# Positional Astronomy: Sunrise, sunset and twilight

{Note: If your browser does not distinguish between "a,b" and "α, β" (the Greek letters "alpha, beta") then I am afraid you will not be able to make much sense of the equations on this page.}

Since refraction affects zenith angle,
it generally changes both the Right Ascension and declination of an object.
It also affects the time the object appears to rise and set.

The standard formula for the altitude of an object is:
sin(α) = sin(δ)sin(φ) + cos(δ) cos(φ) cos(H)

If a = 0° (the object is on horizon, either rising or setting),
then this equation becomes:
cos(H) = - tan(φ) tan(δ)

This gives the semi-diurnal arc H:
the time between the object crossing the horizon, and crossing the meridian.

Knowing the Right Ascension of the object, and its semi-diurnal arc,
we can find the Local Sidereal Time of meridian transit,
and hence calculate its rising and setting times.

However, refraction means that this simplified formula is not accurate,
since the altitude should be, not 0°, but -0°34'.
This is not too important for stars, which are rarely observed close to the horizon.
But it makes an important difference in calculating the times of rising and setting of the Sun.

Furthermore, "sunrise" and "sunset" generally refer to the moment
when the top of the Sun's disc is just on the horizon.
The formula would give us the time of rising or setting
for the centre of the Sun's disc.
So we must also allow for the semi-diameter of the Sun's disc,
which is 16 arc-minutes.

So sunrise and sunset actually occur when the Sun has altitude -0°50'
(34' for refraction, and another 16' for the semi-diameter of the disc).

Since the atmosphere scatters sunlight, the sky does not become dark instantly at sunset;
there is a period of twilight.

During civil twilight, it is still light enough to carry on ordinary activities out-of-doors;
this continues until the Sun's altitude is -6°.
During nautical twilight, it is dark enough to see the brighter stars,
but still light enough to see the horizon, enabling sailors to measure stellar altitudes for navigation;
this continues until the Sun's altitude is -12°.
During astronomical twilight, the sky is still too light for making reliable astronomical observations;
this continues until the Sun's altitude is -18°.
Once the Sun is more than 18° below the horizon, we have astronomical darkness.
The same pattern of twilights repeats, in reverse, before sunrise.

In summer, astronomical twilight will last all night, for any place with latitude above 48.6°.

Exercise:

The Sun is at declination -14°.
What will be its hour angle at sunrise
(the moment the top edge of the Sun first appears over the horizon),
at a latitude of +56°20'?

If the Sun is on the local meridian at 12:03,
what time is sunrise?
and what time is sunset?

And when will astronomical twilight start and finish?

Click here for the answer.

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