Planet Vulcan. Suppose that a planet were discovered between the sun and Mercury, with a circular orbit of radius equal to of the average orbit radius of Mercury. What would be the orbital period of such a planet? (Such a planet was once postulated, in part to explain the precession of Mercury’s orbit. It was even given the name Vulcan, although we now have no evidence that it actually exists. Mercury’s precession has been explained by general relativity.)

Solution 24E Step 1 Suppose, the planet mercury and the unknown planet are moving in an exact circular orbit around the sun. So, in the circular motion of the planet, we can write, the gravitational force = centripetal force That is, F =g c F = GMm/R 2 g Consider, this is the gravitational force acting on mercury by the sun. 2 The centripetal force would be, F = m R c Where, R - radius of the orbit of mercury - Angular velocity of the mercury m - Mass of the mercury M - Mass of the sun 2 2 Therefore, we can equate, GMm/R = m R Or, GM/R = 3 2 We know that, = 2/T Where, T - time period of the oscillation Then, GM/R = 4 /T 2 2 2 3 2 Rearranging, T /R = 4 /GM