How much energy would be necessary to slow down Earth rotation such that a year was 360 days long?
In the same spirit: how much energy would be necessary to make the Earth rotate faster around the Sun such that a year was 360 days long?
(Just for fun: how would you do it practically?)
Answer
The Earth is currently rotating at one revolution per sidereal day. Converting to radians, this is an angular velocity of $\omega_0 = \frac {2\pi}{\text{sidereal day}}$. You want to have 360 solar days per year, or 361 sidereal days per year. That means a rotation rate of one revolution per 1/361 tropical year, or an angular velocity of $\omega_1 = \frac {722\pi}{\text{tropical year}}$.
The rotational kinetic energy of a rotating object is given by $E=I\omega^2$, where $I$ is the moment of inertia about the rotation axis and $\omega$ is the angular velocity. The minimum amount of energy needed to accomplish the desired change is thus $I_\text{earth}({\omega_0}^2 - {\omega_1}^2)$. Using a rather old value for the Earth's moment of inertia about the Earth's rotation axis of $8.034 \times 10^{37}$ kg m2, the minimum amount of energy needed to accomplish this change is $9.8\times10^{27}$ joules (calculation at Wolfram Alpha).
The minimum energy needed to change the Earth's orbit so that a tropical year is 360 days long is to perform a Hohmann transfer. To accomplish this, we need to decrease the Earth's orbit velocity about the Sun by 72.04 m/s (calculation) and then half a year later, decrease the Earth's orbit velocity once again by 72.22 m/s (calculation). Assuming some magic that allows us to convert energy to momentum, the minimum energy needed to accomplish these changes is $3.1\times10^{28}$ joules (calculation).
Either way, that's a lot of energy. Given that all of humanity consumed $5\times10^{20}$ joules during 2010, if we were to do this without any outside help, we'd have to stop consuming energy for about 60 million years. By then we'll have stored enough energy needed to perform either of these tasks, and surely in 60 million years we'll have developed the magic needed to convert energy into momentum.
There are other ways to accomplish the second task. We could use gravity assists to transfer momentum to Jupiter (Korycansky 2001). Capture an asteroid and outfit it with thrusters. Set it into a flyby of Jupiter that drops it to a flyby of Earth. Repeat this, many, many times over. We'll need a bit of fuel to readjust the asteroid's orbit, but the vast majority of the needed energy will come from the Jovian flybys. Eventually (after hundreds of millions of years), the Earth will be in the desired orbit.
Another approach is to build a big (a very, very big) solar array that is somehow maintained in static equilibrium relative to the Earth (McInnes 2002). The center of mass of the sail-Earth system will slowly accelerate (or decelerate, depending on sail orientation), once again eventually moving the Earth.
Note that both of the cited papers address the problem of moving the Earth away from the Sun. We'll need to do that eventually because G-class stars generate ever more energy as they age. The Earth will be completely uninhabitable in less than a billion years if we don't do something. But there's nothing technically wrong with using Korycansky's flybys or McInnes's big solar sail to move the Earth closer to the Sun.
Korycansky, D. G., Laughlin, G., & Adams, F. C. (2001). Astronomical engineering: a strategy for modifying planetary orbits. Astrophysics and Space Science, 275(4), 349-366.
McInnes, C. R. (2002). Astronomical engineering revisited: planetary orbit modification using solar radiation pressure. Astrophysics and Space Science, 282(4), 765-772.
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