Tidal Lock Radius in Habitable Zones

Much is made of finding exoplanets in habitable zones, locations with orbital semi-major axes permitting water in the liquid state. Habitability may be compromised if such bodies become tidally locked, orbiting within the star's tidal lock radius. The illustration below shows the tidal lock radius for various stellar spectral classes:

The diagram shows Mercury within the tidally locked zone although its really in a 3:2 spin-orbit resonance. How is the tidal lock radius calculated?

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I don't know the answer, but presumably it's something like this. You make some assumption about the planet's size, stiffness (i.e., the degree to which it deforms for a given tidal stress), and initial rotation rate. Then for any given choice of $M,R$, it's straightforward to calculate the time scale for tidal locking (say the time for the rotation rate to decline by a factor of 2 or $e$ or something). You also assume some rough age for the star and compare the tidal locking time scale to it. This is sensible as long as the final answer depends only weakly on all those arbitrary choices! – Ted Bunn Jul 20 '11 at 22:15
Ted: The image came from daviddarling.info/images/habzone.gif – Michael Luciuk Jul 20 '11 at 22:26
Thanks! Interestingly, the Web page that contains this image (daviddarling.info/encyclopedia/H/habzone.html) doesn't mention tidal locking, nor does it say where this graphic came from. – Ted Bunn Jul 21 '11 at 2:21

From Wikipedia (which cites the paywalled http://dx.doi.org/10.1006/icar.1996.0117), we get http://en.wikipedia.org/wiki/Tidal_locking#Timescale

Now, a is the semi-major axis (or orbital radius) of the object (I'm not totally sure if the logic changes for elliptical orbits). Anyways, we can easily re-arrange the equation to express a (or the radius) in terms of $t_{lock}$ (or the time) and all the other variables.

Then (using the Wikipedia variable names) we get $$a = (\frac{3t_{lock}Gm_p^2 k_2 r^5}{wIQ})^{1/6}$$

So then if we know t (which is basically the age of the stellar system in question - which we can get a rough idea of since we do have a rough idea of how each star is), then we can get a value of a for that particular t.

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Well done. I discovered another source by S. J. Peale (Rotational Histories of the Natural Satellites, 1977) for tidal locking radius. It's simpler than your WIKI answer but both formulae gave the same result. – Michael Luciuk Jul 21 '11 at 5:27
@Michael all this shows that such a "radius" does not exist. Because of the strong dependence of such radius on the elastic property of the planet to be locked and on time, there is no specific "radius" not even a maximum or minimum radius. – Georg Jul 21 '11 at 12:05
Georg, this is obviously only a highly notional or hypothetical construct. Astrophysicists studying exoplanet habitable zones use estimated planet formation and rotation times and other factors to assess potential tidal locking. The thin tidal locking radius line in the diagram should be thicker and fuzzier. – Michael Luciuk Jul 21 '11 at 14:02
Right. And on the log-log scale of that plot, the amount of fuzziness that would be required, for "reasonable" ranges of planet properties, wouldn't really be all that much. Still, the figure really needs additional explanation to say what parameters were chosen in drawing that line. A Google image search shows this image popping up in a bunch of web pages, but at after a somewhat quick look around I couldn't find the original source or any page with a more detailed caption. – Ted Bunn Jul 21 '11 at 14:24
@TedBunn: I agree. The major uncertainty here is the tidal energy conversion factor Q. And by major I mean ~15 orders of magnitude uncertainty for various solid compositions. There is recent work on astro-ph on omitting Q, if anyones interested. (edit: sorry didn't see this happened 4 years ago) – AtmosphericPrisonEscape Aug 12 '15 at 5:01

You can see the graph used in context, in the paper 'The Habitable zones around main sequence stars 1993' written by James F. Kasting. The graph is on page 124.

The locking formula he uses to calculate the graph comes from a book, a chapter written by Peale in 1977 called 'Rotation histories of the natural satellites'

The formula is Tidal Locking Time in Seconds = $\frac{1}{486}\bigl(\frac{a}{0.027*M^{1/3}}\bigr)^6$

It uses CGS units (centimetres, grams and seconds) where $M$ = Star's Mass (grams) and $a$ = Orbital Distance of Exoplanet (cm)

The above formula assumes the planet has exactly the same properties as Earth. So it has the same shear modulus, radius, density and mass. Its starting rotation is assumed to be 13.5 hours.

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