Problem Set

Pi in the Sky 11

A collage of cartoon spacecraft, asteroids, Earth and rover on a lunar surface surround text that reads π in the sky 11 A Prime Year for Pi.

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

Pi in the Sky 11 poster – Download PDF (for best results, download and print from Adobe Reader)

Pi in the Sky 11 handouts – Download PDF (for best results, download and print from Adobe Reader)

Pi in the Sky 11 answer key – Download PDF

Pi in the Sky 11 answer handouts – Download PDF (also available as a text-only doc)

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 high-definition video encoded in a near-infrared 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.

This animation shows how DSOC's laser signals are sent between the Psyche spacecraft and ground stations on Earth - first as a pointing reference to ensure accurate aiming of the narrow laser signal and then as a data transmission to the receiving station. Credit: NASA/JPL-Caltech/ASU| Watch on YouTube

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.

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. Credit: NASA/Johns Hopkins APL | › Enlarge image

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 ice-covered 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.

An illustration shows the NISAR spacecraft orbiting above Earth.

The NISAR satellite, shown in this artist’s concept, will use advanced radar imaging to provide an unprecedented view of changes to Earth’s land- and ice-covered surfaces. Credit: NASA/JPL-Caltech. | › Full image and caption

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 suitcase-size 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 multi-robot exploration missions. In Moon Mappers, students explore the Moon with pi by determining how far a CADRE rover drives on the Moon’s surface.

A small rover is attached to an elevated rack while two engineers hold their hands out toward the underside of the rover.

Engineers test the system that will lower three small rovers onto the lunar surface as part of the CADRE project. Credit: NASA/JPL-Caltech | › Full image and caption

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

Receiver Riddle

In December 2023, NASA transmitted the first ultra-high-definition 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 15-second 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

An orange cat chasing the squiggly line of a laser pointer is shown in an inset at the end of a laser signal. The signal is travelling from the Psyche spacecraft and pointed ahead of Earth along its orbit.

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

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

Orbit Observation

NISAR is an Earth-orbiting satellite mission designed to measure centimeter-scale movements and other changes of Earth's land- and ice-covered 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

A radar beam extends downward from a spacecraft that is passing over Earth's horizon and coming toward the observer. The beam touches Earth's surface perpendicular to the path of the spacecraft, forming a swath. An inset shows Earth with multiple swaths that wrap around the planet like a ribbon.

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

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 suitcase-size 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