[SPACE] Tidal braking

[Agemegos]

On p. 117 of Space, in Step 30 of the planet generation sequence, we estimate the tidal distortion of a planet as T = (0.46 * M * D) / R^3, (where M is the mass of the star, D is the diameter of the world, and R is the radius of the world's orbit). And the total tidal effect (amount of slowing) is E = (T * A)/ m (where A is the age of the system and m is the mass of the world). That expression expands to E = (0.46 * M * D * A) / (R^3 * m).

Now suppose that the system is only just old enough that the planet has tide-locked. Substituting E = 50 (the critical value for tide-lock), we get (0.46 * M * D * A) / (R^3 * m) = 50, so

A(lock) = (109 * R^3 * m) / (M * D)

That's the time to tide-lock in GURPS Space

Look at the formula for time to tidelock given in Wikipedia. This is

t(lock) = (w * R^6 * I * Q)/(3 * G * M^2 * k * (D/2)^5)

(making the appropriate substitutions for the different notation.). I (moment of inertia of a spherical planet) is approximately = 0.1 m D^2. G, Q, and k are constants, the latter two having to do with the material of the planet. w is the initial rotation rate of the planet.

Collect the constants

t(lock) = (0.1 * Q)/(3 * G * k / 32) * (w * R^6 * D^2 * m)/(M^2 * D^5)

= constant * (w * R^ 6 * m)/(D^3*M^2)

Compare

A(lock) = (109 * R^3 * m) / (M * D)

Something is obviously wrong. I suspect that the authors of Space overlooked the following: that the tide height is proportional to teh tide-raise force divided by the surface gravity; that the tidal torque is proportional not only to the height of the bulges, but to the gravity gradient (0.5*G*M/r^3) and the moment arm (D times the sin of the angle between the bulges and the star-planet line); and that the tendency of the planet to resist rotation, its moment of inertia, is proportion to its mass times the square of its radius.

If I (and Wikipedia) are right, we ought ot replace the current expression for E with

E = (0.46 * A * D^3 * M^2) / (R^6 * m)

Which should make it a lot easier to get planets aroung type K stars that are not tide-locked. Which in turn should be greatly to Zorg's relief.