## Mathematics Problems Dealing with Astrobiology

Although astronomers have considered the possibility of life existing elsewhere in the universe as an open question, only in the last 10 years has this issue been persued aggressively. Since tne debut of Carl Sagan and I.S. Shklovskii's book 'Intelligent Life in the Universe' in 1968 made this topic worthy of serious scientific study, the subject of 'astrobiology' has steadily grown in complexity. NASA's Viking Landers in 1976 were designed with the objective of seeing whether the dry surface of mars still contained vestigal chemistries indicating living systems at work. Although the negative results were a dissappointment, the subject refused to fade away.

Despite the dearth of evidence that other stars even harbored planetary systems, the steady improvement of telescopes and technology first began to uncover huge numbers of 'circumstellar disks' of dust and rocky materials around thousands of young stars in the 1980s and 1990s, to the point where these 'protoplanetary disks' were regarded as an inevitable stage in the formation of normal stars. Following close on the heals of this discovery, sophisticated computer simulations of what these disks become over time made it very clear that we should expect to find plenty of large orbiting bodies - planets - embedded in these circumstellar disks.

By the end of the first decade of the 21st Century, careful studies of other stars using sophisticated photometers and spectroscopes revealed over 350 of these exoplanets orbiting hundreds of nearby stars. Among these were a small number of planets whose orbits allowed the planets to cross directly across the face of their star as viewed from Earth. These transiting exoplanets can come in all sizes, not just the massive jupiter-sized planets so easy to detect by their gravity. They could also be studied to learn about the composition of their atmospheres.

NASA's Kepler mission specializes in detecting planetary transits, and in the comming years will find thousands of these planets, including perhaps a few hundred similar in size to our own Earth.

Meanwhile, astrobiologists study the conditions that lead to living systems, and the tell-tale fingerprints that can give them away. Basic to this is the existence of liquid water, and the chemicals and elements needed to form complex organic molecules. Planetary biospheres can be detected because they generate free oxygen that can be detected in a planetary atmosphere. Advanced life forms that have radio technology can be detected by 'leakage radiation' from thousands of light years away. Both approaches are now in full swing and may reveal profound surprises in the coming decades!

## The Search For Extraterrestrial Life

**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]
(PDF)

**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]
(PDF)

**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]

## 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 ]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)
**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]
(PDF)

**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]
(PDF)

**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]

**Problem 331: Webb Space Telescope: Detecting dwarf planets** The 'JWST' will be launched some time in 2014. 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]

**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]

**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.]

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

**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]

**Problem 160: The Relative Sizes of the Sun and Stars** Students work through a series of comparisons of the relative sizes of the sun compablack 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]

**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

**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]

**Problem 141: Exploring a Dusty Young Star**
Students use Spitzer satellite 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]

## 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 ]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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 ]

**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 ]

**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;]

**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]

**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]

## 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]
(PDF)

**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]
(PDF)

**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]
(PDF)

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

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

**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]

**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]

**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 ]

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

**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]

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

**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]

**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]

**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]

## 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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]
(PDF)

**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]

**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]

**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]

**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]

**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) ]

**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]

**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]

**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]