Starting from these data we tried to find a suitable mathematical algorithm in order to be able to calculate the number of hours of sunlight in each location at any day of the year. On the calendar in each place we found the max and min number of hours of sunlight around resp June 22 and December 22. We took half of the difference as amplitude and half of the sum as the number of hours of sunlight at the equinox moments and the number of the day throughout the year (1 for Jan 1) as x.
Taking a sine curve for the representation of the hours of sunlight happens to be far from reality with greater error the higher the latitude. Unfortunately there is no single formula that can be used to accurately predict times of this phenomenon over an acceptably wide range of dates and places. However there exists a fine book about Astronomical Algorithms for performing a wide variety of celestial calculations. It is written by Jean Meeus, a Belgian astronomer from Bruges. We were given the same better
formula by the Astronomy Applications
Department of the university of Gent, Leuven and Brussels. y(VI) = 12.20 + 24/pi*asin(tan(45.983*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2))) y(SN) = 12.23 + 24/pi*asin(tan(51.168*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2))) y(GR) = 12.35 + 24/pi*asin(tan(59.275*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2))) y(MI) = 12.43 + 24/pi*asin(tan(61.683*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2)))
From the University Almanac
Office in Helsinki we got more information about the hours of sunlight throughout
the year for places situated on the polar circle (66.56°NL) and Utsjoki located
at 69.87°NL. y(UT) = 13 + 24/pi*asin(tan(69.87*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2))) y(POI) = 12.85 + 24/pi*asin(tan(66.56*pi/180)*sin(23.44*pi/180)*sin((x81)*2pi/365.25)*((1(sin(23.44*pi/180)*sin((x81)*2pi/365.25)^2)^(1/2)))
1) The apparent position
of the sun is determined not just by the rotation of the earth about its axis,
but also by the revolution of the earth around the sun. One
of the reasons is that the
plane of the equator is not the same as the plane of the earth's orbit around
the sun, but is offset from it by the angle of obliquity.
3) But that is not all. Also the longest day is longer than the longest
night, and the shortest day is
longer than the shortest night, for the reason that sunrise occurs when the
upper edge of the disk of the Sun appears on the horizon, and sunset is at the
moment when the upper edge disappears below the horizon. These are the
instants of first and last direct sunlight; but at these times the center of
the Sun's disk is already some minutes of arc vertically below the horizon. In addition the
Sun is seen some extra minutes of arc above its actual geometric position on account
of atmospheric refraction. Consequently, the length of every day exceeds the
time that the center of the Sun is geometrically above the horizon by the
intervals of time required for the Sun to move through these extra minutes of arc in altitude at both rising and
setting and this effect shortens the night by the same amount. Sources http://aa.usno.navy.mil/
