Question

In the 19th century, measurements of the precession of the orbits of the planets in the solar system were measured, and measured accurately. Newtonian theory was sufficient to predict the precession in most of the planets, but Mercury’s precession was anomalous: the long axis of its elliptical orbit changes direction by 43”/century (arcseconds per tropical century) faster than the expected speed. One theory that was created to explain this effect was that there was an “anti-Earth” called Vulcan that orbited the sun exactly opposite the Earth. 1 If this theory had been correct, how much different would the orbit of the Earth be from what it is today? Express your answer in terms of the ratio of the difference of the predicted period of the Earth with and without Vulcan with the period of the Earth without it. Do not assume you know T, and use the approximation sqrt(1+?) ≈ 1 + ?/2 or 1/sqrt(1+?) ≈ 1 − ?/2 to finish. [Answer: 1/8(M_Earth/M_Sun)

Answer 1