Cartoon animation of NASA-JPL spacecraft

UPDATE: March 16, 2016 – The answers the the NASA Pi Day Challenge are now available as an illustrated answer key. Download a poster version of the answer key on the "Pi in the Sky 3" activity page.

NASA Pi Day Challenge Answer Key

› Check your answers!


This post was originally published on March 9, 2016

In the News

Pi Day, the informal holiday beloved by math enthusiasts – and even by the math averse – is almost here! March 14 marks the yearly celebration of the mathematical constant (pi), which represents the ratio of a circle’s circumference to its diameter. More than just a number for mathematicians, pi has all sorts of applications in the real world, including on missions developed by NASA’s Jet Propulsion Laboratory. And as a holiday that encourages more than a little creativity – whether it’s making pi-themed pies or reciting from memory as many of the never-ending decimals of pi as possible (the record is 70,030 digits) – it’s a great way to have fun and celebrate the M in STEM.

Why March 14?

Pi is what’s known as an irrational number, meaning its decimal representation never ends and it never repeats. It has been calculated to more than one trillion digits, but 3.14 is often a precise enough approximation, hence the celebration occurring on March 14, or 3/14 (when written in US month/day format). The first known celebration occurred in 1988, and in 2009, the US House of Representatives passed a resolution designating March 14 as Pi Day and encouraging teachers and students to celebrate the day with activities that teach students about pi.

Why It’s Important

Pi Day is lots of fun, but its importance lies in the role that pi plays in the everyday work of scientists and engineers at JPL.

Fred Calef, a geospatial information scientist at JPL, uses pi to make measurements – like perimeter, area and volume – of features on Mars. “I use pi to measure the circularity of features, or how round or compact they are," said Calef. "Craters become more elliptical if the projectile hits the surface at a lower angle, so I use pi to measure how round a crater is to see if it impacted at a low angle.”

"We use pi every day commanding rovers on Mars," said Hallie Gengl, a rover planner for the Mars Exploration Rover Opportunity, "Everything from taking images, turning the wheels, driving around, operating the robotic arm, and even talking to Earth.”

Bryana Henderson, who specializes in planetary ices, uses lasers to explode ice samples and study their composition. "I use pi to calculate the width of my laser beam, which in turn can be used to calculate the amount of energy, or fluence, that hits my ice sample," said Henderson. "A larger fluence equals a bigger explosion in the ice, so this is a very important parameter for us."

The Pi Day Challenge

JPL has released the third installment of its popular Pi Day challenge, which gives students and the public a chance to put their pi skills to the test to solve some of the same problems NASA scientists and engineers do. The set of four illustrated math problems are compiled into a graphic (as well as classroom handouts) designed for students in grade 4 through high school – but fun for all!

Pi in the Sky 3 activity

› Check out this year's Pi Day challenge!

This year’s problem set shows how pi can be used to map the surface of Saturn’s hazy moon Titan, track the Mars Reconnaissance Orbiter as it explores the Red Planet, keep Earth’s satellites powered as Mercury transits the sun, and put the Juno spacecraft into orbit around Jupiter.

“For Pi Day, we like to give students and the public a glimpse into how math is used at JPL through questions that feature current events involving our space missions,” said Ota Lutz, an education specialist at JPL who helped create the problem set. “For instance, to put the Juno spacecraft into orbit around Jupiter on July 4, engineers will have to slow the spacecraft just the right amount. In the Pi Day challenge, students use pi to calculate that change in velocity.”

In the challenge, students will also use pi to calculate how much sunlight is blocked by our solar system’s innermost planet as it passes between Earth and the sun. This year, Pi Day comes just a few months before the May 9 transit of Mercury, making this a timely problem.

On March 16, the answers to all four problems and the steps needed to find those answers will be released in a companion infographic on the Pi Day challenge activity page.

In addition to the Pi Day challenge, JPL is inviting the public to share their Pi Day pictures and stories online. On March 14, JPL will join in on the fun with Pi Day photos and stories from the lab.

› Share Your Pi Day photos and stories

To see a compilation of all 12 Pi Day challenge questions optimized for mobile devices and screen readers, visit: http://www.jpl.nasa.gov/edu/nasapidaychallenge

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TAGS: pi, pi day, math, mathematics

  • Lyle Tavernier
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Artist concept of NASA's Juno spacecraft

In the News

NASA’s Juno mission, the first solar-powered mission to Jupiter, has become the farthest solar-powered spacecraft ever! Juno, and its eight science instruments designed to study the interior of Jupiter, has passed the mark previously held by the European Space Agency’s Rosetta mission and reached a distance of 5.3 astronomical units from the sun (an astronomical unit is equal to the average distance between Earth and the sun – about 149.6 million kilometers). Using only power from the sun, Juno will complete the five-year trip to Jupiter in July 2016 and begin studying the solar system’s most massive world in an attempt to better understand the origins of the planet, and in turn, our solar system.

