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## 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:**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:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**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:**• 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:**• 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**Presentation Summary:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**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**Presentation Summary:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**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:**• 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**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**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**Presentation Summary:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**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**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**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:**• 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**The Inner Solar System:**• Mercury - Mars**The Outer Solar System:**• Jupiter - Neptune**Presentation Summary:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**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:**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**The System Conserves Energy**(Kinetic & potential energies anti-correlated)**Presentation Summary:**• Uses and Definitions • Planetary Setup and Mathematical Model • Asteroid Generation • Code Implementation • Error Analysis • Results and Conclusions**Near Earth Asteroids do not possess significantly different**total energy levels than stable asteroids**Stable Asteroids are harmless because they have spherical**orbits which are difficult to perturb**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**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