# How big does a moon have to be to have the earth locked to it?

I would like a earth-moon system like Pluto-Charon (but with earth instead of Pluto) where both will be tidally locked to each other within something like 3 billion years (before complex life appears). As the moon would probably lock to earth before the opposite it depends only on the mass of the moon so how big has it to be? Is the slowing rate linear to the mass? Is this stable and fast would they spin if it does ?

• The Wikipedia page, which I don't know how to link to, has a entire section dedicated to it. Page: Tidal Locking. Section: Timescales Aug 9, 2018 at 21:17
• I don't really know if this formula applies for the bigger object Aug 9, 2018 at 21:37
• No, the formula applies for both. Aug 9, 2018 at 21:41
• It's an approximation so it may not consider the part of removal due to slowing of the planet spin (the moon's in my case) but I have to consider it here because it's a much bigger term in my case. Aug 9, 2018 at 21:55
• The slowing of a planet or moon's spin is due to tidal locking Aug 9, 2018 at 21:56

In the Earth and Moon system, assuming current theories on the Sun growing into a red giant, the Earth will never become tidally locked to the Moon. As described here, the ~5 degree inclination of the Moon oscillates every 18.6 years stabilizing the Earth's axis relative to the Suns orbital disk. This oscillation will degenerate over astronomical time, but the influence the Moon will have diminished, as it is not in a stable orbit; it is moving away from Earth.

The Pluto-Charon system is completely different. The mass of the Earth is 81 times that of the Moon. The mass of Pluto is less than 9 times that of Charon. The fact that Pluto was acquired by our solar system, instead of created within it, implies a completely different set of equations than natively formed planets and moons. The orbit each other, much like binary star systems, on a very odd inclined elliptical solar orbit.

If the Moon were in a stable orbit around the Earth and there were no other gravitational influences, like the Sun and Jupiter, eventually the Earth-Moon system would become fully tidally locked.

As it stands the mass differential and the Moons increasing orbit around the Earth, imply, in astronomic units, the Moon would have been ejected from Earth's orbit long before the axis of the Earth was in alignment with the Moon; a requirement for tidal lock.

Edit: To answer your questions: Under the assumption we did become tidally locked prior to 3 billion years ago:

Assuming the proper conditions(the biologic components and water arrive at the surface of the Earth at approximately 70% chance given the increased speed of the Moon and the knowledge it did happen), since life is varied, adapting and dynamic, yes life exists. Since the tides as we know them would not exist, life would not evolve to exit the oceans until an event such as an asteroid or comet impact forced them to, thus pushing the evolutionary timeline back at least 500 million years. As to spin, the Earth would slow, a day would be 26+ hours long, the seasons would change(~21 degree axis tilt). A month would be one "day". The Earth would wobble slightly(and increase as the Moon Moves closer) on the solar plane causing a loss of orbital stability and orbital decay crashing us into Venus before humans could appear.

All of the above numbers and conclusion in this edit are approximate educated guesses.

[Second edit] As to minimum masses for tidal lock:

The Moon would have to be larger just to maintain a stable orbit of the Earth(between 3% and 15% larger,depends on the pull of the Sun, Earth and Jupiter to align the Moon on the solar plane). Earth's axis would align marking the end of seasons over the course of several billion years, when tidal lock would occur.

With the current theory on the Moons origin, a larger mass Moon would have already escaped(or been very close to escaping) Earth's gravitational pull and could not achieve tidal lock.(Although the Earth would be spinning slower.)

Additional thought: If the Moon were at 10% of the Earth's mass, I would tidal lock quickly and crash into Earth < a few billion years later.

• Please see edit. Aug 21, 2018 at 1:36
• Ok thanks, so I just have to give up this idea basically Aug 21, 2018 at 14:03
• Don't give up. The scenario I presented is heavily influenced by how things actually happened according to the latest theories. If you remove those restrictions(and make more assumptions), anything is possible. Aug 23, 2018 at 0:01

IRL, bodies become tidally locked because the energy that used to rotate the planet, is used up by something, e.g. pulling seas around. How long this takes depends on the bodies (inertia/mass, distance, makeup, other bodies around).

This means there will be no limit to the size of the bodies. Two objects of 1 kg each, in an otherwise empty universe would orbit each other even if they are a lightyear apart, and eventually tidally lock, just REALLY slowly. It would take about 20 sextillion years, about 12 trillion times the age of our universe, to complete one orbit; tidal locking would take much, much longer.

Eventually, it would happen.

Of course, if there are other objects around, that would change their orbits (depending on the mass and distance of other objects).

• This doesn't really answer OP's question, as he gives a definite time-frame for the tidal locking to occur, and has a few other questions. Aug 10, 2018 at 6:11
• What I want is how big the would have to be to have the locking within 3 billion years, not just some case where it takes extremely long Aug 10, 2018 at 14:52

If I understand your question correctly, I think it's not the size so much as it is the gravitational pull. Increase the gravitational pull to match Earth's where they pull and push each other at the same rate.