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Student Project .

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The NASA Pi Day Challenge

Subject
Math
Time Required
1-2 hrs
A collage of cartoon spacecraft, asteroids, Earth and rover on a lunar surface surround text that reads π NASA Pi Day Challenge. A cat sitting on the π symbol reaches toward a laser pointed in its direction. Stars in the shape of the π symbol fill the bac

Update: March 15, 2024 – The answers to the 2024 NASA Pi Day Challenge are here! If you're stumped on the answers, click the link to view the answer key under each of the problems below. But try not to peek before giving it your best try.

You may already know all about the mathematical constant pi (π) and how it can be used to calculate things like the circumference of a circle or the volume of a sphere. But did you know pi is also used all the time by NASA scientists and engineers to explore other planets?

In this challenge, you can solve some of the same problems NASA scientists and engineers do using pi!

Project Steps

  • Receiver Riddle - New for 2024

    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?

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  • Daring Deflection - New for 2024

    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?

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  • Orbit Observation - New for 2024

    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?

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  • Moon Mappers - New for 2024

    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?

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  • 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?

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  • 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?

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  • 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:

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

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  • 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 a total eclipse?

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  • Lunar Logic

    NASA’s Lunar Flashlight mission will observe and map the location of frost within permanently shadowed craters in the Moon’s south polar region. Knowing how much frost is in these craters and where to find it can help us prepare for extended missions on the Moon, when water will be a valuable resource.

    The spacecraft, a backpack-size cubesat, will collect data during 10 orbits over a two-month period, making repeated measurements over multiple points to map ice in these dark craters. To take measurements, Lunar Flashlight will send infrared laser pulses to the surface of the Moon and measure the signal that is reflected. The amount of light that is reflected back will help scientists determine where the lunar surface is dry and where it contains water-ice.

    At 20 km altitude, the spacecraft's infrared lasers have a radius of 17.5 meters when they reach the surface of the Moon.

    How much area do they cover in a single pulse?

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  • Core Conundrum

    The InSight Mars lander is equipped with several tools to help scientists learn more about the interior of the Red Planet, including a seismometer that detects marsquakes. By measuring the vibrations that travel across the surface of Mars and through its interior layers, scientists were able to accurately measure the size of Mars’ liquid core and estimate its density. Knowing the size and density of Mars' core will help us learn more about how the planet formed, how its magnetic field developed, and what materials make up the core, which will ultimately lead to a better understanding of how Earth and other planets form.

    If Mars' core has a mass of 1.54 x 1023 kg and a radius of 1,830 km, as measured by InSight, what is the density of the core?

    How does that compare to the density of Earth’s core, which ranges from 10 to 13 g/cm3?

    What does that tell us about the makeup of Mars’ core?

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  • Dam Deduction

    Water exiting a hydropower dam is called non-powered or powered outflow. Non-powered outflow exits via a spillway, on top of the dam. Powered outflow, which is used to generate electricity, travels through penstocks, pipes at the bottom of a dam. Powered outflow is usually colder and travels at a higher velocity, so it can disturb sediments, temperatures, and water quality of downstream rivers, especially when it’s a high percentage of the total outflow.

    The SWOT mission, a satellite designed to survey all of Earth’s surface water, including lakes, rivers, oceans, and reservoirs, can help scientists better analyze these impacts.

    A dam has 3 penstocks with diameters of 6.2 meters and a measured total outflow of 1,350 m3/s. If SWOT measured the reservoir’s water depth (H) at 100 m above the penstocks, compute the velocity (m/s) of the powered outflow using V=√(2gH).

    What is the powered outflow if 1 penstock is open?

    Is this a high or low percentage of the total outflow?

    What can this tell you about the potential environmental impacts?

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  • Telescope Tango

    NASA's TESS mission is designed to survey the entire sky in search of exoplanets, or planets orbiting stars other than our Sun. In its two-year primary mission, TESS identified more than 2,600 possible exoplanets and counting.

    To locate exoplanets, the space telescope flies in a highly eccentric elliptical orbit, which had never been attempted before. This orbit, called P/2, minimizes the amount of time that light and heat from Earth and the Moon can interfere with data collection. And it still allows the spacecraft to make close passes by Earth to transmit data about its findings back to scientists. The spacecraft's 13.7 day orbit has an axis of 376,000 km at apogee and an axis of 108,400 km at perigee. Each downlink from TESS takes about three hours to complete.

