# National Aeronautics and Space Administration

## Mathematics Problems Dealing with Astrobiology

Problem 395: Death Stars Some stars create super-flares that are capable of eliminating life on planets that orbit close to the star. Students learn about these flares on common red-dwarf stars and compare them to flares on our own sun [Grade: 6-9 | Topics: Scientific Notation; percentages; rates of change] [Click here]

Problem 392: Exploring the DNA of an organism based upon arsenic. Students estimate the increase in the mass of the DNA from an arsenic-loving bacterium in which phosphorus atoms have been replaced with arsenic. [Grade: 8-10 | Topics: integer math; percentages] [Click here]

Problem 61: Drake's Equation and the Search for Life...sort of! Way back in the 1960's Astronomer Frank Drake invented an equation that helps us estimate how much life, especially the intelligent kind, might exist in our Milky Way. It has been a lively topic of discussion in thousands of college astronomy courses for the last 30 years. In this simplified version, your students will get to review what we now know about the planetary universe, and come up with their own estimates. The real fun is in doing the research to track down plausible values (or their ranges) for the factors that enter into the equation, and then write a defense for the values that they choose. Lots of opportunity to summarize basic astronomical knowledge towards the end of an astronomy course, or chapter. [Grade level: 6-8 | Topics: decimal math; evaluating functions for given values of variables] [Click here]

## The Search for Earth-like Planets

Problem 492: Alpha Centauri Bb - a nearby extrasolar planet? Students plot data for the orbiting planet and determine its orbit period. They use this in a simple formula to determine its distance, then they estimate its surface temperature at this distance. [Grade: 9-12 | Topics: graphing periodic data; finding periods; evaluating simple formulae ] [Click here]

Problem 465: Comparing Planets Orbiting other Stars Students use simple fraction arithmetic to determine the relative sizes of several new planets recently discovered by the Kepler mission, and compare these sizes to that of Jupiter and Earth. [Grade: 3-5 | Topics: scale models; proportions; fractions] [Click here]

Problem 458: Playing Baseball on the Earth-like Planet Kepler-22b! The recently-confirmed Earth-like planet Kepler-22b by the Kepler Observatory is a massive planet orbiting its star in the temperature zone suitable for liquid water. This problem explores the gravity and mass of this planet, and some implications for playing baseball on its surface! [Grade: 8-10 | Topics: scale models; proportions; scientific notation; metric math; Evaluating equations] [Click here]

Problem 441: Exploring the new planet Kepler 16b called 'Tatooine' Using the tangent function, students estimate the angular diameter and separation of the two stars in the Kepler 16 binary system as viewed from the planet's surface...if it had one!! [Grade: 8-10 | Topics: angle measure; tangent] [Click here]

Problem 405: Discovering Earth-like Worlds by their Color Students use recent measurements of the reflected light from solar system bodies to graph their colors and to use this in classifying new planets as Earth-like, moon-like or Jupiter-liike [Grade: 6-8 | Topics: graphing tabular data; interpreting graphical data] [Click here]

Problem 402: Kepler- Earth-like planets by the score! II Students use recent Kepler satellite data summarized in tabular form to estimate the number of planets in the Milky Way galaxy that are about the same size as our Earth, and located in their Habitable Zones were liquid water may exist. [Grade: 6-8 | Topics: Percentage; re-scaling sample sizes] [Click here]

Problem 401: Kepler - Earth-like planets by the score! I Students use recent Kepler satellite data to estimate the number of Earth-like planets in the Milky Way galaxy. [Grade: 6-8 | Topics: Percentage; histograms; Re-scaling sample sizes] [Click here]

Problem 396: Kepler 10b - A matter of gravity Students use the measured properties of the Earth-like planet Kepler 10b to estimate the weight of a human on its surface. [Grade: 8-10 | Topics: Evaluating formulas; mass = density x volume; volume of a sphere; scientific notation] [Click here]

Problem 376: The Earth-like Planet Gliese 518g Students use data for the Gliese 581 planetary system to draw a scaled model of the locations and sizes of the discovered planets. They also identify the location and span of the Habitable Zone for this planetary system. [Grade: 3-5 | Topics: scale models; measurement] [Click here]

Problem 360: Kepler's First Look at 700 Transiting Planets A statistical study of the 700 transits seen during the first 43 days of the mission. [Grade: 6-8 | Topics: Percentages; area of circle] [Click here]

