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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Refraction

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