Lucifers Hammer

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Lucifers Hammer
A Computer Simulation of Asteroid Trajectories

• Derek Mehlhorn
• William Pearl
• Adrienne Upah

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
  Newtons 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 xrcosø yrsinø
• Z coordinate calculated using random angle of
  inclination or declination (/- 5 deg) from the
  plane of the ecliptic zxtanø0

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Asteroid Velocities

• From asteroids mean distance from sun determine
  the period of rotation by Keplers law P2 a3
• From period and distance an average orbital
  velocity can be derived Vave 2?a/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 V4/3 ? 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 e)
• Future work compare to trapezoidal Simpsons

Ref Feynman Lectures on Physics
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Error analysis
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 .0
00233772
Average Z Error .00000069587 .00000069703 .0000006
97618 .000000697913
-Average Error above Computed in Aus from 10
years of data for the Earth
N-body integrator stable and accurate over
thousands of years
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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 NEOs 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