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National Aeronautics and Space Administration S pacM e aIX th i This collection of activities is based on a weekly series of space science problems distributed to thousands of teachers during the 2012- 2013 school year. They were intended for students looking for additional challenges in the math and physical science curriculum in grades 5 through 12. The problems were created to be authentic glimpses of modern science and engineering issues, often involving actual research data. The problems were designed to be ‘one-pagers’ with a Teacher’s Guide and Answer Key as a second page. This compact form was deemed very popular by participating teachers. For more weekly classroom activities about astronomy and space visit the NASA website, http://spacemath.gsfc.nasa.gov Add your email address to our mailing list by contacting Dr. Sten Odenwald at [email protected] Front and back cover credits: Front) Grail Gravity Map of the Moon -Grail NASA/ARC/MIT; Dawn Chorus - RBSP/APL/NASA; Erupting Prominence - SDO/NASA; Location of Curiosity - Curiosity/JPL./NASA; Chelyabinsk Meteor - WWW; LL Pegasi Spiral - NASA/ESA Hubble Space Telescope. Back) U Camalopardalis (Courtesy ESA/Hubble, NASA and H. Olofsson (Onsala Space Observatory) Interior Illustrations: All images are courtesy NASA and specific missions as stated on each page, except for the following: 20) Chelyabinsk Meteor and classroom (chelyabinsk.ru); 32) diffraction figure (Wikipedia); 39) Planet accretion (Alan Brandon, Nature magazine, May 2011); 44) Beatrix Mine (J.D. Myers, University of Wyoming); 53) Mars interior (Uncredited ,TopNews.in); 89) Earth Atmosphere (NOAA); 90, 91) Lonely Cloud (Henriette, The Cloud Appreciation Society, 2005); 101, 103) House covered in snow (The Author); This booklet was created through an education grant NNH06ZDA001N- EPO from NASA's Science Mission Directorate. Space Math http://spacemath.gsfc.nasa.gov Table of Contents ii Grade Page Acknowledgments i Table of Contents ii Mathematics Topic Matrix iv How to use this Book vi Alignment with Standards vii Teacher Comments viii Fun with Gears and Fractions 3-5 1 RBSP - A New Belt for the Van Allen Belts! 3-5 2 Hinode Observes a Solar Eclipse From Space 3-5 3 Curiosity -Telling Time on Mars 3-5 4 SDO-Solar Flares and the Stormy Sun 3-5 5 Landsat - Exploring Water Use in Kansas 6-8 6 RBSP Spacecraft Hears Dawn Chorus in Space - I 6-8 7 RBSP Spacecraft Hears Dawn Chorus in Space - II 6-8 8 RBSP Spacecraft Hears Dawn Chorus in Space - III 6-8 9 SDO Sees Coronal Rain - Estimating Plasma Speed 6-8 10 A Mathematical Model of Magnetic Field Lines - I 6-8 11 A Mathematical Model of Magnetic Field Lines - II 6-8 12 InSight – The Seismographic Station - Wave Arrival Times 6-8 13 Planck Mission Sees Ancient Universe Clearly 6-8 14 Curiosity Discovers an Ancient Mars River - Elementary 6-8 15 Hubble Extreme Deep Field - Counting Galaxies 6-8 16 Grail Spacecraft Create New Craters on the Moon 6-8 17 Space-X Exploring the Launch of the Falcon 9 6-8 18 Fermi Observatory and the Origin of Cosmic Rays 6-8 19 The Frequency of Large Meteorite Impacts 6-8 20 Kepler - Alpha Centauri Bb - A Nearby Extrasolar Planet 6-8 21 Curiosity Rover on the Move! 6-8 22 Curiosity - Exploring Gale Crater 6-8 23 Hubble - The Expanding Gas Shell of U Camelopardalis 6-8 24 Exploring Density, Mass and Volume Across the Universe 6-8 25 Hubble - A Distant Supernova Remnant Discovered 6-8 26 Landsat - NASA Satellite Takes Pictures of the Salton Sea 6-8 27 Chandra - Giant Gas Cloud in System NGC 6240 8-10 28 Fermi - The Remarkable Gamma Ray Burst GRB 130427A 8-10 29 Hubble - Exploring the Most Distant Galaxies 9-12 30 Grail Satellites Create a Gravity Map of the Moon 9-12 31 Curiosity Uses X-Ray Diffraction to Identify Minerals 9-12 32 Hubble - An Enormous Stellar Spiral Pattern in Space 9-12 33 Space Math http://spacemath.gsfc.nasa.gov TableU of Contents (contd) iii Curiosity Discovers an Ancient Martian River - Advanced 9-12 34 Hubble - Pluto's Fifth Moon 9-12 35 The Radioactive Dating of a Star in the Milky Way 9-12 36 Calculus: The Close Encounter to te Sun of Barnards Star 12 37 The Volume of a Lunar Impact Crater 12 38 How to Grow a Planet or a Raindrop! 