Problem 333: Hubble: Seeing a Dwarf Planet Clearly Based on a recent press release, students use the published photos to determine the sizes of the smallest discernible features and compare them to the sizes of the 48-states in the USA. They also estimate the density of Pluto and compare this to densities of familiar substances to create a 'model' of Pluto's composition. A supplementary Inquiry Problem asks students to model the interior in terms of two components and estimate what fraction of Pluto is composed of rock or ice. [Grade: 8-12 | Topics: scales and ratios; volume of sphere; density=mass/volume] [Click here]

Problem 331: Webb Space Telescope: Detecting dwarf planets The 'JWST' will be launched some time in 2021-22. One of its research goals will be to detect new dwarf planets beyond the orbit of Pluto. In this problem, students use three functions to predict how far from the sun a body such as Pluto could be detected, by calculating its temperature and the amount of infrared light it emits. [Grade: 9-12 | Topics: Evaluating square-roots and base-e exponentials] [Click here]

Problem 325: Kepler Spies Five New Planets Students count squares on a Bizarro Star to study the transit of a planet, and determine the diameter of the planet. This demonstrates the basic principle used by NASA's Kepler satellite to search for Earth-sized planets orbiting distant stars. [Grade: 4-6 | Topics: Counting; graphing; area of a square] [Click here]

Problem 213: Kepler: The hunt for Earth-like planets Students compare the area of a star with the area of a planet to determine how the star's light is dimmed when the planet passes across the star as viewed from Earth. This is the basis for the 'transit' method used by NASA's Kepler satellite to detect new planets. [Grade: 6-8 | Topics: Area of circle; ratios; percents.] [Click here]

Problem 197: Hubble Sees a Distant Planet Students study an image of the dust disk around the star Fomalhaunt and determine the orbit period and distance of a newly-discovered planet orbiting this young star. [Grade: 6-10| Topics: Calculating image scales; Circle circumferences; Unit conversions; distance-speed-time] [Click here]

Problem 168: Fitting Periodic Functions - Distant Planets Students work with data from a newly-discovered extra-solar planet to determine its orbit period and other parameters of a mathematical model. [Grade: 9-12 | Topics: trigonometry; functions; algebra] [Click here]

Problem 160: The Relative Sizes of the Sun and Stars Students work through a series of comparisons of the relative sizes of the sun compared to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models] [Click here]

Problem 156: Spectral Classification of Stars Students use actual star spectra to classify them into specific spectral types according to a standard ruberic. [Grade: 5-8 | Topics: Working with patterns in data; simple sorting logic] [Click here]

Problem 155: Tidal Forces: Let 'er rip! Students explore tidal forces and how satellites are destroyed by coming too close to their planet. [Grade: 7-10| Topics: Algebra; number substitution] [Click here]

Problem 141: Exploring a Dusty Young Star Students use Spitzer Space Telescope data to learn about how dust emits infrared light and calculate the mass of dust grains from a young star in the nebula NGC-7129. [Grade: 4 - 7 | Topics: Algebra I; multiplication, division; scientific notation] [Click here]

## Planetary Atmospheres and Composition

Problem 545:Measuring Atmospheric Trace Gases Using Parts Per Million Students convert from percentage units to parts-per-million and compare trace gases in the atmospheres of various planets. [Grade: 6-8 | Topics: percentages; unit conversions ] [Click here]

Problem 544:The Composition of Planetary Atmospheres Students study the composition of planetary atmospheres and compare the amounts of certain compounds in them [Grade: 6-8 | Topics: Pie graphs; percentages; scientific notation] [Click here]

Problem 391: Investigating the atmosphere of Super-Earth GJ-1214b Students investigate a simple model for the interior of an exoplanet to estimate the thickness of its atmosphere given the mass size and density of the planet. [Grade: 6-8 | Topics: graphing functions; evaluating functions for given values; volume of a sphere; mass = densityxvolume] [Click here]

Problem 352: Exponential Functions and Atmospheric 'Scale heights' A study of the way a planet's atmosphere changes as its temperature is changed using exponential functions. [Grade: 9-12 | Topics: Scientific Notation; evaluating exponential functions; Optional calculus] [Click here]

