# Moon's orbit around the Sun

The Earth revolves around the Sun and the Moon revolves around the Earth. Out of curiosity I started thinking about the orbit of the Moon around the Sun and expected (assumed) it to be as follows:

But on Wikipedia and some other sites I found out that the orbit is actually like this:

I have 3 questions:

1. What is the reason for this difference between assumed and actual path variation?
2. Has this path been like this since the formation of the Moon?
3. Do natural satellites of other planets also follow the same orbit around the Sun?

After further searching I found out a better, easier explanation regarding the orbit path on YouTube for those interested, be sure to check it out.

• In addition to what @LDC3 said, your scale is way off. Possibly you assumed 12 loops around a circular path would roughly mirror the Moon's path around the Sun? If you try tracing a path where the moon revolves 12 times in the circular path but much more closely to the path, you will get a more similar result. – Mitch Goshorn Jun 13 '15 at 15:57
• One way to think about it is the Earth orbits the sun at 30 KM per second, the Moon orbits the earth at 1 KM per second, so the moon is always moving around the sun at at least 29 KM per second. Another way to look at it is the Moon's orbital diameter around the earth is about 1/2 million miles - a tiny part of the 93 million miles it's from the sun. - taking those facts together, it's not surprising that it has a near circular orbit around the sun, but it's more correct to say the moon orbits the earth cause it's within a stable orbit inside earth's Hill-sphere. – userLTK Jun 13 '15 at 21:37
• Link to Hill Sphere en.wikipedia.org/wiki/Hill_sphere (@userLTK) – Eubie Drew Nov 8 '15 at 20:35
• Also, both pictures are incorrect because orbit of the Moon about the Sun doesn’t form a closed curve after one revolution. If it had, we’d use lunar calendars to define months of the year. – Incnis Mrsi Sep 10 '16 at 22:05
• slightly related: Why is there no concavity in the orbit of the moon around the Sun? – uhoh May 10 at 5:51

What is the reason for this difference between assumed and actual path variation?

Even your second image isn't correct. Imagine zooming in on a small portion of the Moon's orbit about the Sun, for example, one full moon to the next, with the Sun zoomed out of the picture. Now imagine drawing a line segment from one outer cusp (full moon) to the next. In both of your images, that line segment crosses outside of the curve. In other words, both of your curves are concave.

Compare that to the he Moon's orbit about the Sun. This is a convex curve. If you pick any two points on that curve and draw a line segment between them, the entirety of that segment will be on or inside the curve. The reason the Moon's orbit about the Sun is convex is because the gravitational force exerted by the Sun on the Moon is more than twice than exerted by the Earth on the Moon. The orbit would be concave if the Moon was closer to the Earth than 259000 km (about 40.6 Earth radii). Since the Moon orbits at about 385000 km (about 60.4 Earth radii), the Moon's orbit about the Sun is convex.

Whether the orbit of a moon about the Sun is non-simple (first image in the question), simple/concave (second image in the question), or simple/convex (Moon's orbit about the Sun), the deviations from an ellipse are tiny. With regard to the Earth-Moon system, the deviations are so very small that at the plotted resolution (288x288 pixels), the orbits of the Earth, the Earth-Moon barycenter, and the Moon about the Sun will be right on top of one another. The reason the variations are so small (less than one pixel at 288x288 pixels) is because of the huge ratio of the size of Earth/Moon orbit about the Sun compared to the size of the Moon's orbit about the Earth.

Those backward loops in your first image don't happen for any object orbiting the Earth. That would require an orbital velocity about the Earth greater than the Earth's orbital velocity about the Sun. The Earth's orbital velocity about the Sun is about 30 km/sec, considerably more than the orbital velocity of an object in low Earth orbit is about 7.8 km/sec.

Has this path been like this since the formation of the Moon?

No. The Moon formed at four to six Earth radii, far less than the 40.6 Earth radii figure cited above. The Moon's orbit initially looked like your second image.

Do Natural Satellites of other planets also follow the same orbit around the Sun?

The massive planets are much further from the Sun than is the Earth and are much more massive than is the Earth. The orbits of most of the moons of Jupiter about the Sun are concave rather than convex. Only the outermost moons of Jupiter have convex orbits about the Sun. A few of Jupiter's innermost moons (Metis, Adrastea, Amalthea, Thebe, Io, and Europa) exhibit the retrograde motion depicted in your first image.

With regard to moons whose orbit about the Sun is convex, the distances that correspond to the 259000 km value for the Earth are 129000 km for Mars, 24.1 million kilometers for Jupiter, 24.2 million kilometers for Saturn, 19.0 million kilometers for Uranus, and 32.3 million kilometers for Neptune. Both of Mars' moons orbit close-in. However, all four of the giant planets have moons whose semi-major axis orbit fall outside the corresponding limit.

