Is the difference in time between the sideral and synodyc month - constant?

According what I read, the sideral month isn't constant (here)1, as well as the synodic month that isn't constant (here)2. Now, as far as I understood, because the sideral month isn't constant therefore the synodic month isn't constant too.

My question is: Is the difference in time between the sideral and synodyc month (around 30 degrees) - constant, or it may be various too?

The information above was taken from here:

1) synodic month. (n.d.) The Great Soviet Encyclopedia, 3rd Edition. (1970-1979).

2) sidereal month. (n.d.) The Great Soviet Encyclopedia, 3rd Edition. (1970-1979).

• Since you are interested in the details of the Moon's motion, you may also be interested to learn about the phenomenon known as the lunar standstill. And for even more details, take a look at lunar theory. Apr 22 '20 at 17:42

The time difference (synodic month) - (sidereal month) will not be constant. The reason for the variation is due to the eccentricity of the Moon's orbit: the Moon moves faster at perigee (closest to the Earth) than when at apogee (farthest from the Earth). If the Moon is near perigee around the time of New Moon, it takes less time for the Moon to "catch up" with the Sun, so the synodic month is shorter.

Also, the Moon's perigee and apogee distance changes from month to month due to the influence of the Sun. This also affects the speed of the Moon in its orbit. (The apparent speed of the Sun also changes because of the Earth's elliptical orbit around the Sun, but the variation is smaller.)

It may be helpful to take a "geocentric" view; that is, look at the motion of the Sun and Moon on the sky. In the following figure, a rare annular eclipse occurs with the Sun and Moon occulting the star Regulus ($$\alpha$$ Leonis). The days listed are the average sidereal and synodic periods. After 27.32 days (one sidereal month), the Moon is again aligned with Regulus. The Sun is farther east in the sky, so it takes a few more days for the Moon to catch up with the Sun and complete a synodic month (39.53 days). Note: The distances travelled are approximate in this figure. The apparent size of the Sun and Moon are greatly exaggerated!

The two months are related by the speed of each object:

• average speed of Moon = 360 degrees/27.32 days
• average speed of Sun = 360 degrees/365.26 days
• distance travelled by the Moon in one synodic month = T*360/27.32
• distance travelled by the Sun in one synodic month = T*360/365.26

The two distances are not equal: the Moon moves an extra 360 degrees. So the length of the average synodic month, T, can be calculated by solving this equation: $$T*\frac{360}{27.32}-360=T*\frac{360}{365.26}$$

It should be apparent from this equation that if the speed of the Moon varies, then the duration of the synodic month T will change as well.

• It may be worth mentioning that the lunar apsidal precession is rather rapid, with a period of only 8.85 years. Apr 22 '20 at 12:51
• A great answer. Thank you. Do we have a way to know the range? (Maximal and minimal time it may take) Apr 22 '20 at 13:47
• The synodic period is easy. Get a list of date & time of New Moon for a year and subtract the values. The sidereal period is more time consuming. It would require finding the date & time when the Moon returns to the same ecliptic longitude (such as 0 deg), and then subtract the two date/time values. A site like JPL Horizons could be used to produce a table from which the date/time can be interpolated. Apr 22 '20 at 15:11
• I have no problem to substract, but my problem is to find 1 year exact time for the begining and the end of thenew moon. Do you where I can get this information from? Apr 22 '20 at 18:35
• @RecklessGlacier Try timeanddate.com/moon/phases Apr 23 '20 at 1:06

I used Skyfield to find the actual lengths of synodic and sidereal months from 2010 through 2029 according to JPL ephemeris DE430. For each synodic month, I chose the sidereal month having the same midpoint in time. Standard deviations were 0.18 day in synodic month length, 0.07 day in sidereal month length, and 0.19 day in the difference between them. The blue dots represent synodic months measured from new moon to new moon or full moon to full moon. The red dots represent months beginning and ending at first quarter or last quarter. Points on the gray curves are reckoned for other phases.

If the difference between synodic and sidereal month lengths were constant, all the points would lie on the dashed diagonal line.