PhysicsLAB: Test Scenario: Exoplanet Data

Early on February 23rd, 2017, astronomers announced that they have identified a nearby solar system [39 light-years away] hosting seven Earth-sized planets. Most intriguing: three planets that orbit its central dwarf star — known as TRAPPIST-1 which is not much larger than Jupiter — may even be within a habitable zone which means they fall within a region that could support life as we know it.

Refer to the following information for the next six questions.

A student is given the following set of orbital data for the exoplanets in the Trappist-1 system.223.8

(a) Write an algebraic expression for the gravitational force between TRAPPIST-1, of mass, M_T, and any one of its orbiting planets, of mass, m_sat.

(b) Use your expression from (a) and the assumption of circular orbits to derive an equation for the orbital radius R_AV of a planet as a function of its orbital period T.

(c) Kepler's 3rd Law, the Law of Periods, states that all satellites orbiting the same central body have a common R³/T² value called Kepler's Constant. The AU values in the following chart have been converted into meters and then cubed; while the period values have been converted into seconds and then squared.

Calculate Kepler's constant for each planet.

What is the average Kepler Constant for this system of planets?

Based on your average value for Kepler's constant, what should be the mass of Trappist-1?

If the reported value for MT is 1.796 x 10²⁹ kg, what is your percent error

Why did the data have to be changed into SI units in order to calculate the mass?