Problem Set
Daring Deflection: 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 11th set, students use pi to calculate the change in an asteroid's orbit after it was impacted by a spacecraft.
Materials
Background
Daring Deflection
In 2022, NASA crashed a spacecraft into the asteroid Dimorphos in an attempt to alter its orbit. The mission, known as the Double Asteroid Redirection Test, or DART, took place at an asteroid that posed no threat to our planet. Rather, it was an ideal target for NASA to test an important element of its planetary defense plan. DART was designed as a kinetic impactor, meaning it transferred its momentum and kinetic energy to Dimorphos upon impact, altering the asteroid's orbit. In Daring Deflection, use pi to determine the shape of Dimorphos’ orbit after DART crashed into it.
Procedures
Daring Deflection
The asteroid Dimorphos has a mass of about 4.3 billion kg and orbits the larger Didymos asteroid, which has a mass of 560 billion kg. In 2022, the DART spacecraft impacted Dimorphos to see if it was possible to change its orbit.
Before the impact, Dimorphos orbited Didymos every 11 hours and 55 minutes at a distance of 1.16 km in a nearly circular orbit with an eccentricity (e) of 0. After impact, Dimorphos orbited Didymos every 11 hours and 23 minutes with an eccentricity of 0.02. Use Kepler’s third law to calculate the semimajor axis (a) of the new orbit, given that T = 2π√(a^{3}/GM).
T = orbital period in seconds
a = semimajor axis in meters
G = gravitational constant (6.674×10^{−11} N⋅m^{2}/kg^{2})
M = total mass of the binary system
Use the semimajor axis and eccentricity to calculate Dimorphos’ farthest distance from Didymos (apoapsis = a(1+e)) and closest distance to Didymos (periapsis = a(1e)). How do these differ from the circular orbit?
Assessment
Extensions
Participate
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Downloads
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