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I'm curious as to what the Moon's orbit around the Sun looks like. If there's an answer, what's the intuition for it? Here are some things I'm assuming when trying to tackle this question:

  1. The Moon's orbit must be concave toward the Sun.

  2. The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.

  3. For an observer on the Earth, the Moon appears to orbit the Earth roughly $13$ times a year.

  4. The Earth, the Moon, and the Sun remain in the same plane.

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    $\begingroup$ I am curious also : what do you think it looks like? $\endgroup$ – sammy gerbil Jul 5 '16 at 21:35
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    $\begingroup$ You mean this: blogs.discovermagazine.com/badastronomy/2008/09/29/…? $\endgroup$ – CuriousOne Jul 5 '16 at 21:45
  • $\begingroup$ It looks just like the Earth's orbit around the Sun, if viewed from a sufficient distance. $\endgroup$ – davidbak Jul 6 '16 at 0:19
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    $\begingroup$ @CuriousOne: Phil Plait was guilty of bad astronomy in that blog. It doesn't happen very often, but it did in this case. His "This" figure is a greatly exaggerated view of the orbit of a vehicle in low Earth orbit about the Sun, but not the Moon. There are no cusps, there isn't anything cusp-like. You know what a convex closed curve looks like. That figure is not a convex curve. $\endgroup$ – David Hammen Jul 6 '16 at 1:46
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    $\begingroup$ @CuriousOne -- The right scale would result in a curve that is everywhere convex and very close to a nearly circular ellipse. Pick any two points on the curve and connect them with a line segment. Except for the endpoints, every point on that line segment will be in the interior of the curve. $\endgroup$ – David Hammen Jul 6 '16 at 1:57
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Incorrect Path

I'm curious as to what does the moon's orbit around the sun looks like?

One might think the orbit (in the sun's rest frame) follows the path of an epitrochoid.

A (very) over exaggerated view of this motion (for unrealistic parameters, thus, not an accurate representation) can be seen in the following animation: over exaggerated orbit of the moon about the earth, about the sun

Note that if you change the ratio of the different radii to values that are to scale, then the plot would look more like an epicycloid as in the following example animation. The orbit is more realistic but still exaggerated because it would be impossible to show half an orbit to scale.

more realistic view, but still wrong

The correct result is shown in the zoomed-in view of David Hammen's post above, but the orbit can still be approximated by an epicycloid.

Update/Correction

The above animations are flawed because the little red dot orbits as fast as the blue circle rotates as it "rolls" around the large red circle with no slippage. For a realistic Earth-moon system, there should be a lag between the rotation of the blue circle and the red dot, as if the blue circle were "slipping." Or equivalently, one would not use a rigid axis connecting the center of the blue circle and the red dot. This would result in there never being a negative velocity of the moon relative to the sun in the fashion shown by the epicycloid path in the 2nd animation above.

Correct (still exaggerated) Path

The correct path uses two different orbit rates, one for the Earth about the sun (i.e., 1 year) and one for the Moon about the Earth (i.e., ~27 days). In the following example, which is the correct (still exaggerated) orbital motion, I exaggerated the ratio of the astronomical unit to Earth radius by a factor of 100 and increased the moon's orbit by a factor of four to help make the visualization more obvious.

correct (still exaggerated) orbital path of moon

Correct Path

The following example does not exaggerate the orbital periods relative to each other and is a zoom-in of the above graphic (oddly the GIF, created through the same methods, does not loop on my screen).

correct (not exaggerated) orbital path of moon

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    $\begingroup$ I'm afraid this is not quite right because the moon's orbit doesn't have loops. math.nus.edu.sg/aslaksen/teaching/convex.html $\endgroup$ – Diracology Jul 5 '16 at 23:19
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    $\begingroup$ This is completely incorrect. The Moon's orbit about the Sun is convex. $\endgroup$ – David Hammen Jul 6 '16 at 1:23
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    $\begingroup$ With your "correct version" (v6) of the animation this answer becomes very nice. $\endgroup$ – rob Jul 6 '16 at 21:10
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    $\begingroup$ WOHOOO fancy gifs. The best way of communicating science is, and always will be fancy gifs. $\endgroup$ – Ander Biguri Jul 7 '16 at 9:40
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    $\begingroup$ @AnderBiguri -- No it's not. Saying that animated gifs are the best way of communicating science is exclusive; some people cannot see. Moreover, the animation gives an incorrect view of things. The shown curve is not convex. The correct view would be a curve that is so close to elliptical that the difference cannot be seen. $\endgroup$ – David Hammen Jul 10 '16 at 13:53
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  1. The Moon's orbit must be concave toward the Sun.

