Question

Scientists once hypothesized the existence of a planet called Vulcan to explain Mercury’s precession. Vulcan is supposed to be between Mercury and the Sun with a solar distance equal to 2/3 of that of Mercury. What would be its supposed period?

Answer 1

According to Kepler's third law:

$$\frac{T_M^2}{T_V^2}=\frac{a_M^3}{a_V^3},\\\space\\ T_V=\sqrt{T^2_M\frac{a_V^3}{a_M^3}}=T_M\sqrt{\bigg(\frac{2a_M}{3a_M}\bigg)^3},\\\space\\ T_V=87.97\sqrt{(2/3)^3}=58.65\text{ d}.$$