What Made It Possible

Just as a bright source of light dims as you move away from it, sunlight becomes less intense the farther a spacecraft travels from the sun, limiting the amount of power that can be generated using solar cells. Previous missions that visited Jupiter, like Galileo, Voyager 1 and Voyager 2, couldn’t use solar power and instead used radioisotope thermoelectric generators (RTGs) to supply power.

Advances in solar panel efficiency along with improvements in the way spacecraft and their instruments use power have recently made solar power a viable option for spacecraft heading as far as Jupiter – though going beyond will require further technological advances.

Engineers designed Juno with three massive solar panels, each nearly 30 feet long. Combined, they provide Juno with 49.7 m2 of active solar cells. Once it reaches Jupiter, Juno will generate more than 400 watts of power, which may not sound like a lot, but it’s an impressive feat at so great a distance. For comparison, Juno’s solar panels can generate about 14 kilowatts near Earth.

NASA's Juno spacecraft being prepped for launch
Technicians stow for launch a solar array on NASA's Juno spacecraft. Each of Juno's three solar arrays is 9 feet (2.7 meters wide), by 29 feet (8.9 meters long). Image credit: NASA/JPL-Caltech/KSC

Teach It

Juno's record-setting achievement translates into a powerful lesson in exponents.

> Get the problem set!

Middle school students and other students working with exponents will find challenging, real-world applications related to the work being done here at NASA while addressing four Common Core Math standards:

  • Grade 6: Expressions and Equations A.1 - "Write and evaluate numerical expressions involving whole-number exponents."
  • Grade 6: Expressions and Equations A.2 - "Write, read, and evaluate expressions in which letters stand for numbers."
  • Grade 6: Expressions and Equations A.2.C - "Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)."
  • Grade 8: Expressions and Equations A.1 - "Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27."

Explore More!

  • Juno mission website - News, resources and updates on NASA's mission to Jupiter.

  • Eyes on the Solar System - Take a virtual journey to Jupiter with Juno (scroll to "Solar System Tours" and click on Juno).

  • To Jupiter with JunoCam! - Find out how classrooms can participate in the Juno mission to Jupiter using the spacecraft's on-board educational camera.

  • Infographic: Solar Power Explorers - This graphic shows how NASA’s Juno mission to Jupiter became the most distant solar-powered explorer and influenced the future of space exploration powered by the sun.

TAGS: Juno, Jupiter, Exponents, Math, Lesson, Activity, Teachable Moment

  • Lyle Tavernier
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Saturn's moon Enceladus

In the News

Saturn’s icy moon Enceladus has been making news lately, and it could make even bigger news soon! In September, scientists confirmed that there was a global ocean underneath Enceladus’ thick icy shell. That was just the latest in a long history of exciting finds dating back to the beginning of NASA’s Cassini-Huygens Mission to Saturn in 2004 that have helped scientists to better understand this fascinating world!

Even while Cassini was still on its way to Saturn, its Cosmic Dust Analyzer detected microscopic grains of silica (tiny grains of sand). On Earth, grains of silica similar in size to those detected near Saturn form when hydrothermal activity -- the processes involving heated water beneath Earth’s surface or ocean -- causes salty water to chemically interact with rocky material to form silica. But where were these grains coming from in the space around Saturn?

In 2005, scientists were surprised to find out that Enceladus’ south pole is both warmer than expected and warmer than the surrounding areas, suggesting there is a heat source inside Enceladus. Not only that, but they also discovered long parallel cracks in the ice on Enceladus’ south pole. The young age of these cracks, nicknamed Tiger Stripes, meant that Saturn’s icy moon is a geologically active place.


Color image of the cracks, or Tiger Stripes, on the South Pole of Saturn's moon Enceladus
This enhanced color view of Saturn's moon Enceladus shows the south polar terrain, where jets of material spray out form long cracks called Tiger Stripes. Image credit: NASA/JPL-Caltech/Space Science Institute | Full image and caption


Heat map of Saturn's moon Enceladus
This image shows the infrared (heat) radiation at the south pole of Saturn's moon Enceladus, including the dramatic warm spot centered on the pole near the moon's Tiger Stripes feature. The data were taken during the spacecraft's third flyby of Enceladus on July 14, 2005. Image credit: NASA/JPL-Caltech/Space Science Institute | Full image and caption

Another piece of this puzzle was put in place with the discovery of jets of material spraying out of the Tiger Stripes. Studies have shown these jets are composed of mostly of water vapor, tiny ice particles and small amounts of other material (for example, microscopic silica grains). Together, over 100 jets make up a feature called a plume. Investigating further, scientists have hypothesized that these silica grains are the result of hydrothermal activity on the ocean floor below Enceladus’ icy crust.


