# Braking the earth

Classical Mechanics Level 5

The earth goes around the sun, but this motion is not actually a constant. Bodies that orbit each other emit gravitational radiation. This is radiation of literal waves of gravity, just as a lightbulb emits waves of light. The total emitted power of the radiation for two bodies orbiting around their common center of mass in circular orbits is

$$P=\frac {32G^4m_1^2m_2^2(m_1+m_2)} {5c^5r^5}$$

where G is Newton's constant, $$m_1,m_2$$ are the masses of the bodies, c is the speed of light, and r is the distance between the two bodies.

Given that the earth-sun system is radiating gravitational waves, what's the magnitude of the current tangential acceleration of the earth in $$m/s^2$$?

Details and assumptions

• Assume the earth is in a circular orbit about the sun and that the sun remains perfectly stationary. The orbit will decay over time, but the rate of decay is so slow that using the circular approximation is valid for this problem.
• The mass of the earth is $$6 \times 10^{24}$$ kg and the mass of the sun is $$2 \times 10^{30}$$ kg.
• The distance from the earth to the sun is $$1.5 \times 10^{11}$$ m.
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