Problem Set

Pi in the Sky 10

Overview

In the 10th installment of the Pi in the Sky illustrated problem set, students use pi to calculate the size of a rock sample collected by the Perseverance Mars rover, compare the sensitivity of two space telescopes, determine the composition of an asteroid, and find out how much of the Sun will be covered during a solar eclipse.

Materials

Pi in the Sky 10 poster – Download PDF (for best results, download and print from Adobe Reader)

Pi in the Sky 10 handouts – Download PDF (for best results, download and print from Adobe Reader)

Pi in the Sky 10 answer key – Download PDF

Pi in the Sky 10 answer handouts – Download PDF (also available as a text-only doc)

"Pi in the Sky" Student Challenge (mobile, tablet and screen-reader friendly)

Background

This illustration shows a concept for multiple robots that would team up to ferry to Earth samples of rocks and soil being collected from the Martian surface by NASA's Mars Perseverance rover. Image credit: NASA/JPL-Caltech | › Full image and caption

Image from animation comparing the relative sizes of James Webb's primary mirror to Hubble's primary mirror. Credit: NASA/Goddard Space Flight Center . | › Full animation

This illustration depicts the metal-rich asteroid Psyche, which is located in the main asteroid belt between Mars and Jupiter. Credits: NASA/JPL-Caltech/ASU | + Full image and caption

This image sequence shows an annular solar eclipse from May 2012. The bottom right frame illustrates the distinctive ring, or "annulus," of such eclipses. A similar eclipse will be visible from the South Pacific on May 10, 2013. Credits: Brocken Inaglory, CC BY-SA 3.0, via Wikimedia Commons | + Expand image

Tubular Tally

NASA’s Mars rover, Perseverance, was designed to collect rock samples that will eventually be brought to Earth by a future mission. Sending objects from Mars to Earth is very difficult and something we've never done before. To keep the rock cores pristine on the journey to Earth, the rover hermetically seals them inside a specially designed sample tube. Once the samples are brought to Earth, scientists will be able to study them more closely with equipment that is too large to make the trip to Mars. In Tubular Tally, students use pi to determine the volume of a rock sample collected in a single tube.

Rad Reflection

When NASA launched the Hubble Space Telescope in 1990, scientists hoped that the telescope, with its large mirror and sensitivity to ultraviolet, visible, and near-infrared light, would unlock secrets of the universe from an orbit high above the atmosphere. Indeed, their hope became reality. Hubble’s discoveries, which are made possible in part by its mirror, rewrote astronomy textbooks. In 2022, the next great observatory, the James Webb Space Telescope, began exploring the infrared universe with an even larger mirror from a location beyond the orbit of the Moon. In Rad Reflection, students use pi to gain a new understanding of our ability to peer deep into the cosmos by comparing the area of Hubble’s primary mirror with the one on Webb .

Metal Math

Orbiting the Sun between Mars and Jupiter, the asteroid (16) Psyche is of particular interest to scientists because its surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. By sending a spacecraft to study Psyche up close, scientists hope to learn more about terrestrial planet cores and our solar system’s history. That's where NASA's Psyche comes in. The mission will use specialized tools to study Psyche's composition from orbit. Determining how much metal exists on the asteroid is one of the key objectives of the mission. In Metal Math, students will do their own investigation of the asteroid's makeup, using pi to calculate the approximate density of Psyche and compare that to the density of known terrestrial materials.

Eclipsing Enigma

On Oct. 14, 2023, a solar eclipse will be visible across North and South America, as the Moon passes between Earth and the Sun, blocking the Sun's light from our perspective. Because Earth’s orbit around the Sun and the Moon’s orbit around Earth are not perfect circles, the distances between them change throughout their orbits. Depending on those distances, the Sun's disk area might be fully or only partially blocked during a solar eclipse. In Eclipsing Enigma, students get a sneak peek at what to expect in October by using pi to determine how much of the Sun’s disk will be eclipsed by the Moon and whether to expect a total or annular eclipse.

Teachable Moments: 10 Years of NASA's Pi Day Challenge

Learn more about pi, the history of Pi Day before, and the science behind the 2023 NASA Pi Day Challenge.

Procedures

Tubular Tally

The Perseverance Mars rover is designed to collect rock samples that will eventually be brought to Earth for further study. This would be the first time we've ever brought back samples from Mars! After scientists identify an interesting rock they would like the rover to collect, Perseverance uses a special coring bit to drill out a rock cylinder 13 mm in diameter. As the rover drills, the rock core moves into one of 38 available tubes that will store the rock sample – sealed until it is opened one day in a lab on Earth.

If the coring bit collects a rock cylinder 60 mm in length, what is the volume of the rock in the sample tube?

