## Astronomy and Space Science Applications Featuring Geometry

## Properties of Angles and Angular Measure

**Problem 299: Changing Perspectives on the Sun's Diameter** Students compare two images of the sun taken by the SOHO satellite to measure the apparent diameter
change from different earth obit locations in the winter and summer.
[Grade: 6-8 | Topics: measurement; parallax; metric units; percentage change]

**Problem 296: Getting an Angle on the Sun and Moon** Students explore angular size and scale by comparing two images of the sun and moon which have identical
angular size, but vastly different scales.
[Grade: 8-10 | Topics: Geometry; angle measure; scale; proportion]

**Problem 196: Angular Size and velocity-** Students study a spectacular photo of the ISS passing across the face of the sun, and work out the angular sizes and speeds of the transit to figure out how long the event took in order to photograph it.
[Grade: 8-10| Topics: Geometry; Angle measurement]

**Problem 118 **** An Application of the Parallax Effect **
The STEREO mission views the sun from two different locations in space. By combining this data, the parallax
effect can be used to determine how far above the solar surface various active regions are located. Students use the Pythagorean Theorem, a bit of
geometry, and some actual STEREO data to estimate the height of Active Region AR-978.
[Grade: 8-10 | Topics:Pythagorean Theorem; square-root; solving for variables]

**Problem 250: The Most Important Equation in Astronomy** Students learn about how an instrument's ability to see details depends on its size and its operating wavelength - the key to
designing any telescope or camera.
[Grade: 8-10 | Topics: geometry, angle measure, scientific notation]

**Problem 196: Angular Size and velocity-** Students study a spectacular photo of the ISS passing across the face of the sun, and work out the angular sizes and speeds of the transit to figure out how long the event took in order to photograph it.
[Grade: 8-10| Topics: Geometry; Angle measurement]

**Problem 144 **** Exploring Angular Size **
Students examine the concept of angular size and how it relates to the physical size of
an object and its distance. A Chandra Satellite x-ray image of the star cluster NGC-6266 is used, along with its distance, to
determine how far apart the stars are based on their angular separations.
[Grade: 7 - 10 | Topics:Scientific Notation; degree measurement; physical size=distance x angular size.]

## Properties of Triangles

**Problem 92 ****A Lunar Transit of the Sun from Space - **
One of the STEREO satellites observed the disk of the moon pass across the sun. Students will use simple geometry to determine how far
the satellite was from the moon and Earth at the time the photograph was taken.
[Grade level: 8-10 | Topics: Geometry; parallax; arithmetic]

**Problem 11 ****How high is an aurora? ** Students use the properties of a triangle to
determine how high up aurora are. They also learn about the parallax
method for finding distances to remote objects.
[Grade: 5 - 8 | Topics: Geometery; angle measure]

**Problem 298: Seeing Solar Storms in STEREO - II** Students explore the geometry of stereo viewing by studying a solar storm viewed from two satellites.
[Grade: 10-12 | Topics: Geometry; Trigonometry]

**Problem 118 **** An Application of the Parallax Effect **
The STEREO mission views the sun from two different locations in space. By combining this data, the parallax
effect can be used to determine how far above the solar surface various active regions are located. Students use the Pythagorean Theorem, a bit of
geometry, and some actual STEREO data to estimate the height of Active Region AR-978.
[Grade: 8-10 | Topics:Pythagorean Theorem; square-root; solving for variables]

**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 44 ****Interstellar Distances with the Pythagorean Theorem - **
If you select any two stars in the sky and calculate how far apart they are,
you may discover that even stars that appear to be far apart are actually close neighbors in space. This activity
lets students use the Pythagorean distance formula in 3-dimensions to explore stellar
distances for a collection of bright stars, first as seen from Earth and then as seen from a planet
orbiting the star Polaris. Requires a calculator and some familiarity with algebra and square-roots.
[Grade level: 9-11 | Topics: Decimal math; Pythagorean Theorem; square root]

**Problem 19 ****An Application of the Pythagorean
Theorem ** Students learn that the
Pythagorean Theorem is more than a geometric concept. Scientists use it
all the time when calculating lengths, speeds or other quantities. This
problem is an introduction to magnetism, which is a '3-dimensional
vector', and how to calculate magnetic strengths using the Pythagorean
Theorem. [Grade: 8 - 10 | Topics: Squares and square-roots; Pythagorean Theorem in 3-D]

## Areas and Volumes

**Problem 16 ****Solar Power and Satellite Design ** Students perform simple surface area calculations to
determine how much solar power a satellite can generate, compared to the
satellite's needs. [Grade: 6 - 8 | Topics: Area of irregular polygons]

**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 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 316: Counting Craters on the Hubble Space Telescope** Students count craters on a piece of the Wide Field Planetary
Camera recovered from the Hubble Space Telescope in 2009. They determine the cratering rate and use this to predict
how many impacts the solar panels on the International Space Station experiences each day.
[Grade: 6-9 | Topics: Counting; Area; density]

**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 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 272: Spitzer Telescope Discovers New Ring of Saturn! ** Students calculate the volume of the ring and compare it to the volume of Earth to check a news release figure that
claims 1 billion Earths could fit inside the new ring.
[Grade: 8-9 | Topics: Geometry, Algebra, volumn, scientific notation]

**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 177: Lunar Cratering: Probability and Odds-** Students work with crater counting to estimate the area coveblack by craters and how to convert this into impact probabilities.
[Grade: 4-7| Topics: Area; probability]

