The moon's orbit is increasing 3.8 cm per year. How much does that warm the earth?
Prefer an answer in watts, or watts per unit area. Thx.
This is conservation of angular momentum and energy.
Orbital semi-major Axis = a = 384,399 km
da/dt = 3.5 cm/yr
m = 7.347 7 x10^22 kg
Period = P.m = 27.321 582 days
M.e = 5.97 x10^24 kg
r.e = 6370 km
Rotation earth = P.e = 23 hr, 56 min.
Angular momentum moon
L.m = a *mv
v.m^2/a = G M/a^2
v.m = √ (GM/a)
L.m = m √ (a GM)
dL.m/dt = d L.m/da * da/dt
dL.m/dt = m 1/2 √ ((GM/a) da/dt
The increase in the moon's angular momentum will be offset by a decrease in the Earth's angular momentum.
dL.e/dt + dL.m/dt = 0 ..... Conservation of angular momentum
dLe/dt = - dL.m/dt
L.e = I ω
dL.e/dt = dL.e/dω *dω/dt
dL.e/dt = I *d&omega/dt
dLe/dt = -dL.m/dt
I dω/dt = - m 1/2 √ ((GM/a) da/dt
dω/dt = - m (1/2da/dt) √(GM/a) /I
Earth's rotational energy is
E = 1/2 I ω^2
dE/dt = I ω * dω/dt
dE/dt =( I ω) (- m (1/2da/dt) √(GM/a) /I )
dE/dt = -1/2 mω √(GM/a) da/dt
dE/dt = - 3.015x10^12 watts
This loss of energy will go into the increase in the moon's orbit and the remainder will be heating of the earth. Energy to increase orbit of the moon, per Bekki, below, is
Power = dE/dt = (dE/da)(da/dt)
= (GMm / 2a^2)(da/dt)
= 1.099x10^11 watts
Ergo: the amount heating the earth is roughly 2.9 x10^12 watts
I am at a loss regarding the difference between Wiki, per cite of Captain M, and my answer
Excellent answers by both Captain M and M. Bekki. I'd like to give best answer to both of them. But, I think I'll hold off a day.
I used 3.5 cm/yr instead of 3.8 cm per year (3.8 is the correct value)
This increases the power as a result of the earth slowing down to
3.3x^12 Watts, the amount going to increase the moon's orbit to 1.2x10^11 Watts, and the amount heating the earth to 3.1x10^12 Watts
I looked at the abstract to the reference cited in Wiki. http://www.sciencedirect.com/science?_ob... It says the 3.75x10^12 watts is from tidal acceleration from astronomical sources. Id. This also includes the sun. The abstract also states that only 2.5x10^12 is from the moon. Id.
I hadn't considered how the sun would also cause a gradual increase in the moon's orbit. So my calculation of 3.3 will be just a tad high for the amount caused solely by the moon. (I do get the 30:1 ratio cited by Wiki for the energy going into increasing the moon's orbit and the energy going into tidal heating on earth, which is good confirmation). ;-)