Movie of the Plume on Saturn's moon Enceladus
Jets of icy particles burst from Saturn’s moon Enceladus in this brief movie sequence of four images taken on Nov. 27, 2005. Credit: NASA/JPL-Caltech/Space Science Institute | Full image and caption

On October 28, Cassini will fly right through the plume jetting out of Enceladus’ south pole at an altitude of only 49 kilometers (30 miles) – closer than any previous passes directly through the plume! This is an exciting moment in the mission -- one that allows science teams to use a combination of tools on board the spacecraft to strengthen previous findings and potentially make new discoveries.

Why It's Important

Cassini will use its Cosmic Dust Analyzer to study the solid plume particles and an instrument called the Ion and Neutral Mass Spectrometer to “sniff” the gas vapor in order to determine the composition of the jets. Specifically, the latter instrument is looking for H2, or molecular hydrogen. Finding H2 in the plume will strengthen the evidence that hydrothermal activity is occurring on Enceladus’ ocean floor. And the amount of H2 in the plume, will tell scientists just how much activity is happening.

In addition to indicating that hydrothermal activity is taking place, figuring out the amount of hydrothermal activity will give scientists a good indication of how much internal energy there is deep inside Enceladus.

That Cassini is making a pass through the plume at such a low, 49-kilometer-high altitude is also important. Organic compounds -- substances formed when carbon bonds with hydrogen, nitrogen, oxygen, phosphorus or sulfur -- tend to be heavy and would fall out of the plume before reaching the heights of Cassini’s previous, higher altitude flybys and be undetected. Organic compounds are the building blocks of life on Earth. Without them, life as we know it wouldn’t exist. If they are present in Enceladus’ oceans, they could be detected when Cassini passes through the plume on this encounter.

Perhaps more important, though, are the implications of finding hydrothermal activity somewhere other than Earth. It was once believed that all forms of life needed sunlight as a source of energy, but in 1977, the first hydrothermal vent -- essentially an underwater geyser of hot, mineral-rich water -- was discovered and it was teeming with life. The organisms were using the heat and minerals as a source of energy! Some scientists have hypothesized that hydrothermal vents could be where life on our planet first took hold and could represent environments in the solar system with the necessary ingredients to support life.

Teach It

Here are a handful of lessons and resources you can use to teach key concepts related to the October 28 Enceladus flyby and help your students feel connected to this exciting moment in science at Saturn.

Modeling

Standard(s):

  • NGSS 5-ESS2-1 - Develop a model using an example to describe ways the geosphere, biosphere, hydrosphere, and/or atmosphere interact.

Activity:

Because scientists can’t dig beneath the ice and see what’s below, they rely on creating models that show what is happening beneath the surface. A model helps us imagine what can’t be seen and explains the things that we can see and measure. A model could be a drawing, a diagram or a computer simulation. For this model, students will draw a cut away model of Enceladus and iterate, or improve, their model as you provide more description, just as scientists improved their models as they learned more about Enceladus.

  1. Tell students there is a moon around Saturn. They should draw a moon (likely a circle, half-circle, or arc, depending on how big you want the drawing to be).

  2. Explain to students that the moon is covered in a shell of ice (students will need to modify their model by drawing a layer of ice). Thus far, everything students are modeling is observable by looking at the moon.

  3. Share with students that temperature measurements of the south pole revealed spots that are warmer than the rest of the moon’s surface. Ask students to brainstorm possible sources of heat at the south pole and explain what might happen to ice near a heat source. Based on this new information, and what they think might be causing the heat, allow them to modify their drawing. (Depending on what students brainstorm, their drawing might now include volcanoes, hot spots, magma, hydrothermal vents and a pool of liquid water beneath the ice).

  4. The next piece of information the students will need to incorporate into their drawing is that there are large cracks in the ice over the warmer south-pole region.

  5. Explain that students have now received images that show jets expelling material from the cracks. They will need to incorporate this new data and add it to their drawing.

  6. Tell students that by studying the gravity of the moon, scientists now believe there is an ocean covering the whole surface of the moon beneath the ice. Ask students to share how they would represent that in the model. Allow them to modify their drawing.

  7. Show students the following image depicting a model of Enceladus:

    Saturn's moon Enceladus global ocean model

    This model shows what scientists believe the interior of Enceladus may look like. Have students compare it to what they drew and note similarities and differences.

