Since the true zenith distance of a star or planet is measured from the centre of the earth, the observed zenith distance will vary slightly according to the location of the observer. However because of its size, in the case of the sun the effect is minimal, at most about 8.8 seconds of arc, and again has little effect on the azimuth for our purposes. Corrections for Refraction and Parallax are really only important in navigation, when trying to locate your position on the earth by observing the heavens.
It is always my feeling that one should do this work as accurately as the tools available can allow. As we are not suggesting the use of theodolites or transits, we do not believe anyone using the best Suunto hand held compass can get to within half a degree of arc. So we have not included refraction or parallax in our calculations.
Calculating the Major and Minor Standstills of the Moon
The Moon deviates from the ecliptic by as much as 5 degrees 08 minutes (5.15 degrees), thus making a total deviation in declination of 11.2 degrees. It takes 18.67 years for the Moon to go from one extreme to the other and back again. This 18.67 year cycle is called the Metonic Cycle after Meton, the Greek who supposedly first identified it. The Full Moons' rises and sets mirror the Sun's throughout the year. The Full Moon closest to the Winter Solstice rises around the point where the Summer Solstice Sun rises - and vice-versa, the Full Moon nearest the Summer Solstice rises around the point on the horizon where the Winter Solstice Sun rises. The Major Standstill is as far outside the Solstice rise/azimuth for that latitude as possible. The Minor Standstill is as far inside the Solstice Sun's path (ecliptic) as the Moon deviates.
Thus the Northern Major Standstill of the Moon, given a level horizon, will be at an declination of 5.15 degrees less than the declination for the Summer Solstice Sunrise for that latitude. The Northern Minor Standstill of the Moon, given a level horizon, will be at an declination of 5.15 degrees more than the declination for the Summer Solstice Sunrise for that latitude. And mirroring that, the Southern Major Standstill of the Moon, given a level horizon, will be at an declination of 5.15 degrees more than the declination for the Winter Solstice Sunrise for that latitude. The Southern Minor Standstill of the Moon, given a level horizon, will be at an declination of 5.15 degrees less than the declination for the Summer Solstice Sunrise for that latitude. The Major Standstill Moon sets work in the same ways. See the section on this in Orthographic Projection.
I wish to thank Fred VandenBergh, Grahame Gardner, Victor Rejis and Kevin Kilburn for their help in preparing these Sunfinder pages.