The Moon's orbit with respect to the Sun is always convex. This is easily proven by comparing the minimum possible gravitational acceleration of the Moon toward the Sun (5.7 mm/s2) and the maximum possible gravitational acceleration of the Moon toward the Earth (3.1 mm/s2). The acceleration vector, and hence the curvature, is always inward.

This means the Moon's orbit about the Sun doesn't look like either of the two images below:

Two images of the Moon's orbit about the Sun, courtesy of Phil Plait (who almost always is very good, but not in this case). One is labeled "Not This!", the other "This!". The left image, labeled "Not This!", depicts an epicycloid with overlapping leaves. That is a correct view for the innermost moons of Jupiter, but not for anything that orbits the Earth. The right image, labeled "This", depicts a less extreme epicycloid with cusps. There are no cusps in the Moon's orbit about the Sun.
(Source: https://blogs.discovermagazine.com/badastronomy/2008/09/29/the-moon-that-went-up-a-hill-but-came-down-a-planet/#.V3xtsDcgtOo)

Phil Plait is normally very good. This is one of those places where he was bad. The Moon's orbit about the Sun instead looks like this:
An image of a curve that is nearly circular.
(Source: http://www.math.nus.edu.sg/aslaksen/teaching/convex.html)

A close-up view: A zoomed-in view of a small part of the Moon's orbit about the Sun. Even zoomed in, the Moon's path remains convex. The difference between the Earth's and Moon's paths about the Sun is very small.
(Source: http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/)

  1. The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.

Yes and no. The “yes” part is that the dominant feature of the Moon's orbit about the Sun is that the Moon orbits the Sun with the Earth. Ignoring the Moon's acceleration toward the Earth, the Moon's acceleration toward the Sun is greatest now in early July when the Earth/Moon system is closest to the Sun than at any other time.

The “no” part: The Moon is moving slightly away from the Sun when the Moon is in the second and third quarters. This is when the acceleration toward the Earth coincides with the acceleration toward the Sun. In the fourth and first quarters, the acceleration toward the Earth is directed against the acceleration toward the Sun, slowing the Moon down a bit with respect to the Sun.

For an observer on the Earth, the Moon appears to orbit the Earth roughly 13 times a year.

This is correct. More on this below.

The Earth, the Moon, and the Sun remain in the same plane.

This is incorrect. As noted by Diracology in his answer, the Moon's orbit about the Earth is slightly inclined with respect to the Earth's orbit about the Sun.


Note that one of the sites I referenced (http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/) asked whether the Moon orbits the Earth or the Sun. That blog erroneously came up with the answer that the Moon orbits the Sun rather than the Earth. Looking at acceleration or the shape of the orbit is the wrong way to look at the question “Does object A orbit object X or object Y?”

One answer to the question of “does the Moon orbit the Earth or the Sun?” is “Yes!” "Orbit" is not a mutually exclusive term. If one insists on a singular answer, the right way to look at things is via energy rather than acceleration. Since the Moon is gravitationally bound to the Earth, the Moon does indeed orbit the Earth.

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    $\begingroup$ "Phil Plait is normally very good. This is one of those places where he was bad." - to be fair, he did say this: "That diagram is exaggerated; on that scale the Moon’s combined path around the Earth and Sun would look pretty much like a circle". $\endgroup$ – JBentley Jul 6 '16 at 7:00
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    $\begingroup$ The Moon orbits neither the Earth nor the Sun. It orbits the barycenter of the Earth-Moon system. The Earth-Moon system orbits the barycenter of the entire solar system. $\endgroup$ – user57109 Jul 6 '16 at 16:55
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    $\begingroup$ The close-up view really helps here. $\endgroup$ – shadowtalker Jul 6 '16 at 18:34
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    $\begingroup$ @R.. the lunar period is 28 days, and there are 365 days in a year, so 365*2/28=25 times. $\endgroup$ – Davidmh Jul 7 '16 at 11:29
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    $\begingroup$ Does any moon in the Solar system describe loops in an epitrochoid? Like fast Io around Jupiter, or Charon around slow Pluto? $\endgroup$ – LocalFluff Feb 10 '18 at 12:17
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Before answering let me mention that there is a terrific free applet showing the orbits, including the velocity vectors of the system Sun/Earth/Moon:

