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

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?

• Just to be clear: it's our expectation, based on orbital radii and assumed ages of the planets, that they're tide-locked. We have no way of observing this at present. – Carl Witthoft Feb 27 '17 at 15:25

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.

• Better if you explain what $Q$ and $k_2$ are. – ProfRob Feb 26 '17 at 8:39
• Do you mean that Earth (and every body) will be eventually locked to the Sun in a certain time? – Carlos Vázquez Monzón Feb 26 '17 at 14:07
• No, because the sun will destroy the Earth by becoming a red giant, and then a white dwarf long before the Earth will become tidally locked. Theoretically, if two bodies are orbiting, then eventually they will become mutually tidally locked, but this may take trillions of years (if the bodies are distant) – James K Feb 26 '17 at 14:20
• I see. Was the time value for the planets of the Solar System calculated, or is there some estimation of some kind? – Carlos Vázquez Monzón Feb 26 '17 at 17:47