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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

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

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

We generate a realistic range of densities that result in a distribution of asteroid masses

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

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

The System Conserves Energy

(Kinetic & potential energies anti-correlated)

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

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