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CIS 110

N-Body Simulation
Programming Assignment

In 1687 Sir Isaac Newton formulated the principles governing the the motion of two particles under the influence of their mutual gravitational attraction in his famous Principia. However, Newton was unable to solve the problem for three particles. Indeed, in general, systems of three or more particles can only be solved numerically. Your challenge is to write a program to simulate the motion of N particles in the plane, mutually affected by gravitational forces, and animate the results. Such methods are widely used in cosmology, semiconductors, and fluid dynamics to study complex physical systems. Scientists also apply the same techniques to other pairwise interactions including Coulombic, Biot-Savart, and van der Waals.

The physics. We review the equations governing the motion of the particles, according to Newton's laws of motion and gravitation. Don't worry if your physics is a bit rusty; all of the necessary formulas are included below. We'll assume for now that the position (px, py) and velocity (vx, vy) of each particle is known. In order to model the dynamics of the system, we must know the net force exerted on each particle.

The numerics.  We use the leapfrog finite difference approximation scheme to numerically integrate the above equations: this is the basis for most astrophysical simulations of gravitational systems. In the leapfrog scheme, we discretize time, and update the time variable t in increments of the time quantum Δt (measured in seconds). We maintain the position (px, py) and velocity (vx, vy) of each particle at each time step. The steps below illustrate how to evolve the positions and velocities of the particles.

  1. For each particle: Calculate the net force (Fx, Fy) at the current time t acting on that particle using Newton's law of gravitation and the principle of superposition. Note that force is a vector (i.e., it has direction). In particular, be aware that Δx and Δy are signed (positive or negative). In the diagram above, when you compute the force the sun exerts on the earth, the sun is pulling the earth up (Δy positive) and to the right (Δx positive).

  2. For each particle:

    1. Calculate its acceleration (ax, ay) at time t using the net force computed in Step 1 and Newton's second law of motion: ax = Fx / m, ay = Fy / m.

    2. Calculate its new velocity (vx, vy) at the next time step by using the acceleration computed in Step 2a and the velocity from the old time step: Assuming the acceleration remains constant in this interval, the new velocity is (vx + Δt ax, vy + Δt ay).

    3. Calculate its new position (px, py) at time t + Δt by using the velocity computed in Step 2b and its old position at time t: Assuming the velocity remains constant in this interval, the new position is (px + Δt vx, py + Δt vy).

  3. For each particle: Draw it using the position computed in Step 2.
The simulation is more accurate when Δt is very small, but this comes at the price of more computation.

Creating an animation. Draw each particle at its current position using standard drawing, and repeat this process at each time step until a designated stopping time. By displaying this sequence of snapshots (or frames) in rapid succession, you will create the illusion of movement. After each time step (i) draw the background image starfield.jpg, (ii) redraw all the bodies in their new positions, and (iii) control the animation speed using StdDraw.show().

Input format. The input format is a text file that contains the information for a particular universe. The first value is an integer N which represents the number of particles. The second value is a real number R which represents the radius of the universe: assume all particles will have x- and y-coordinates that remain between -R and R. Finally, there are N rows, and each row contains 6 values. The first two values are the x- and y-coordinates of the initial position; the second two values are the x- and y-components of the initial velocity; the third value is the mass; the last value is a String that is the name of an image file used to display the particle. As an example, planets.txt contains data for our solar system (in SI units).

% more planets.txt
 1.4960e+11  0.0000e+00  0.0000e+00  2.9800e+04  5.9740e+24    earth.gif
 2.2790e+11  0.0000e+00  0.0000e+00  2.4100e+04  6.4190e+23     mars.gif
 5.7900e+10  0.0000e+00  0.0000e+00  4.7900e+04  3.3020e+23  mercury.gif
 0.0000e+00  0.0000e+00  0.0000e+00  0.0000e+00  1.9890e+30      sun.gif
 1.0820e+11  0.0000e+00  0.0000e+00  3.5000e+04  4.8690e+24    venus.gif
Note: On Windows, you should use the command type planets.txt to print out the contents of the file planets.txt.

Your program.   Write a program NBody.java that:

Maintain several parallel arrays to store the data. To make the computer simulation, write a loop that repeatedly updates the position and velocity of the particles. Before plotting, use StdDraw.setXscale(-R, +R) and StdDraw.setYscale(-R, +R) to scale the physics coordinates to the screen coordinates.

Optional finishing touch. For a finishing touch, play the theme to 2001: A Space Odyssey using StdAudio and the file 2001.mid. It's a one-liner using the method StdAudio.play(). If you have trouble doing this, make sure you note it in your readme_nbody.txt.

Compiling and executing your program. If you used the introcs installer to set up Java and Dr. Java in Assignment 0, you should be fine. Otherwise, you made need to follow these instructions [ Windows · Mac · Linux ] to configure the command line on your system. To compile your program from the command line, type:

% javac NBody.java

in your terminal application. To execute your program from the command line, redirecting from the file planets.txt to standard input, type:

% java NBody 157788000.0 25000.0 < planets.txt
 1.4925e+11 -1.0467e+10  2.0872e+03  2.9723e+04  5.9740e+24    earth.gif
-1.1055e+11 -1.9868e+11  2.1060e+04 -1.1827e+04  6.4190e+23     mars.gif
-1.1708e+10 -5.7384e+10  4.6276e+04 -9.9541e+03  3.3020e+23  mercury.gif
 2.1709e+05  3.0029e+07  4.5087e-02  5.1823e-02  1.9890e+30      sun.gif
 6.9283e+10  8.2658e+10 -2.6894e+04  2.2585e+04  4.8690e+24    venus.gif

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Getting started. To get started, create an nbody folder for you assignment, download the input data files here and unzip them into your nbody folder.. It contains our standard libraries that you will need for input and output (StdIn.java, StdOut.java, StdDraw.java, and StdAudio.java). It also includes many sample universes (such as planets.txt) that you can use to test your program, along with image files of the planets. Finally, download the readme.txt template.

Checklist. Make sure to read the Checklist very carefully. This assignment is far more challenging that the first two, and it is important to work through it step by step.

Submission. Submit NBody.java. Also submit a readme_nbody.txt file and answer the questions.

Extra credit. Submit a zip file containing an alternate universe (in our input format) along with the necessary image files. If its behavior is sufficiently interesting, we'll award extra credit. Your submission must be in a zip file, even if there are no images, so that our grading scripts can handle it correctly.

Challenge for the bored. There are limitless opportunities for additional excitement and discovery here. Try adding other features, such as supporting elastic or inelastic collisions. Or, make the simulation three-dimensional by doing calculations for x-, y-, and z-coordinates, then using the z-coordinate to vary the sizes of the planets. Add a rocket ship that launches from one planet and has to land on another. Allow the rocket ship to exert force with the consumption of fuel.

Copyright © 1999–2010, Robert Sedgewick and Kevin Wayne.