Problem 335: Methane Lakes on Titan Students use a recent Cassini radar image of the surface of Titan to estimate how much methane is present in the lakes that fill the image, and compare the volume to that of the fresh water lake, Lake Tahoe. [Grade: 6-8 | Topics: estimating irregular areas; calculating volume from area x height; scaled images ] [Click here]

Problem 332: Hubble: The Changing Atmosphere of Pluto Based on a recent press release, students determine the aphelion and perihelion of Pluto's elliptical orbit using the properties of ellipses, then calculate the temperature of Pluto at these distances to estimate the thickness of Pluto's atmosphere and its changes during its orbit around the sun. [Grade: 10-12 | Topics: properties of ellipses; evaluating an algebraic function ] [Click here]

Problem 278: Spitzer Studies the Distant Planet Osiris Students learn about the density of the planet HD209458b, also called Osiris, and compare it to that of Jupiter. [Grade: 8-10 | Topics: Spherical volumes; density; Scientific Notation;] [Click here]

Problem 181: Extracting Oxygen from Moon Rocks Students use a chemical equation to estimate how much oxygen can be liberated from a sample of lunar soil. [Grade: 9-11| Topics: ratios; scientific notation; unit conversions] [Click here]

Problem 124: The Moon's Atmosphere! Students learn about the moon's very thin atmosphere by calculating its total mass in kilograms using the volume of a spherical shell and the measured density. [Grade: 8-10 | Topics:volume of sphere, shell; density-mass-volume; unit conversions] [Click here]

## Water and Habitable Zones

Problem 403: The Goldilocks Planets - Not too hot or cold Students use a table of the planets discovered by the Kepler satellite, and estimate the number of planets in our Milky Way galaxy that are about the same size as Earth and located in their Habitable Zones. They estimate the average temperature of the planets, and study their tabulated properties using histograms. [Grade: 6-8 | Topics: Averaging; histogramming] [Click here]

Problem 350: Estimating the Temperatures of Exoplanets Students review the basic properties of ellipses by exploring the orbits of newly-discovered planets orbiting other stars. They also use a simple formula to determine the temperatures of the planets from their orbits.[Grade: 9-12 | Topics: Equation of ellipse; evaluating functions] [Click here]

Problem 349: Exoplanet Orbits and the Properties of Ellipses Given the formula for the orbits of newly-discovered planets, students determine the basic properties of the elliptical orbits for the planets. [Grade: 9-12 | Topics: Properties of ellipses] [Click here]

Problem 338: Asteroids and Ice Students calculate how much ice may be present on the asteroid 24-Themis based on discoveries by NASA [Grade: 9-12 | Topics: mass=densityxvolume; volume of a spherical shell] [Click here]

Problem 292: How Hot is That Planet? Students use a simple function to estimate the temperature of an exoplanet called CoRot-7b. [Grade: 8-10 | Topics: Algebra II; Evaluating Power functions] [Click here]

Problem 287: LCROSS Sees Water on the Moon Students use information about the plume created by the LCROSS impactor to estimate the (lower-limit) concentration of water in the lunar regolith in a shadowed crater. [Grade: 9-12 | Topics: Geometry; volumes; mass=density x volume] [Click here]

Problem 275: Water on the Moon! Students estimate the amount of water on the moon using data from Deep Impact/EPOXI and NASA's Moon Minerology Mapper experiment on the Chandrayaan-1 spacecraft. [Grade: 8-10 | Topics: Geometry, Spherical volumes and surface areas, Scientific notation] [Click here]

Problem 267: Identifying Materials by their Reflectivity The reflectivity of a material can be used to identify it. This is important when surveying the lunar surface for minerals, and also in creating 'green' living environments on Earth. [Grade: 6-8 | Topics: percentage, interpreting tabular data, area ] [Click here]

Problem 264: Water on Planetary Surfaces Students work with watts and Joules to study melting ice. [Grade: 8-10 | Topics: unit conversion, rates] [Click here]

Problem 263: Ice or Water? Whether a planetary surface contains ice or liquid water depends on how much heat is available. Students explore the concepts of Specific heat and Latent Heat of Fusion to better understand the and quantify the energy required for liquid water to exist under various conditions. [Grade: 8-10 | Topics: unit conversion, scientific notation] [Click here]

