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, solutions to systems of three or more particles must be approximated via numerical simulations. 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 (r_x, r_y) and velocity (v_x, v_y) of each particle is known. In order to model the dynamics of the system, we must know the net force exerted on each particle.

• Pairwise force. Newton's law of universal gravitation asserts that the strength of the gravitational force between two particles is given by the product of their masses divided by the square of the distance between them, scaled by the gravitational constant G (6.67 × 10^-11 N m² / kg²). The pull of one particle towards another acts on the line between them. Since we are using Cartesian coordinates to represent the position of a particle, it is convenient to break up the force into its x- and y-components (F_x, F_y) as illustrated below.
  
• Net force. The principle of superposition says that the net force acting on a particle in the x- or y-direction is the sum of the pairwise forces acting on the particle in that direction.

• Acceleration. Newton's second law of motion postulates that the accelerations in the x- and y-directions are given by: a_x = F_x / m, a_y = F_y / m.

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 (r_x, r_y) and velocity (v_x, v_y) 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 (F_x, F_y) 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 (a_x, a_y) at time t using the net force computed in Step 1 and Newton's second law of motion: a_x = F_x / m, a_y = F_y / m.

   2. Calculate its new velocity (v_x, v_y) 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 (v_x + Δt a_x, v_y + Δt a_y).

   3. Calculate its new position (r_x, r_y) 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 (r_x + Δt v_x, r_y + Δt v_y).

3. For each particle: Draw it using the position computed in Step 2.

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 next pair of values are the x- and y-components of the initial velocity; the fifth 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
5
2.50e+11
 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

You should read in exactly as many rows of body information as are indicated by N, the first value in the file.

Your program.   Write a program NBody.java that:

• Reads two double command-line arguments T and Δt.

• Reads in the universe from standard input using StdIn.

• Simulates the universe, starting at time t = 0.0, and continuing as long as t < T, using the leapfrog scheme described above.

• Animates the results to standard drawing using StdDraw.

• Prints the state of the universe at the end of the simulation (in the same format as the input file) to standard output using StdOut.

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.txt.

Compiling and executing your program. To compile your program from the command line, type:

% javac NBody.java

in your terminal application (e.g., Command Prompt or Terminal). 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

5
2.50e11
 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. Before you begin coding, do the following:

• Using the command line. If you previously used our Windows installer or Mac OS X installer, your command line is ready to go; steps 4–6 instruct you on how to compile and execute a program from the command line.

• Using the standard libraries. To use our standard libraries, you need to have stdlib.jar in both your DrJava classpath and your CLASSPATH environment variable. If you previously used our Windows installer or Mac OS X installer, this has already been done. The standard libraries include StdIn (for reading data from standard input), StdOut (for writing data to standard output), StdDraw (for drawing results to standard draw), and StdAudio (for sending sound to standard audio).

• Download the data files. Download the nbody subdirectory from the COS 126 ftp site. The whole directory is available as a .zip file, if you prefer. If you are downloading files one at a time, you only need these ones: planets.txt, the gifs (1 2 3 4 5), starfield.jpg, and 2001.mid. (It's a good idea to review if you understand what each of these files is for.) The complete directory contains many sample universes beyond planets.txt, along with the associated image files. It also includes the readme.txt template.

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

Extra credit. Submit an alternate universe (in our input format) along with the necessary image files. If its behavior is sufficiently interesting, we'll award extra credit.

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.