Celestia Forums

Is there a way to figure out roughly how long it would take a planet in an otherwise "habitable" orbit around a red dwarf star to brake enough that 1 side of the planet is always facing the sun? And (given the same primary) how long it would take for a double planet or earth-moon type system to degrade to the point where one or both worlds are ripped apart by tidal stresses? So far I've been using numbers for my current project that "feel right", but I'd like to have something to check my numbers against.

Thanks in advance,

Andrew

There are some quite complex equations for this. The best on-line source for tidal evolution equations I've seen is our own dear Evil Doctor Ganymede's thesis - I've lost the URL, but I'm sure he'll step in in a jiffy with a link. Barnes and O'Brien's Stability of satellites around close-in extrasolar giant planets also gives the necessary equations (which are generally applicable, not just to giants), but beware that there are serious misprints in their equations 10 and 11.
The hassle is that these equations need planetary bulk parameters that are hard to find - the tidal dissipation factor, moment of inertia factor and Love number k2 for the bodies concerned.
A while back I dug these out for terrestrials and gas giants, and then condensed all the constants and fixed parameters out of the tidal spindown equation to yield the following for terrestrial worlds:

T = [150000 * Mp * a^6]/[RotPer0 * Ms^2 *r^3]

Where T is the spindown time in millions of years; RotPer0 is the initial rotation period of the terrestrial world, in hours; Mp is the mass of the planet in Earth masses; Ms is the mass of the star in solar masses; a is the semimajor axis of the planet's orbit in AU; and r is the planet's radius in Earth radii.
The rotation period at formation is obviously tricky ... I think estimates for the Earth are about 5 hours.
(Note that you can't use this equation for gas giants - the various bulk parameters are very different for such worlds.)

If you plug in likely values for red dwarfs, you'll find tidal spindown in orbits with Earth-like insolation takes place very quickly - of the order of a million years.

My understanding is that double worlds with mass ratios in the vicinity of 1:1 are predicted to "save" themselves by locking into the mutually synchronous state - this prevents the primary synchronizing on the star, and so avoids the tidal decay of the secondary's orbit.

Grant

Hmmm...

Using your equation, I plugged in values for Struve 2398 A and the hypothetical planet:

Planet Mass: 1.059 Earths
Planet Radius: 1.11 Earths
Star Mass: 0.2151 Sols
Semimajor Axis: 0.0545 AU
Starting Rotation Rate: 5 hours

Time until the planet becomes tidally locked = 12,900 years. Eep!

It's an order-of-magnitude estimate, but that's telling you that the thing's going to effectively form synchronous. The a^6 term is a killer.

Grant

Thinks ...
Or are you worried that the equation is wrong? I probably should have said that there are ten-fold variations in estimated tidal dissipation factor across the terrestrial planets in our solar system - so it really is order-of-magnitude stuff.
But as a quick check against published data: tidal spindown time for the Earth from my equation, 30 billion years; tidal spindown time for the Earth from the textbook Solar system dynamics, 50 billion years. (They start from an initial rotation of 10 hours but use a lower value for k2 than I used in building my formula, and the two effects counteract each other.)
Reduce the mass of the Sun by a factor of 5, and you increase spindown by a factor of 25. But shove the Earth 20 times closer to the Sun, and you reduce the spindown time by a factor of 64 million. Put these factors together and you have something like your red dwarf system, with a spindown estimate in the tens of thousands of years.

Grant

Heh. I was going to suggest my thesis, but then I thought that might be a bit too detailed...

But since Grant mentioned it, yes - it's in CH4 of my thesis, which is a 5.2MB download. Specifically, to sort out tidal braking from stars, you'll need to look at section 4.2.3 (Tidal Evolution Equations) and you need equation (4.24) which is at the top of page 63.
(you may wonder why I put solar tide despinning equations in a thesis about the satellites of jupiter. Well, it was so I could have all my handy orbital evolution equations in one place for world-building ). But you're best off reading all of CH4 if you want the full background, I start pretty much from scratch about how orbital evolution works and hopefully it's fairly readable if you have some science background.

Figuring evolution out for a planet-moon system orbiting close enough to a star to be affected by tidal braking from that is a nightmare though (in fact, I couldn't figure out how to do it, since terms in both sets of equations are changing at the same time and it's all horribly interlinked). I do know that large moons aren't likely to exist in worlds that are much closer to Sol than we are. I figured out that the moon was actually about 100,000km closer than it should be if it was just evolving outwards because of earth-moon tides alone - if you rewind the evolution back according to those equations then it ends up being at earth's surface about 2 billion years ago. That could be down to the influence of the solar tides too, or it could be down to its origin in a giant impact (or it could be soething else entirely).

Oh, I'm not worried about the equation, grant... but that small a spindown time tells me that much of the geological, climate, and biological history I devised for this world will need to be torn down and rebuilt from scratch.

Initial assumption was that the planet would have a large (somewhere between the Moon and Mars in mass) satellite, either captured or co-formed, that would have kept it in a semistable lock to each other. Over time (which I foolishly assumed would be a few billion years) tidal braking from the star would draw the satellite closer until it breached the planet's Roche limit, destroying the satellite, plunging the planet into an impact winter for an aeon or two, and creating a beautiful ring system.

Given that it took 65 million years for life to recover from the K-T extinction and for a sophont to arise here on Earth, I figured that I'd need a similar length on time for my planet - but not so long that the planet gets into a 1:1 lock with its star and becomes uninhabitable by anything more complex than a microbe.

Given the short time for spindown with no satellite to muck things up, I'm guessing that any satellite would have met a tidal doom early in its formation, and that the ring system would have dissipated a few million years later. In a cosmic eyeblink, the planet is locked for the remainder of its lifespan. Bummer... but since we can't change the universe, "Struve Prime" will have to be relegated to the trash bin.

Yeah, I figure that if you put a planet identical to Earth at 0.2 AU from Struve 2398A and give it an initial rotation rate of 10 hours, it'd be tide-locked to the star within about 19 million years of its formation.

That's what you get if you want a habitable planet around an M dwarf - they're always going to be tide-locked and moonless.

Question, if you have a planet orbiting around a pair of stars which form a close binary e.g. CM Draconis or Delta Trianguli, can you just put the combined mass of the two stars into the equation and still get an answer in the right sort of region, or does it vary massively?

chaos syndrome wrote:Question, if you have a planet orbiting around a pair of stars which form a close binary e.g. CM Draconis or Delta Trianguli, can you just put the combined mass of the two stars into the equation and still get an answer in the right sort of region ...

That should do it - by the time you're far enough out to have a stable orbit around the pair, they're beginning to look quite a lot like a point mass from the planet's gravitational perspective.

Grant