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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

- 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

- 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

- 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: ratio of small to large asteroids

- 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: ratio of small to large asteroids

- Uses and Definitions
- Planetary Setup and Mathematical Model
- Asteroid Generation
- Code Implementation
- Error Analysis
- Results and Conclusions

The Code Modules: ratio of small to large asteroids

- 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: ratio of small to large asteroids

- 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: ratio of small to large asteroids

- 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: ratio of small to large asteroids

- 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: ratio of small to large asteroids
- Mercury - Mars

- The Outer Solar System: ratio of small to large asteroids
- Jupiter - Neptune

An eccentric yet stable Near Earth Asteroid ratio of small to large asteroids

Presentation Summary: ratio of small to large asteroids

- Uses and Definitions
- Planetary Setup and Mathematical Model
- Asteroid Generation
- Code Implementation
- Error Analysis
- Results and Conclusions

Integration Method: ratio of small to large asteroids

- “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: ratio of small to large asteroids

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 ratio of small to large asteroids

(Kinetic & potential energies anti-correlated)

Inter-asteroid forces can for the most part be ignored ratio of small to large asteroids

Presentation Summary: ratio of small to large asteroids

- 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

Conclusions: Asteroid

- 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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