    While TESS actually moves at different speeds throughout its orbit – from 0.5 km/s at apogee to 4 km/s at perigee – if its velocity stayed uniform, how many kilometers would TESS need to travel to successfully transmit its data?

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  • Sample Science

    NASA’s OSIRIS-REx mission was designed to travel to an asteroid called Bennu and bring a small sample back to Earth for further study. To achieve its mission, the spacecraft needed to make contact with 26 cm2 of asteroid Bennu’s surface and collect millimeter-size particles using its "contact-pad samplers." These are 1.5-centimeter diameter circular pads of Velcro-like stainless steel. There are 24 pads on the mechanism designed to collect the samples.

    How many pads needed to make contact with Bennu's surface to meet the mission requirement?

    If all 24 pads contacted Bennu, how much asteroid surface area would the contact pads sample?

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  • Whirling Wonder

    Joining the Perseverance rover on Mars is a small helicopter named Ingenuity. With twin counter-rotating blades spanning 1.2 meters, Ingenuity is a test of new technology and is designed to achieve the first powered flight on another world.

    Despite Mars having less gravity than Earth, the atmosphere on the Red Planet is much thinner than it is here on our home planet. This makes it challenging to lift off the ground on Mars. To generate enough lift for Ingenuity, engineers determined that the helicopter's blades need to rotate at approximately 250 radians per second on Mars.

    How fast – in rotations per minute – do Ingenuity’s blades spin?

    How does that compare to a typical helicopter on Earth with blades that spin at 500 rotations per minute?

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  • Signal Solution

    As more and more data are collected and transmitted through space, NASA needs new technologies to communicate faster and more efficiently with its spacecraft. One such technology is called Deep Space Optical Communications, or DSOC, which uses near-infrared light instead of radio waves to transmit a signal. This allows us to use a higher frequency (shorter wavelength), so more data can be transmitted per second.

    The twin Voyager spacecraft launched in 1977 use a 12.5 Watt transmitter paired with a parabolic reflector that creates a circular radio signal with a diameter roughly 0.5 degrees wide. A DSOC system would use a 4 Watt transmitter on a flight laser transceiver, producing a light signal with a diameter of 0.0009 degrees.

    If Voyager and a DSOC-equipped spacecraft were both placed 124 AU from Earth (1 AU = 150,000,000 km) what fraction of each original wattage would be received by a 70m antenna back on Earth?

    By what factor is DSOC more effective?

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  • Force Field

    Every day, Earth is showered in radiation from the Sun. The Sun also emits charged particles almost entirely in the form of ionized hydrogen and helium. These ions travel at speeds of about 400 km per second but rarely reach Earth's surface. That’s because they are deflected by Earth’s magnetic field due to the Lorentz force, given by the equation: F = qvBsinθ F = force (N) q = charge of the particle in coulombs (C) v = velocity of the particle in meters per second (m/s) B = the magnetic flux density of Earth’s magnetic field in teslas (T) θ in radians.

    The charged particles can't cross Earth's magnetic field, so they follow it to Earth's North and South poles. The resulting concentration of charged particles is what creates auroras.

    If Earth’s magnetic flux density is 60µT, what force would a hydrogen ion observe at π/4 radians from the equator? What about at the North Pole (π/2 radians)?

    Does the relative magnetic field agree or disagree with what you’d expect about the location of auroras?

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  • 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?

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  • 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.

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  • 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

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  • 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, led 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)

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  • Deadly Dust

    In the summer of 2018, a large dust storm enshrouded Mars, blocking visibility over a large portion of the planet. The thick dust covered almost all of the Mars surface, blocking the vital sunlight that NASA’s solar-powered Opportunity rover needed to survive. In fact, the storm was so intense and lasted for so long that Opportunity, which had spent 14.5 years traveling around the Red Planet, never managed to regain consciousness and the mission had to come to an end.

    During the height of the storm, only the upper caldera of one of the solar system’s largest volcanos, Olympus Mons, peeked out above the dust cloud. The diameter of Olympus Mons’ caldera is approximately 70 km.

    What percent of the Mars surface was covered in dust at that time?

