What does the Moon's orbit around the Sun look like?

Question

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:

The Moon's orbit must be concave toward the Sun.
The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.
For an observer on the Earth, the Moon appears to orbit the Earth roughly $13$ times a year.
The Earth, the Moon, and the Sun remain in the same plane.

You mean this: http://blogs.discovermagazine.com/badastronomy/2008/09/29/the-moon-that-went-up-a-hill-but-came-down-a-planet/#.V3wqZyMkpmA? — CuriousOne, Jul 05 '16 at 21:45
It looks just like the Earth's orbit around the Sun, if viewed from a sufficient distance. — davidbak, Jul 06 '16 at 00:19
@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. — David Hammen, Jul 06 '16 at 01:46
@DavidHammen: I wasn't implying that the detailed orbit looks exactly like that, only trying to establish if the OP was looking for something like this in general. I simply could not find a drawing with the right scale (150 million km to 380,000 km is kind of a poor ratio for the purposes of visualization). — CuriousOne, Jul 06 '16 at 01:53
@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. — David Hammen, Jul 06 '16 at 01:57
@DavidHammen: This may satisfy your desire for detail more: https://www.wired.com/images_blogs/wiredscience/2012/12/earthmoonpath.jpeg :-) — CuriousOne, Jul 06 '16 at 02:02
@DavidHammen: I am having a bit of trouble with the convexity. The lunar orbit is a few degrees inclined against Earth's orbit, so the trajectory is a 3 dimensional curve. How do we define the "interior" for those, again? If I imagine the projection of orbit onto the celestial sphere, it should go above and beyond the ecliptic, right? That curve certainly has secant lines that are above and below the ecliptic, but how do I define convexity in this case? It seems that the secants should wind like a corkscrew around the orbit. Other than that I agree for the case of coplanar orbits. — CuriousOne, Jul 06 '16 at 02:54
@CuriousOne -- One way to look at it is the inner product of the curvature $\partial T / \partial s$ with the vector from the Moon to the Sun. It is always positive. — David Hammen, Jul 06 '16 at 03:15

honeste_vivere · Answer 1 · 2020-02-18T16:24:31.423

59

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:

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.

The correct result is shown in the zoomed-in view of David Hammen's post above.

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.

Incorrect (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 still an incorrect (i.e., exaggerated) orbital motion but much better approximation, 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 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). Here the orbit is always convex with respect to the sun, as David has correctly stated.

