The sunrise equation or sunset equation can be used to derive the time of
sunrise
Sunrise (or sunup) is the moment when the upper rim of the Sun appears on the horizon in the morning. The term can also refer to the entire process of the solar disk crossing the horizon and its accompanying atmospheric effects.
Terminology
A ...
or
sunset
Sunset, also known as sundown, is the daily disappearance of the Sun below the horizon due to Earth's rotation. As viewed from everywhere on Earth (except the North and South poles), the equinox Sun sets due west at the moment of both the spri ...
for any
solar declination and
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
in terms of local solar time when sunrise and sunset actually occur.
Formulation
It is formulated as:
:
where:
:
is the
solar hour angle at either
sunrise
Sunrise (or sunup) is the moment when the upper rim of the Sun appears on the horizon in the morning. The term can also refer to the entire process of the solar disk crossing the horizon and its accompanying atmospheric effects.
Terminology
A ...
(when negative value is taken) or
sunset
Sunset, also known as sundown, is the daily disappearance of the Sun below the horizon due to Earth's rotation. As viewed from everywhere on Earth (except the North and South poles), the equinox Sun sets due west at the moment of both the spri ...
(when positive value is taken);
:
is the
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
of the observer on the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
;
:
is the sun
declination
In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
.
Principles
The
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
rotates at an
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of 15°/hour. Therefore, the expression
, where
is in degree, gives the interval of time in hours from
sunrise
Sunrise (or sunup) is the moment when the upper rim of the Sun appears on the horizon in the morning. The term can also refer to the entire process of the solar disk crossing the horizon and its accompanying atmospheric effects.
Terminology
A ...
to local
solar noon
Noon (or midday) is 12 o'clock in the daytime. It is written as 12 noon, 12:00 m. (for meridiem, literally 12:00 noon), 12 p.m. (for post meridiem, literally "after noon"), 12 pm, or 12:00 (using a 24-hour clock) or 1200 ( military time).
Sol ...
or from local solar noon to
sunset
Sunset, also known as sundown, is the daily disappearance of the Sun below the horizon due to Earth's rotation. As viewed from everywhere on Earth (except the North and South poles), the equinox Sun sets due west at the moment of both the spri ...
.
The sign convention is typically that the observer latitude
is 0 at the
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
, positive for the
Northern Hemisphere
The Northern Hemisphere is the half of Earth that is north of the Equator. For other planets in the Solar System, north is defined as being in the same celestial hemisphere relative to the invariable plane of the solar system as Earth's N ...
and negative for the
Southern Hemisphere, and the solar declination
is 0 at the vernal and autumnal
equinoxes
A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and s ...
when the sun is exactly above the equator, positive during the Northern Hemisphere summer and negative during the Northern Hemisphere winter.
The expression above is always applicable for latitudes between the
Arctic Circle
The Arctic Circle is one of the two polar circles, and the most northerly of the five major circles of latitude as shown on maps of Earth. Its southern equivalent is the Antarctic Circle.
The Arctic Circle marks the southernmost latitude at ...
and
Antarctic Circle. North of the Arctic Circle or south of the Antarctic Circle, there is at least one day of the year with no sunrise or sunset. Formally, there is a sunrise or sunset when
during the Northern Hemisphere summer, and when
during the Southern Hemisphere winter. For locations outside these latitudes, it is either 24-hour
day
A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two so ...
time or 24-hour
nighttime
Night (also described as night time, unconventionally spelled as "nite") is the period of ambient darkness from sunset to sunrise during each 24-hour day, when the Sun is below the horizon. The exact time when night begins and ends depends on ...
.
Expressions for the solar hour angle
In the equation given at the beginning, the cosine function on the left side gives results in the range
1, 1
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
but the value of the expression on the right side is in the range