Particle Travel Rate

Standard(s):

  • CCSS.MATH 6.RP.A.3.B - - Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Problem:

Based on the size of the silica grains (6 to 9 nanometers), scientists think they spend anywhere from several months to a few years (a longer time than that means the grains would be larger) traveling from hydrothermal vents to space, a distance of 40 to 50 km.

  1. What rate (in km/day) are the particles traveling if it takes them 6 months to travel 50 km (assume 182 days)?

    50 km ÷ 182 days = 0.27 km/day

  2. What rate are they traveling if it takes two years to travel 40 km?

    40 km ÷ 730 days = 0.05 km/day

  3. Do you think the particles in each example traveled at the same speed the entire time they moved?

  4. Why might the particle rate vary?

  5. At what point in their journey might particles have been traveling at the highest rate?

Plume Data

Standard(s):

  • CCSS.MATH 6.RP.A.3.B - Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
  • CCSS.MATH 8.G.B.7 - Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Problem:

Cassini will be flying past Enceladus at a staggering 8.5 km per second (19,014 mph). At an altitude of 49 km, the plume is estimated to be approximately 130 km across.

  • How long will Cassini have to capture particles and record data while within the plume?

    130 km ÷ 8.5 km/sec ≈ 15 seconds

  • If Cassini is 49 km above the surface of Enceladus at the center of the plume, what is its altitude as it enters and exits the plume (the radius of Enceladus is 252.1 km)?

    252.1 km + 49 km = 301.1 km
    (301.1 km)2 + (65 km)2 ≈ 95,000 km2
    √(95,000 km2) ≈ 308 km
    ≈ 308 km – 252.1 km ≈ 56 km

  • This information can help scientists determine where in the plume heavy particles may fall out if they are not detected on the edge of the plume but are detected closer to the middle of the plume. It is also important because the Cosmic Dust Analyzer uses a high-rate detector that can count impacting particles at over 10,000 parts per second to tell us how much material is being sprayed out.

Volume of Enceladus’ Ocean

Standard(s):

  • CCSS.MATH 8.G.C.9 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
  • CCSS.MATH HSG.GMD.A.3 - Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Problem:

Gravity field measurements of Enceladus and the wobble in its orbital motion show a 10 km deep ocean beneath a layer of ice estimated to be between 30 km and 40 km thick. If the mean radius of Enceladus is 252.1 km, what is the minimum and maximum volume of water contained within its ocean?

Volume of a sphere = 43πr3

Minimum volume with a 40 km thick crust
43 π212.1 km3 - 43π202.1 km3 ≈ 40,000,000 km3 – 35,000,000 km3 ≈ 5,000,000 km3

Maximum volume with a 30 km thick crust
43 π222.1 km343 π212.1 km3 ≈ 46,000,000 km3 – 40,000,000 km3 ≈ 6,000,000 km3

This is important because if scientists know how much water is in the ocean and how much vapor is escaping through the plume, they can make estimates about how long the plume has existed -- or could continue to exist.

Download the Full Problem Set

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TAGS: Enceladus, moon, Saturn, Cassini, flyby, spacecraft, plume, jets, geysers, science, math

  • Lyle Tavernier
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Illustration of Kepler 452b

In the News

Twenty years after the first discovery of a planet orbiting another sun-like star, scientists have discovered the most Earth-like exoplanet ever: Kepler-452b. Located in the habitable zone of a star very much like our sun, Kepler-452b is only about 60 percent wider than Earth.

What makes it the most Earth-like exoplanet ever discovered?

First a couple definitions: An exoplanet is simply a planet that orbits another star. And the habitable zone? That’s the area around a star in which water has the potential to be liquid -- not so close to the star that all water would evaporate, and not so far that all water would freeze. Think about Goldilocks eating porridge. The habitable zone is not too hot, and not too cold. It’s just right.

Okay, back to Kepler-452b. Out of more than a thousand exoplanets that NASA’s Kepler spacecraft has detected, only 12 have been found in the habitable zone of their stars and are smaller than twice the size of Earth, making Earth-like planets a rarity. Until this discovery, all of them have orbited stars that are smaller and cooler than our sun.


