Requirements for a satellite/planet to be tidally locked to a planet/star

Question

The Moon is tidally locked to the Earth, the Four Galilean are tidally locked as well, and the recently found planetary sistem TRAPPIST-1 has seven tidally locked planets, but Venus or Mercury are not. Why? Is there any conditions or analytic formula which says if a small body will be eventually tidally locked to the major body?

Answer 1

Wikipedia gives the formula

$$t_{\text{lock}} \approx \frac{\omega a^6 I Q}{3 G m_p^2 k_2 R^5}$$

where

• $\omega$ is the initial spin rate expressed in radians per second,
• $a$ is the semi-major axis of the motion of the satellite around the central body (given by the average of the periapsis and apoapsis distances),
• $I\approx 0.4 m_s R^2$ is the moment of inertia of the satellite, where $m_s$ is the mass of the satellite and $R$ is the mean radius of the satellite,
• $Q$ is the dissipation function of the satellite,
• $G$ is the gravitational constant,
• $m_p$ is the mass of the central body, and
• $k_2$ is the tidal Love number of the satellite." source

Notice that the semi-major-axis is the power of 6. A small change in the orbital distance can have a very large effect on whether the body will become locked.

Also note the terms $Q$ (a measure of how elastic the body is, and hence how much energy is lost in distortion source) and $k_2$ (a measure of the rigidity of the body: how much it is distorted by tides source), which are both hard to measure. The 7 detected planets in the Trappist-1 system are all much closer to their star than any of the sun's planets and for reasonable values of $Q$ and $k_2$, the planets will be tidally locked within a few million years, much as the moons of Jupiter are tidally locked.

In our solar system, mercury is asynchronously locked with the sun (on a 3:2 resonance) The Earth is not locked: it is too far from the sun, and the moon has a big effect. Mars is much too far out, and Venus is odd.