What If the Moon and Earth Were Twin Planets?

[Stan52]

Here's a good exercise for you math whizzes (no, I don't know the answer):

How stable (and what shape) would the orbit (around the Sun) of the Earth be if the moon was equal to the Earth in size and mass, but correspondingly further away so that the tidal effect would be the same as it is now?
1: how far apart would that be?
2: would the orbit be disrupted by the Sun or Venus? (assume the rest of the solar system unchanged)
3: how likely for such a structure to develop somewhere?
4: would these twin planets be rotation-locked (same side always facing each other)? (This would make the "day" equal to their "month": how long would that be?)
5: considering the unusual size our moon actually is compared to its planet, how much of those effects are present as it is?

 

[Stan52]

Hello out there, I didn't think this was a dumb question. Maybe it is. If the moon was the same size as Earth and orbited far enough away to not cause greater tides, both probably would have such eccentric solar orbits that it would not last long, hence would never get started. Even if Venus wasn't there. Either that or this has stumped you guys with the calculators. Which is it?

 

[gabee]

If the moon was as big as the Earth it probably wouldn't have a stable orbit around the earth--instead, it would orbit the sun as our Earth does.

 

[Stan52]

Probably, yes. But stretching our imagination as an exercise here ----

True, that is the most likely. But some have remarked how unlikely it is for our Earth's satellite to be so large for its planet. So the question naturally should be posed, what are the calculable limits of such occurrence? Is a twin planet as I posited impossible (not merely improbable)? It's just an exploration of limits, a mind exercise. We will have to travel to other systems no doubt before observing any such structure directly. Of course twin asteroids are already known, but they are much smaller and orbit one another much more closely. Starting from there and incrementing mass and separation, at what point do we reach the impossible?

Maybe it's not worth many ticks of compute time but surely a few.

 

[Tzemach]

If the moon were as large as the Earth it would no longer be a moon but a planet in its own right. The Suns gravity would have a far greater effect on it than could the Earth. Remember gravitational force is inversely proportional to the square of the distance. If it were close enough for the Earth to have dominance over the Sun's gravity they would have to collide, any safe distance and the solar gravity must dominate.

 

[Janus]

If the moon were as large as the Earth it would no longer be a moon but a planet in its own right. The Suns gravity would have a far greater effect on it than could the Earth.

Um, this is already true for the Moon. The Sun's gravitational grip on the moon is just about twice that of the Earth's.

Remember gravitational force is inversely proportional to the square of the distance. If it were close enough for the Earth to have dominance over the Sun's gravity they would have to collide, any safe distance and the solar gravity must dominate.

 

[gabee]

Um, this is already true for the Moon. The Sun's gravitational grip on the moon is just about twice that of the Earth's.

*duh* I didn't know that. :X

http://www.math.nus.edu.sg/aslaksen/teaching/convex.html

Interesting, this might help explain the likelihood of such a situation, since as you said the moon is already rather large. It seems to me that a twin planet system would be possible given that it orbited the sun in the right way.

 

[Stan52]

??How does that work?

Quote: The Sun's gravitational grip on the moon is just about twice that of the Earth's.

Pardon my ignorance (I am a product of backward Alabama schools of the 60's) but wouldn't that mean the moon's orbit would be very non-circular? On the sun side, wouldn't it be pulled completely loose of Earth's grip? And even if not, that would still greatly affect the orbital shape. Sorry, this info seems to have been completely absent from my high school lessons. Most of my reading since then has been about other areas of science. What astronomy I've stumbled across over the years seems to be about other planets and stars, very little about the moon.
I assume the path the Earth traces around the sun cannot be a smooth ellipse, with this body swinging around it every month. Just how much deviation from a moonless path is it? This published anywhere?

Edit: The post above made it in while I was typing. Went to the math.nus.edu.sq site. Veddy interestink!

 

[russ_watters]

Quote: The Sun's gravitational grip on the moon is just about twice that of the Earth's.

