In the 19th century, measurements of the precession of the orbits of the planets in the solarsystem were performed, and preformed to a new standard of precision that allowedpredictions to be made from deviations from gravitational theory. Newtonian gravitationwas sufficient to predict the precession in most of the planets, but Mercury’s precession wasanomalous: the long axis of its elliptical orbit changes direction by 43”/century (arcsecondsper tropical century) faster than the expected speed. One theory that was created to explainthis effect was that there was an “anti-Earth” called Vulcan that orbited the sun exactlyopposite the Earth. 1 If this theory had been correct, how much different would the orbit of the Earth be fromwhat it is today? Express your answer in terms of the ratio of the difference of the predictedperiod of the Earth with and without Vulcan to the period of the Earth without thehypothetical planet. Some assumptions will be necessary to get a nice answer:(i) Do not assume you know the period of the Earth T, that’s what you’re trying tomeasure,(ii) Assume that the Sun does not move, and(iii) Use the approximation (1+x)^(1/2) ≈ 1+(x/2) or 1/((1+x)^1/2) ≈ 1−(x/2) to finish your execution. [Answer: (1/8)*(M⊕/M⊙)] I just need to know how to get to the answer
In the 19th century, measurements of the precession of the orbits of the planets in the solar
system were performed, and preformed to a new standard of precision that allowed
predictions to be made from deviations from gravitational theory. Newtonian gravitation
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 to the period of the Earth without the
hypothetical planet. Some assumptions will be necessary to get a nice answer:
(i) Do not assume you know the period of the Earth T, that’s what you’re trying to
measure,
(ii) Assume that the Sun does not move, and
(iii) Use the approximation (1+x)^(1/2) ≈ 1+(x/2) or
1/((1+x)^1/2) ≈ 1−(x/2) to finish your execution.
[Answer: (1/8)*(M⊕/M⊙)]
I just need to know how to get to the answer
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