Synchronous Moon Stability: Conditions & Twin Planet Distance

[shelanachium]

I have been toying with an SF scenario in which an Earth-like planet has a synchronous moon. Among the consequences are regular eclipses every spring and autumn, poor visibility of the stars on the side facing the moon except during eclipses, and no need for chronometers to determine your longitude on the side facing the moon (and perhaps a great reluctance among sailors to visit the averted side).

However I have read that such systems are not stable unless the moon has 10% or more of the mass of its planet, so I'd have to make my moon as big as Mars. Is this true?

What are the conditions for stability of a mutually synchronous pair of bodies in the gravitational field of a star?

And also, suppose the Moon were a twin of Earth. How far away could it be without escaping into its own orbit? (and probably ending up colliding with Earth - end of story with a very big bang!)

 

[DaveC426913]

Did you know that Pluto/Charon are both tidally locked to each other? It is hypothesized that, during the "warm season" they share their atmospheres.

 

[shelanachium]

Pluto and Charon

Knew about them - but also very far from Sun so negligible solar influence on their mutual orbit. They also fit the criterion I read of (can't remember where) of the pair having a mass ratio 10 or less:1.

I did hear of an SF story where two Earthlike planets were in mutually synchronous 24-hour orbit (requires centre to centre about 36,000 miles (58,000km)). In this tale the atmospheres interacted enough for balloon travel between them (a likely story!). Wonder what the mutual magnetospheric influence would be?

 

[DaveC426913]

I did hear of an SF story where two Earthlike planets were in mutually synchronous 24-hour orbit (requires centre to centre about 36,000 miles (58,000km)).

36,000 miles is getting awfully close to the Roche limit.

 

[shelanachium]

Thought you'd wrecked my whole scenario. But checked on Wikipedia: Roche limit for Earth and moon-sized body is 18,000 km even in the worst case (fluid body). My synchronous moon would be at about 40,000 km so should be fine. Still can't find anybody to tell me if the system is stable! (58,000 km is the separation if both bodies are Earthlike)

I hate impossibilities in 'hard' SF, though I loved Le Guin's 'The Dispossessed' so much that I forgave her having an only slightly subterrestrial moon Anarres to her planet Urras - and that round a low-mass star, Tau Ceti. Either impossible tides, impossibly long days or system unstable - it was clear Anarres was far enough away from Urras to be comparable to our own moon in apparent brightness, and the pair were not synchronous.

The great Asimov fell down badly in 'Nemesis'. He wanted an Earth-like planet of a red dwarf. Aware that to have Earth-like temperatures and light-levels, such a planet would normally be tidally locked by its star, he made it orbit a close superjovian planet of the red dwarf in 24 hours or so, so that it would be tidally locked to the superjovian instead. Doesn't, however, stop the star causing horrendous tides! System probably not stable as a result.

 

[LURCH]

10% does not make the moon Mars-sized. Earth's moon is more than 25% the size Earth.

 

[Janus]

10% does not make the moon Mars-sized. Earth's moon is more than 25% the size Earth.

I believe he was considering it in terms of mass. Mars has 1/10 the mass of the Earth and The moon 1/81.

 

[FTL_Diesel]

I'm not clear why you need the moon to be at least 10% the mass of the planet for a tidally locked system to be dynamically stable. The Earth and Moon are heading towards this scenario themselves (albeit in several billion years).

Or is it because you need enough mass in the moon to make the synchronization timescale reasonable? If this is fiction, then can't we just declare the relative rotations at the time of formation to be close, and therefore by the time of the story, the rotations have locked themselves down?

 

[DaveC426913]

Or is it because you need enough mass in the moon to make the synchronization timescale reasonable? If this is fiction, then can't we just declare the relative rotations at the time of formation to be close, and therefore by the time of the story, the rotations have locked themselves down?

Then you'd be dealing with a cold, solid, dead planet.

 

[FTL_Diesel]

Then you'd be dealing with a cold, solid, dead planet.

How do you mean?

 

[DaveC426913]

How do you mean?

After another few billion years, by the time Earth is locked to the Moon, its core will have gone cold and solid. No tectonics, no geothermal activity, no radioactives, no crustal refresh, etc.

 

[shelanachium]

Many thanks for all your comments, but nobody has yet answered IS MY SCENARIO STABLE IN THE LONG TERM? Or will the orbit rapidly expand or contract, resulting either in merger of the bodies or their increasing separation? I suspect this because moons like Phobos which orbit closer than the synchronous distance are destined to merge with their planets, and those further out (like our own moon) are moving away.

This suggests the synchronous orbit is on a knife-edge, like a pencil stood on its tip -any small perturbation in either direction rapidly accelerates. On the other hand, one suspects that if an orbit were ever mutually synchronous, it would remain so even if changing in size, unless the change is very rapid indeed.


I gather the Moon was pretty close to the Earth when first formed, though probably not close enough for mutual synchrony, especially as Earth's rotation was much faster then.

