Stability of a binary planet system

[Somes J]

Hello. I have a question concerning the stability of a hypothetical binary planet system. I'm sorry if this properly belongs in general astronomy instead of astrophysics, but it does concern orbital mechanics.

The hypothetical system is an Earthlike planet orbiting a relatively bright K class star at a distance of maybe .5-.7 AU. The planet is roughly Earth-sized and it has a very large moon, around 1/3 the mass of Earth (the idea here is that the moon would also be capable of supporting an Earthlike environment - basically this is the old Earthlike double planet concept). The moon would orbit the planet at a distance of around 40-50,000 km, with a period of around 30-35 hours, and both the planet and moon would be tidally locked to each other (like Pluto and Charon in our solar system). The planet's axis would be inclined at roughly 20-25 degrees (like Earth's) and the moon would orbit in the plane of the planet's equator. The planet would have no other natural satellites.

Now, the issue that's troubling me is tides. There would be huge tides (thousands of times ours), obviously, but since the planets are mutually tidelocked they'd be fixed so I imagine they'd only show up as a distortion in both worlds' overall shape, as generally unnoticed as the equatorial bulge on Earth. But there might also be variable tides, created by the eccentricity of the moon's orbit and any precession-type motions of the axis of the planet relative to the moon's orbital plane etc. Given how close together these worlds are even relatively small secondary variable tides could be catastrophic at the surface (oceans being raised many meters and penetrating deep inland etc.).

Say the moon starts out with a very very low eccentricity, like Neptune's moon Triton. Both planets may have been terraformed at some point in the deep geologic past, so maybe the aliens adjusted the orbit or something if that's necessary to justify it. But that was long, long ago, so my concern is would such an extremely circular orbit naturally stay extremely circular? From what I've heard strong tides tend to circularize orbits, and I've read that tidal dissipation scales by radius^6 whereas the strength of the tides only scales by radius^3, so based on that I'd think the near-perfectly circular orbit would be quite stable (the two worlds would gradually draw closer together as the solar tide took angular momentum out of the arrangement but it shouldn't become much more eccentric) but my knowledge of the relevant physics is extremely limited. I'm only an aspiring science fiction writer, not a professional astronomer.

Similarly I'd imagine that the huge tidal forces involved would keep the rotational axis of both the planet and the moon very close to 90 degrees to the plane of the moon's orbit, but again I lack the knowledge to be sure.

Would the inclined plane of the moon's orbit relative to the plane of the planet's orbit around the sun be a complication?

Basically I'm asking whether these worlds could realistically stay nice and Earthlike without artificial intervention or whether, once the terraformers were gone, we'd be back to looking at some sort of Io-like hellhole with massive ocean tides in a geologic instant. And if the latter, what could I do to make such an arrangement more plausibly stable?

Also, suppose we had a similar scenario but the moon was smaller, more like our moon or Mercury. Would that also be stable?

Sorry if this was a little long. Thanks, it'd be a huge help if I could get an answer.

 

[higginsdj]

I'm no astrophysicist but there seem to be a few issues here. 1, if they are mutually tidally locked then the 'moon' will not have an eccentric orbit (or any orbit around the 'earth' for that matter). 2, if the moon is that large then both 'earth' and 'moon' will be orbiting a barycentre at some point well between the 2. 3, the variability of the tides will come from the sun as the pair spin on their barycentre, not the 'moon'.

You need to be careful with your wording to avoid confusion - ie "the planets orbit around the sun" - as the systems are mutually tideally locked is the "systems orbit around the sun" and that is based on the barycentre of the systems spin rotation.

Note that 'stable' is a relative term (relative to time)

Cheers

David

 

[Somes J]

Yes, I suppose I should have made reference to both worlds orbiting a common barycenter rather than a moon orbiting a planet.

 

[higginsdj]

and since they are tidally locked to each other spinning around a common barycentre there is no variability in tide from within this system.

 

[Somes J]

and since they are tidally locked to each other spinning around a common barycentre there is no variability in tide from within this system.

If the orbit was eccentric the tide would be stronger at perigee than at apogee, and the moon would move (slightly) back and forth in the sky as it pulled ahead of the planet's rotation at perigee and fell behind it at apogee. This is what I'm worried about in regards to whether the planet would have huge tides; even with a small eccentricity (say .01) I get a variance in tide between apogee and perigee much greater than our lunar tide.

From what I've read tidal effects tend to circularize orbits, and tidal dissipation scales by inverse radius^6 with distance while tides scale with inverse radius^3, so by that I would think the trend would be for any eccentricity introduced to the orbit to be dissipated away before such tides could become extreme, but my knowledge of this subject is extremely limited.

 

[higginsdj]

There is no eccentricity of orbit, the moon and Earth are tidally locked to each other ie neither is orbiting the other - consider them a single 'object' spinning on a axis that is the barycentre of the system.