› Learn more about the Mars Perseverance rover

The center of the image shows an illustration of the Perseverance Mars rover with its robotic arm stretched out to the right, touching a rock on the ground. An inset in the lower right shows a cutaway of a coring bit filled with a rock sample measuring 13 mm wide and 60 mm tall. In the lower left a sealed sample tube is shown on the ground next to the rover. A thought bubble showing hands holding a sample tube is coming from the top of the rover's mast. The text of the Tubular Tally problem is shown on the left.

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

Rad Reflection

The James Webb Space Telescope was designed to look back at some of the earliest galaxies in the universe. To capture light from these distant and faint objects, the telescope must be very sensitive. Webb uses 18 hexagonal mirrors that combine to form a massive primary mirror with a surface area of 26.4 m². This large mirror allows the telescope to collect incredibly faint infrared light and reflect it onto four onboard science instruments, like the Mid-Infrared Instrument, or MIRI. This science instrument can reveal stars hidden within gas and dust clouds and tell scientists about the materials that make up distant galaxies.

Launched in 1990, the Hubble Space Telescope changed our understanding of the universe when it began operations using a primary mirror that had a diameter of just 2.4 meters.

How much bigger is the area of Webb’s primary mirror than Hubble’s?

› Learn more about the Webb Telescope

In the upper left of the image is an illustration of the James Webb Space Telescope. In the lower left is an illustration of the Hubble Space Telescope. To the right of each telescope is an arrow pointing to a face-on view of its primary mirror. Next to the Webb telescope is a primary mirror composed of 18 hexagonal gold-plated mirrors arranged in a roughly circular shape. Above the mirror is text stating the surface area is 26.4 square meters. Next to the Hubble telescope is a round primary mirror with text stating the diameter is 2.4 meters. Both telescopes are in front of an illustrated star field. On the right hand side of the image is a view of space containing stars, spiral galaxies, and elliptical galaxies. Some of the galaxies are warped as a result of gravitational lensing. The text of the Rad Reflection problem is shown on the right.

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

NASA's Universe of Learning materials are based upon work supported by NASA under award number NNX16AC65A to the Space Telescope Science Institute, working in partnership with Caltech/IPAC, Center for Astrophysics | Harvard & Smithsonian, and the Jet Propulsion Laboratory.

Metal Math

Asteroid (16) Psyche is of particular interest to scientists because ground-based observations indicate that the surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. If Psyche consists of a large amount of metal, it might resemble a planetary core from which we could learn about terrestrial planet core formation. Determining how much metal exists on the asteroid is one of the goals of NASA’s Psyche mission, which will use specialized tools to study the asteroid's composition from orbit.

Psyche has a roughly triaxial ellipsoid shape with axes of about 290 km, 245 km, and 170 km. Its mass, as estimated from its gravitational effects on nearby bodies such as Mars, is about 2.7 x 10¹⁹ kg. Use the formula for volume, V = 4/3 πabc, where a, b, and c are the lengths of the semi-axes, to compute Psyche's approximate density.

Based on the average density of terrestrial materials (listed below), does Psyche's density support the observations indicating the presence of metal?

Average density of terrestrial materials

ice: 917 kg/m³
water: 997 kg/m³
rock: 1,600 - 3,500 kg/m³
metal: 534 - 22,590 kg/m³

› Learn more about the Psyche mission

In the lower right of the image is an illustration of a telescope inside an observation dome. Next to the telescope is a computer screen displaying text that reads 'analyzing asteroid makeup...comparing density to...ice: 917 kg/m^3, water: 997 kg/m^3, rock: 1600-3500 kg/m^3, metal: 534-22,590 kg/m^3. Result.' In the center of the image is an inset of an illustrated asteroid (16) Psyche. The asteroid has the upper right quarter cut out to show internal semi-major axis lines. On the right of the image is the Psyche spacecraft in front of a field of stars. The text of the Metal Math problem is shown on the left.

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

Eclipsing Enigma

A solar eclipse occurs when the Moon passes between Earth and the Sun, fully or partially blocking the Sun's light from our perspective. Because Earth’s orbit around the Sun and the Moon’s orbit around Earth are not perfect circles, the distances between them change throughout their orbits. During a total eclipse, the distances are such that the Moon covers all of the Sun's disk area. When the Moon is farther from Earth during an eclipse, it leaves a glowing ring of sunlight shining around the Moon, resulting in an annular eclipse.

On Oct. 14, 2023, a solar eclipse will be visible across North and South America. The Sun, with a radius of 695,700 km, will be 148,523,036 km from Earth. The Moon, with a radius of 1,737 km, will be 388,901 km from Earth.

What percentage of the Sun’s disk area will be obscured by the Moon? Will the eclipse be an annular eclipse or total eclipse?

› Learn more about the 2023 eclipse