# National Aeronautics and Space Administration

## Astronomy and Space Science Problems Involving Calculus

Problem 673:VAB - An Improved Model for Van Allen Belt Radiation Dose Students use a detailed model of the path of a satellite and the radiation dose rate along the path to calculate the total radiation dose to the spacecraft. [Grade: 9-11 | Topics: Polynomial equations; trigonometric equations; composite finctions f(f(x)); estimating areas under curves] (PDF)

Problem 672:VAB - Modeling the Radiation Dose of the Van Allen Probes Students create a simple mathematical model of the radiation exposure to the VABP satellites as they travel through the Van Allen belts. [Grade: 11-12 | Topics: Parametric equations;composite functions f(g(x)); integral calculus ] (PDF)

Problem 668: Meteor Impacts � How Much Stuff? Students integrate a powerlaw function to estimate the number of tons of meteoritic debris that Earth collects every year. [Grade: 12 | Topics: Integral calculus] (PDF)

Problem 496: How to Grow a Planet or a Rain Drop
Students use calculus to slove for the growth in mass of a body, and solve the equation for the case of a raindrop and a planet like Earth. [Grade: 12 | Topics: Solving a simple differential equation.] (PDF)

Problem 495: The Volume of a Lunar Impact Crater
Students use calculus to determine the volume of a crater whose depth is defined by a fourth-order polynomial [Grade: 12 | Topics: Integration involving vollumes of rotation] (PDF)

Problem 494: The Close Encounter to the Sun of Barnards Star
Students use parametric equations and calculus to determine the linear equation for the path of Barnards Star, and then determine when the minimum distance to the sun occurs [Grade: 12 | Topics: Derivitives and minimization] (PDF)

Problem 383: Estimating the mass of Comet Hartley 2 using calculus.
Students use a recent image of the nucleus of Comet Hartley 2 taken by the Deep Impact/EPOXI camera and a shape function described by a fourth-order polynomial to calculate the volume of the comet's head using integral calculus. to estimate the volume of the comets nucleus, and its total mass, [Grade: 12 | Topics: Volume integral using disk method; scale model; scientific notation; unit conversion] (PDF)

Problem 328: WISE: F(x)G(x): A Tale of Two Functions Students use WISE satellite data to study a practical application of the product of two functions by graphing them individually, and their product. A calculus-level problem is included for advanced students. [Grade: 10-12 | Topics: Power-law functions; domain and range; graphing; areas under curves; integration]

Problem 327: WISE: Exploring Power-law Functions Using WISE Data Based on a recent press release of the 'First Light' image taken with NASA's new WISE satellite, students explore a practical application of a power law function to count the number of stars in the sky. An additional calculus-level problem is included for advanced students. [Grade: 10-12 | Topics: areas; functions; histograms; unit conversion; power-laws; integration]

Problem 326: Hubble Spies Colliding Asteroids Based on a recent press release, students calculate how often asteroids collide in the Asteroid belt using a simple formula. Students estimate belt volume, and asteroid speeds to determine the number of years between collisions. They also investigate how the collision time depends on the particular assumptions they made. An 'extra' integration problem is also provided for calculus students. [Grade: 8-12 | Topics: Volume of a thin disk; Algebra 1; Evaluating a definite integral; power-law]

Problem 324: Deep Impact Comet Flyby The Deep Impact spacecraft flew by the Comet Tempel-1 in 2005. Students determine the form of a function that predicts the changing apparent size of the comet as viewed from the spacecraft along its trajectory. [Grade: 9-12 | Topics: Algebra, geometry, differential calculus]

Problem 323: How Many Quasars are There? Students use a piecewise function that estimates how many quasars are found in a given area of the sky. The function is integrated to determine the estimated total number of quasars across the entire sky. [Grade: 11-12 | Topics: Piecewise functions; integral calculus]

Problem 322: Rotation Velocity of a Galaxy Students examine a simple model of the rotation of a galaxy to investigate how fast stars orbit the centers of galaxies in systems such as the Milky Way and Messier-101. [Grade: 10-12 | Topics: Algebra, limiting form of functions; derivitives]

Problem 321: Lunar Crater Frequency Distributions Students use an image from the LRO satellite of the Apollo-11 landing area, along with a power-law model of cratering, to determine what fraction of the landin garea was safe to land upon. [Grade: 11-12 | Topics: Integral calculus]

Problem 318: The Internal Density and Mass of the Sun Students use a simple, spherically symmetric, density profile to determine the mass of the sun using integral calculus. [Grade: 11-12 | Topics: Algebra II; Polynomials; integral calculus]

