Answer to Question #275029 in Astronomy | Astrophysics for Isaiah

Question

.The planet Vulcan was postulated to exist in an attempt to explain the precession of Mercury. How long will Vulcan make one complete revolution around the sun if its distance from the sun is two-thrids that of Mercury?

Answer 1

Explanations & Calculations

• Apply Newton's law of gravitation for both planets.

$\qquad\qquad \begin{aligned} \small \frac{GM_sm_m}{r^2}&=\small m_m.r.\omega^2=m_m.r.\frac{4\pi^2}{T_m^2}\\ \small GM_s&=\small r^3.\frac{4\pi^2}{T^2_m}\cdots(1)\\ \\ \small \frac{GM_sm_v}{(\frac{2r}{3})^2}&=\small m_v.(\frac{2r}{3}).\frac{4\pi^2}{T_v^2}\\ \small GM_s&=\small (\frac{2r}{3})^3.\frac{4\pi^2}{T_v^2}\cdots(2)\\ \\ \small (1)=(2),\\ \small r^3.\frac{4\pi^2}{T_m^2}&=\small (\frac{2}{3})^3.r^3.\frac{4\pi^2}{T^2_v}\\ \small T_v&=\small \Big(\frac{2}{3}\Big)^\frac{3}{2}.T_m \end{aligned}$

• Once the period of Mercury is known, the period of the planet Vulkan can be calculated.