The Average Motion of the Moon
The Moon moves around the Earth in an approximately circular orbit, going once around us in approximately 27.3 days, or one sidereal period of revolution. As it does so, its position changes, relative to the stars.
Since there are 360 degrees in a circle, the Moon moves 360 / 27.3 or 13.2 degrees per day relative to the stars, which is just over half a degree per hour, and approximately equal to its apparent size. This means that from night to night, the Moon moves a little more than one hand-width to the East (the direction of its motion around the Earth) relative to the stars, and from hour to hour, it moves about one diameter to the East, among the stars.
An approximate representation of the motion of the Moon around the Earth. Moving once around in 27.3 days, its average movement is about 13.2 degrees per day, or 92 degrees per week. (As is usual in such diagrams, the sizes of the Earth and Moon are exagerrated, in comparison to their distance.)
The apparent motion of the Moon, from night to night.
Each night, it moves around 13 degrees, or about 26 diameters.
The apparent motion of the Moon, from hour to hour; each hour, it moves about one diameter to the East.
Above: The crescent Moon, Venus, Jupiter and Spica as seen near Quebec on September 6, 2005
Below: The crescent Moon, Venus, and Jupiter as seen near Los Angeles on September 6, 2005
Credits: Quebec, Jay Ouellet, apod050909; Los Angeles, Sheri Seligman
Although these pictures were taken at about the same time of evening, the lower picture, taken three time zones to the west, was taken nearly three hours later, so the Moon's position is shifted three diameters to the left as a result of its orbital motion during that period. (There is also a slight difference in the alignment of the objects, because the diurnal paths of celestial bodies are more horizontal at the more northern latitude of Quebec.)
The Moon's Westward Motion Across the Sky
Although the Moon is moving eastward around the Earth, the Earth is also turning to the east, and much faster, for it goes all the way around its axis of rotation in just under a day. As a result, although the Moon is moving to the east relative to the stars, the much faster westward motion of the sky is carrying it to the west, and so, despite its eastward motion relative to the center of the Earth, it rises in the east, and sets in the west.
The stars go once around the sky in 23 hours 56 minutes (approximately), so the Moon, moving more slowly to the west, takes longer than this. Since its eastward motion averages 13.2 degrees per day, and the Earth takes 4 minutes to rotate through one degree, it takes about 53 minutes (13.2 times 4) for the Earth to rotate through this extra angle; which means that, on the average, the Moon crosses the sky once every 24 hours and 49 minutes (53 minutes longer than the stellar "day"). As a result, it rises (and sets) later and later, every day, until -- after about 27 days, when it has gone once around the sky, relative to the stars -- it is back in its original position, rising and setting at its original time(s).
The Moon's eastward motion is much slower than the sky's westward motion. So, though moving to the east from day to day, it still has a net motion toward the west, each day. This means that it still rises in the east, and sets in the west, like the stars, but a little later, each day
The Moon's Variation in Distance and Speed
Although the Moon has an average motion of 13.2 degrees per day, this motion varies, for two reasons. First, the orbit of the Moon is an ellipse, and is not centered on the center of the Earth, but on a point about 12000 miles from the center of the Earth. As a result, during each orbit, the Moon's distance varies by twice that 12000 miles. During half the orbit, it is approaching us, and during the other half, it is moving away from us. During the half orbit that it is approaching us, our mutual gravitational pull accelerates the Moon, causing it to move faster and faster, until at the closest point in the orbit, or perigee, it is moving about 6% faster than its average motion. Similarly, during the half orbit that it is receding from us, our mutual gravitational pull decelerates the Moon, causing it to move slower and slower, until at the furthest point in the orbit, or apogee, it is moving about 6% slower than its average motion. In addition to these actual changes in velocity, there is an apparent change caused simply by its being nearer or further; when it is closer, any motion that it has looks faster, in angular terms, than when it is further away. This effect causes another 6% apparent increase or decrease in velocity, in addition to the actual change.
In other words, as the Moon approaches perigee, its angular speed among the stars will appear to increase by 12% of its average speed, half of that change being due to its lesser distance, and half being due to an actual increase in speed; and as it approaches apogee, its angular speed among the stars will appear to decrease by 12% of its average speed, half of that change being due to its greater distance, and half being due to an actual decrease in speed. Since 12% of 13.2 degrees per day is 1.6 degrees per day, the daily motion of the Moon to the east can vary from as little as 11.6 degrees per day near apogee, to as much as 14.8 degrees per day, near perigee.
An image showing the apparent size of the Moon at apogee (on the left), and at perigee (on the right). (The change in size is not so obvious when the time between the extremes is two weeks, as it is when images are placed side by side.) When at apogee, the Moon will appear to move less than 12 degrees per day to the East among the stars, whereas at perigee, it will appear to move nearly 15 degrees per day. (António Cidadão, apod041021)
Other topics to be covered as time permits:
(need to discuss rate of motion, rising/setting later each day (can leave effects due to N/S motion until later on, or perhaps only cover in more detailed pages intended for lab students), effects of eccentricity, and precessional motions of orbital and plane and line of nodes -- any other topics?)
(notes from Fall 2004; need to compare to above, and expand or delete)
The Moon moves one diameter, or half a degree, per hour, or about 26 diameters, or 13 degrees, per day (27 1/3 days, its orbital period, divided into 360 degrees, or once around). (Refer to the page on Tycho Brahe's astronomical accomplishments, for a more detailed discussion of the parallax of the Moon?)
THE SUN FOLLOWS ALMOST EXACTLY THE SAME PATH, but only moves two diameters (again, about half a degree per diameter, or one degree) PER DAY (365 1/4 days, our orbital period, divided into 360 degrees -- which is probably about the same number, specifically because of our orbital period).
It is easier to see the motion of the Moon, than of the Sun, for two reasons -- it is much faster, AND you can see stars, when the Moon is near them, but NOT when the Sun is near them. However, it is possible, by measuring the right ascension and declination of the Sun, to see that it does follow almost exactly the same path as the Moon, but much more slowly.
(Other planets follow similar paths, but have more complicated motions, involving retrograde loops and esses. This is because the motion of the Moon only involves one motion -- its own; and the motion of the Sun only involves one motion -- ours, around the Sun; but the motion of another planet involves two motions -- its and our motion around the Sun (see discussion of retrograde motion, in the book, and on the web site).