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


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.



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.


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

A spacecraft flies toward a small asteroid orbiting a larger asteroid along partially overlapping paths with different eccentricities labeled original orbit and new orbit. Extending down from the new orbit, past the center of the large central asteroid, is a line labeled semi-major axis. An inset wrapping around the small asteroid and the portion of the orbits that overlaps is labeled DART Impact – September 2022.

Image credit: NASA/JPL-Caltech | + Expand image


Illustrated answer key for the Daring Deflection problem.

Image credit: NASA/JPL-Caltech | + Expand image

Download text-only answer key (Google Docs)



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