Problem 305: From Asteroids to Planetoids 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. [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. [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. [Grade: 11-12 | Topics: Integral Calculus]

Problem 271: A Simple Model for Atmospheric Carbon Dioxide Students work with the known sources of increasing and decreasiong carbon dioxide to create a simple model of the rate of change of atmospheric carbon dioxide. [Grade: 10-12 | Topics: Algebra I, rates of change, differential calculus]

Problem 266: The Ares-V Cargo Rocket Students work with the equations for thrust and fuel loss to determine the acceleration curve of the Ares-v during launch. [Grade: 11-12 | Topics: Algebra II, properties of functions, differential calculus, Excel Spreadsheet]

Problem 265: Estimating Maximum Cell Sizes Students estimate tyhe maximum size of spherical cells based on the rates with which they create waste and remove it through their cell walls. [Grade: 11-12 | Topics: differential calculus, unit conversion]

Problem 234: Calculating Arc Lengths of Simple Functions- Students work with the differential form of the Pythagorean Theorem to determine the basic integral formula for arc length, then evaluate it for a parabola, logrithmic spiral and normal spiral. They evaluate the length of the spiral track on a CDrom. [Grade: 11-12 | Topics: Calculus; differential; integral, U-substitutions; significant figures.]

Problem 225: Areas Under Curves; An astronomical perspective- Students work with a bar graph of the number of planet discoveries since 1995 to evaluate the total discoveries, as areas under the graph, for various combinations of time periods. [Grade: 6-8 | Topics: Adding areas in bar graphs.]

Problem 208: Optimization- Students determine the optimal dimensions of an hexagonal satellite to maximize its surface area given its desiblack volume. [Grade: 9-12| Topics: Calculus; differentiation.]

Problem 202: The Dawn Mission - Ion Rockets and Spiral Orbits- Students determine the shape of the trajectory taken by a spacecraft using a constant-thrust ion motor using differential and integral calculus for arc lengths. [Grade: 9-12| Topics: Calculus - Arc lengths.]

Problem 193: Fluid Level in a Spherical Tank - Students explore the relationship between volume, and the height of fluid in a spherical tank as fluid is being drained at a constant rate. [Grade: 10-12| Topics: Algebra, differential calculus, related rates]

Problem 192: The Big Bang - Cosmic Expansion - Students explore the expansion of the universe pblackicted by Big Bang cosmology [Grade: 10-12| Topics: Algebra, Integral Calculus]

Problem 191: Why are hot things black? - Students explore the Planck Function using graphing skills, and calculus for experts, to determine the relationship between temperature and peak wavelength. [Grade: 10-12| Topics: Algebra, graphing, differential calculus]

Problem 190: Modeling a Planetary Nebula - Students use calculus to create a mathematical model of a planetary nebula [Grade: 10-12| Topics: Algebra, Integral calculus]

Problem 187: Differentiation- Students explore partial derivatives by calculating rates of change in simple equations taken from astrophysics. [Grade: 11-12| Topics: differentiation; algebra]

Problem 186: Collapsing Gas Clouds and Stability- Students use the derivative to find an extremum of an equation governing the pressure balance of an interstellar cloud. [Grade: 11-12| Topics: differentiation; finding extrema; partial derivitives]

Problem 184: The Ant and the Turntable: Frames of reference - Students pblackict the motion of an ant crawling from the center of a spinning CDrom to the edge. They also use calculus to estimate the length of the spiral path seen by a stationary observer. [Grade: 11-12| Topics: integration; parametric equations; polar coordinates]

Problem 183: Calculating Arclengths of Simple Functions- Students determine the basic equation for arclength and its integral, and evaluate it for simple polar functions. [Grade: 11-12| Topics: calculus; integration; parametric equations]

Problem 169: The Limiting Behavior of Functions- Students work with two complex formulae to determine their limiting behavior as the independent variables approach infinity and zero. [Grade: 9-12 | Topics: Algebra II, pre-calculus]

Problem 157: Space Shuttle Launch Trajectory - I - Students use the parametric equation for the altitude and range for an actual Shuttle launch to determine the speed and acceleration of the Shuttle during launch and orbit insertionh [Grade: 11-12 | Topics: Algebra; Calculus; Parametric Equations; Differentiation

Problem 89 : Atmospheric Shielding from Radiation- III - This is Part III of a 3-part problem on atmospheric shielding. Students use exponential functions to model the density of a planetary atmosphere, then evaluate a definite integral to calculate the total radiation shielding in the zenith (straight overhead) direction for Earth and Mars. [Grade level: 11-12 | Topics: Evaluating an integral, working with exponential functions]

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) ]