Early Moon's orbit - Question

[Althistorybuff]

From what I gather, the early moon was 4 or so radii away from Earth after formation/the big impact (3 being minimum without tearing apart via gravity and 5 being the maximum the debris would have been spread).

 

Now it is 60 Radii, moving at about 1KM per second. I assume it would have been faster after the Earth/Thea impact. Any idea how much?

 

The main reason for the move was tidal friction from Earths ocean "pushing" forward which transferred momentum to the move and therefore out by about 3CM a year.

 

However, there was is also a theory that the Earth's tilt was up to 60 degrees and that also resulted in an early and quick movement outward by the moon early in its existence. (Matija Cuk)

 

The moon will keep moving out until that angular momentum is transferred to the point where Earth and the Moon are "tidally locked".

 

Presumably, the moon's orbit early in its existence was quicker due to this shorter orbit, though I'm not sure of projected actual speed.

Question:

 

If Earth didn't have this "tilt" early on that gravitational interaction with the moon largely corrected (at the expense of the moon making a big and fast leap out), would the "tidal locking" of the earth to the moon have already occurred?

 

If the moon was 15 times closer (I don't know the speed back then so I can't calculate the "month"), would it be possible that the Earth's day/month would have already been "tidally locked" at something closer to what we know as 24 hours? That the moon would orbit so closely that the month would be far closer to what we know as a day than, in the future, our "day" will be closer to what we think of as a month?

 

I know that the Earth rotated faster prior to the impact, maybe 6 hour rotations immediately after (and probably before, I assume).

 

If Earth rotated 24 hours (as we know it) PRIOR to the collision and the moon was much, much closer, and this "tilt" issue wasn't involved, how quickly could the Earth/moon Day/Month be determined.

I'm trying to figure out if, after the "Thea" collision, that Earth and the Moon might have found a much earlier, and short, "day/month" combo.

 

Am I missing some large factor?

 

Obviously, the tides would much, much higher/worse and the moon would be far bigger in the night sky.

 

I ran some middle school math. Assuming the moon was formed at 1/10th the current, average distance, this means 2 x 3.14 x 38,500 (actual is 385,000) KM is 241,780 KM.

At the current 1.022 KM per second (again, I'm guessing it would be faster back then as the Moon's speed is slower as it moves out), this would mean 65.7154 hour long month?

 

Double the speed and you get pretty close to 33 hour month.

If the Earth had been rotating slower back then and the impact maybe slowed it a bit more, would the moon have remained in place at 6 radii?

Edited March 23, 2017 by Althistorybuff

[swansont]

From what I gather, the early moon was 4 or so radii away from Earth after formation/the big impact (3 being minimum without tearing apart via gravity and 5 being the maximum the debris would have been spread).

 

Now it is 60 Radii, moving at about 1KM per second. I assume it would have been faster after the Earth/Thea impact. Any idea how much?

It's simple to calculate. For a circular orbit, the gravitational force is centripetal, so GMm/r^2 = mv^2/r

 

GM/r = v^2

 

M is the mass of the earth

[Sriman Dutta]

Moon's orbit is not exactly circular. It's a bit elliptic with earth in one of the foci. So Vis-Viva equation will help.

[swansont]

Moon's orbit is not exactly circular. It's a bit elliptic with earth in one of the foci. So Vis-Viva equation will help.

 

But close enough to get a good approximation of the speed by assuming it is

[AbnormallyHonest]

The moons orbit is a period that is "synched" with its revolution around the earth. So a period of 27-29 days or so. If it's tidally locked this revolution is just a consequence of that, and not a result of angular momentum. So what happened to that angular momentum? The momentum that should cause the rotation has been transferred somewhere else, probably the kick that began its orbit to move slightly out of place. Earths tidal friction explains where the acceleration comes from, but a mean central force as Newton predicted does not allow that bulge to pull discriminately on the moon. A preservation of its orbital period due to resonance does. The force from the angular momentum of the moons rotation can not be dismissed until the moons rotation is completely stopped. At the point of tidal locking, there is an abundance of force otherwise its tidal lock is merely just a coincidental progression... and it's maintainence is due to some other force we do not understand.

[Althistorybuff]

Sorry, I'm having trouble figuring out this gravitational/orbital speed equation.

 

G= ?

 

M=Mass of Earth -

 

m= mass of moon?

r = radius of the moon to center of earth?

[Janus]

Sorry, I'm having trouble figuring out this gravitational/orbital speed equation.

 

G= ?

The gravitational constant in the Kg-meter-sec system it has a value of ~6.673x10^-11

 

 

M=Mass of Earth -

 

Yes

 

 

m= mass of moon?

 

Yes

 

 

r = radius of the moon to center of earth?

 

 

Yes.

 

A simple rule is that when orbiting the same body, a satellite's orbital speed at different orbital radii is inversely proportional to the square root of the orbital radius. (divide the radius by 10 and the orbital speed increases by a factor of 3.16)

Orbital period changes by the square root of the radius cubed. ( dividing an orbit's radius by ten gives you ~1/31.6 the period.)

Edited March 28, 2017 by Janus

[Althistorybuff]

Thanks.

[Enthalpy]

The square of the period varies like the cube of the orbital radius (or the big axis if elliptic).

[Althistorybuff]

So the moon is about 384,000 KM from earth: square root is 619

 

If it were 10x closer, square root is 195.

 

So roughly, it would be 3.15x faster.

 

Naturally, if we had different masses of each body, that would change a great deal. I looked up IO, which orbits Jupiter at roughly the same distance as the moon does from Earth and it moves 17X faster.