Tried 'mutually synchronous orbits' on Google. All either Pluto/Charon or references to these postings! As I said, in the case of Pluto/Charon solar tidal influence is neglible (proportional to 1/r^3), which it would not be in a synchronous pair in an Earthlike orbit.

 

[DaveC426913]

Many thanks for all your comments, but nobody has yet answered IS MY SCENARIO STABLE IN THE LONG TERM?

We don't kn...
We don't knnn...
We don't KNNNN$@&$%*...

Sorry, can't do it.

Did you know Pluto and Charon are synchronous?

 

[Janus]

Many thanks for all your comments, but nobody has yet answered IS MY SCENARIO STABLE IN THE LONG TERM? Or will the orbit rapidly expand or contract, resulting either in merger of the bodies or their increasing separation? I suspect this because moons like Phobos which orbit closer than the synchronous distance are destined to merge with their planets, and those further out (like our own moon) are moving away.

This suggests the synchronous orbit is on a knife-edge, like a pencil stood on its tip -any small perturbation in either direction rapidly accelerates. On the other hand, one suspects that if an orbit were ever mutually synchronous, it would remain so even if changing in size, unless the change is very rapid indeed.


I gather the Moon was pretty close to the Earth when first formed, though probably not close enough for mutual synchrony, especially as Earth's rotation was much faster then.

Tried 'mutually synchronous orbits' on Google. All either Pluto/Charon or references to these postings! As I said, in the case of Pluto/Charon solar tidal influence is neglible (proportional to 1/r^3), which it would not be in a synchronous pair in an Earthlike orbit.

Here's the deal. If a satellite has an orbital period greater than the rotational period of the primary, it will recede due to tidal action while the primary slows its rotation. In this manner they will approach and eventually achieve mutual tidal lock.

If a satellite has an orbital period less than the primary's rotational period, the satellite will approach the primary while the primary speeds up it rotation, until the satelllite collides with the primary, or if the satellite is large enough, torn apart by tidal forces and forms a ring system.

Now is mutual tidal lock stable in the long term? I guess this depends on what you mean by "long term". Long term on a human scale, or long term on a astronomical scale?

Once mutual lock is attained, there are other influences to take into account. The Sun's tidal effect on the primary for example. The Sun will try to tidally lock the primary to itself. As it does so, it tries to slow the primary's rate of rotation. This would cause the satellite to orbit faster than the primary's rotation and fall in nearer the primary, which in turn opposes the slowing of the primary's rotation.

This is likely where the 1:10 ratio you read about comes in. If the satellite is too small, its tidal effect of the primary will be to weak to effectively oppose the slowing of the primary's rotation by the Sun. The primary will slow, the satellite will drop to a lower and faster orbit and the two will drift out of mutual tidal lock.

If it is large enough, it will still drop into a lower, faster orbit, but will keep the primary locked to it. (like the Earth keeps the Moon tidally locked even as the Moon recedes to higher and slower orbits.)

As far as time scales go. The Moon presently recedes at a rate of 4 cm per year, due to Earth-Moon tidal interaction. The Sun's tidal effect on the Earth is about half that. So at a rough estimate we might say that the Sun could cause a decrease in our Mutually locked pair's distance of 2 cm/year.

If our pair started in mutual lock while separated by 100,000 km( period 87 hrs). it would take about 2.5 billion years to approach to a distance of 50,000 km (giving it a period of ~30 hrs. ). It would take an additional 1.6 billion years to near Roche limit distance and break up.

Granted, these are very rough estimates.

 

[LURCH]

After another few billion years, by the time Earth is locked to the Moon, its core will have gone cold and solid. No tectonics, no geothermal activity, no radioactives, no crustal refresh, etc.

But this is caused by age, not by being tidally-locked. If the Earth-Moon system started out synchronous from the start, the Earth's core heat would be essentially identical to what it is today, in terms of tectonic activity and so forth.

 

[shelanachium]

Many thanks Janus, we seem to be getting somewhere at last! Looks like my moon got to be Mars-sized, then! And planet started with slower rotation (reverse of Earth) and has been slowly speeding up as moon approaches. If 30hr days at present the guys there don't need to worry quite yet!

 

[Janus]

Many thanks Janus, we seem to be getting somewhere at last! Looks like my moon got to be Mars-sized, then!

Probably. But not for reasons of stability.
After I posted I got to thinking about the moon's size. It occurred to me that it wasn't the planet:moon mass ratio that counted. What counts is the Sun:moon mass ratio as compared to the Sun:moon distance ratio.

What we want is a situation where the tidal torque exerted by the moon on the planet is at least as large as the tidal torque exerted by the Sun on the planet. This way the moon can keep the planet tidally locked to itself as its orbit decreases.

