Formulae

The actual day of year and the latitude (0 deg at the equator to 90 deg at the North pole) both influence the length of the day.

The perceived way of the sun around the planet can be viewed at as the boundary circle of the planet's disc. However, this constellation (in which the sun apparently circles along the disc's boundary) applies only at equinoxes and only at the North pole. The further away one is from the North pole (towards the equator), the more the surrounding circle is tilted along the West-East axis, until it is completely upright (perpendicular to the planet's disc) at the equator.

Furthermore, there is also a shift of the circle away from the disc, along the obliquity of the ecliptic (connecting the centers of the two circles at an angle of 23.439 deg ). This shift can be "upwards" (max. distance at the summer solstice) or "downwards" (max. distance at the winter solstice) depending on the actual latitude.

The following image shows the tilted and shifted solar circle for the Winter Solstice at 45 deg North. It is only the part b out of the whole circle in which the sun in visible: when continueing its path on the blue line it is night (but see the part titled #B Twilight below).

/img/lod_fig01.jpg

Solar Circle for the Summer Solstice at 45 deg North

Fig. 1: Solar Circle for the Summer Solstice at 45 deg North

The following table calculates the exposed part b in relation to the whole circle. The formulas mention 3 parameters, which signify:

Axis: Obliquity of the ecliptic (as the rotation axis of the Earth is not perpendicular to its orbital plane, the equatorial plane is not parallel to the ecliptic plane, but makes an angle of 23.439 deg ); for our purposes this is a constant value, it changes slowly only within thousands of years. Lat: Latitude of the observer (0 deg at the equator, 90 deg at the Northpole). Day: Day of year (1st year 0...364, from 365 add 0.25 for every completed year within the Great Year consisting of 4 years, i.e. 365.25 etc.). Note, that the day of year does not start with the astronomically quite arbitrary January 1st, but with the day of the winter solstice in the first year a four years cycle. Thanks to David X. Callaway to point this out early in the text to avoid confusion.

Note: The expression "observer" in the remarks refers to a hypothetical observer located on the center of the planet's "disc".

Angle between observer and sun's zenith: Thanks to Andrew Green for spotting an error which was introduced while translating from HTML to XLM in formulas #m_eq_1 (1) and #m_eq_9 (9) .

z = 90 - Lat - cos pi times nbsp Day 182.625 nbsp times nbsp Axis

Latitude of observer:

c = -Lat

Angle between solar disc and sun's zenith:

a = z - c

Distance from observer to sun's zenith:

d = nbsp 1 sin(a)

Distance from observer to the center of the sun's circle:

t = cos(a)d

Exposed radius part between sun's zenith and sun's circle:

m = 1 + tan(c)t

Adjust range:

if m is negative, then the sun never appears the whole day long (polar winter): m must be adjusted to 0 (the sun can not shine less than 0 hours). if m is larger than 2, the "sun circle" does not intersect with the planet's surface and the sun is shining the whole day (polar summer): m must be adjusted to 2 (the sun can not shine for more than 24 hours).

Angle between center of sun's disc and sunrise or sunset point on the solar circle (not the planet's disc), resp.:

f = arccos(1 - m)

Exposed fraction of the sun's circle (0=never...1 = whole day):

b = nbsp f 180