Lucifer s hammer
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Lucifer’s Hammer. A Computer Simulation of Asteroid Trajectories. Derek Mehlhorn William Pearl Adrienne Upah. Team 34 Albuquerque Academy. Project Objective:.

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Lucifer s hammer

Lucifer’s Hammer

A Computer Simulation of Asteroid Trajectories

Derek Mehlhorn

William Pearl

Adrienne Upah

Team 34

Albuquerque Academy


Project objective

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


Presentation summary

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


Definitions

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


The asteroid belt

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


Planetary motion and initialization

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


Asteroid positions

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


Asteroid velocities

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 = 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


Other asteroid characteristics

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


Event checking and handling

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


The code modules

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


Parallel implementation

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


Integration method

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 & Simpson’s

Ref: Feynman Lectures on Physics


Error analysis

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

.000233772

Average Z Error

.00000069587

.00000069703

.000000697618

.000000697913

-Average Error above Computed in Au’s 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


Conclusions

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


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