Assume that a planet just like Earth orbits the bright star named Sirius. If this Earth-like planet orbits with a semimajor axis of 1 AU and an orbital period of 7 months, what is the mass of Sirius? A) 3 M _$\odot$ B) 12 M _$\odot$ C) 8 M _$\odot$ D) 5 M _$\odot$ E) 17 M _$\odot$

Question

Quizplus · Accepted Answer

We can use Kepler's third law of planetary motion to solve for the mass of Sirius: $\frac{T^2}{a^3}=\frac{4\pi^2}{GM}$ where T is the orbital period, a is the semimajor axis, G is the gravitational constant, and M is the mass of Sirius. Substituting the given values, we have: $\frac{(7/12)^2}{(1 ext{ AU})^3}=\frac{4\pi^2}{G M}$ Solving for M, we get: $M = \frac{4\pi^2 a^3}{G T^2} = \frac{4\pi^2 (1 ext{ AU})^3}{G (7/12)^2}$ Using the value for the gravitational constant, G = $6.674 imes 10^{-11}$ N(m/kg)$^2$, we get: $M \approx 3.93 imes 10^{30}$ kg, which is equivalent to about 3 solar masses: 3 M$_\odot$. The closest choice to this answer is (A), which is 3 MF2S2S2W1_F1F1F10.

Assume That a Planet Just Like Earth Orbits the Bright ⊙\odot⊙

Assume That a Planet Just Like Earth Orbits the Bright $\odot$