Problem Set
Pi in the Sky 7
Overview
In the seventh installment of the "Pi in the Sky" illustrated problem set, students use the mathematical constant pi to solve real-world science and engineering problems. Students will use pi to compare the sizes of Mars landing areas, calculate the length of a year for one of the most distant objects in the solar system, measure the depth of the ocean from an airplane, and determine the diameter of a distant debris disk.Materials
Background
![Overhead view of Mars with a comparison of the smaller landing ellipse made possible by Range Trigger technology](https://mars.nasa.gov/imgs/mars2020/Range-Trigger-Rover-Landing-Site.jpg)
A new Mars landing technique called Range Trigger is making it possible to reduce the size of the ellipse where spacecraft touch down. Image credit: NASA/JPL-Caltech | › Full image and caption
![](https://solarsystem.nasa.gov/system/resources/detail_files/2449_mu69_med.jpg)
This composite image of the Kuiper Belt object Arrokoth was compiled from data obtained by NASA's New Horizons spacecraft as it flew by the object on Jan. 1, 2019. Image credit: NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute/Roman Tkachenko | › Full image and caption
![Diagram of an airplane flying over a section of ocean with an example of the spectral data that CORAL collects](https://www.jpl.nasa.gov/edu/images/activities/coral_diagram.jpg)
Flying onboard a Gulfstream IV aircraft, CORAL records the spectra of light reflected from the ocean below to study the composition and health of Earth's coral reefs. Image credit: NASA | + Expand image
![Rays of bright orange and red shoot out diagonally from a blue circle surrounding the star Beta Pictoris](https://exoplanets.nasa.gov/system/resources/detail_files/301_eso0842a.jpg)
The star Beta Pictoris and its surrounding debris disk are shown in near-infrared light in this composite image. The outer part of the image shows the reflected light on the dust disc. Image credit: ESO/A.-M. Lagrange et al. | › Full image and caption
Mars Maneuver
Long before a Mars rover touches down on the Red Planet, scientists and engineers must determine where to land. Rather than choosing a specific landing spot, NASA selects an area known as a landing ellipse. A Mars rover could land anywhere within this ellipse. Choosing where the landing ellipse is located requires compromising between getting as close as possible to interesting science targets and avoiding hazards like steep slopes and large boulders, which could quickly bring a mission to its end. In the Mars Maneuver problem, students use pi to see how new technologies have reduced the size of landing ellipses from one Mars rover mission to the next.
Cold Case
In January 2019, NASA's New Horizons spacecraft sped past Arrokoth, a frigid, primitive object that orbits within the Kuiper Belt, a doughnut-shaped ring of icy bodies beyond the orbit of Neptune. Arrokoth is the most distant Kuiper Belt object to be visited by a spacecraft and only the second object in the region to have been explored up close. To get New Horizons to Arrokoth, mission navigators needed to know the orbital properties of the object, such as its speed, distance from the Sun, and the tilt and shape of its orbit. This information is also important for scientists studying the object. In the Cold Case problem, students can use pi to determine how long it takes the distant object to make one trip around the Sun.
Coral Calculus
Coral reefs provide food and shelter to many ocean species and protect coastal communities against extreme weather events. Ocean warming, invasive species, pollutants, and acidification caused by climate change can harm the tiny living coral organisms responsible for building coral reefs. To better understand the health of Earth's coral reefs, NASA's COral Reef Airborne Laboratory, or CORAL, mission maps them from the air using spectroscopy, studying how light interacts with the reefs. To make accurate maps, CORAL must be able to differentiate among coral, algae and sand on the ocean floor from an airplane. And to do that, it needs to calculate the depth of the ocean at every point it maps by measuring how much sunlight passes through the ocean and is reflected upward from the ocean floor. In Coral Calculus, students use pi to measure the water depth of an area mapped by the CORAL mission and help scientists better understand the status of Earth's coral reefs.
Planet Pinpointer
Our galaxy contains billions of stars, many of which are likely home to exoplanets – planets outside our solar system. So how do scientists decide where to look for these worlds? Using data gathered by NASA's Spitzer Space Telescope, researchers found that they're more likely to find giant exoplanets around young stars surrounded by debris disks, which are made up of material similar to what's found in the asteroid belt and Kuiper Belt in our solar system. Sure enough, after discovering a debris disk around the star Beta Pictoris, researchers later confirmed that it is home to at least two giant exoplanets. Learning more about Beta Pictoris' debris disk could give scientists insight into the formation of these giant worlds. In Planet Pinpointer, put yourself in the role of a NASA scientist to learn more about Beta Pictoris' debris disk, using pi to calculate the distance across it.
Procedures
Mars Maneuver
When we plan where to land a spacecraft on Mars, we don’t choose a specific spot, but a larger area called a landing ellipse. It's like choosing a parking lot rather than a parking spot. To choose a landing ellipse, we have to compromise between getting as close as possible to interesting science targets and avoiding hazards. As we've created new technology to help direct spacecraft, landing ellipses have gotten smaller and smaller. That means that we're able to land in places we couldn't before and get closer to the stuff we want to study.