12 39 A Mathematical Model of a Magnetic Field Line - III 12 40 Special NASA Mission Section - InSight Scheduling Events in Time for Launch 5-7 41 Exploring a Possible InSight Landing Area on Mars 6-8 42 Comparing the InSight Landing Area to a City Block! 6-8 43 Exploring Temperature Change in Earth's Outer Crust 6-8 44 Exploring the InSight Lander Telemetry Data Flow 6-8 45 Seeing the Martian Surface with IDC 6-8 46 Telling Time on Mars - Earth Days and Mars Sols 6-8 47 The InSight Seismographic Station Solar Power System 6-8 48 The Work Area In Front of the Lander 6-8 49 Marsquake Energy and the Moment Magnitude Scale 8-10 50 Exploring Logarithms and the Richter Magnitude Scale 8-10 51 The Distance to the Martian Horizon 8-10 52 Exploring the Interior of Mars with Spheres and Shells 8-10 53 Exploring the Mass of Mars 8-10 54 Exploring Impacts and Quakes on Mars 8-10 55 Comparing the Heat Output of Mars and Earth 8-10 56 Exploring Heat Flow and Insulation 8-10 57 Radio Communications with Earth -The Earth-Sun Angle 8-10 58 Estimating the Mass of a Martian Dust Devil! 8-10 59 The InSight Seismographic Station - Wave arrival times 8-10 60 Special NASA Mission Section - Juno Solar Electricity for a New Spacecraft 5-7 61 Investigating Juno's Elliptical Transfer Orbit 8-10 62 The Launch of the Juno Spacecraft 8-10 63 The Launch of the Juno Spacecraft - Ascent to Orbit 8-10 64 Solar Energy and the Distance of Juno from the Sun 10-12 65 The Speed of the Juno Spacecraft 10-12 66 Investigating the Elliptical orbit of Juno Around Jupiter 10-12 67 Investigating the Radiation Belts Around Jupiter 10-12 68 Special NASA Mission Section - MMS The MMS Satellites and Solar Power 5-7 69 Some Facts About the MMS Atlas Rocket 5-7 70 Launching the MMS Satellite Constellation into Orbit 6-8 71 Space Math http://spacemath.gsfc.nasa.gov Table of Contents (contd) iv The MMS Launchpad at Cape Canaveral 6-8 72 Stacking Four MMS Satellites Inside the Atlas Nosecone 6-8 73 Volume and Surface Area of an Octagonal MMS Satellite 8-10 74 The MMS Payload Up Close 8-10 75 The Acceleration Curve for the MMS Launch 8-10 76 The Orbit of the MMS Satellite Constellation 8-10 77 The Orbit of the MMS Satellite Constellation - Advanced 10-12 78 Radiation Belts Storm Probes - RBSP Working with Areas of Rectangles and Circles 3-5 79 The RBSP Satellite - Working with Octagons 6-8 80 Radiation Belts Storm Probes - Telemetry Math 6-8 81 Exploring Gas Density in Space 6-8 82 Electricity from Sunlight 6-8 83 Exploring the Outer Atmosphere - Gas Density 6-8 84 Measuring the Mass of Earth with the RBSP Satellites 8-10 85 Exploring the Donut-Shaped van Allen Belts 8-10 86 Estimating the Total Mass of the van Allen Belts 8-10 87 Measuring Earth's Magnetic Field in Space 8-10 88 Exploring the Density of Gas in the Atmosphere 8-10 89 The Science of Clouds- S'COOL Using Proportions to Estimate the Height of a Cloud 6-8 90 Using Proportions to Estimate the Size of a Cloud 6-8 91 Estimating the Mass of a Cloud 6-8 92 Working with Rainfall Rates and Water Volume 6-8 93 Cloud Droplets and Rain Drops 6-8 94 How Clouds Form - Working with Rates of Change 8-10 95 Cloud Cover and Solar Radiation 8-10 96 Cloud Cover, Albedo, Transmission and Opacity 8-10 97 The History of Winter Graphing a Snowflake Using Symmetry 6 98 The Surface