Pardon my ignorance (I am a product of backward Alabama schools of the 60's) but wouldn't that mean the moon's orbit would be very non-circular? On the sun side, wouldn't it be pulled completely loose of Earth's grip?

Why? Remove the Earth and the moon's orbit around it from the picture and what happens...? Answer: nothing. The moon would continue in a stable orbit around the sun. It's not like the Earth drags the moon with it around the sun.

 

[Janus]

Here's a good exercise for you math whizzes (no, I don't know the answer):

How stable (and what shape) would the orbit (around the Sun) of the Earth be if the moon was equal to the Earth in size and mass, but correspondingly further away so that the tidal effect would be the same as it is now?
1: how far apart would that be?

Just about 4.3 times the present distance of the Sun. while gravitational force decreases by the square of the distance, tidal forces decrease by the cube of the distance.

2: would the orbit be disrupted by the Sun or Venus? (assume the rest of the solar system unchanged)

I wouldn't expect much effect from Venus. The Sun will have an effect. In fact, the Sun will very nearly pull them apart. There is something known as the Hill sphere, it is the maximum distance that a moon can be from its planet without being pulled away by the Sun. Usually you'll hear that the Earth's Hill sphere has a radius of about 1,496,000 km or about 3.9 times the distance of the Moon. So, at first, it would seem that the distance I gave above would be outside that distance. However, this stated Hill sphere radius is for when the moon is guite small as compared to the planet. Having two bodies the same size changes this slightly, and moves the Hill sphere out to about 4.45 times the distance of the Moon, which our twin Earth scenerio just fits inside of. The tidal effect of the Sun will however tend to elongate the planets' mutual orbit along the line joining them to the Sun. (The Sun already does this to a certain degree to the Moon's orbit around the Earth)

3: how likely for such a structure to develop somewhere?

Unknown. Though I wouldn't expect the chance to be high.

4: would these twin planets be rotation-locked (same side always facing each other)? (This would make the "day" equal to their "month": how long would that be?)

Assuming that both bodies started with the same rotation rate as the Earth did and have existed for the same amount of time as the Earth-moon system, they shouldn't be tidally locked. Remember, this locking is due to the tidal force, and we have removed these planets to a distance where their tidal effect on each other equals that of the moon on the Earth now. Since the Moon has not been able to lock the Earth to it yet, These planets shoudn't have either.

5: considering the unusual size our moon actually is compared to its planet, how much of those effects are present as it is?

One of the most telling effects is that one planet would not orbit the center of the other, but they would orbit a barycenter located halfway between the two. Likewise, the Earth and Moon orbit a common barycenter, in this case one just a little under the surface of the Earth.

One interesting point is that the wach planet, as viewed from the other would be appear to be almost the same size as the moon now apprears from the Earth. The only reason that they wouldn't be exactly the same size is that the planets would be denser than the Moon. (if they were of the same density as the Moon, they would look to be exactly the same size as the moon does now.)

 

[MadScientist 1000]

Simple answer: We'd all be underwater.

 

[Stan52]

Barycenter

Wow, that's a new thought: the Earth orbits the sun yearly, rotates on its axis daily, AND orbits a "barycenter" monthly. No wonder I get dizzy at times! Just thinking about it! :yuck:

 

[Tzemach]

Even better look up exercises involving three body dynamics or the Trojan planets, the planets that revolve in Jupiter's orbit, held in place by the balance between the gravity of Jupiter and the sun. Depending on the size of the various bodies there is a point in an orbit where objects can be held in orbit by the gravity of two objects.

 

[MadScientist 1000]

Even better look up exercises involving three body dynamics or the Trojan planets, the planets that revolve in Jupiter's orbit, held in place by the balance between the gravity of Jupiter and the sun. Depending on the size of the various bodies there is a point in an orbit where objects can be held in orbit by the gravity of two objects.

Or an infinite number of objects...