Problem Set
Metal Math: A ‘Pi in the Sky’ Math Challenge
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 10th set, students use pi to calculate the approximate density of the asteroid (16) Psyche and compare that to the density of known terrestrial materials.
Materials
Background
Metal Math
Orbiting the Sun between Mars and Jupiter, the asteroid (16) Psyche is of particular interest to scientists because its surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. By sending a spacecraft to study Psyche up close, scientists hope to learn more about terrestrial planet cores and our solar system’s history. That's where NASA's Psyche comes in. The mission will use specialized tools to study Psyche's composition from orbit. Determining how much metal exists on the asteroid is one of the key objectives of the mission. In Metal Math, students will do their own investigation of the asteroid's makeup, using pi to calculate the approximate density of Psyche and compare that to the density of known terrestrial materials.
Procedures
Asteroid (16) Psyche is of particular interest to scientists because groundbased observations indicate that the surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. If Psyche consists of a large amount of metal, it might resemble a planetary core from which we could learn about terrestrial planet core formation. Determining how much metal exists on the asteroid is one of the goals of NASA’s Psyche mission, which will use specialized tools to study the asteroid's composition from orbit.
Psyche has a roughly triaxial ellipsoid shape with axes of about 290 km, 245 km, and 170 km. Its mass, as estimated from its gravitational effects on nearby bodies such as Mars, is about 2.7 x 10^{19} kg. Use the formula for volume, V = 4/3 πabc, where a, b, and c are the lengths of the semiaxes, to compute Psyche's approximate density.
Based on the average density of terrestrial materials (listed below), does Psyche's density support the observations indicating the presence of metal?
Average density of terrestrial materials
 ice: 917 kg/m^{3}
 water: 997 kg/m^{3}
 rock: 1,600  3,500 kg/m^{3}
 metal: 534  22,590 kg/m^{3}
Assessment
Extensions
Participate
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Downloads
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