Problem 222: Kelvin Temperatures and Very Cold Things Students convert from Celsius to Fahrenheit and to Kelvin using three linear equations. [Grade: 5-8 | Topics: Evaluating simple linear equations for given values.] [Click here]

Problem 189: Stellar Temperature, Size and Power Students work with a basic equation to explore the relationship between temperature, surface area and power for a selection of stars. [Grade: 8-10| Topics: Algebra] [Click here]

Problem 170: Measuring Star Temperatures Students use a simple formula to determine the temperatures of stars, and to use a template curve to analyze data for a specific star to estimate its temperature. [Grade: 6-8 | Topics: algebra, graph analysis] [Click here]

Problem 121: Ice on Mercury? Since the 1990's, radio astronomers have mapped Mercury. An outstanding curiosity is that in the polar regions, some craters appear to have 'anomalous reflectivity' in the shadowed areas of these craters. One interpretation is that this is caused by sub-surface ice. The MESSENGER spacecraft hopes to explore this issue in the next few years. In this activity, students will measure the surface areas of these potential ice deposits an calculate the volume of water that they imply. [Grade: 8-10 | Topics:Area of a circle; volume, density, unit conversion] [Click here]

## Planetary Formation and Properties

Problem 543:Timeline for Planet Formation Students calculate time intervals in millions and billions of years from a timeline of events [Grade: 3-5 | Topics: time calculations; integers] [Click here]

Problem 542:The Late Heavy Bombardment Era Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth durung this era. [Grade: 6-8 | Topics: volume of spheres; proportions] [Click here]

Problem 541:How to Build a Planet Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres. [Grade: 3-5 | Topics: graphing; counting] [Click here]

Problem 353: Dwarf Planets and Kepler's Third Law Students plot the distance versus period relationship for planets and minor bodies in the solar system and fit it to two functions to determine Kepler's Third Law. [Grade: 9-12 | Topics: Fitting functions to data; Evaluating a polynomial] [Click here]

Problem 305: From Asteroids to Planets Students explore how long it takes to form a small planet from a collection of asteroids in a planet-forming disk of matter orbiting a star based on a very simple physical model. [Grade: 11-12 | Topics: Integral calculus] [Click here]

Problem 304: From Dust Balls to Asteroids Students calculate how long it takes to form an asteroid-sized body using a simple differential equation based on a very simple physical model. [Grade: 11-12 | Topics: Integral Calculus] [Click here]

Problem 303: From Dust Grains to Dust Balls Students create a model of how dust grains grow to centimeter-sized dust balls as part of forming a planet based on a very simple physical model. [Grade: 11-12 | Topics: Integral Calculus] [Click here]

Problem 302: How to Build a Planet from the Inside Out Students model a planet using a spherical core and shell with different densities. The goal is to create a planet of the right size, and with the correct mass using common planet building materials. [Grade: 9-11 | Topics: Geometry; volume; scientific notation; mass=density x volume] [Click here]

Problem 84: Beyond the Blue Horizon How far is it to the horizon? Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: Differential calculus) ] [Click here]

Problem 60: When is a planet not a planet? In 2003, Dr. Michael Brown and his colleagues at CalTech discovered an object nearly 30% larger than Pluto, which is designated as 2003UB313. It is also known unofficially as Xenia, after the famous TV Warrior Princess! Is 2003UB313 really a planet? In this activity, students will examine this topic by surveying various internet resources that attempt to define the astronomical term 'planet'. How do astronomers actually assign names to astronomical objects? Does 2003UB313 deserve to be called a planet, or should it be classified as something else? What would the new classification mean for asteroids such as Ceres, or objects such as Sedna, Quaoar and Varuna? [Grade level: 6-8 | Topics: Non-mathematical essay; reading to be informed] [Click here]

Problem 59: Getting A Round in the Solar System! How big does a body have to be before it becomes round? In this activity, students examine images of asteroids and planetary moons to determine the critical size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives this activity can be expanded to include dozens of small bodies in the solar system to enlarge the research data for this problem. Only a few example images are provided, but these are enough for the student to get a rough answer! [Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing] [Click here]

Problem 8: Making a Model Planet Students use the formula for a sphere, and the concept of density, to make a mathematical model of a planet based on its mass, radius and the density of several possible materials (ice, silicate rock, iron, basalt). [Grade: 7 - 9 | Topics: Volume of sphere; mass = density x volume; decimal math; scientific notation] [Click here]