Hurried Thoughts: Late Tidal-Locking Planets

This is a concept I came across while trying to work out the probable maximum tidal range for a mature, habitable planet for Part IVa (it came out to around 3-8 meters). I mentioned it in a quick aside there and I'll bring it up again in the future, but it intrigues me enough that I thought it deserved highlighting before then.

 

You may remember that habitable-zone planets of small M-type stars are likely to tidal-lock soon after they form, typically in a 1:1 spin-orbit resonance such that one side constantly faces the star. Earth and similar HZ planets of G-type stars are far enough from their stars that tidal braking is very slow and so they shouldn't expect to tidal-lock within their habitable lifetimes. But somewhere in between, within the range of K-type stars, there must reasonably be HZ planets that do eventually tidal-lock but not for billions of years after they form.

 

To investigate this, here's the formula for the total time to tidal-locking I put together in Part IVa:

 

 

t = time to tidal-lock (billion years)

Q = dissipation factor (~100 for typical solid planet of Earthlike size)

m = mass of planet (Earth masses)

a = semimajor axis (AU)

P = initial rotation period (hours)

M = mass of other body (sun masses)

r =  radius of planet (Earth radii)

On its own this doesn't tell us much about what happens to the planet while it's tidal-locking, but note that it depends in part on the initial rotation period; the shorter the rotation period (and thus faster the rotation and greater the planet's angular momentum), the more tidal braking is required to spin down the planet. As it spins down, the rotation period increases; so at any time in the process, you can plug that new rotation period back into the formula as the new "initial" rotation and get the remaining time to tidal-locking; you can also rearrange the formula to give you the rotation period at a given remaining time until tidal-locking. In either case, you can chart out the total evolution of the planet's spin backwards from the point of tidal-locking and then offset that by the age of the planet at that point (i.e., time to tidal-locking from the actual initial rotation).

So, if we take an Earthlike but moonless planet (with a dissipation factor of 100) orbiting at 1 AU from a sunlike star and give it an initial rotation period of 12 hours, that gives it a comfortable 200 billion years until tidal-locking, with the rotation period barely affected in the first 10 billion.

 

Dotted red line indicates time of tidal-locking; and note the logarithmic vertical scale.

If we take a star half our sun's mass and put an Earthlike planet in a roughly Earth-analagous orbit just past the inner edge of the conservative habitable zone at 0.196 AU, tidal-locking takes under 300 million years even with blisteringly fast 2-hour initial rotation.

 

So, of course, there must planets between these extremes. And, indeed, if we take a star of 0.665 solar masses and place an Earthlike planet at 0.3681 AU (again, a roughly Earth-analagous inner-HZ orbit) with a brisk but reasonable initial rotation of 3 hours, we get tidal-locking in almost exactly 4.5 billion years.

 

But note here how gradually the period increases at first before rocketing up just before locking. It takes over 1.1 billion years to slow the rotation to 4-hours days, and when the planet is 4 billion years old it has only slowed to 27-hour days; if we imagine that life evolves here at about the same pace as on Earth, the first complex flora and fauna will emerge on a planet with days and a climate broadly similar to what we have on Earth today (though with years less than 1/3 as long).

The pace of spindown increases thereafter; 48-hour days are reached when the planet is 4.22 billion years old, 280 million years before final tidal-locking; 100-hour days at 135 million years before tidal-locking; and 240-hour days at a mere 56 million years before tidal-locking.

 

You can look at my recent post on climates of planets with varying day length to get a sense of how the climate will shift over this period, though they are for a planet of a different star with a longer year and with Earthlike obliquity; we'd expect any initial obliquity this planet had to also reduce over time from tidal effects, though I couldn't say at what rate. At any rate, the general trend appears to be that the deserts would move towards the poles, with the tropics becoming increasingly humid but also cooler and with greater swings in day and night temperature that life would have to adapt to. Then, the planet would transition between that climate and the wholly different climate of a tidal-locked planet in less time than has passed on Earth since the end of the Cretaceous period. One can imagine the gradual mass extinction that would cause and the rapid adaptation required to survive the ever-shifting climate.

In short, I think it would make for a pretty cool speculative evolution setting and encourage anyone interested to run with it.

Now, this is a fairly simplistic model of tidal evolution. The tidal dissipation (Q) factor is covering for a lot of complex tidal mechanics, though it doesn't necessarily matter much if we can't estimate it well; if we shifted the Q factor to 12, closer to what it is on modern Earth, the tidal spindown would take only about 1/10 as long for this planet but we can still get basically the same scenario to play out if we shift the planet to orbiting 0.56 AU from a star 0.81 times the mass of the sun.

 

But the Q factor has also likely shifted over Earth's history as the continents have drifted and blocked or corralled the tides in different ways. We could take advantage of this for even more rapid spindown: if we supposed our previous planet had a Q factor of 100 for 4 billion years and then quickly dropped to 12, the spindown from 24-hour days to tidal-locking could take less than 100 million years. But slowing rotation will probably also create varying tidal resonances that could alter the Q factor, so I can't be quite sure what will happen in those final years.

 

Mass and composition similarly alter the rate of spindown, with more massive and dense planets taking longer to tidal-lock, but again this can be compensated for by shifting to a slightly different star or orbit. It seems we can be fairly confident that something like this overall scenario will play out for some combination of parameters, even if we can't be totally sure what those number will be.

Incidentally, the example planet I've been building in the main series, Teacup Ae, falls within this range of planets we've discussed here, orbiting 0.45 AU from a star 0.7 times the mass of the sun. I've established it to have 34-hour days at its "current" age of 6 billion years old; going by the model here and assuming an average Q factor of ~100 (along with its parameters of 0.8 Earth masses and 0.96 Earth radii), that would imply its initial rotation was a tad over 5 hours and it has significantly spun down over its history; 500 million years ago it had days about as long as Earth, in another 500 million they will roughly double in length, and in a bit over a billion years it will fully tidal-lock. The planet will still be well within the habitable zone and should still be geologically active, so it could have a substantial period as a habitable tidal-locked planet in its future.

Teacup Ae does have a large moon as well, but it's smaller than our moon, and if considered alone only spins down the planet about 1/10 as fast as the star, so it probably doesn't change the results much. There are some better models available for describing the tidal evolution of star-planet-moon systems, but so far I haven't had much luck in properly implementing them; if I do get them working I'll probably put the model in a supplement post like I did with StarPasta.

 

In the meantime, I've put this simple model into a spreadsheet with some other helpful calculations, so you can play around with it yourself (the chart is a bit wonky on the google sheets version, you may have to play around with it or download the file to get it working properly).

 

Buy me a cup of tea (on Patreon)