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  • Cloud Computing

    The MISR instrument on NASA’s Terra satellite has nine cameras that view Earth from different angles to study features on the surface and in the atmosphere in 3D. One of MISR’s tasks is to collect measurements of clouds, which are full of liquid water or ice. Scientists can use the measurements to estimate how much water is in a cloud. Imagine MISR flies over a cloud that from directly overhead looks like a circle, 10 km across. From the side, it looks like a soup can, indicating it’s roughly the shape of a right cylinder.

    Given that the cloud’s top and height measure 16 km combined, calculate the approximate volume of the cloud in cubic kilometers.

    Given the liquid water content of a typical puffy cumulus cloud, calculate the total amount of water in the cloud.

    If all the water in the cloud fell as rain, how many Olympic size swimming pools could it fill?

    Cumulus cloud liquid water content = 500,000 kg/km3 Olympic swimming pool volume = 2,500 m3 Water density = 1,000 kg/m3

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  • Storm Spotter

    Jupiter’s well known Great Red Spot is shrinking and someday may disappear entirely. Continuously observed since the 1830s, this massive storm was once more than three times the diameter of Earth.

    When the twin Voyager spacecraft flew by Jupiter in 1979, they sent back images of the Great Red Spot. At that time, the storm measured 24,700 km wide by 13,300 km tall. When scientists measured the storm again in 2018, using images from the Hubble Space Telescope, their estimates were 16,500 km wide by 11,400 km tall.

    Given these measurements, how does the current width of the Great Red Spot compare to the diameter of Earth?

    By what percent did the area of the Great Red Spot shrink from 1979 to 2018?

    The formula for the area of an ellipse is πab.

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  • Icy Intel

    Scientists at JPL study ices found in space to understand what they’re made of and how chemical processes unfold in cold environments. To find out what molecules are produced when sunlight or solar wind hits a comet, scientists place a piece of simulated comet ice in a vacuum to expose it to conditions that exist in space. Then, they aim an infrared laser at the sample to produce a plume that can be analyzed. Scientists have found that when simple molecules are exposed to light or electrons, they can transform into more complex molecules – even ones considered key to life’s formation!

    Scientists need to know how much energy is hitting the sample in a given area. This is called “fluence.” Enough of it will explode the ice so the sample can be analyzed. Peak fluence is found by dividing the laser’s total optical pulse energy by πw2/2, where w is the radius of the beam.

    Using a beam that has a radius of 125.0 µm and a total optical pulse energy of 0.30 mJ, what is the laser’s peak fluence in J/cm2?

    If the optics used to aim and focus the laser reduce its energy by 27% before it hits the sample, will this beam be sufficient to examine a sample that needs a peak fluence of 1.0 J/cm2 to explode?

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  • Solar Sleuth

    Exoplanets are worlds that orbit other stars. Using the Kepler Space Telescope, scientists can study distant stars and search for the exoplanets around them. When Kepler measures repeated dips in the brightness of a star, it can mean that an exoplanet is passing in front of that star from Kepler’s point of view. Scientists can then determine the size of the exoplanet based on how much the star’s light dipped when the planet passed in front of it. This dip in brightness detected by Kepler is expressed as a percentage of the star’s light that is blocked by the planet – with large planets blocking out more of the star’s light and small, Earth-size planets blocking less. This percentage equals the ratio of the area of the planet’s disk to the area of the star’s disk.

    If the Kepler detects a 0.042% drop in brightness from the star Kepler-186, which has a disk area of 416,000,000,000 km2, what is the radius of the exoplanet, known as Kepler-186f?

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  • Helium Heist

    With a radius of 70,000 km, Jupiter is our solar system’s most massive planet. About 10% of the volume from Jupiter’s cloud tops to 20,000 km below is helium, with the rest being mostly hydrogen. Circulation in this molecular hydrogen layer causes some of that helium to be depleted as it moves into the liquid metallic hydrogen layer beneath. The tremendous pressure inside Jupiter condenses helium into droplets that fall like rain through the less dense liquid metallic hydrogen. The presence of helium rain inside Jupiter helps explain why scientists observe less helium in the clouds than expected.

    If 10% of the helium volume in Jupiter’s molecular hydrogen layer has been rained out since the planet formed, what is the volume in cubic km that has rained out?

    Given that Earth’s radius is 6,371 km, about how many Earth-size spheres of helium have been rained out?