https://phet.colorado.edu/en/simulation/gravity-and-orbits

It is in java so pretty easy to download and use.

The moon's orbit must be concave toward the sun.

The Moon' orbit around the Sun is a composition of the Moon'orbit around the center of mass of the system Earth/Moon and the orbit of this center of mass around the Sun.

The moon speeds up as it goes toward the sun, and it slows down as it moves away.

Actually is the other way around (in the Sun's frame of reference). The moon slows down when it moves towards the Sun and speeds up when it moves away. You can see this by doing a velocity composition. When The bodies are ordered as Sun--Earth--Moon the velocities of the Earth (approximately the velocity of the center of mass of Earth/Moon) and the velocity of the Moon are approximately parallel and the resultant is a large vector. When they are ordered as Sun--Moon--Earth, the velocities are anti-parallel and the resulting one is a small vector.

For an observer on the earth, the moon appears to orbit the earth, roughly 13 times a year.

Right.

The Earth, the moon, and the sun remain in the same plane.

No. The plane of the Moon's orbit is slightly different than the plane of the Earth's orbit. The difference is about $5^\circ~.$ That is why not every new and full moon is an eclipse (solar and moon eclipses, respectively). Only twice a month (28 day actually) the moon intersects the plane of the Earth's orbit. If this coincides with the alignment Sun--Moon--Earth we have a solar eclipse.

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    $\begingroup$ I think part of the confusion regarding whether the moon speeds up or slows down is related to frames of reference. If you are in the Earth's frame of reference, then you can't add "the velocities of the Earth and the velocity of the moon". If you're in the moon's frame of reference, then the moon's velocity is always 0. It is probably better to look at the acceleration vectors, as the answer by @DavidHammen does... continued... $\endgroup$ – craq Jul 6 '16 at 11:12
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    $\begingroup$ ...continued... Acceleration vectors have parallel components when the moon is further from the sun than the Earth, and anti-parallel components when the moon is closer to the sun than the Earth. So the maximum acceleration occurs when the moon is furthest from the sun. (Whether maximum acceleration corresponds to a speed increase or a speed decrease a.k.a. braking will depend on your frame of reference.) $\endgroup$ – craq Jul 6 '16 at 11:12
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    $\begingroup$ @craq I agree with you. I shall mention in my post that I am considering the Sun's frame of reference. $\endgroup$ – Diracology Jul 6 '16 at 11:55
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The orbital speed of the earth around the sun is about 30 km/s, whereas the orbital speed of the moon around the earth is about 1 km/s.

From this it follows that at no point of its path around the sun the moon will ever show a backwards motion.

The path is similar to the trajectory of a point (moon) on the perimeter of a (somewhat sliding) wheel rolling around a large circle.

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The moon is dynamically and almost rigidly (but for librations) tied to the earth by means of an invisible solid rod, as it were. That is why we inhabitants of earth do not get to see the far side of the moon from our base.

Solar eclipse occurs when the invisible rod places the moon between earth and sun casting a tiny shadow on earth.

Lunar eclipse occurs when the invisible rod places the moon behind the earth in earth's shadow.

Honeste_vivere's animation of moon's motion around sun is well depicted except that there are 10 months to a year instead of 12 making the loops less densely packed than they ought to be.

MoonPositions_inEclipses

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    $\begingroup$ Is this some really odd conspiracy theory? $\endgroup$ – SMS von der Tann Jul 6 '16 at 12:15
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    $\begingroup$ We can use the free energy from the invisible rod to stop governments putting chemicals in our water supplies! $\endgroup$ – Michal Paszkiewicz Jul 6 '16 at 13:33

protected by Qmechanic Jul 10 '16 at 15:15

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