Tidal forces is directly proportional to mass and inversely proportional to the cube of the distance.

so what we want is a situation where

\frac{M_{moon}}{D_{planet-moon}^3} > \frac{M_{sun}}{D_{planet-sun}^3}

or

\frac{M_{moon}}{M_{sun}} > \frac{D_{planet-moon}^3}{D_{planet-sun}^3}

Given a one solar mass sun with the planet orbiting at an Earth equivalent distance,and the moon orbiting at 100,000 km, this works out to a minimum mass for the moon of about 6e20 kg, or about 1/120 the mass of the Moon. This is what it would take to maintain the mutual tidal lock after it was achieved.

That being said, it probably isn't sufficient to achieve mutual lock in the first place, in a reasonable time period. For this, the planet-moon mass ratio comes into play. Unless by some rare coincidence, the moon's orbit and planet's rotation were almost matched to begin with, you are going to need a large mass moon to pull the planet into tidal lock quickly enough to do you any good.

And planet started with slower rotation (reverse of Earth) and has been slowly speeding up as moon approaches. If 30hr days at present the guys there don't need to worry quite yet!

 

[shelanachium]

Thanks. More to think about. Like if the orbit starts large it is likely to be and remain nearly coplanar with the ecliptic, as is that of our own Moon. You end up with an aseasonal planet. And I so wanted my people to sing of the beauties of Spring and Autumn, and of their accompanying eclipses and only REAL chance to see the stars in all their glory! But what the Hell, only scientific pedants like me would bother about minor impossibilities!

Also many think our own Earth's situation extremely unlikely, so I can myself allow the improbable, but not the impossible.

Read for example Simon Conway Morris, with whom I have discussed such matters, in his book 'Life's Solution - Inevitable Humans in a Lonely Universe' - Cambridge University Press 2003 - he is a leading evolutionary palaeontologist. He is however also a devout though non-fundamentalist Christian which may give him an agenda external to Science, though I suspect he would deny it.

 

[DaveC426913]

But this is caused by age, not by being tidally-locked.

Yes. That's what I'm saying.


If the story is set in a time far enoguh that the planets are locked, then it will be several billion years in the system's old age, thus the planets will have gone cold from age.

 

[LURCH]

I believe the idea is to set the story on a planet tidally locked to its satellite from the earliest part of the system's formation.

 

[shelanachium]

It is now.

 

[FTL_Diesel]

If the story is set in a time far enoguh that the planets are locked, then it will be several billion years in the system's old age, thus the planets will have gone cold from age.

But that depends on the initial conditions of the system, as well as a host of other factors (like the q-factor, for one). If this is fiction, then we can easily come up with a scenario where tidal locking has occurred by a (tectonically) reasonable date.

No?

 

[shelanachium]

Thanks guys (and gals?) for all your feedback. If Janus is right the system can evolve in a reasonable time (few billion years). Only remaining worry is that if orbit began quite large (e.g. 100,000km+) it might have been nearly coplanar with the ecliptic like that of our own moon. It would then have remained so giving an aseasonal planet, which I wish to avoid.

I know I can make it seasonal by having a strongly eccentric orbit, but that creates its own problems such as very much greater tidal effect of Sun around perihelion.

 

[Janus]

There's another problem: regression of the nodes.

From my understanding, you want the eclipses to happen the same time every year, correct? Your orbit will be tilted to the ecliptic. (otherwise you get eclipses every day.) In such a situation, the Sun produces a torque on the orbit which causes it to precess. This will cause the ascending and descending nodes to shift. This same effect causes our own Moon's nodes to shift in a cycle that lasts about 18 yrs. This is why we don't get eclipses at the same time of the year every time.

Geostationary communication satellites have the same problem with a cycle of 54 years. (which is why they have to use station keeping to prevent the drift.)

Unfortunately, even a 54 year cycle would mean that the dates of the eclipses would shift significantly from year to year.

 

[LURCH]

Well, what about a planet with an orbit that is slightly inclined off of the ecliptic, with a moon whose orbit is parrllel to the ecliptic. Taht way, you can get seasons without axial tilt, and twice a year (as the planet passes through the ecliptic plane) you ge eclipses. Now that I say it, though, I am forced to wonder; is there any difference between what I've just described and the current situation on Earth?

BTW, there is another factor to consider. If the days are roughly the same length as Earth-days, I don't think you can get one eclipse per year. Would your story still work if there were a season of the year when eclipses happened every day? From the problems encountered by comm-sats every year, I would guess this would be a period of several days to a week.

 

[Janus]

Well, what about a planet with an orbit that is slightly inclined off of the ecliptic, with a moon whose orbit is parrllel to the ecliptic. Taht way, you can get seasons without axial tilt, and twice a year (as the planet passes through the ecliptic plane) you ge eclipses. Now that I say it, though, I am forced to wonder; is there any difference between what I've just described and the current situation on Earth?

No difference. In this case, the "ecliptic" is always defined by the planet's own orbit. IOW the planet always has a "inclination" of zero.