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 to find the "habitable zone" around a distant star, TRAPPIST-1, and determine which of its planets are in that zone.
NASA's Spitzer Space Telescope has revealed the first known system of seven Earth-size planets around a single star. Three of these planets are firmly located in the habitable zone, the area around the parent star where a rocky planet is most likely to have liquid water.
The discovery sets a new record for greatest number of habitable-zone planets found around a single star outside our solar system. All of these seven planets could have liquid water – key to life as we know it – under the right atmospheric conditions, but the chances are highest with the three in the habitable zone.
At about 40 light-years (235 trillion miles) from Earth, the system of planets is relatively close to us, in the constellation Aquarius. Because they are located outside of our solar system, these planets are scientifically known as exoplanets.
This exoplanet system is called TRAPPIST-1, named for The Transiting Planets and Planetesimals Small Telescope (TRAPPIST) in Chile. In May 2016, researchers using TRAPPIST announced they had discovered three planets in the system. Assisted by several ground-based telescopes, including the European Southern Observatory's Very Large Telescope, Spitzer confirmed the existence of two of these planets and discovered five additional ones, increasing the number of known planets in the system to seven.
Based on their densities, all of the TRAPPIST-1 planets are likely to be rocky. Further observations will not only help determine whether they are rich in water, but also possibly reveal whether any could have liquid water on their surfaces. The mass of the seventh and farthest exoplanet has not yet been estimated – scientists believe it could be an icy, "snowball-like" world, but further observations are needed.
Scientists can learn a lot about planets beyond our solar system by studying their stars. They can calculate an exoplanet’s orbital period by measuring how often its star dims as the planet passes by. They can even find potentially habitable worlds with a few key details. The star’s temperature and luminosity, which are related to its mass, define its habitable zone, the area where liquid water can exist. And the bond albedo, or percentage of light reflected by the exoplanet, helps estimate its temperature.
Scientists recently discovered seven Earth-like planets orbiting the star TRAPPIST-1. Given TRAPPIST-1’s measurements below, what are the inner and outer radii (r), in AU, of its habitable zone? Use the formula below.
- L* (star luminosity) = 2.0097x1023 watts
- µcb (star gravitational parameter) = 1.06198x1019 m3/s2
- σ (Stefan-Boltzmann constant) = 5.67×10-8 Wm-2K-4
- T (planetary temperature) = 192-295 K
- A (planetary bond albedo) = 0.3
Given the orbital periods (Tp), for TRAPPIST-1’s planets below, which are in the habitable zone? Use Kepler’s third law below to find the semi-major axis of each orbit (ap).
- TRAPPIST-1b = 1.51087081 days
- TRAPPIST-1c = 2.4218233 days
- TRAPPIST-1d = 4.049610 days
- TRAPPIST-1e = 6.099615 days
- TRAPPIST-1f = 9.206690 days
- TRAPPIST-1g = 12.35294 days
- TRAPPIST-1h = 20 days
ExtensionsPi Day Challenges
- Pi in the Sky
- Pi in the Sky 2
- Pi in the Sky 3
- Pi in the Sky 4
- Pi in the Sky Challenge (slideshow for students)