Astronomy Principles and Practice Fourth Edition.pdf

The Moon’s orbit: practical projects 437
Figure 24.26. Determination of the Moon’s distance by measuring the effect of diurnal parallax on the Moon’s
sidereal position.
24.6.1 Measuring the Moon’s distance
Suppose that at full moon, an observer notes carefully the position of the Moon on a star atlas every
hour, say from four hours before midnight to four hours after midnight. The difficulty of seeing stars
relatively close to the bright Moon can be solved by holding a small disc at arm’s length over the Moon
to eclipse most of its light. Then the Moon’s observed shift in sidereal position will be caused by (i)
the Moon’s motion in its orbit and (ii) the movement of the observer due to the Earth’s rotation.
(a) The parallactic shift
In figure 24.26 the observer in latitude φ ◦ N is moved from O 1 to O 2 during the night’s observations,
of duration t hours. Forgetting for the moment the Moon’s orbital movement, let it be at M while
observations are being made. Then, if δ is the Moon’s declination, and p is the parallactic angle moved
through by the Moon during t hours, we may write
p = α p cos δ (24.10)
where α p is the parallactic displacement in right ascension. It may be noted that, as the observer
moves along a small circle parallel to the equator, there is no corresponding parallactic shift in
declination.
Let the Moon be on the meridian at a time halfway through the period of observation. Then, since
p is a small angle,
p = O 1 O 2 /D (24.11)
D being the Moon’s distance from the observer.
By equations (24.10) and (24.11),
D = O 1 O 2
α p cos δ . (24.12)