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  • Quake Quandary

    During a seismic event on Mars, or a “marsquake,” a type of seismic wave called surface waves travel outward from the epicenter, across the planet in all directions. Scientists expect these surface waves to arrive at NASA’s InSight lander, designed to study the quakes, at three different times: R1, when the first wave arrives, having traveled the shortest distance from the epicenter to the lander; R2, when the second wave arrives, having traveled the other way around Mars; and R3, when the first wave again impacts the lander, having traveled all the way around Mars. Let’s imagine InSight records marsquake waves at the following Earth times:

    R1 = 08:38:09.4 UTC

    R2 = 10:04:48.2 UTC

    R3 = 10:25:43.0 UTC

    *Note times are in UTC, which is written in hh:mm:ss format.

    Use the formulas below to determine the velocity (U) in rad/s of the surface wave, the distance in radians on the sphere from InSight to the epicenter (Δ), and the time the marsquake occurred (t0).

    Quake Quandary Formulas

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  • Asteroid Ace

    Asteroid 'Oumuamua is a uniquely-shaped interstellar object discovered in October 2017. It’s the first visitor from outside our solar system to be detected. Preliminary analyses indicate that 'Oumuamua is quite elongated, about 10 times as long as it is wide. It was first detected after it had passed Earth at a high speed on its journey out of our solar system, traveling at about 85,700 miles per hour.

    So scientists could make detailed observations of the interstellar visitor before it sped too far away, they had to quickly re-plan their schedules. By monitoring how the brightness of the asteroid fluctuated as it spun on its axis, scientists estimate that 'Oumuamua rotates once every 7.3 hours.

    Given these findings, what's the angular rotation rate of asteroid 'Oumuamua in rad/s?

    How does this compare with Earth's rotation rate?

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  • Crater Curiosity

    Craters form when an object hits the surface of a planet or other body. The impact creates a round impression surrounded by material, called ejecta, that gets blasted out of the crater. Scientists study ejecta because it contains clues about what’s below a planet’s surface. When an object hits Mars at an angle under 20 degrees, the crater is less circular and the ejecta settles in a butterfly shape. Some areas around the crater contain no blast material. Finding craters that formed this way can help scientists understand how meteor impacts change the surface of a planet. To do this, they measure a crater’s circularity ratio. If the ratio is less than 0.925, it suggests that an object impacted at an angle under 20 degrees and created a butterfly ejecta pattern.

    Using the circularity ratio formula, 4πA /p2, determine which of the following craters would have the butterfly ejecta pattern.

    Aveiro crater (shown in the front of the above illustration)

    • A (area) = 67 km2
    • P (perimeter) = 30 km

    Unnamed crater (shown in the back of the above illustration)

    • A (area) = 32 km2
    • P (perimeter) = 21 km

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  • Epic Eclipse

    When sunlight hits the moon (which has a radius of 1,738 km), a cone-shaped shadow is created. During the total solar eclipse on August 21, 2017, the distance from the center of the moon to the center of Earth (which has a radius of 6,378 km) will be 372,027 km. On that day, if the moon’s shadow were not intersected by the surface of Earth, it would extend 377,700 km from the moon to its vertex.

    Viewers on Earth who want to witness the eclipse will have to be at a location inside this shadow as it passes over Earth to see the eclipse at totality. What is the approximate surface area of Earth that will be covered by the disc of the moon’s shadow at any one time during the eclipse?

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  • Finale Fanfare

    In 2017, after more than 12 years at Saturn, the Cassini mission will come to an end with a plunge into Saturn. The finale is designed to keep Cassini from impacting and possibly contaminating any of Saturn’s scientifically intriguing moons. First, mission operators have planned a daring series of orbits that will take Cassini closer to Saturn than ever before. Cassini will use the gravity of Saturn’s moon Titan to alter its trajectory and fly into the gap between Saturn and its rings. It all begins with a flyby of Titan on April 22, putting Cassini on a new orbital path whose first apoapsis is on April 23. Then, it will complete 22 elliptical orbits with an average periapsis altitude of 63,022 km and an average apoapsis altitude of 1,274,828 km. A final flyby of Titan will place Cassini on a half-orbit trajectory for Saturn impact.

    Use Kepler’s third law below to find approximately how many days each orbit will take. Approximately what day will Cassini dive into Saturn’s atmosphere?

