Will the Earth ever show synchronous rotation? Why and when?

Question

In this answer to the question "why does the moon have the same rotation and revolution periods?", we read:

The mass and speed of rotation of the Earth influence the moon in that some of its rotational energy is actually transferred to the moon. (...) Many of the moons in the solar system have also reached this point of equilibrium. In Jupiter, the moons Amalthea, Thebe, Io, Ganymede, Callista, and Europa, all have identical rotational and revolutionary periods.

My two questions are:

1. Can we show the mechanics of this quantitatively?

2. Does the system Earth-Sun experience the same phenomenon, in some measure? Will then the earth slow down its rotation speed until it will show always the same face to the sunlight? In this last case, on which timescale?

Answer 1

The same mechanics that applies to the earth-moon system should apply to the sun-earth system. One might worry about the fact that the sun is a not a solid but is a plasma, furthermore there are other planets in the solar system. Fortunately it is mostly the satellite that determines the spin-down time, and the Earth is not too different from the moon.

See the wiki page on tidal locking for more on the mechanics and an equation for the spin-down time. When making these sort of estimates it is usually best to focus on the most important factors. In this case the obvious difference between the systems is the masses and the orbital distance. Let $$r_1=m_\mathrm{sun}/m_\mathrm{earth}\approx 3\times 10^5$$ and $$r_2=R_\mathrm{sun-earth}/R_\mathrm{earth-moon}\approx 3\times 10^2$$ and assume everything else between the sun-earth and earth-moon systems is identical. Then, using the formula for tidal-locking timescale on the wiki page, the timescale for earth to become tidally locked is approximately $$t_\mathrm{sun-earth}\approx \frac{r_1^6}{r_2^2}t_\mathrm{earth-moon}\approx 10^4t_\mathrm{earth-moon}$$ Determining $t_\mathrm{earth-moon}$ is estimated to be roughly 15 million years (from stackexchange). So the time for earth to spin down is roughly 100 billion years. This is much longer than the timescale for the sun to burn out, 5 billion years (from wiki).