Area of a Snowflake 6 99 Snowflake Growth Rates and Surface Area 6 100 Snow to Water Ratios 7 101 Snow Density and Volume 7 102 Snow Density, mass and Roof Failure 7 103 Exploring Energy and Temperature 8 104 Exploring Temperature and States of Matter 8 105 What is a Snowball's Chance on Mars? 8 106 Useful Links 107 Author Comments 108 Space Math http://spacemath.gsfc.nasa.gov v MathematicsU Topic Matrix Topic Problem Numbers 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 Inquiry Technology, rulers Numbers, patterns, X X X X X X X percentages Averages X Time, distance, speed X X X X X X X X X Density, mass, X X X X volume Areas and volumes X X X X X X Scale Drawings, X X X X X X X X X X X X X X X X X X X proportions Geometry, Pythagorean X X X X X X X X X Theorem Scientific X X X X X X X X Notation Unit Conversions X X X X X Fractions X Graph or Table X X X X X X X Analysis Solving for X X X Evaluating Fns X X X X X X Modeling X X X X Probability X Rates/Slopes X Logarithmic Fns Polynomials X Power Fns Exponential Fns Conics Piecewise Fns Trigonometry Integration Differentiation Limits Space Math http://spacemath.gsfc.nasa.gov Mathematics Topic Matrix (cont'd) vi Topic Problem Numbers 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 Inquiry Technology, rulers Numbers, patterns, X X X X percentages Averages Time, distance, speed X X X X X Density, mass, X X volume Areas and volumes X X X X X Scale drawings X X X X X X X X X Geometry, X X X X X X X X Pythagorean Theorem Scientific Notation X X X X X X X X X X Unit X X X X X X Conversions Fractions Graph or Table Analysis X X X X Solving for X X X X X Evaluating Fns X X X X X X X X X X Modeling X X X X X X X X X X Probability Rates/Slopes X X X X X Logarithmic X X Fns Polynomials X Power Fns Exponential Fns X Conics X Piecewise Fns Trigonometry X X X Integration X X Differentiation X X X Limits Space Math http://spacemath.gsfc.nasa.gov
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  • Monte Carlo Simulations of GRB Afterglows

    Monte Carlo Simulations of GRB Afterglows

    ABSTRACT WARREN III, DONALD CAMERON. Monte Carlo Simulations of Efficient Shock Acceleration during the Afterglow Phase of Gamma-Ray Bursts. (Under the direction of Donald Ellison.) Gamma-ray bursts (GRBs) signal the violent death of massive stars, and are the brightest ex- plosions in the Universe since the Big Bang itself. Their afterglows are relics of the phenomenal amounts of energy released in the blast, and are visible from radio to X-ray wavelengths up to years after the event. The relativistic jet that is responsible for the GRB drives a strong shock into the circumburst medium that gives rise to the afterglow. The afterglows are thus intimately related to the GRB and its mechanism of origin, so studying the afterglow can offer a great deal of insight into the physics of these extraordinary objects. Afterglows are studied using their photon emission, which cannot be understood without a model for how they generate cosmic rays (CRs)—subatomic particles at energies much higher than the local plasma temperature. The current leading mechanism for converting the bulk energy of shock fronts into energetic particles is diffusive shock acceleration (DSA), in which charged particles gain energy by randomly scattering back and forth across the shock many times. DSA is well-understood in the non-relativistic case—where the shock speed is much lower than the speed of light—and thoroughly-studied (but with greater difficulty) in the relativistic case. At both limits of speed, DSA can be extremely efficient, placing significant amounts of energy into CRs. This must, in turn, affect the structure of the shock, as the presence of the CRs upstream of the shock acts to modify the incoming plasma flow.