In 2012, the Curiosity rover used its sky crane landing system to touch down in a 20 km by 7 km ellipse. When the Perseverance rover lands on Feb. 18, 2021, it will use the same system along with a new technique called Range Trigger that will allow the spacecraft to land in the smallest ellipse yet, measuring just 13 km by 7 km. What percentage of Curiosity's landing ellipse is Perseverance's landing ellipse?
› Learn more about the Mars rover Perseverance
![Illustration (split-screen) of Perseverance landing on Mars in a smaller landing ellipse compared with Curiosity's.](https://www.jpl.nasa.gov/edu/images/activities/piday2020_mars.png)
Image credit: NASA/JPL-Caltech | + Expand image
Cold Case
In January 2019, NASA's New Horizons spacecraft flew within 3,538 km of the most distant and primitive object explored up-close by a spacecraft. The object was originally known as 2014 MU69, but it was later renamed Arrokoth. It looks like a partially flattened, reddish snowman and is made up of two objects that merged into one. Found 6.6 billion km from Earth, Arrokoth is a small “Cold Classical” Kuiper Belt object, meaning it orbits the Sun in a nearly circular path and has a low orbital inclination. Cold Classical objects make up about one-third of the Kuiper Belt.
One reason scientists are interested in studying Arrokoth and other Kuiper Belt objects is that they are thought to be well preserved, frozen samples of what the outer solar system was like at its birth, more than 4.5 billion years ago. Learn a bit more about Arrokoth by calculating how long it takes the object to make one trip around the Sun.
› Learn more about the Kuiper Belt object Arrokoth
![Illustration of Arrokoth orbiting the Sun with New Horizons flying by.](https://www.jpl.nasa.gov/edu/images/activities/piday2020_mu69.png)
Image credit: NASA/JPL-Caltech | + Expand image
Coral Calculus
Flying aboard an aircraft, NASA’s CORAL mission uses spectroscopy to study the health of coral reefs and the threats they face. To differentiate among coral, algae and sand on the ocean floor, CORAL computes the depth of every point it maps. The water’s depth can be determined using the “absorption coefficient,” indicating how much light is absorbed through a given depth of water.
Imagine CORAL collects a light measurement reflected by white sand covered by an unknown depth of water that is 76% in the blue and 4.5% in the red. Using the formulas below, calculate the water’s depth. Note that sunlight passes through the water twice: when traveling from the Sun to the ocean floor and when reflecting up to the aircraft.
absorption coefficient, α = (4πk)/λ
k = coefficient of the imaginary number portion of the refractive index
λ = wavelength (meters) of light observed
Beer-Lambert law, T = e(-α•d)
T = observed reflectance, or transmittance
(T), of light through a distance (d) of water
Refractive Indices:
- Water in the blue wavelength (450 nm) = 1.3369 + 1.01E-09i
- Water in the red wavelength (650 nm) = 1.3314 + 1.60E-08i
› Learn more about the CORAL mission
![Illustration of an airplane flying over the ocean with coral shown below the water's surface. A spectral ray expands from the airplane down to the ocean floor](https://www.jpl.nasa.gov/edu/images/activities/piday2020_coral.png)
Image credit: NASA/JPL-Caltech | + Expand image
Planet Pinpointer
Our galaxy contains billions of stars, most of which are likely home to exoplanets – planets outside our solar system. How do scientists decide where to look for these worlds? Researchers looking at data from NASA's Spitzer Space Telescope found that giant exoplanets tend to exist around young stars surrounded by a disk of debris. A prominent debris disk around the star Beta Pictoris, which is 6 x 1014 km away from Earth, lead scientists to discover two exoplanets several times bigger than Jupiter orbiting the star! Learning more about the debris disk could give scientists insight into the formation of these giant worlds.
Given the angle of the disk's apparent size is 169 arcseconds, determine the actual distance across it using the formula for small angle approximation, below. (An arcsecond is 1/3,600 of a degree.)
D = dθ
D = distance across the debris disk (km)
d = distance to Beta Pictoris (km)
θ = angle of apparent size (radians)
![Illustration of two planets orbiting the star Beta Pictoris surrounded by a disk of debris. An inset illustration of the Spitzer space telescope shows a triangular beam representing the spacecraft's view of the disk](https://www.jpl.nasa.gov/edu/images/activities/piday2020_spitzer.png)
Image credit: NASA/JPL-Caltech | + Expand image
![Infographic of all of the Pi in the Sky 7 graphics and problems](https://www.jpl.nasa.gov/edu/images/activities/piday2020_infographic.png)
Image credit: NASA/JPL-Caltech | + Expand image
Assessment
![Illustrated answer key for the Pi in the Sky 7 Math Problem Set](https://www.jpl.nasa.gov/edu/images/activities/piday2020_answers.png)
Image credit: NASA/JPL-Caltech | + Expand image
Extensions
Participate
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Pi Day Challenge Lessons
Here's everything you need to bring the NASA Pi Day Challenge into the classroom.
