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2.8 Range of the sun measured along the sky

Frank L. Preuss

Information is given here about the direction of a sunrise in relation to the east direction and the direction of a sunset in relation to the west direction.

That point, at which in summer at the time of the solstice the sun rises at the horizon, is called AS (Aufgang im Sommer). That point, at which in winter at the time of the solstice the sun rises at the horizon, is called AW (Aufgang im Winter).

That point, at which in summer at the time of the solstice the sun sets at the horizon, is called US (Untergang im Sommer). That point, at which in winter at the time of the solstice the sun sets at the horizon, is called UW (Untergang im Winter).

The difference the sun travels during a day from sunrise to sunset is measured in angles and is called L for length.

The formula for the longest day is

L = 2 arccos(-sin(ε)/cos(|φ|))

This is therefore the time the sun travels from AS to US.

This is, on the longest day, the distance the sun travels, measured along the horizon.

It is the distance between the point where the sun rises and the point where the sun sets.

To obtain the shortest angle, the minus sign is removed from the formula:

L = 2 arccos(sin(ε)/cos(|φ|))

This is therefore the time the sun travels from AW to UW.

This is, on the shortest day, the distance the sun travels, measured along the horizon.

ε = obliquity of the ecliptic = 23.44 degrees

φ = geographical latitude

The vertical lines around the φ mean that one should omit any minus sign from the value.

Obliquity of the ecliptic.

The angle at which the celestial equator intersects the ecliptic. In 1988 this angle was slowly decreasing by 0.47 arc seconds a year, due to precession and nutation. It varies between 21^{o}55' and 24^{o}180'. Its value in 2000 is about 23^{o}26'34".

Precession of the equinoxes.

The westward motion of the equinoxes caused mainly by the attraction of the sun and the moon on the equatorial bulge of the earth. This *luni-solar* precession together with the smaller planetary precession combine to give the general precession amounting to 52.27" per annum. The equinoxes thus make one complete revolution of the ecliptic in 25 800 years, and the earth's pole turns in a small circle of radius 23^{o}27' about the pole of the ecliptic, thus changing the coordinates of the stars.

Nutation.

An oscillation of the earth's pole about the mean position. It has a period of about 19 years, and is superimposed on the precessional movement.

Celestial equator.

The great circle in which the plane of the earth's equator cuts the celestial sphere; the primary circle to which the coordinates, right ascension and declination are referred.

Ecliptic.

The great circle in which the plane containing the centres of the earth and sun cuts the celestial sphere; hence, the apparent path of the sun's annual motion through the fixed stars.

In the now following table, in column 9, the values of columns 7 and 8 are added and then divided by two to check if they result in a mean length of day of 180 degrees.

In column 10, the difference of the angle is given between AS and AW or between US and UW. This is therefore the angle the sunrise or the sunset travels during the year on the horizon between the solstices. It is the angle for the longest day minus the angle for the shortest day divided by two.

In column 11 half the value of column 10 is given. It is the angle between the east direction and the sunrise or the west direction and the sunset at solstices.

The more the location is away from the equator and comes closer to the polar circle the colder it gets, but the longer the days are and the shorter the nights - in summer.

And another result is that the nights are starting to not really getting completely dark, as this a already the case in the north of Germany in summer.

Longest day:

L = 2 arccos(-sin(ε)/cos(|φ|))

Shortest day:

L = 2 arccos(sin(ε)/cos(|φ|))

The following table results from the above formulas:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Latitude |
Obli-quity of the eclip-tic |
Values |
Length of time measured as angle |
|||||||

Des-crip-tion |
φ |
ε |
sin(ε) |
cos(φ) |
sin(ε)/
cos(φ) |
Length of long-est day: be-tween AS and US |
Length of short-est day: be-tween AW and UW |
Mean length of day |
Be-tween AS and AW or US and UW |
Be-tween AS or AW or US or UW and east west axis |

de-grees |
de-grees |
de-grees |
de-grees |
de-grees |
de-grees |
de-grees |
||||

Equa-tor | 0 | 23.44 | 0.3978 | 1.0000 | 0.3978 | 226.9 | 133.1 | 180.0 | 46.9 | 23.4 |

10 | 23.44 | 0.3978 | 0.9848 | 0.4039 | 227.6 | 132.4 | 180.0 | 47.6 | 23.8 | |

20 | 23.44 | 0.3978 | 0.9397 | 0.4233 | 230.1 | 129.9 | 180.0 | 50.1 | 25.0 | |

Tropic | 23.44 | 23.44 | 0.3978 | 0.9175 | 0.4336 | 231.4 | 128.6 | 180.0 | 51.4 | 25.7 |

30 | 23.44 | 0.3978 | 0.8660 | 0.4593 | 234.7 | 125.3 | 180.0 | 54.7 | 27.3 | |

40 | 23.44 | 0.3978 | 0.7660 | 0.5193 | 242.6 | 117.4 | 180.0 | 62.6 | 31.3 | |

50 | 23.44 | 0.3978 | 0.6428 | 0.6188 | 256.5 | 103.5 | 180.0 | 76.5 | 38.2 | |

60 | 23.44 | 0.3978 | 0.5000 | 0.7956 | 285.4 | 74.6 | 180.0 | 105.4 | 52.7 | |

66 | 23.44 | 0.3978 | 0.4067 | 0.9780 | 335.9 | 24.1 | 180.0 | 155.9 | 78.0 | |

Polar circle | 66.56 | 23.44 | 0.3978 | 0.3978 | 1.0000 | |||||

70 | 23.44 | 0.3978 | 0.3420 | 1.1631 | ||||||

80 | 23.44 | 0.3978 | 0.1736 | 2.2908 | ||||||

Pole | 90 | 23.44 | 0.3978 | 0.0000 |

This is the end of "2.7 Range of the sun measured on the horizon"

To the German version of this chapter:
2.7 Bereich der Sonne horizontal gemessen

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2.6 Angle of sun orbit

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2.8 Range of the sun measured along the sky

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