edited Feb 18 '20 at 16:24

answered Jul 05 '16 at 22:45

honeste_vivere

15,592

15

I'm afraid this is not quite right because the moon's orbit doesn't have loops. http://www.math.nus.edu.sg/aslaksen/teaching/convex.html – Diracology Jul 05 '16 at 23:19
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This is completely incorrect. The Moon's orbit about the Sun is convex. – David Hammen Jul 06 '16 at 01:23
@DavidHammen - I added a more realistic animation and clarified that the results are over exaggerated but even in your zoomed-in picture, the orbit would still be an epicycloid, would it not? It's just that the number of cycloid motions would be far fewer and much less exaggerated than what is shown in my animations. – honeste_vivere Jul 06 '16 at 12:16
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@honeste_vivere As far as I can see, it would not. Because of the speeds of Earth and Moon, the Moon would relative to the Sun never move backwards. This fact makes the loops that you show impossible. – Steeven Jul 06 '16 at 12:38
@Steeven - I agree, the in/out path would not be as sharp as is shown in my animation. However, the moon orbits the Earth and the Earth orbits the sun. Your statement seems to suggest that the moon always stays on the anti-sunward side of the Earth for it went to the sunward side, it would need to retreat at some point. So how are there solar eclipses? I do not follow but I assume I have missed something... – honeste_vivere Jul 06 '16 at 12:56
No, no, it is much simpler than that. From the loops on your animation, you assume that at some point in time, the Moon will have negative velocity relative to the Sun. That is the only way that the path can cross itself and create a loop. And this is what does not happen, as far as I can see. Maybe, if you can redo your first animation here with the proper speeds scaled to fit each other. Speeds are given in @aventurin's answer. – Steeven Jul 06 '16 at 13:06
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@Steeven - Okay, I see and have added a correction. What I screwed up was to assume that there was no lag between the rotation of the blue circle and the red dot (i.e., kind of like having a rigid rod connecting the center of the blue circle and the red dot). Thanks for the clarification. – honeste_vivere Jul 06 '16 at 13:40
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@DavidHammen - I added a much more realistic animation now. – honeste_vivere Jul 06 '16 at 19:39
@Diracology - I fixed my animations and corrected some statements. – honeste_vivere Jul 06 '16 at 19:40
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With your "correct version" (v6) of the animation this answer becomes very nice. – rob Jul 06 '16 at 21:10
@rob - Yeah, unfortunately, it took me several iterations to realize just how slowly the moon orbits the Earth relative to its motion around the sun ;P – honeste_vivere Jul 06 '16 at 23:58
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WOHOOO fancy gifs. The best way of communicating science is, and always will be fancy gifs. – Ander Biguri Jul 07 '16 at 09:40
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@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. – David Hammen Jul 10 '16 at 13:53
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@DavidHammen I was just giving some good vives to someone that did a good effort .... So easy to offend someone in the internetz :( – Ander Biguri Jul 10 '16 at 23:25
@AnderBiguri - David is correct, all the animations are over exaggerated for viewing purposes but not physically correct. I created one that was accurate and it looked just like David's correct plot, which almost follows what the Earth's orbit looks like. People really should be voting for David's post more, as he was one of the people that helped to correct my initial mistakes/errors. – honeste_vivere Jul 11 '16 at 12:31
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@honeste_vivere yes I was giving props for the gifs! Just make fun gifs, thats what I meant! nothing more! – Ander Biguri Jul 11 '16 at 12:40
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@AnderBiguri - No worries, my comment was meant for all users not just yourself. I noticed that I had somehow gotten more votes than David and was a little concerned since he posted first and corrected my first few versions. Alex should accept David's answer, not mine for accuracy and future readers benefit. – honeste_vivere Jul 11 '16 at 12:52

score 49 · Answer 2 · edited Mar 02 '21 at 22:58

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:

^{(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:

^{(Source: http://www.math.nus.edu.sg/aslaksen/teaching/convex.html)}

A close-up view:
^{(Source: http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/)}

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.

"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". — JBentley, Jul 06 '16 at 07:00
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. — , Jul 06 '16 at 16:55
Bonus question: how often do the earth's and moon's orbits cross? — R.. GitHub STOP HELPING ICE, Jul 07 '16 at 02:37
@R.. the lunar period is 28 days, and there are 365 days in a year, so 365*2/28=25 times. — Davidmh, Jul 07 '16 at 11:29
"This is easily proven...". At first I thought you meant that it follows from the difference in accelerations alone, but I guess you are assuming facts about the actual earth-moon-sun system — doetoe, Jul 10 '16 at 15:05
Does any moon in the Solar system describe loops in an epitrochoid? Like fast Io around Jupiter, or Charon around slow Pluto? — LocalFluff, Feb 10 '18 at 12:17

Diracology · Answer 3 · 2016-07-06T11:59:00.157

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.

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... — craq, Jul 06 '16 at 11:12
...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.) — craq, Jul 06 '16 at 11:12
@craq I agree with you. I shall mention in my post that I am considering the Sun's frame of reference. — Diracology, Jul 06 '16 at 11:55

score 10 · Answer 4 · answered Jul 05 '16 at 22:08

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.

What does the Moon's orbit around the Sun look like?

4 Answers4

Incorrect Path

Update/Correction

Incorrect (still exaggerated) Path

Correct Path

Linked