# For two planets orbiting the sun, do you see why it makes sense to speak of the planets as revolving around each other?

I offer this question as an introduction to a useful discovery that I made that has scant reference in the literature. I will show how it makes it easy to calculate the number of times the Earth revolves around the sun in a year and to determine how long it takes for the moon to revolve around the Earth.

If this question appears nonsensical, consider the following. Suppose the planet in the inner orbit was stationary. Then it would certainly make sense to speak of the other planet as revolving around it. Now introduce some motion to the inner planet. Shouldn’t we still be able to speak of the planets as revolving around each other? How would you define the amount by which this revolution occurs?

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No, technically they do not revolve around each other. They both revolve around a third object.

marinelife (62485)

OK, but consider the opposite scenario, where the outer planet is stationary. Now neither is orbiting the other. Then consider, what does “stationary” actually mean in this kind of system? Stationary relative to what?

thorninmud (20495)

Planets on an orbit closer to the sun go around the sun quicker than those in an orbit further out.

So your example would never work. The only thing you could do, is make a very complicated mathemetical model with one of the planets as a stationary center.

whitenoise (14152)

If a planet is stationary, it will start falling to the sun.

whitenoise (14152)

You are starting to realize something that I have known for years:

I am stationary, and the universe revolves around me.

I admit that the motion may get pretty complicated when I get into my car and drive around the block, so I apologize for any dizziness you may feel from time to time.

CWOTUS (26102)

@CWOTUS I was wondering why I could never figure out quite where I was!

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zenvelo (39284)

@CWOTUS :)

whitenoise (14152)

Well, this is of course exactly what people used to think: that the rest of the solar system revolved around the Earth. That’s why the Zodiac has a name. But since we know that the motion of the planets around the Sun is governed by gravity, it’s a lot easier (and more elegant) to make calculations with reference to orbit of each planet around the sun, and it’s conceptually easier to grasp as well.

glacial (12145)

Interesting question. If both planets are revolving around the same sun and each is revolving on its own axis, aren’t they, though not technically at least in a sense, revolving around each other?

Pachy (18610)

Spatially, they can be seen to do so… but the mechanical reality is that each planet is orbiting the sun.

glacial (12145)

@glacial, agreed.

Pachy (18610)

@glacial, What if the motion was somehow the same, but the sun was not present?

I see that I am going to have to do some convincing. I propose the following very simple definition for revolution of two objects relative to one another in two dimensions.

Select a reference line. We could select a pair of coordinate axes, but we only need one line. Now draw the line connecting the centers of the two objects. The revolution of one object with respect to another is just the change in angle made by the line joining the centers with respect to the reference line.

In the simple case where one object revolves around a stationary object, you can see where the definition I am giving is in agreement with the usual notion of revolution. Notice though that revolution is relative. We do not need to specify which object revolves around which.

The usefulness of the definition is that a revolution is now just the rotation of a particular line. We can now speak of both rotations of objects around their axes and revolutions as being types of rotation.

And the usefulness of this observation relates to my great rediscovery – the law of relative rotation: The difference between the perceived number of rotations of two objects remains unchanged when the observer moves to a rotating platform.

Once I can get a few people on board with this notion, I will show how easy it makes the calculation of the number of yearly rotations of the Earth and the time it takes for the moon to revolve around the Earth.

You don’t need to convince me that the universe can be looked at from that perspective, as a thought experiment. I’m not even arguing that you can’t make the calculations that way. It was done that way for a long time, under the Ptolemaic model among others. Some even managed to find ways to account for retrograde motion (which I assume you’re also doing with your thought experiment) – but it’s messier than reality, and well… not real.

glacial (12145)

Planets orbiting a sun actually orbit the centre of gravity of the system. They appear to orbit the sun only because the sun’s gravity predominates. They are all revolving around each other. The ‘law of relative rotation’ seems trivial. The number of observed rotations may differ depending on the rotation of the observer but the change will be exactly the same for each object and so the difference will be the same.

flutherother (34332)

@LostInParadise It would seem your misperception arises from a lack of comprehension of the Universal Laws of Gravitation and Newtonian physics. Lacking adequate knowledge, humans in ancient times insisted on what it seemed they saw happening in the night sky.

Dr_Lawrence (20014)

@flutherother, I agree that the law of relative rotation was overhyped, but look how useful it is.

Suppose we want to determine the number of times the Earth rotates in a year.
For an observer in the Northern Hemisphere, the sun appears to revolve 365¼ times per year and the Earth does not appear to rotate.
revolutions – rotations = 365¼, taking clockwise as positive.

Now observe the same situation from space above the Northern Hemisphere. The earth revolves once counterclockwise.
So we get:
revolutions – rotations =( -1) – rotations = 365¼. Number of rotations = -366¼, meaning the Earth rotates 366¼ times counterclockwise. Notice how nicely the relativistic definition of revolution fits in.

We can do a similar calculation for moon revolutions. A lunar month is a complete phase cycle of the moon and it occurs every 29.5 days. The sun and the moon both appear to revolve around the Earth. The sun appears to revolve faster, so every 29.5 days the sun gains a lap on the moon. The difference in their revolutions is therefore 365¼/ 29.5 = 12.38,. sun revolutions – moon revolutions = 12.38

From outer space we get:
-1 – moon revolutions = 12.38. Moon revolutions = -13.38. Time for moon to revolve = 365¼ / 13.38 = 27.3 days.

One more example, a great recreational math problem. Imagine taking two coins of the same denomination and rotating one around the circumference of the other. How many times does it rotate? I, like most people guessed wrong. I invite you to try this on your own before reading further.

One neat way of finding the answer is to first imagine the two coins as fixed gears spinning on their spindles. As one turns once clockwise, the other turns once counterclockwise. In this case, the numbers for rotations and revolutions is -1, 1 and 0. The difference in rotations is 1 – (-1) = 2. To duplicate the setup for the problem, imagine a small observer on the coin going counterclockwise. The difference in rotations is the same, so the observer will see 2 – 0 rotations. The number of revolutions is will be between 0 and 2, so it will be 1, which makes sense, because it is like the apparent revolution of the sun due to the rotation of the Earth.

Now this makes me wonder if you actually know how often in a year our planet rotates around its axis?

(Given that we have approximately 365.25 days in a year.)

whitenoise (14152)

I guessed the coin would rotate once, but I was wrong. Similarly you would think the Earth rotates 365.25 times in a year but in fact it is 366.25 times. It has to turn a little more than a full revolution each day to compensate for the movement along its orbit. I’m still not convinced of the value of this law however.

flutherother (34332)

@flutherother
You’re right… it is 366.25 times.

Actually 364.25 could also have been possible, had the earth spun in the other direction.

This is the difference between a synodic day and a sidereal day.

A synodic or solar day is the period it takes for a planet to rotate once in relation to the sun it is in orbit around. A sidereal day is one complete rotation in relation to the stars outside its solar system.

Earth’s sidereal day takes about 23 hours, 56 minutes, on average.

whitenoise (14152)

The difference between a sidereal day and a synodic day is one way of explaining why the night sky changes. All the constellations appear overhead every time the Earth rotates, but the only ones that can be seen are the ones that appear at night. Because the two types of day are out of sync, the view of the constellations changes with every rotation.

I found this article, which is basically the application of the principle of relative rotation. Notice that in the top row of the table all the entries are the same value, meaning that differences will preserved in the second row after the first row is added.