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I've come across a number of sources (example) saying some variation of this:

The moon's orbital path around Earth is tilted by 5.1 degrees with respect to the orbit of the Earth around the sun.

This isn't strictly true though, right? Per this article, the angle of the moon's orbit varies by something like +/-5 degrees. So really we should say "The moon's orbital path around Earth is tilted by up to 5.1 degrees".

Is this a correct understanding?

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    $\begingroup$ I think you're misreading the lunar standstill article. It's just saying that the Moon's ~5.1° inclination to the ecliptic (which is inclined by ~23.5° to the equator) means that the Moon's maximum declination range varies over the nodal precession cycle from ±(23.5°-5.1°) to ±(23.5+5.1°). I have some info on the lunar declination & standstill here: astronomy.stackexchange.com/a/50974/16685 That answer is in response to a kind of complicated question, so some of the info & graphs may be a little confusing. ;) $\endgroup$
    – PM 2Ring
    Mar 28 at 1:12
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    $\begingroup$ The Moon's inclination to the ecliptic varies from ~5° to ~5.3° in a rather complicated way. I have graphs of the inclination and other lunar orbital elements here: astronomy.stackexchange.com/a/55112/16685 $\endgroup$
    – PM 2Ring
    Mar 28 at 1:16

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Not quite.

The angle of the Moon's orbital plane with respect to the ecliptic is fairly constant at ~5.1° with a bit of variation as pointed out by @PM2Ring in their comment. The orbit and the plane defined by it does not change (much).

Over the duration of one month the Moon moves along its orbit. Thus its position changes during these days from a position about 5.1 degrees below the ecliptic to on on the ecliptic 7 days later to one 5.1 degrees above 14 days later - and back to -5.1° when the orbit is completed one month later.

Thus:

The Moon's orbital path around Earth is tilted by 5.1 degrees.

The Moon's position varies up to 5.1 degrees below and above the ecliptic.

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Is this a correct understanding?

Not exactly. The Sun is a huge perturber of the Moon's orbit. From a heliocentric perspective, the gravitational acceleration of the Moon toward the Sun is over twice that of the Moon toward the Earth. From a geocentric perspective, the Sun is still a huge huge perturber of the Moon's orbit about the Earth. It wasn't until the early 20th century that a fairly accurate description of the Moon's orbit was developed, primarily due to this huge perturbation. The other planets and the Earth's not-quite spherical shape also perturb the Moon's orbit, but to a much lesser extent than the Sun.

That said, the inclination of the Moon's orbit with respect to the ecliptic is fairly constant. Below is a plot of the Moon's instantaneous (osculating) inclination:

Lunar orbit inclination for 2008 to 2010
Source: Eclipses and the Moon's orbit at nasa.gov

Note that the inclination varies from less than 5.0° to about 5.3°, with a mean of 5.145°. So, not quite constant, but still, fairly constant.

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