Graphic showing habitable zone planets

Twelve Exoplanet discoveries from Kepler that are less than twice the size of Earth and reside in the habitable zone of their host star. The sizes of the exoplanets are represented by the size of each sphere. These are arranged by size from left to right, and by the type of star they orbit, from the M stars that are significantly cooler and smaller than the sun, to the K stars that are somewhat cooler and smaller than the sun, to the G stars that include the sun. The sizes of the planets are enlarged by 25X compared to the stars. The Earth is shown for reference. Image credit: NASA/JPL-Caltech/R. Hurt

Graphic showing habitable zone planets

The sweep of NASA Kepler mission’s search for small, habitable planets in the last six years. The first planet smaller than Earth, Kepler-20e, was discovered in December 2011 orbiting a Sun-like star slightly cooler and smaller than our sun every six days. But it is scorching hot and unable to maintain an atmosphere or a liquid water ocean. Kepler-22b was announced in the same month, as the first planet in the habitable zone of a sun-like star, but is more than twice the size of Earth and therefore unlikely to have a solid surface. Kepler-186f was discovered in April 2014 and is the first Earth-size planet found in the habitable zone of a small, cool M dwarf about half the size and mass of our sun. Kepler-452b is the first near-Earth-Size planet in the habitable zone of a star very similar to the sun. Image credits: NASA Ames/W. Stenzel

Kepler-452b is the first to be discovered orbiting a star that is about the same size and temperature as our sun. Not only that, but it orbits at nearly the same distance from its star as Earth does from our sun! Conditions on Kepler-452b could be similar to conditions here on Earth and the light you would feel there would be much like the sunlight you feel here on Earth. Scientists believe that Kepler-452b has been in the habitable zone for around six billion years -- longer than Earth has even existed!


Graphic comparing our solar system and Kepler-452b's system

This size and scale of the Kepler-452 system compared alongside the Kepler-186 system and the solar system. Kepler-186 is a miniature solar system that would fit entirely inside the orbit of Mercury. The habitable zone of Kepler-186 is very small compared to that of Kepler-452 or the sun because it is a much smaller, cooler star. The size and extent of the habitable zone of Kepler-452 is nearly the same as that of the sun, but is slightly bigger because Kepler-452 is somewhat older, bigger and brighter. The size of the orbit of Kepler-452b is nearly the same as that of the Earth at 1.05 AU. Kepler-452b orbits its star once every 385 days.Image credit: NASA/JPL-Caltech/R. Hurt

How They Did It

The Kepler spacecraft, named for mathematician and astronomer Johannes Kepler, has been working since 2009 to find distant worlds like Kepler-452b. It does so by looking at more than 100,000 stars near the constellation Cygnus. If one of those stars dims temporarily, it could be that an object passed between the spacecraft and the star. If it dims with a repeatable pattern, there’s a good chance an exoplanet is passing by again and again as it orbits the star. The repeated dimming around one of those stars is what led to the discovery of Kepler-452b.


Kepler measures the brightness of stars. The data will look like an EKG showing the heart beat. Whenever a planet passes in front of its parent star as viewed from the spacecraft, a tiny pulse or beat is produced. From the repeated beats we can detect and verify the existence of Earth-size planets and learn about the orbit and size of the planet.Video credit: NASA Ames and Dana Berry

Teach It

This exciting discovery provides opportunities for students to practice math skills in upper elementary and middle school, and gives high school students a practical application of Kepler’s third law of planetary motion. Take a look below to see where these might fit into your curriculum.

Upper Elementary and Middle School

After learning about Earth’s cousin, students might wonder about a trip to this world. Scientists have calculated the distance between Earth and Kepler-452b at 1,400 light years. A light year is a measure of distance that shows how far light travels in one year. It’s equal to about 10 trillion kilometers (six trillion miles) or, to be more precise, 9,461,000,000,000 kilometers (5,878,000,000,000 miles). Ask students to calculate the distance between Earth and Kepler-452b at various levels of precision, depending on what they are prepared for or learning. For an added challenge, have them determine how long it would take a fast moving spacecraft like Voyager 1 traveling at 61,000 kph (38,000 mph) to reach this new world.

Note: Due to the approximations of spacecraft speed and light year distance used for these problems in both standard and metric units, there is a variation among the answers.

    Distance: 10 trillion km x 1,400 = 14,000 trillion km (that’s 14,000,000,000,000,000 kilometers!)

    Travel time: 14,000 trillion km ÷ 61,000 kph ÷ 24 ÷ 365 ≈ 26,000,000 years

    Distance: 6 trillion miles x 1,400 = 8,400 trillion miles (that’s 8,400,000,000,000,000 miles!)