    Kepler's third law formula
    • µcb (gravitational parameter of Saturn) = 3.7931187 x 1016 m3/s2
    • aSC = semi-major axis of Cassini's orbit
    • TSC = orbital period of Cassini

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  • Habitable Hunt

    Scientists can learn a lot about planets beyond our solar system by studying their stars. They can calculate an exoplanet’s orbital period by measuring how often its star dims as the planet passes by. They can even find potentially habitable worlds with a few key details. The star’s temperature and luminosity, which are related to its mass, define its habitable zone, the area where liquid water can exist. And the bond albedo, or percentage of light reflected by the exoplanet, helps estimate its temperature.

    Scientists recently discovered seven Earth-like planets orbiting the star TRAPPIST-1. Given TRAPPIST-1’s measurements below, what are the inner and outer radii (r), in AU, of its habitable zone? Use the formula below.

    Habitable Zone formula

    TRAPPIST-1 system:

    • L* (star luminosity) = 2.0097x1023 watts
    • µcb (star gravitational parameter) = 1.06198x1019 m3/s2
    • σ (Stefan-Boltzmann constant) = 5.67×10-8 Wm-2K-4
    • T (planetary temperature) = 192-295 K
    • A (planetary bond albedo) = 0.3

    Given the orbital periods (Tp), for TRAPPIST-1’s planets below, which are in the habitable zone? Use Kepler’s third law below to find the semi-major axis of each orbit (ap).

    Kepler's third law formula

    Orbital periods:

    • TRAPPIST-1b = 1.51087081 days
    • TRAPPIST-1c = 2.4218233 days
    • TRAPPIST-1d = 4.049610 days
    • TRAPPIST-1e = 6.099615 days
    • TRAPPIST-1f = 9.206690 days
    • TRAPPIST-1g = 12.35294 days
    • TRAPPIST-1h = 20 days

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  • Hazy Halo

    With its methane lakes and hazy atmosphere reminiscent of a primordial Earth, Saturn’s moon Titan is an intriguing world – and one that scientists believe may harbor ingredients for life. Though spacecraft have studied Titan up close, and the Cassini mission sent a probe to the surface, much of the moon remains a mystery because a dense, 600-km thick atmosphere masks its rocky surface. To study Titan in more detail, scientists have proposed developing a spacecraft to map the surface of this mysterious moon.

    Given Titan’s radius of 2,575 km, what percentage of the moon’s makeup by volume is atmospheric haze?

    If scientists hope to create a global map of Titan, what is the surface area that a future spacecraft would need to map? Downloads & Related Links

  • Round Recon

    The Mars Reconnaissance Orbiter (MRO) has been zipping around Mars since 2006, collecting data and images that have led to exciting discoveries about the Red Planet. So scientists can get the data and images they need from MRO, they must know when the spacecraft (traveling in a near-circular, near-polar orbit at an average speed of 3.42 km per second) will reach certain locations around Mars.

    Given that Mars has a polar diameter of 6,752 km and MRO comes as close to the planet as 255 km at the south pole and 320 km at the north pole, how far does MRO travel in one orbit*?

    How long does it take MRO to complete one orbit?

    How many orbits does MRO complete in one Earth day?

    *MRO’s orbit is near enough to circular that the formulas for circles can be used.

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  • Sun Screen

    A transit occurs when a planet passes in front of the disk of a star. As seen from Earth, only Mercury and Venus transit our star, the sun. During a transit, there is a slight dip in the amount of solar energy reaching Earth, which can be found using this equation:

    B% = 100 (

    πr^2/πR^2

    B = percentage drop in the brightness of the sun

    r = the radius of the planet as it appears from Earth (in arcseconds)

    R = the radius of the sun as it appears from Earth (in arcseconds)

    With many solar-powered satellites orbiting Earth, it’s important to know what impact a dip in solar energy might have.

    If 1,360.8 w/m2 of solar energy reaches the top of Earth’s atmosphere, how many fewer watts reach Earth when Mercury (diameter = 12 arcseconds) transits the sun (diameter = 1,909 arcseconds)?

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  • Gravity Grab

    The Juno spacecraft is hurtling toward Jupiter. At closest approach, it will reach a velocity of 57.98 km per second relative to the planet. To get into orbit around Jupiter, Juno will have to brake at just the right time to be pulled in by Jupiter’s gravity or miss its target completely.

    By how much does Juno need to change its velocity relative to Jupiter to get into a 53.5-day orbit around the planet?

