The actual day of year and the latitude (0deg at the equator to 90deg 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 23.439deg). 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 45deg 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).

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

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) .

Latitude of observer:

Angle between solar disc and sun's zenith:

Distance from observer to sun's zenith:

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

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

Adjust range:

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

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

To get the number of hours the sun shines at the given Day at the given Latitude Lat, b needs to be multiplied by 24.

The calculation of a and m can be simplified to:

and

Because cos / sin = cot = 1 / tan, a and m can be merged into:

Since tan(x) = cot(90 - x) = 1 / tan(90 - x), the division can thankfully be converted into a multiplication. Also, tan(-Lat) is equivalent with -tan(Lat):

The expression pi/182.625 can be precalculated and saved as a constant j:

This reduces the calculation of m prior the correction of out-of-range values to 3 multiplications and 1 addition:

(Note, that the argument of the cos function is in radians, whereas the arguments of the tan functions are in degrees.

Thanks to Kim Mackay for pointing this out.

Adjust the limits of m to be between 0...2; then

completes the calculation.

Notice, that depending on what your plotting software accepts (deg/rad), you might need to modify the b formula slightly. For example would you use ARCCOS(1-m)/(2*PI())*360/180 in Microsoft Excel, which simplifies to ARCCOS(1-m)/PI().

Thanks to reader justanote for this observation.

Above formulas for the Length of Day b produce the following graphs over a whole year, shown for the latitudes at 0deg, 10deg ... 90deg North:

When the planet's so far flat disc is given some height h, then twilight is defined as the part e of the solar arc.

The twilight angle (sun below horizon), as per above definition:

Thickness of the planet's disc:

The angle v is identical with the Latitude. This is true along the whole radius of the solar circle, particularly also where the distance between the solar circle and the surface of the planet disc is h:

Knowing the angle v, the radius fraction i (an extention of the radius fraction m) can be calculated:

The whole radius fraction m+i defines the point, at which the planet disc's lower surface is crossed by the solar circle. The value m is the same as calculated above. Note, however, that its uncorrected value must be used:

Adjust range: 0...2 is the valid range (see comments in the formula table of the preceding section). Note, that the m part (before adjustment) is the same as in the previous section, but range adjustment may not happen before the addition of the i term.

Angle between center of sun's disc and lower twilight point on the solar circle (not the planet's disc):

Exposed fraction of the sun's circle (0=never...1=whole day). The arc describes the daytime plus both twilight zones (b+2e):

n can not be simplified any more:

with the constants

This is the calculation of the twilight arc (comprising both twilight durations and the daylength).

Then

completes the calculation.

Some individual twilight durations e (dusk or dawn) are given in the following tables. The tables' cells give the duration in hours.

Also note, that at and near the Pole there are phases with no twilight, because the sun is present all the day, or circles too far below the horizon. The values are given as 0, because half the difference between the daylength b and the arc comprising the day length and the 2 twilights ((b+2e - b)/2) are presented. These values are identical at the pole (and near it), i.e. 0 around the winter solstice, and 1 around the summer solstice (0 and 24 hrs., resp.)

The following table shows the twilight duration for the civil twilight (6deg rather than 12deg).