Problem Set

Solar Sleuth: A 'Pi in the Sky' Math Challenge

Solar Sleuth, Pi in the Sky Math Problem – Illustration of Kepler-186f orbiting its star

Overview

The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the set, students use the mathematical constant pi and data from the Kepler space telescope to find the size of a planet outside our solar system.

Materials

Pi in the Sky 5: Solar Sleuth worksheet – Download PDF

Pi in the Sky 5 answer key – Download PDF

Background

The centuries-old quest for other worlds like our Earth has been rejuvenated by the intense excitement and popular interest surrounding the discovery of hundreds of planets orbiting other stars. There is now clear evidence for substantial numbers of three types of exoplanets; gas giants, hot-super-Earths in short period orbits, and ice giants. The challenge now is to find terrestrial planets (i.e., those one half to twice the size of the Earth), especially those in the habitable zone of their stars where liquid water might exist on the surface of the planet.

When a planet passes in front of a star as viewed from Earth, the event is called a “transit”. On Earth, we can observe an occasional Venus or Mercury transit. These events are seen as a small black dot creeping across the Sun—Venus or Mercury blocks sunlight as the planet moves between the Sun and us. The Kepler Space Telescope finds planets by looking for tiny dips in the brightness of a star when a planet crosses in front of it—we say the planet transits the star.

Once detected, the planet's orbital size can be calculated from the period (how long it takes the planet to orbit once around the star) and the mass of the star using Kepler's Third Law of planetary motion. The size of the planet is found from the depth of the transit (how much the brightness of the star drops) and the size of the star. From the orbital size and the temperature of the star, the planet's characteristic temperature can be calculated. From this the question of whether or not the planet is habitable (not necessarily inhabited) can be answered.

Procedures

Exoplanets are worlds that orbit other stars. Using the Kepler Space Telescope, scientists can study distant stars and search for the exoplanets around them. When Kepler measures repeated dips in the brightness of a star, it can mean that an exoplanet is passing in front of that star from Kepler’s point of view. Scientists can then determine the size of the exoplanet based on how much the star’s light dipped when the planet passed in front of it. This dip in brightness detected by Kepler is expressed as a percentage of the star’s light that is blocked by the planet – with large planets blocking out more of the star’s light and small, Earth-size planets blocking less. This percentage equals the ratio of the area of the planet’s disk to the area of the star’s disk. If the Kepler detects a 0.042% drop in brightness from the star Kepler-186, which has a disk area of 416,000,000,000 km2, what is the radius of the exoplanet, known as Kepler-186f?