Problem Set

Daring Deflection: A ‘Pi in the Sky’ Math Challenge

A spacecraft flies toward a small asteroid orbiting a larger asteroid along partially overlapping paths with different eccentricities.

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

Pi in the Sky 11: Daring Deflection worksheet – download PDF (for best results, download and print from Adobe Reader)

Pi in the Sky 11: Daring Deflection answer key – download PDF (also available as a text-only doc)

Background

An animation shows the surface of an asteroid getting closer and closer. In the last several frames, the animation slows and details of the rocky surface come into view.

This image shows the final minutes of images leading up to the DART spacecraft's intentional collision with asteroid Dimorphos. Credits: NASA/Johns Hopkins APL | › Enlarge image

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.

Teachable Moments

A Prime Year for NASA's Pi Day Challenge

Learn more about about pi, the history of Pi Day, and the science behind the 2024 NASA Pi Day Challenge.

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 semi-major axis (a) of the new orbit, given that T = 2π√(a³/GM).

T = orbital period in seconds
a = semi-major axis in meters
G = gravitational constant (6.674×10⁻¹¹ N⋅m²/kg²)
M = total mass of the binary system

Use the semi-major axis and eccentricity to calculate Dimorphos’ farthest distance from Didymos (apoapsis = a(1+e)) and closest distance to Didymos (periapsis = a(1-e)). How do these differ from the circular orbit?

› Learn more about the DART mission