Problem Set
Pi in the Sky 11
Overview
In the 11th installment of the Pi in the Sky illustrated problem set, students use pi to calculate the change in an asteroid's orbit, determine how much data an Earth orbiting satellite will collect, map the surface of the Moon, and measure Earth's movement in orbit around the Sun.
Materials
Background
Receiver Riddle
In December 2023, NASA tested a new way to communicate with distant spacecraft using technology called Deep Space Optical Communications, or DSOC. From 19,000,000 miles (30,199,000 km) away, the Psyche spacecraft beamed a highdefinition video encoded in a nearinfrared laser to Earth. The video, showing a cat named Taters chasing a laser, traveled at the speed of light, where it was received at Caltech’s Palomar Observatory. Because of the great distance the laser had to travel, the team needed to aim the transmission at where Earth would be when the signal arrived. In Receiver Riddle, use pi to determine where along Earth's orbit the team needed to aim the laser so that it could be received at the Observatory at the correct moment.
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.
Orbit Observation
The NISAR mission is an Earth orbiting satellite designed to study our planet's changing ecosystems. It will collect data about Earth's land and icecovered surfaces approximately every 6 days, allowing scientists to study changes at the centimeter scale – an unprecedented level of detail. To achieve this feat, NISAR will collect massive amounts of data. In Orbit Observation, students use pi to calculate how much data the NISAR spacecraft captures during each orbit of Earth.
Moon Mappers
The CADRE project aims to land a team of mini rovers on the Moon in 2025 as a test of new exploration technology. Three suitcasesize rovers, each working mostly autonomously, will communicate with each other and a base station on their lunar lander to simultaneously measure data from different locations. If successful, the project could open the door for future multirobot exploration missions. In Moon Mappers, students explore the Moon with pi by determining how far a CADRE rover drives on the Moon’s surface.
Procedures
Receiver Riddle
In December 2023, NASA transmitted the first ultrahighdefinition video from deep space using new technology known as Deep Space Optical Communications, or DSOC. DSOC uses an infrared laser to transmit data at a much higher rate than current radio transmitters. The 15second video, featuring a cat chasing a laser, was beamed to Earth from the Psyche spacecraft at a rate faster than many terrestrial internet connections.
DSOC’s transmission had to travel 30,199,000 km to reach Earth. Even traveling at the speed of light, that takes a long time! And all that time, Earth was still moving along its orbit. That meant that the team needed to aim the laser transmission at where Earth would be when the signal arrived.
Given this, how many kilometers ahead along Earth’s orbit did the team need to aim the laser?
› Learn more about Deep Space Optical Communications
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?
› Learn more about the DART mission
Orbit Observation
NISAR is an Earthorbiting satellite mission designed to measure centimeterscale movements and other changes of Earth's land and icecovered surfaces twice every 12 days – a scale of coverage and sampling never before achieved.
Using a technique called Synthetic Aperture Radar, NISAR will produce more than 85 terabytes of data products every day (1 TB = 1,000 gigabytes) that will allow scientists to better monitor and mitigate natural disasters and understand the effects of climate change.
NISAR has an imaging swath of 240 kilometers, but the ground track spacing is 231 km to allow overlap between swaths. Given that Earth’s radius is 6,371 km, how many orbits are executed in one day? How much data is produced per orbit on average?
› Learn more about the NISAR mission
Moon Mappers
NASA’s CADRE project is made up of a network of three small rovers. The rovers are designed to work together to create a 3D map of a scientifically intriguing area of the Moon's surface known as Reiner Gamma. Communicating with each other and a base station aboard a lunar lander, the rovers will be largely autonomous, making decisions and acting without the need for constant human intervention.
Each suitcasesize rover has a field of view that is about π/2 radians wide, and its sensors can accurately map as far as 2 meters ahead. Assuming the rovers drive in a “lawnmower” pattern, how far does each rover have to drive to survey its portion of a 20 m x 20 m square of the Moon’s surface?
› Learn more about the CADRE project
Assessment
Extensions
Pi Day Resources

Pi in the Sky Lessons
Here's everything you need to bring the NASA Pi Day Challenge into the classroom.
Grades 412
Time Varies

NASA Pi Day Challenge
The entire NASA Pi Day Challenge collection can be found in one, handy collection for students.
Grades 412
Time Varies

How Many Decimals of Pi Do We Really Need?
While you may have memorized more than 70,000 digits of pi, world record holders, a JPL engineer explains why you really only need a tiny fraction of that for most calculations.

18 Ways NASA Uses Pi
Whether it's sending spacecraft to other planets, driving rovers on Mars, finding out what planets are made of or how deep alien oceans are, pi takes us far at NASA. Find out how pi helps us explore space.

10 Ways to Celebrate Pi Day With NASA on March 14
Find out what makes pi so special, how it’s used to explore space, and how you can join the celebration with resources from NASA.

Infographic: Planet Pi
This poster shows some of the ways NASA scientists and engineers use the mathematical constant pi (3.14) and includes common pi formulas.

Downloads
Can't get enough pi? Download this year's NASA Pi Day Challenge graphics, including mobile phone and desktop backgrounds:
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Related Activities for Students

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Follow these easy instructions to draw and decorate your own model of the Psyche spacecraft.
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Recursos en español
Facts and Figures
Websites
 NISAR Mission
 CADRE Project
 Deep Space Optical Communications
 Psyche Mission
 DART Mission
 Asteroid Watch
Articles
 How NASA Studies and Tracks Asteroids Near and Far
 NASA Cat Video Explained
 Article for Kids: Asteroid or Meteor: What's the Difference?
 Article for Kids: What Is an Asteroid?
Videos
 The Video NASA’s Laser Communications Experiment Streamed From Deep Space
 NASA's DART Mission Confirms Crashing Spacecraft into Asteroids Can Deflect Them