**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]

**Problem 96 ****Hinode Satellite Power - **
Students will study the design of the Hinode solar satellite and calculate how much power it can generate from its solar panels.
[Grade level: 6-8 | Topics:area of rectangle,area of cylinder, unit conversion]

**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 2 ****Satellite Surface Area **Students calculate the surface area of an octagonal
cylinder and calculate the power it would yield from solar cells covering
its surface. [Grade: 7 - 9 | Topics: surface areas; hexagone; decimal math]

**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 283: Chandra Observatory Sees the Atmosphere of a Neutron Star ** Students determine the mass of the carbon atmosphere of the neutron star Cas-A.
[Grade: 8-10 | Topics: Volume of spherical shell; mass = density x volume]

**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 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 272: Spitzer Telescope Discovers New Ring of Saturn! ** Students calculate the volume of the ring and compare it to the volume of Earth to check a news release figure that
claims 1 billion Earths could fit inside the new ring.
[Grade: 8-9 | Topics: Geometry, Algebra, volumn, scientific notation]

**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]

**Problem 115 ****A Mathematical Model of the Sun **
Students will use the formula for a sphere and a shell to calculate the mass of the
sun for various choices of its density. The goal is to reproduce the measured mass and radius of the sun by
a careful selection of its density in a core region and a shell region. Students will manipulate the values for density and shell size to
achieve the correct total mass. This can be done by hand, or by programming
an Excel spreadsheet.
[Grade: 8-10 | Topics: scientific notation; volume of a sphere and a spherical shell; density, mass and volume.]

**Problem 104 ****Loopy Sunspots! **
Students will analyze data from the Hinode satellite to determine the volume and mass of a
magnetic loop above a sunspot. From the calculated volume, based on the formula for the volume of a cylinder, they will use the density of the plasma determined by
the Hinode satellite to determine the mass in tons of the magnetically trapped material.
[Grade: 9-11 | Topics:image scales; cylinder volume
calculation; scientific notation; unit conversions]

## Similar Triangles and Scales

**Problem 295: Details from an Exploding Star** Students work with an image from the Hubble Space Telescope of the Crab Nebula to calculate scales and sizes of various features.
[Grade: 6-9 | Topics: Scale; measurement; metric units]

**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 103 ****The Mysterious Solar Micro-Flares! **
Students will analyze an image taken by the Hinode
solar satellite to determine the scale of the image in kilometers per millimeter,
then use this to determine the sizes of solar micro-flares.
From the number of micro-flares that they count in the image,
the area of the image in square kilometers, and the surface area
of a spherical sun, they will calculate the total number
of micro-flares on the solar surface.
[Grade: 6-9 | Topics:image scales; area calculation; unit conversions]

**Problem 290: The Apollo-11 Landing Area at High Resolution** Students use recent images made by the LRO satellite to estimate distances, crater sizes, and how many tons of
TNT were needed to create some of the craters by meteor impact.
[Grade: 9-12 | Topics: metric measurement; scaling; A = B/C]

**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 241: Angular Size and Similar Triangles** A critical concept in astronomy is angular size, measured in degrees, minutes or arc-seconds. This is a
review of the basic properties of similar triangles for a fixed angle.
[Grade: 8-10 | Topics: geometry, similar triangles, proportions]

**Problem 259: Mare Nubium And Las Vegas ** Students compare two satellite images taken at the same resolution to appreciate how large lunar features are compared to more familiar objects.
[Grade: 6-8 | Topics: scale, proportion, ratio]

**Problem 258: LRO's First Image of Mare Nubium ** Students examine the first image of this lunar region using the high-resolution camera image provided by the Lunar Reconnaissance Orbiter.
[Grade: 6-8 | Topics: scale, ratio, proportion]

**Problem 257: LRO and the APollo-11 Landing Site** Students examine a map of the Apollo-11 landing area and determine how well various features
will be visible to the Lunar Reconnaissance Orbiter high-resolution camera.
[Grade: 6-8 | Topics: scale, proportion, ratios]

**Problem 256: A High-resolution Satellite Photo ** Students examine a satelite photo of the Tennessee Court House from the GEO-1 satellite and determine the sizes of familiar features in the image.
[Grade: 6-8 | Topics: scale, ratios, proportions' angle measure, triangle geometry]

**Problem 255: Temple-1 - Closeup of a Comet ** Students examine an image of the Comet Temple-1 taken by the Dawn spacecraft to determine feature sizes and other details.
[Grade: 6-8 | Topics: scales, proportions ]

**Problem 240: The Eagle Nebula Close-up** Students measure a Hubble image of the famous Eagle Nebula 'Pillars of Creation' to determine the sizes of
arious features compared to our solar system
[Grade: 6-8 | Topics: scale, proportion, angle measure]

**Problem 237: The Martian Dust Devils** Students determine the speed and acceleration of a martian dust devil from time laps images and information about the scale of the image.
[Grade: 6-8 | Topics: scales; Determining speed from sequential images; V = D/T

**Problem 236: LRO Sees Apollo-11 on the Moon! ** Students use the latest image from the Lunar Reconnaissance Orbiter of the Apollo-11 landing
site to explore lunar features at 1-meter resolution, and determine the solar elevation angle.
[Grade: 6-8 | Topics: scale; ratios; angle measure; right triangles]

**Problem 234: The Hand of Chandra** Students use an image from the Chandra Observatory to measure a pulsar ejecting a cloud of gas.
[Grade: 6-8 | Topics: Scientific Notation; proportions; angle measure]

## Trigonometry

**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]