Chapter 3 – The Moon (The Moon’s Phases)Posted: March 20, 2013
The Moon’s Phases, the phases of the moon over a monthly cycle
Phases of the Moon, plots the Earth and Moon in orbit around each other with a full description of each phase
UDG Moon’s Phases, using the UDG character set.
The lifegiving Sun gets a raw deal in the public eye and the reason is not too difficult to find — the weather. The Sun completely dominates the scene and man has absolutely no control over these events.
In contrast the Moon has no such public relations problems. It is probably the first object in the sky to set a young mind wondering — ‘What is it?’ — ‘Where is it?’. For many the Moon is the embodiment of all that is mysterious and strange beyond the cosy confines of home and yet it is just the stepping stone to the exploration of a beautiful universe. Even if we can never visit these places in person we can learn to understand them and perhaps view them aided with modest binoculars or a small telescope. The Moon is a perfect starting point.
The Moon’s Phases
The Moon is our nearest neighbour in space and the only world (apart from Earth) that man has set foot upon — during the late 1960s with the Apollo missions. The program that follows is one of the shortest in this book but the Spectrum graphics more than compensate for its brevity. The complete phases of the Moon from new moon to full moon and back again (via a shrinking crescent) to new moon are shown, covering one monthly cycle. Figures 3.1 and 3.2 are typical examples.
The Moon one day old
The Moon after ‘Full’
The Moon’s appearance
Artists are renowned for including the crescent moon in their works and invariably getting its appearance completely wrong! The phases of the Moon are measured from new moon to the next new moon, and in reality the Moon is invisible at this stage due to its apparent proximity to the Sun. The only true new moon that can be seen is when it passes directly in front of the Sun during an eclipse of the Sun, so that the Moon’s silhouette is revealed.
The commonly ascribed new moon is when the crescent first becomes evident, about two days later, in a bright western sky as a letter C backwards. The crescent grows in the days that follow, becoming brighter as more of the lunar surface is illuminated as it moves to the left away from the Western skyline.
About fourteen days later the Moon appears as a very bright circular disc due south at midnight — full moon. It is now directly opposite to the Sun (hidden below the northern horizon). Continuing to move to the left amongst the starry background, its phase diminishes until fourteen days later it is lost in the bright dawn sky having shrunk to a fine crescent again — this time like a true letter C (ie not reversed). Astronomers find it convenient to regard the crescent phase as a bow — an arrow shot from it would pierce the Sun. Artists please note!
The program uses a simple FOR/NEXT d loop to DRAW the two arcs which represent each phase. Line 80 first PLOTs a reference point on which to start the DRAWing and then DRAWs the bright ‘limb’ or edge, that is nearest to the Sun’s direction. Line 70 uses a conditional PRINT statement to identify this limb with the word ‘sunward’. Line 110 then DRAWs the ‘terminator’ or sunrise/sunset line on the Moon’s surface.
Because the Moon is a near-perfect spherical globe, the limb is DRAWn with the value PI (via the variable P in Line 40) as a semicircle. Line 50 renders this value as negative after the fourteenth day (full moon) so that this initial arc is DRAWn on the left of the screen instead of on the right, and in the opposite curvature. The formula in Line 110 controls the terminator line from semicircular (new or full moon) to a straight line (first or last quarter moon). Lines 90 and 100 ensure the terminator arc is DRAWn with a positive (to right) or negative (to left) curvature.
The Moon is effectively a globe and, although the terminator line advances at about 13° each day (28 days x 12.86° = 360°) as seen from Earth, foreshortening occurs so that the daily motion of this line is not constant. It appears to advance more rapidly about the time of first or last quarter moon whilst in the centre of the disc. About new or full moon the terminator is seen at such acute angles that the actual movement is much compressed — to the point where the precise occasion of full moon can be misjudged by at least a day or so.
This program mimics this effect solely by the formula in Line 110. As an experiment, try amending the variable x in Line 30 and the value in the formula (the last part of the expression), ie 25.
The program is intended to indicate the varying phases of the Moon and no particular accuracy is claimed for the precise shape of the terminator line or for the timing interval of the monthly cycle. These two aspects are unrelated and are explained below.
The true profile of the terminator line is an ellipse but the Spectrum’s DRAW command is used instead (where the third part of the expression produces the curvature) to ensure rapid results. The errors produced by the arc when compared with the ellipse are not excessive. The major part of the error, such as it is, occurs adjacent to the polar regions of the DRAWing from a latitude of about 60° to 75° and then only for a few days about first or last quarter moon. This compromise was deemed acceptable.
The FOR/NEXT d loop in Line 40 accounts for a 28-day period to represent the monthly cycle of phases. This period is precisely divisible by four, so that each quarter moon is displayed. In reality the Moon’s ‘mean synodic period’ (average from new moon to new moon) is 29d 12h 44m or about 29.53 days, which means that the program RUNs about 5% fast on the real thing.
9 REM ***********************
10 REM Moon’s Phase
11 REM ***********************
30 LET x=2.5: LET r=PI/180
40 FOR d=0 TO 28: LET p=PI
50 CLS : IF d>14 THEN LET p=-p
60 PRINT “Moon’s Phase-day “;d
70 PRINT AT 12,(25 AND d7 THEN LET b=b-14
110 DRAW 0,-130,x*ATN (r*b*25)
120 PAUSE 50: NEXT d
9900 REM ***********************
9990 SAVE “phaseM”