• please explain: "Those backward loops in your first image don't happen for any object orbiting the Earth. That would require an orbital velocity about the Earth greater than the Earth's orbital velocity about the Sun." – model_checker Sep 17 '16 at 19:42
• No object in a closed orbit around the Earth can have a velocity relative to the Earth of more than about 11.2 km/s (escape velocity). Otherwise its orbit would not close. So even when between the Earth and the Sun, it will still be moving relative to the Sun at at least (30-11.2) km/s in the same direction that the Earth is. So it never "loops back". – Steve Linton Jul 12 '18 at 14:02

Not an answer, but I thought this was a good slice of a picture of the Moon's orbit around the sun.

• That's the true picture. The Moon's orbit about the Sun is a convex curve. It has no loops (first image in the question), not even any dimples (second image in the question). Regarding the question raised in the linked article, Does the Moon orbit the Sun or the Earth?, the answer is yes. It's not an either-or question. – David Hammen Jun 14 '15 at 12:08

This is a very old question, and already has great answers, including a diagram drawn to scale. I just want to add a very simple analogy that shows how both pictures in the question are wrong (the second one is less so than the first, if we admit the dubious degree of wrong). Below is how I explained the Moon-Sun motion to some friends, and they grasped it right away, so I still hope my addtion may be helpful.

Think of the Olympic sport of long track speed-skating. Two competitors are running close to each other at a great speed on almost concentric circles (let's make this track round for simplicity, not oval like in real sport). Since one of the skaters is running a bit longer track, the rules require they switch lanes on every lap, so no one of the two is at a disadvantage.

Image credit: Wikipedia File:Jan_Smeekens_(23-02-2008).jpg

Suddenly, the Olympic committee changes the rules. The skaters must now run not a 400m track, but a whopping 10km long one (a circle about 3.2km in diameter), and change tracks not once per lap, but 13 times. The stadium is large, and the curve of the track is so slight, that the outer runner during the change still traces a convex curve when moving towards the inner track at 0.2 m/s while maintaining 15 m/s forward speed along the track [Note: these two figures, unlike others, are inconsequential for the analogy].

Now, say one of the guys is not quite consistent. He speeds up a bit, then slows down a bit, but still catches up with the second. In fact, it happens so that when he moves outside, he's behind the second, but when moving to the inside track, he's ahead. Imagine what you would see from a drone camera following the runners from above. Wow, they orbit each other! And neither of the paths is concave, like on the second picture, nor are there any backward-going loops like that on the first picture. They always go forward, to bring home that medal!

The numbers above are approximating Earth and Moon orbits. Earth orbit around the Sun is ~400 times larger than that of the Moon around the Earth, as is a standard 2-lane track width of 4m is 1/400 of the 1.6 km radius of our supertrack. The Moon makes about 13 turns around the Earth in one Earth's year. Of course, in a better analogy, the Earth runs in the middle of the track, deviating only slightly, and the Moon is rhythmically bobbing between the innermost and outermost edges. There are also no abrupt lane changes at set points for the Moon (and this smooths out the curve even more away from concavity compared to skaters paths). But since we are in a thought experiment, let's pretend the Olympic committee has an exception for this peculiar pair.

What is the reason for this difference between assumed and actual path variation?

The orbit of a moon around the sun depends on the time to orbit the planet and the planet's time to orbit the sun.

In the case where the moon takes a long time to orbit the planet (like the Earth-Moon), the orbit just wiggles along the circle.

In the case where the moon has a short orbital period compared to the planets year (like Jupiter-Io), the path is like you drew in the first figure.

Has this path been like this since the formation of the Moon?

For the Earth-Moon system, ...
Yes, it has always been that way.

Do Natural Satellites of other planets also follow the same orbit around the Sun?

No.

• I think, natural satellites of other planets do in fact follow similar orbits. Not the same obviously, but similar. – userLTK Jun 13 '15 at 21:32
• @userLTK Jupiter's orbital speed is about 13.07 km/s and Io's orbital speed is 17.334 km/s. This means that Io moves faster backwards than Jupiter moves forward. Europa has an orbital speed of 13.740 km/s, which is just faster than what is needed to go backwards. – LDC3 Jun 13 '15 at 22:03
• Good points. As a counter argument, Phobos for example has an orbital speed of 2 km/s and Mars an orbital speed of 24 KM/s. And (I think, perhaps I made a bad assumption, there may not be many), but I think, some of the more distant moons of Jupiter, Saturn, Uranus could have a more circular orbit around the sun. – userLTK Jun 13 '15 at 22:24
• @userLTK Yes, the larger radius of the moons' orbits will result with an orbit like Chad's second figure. – LDC3 Jun 13 '15 at 22:37

[NOTE: The animated GIFs are too large to copy into this post, but the URLs should work]

EDIT: all 3 animated GIFs are now available on youtube:

END EDIT

The image above would apply if:

• the Earth and Moon both had circular orbits (approximately true)

• the Moon's siderial period was exactly 1/12th year (approximately true)

• the Earth's distance from the Sun was 150 (million km) (approximately true)

• the Moon's distance from the Earth was 30 (million km) (completely false)

Here, you see the loop-de-loop orbit you originally expected.

Now, what if we reduce the distance to 10 million km (still very large):

As you can see, the loop-de-loops are gone, although the orbit still has some "sharp points", which we don't see in the Moon's real orbit.