    Travel time: 8,400 trillion miles ÷ 38,000 mph ÷ 24 ÷ 365 ≈ 25,000,000 years

or more precisely…

    Distance: 9,461,000,000,000 km x 1,400 = 13,245,400,000,000,000 km

    Travel time: 13,245,400,000,000,000 km ÷ 61,000 kph ÷ 24 ÷ 365 ≈ 25,000,000 years

    Distance: 5,878,000,000,000 miles x 1,400 = 8,229,200,000,000,000 miles

    Travel time: 8,229,200,000,000,000 miles ÷ 38,000 mph ÷ 24 ÷ 365 ≈ 25,000,000 years

or using exponents and powers of 10…

    Distance: 9.461 x 1012 x km x 1.4 x 103 = 1.32454 x 1016 km

    Travel time: 1.32454 x 1016 km ÷ 6.1 x 104 kph ÷ 2.4 x 101 ÷ 3.65 x 102 ≈ 2.5 x 107 years

    Distance: 5.878 x 1012 miles x 1.4 x 103 = 8.2292 x 1015 miles

    Travel time: 8.2292 x 1015 miles ÷ 3.8 x 104 mph ÷ 2.4 x 101 ÷ 3.65 x 102 ≈ 2.5 x 107 years

Middle and High School

The time between detected periods of dimming, the duration of the dimming, and the amount of dimming, combined with a little math, can be used to calculate a great deal of information about an exoplanet, such as the length of its orbital period (year), the distance from its star, and its size.

Kepler-452b has an orbital period of 384.84 days -- very similar to Earth’s 365.25 days. Students can use the orbital period to find the distance from its star in astronomical units. An astronomical unit is the average distance between Earth and our Sun, about 150 million kilometers (93 million miles).

Kepler’s 3rd law states that the square of the orbital period is proportional to the cube of the semi-major axis of an ellipse about the sun. For planets orbiting other stars, we can use R = ∛(T2 ∙ Ms)  where R = semi-major axis, T = orbital period in Earth years, and Ms = the mass of the star relative to our sun (the star that Kepler-452b orbits has been measured to be 1.037 times the mass of our sun).

    T = 384.84 ÷ 365.25 = 1.05

    R = ∛(1.052 ∙ 1.037)

    R = ∛1.143 = 1.05 AU

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TAGS: Exoplanets, Kepler, Kepler-452b, Habitable Zone, Math, Activities, Classroom Activities, Resources

  • Lyle Tavernier
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In the News

We visited Pluto!

On July 14, 2015 at 4:49 a.m. PDT, NASA's New Horizons spacecraft sped past Pluto -- a destination that took nearly nine and a half years to reach -- and collected scientific data along with images of the dwarf planet.

Pluto, famous for once being the ninth planet, was reclassified as a dwarf planet in 2006 after new information emerged about the outer reaches of our solar system. Worlds similar to Pluto were discovered in the region of our solar system known as the Kuiper Belt. The Kuiper Belt --named for astronomer Gerard Kuiper --is a doughnut-shaped area beyond the orbit of Neptune that is home to Pluto, other dwarf planets such as Eris, Makemake, and Haumaea, as well as hundreds of thousands of other large icy bodies, and perhaps trillions of comets orbiting our sun. Over the next several years, the New Horizons spacecraft is expected to visit one to two more Kuiper Belt objects.

Even though it will take 16 months for New Horizons to return all the Pluto science data to Earth, we have already made some interesting and important discoveries about Pluto.

Why It's Important

Through careful measurements of new images, scientists have determined that Pluto is actually larger than previously thought: 2,370 kilometers in diameter.  This is important information for scientists because it helps them understand the composition of Pluto. Because of the orbital interactions between Pluto and its moon Charon, Pluto’s mass is well known and understood. Having a more precise diameter gives scientists the ability to more accurately calculate the average density. A greater diameter means Pluto’s density is less than we thought. If you do the math, you’ll see that Pluto’s calculated density dropped from 2,051 kg/m3 to 1,879 kg/m3 with this new finding. Most rock has a density between 2000-3000 kg/m3 and ice at very cold temperatures has a density of 927 kg/m3, so we can conclude that Pluto is a bit more icy than previously believed. In addition to helping scientists calculate the density of Pluto, this measurement confirms Pluto as the largest known object in the Kuiper Belt!

Teach It

We’ve provided some math problems (and answers) for you to use in the classroom. They’re a great way to provide students with real-world examples of how the math they’re learning in class is used by scientists. There are also some additional resources below that you can use to integrate the Pluto flyby into your lessons, or use the flyby as a lesson opener!

Pluto Math Problems

  1. Find the radius(r) of Pluto.
    2,370 kilometers ÷ 2 = 1,185 km

  2. Find the circumference of Pluto.
    C = 2 π r = 7,446 km

  3. Find the surface area of Pluto.
    SA = 4 π r2 = 17,646,012 km2

  4. Find the volume of Pluto.
    4/3 π r3  = 6,970,174,651 km3

  5. Find the density of Pluto in kg/m3.
    mass/volume
    Pluto mass = 1.31 × 1022kg
    Convert volume in km3 to m3: 6,970,174,651 × 1,000,000,000 = 6.970174651 × 1018m3
    1.31 × 1022kg / 6.970174651 × 1018m3 = 1,879 kg/m3

  6. How does this new density calculation compare to the previous calculation (2051 kg/m3) when Pluto’s diameter was thought to be 2,302 km?

