# What would the orbit diameter of the Moon be for an 'eclipse free' orbit period?

Let's see if I'm looking at this properly. In a simplistic way (ignoring orbit precession and other minor effects) if the Moon had a period of 365.25 days / 12 (30.438 days) and Moon was positioned in its orbit such that it is not on the line of Sun/Earth alignment described by the intersection of the ecliptic plane and the plane of the Moon's orbit... a solar eclipse would 'never' occur. I think this is reflected in Orbit of the moon so that there are no eclipses

I've tried to calculate the distance to the Moon for this but think I keep screwing up the units and getting nutty distances.

I've been using:

Gravitational attraction force
$$F = G (M_1 M_2)/r^2$$

Centrifugal force
$$F = M_2 r \omega^2$$

Setting them equal to each other
$$G (M_1 M_2)/r^2 = M_2 r\omega^2$$

With angular velocity of the moon ($$\omega$$) such that its period is 30.438 days (1/12 of Earth’s orbital period) what would the distance $$r$$ to the moon be?

Or maybe I've got this all wrong?

We really shouldn't ignore the precession, since the Moon completes a precession cycle in only ~18.6 years. The plane of the lunar orbit is inclined by ~5.14° to the ecliptic, and Earth's equatorial plane is inclined by ~23.44° to the ecliptic, so it's inevitable that the Earth-Moon system wobbles a bit. ;) BTW, the line of intersection of the lunar orbital plane with the ecliptic is called the line of nodes.

As I've said numerous times, the Moon's orbit is only roughly a Kepler orbit. But let's keep things simple, and ignore precession, etc, and assume the orbit is circular, and that it obeys the simple orbit equations.

I gather that your plan is to make a whole number of months in a year, so that the Full and New Moons always occur at the same position relative to the line of nodes, and that those positions are ~90° away from the ascending and descending node points.

The period of the Moon's phases is the Moon's synodic period. We can't use that period in the orbit radius equation, we need to use the sidereal period. But there's a simple equation connecting those periods. Let $$Y$$ be the length of the sidereal year, $$T$$ the sidereal month, and $$S$$ the synodic month. Then $$\frac1 S = \frac1 T - \frac1 Y$$

You want $$12S = Y$$ exactly, so $$13T = Y$$.

Now $$Y \approx 365.256363$$ days, so $$S \approx 30.438030$$ days and $$T \approx 28.096643$$ days.

The equation relating the orbit radius $$r$$ to its period is $$4\pi^2 r^3 = \mu T^2$$ where $$\mu = GM$$ is the gravitational parameter . For the Earth-Moon system we should use the combined mass of the system. JPL has the latest gravitational parameter estimates on its Astrodynamic Parameters page.

Using that equation with the true sidereal month length (~$$27.321662$$ days) gives $$384747.962$$ km for the radius, which is only a bit larger than the true mean orbit radius, $$384399$$ km.

Using it with our larger sidereal month gives a radius of $$391989.604$$ km.

• Thank you so much! I didn't realize that Moon's precession period is so short. It would seem to be a trivial matter to crank that factor into the calculation. I wonder how much precession alone would contribute to the difference. As the answer stands now there is a 1.97 % increase in orbital radius. By the way... I wasn't considering things in terms of "months" but rather sticking close to present earth/moon system size. Similar to the way @Woody did in his answer to prior question. Woody provided excellent diagram of system there. Commented Apr 10 at 13:34
• As a final note... as I see it, Moon could be just a few degrees off of the line of nodes and remain eclipse free. Does not need to be 90 deg off. Commented Apr 10 at 13:34
• @BradV Each eclipse season lasts about a month. There are always 2 & sometimes 3 eclipses per season. So the distance needs to be more than just a few degrees. Including precession is tricky: the line of nodes precesses backwards relative to the orbital direction. Commented Apr 10 at 13:46
• The apsidal precession is even faster. It makes it even more complicated because it affects where the Moon's motion speeds up and slows down. See en.wikipedia.org/wiki/Lunar_precession and en.wikipedia.org/wiki/Lunar_month for the lengths of the various basic periods. Commented Apr 10 at 13:49