Do you mean the effect the sun may have on the individual components of this system as they spin around the barycentre? The sun will pull at each component and the pull will vary in a small way changing the distance between the moon and Earth dependent on the orientation of the system in relation to the sun at a point in the orbit and how fast the system spins.

 

[Somes J]

There is no eccentricity of orbit, the moon and Earth are tidally locked to each other ie neither is orbiting the other - consider them a single 'object' spinning on a axis that is the barycentre of the system.

I thought perfectly circular orbits were basically impossible? Also Pluto and Charon are tidally locked to each other and http://ssd.jpl.nasa.gov/?sat_elem".

Do you mean the effect the sun may have on the individual components of this system as they spin around the barycentre?

Basically, I have the assumption based on my very limited knowledge that it's reasonable to think that the orbit of the two worlds around the barycenter will stay very close to a perfect circular over mega year/giga year timescales in a realistic situation, and I'm asking whether this assumption would actually be realistic if this system was real.

Basically I'm trying to figure out whether my fictional planet is realistic or I need to go back to the drawing board.

 

[higginsdj]

OK, granted that nothing is 'perfect' as in Pluto and Charon can be considered to be tidally locked though it's not a perfect system. This system is impacted by other objects in orbit. e = 0.0022 is still very small, an 0.4% variation from a circular orbit.

BUT these are barycentre orbital elements - ie the shape of the 'orbit' around the barycentre and NOT around the 'parent'.

 

[D H]

I'm no astrophysicist but there seem to be a few issues here. 1, if they are mutually tidally locked then the 'moon' will not have an eccentric orbit (or any orbit around the 'earth' for that matter).

The Moon is tidally locked to the Earth and yet the Moon is orbiting the Earth and the eccentricity of the orbit is 0.0549.

2, if the moon is that large then both 'earth' and 'moon' will be orbiting a barycentre at some point well between the 2.

So what? The Earth-Moon barycenter is currently about 2/3 of an Earth radius from the center of the Earth.

3, the variability of the tides will come from the sun as the pair spin on their barycentre, not the 'moon'.

The Moon is responsible for about 2/3 of the tides on the Earth, the Sun accounting for only 1/3.

 

[higginsdj]

1. The moon still orbits the Earth (~28 days), its spin has been tidally locked - its not mutual. Note that my understanding of mutual here is that the moon presents the same face to Earth and the Earth would present the same face to the moon ie they are synchronised.

2. I was simply referencing the dynamics of the system.

3. Yes, because the moon is orbiting our earth. In a mutual, as I describe above, system the components orbit the barycentre and not each other.

 

[D H]

The moon still orbits the Earth (~28 days), its spin has been tidally locked - its not mutual.

Yes, because the moon is orbiting our earth.

What does this have to do with your earlier statement (which was absolutely wrong),

if they are mutually tidally locked then the 'moon' will not have an eccentric orbit (or any orbit around the 'earth' for that matter).

Just because a pair of objects are mutually tidally locked to one another does not mean that the objects are no longer orbiting one another.

The Moon has come pretty close to making the Earth be tidally locked to the Moon. The current working hypothesis regarding the formation of the Moon is the giant impact hypothesis. Per this hypothesis, a Mars-sized body collided with the Earth about 100 million years after the Earth finished forming. The Moon coalesced from the ejecta at a distance of four to six Earth radii from the center of the Earth. The length of the Earth' day after this collision was four to six hours. In other words, the Moon has slowed the Earth's initial rotation rate by 75% or more. It would have taken much of a change to make in the collision or in the formation of the Moon to have resulted in an mutually locked planet-moon system after a few billion years. That mutually locked planet and moon would still be orbiting one another.Responding to the original post,

Now, the issue that's troubling me is tides. There would be huge tides (thousands of times ours), obviously, but since the planets are mutually tidelocked they'd be fixed so I imagine they'd only show up as a distortion in both worlds' overall shape, as generally unnoticed as the equatorial bulge on Earth. But there might also be variable tides, created by the eccentricity of the moon's orbit and any precession-type motions of the axis of the planet relative to the moon's orbital plane etc. Given how close together these worlds are even relatively small secondary variable tides could be catastrophic at the surface (oceans being raised many meters and penetrating deep inland etc.).

The tides would not necessarily be huge. Tidal forces are proportional to mass and inversely proportional to the cube of the distance. This is why the Moon is responsible for about 2/3 of the tides on the Earth even though the Sun is 27 million times more massive than is the Moon. However, to have a livable planet you will want your moon orbiting considerably closer to the planet than the distance between the Earth and the Moon so that days are not 700+ hours long. That means that the tides are going to be rather large.

