# Positional Astronomy: The Moon

The Moon always keeps the same face turned towards the Earth.
How much we see of that face depends on the direction of the Sun:
we only see the part which is illuminated by sunlight,
as shown in the diagrams below.

The interval between one New Moon and the next is the synodic month;
it averages 29.53 days.
However, because the Earth is orbiting the Sun,
the Moon will return to the same place relative to the stars
in a shorter interval – the sidereal month;
this averages 27.32 days.
The diagram shows the Moon at Full, and lined up with a certain star, at time t0 .
It is lined up with that star again at time t1 , after one sidereal month,
but it doesn't reach Full again until time t2 , after one synodic month.

The Moon's actual motion is extremely complicated:
it orbits the Earth in an elliptical orbit,
tipped at an angle to the Earth's own orbital plane (the ecliptic);
and its orbit is constantly being perturbed
by the gravitational influence of the Sun.

This is not the place for the full theory of the Moon's motion.
However, there are some simple approximations
which help to determine when the Moon will or will not be visible.

1. When will the Moon transit the meridian?

At New Moon, the Moon lies in the same direction as the Sun.
(Owing to the tilt of the Moon's orbit,
it does not generally pass directly in front of the Sun.)
The Moon then moves eastwards, relative to the Sun.
It moves 360° in 29.53 days,
at which time it is lined up with the Sun again for the next New Moon.
Thus it moves about 12.2° each day, relative to the Sun;
which corresponds to lagging behind the Sun, as it crosses the sky,
by about 48.8 minutes of time each day.

So, if you know the "age" of the Moon
(that is, how many days since the last New Moon),
you can calculate how much later the Moon will cross the sky, compared to the Sun.

The Sun crosses the meridian at noon
(you can be more precise than this,
if you know your longitude and the Equation of Time).
So you can calculate the time at which the Moon will cross the meridian.

The result will not be very accurate,
since the Moon's motion is not uniform,
but should be correct to within an hour.

Exercise:

The Moon was New on December 25th, 2000.
At St.Andrews, on January 1st 2001,
the Sun crossed the meridian at 12:15.
At what time did the Moon cross the meridian?

2. When will the Moon rise and set?

If the Moon were always on the celestial equator,
it would always rise 6 hours before transit,
and set 6 hours after transit.

We know that the Sun does not keep to the celestial equator.
It lies on the equator at the equinoxes, in March and September,
but its declination varies between 23.4°N in June and 23.4°S in December.
At the equinoxes, anywhere in the world,
the Sun rises due east, 6 hours before noon, and sets due west, 6 hours after noon.
But at the summer (June) solstice, in latitudes north of 66.6°N,
the Sun never sets at all! (This defines the Arctic Circle.)
In latitudes around 58°N,
the Sun rises in the north-east, about 9 hours before noon,
and sets in the north-west, about 9 hours after noon.
Similarly, at the winter (December) solstice,
in the Arctic Circle the Sun never rises;
in latitudes around 58°N,
the Sun rises in the south-east, about 3 hours before noon,
and sets in the south-west, about 3 hours after noon.

Now, the Moon follows roughly the same path as the Sun
(ignoring its orbital tilt)
but it takes only a month to trace the path which the Sun takes in a year.

The Sun moves about 1° a day (360° in 365.25 days)
The Moon lags behind the Sun by about 12.2° a day,
so you can work out the date on which the Sun will be at the point where the Moon now is.

This means you can estimate roughly how long the Moon will be above the horizon.

Having already calculated the time at which it will cross the meridian,
you can now estimate its rising and setting times.

This will not be very accurate.
But it should be sufficient to determine, for example,
whether a particular night's observing will be affected by moonlight.

Exercise:

Given the data in the previous exercise,
estimate the times of moonrise and moonset
at St.Andrews on January 1st 2001.

And here's an example where you can deduce a great deal
from very little information:

Exercise:

At a point with latitude near 58°N,
the Last-Quarter Moon is seen rising in the north-east.

What time of day is it?
What time of year is it?