A star is seen to have two transiting planets. Planet 1 transits every 3 months, and Planet 2 transits every 15 months. What can we infer about their orbits? (Hint: Review the use of Newton's version of Kepler's Third law from Ch. 4)
A)Planet 2's semimajor axis is about 3 times smaller than Planet 1's.
B)Planet 2's semimajor axis is about 3 times larger than Planet 1's.
C)Planet 2's semimajor axis is about 5 times larger than Planet 1's.
D)Planet 2's semimajor axis is about 5 times smaller than Planet 1's.

Question

Quizplus · Accepted Answer

According to Kepler's Third Law, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This means that if Planet 2 takes 5 times longer to orbit the star than Planet 1 (15 months vs. 3 months), then the semi-major axis of Planet 2's orbit is larger. The relationship between the periods (T) and the semi-major axes (a) can be expressed as $T_2^2/T_1^2 = a_2^3/a_1^3$. Solving for the ratio of the semi-major axes gives $a_2/a_1 = (T_2/T_1)^{2/3}$, and substituting $T_2/T_1 = 5$ yields $a_2/a_1 = 5^{2/3}$, which is approximately equal to 2.92, or about 3 times larger.

A Star Is Seen to Have Two Transiting Planets