Grades 4-12
Time Varies
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Slideshow: NASA Pi Day Challenge
The entire NASA Pi Day Challenge collection can be found in one, handy slideshow for students.
Grades 4-12
Time Varies
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Pi Day: What’s Going ’Round
Tell us what you’re up to this Pi Day and share your stories and photos with NASA.
Join the conversation and share your Pi Day Challenge answers with @NASAJPL_Edu on social media using the hashtag #NASAPiDayChallenge
Blogs and Features
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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.
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Slideshow: 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.
Related Lessons for Educators
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Rover Lessons
Explore a collection of standards-aligned STEM lessons all about rovers.
Grades K-12
Time Varies
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Touchdown
Students design and build a shock-absorbing system that will protect two "astronauts" when they land.
Grades 3-8
Time 30 mins - 1 hr
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On Target
Students modify a paper cup so it can zip down a line and drop a marble onto a target.
Grades 6-12
Time 30 mins - 1 hr
-
Solar System Scale Models
Explore a collection of standards-aligned STEM lessons all about the size and scale of our solar system.
Grades 1-12
Time Varies
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Modeling an Asteroid
Lead a discussion about asteroids and their physical properties, then have students mold their own asteroids out of clay.
Grades 3-5
Time 30 mins - 1 hr
-
Math Rocks: A Lesson in Asteroid Dynamics
Students use math to investigate a real-life asteroid impact.
Grades 8-12
Time 30 mins - 1 hr
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Asteroid Ace: A 'Pi in the Sky' Math Challenge
Students use pi to calculate the rotation rate of an asteroid from another solar system in this illustrated math problem.
Grades 11-12
Time < 30 mins
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Climate Change Lessons
Explore a collection of standards-aligned STEM lessons all about Earth's changing climate.
Grades K-12
Time Varies
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Using Light to Study Planets
Students build a spectrometer using basic materials as a model for how NASA uses spectroscopy to determine the nature of elements found on Earth and other planets.
Grades 6-11
Time < 2 hrs
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Solar Sleuth: A 'Pi in the Sky' Math Challenge
In this illustrated math problem, students use pi and data from the Kepler space telescope to find the size of a planet outside our solar system.
Grades 6-9
Time < 30 mins
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Exploring Exoplanets with Kepler
Students use math concepts related to transits to discover real-world data about Mercury, Venus and planets outside our solar system.
Grades 6-12
Time 30 mins - 1 hr
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Habitable Hunt: A 'Pi in the Sky' Math Challenge
In this illustrated math problem, students use the mathematical constant pi to find the "habitable zone" around a distant star and determine which of its planets are in that zone.
Grades 11-12
Time < 30 mins
Related Activities for Students
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Make a Moon or Mars Rover Game
Create a Moon or Mars exploration game using Scratch, a visual programming language. Think like NASA space-mission planners to design your game!
Type Project
Subject Technology
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Make a Cardboard Rover
Build a rubber-band-powered rover that can scramble across a room.
Type Project
Subject Engineering
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Mars in a Minute: How Do You Choose a Landing Site?
So, you want to study Mars with a lander or rover – but where exactly do you send it? Learn how scientists and engineers tackle the question of where to land on Mars in this 60-second video.
Type Video
Subject Engineering
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Mars in a Minute: How Do You Land on Mars?
Getting a spacecraft to Mars is one thing. Getting it safely to the ground is a whole other challenge! This 60-second video from NASA's Jet Propulsion Laboratory explains three ways to land on the surface of the Red Planet.
Type Video
Subject Engineering
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What's That Space Rock?
Find out how to tell the difference between asteroids, comets, meteors, meteorites and other bodies in our solar system.
Type Slideshow
Subject Science
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Mars in a Minute: How Long is a Year on Mars?
How long is does it take Mars to make one trip around the Sun and why is one Earth year shorter? Find out in one minute!
Type Video
Subject Science
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Space Place in a Snap: The Solar System's Formation
Find out how our solar system formed and how it came to be the busy place it is today.
Type Video
Subject Science
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What Is the Kuiper Belt?
Learn about the Kuiper Belt and some of its famous members, Kuiper Belt Objects.
Type Article
Subject Science
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Coral Bleaching Simulator
Adjust water temperature and pollution levels in this simulator to see what happens to a coral reef depending on the conditions you choose!
Type Interactive
Subject Science
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Ocean Worlds
Where might oceans – and living things – exist beyond Earth? Scientists have their eyes on these places in our own solar system.
Type Slideshow
Subject Science
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NASA's Earth Minute: Mission to Earth?
NASA doesn't just explore outer space! It studies Earth, too, with a fleet of spacecraft and scientists far and wide.
Type Video
Subject Science
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NASA's Earth Minute: Earth Has a Fever
Why is Earth getting hotter and what does that mean for us?
Type Video
Subject Science
NOAA Video Series: Coral Comeback
- Article: Giant Exoplanet Hunters: Look for Debris Disks
- Video: The Evolution of a Planet-Forming Disk
- Video: Birth of "Phoenix" Planets?
Multimedia
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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.
- Posters: Exoplanet Travel Bureau