Tidal Lock Radius in Habitable Zones

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 rearrange 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.


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.

Tags:

Tidal Effect