Explore More

Take a look at some of the lessons, videos, activities and interactives related to Pluto. They’re a great way to engage students in STEM and learning more about their solar system!

  • Video: What is a Dwarf Planet? (K-12)
    Dwarf planets are a lot like regular planets. What’s the big difference? Find out in 60 seconds.

  • Activity: Solar System Bead Activity (4-8)
    The solar system is big, and Pluto is way out there! Students calculate scale distances to create a model of objects in our solar system.
    Next Generation Science Standards: MS-ESS1-3
    Common Core Math: 4.MD.A.2, 5.NBT.B.7

  • Activity: How Far? How Faint? (9-12)
    Calculate how much light Pluto receives from the sun, compared to Earth.
    Common Core Math: HSF.IF.C.7.E

  • Resource: Pluto Facts and Figures
    Get lots of facts and figures about this dwarf planet in the Kuiper belt!

  • Interactive: Eyes on Pluto
    Ride along with New Horizons in this simulation of its closest approach to Pluto!

  • Participate: Pluto Time
    Though Pluto is a distant world with very different characteristics from Earth, for just a moment near dawn and dusk each day, you can experience “Pluto Time.” This is when the amount of light reaching Earth matches that of noon on Pluto. Find out exactly when Pluto Time happens in your area and share your photos online!

  • News and Images: NASA New Horizons Website
    Get the latest news and images from NASA's New Horizons mission.


TAGS: Pluto, New Horizons, Math, Teachable Moment

  • Lyle Tavernier
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Pi in the Sky Infographic


In honor of Pi Day, March 14 (or 3.14), 2013, the JPL Education Office has released an infographic highlighting some of the ways scientists and engineers at the laboratory use pi in their daily work. For example, scientists can use pi (along with mass and radius) to calculate the density of an asteroid and its material makeup.

The infographic also features a pi- and planetary-themed mathematics challenge for students that asks them to find various measurements for a fictional "Planet Pi," which appropriately has a circumference of 314,152 miles.

The infographic is available on the JPL Infographics website and as a full-resolution download below.

"Planet Pi" Downloads:

TAGS: Pi Day, Infographics, Math, K-12, Downloads, Posters

  • Kim Orr
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Andrew Crawford and Todd Barber in mission control at JPL

There are certain people in life that exude such enthusiasm and passion, that you are helpless to escape their tractor beam of positivity. This summer, during my second internship at NASA's Jet Propulsion Laboratory in Pasadena, Calif., I was fortunate enough to meet and befriend just that kind of person. Todd Barber has such passion and talent for exploring and doing what humans do best: pushing the boundaries of what humankind is capable of. As the lead propulsion engineer for the Cassini spacecraft, he was at the controls for the tricky insertion of the spacecraft into Saturn's orbit and recently was at the helm for the cruise stage of the Mars Science Laboratory's Curiosity rover during its "seven minutes of terror" descent through the Martian atmosphere. As if that's not interesting enough, during his free time, he goes on math-based adventures, inspires students through NASA outreach activities and plays in at least two jazz bands. I caught up with Todd just days before he helped successfully land NASA's Curiosity rover on Mars to learn a bit about his path to success and how he inspires others.

How did you get to be the lead propulsion engineer for Cassini, one of my personal favorite missions?

"I started here 22 years ago. Often when they bring you in, they put you on mission operations, flying something, because you get to interface with all the other sub-system engineers and what they do, including thermal, power, communications, propulsion, essentially a boot camp. I started at JPL on the Galileo mission, doing mission ops. I loved mission ops so much, and I had a penchant for ops work. I think I'm a planetary science geek. The Voyager missions are what grabbed me. My way to contribute to planetary science is by being a propulsion engineer. I didn't know what missions I'd work on after Galileo, and luckily a Cassini engineer moved on. Six months before launch, they sought me out, and I've been on Cassini ever since. I get to work on other missions part-time, including the Mars Exploration Rovers, Deep Impact and a Mars airplane study."

"The Cassini job opened up, and my mentor, Dick Cowley, was the lead propulsion engineer at the time, but near retirement. He worked on Apollo -- the F1 engine -- so this guy was the propulsion guru. I picked his brain every day until he retired, and then one day, he said, 'I think it's time we make you the lead propulsion engineer.' This was a couple years before Saturn orbit insertion, and I had the privilege of presenting to the review boards, getting barbecued at the flight reviews. But all went smoothly and we had a marvelous orbit insertion, and now the Cassini spacecraft is still sending amazing images and data home every day. I like to say that if everything goes well, I'll be 51 years old after these missions and wonder what I'll do when I grow up, because I've been having so much fun on these outer planet missions!"

