Expert: Jayendra Upadhye - 1/19/2005

Question
Hi, I'm a university student taking first year astronomy. so far we've been looking at the foundations of astronomy. I've come across an interesting question, that I've been really having trouble getting my head around. We were discussing the synodic month and the moon;s orbital period.  
The question I have is what would the lenght of a synodic month change to if the Moon's sidereal orbital period were one week in solar days ? and if it were one sidereal year?
Now i know that for the sidereal year it should technically be infinite, but I cant really understand why. For the week, the answer is about 7.14 days, however in class I didn't really understand the calculations accompanied with this answer. if you could help me explain these two things, I would greatly appreciate it!

Thanks a bunch!
Lara

Answer
Hi,
Lara, definitions first!
1 - synodic month:- (thanks to google)
The period of time between two successive identical phases of the moon; this is the lunar month.. 29.53 days

2 - sidereal month:- (thanks to google)
Average period of the revolution of the Moon around the Earth with respect to a fixed star, equal to 27.321,661 mean solar days.
or
The average period of revolution of the Moon around the Earth in reference to a fixed star, equal to 27 days, 7 hours, 43 minutes in units of mean solar time.

3 - sidereal earth year:-
the time required for the Earth to complete an exactly 360° orbit around the Sun as measured with respect to the stars = 365.2564 mean solar days (contrast with tropical year).
[31558152.96 seconds]

Now, after some simple geometrical analysis, i have found that the formula for synodic month (unfortunately i cant draw figures here, contact me at jupadhye@hotmail.com, and i will mail you the answer with attached figures., in case you need them), works out as under.
Time (synodic) in seconds = {(2*pi)/(angular vel of moon - angular velocity of earth)}
let us work out and check the formula for existing values.
existing angular velocity of earth
W(e) = 2*pi/[number of seconds in its sidereal year,its sidereal orbital period]
W(e) = [1.9909 * 10 ^ (-7)] radians/second
W(m) = 2*pi/[number of seconds in its sidereal year,its sidereal orbital period]
W(m) = 2.6618 * 10 ^ (-6) radians/second

the current synodic month = 2*pi/[W(m)-W(e)] = 29.5298 days
Remarkable accuracy of this formula! google defines synodic month as 29.53 days! [just type "define:synodic month" without the quotes into google]

So now we are ready to apply this formula to your "hypothetical" situations.

case 1 - sidereal month = 1 sidereal week.
new W(m) = 2*pi/(7 *3600 *24) = 1.03888 * 10^(-5) radians/sec.
new synodic month = 2*pi/[1.03888 * 10^(-5) - 1.9909 * 10 ^ (-7)]
new synodic month = 7.136 days (as you said 7.14)
So case 1 too works out for this formula!

Case 2 - 1 sidereal month = 1 sidereal year
New W(m) matches W(e)
so the term in the denominator for synodic month =
2*pi/[W(m) - W(e)], becomes zero
and (2*pi)/zero WILL be infinity!

The geometry part is easy to visualise.
The moon and earth "turn" in the same direction, in their respective orbits.
So when we have a full moon, the sun, earth moon form a straight line.
SAY this line lies on the x axis at our first observation. (starting point).
After a synodic month, by definitions, all 3 are alligned again!
But that means that the moon not only traversed 2*pi radians of a complete cirecle, it traversed an additional angle forward, that exactly matches the angle slowly traversed by the earth in that much time! That is why they are alligned again in the first place!
Thus the first equation is
1 - Angle traversed by earth in one synodic month = aNGLE covered by the moon in ONE SYNODIC MONTH - (2*pi)
or [W(m) * T] - (2*pi) = W(e)* T
where T is synodic month in seconds.
Therefore second equation becomes..
2 - The synodic month T = 2*pi/[W(m) - W(e)] !

Now THAT should be clear, ny itself.
As the earth advances in its own orbit, the direction of the new allignment advances by angle W(e)*T.
The moon has to thus travel this additinal angle in order to allign itself and complete the synodic month.

Had the moon's direction been retrograde, the equation would change.
I leave it to you as an interesting excercise.
try it out with friends.

Pls rate this answer, as i really need to know if it helped.

Jayen