If we reduce the distance to 3 million km, we get something closer to what expect:

Here, we have a wobbly circle, closer to what actually happens.

Of course, the Moon's actual distance from the Earth is only 0.35 million km, so the actual wobbles are much smaller. I tried to do a graphic of those, but the two orbits ended up on top of each other.

The moon's orbital path around the sun is a monthly semi-circle called a cycloid in geometric terms.

Draw a circle representing the moon's orbit. This is the rolling circle (forget the earth). Touching this circle draw an arc to represent the sun's extended circumference. This will be the base circle.

Where a line drawn between the two centers cuts the circles is the start point of the cycloid, named the new moon which is the closest point to the sun. As this point on the rolling circle moves around the base circle it will come, after one revolution back to the base curve. The path traced out will be a semi-circle and the distance between the two points will be a month on the baseline circle.

• Take a look at some of the existing answers. The "rolling circle" model doesn't give the correct shape. – James K Apr 5 at 8:12
• The model is a series of semi-circles whose start and finish points are equal to the circumference of the moon's orbit. The extended circumference of the sun is where it touches the moon's orbit at the new moon phase. Jim V – Jim Veryard Apr 6 at 0:12
• Yes, and that model is not useful. The path of the moon isn't like that. See other answers. – James K Apr 6 at 8:57
• The orbit is a simple exercise in practical geometry. – Jim Veryard Apr 7 at 7:36

The Moon’s path around the Sun IS in the form of an epicycloidal curve.

The dictionary explanation of epicycloid is: ─ A curve traced by a point (the moon) on the circumference of a circle (moon’s orbit), rolling on the exterior of another circle (the arc of the extended radius of the sun)

Where the point touches the other circle is the point of the new moon, as it revolves 360° to touch the other circle again is the next new moon. The elapsed time is one solar month.

The Moon is a part of a two orbiting systems.

1. The Moon in orbit around the Earth and

2. The Moon’s orbit around the Sun

When the rolling circle is modified to suit the elliptical orbit of the moon the curve flattens slightly between the new moon & full moon and again between the full moon & new moon.

• The Moon’s path around the Sun IS NOT in the form of an epicycloidal curve. I'll ignore that the Earth-Moon barycenter orbits the Sun in an ellipse rather than a circle. I'll ignore that the Moon orbits the Earth in an ellipse rather than a circle. I'll ignore that the Moon's orbit about the Earth is inclined with respect to the Earth-Moon barycenter's orbit about the Sun (so the Moon's orbit about the Sun is not even a plane figure). I'll ignore all that. All one has to to do is to look at the ratio of the radii. (Continued) – David Hammen May 12 at 23:45
• A quintessential feature if anything remotely epicycloidal is that the object in question rolls without slipping along a closed path. For this to happen with a circular object of radius $r$ rolling along a (larger) circular path of radius $R$ is that the object must rotate $R/r$ times for every orbit about the larger circular path. In the case of the Earth-Moon system, that would mean the Moon would have to orbit the Earth about 389 times per year. That obviously is not the case. The Moon's path about the Sun is not anywhere close to an epicycloidal curve. – David Hammen May 12 at 23:50

Earth, Moon and Sun relationships and the Moon’s movement around the Sun.

1. An orbit is a modified helical path through the solar system. The depiction of an orbit as a complete circle is incorrect when viewing the solar system. When an orbit is around a parent body (Moon/Earth, or Earth/Sun) it is closed and ovoid in shape (2 Dimension). When it is a part of the solar system it becomes open and a never-ending spiral (3 Dimension), it’s orbital path still being ovoid.
2. The Moon cannot spin on its own axis, as many astronomical publications suggest because it doesn’t spin, maybe it used to but it doesn’t now. It’s just a blob fixed radially to the Earth’s axis, slowly moving around the Earth once every solar month dropping back its position daily by approx. 1/30th of the orbital circumference. One Moon orbit plus 29 Earth spins gives a total of 30 passes every solar month. The Moon’s orbit will always touch the extended radius of the Sun.
3. The Earth spins inside the Moon’s orbit approx. 29 times every solar month and advances in its own orbit around the Sun approx. 30 degrees every solar month therefore the orbit of the Moon advances with the Earth.
4. The Moon is a part of two systems.

a. In its own orbit around the Earth being its Earth system and,

b. The orbit of the Moon in its movement around the Sun being its Sun system.

5. The Moon’s movement around the Sun is a slightly modified epicycloidal curve. By constructing this curve, it is possible to track the Moon around the Sun.

6. If you construct the epicycloid and where the two circles touch (the rolling circle as the Moon’s orbit and the base circle as the Sun’s extended radius) and call this point ‘NM’ then moving the rolling circle 1/29th of its circumference along the base circle then ‘NM’ will have moved away from the base circle but still within the Moon’s orbit. This is the first point on the epicycloid. Do this another 28 times and ‘NM’ will be back on the base circle about 30° later. These are the positions of successive New Moons. This will happen approximately 12 times a year. The ratio of the two circles should be about 12 to 1. Epicycloid constructions may be found on the internet. Thursday, 21 May 2020