What exactly does a lead propulsion engineer for Cassini do?

"I am responsible for the health and safety of the propulsion system. We are like 'space plumbers.' All the engines, regulators, tanks, valves, we are responsible for all these systems. The Cassini propulsion system is the most complex propulsion system ever flown by JPL. It's a plumber's nightmare! My job also includes figuring out the gas mileage of the spacecraft -- bean counting the propellant, if you will -- and that is getting very important now as our monopropellant gas tank on the spacecraft is half full. (I'm an optimist.)"

What happens to Cassini when it runs out of its propellant?

"At the end of the mission, we will cross inside Saturn's rings, getting views and pictures we've never seen before, and then go out in a blaze of glory. On September 15, 2017, Cassini will be crushed and vaporized by Saturn's atmosphere."

Why not save the spacecraft?

"Actually, it's for astrobiological reasons. We would never want to accidently crash the spacecraft into Enceladus or Titan, two of Saturn's moons, and possibly contaminate them. Even though it's hard to imagine 'hearty spores' could have hitched a ride on the spacecraft from Earth that are just waiting for liquid water, we would never want that to happen before we can answer the most fundamental question, 'Did indigenous life start on its own?' It's called 'forward planetary protection.' The opposite is true as well, which is called 'backward planetary protection,' where if you bring things back to Earth from other planets, you would never want to bring back something that could wipe out humanity. We want to be as careful as we can."

Do you have to dodge asteroid belts, space junk, etc.?

"Yes. We have to worry about Saturn's ring plane, made up of dust and small particles. The rings are very thin, like a phonograph record, but huge across. There's a ton of dust and particles that make up the rings, and sometimes we have to turn the spacecraft around with the antenna facing forward, which acts as a giant dust shield. The dust can damage the engines, so we built a dust cover similar to a baby stroller sun shield that folds down. We've had to use it 70 times on the mission, and the engine is still working great! Our engine has a wonderful heritage, as it was the same reaction control thruster that carried Neil Armstrong and Buzz Aldrin on Apollo!"

Where do you think we should explore next?

"Well, I love the outer planets, I would love to work a Neptune orbiter and would love to go to Saturn's moons Titan and Enceladus, or Jupiter's moon Europa. There are very complex propulsion trajectories to get there, but the payoff could be huge!"

Were you an intern before you worked here at JPL?

"No, I was not. I actually interviewed here three times and got turned down twice. The third time was a charm. After getting turned down the first time after getting my undergrad in aerospace engineering, I went back and got my masters in aerospace engineering at MIT, and then I got my dream job. I remember the summer of my 8th grade year, I saw a National Geographic with the Voyager spacecraft flying by Jupiter and Saturn and was hooked. I said, 'I have to work there at JPL!'"

I know that you are very involved with NASA's educational outreach and are constantly inspiring and trying to get kids involved with NASA and JPL. What do you feel is the most vital thing we need to tell young people today to get them inspired?

"Well, I think the most vital thing is showing them that math and science are cool. Math and science are not generally thought of as cool, but when you see math and science in action at NASA, it's the coolest -- especially during our missions such as the Mars Science Laboratory. For the U.S. to stay competitive, I think we really need to focus on new innovation, for that's what made this country great!"

"I've had strange hobbies that have helped me apply math and science. One of my hobbies is 'confluence hunting,' basically geeks with GPS devices trying to find and visit integer (whole counting numbers) latitude and longitude intersections [across the globe]. There are 850 confluence points in the continental United States, and I visited 17 of the last 50. They gave me the nickname 'The Closer' on the website."

Do you think innovation is a learned trait, or something people are just naturally good at?

"I'm here because of great teachers. I'm the first to sing the praises of teachers who got me fired me up about math and science. I went to public schools and got really lucky that my teachers were world class and fired up on math and science, and I'm still in touch with them. They follow all my missions and tell the kids that I was once a former student of theirs and to shoot for the stars."

"I had two dreams as a kid, to work for NASA, and to compose music. I'm so happy that they both came true, and I feel like I have an explorer's spirit, and need to share that with others."

Speaking of music, I hear that you are in a Jazz band called "The Big Band Theory," and that you have a minor from MIT in music? Well, I play the classical violin, and I was wondering if we could jam sometime?

"Haha, absolutely! I'm in two bands, The Big Band Theory and The Jazz Propulsion Band, and I'm always playing music. Let's make that happen!"

TAGS: Todd Barber, Propulsion, Engineering, Cassini, Math

  • Andrew Crawford
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