    Use these equations to approximate a solution assuming Juno could instantaneously decelerate at perijove:

    T = 2π√(

    a^3/µ

    E =

    (-µ/2a) = (v^2/2) - (µ/r)

    T = orbital period (in seconds)

    E = total orbital energy

    a = semi-major axis of the orbit (in km)

    µ = gravitational parameter for Jupiter (126,686,536 km3/sec2)

    v = velocity of Juno relative to Jupiter after deceleration

    r = radius of Juno at perijove (76,006 km)

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  • Mars Marathon

    The Mars Exploration Rover Opportunity has been driving on the Red Planet for more than 11 years -- not bad for a mission only planned to last for three months! Opportunity has already beat the off-Earth driving distance record of 39 kilometers and is approaching a marathon distance: 42.195 kilometers.

    When Opportunity reaches the marathon mark, how many times will its 25-centimeter diameter wheels have rotated?

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  • Pixel Puzzler

    The Dawn spacecraft is orbiting Ceres -- a nearly spherical dwarf planet with an average radius of 475 kilometers -- in a perfectly circular polar orbit. While in orbit, Dawn will snap images of Ceres’ surface to piece together a global map. From its lowest altitude orbit of 370 kilometers, Dawn’s camera can see a patch of Ceres about 26 kilometers on a side.

    Assuming no overlap in the images, how many photographs would Dawn have to take to fully map the surface of Ceres?

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  • Frozen Formula

    Scientists have good reason to believe that Jupiter’s moon Europa has a liquid ocean wedged between its ice shell and a rocky sea floor. Though it has a known radius of 1,561 kilometers — slightly smaller than Earth’s moon — uncertainty exists about the exact thickness of Europa’s ice shell and the depth of its ocean.

    Assuming Europa’s ice shell is between 2 and 30 kilometers thick and its ocean is between 3.5 and 100 kilometers deep, what is the minimum and maximum volume of its ocean?

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  • Hear Here

    The twin Voyager spacecraft, which launched in 1977, are the most distant human-made objects in space. It takes more than 18 hours for a signal from the 12.5-watt X-band transmitter on Voyager 1 to reach Earth, nearly 131 astronomical units away (one astronomical unit, AU, is equal to about 150,000,000 kilometers). The Voyager high-gain antenna, a circular parabolic reflector, transmits a circular radio signal about 0.5 degrees wide.

    At the current distance, what fraction of the Voyager 1 radio beam is received on Earth by a 70-meter-diameter antenna at NASA’s Deep Space Network (DSN)?

    How many of the original 12.5 watts are received by the DSN antenna?

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  • Satellite Solver

    The Soil Moisture Active Passive, or SMAP, satellite is designed to image 1,000-kilometer-wide swaths of Earth from a near-polar, sun-synchronous orbit 685 kilometers above Earth’s surface.

    How many days will it take SMAP to image all of Earth’s surface?

    *You may disregard any overlap that may occur

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  • Roving Riddle

    The Curiosity Mars rover doesn’t have an odometer like those found in cars, so rover drivers calculate how far the rover has driven based on wheel rotations. Since landing on Mars in August 2012, Curiosity’s 50-centimeter-diameter wheels have rotated 3689.2 times in 568 sols (Martian days).

    How many kilometers has Curiosity traveled?

    Loose sand, dirt, slopes and rocks can influence the rover’s progress, so engineers use a technique called visual odometry to determine how much Curiosity’s wheels are slipping. On a steep slope covered in loose dirt, engineers note that the distance between the rover’s visual odometry markers is only 143 centimeters.

    What percent are Curiosity’s wheels slipping with each rotation?

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  • Jupiter Jockey

    NASA's Juno spacecraft orbits Jupiter in a highly eccentric orbit, allowing very close passes of the spacecraft to the planet. In one orbit, Juno gets as close as 75,800 kilometers (perijove) to Jupiter and passes as far as 2,771,000 kilometers (apojove) from Jupiter.

    How many kilometers does Juno travel in one orbit?

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  • Flying Formula

    The Cassini spacecraft was launched to Saturn with its 28-inch spherical hydrazine tank filled to 69 percent of its volume with hydrazine. After many years of studying Saturn, 82 kilograms of hydrazine were used to maneuver around the ringed planet.

    Given the density of hydrazine is 1.02 grams/cubic centimeter, how much fuel remained in the tank at this time?

    *Assume no fuel is sitting in the fuel lines

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Lesson Last Updated: Oct. 11, 2024