1 lucifer’s hammer derek mehlhorn william pearl adrienne upah a computer simulation of asteroid...
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Lucifer’s Hammer
Derek Mehlhorn
William Pearl
Adrienne Upah
A Computer Simulation of Asteroid Trajectories
Team 34
Albuquerque Academy
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Project Objective:
To model and observe Near Earth Objects (asteroids which come within 1.3 Au of the Sun) by simulating orbital motion using N-body gravitational interactions as well as Kepler and Newton’s laws of motion
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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Uses:
• Evaluating the probability of a space borne entity becoming a threat
• Plotting the course of satellites and probes (including “slingshot” maneuvers)
• Modeling comet and asteroid trajectories
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Definitions:
• 2-Body calculations: determining gravitational forces assuming that the sun is the only body interacting with a given body
• N-Body calculations: determining gravitational interactions between ‘N’ objects
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The Asteroid Belt:
• A large concentration of asteroids mainly located between the orbits of Mars and Jupiter
• Contains over 10,000 recorded asteroids over 1 km in radius
• Contains as many as half a million asteroids over 1/2 km in radius
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Diagram of Initial Asteroid Distribution
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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Planetary Motion and Initialization:
• Mathematical model1 used to accurately predict planetary positions on any given day– Derive initial velocities from change in positions
• Motion determined by calculating acceleration due to sum of the gravitational forces
• Integration of acceleration to find velocity and then position 1Courtesy of NASA
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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Asteroid Positions:
• User defines total number of asteroid desired
• Random distance from the Sun determined
• Random angle between 0 and 360 degrees determined• X and Y coordinates calculated from mean distance
from to sun and angle; x=rcosø y=rsinø
• Z coordinate calculated using random angle of inclination or declination (+/- 5 deg) from the plane of the ecliptic; z=xtanø0
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Asteroid Velocities:
• From asteroid’s mean distance from sun determine the period of rotation by Kepler’s law: P2 = a3
• From period and distance an average orbital velocity can be derived: Vave = 2a/P
• Orbital velocity is divided into x, y components :– Divide velocity into components, thus producing spherical to
mildly elliptic orbits– Randomly perturb velocity components varied by +/-
10% proportionally to create highly eccentric and abnormal orbits
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A Mixed Plot of Stable and Unstable Asteroids
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Other Asteroid Characteristics:
• Random radius determined between 1 and 500 km
• Measured density of Eros: 2.5 gm/cm3 +/- .8
• Asteroids assigned a density between 1.7 and 3.3 gm/cm3
• Volume determined assuming asteroids are perfect spheres: V=4/3 r3
• Mass derived from volume and density
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We generate a realistic range of densities that result in a distribution of asteroid masses
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As per empirical data, our asteroid belt possesses a high ratio of small to large asteroids
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Event Checking and Handling:
• Asteroid positions are checked at each time step :– Collisions with planets result in asteroid node deletions – Collisions between asteroids are considered purely elastic
• New velocities are determined assuming that momentum and kinetic energy are conserved
– Distance from Sun checked and flags marked accordingly
• Asteroids flags are checked and position information output accordingly
• Planet information printed every time step
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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The Code Modules:
• The Parameter Class: para.h– Uses mathematical model to obtain realistic initial positions and
velocities for each planet
• The Planet Class: planet.h– Creates orbital objects (planets and asteroids) whose motion is
determined through N-body calculations
• starter.cpp– Used to test the parameter class
• main.cpp (parallelized using MPI)– Implements the Planet class to create and run the simulation
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Master Node Operations:
• Implements a mathematical model for predicting planetary positions and starting variables
• Determines planetary positions through N-body calculations
• Writes positions to output files
• Broadcasts planetary positions to slave nodes
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Slave Node Operations:
• Randomly generate a specified number of asteroids on each node that are stored within a linked list.
• Receive and use planetary data to determine individual asteroid motion through N-body calculations (relative to the planets)
• Check (“on node”) asteroid positions for collisions and interesting orbital characteristics
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Parallel Implementation:
• Two processor tests run on Pi
• Scalability tested through 5 nodes using the Blue Mountain Super Computer
• A number of limited time (~100 years) large asteroid population (~10000) completed
• Several larger runs (~10000 years) attempted but limited by storage space– runs completed using 20 processors
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The Inner Solar System:•Mercury - Mars
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The Outer Solar System:•Jupiter - Neptune
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An eccentric yet stable Near Earth Asteroid
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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Integration Method:
• “Leap frog method” – positions and forces centered on time step– velocities centered on 1/2 time step
• Method conserves energy
• Resolution convergence confirmed (vary )
• Future work: compare to trapezoidal & Simpson’s
Ref: Feynman Lectures on Physics
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Error analysis:
N-body integrator stable and accurate over thousands of years
-Average Error above Computed in Au’s from 10 years of data for the Earth
Time Step Length
1 Day
1/2 Day
1/4 Day
1/8 Day
Average X Error
.000313884
.000246115
.000229601
.000225648
Average Y Error
.000322463
.000254278
.00023773
.000233772
Average Z Error
.00000069587
.00000069703
.000000697618
.000000697913
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The System Conserves Energy
(Kinetic & potential energies anti-correlated)
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Inter-asteroid forces can for the most part be ignored
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Presentation Summary:
• Uses and Definitions
• Planetary Setup and Mathematical Model
• Asteroid Generation
• Code Implementation
• Error Analysis
• Results and Conclusions
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Near Earth Asteroids do not possess significantly different total energy levels than stable asteroids
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Stable Asteroids are harmless because they have spherical orbits which are difficult to perturb
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Near Earth Asteroids are dangerous because of they have eccentric orbits which can be easily perturbed
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Real space plot of an eccentric and perturbed Near Earth Asteroid
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Conclusions:
• Although NEO’s have eccentric orbits that are easily perturbed, they are not less bound to the Solar System
• Regular asteroids pose little or no threat to the earth because of their spherical and predictable orbits
• Near Earth Objects present a large threat of collision because of their eccentricity and their susceptibility to perturbations