Regarding tides that result from eccentricity or a misalignment between the planet's axis of rotation and the orbital axis: The eccentricity and misalignment have to be fairly small for a body to become tidally locked to another. Too high an eccentricity and you'll get something like Mercury, which rotates three times (sidereal rate) for every two orbits. (Aside: A planet and a large moon each in a 3:2 spin-orbit resonance might make for an interesting sci-fi setting.)

If the eccentricity and misalignment are sufficiently small, the tidal deformations will eventually result in a planet and moon whose orbits about one another are nearly circular and whose rotational axes are more or less aligned with the orbit axis. The coupling with axis misalignment is (I think) considerably weaker than is the coupling with eccentricity.

If the planet and moon are orbiting one another in perfectly circular orbits, and if the rotational axes and orbit axis are perfectly aligned, you are right in that the moon would induce permanent tides on the planet: The tides would manifest as a slight change in the shape of the planet.

Basically I'm asking whether these worlds could realistically stay nice and Earthlike without artificial intervention or whether, once the terraformers were gone, we'd be back to looking at some sort of Io-like hellhole with massive ocean tides in a geologic instant. And if the latter, what could I do to make such an arrangement more plausibly stable?

Over the very long term, this situation is not stable. With a planet and moon tidally locked to one another, the tides raised on the planet by the star will act to slow the planet's rotation rate. This will make the planet rotate slightly slower than the orbital period, and this in turn will cause the moon to move closer to the planet.

However, as you noted, tidal locking time is inversely proportional to R⁶. When I said "very long term" I meant very, very long term: Longer than the lifespan of the star for the planet to be at a habitable distance from the star. Not something to be worried about in your sci-fi setting.

Precession might be a bit more interesting.

 

[higginsdj]

What does this have to do with your earlier statement (which was absolutely wrong),

Just because a pair of objects are mutually tidally locked to one another does not mean that the objects are no longer orbiting one another.

The Moon has come pretty close to making the Earth be tidally locked to the Moon. That mutually locked planet and moon would still be orbiting one another.

Did you read my definition of mutually tidally locked? Is that what you think is wrong? If the moon (not our moon) and the hyperthetical Earth (not our earth) were locked such that the moon showed the same face to the Earth and the Earth showed the same face to the moon then the moon is acting like a geostationary satellite ie synchronised - how is this orbiting? (or perhaps the issue is the use of the word 'orbiting').

As it is locked in place over the same point of the earth, it caused a bulge at the time it became locked but no tides will occur since there is no movement from this locked position. In my view the orbjects are then orbiting the barycentre and not each other.

Please note that I am not trying to be difficult here - just trying to understand why you think I am wrong. (or where I have gone wrong in my understanding)

 

[D H]

Did you read my definition of mutually tidally locked? Is that what you think is wrong?

What definition? All I saw was the statement to which I took exception (emphasis mine): "if they are mutually tidally locked then the 'moon' will not have an eccentric orbit (or any orbit around the 'earth' for that matter)."

If the moon (not our moon) and the hyperthetical Earth (not our earth) were locked such that the moon showed the same face to the Earth and the Earth showed the same face to the moon then the moon is acting like a geostationary satellite ie synchronised - how is this orbiting? (or perhaps the issue is the use of the word 'orbiting').

Are you truly saying that geosynchronous satellites are not orbiting the Earth? If that is the case, you have a rather non-standard meaning of the word "orbit".

As it is locked in place over the same point of the earth, it caused a bulge at the time it became locked but no tides will occur since there is no movement from this locked position.

Another name for that is a permanent tide. See post #11.

In my view the orbjects are then orbiting the barycentre and not each other.

A better view is that they are orbiting one another. Two objects are orbiting one another if they are stably bound to one another gravitationally (i.e., well within the Hill sphere). How the objects are rotating has nothing to do with whether they are orbiting one another.

 

[higginsdj]

OK, so the whole issue is my 'understanding'/'use' of Orbit (and yes in hindsight it was an incorrect use of the term). I am a layman so was using it in reference to the 'moon' changing it's place in reference to the 'Earth's' surface and thus its impact on tides. And I dare say my use of tides may differ from yours/(norm/accepted) as I used it in terms of some variability - ie no variability no tides.

Cheers

David

 

[D H]

And I dare say my use of tides may differ from yours/(norm/accepted) as I used it in terms of some variability - ie no variability no tides.

The modern theory of tides (not so modern actually; tidal theory was pretty much nailed down in the late 19^th / early 20^th centuries by multiple people, key ones being William Thompson, George Darwin, A.E.H. Love, and A.T. Doodson) is to look at things in terms of forcings and in the frequency domain. Multiple terms arise with periods of close to 1/2 day, another set of terms arise with periods close to 1 day, and so on. A constant term (infinite period / zero frequency) can arise when doing Fourier analysis. This is the case for the tides, relative to a hypothetical Earth with no sun or moon present. This zero frequency component is called the permanent tide (google that term).