In 1687 Sir Isaac Newton formulated the principles governing 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.

Program specification. Write a program NBody.java that:

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

• Reads in the universe from standard input using StdIn, using several parallel arrays to store the data.

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

• 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.

Reading in the universe. The input format is a text file that contains the information for a particular universe (in SI units). 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, used to determine the scaling of the drawing window. 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 own solar system (up to Mars):

% 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.

Simulating the universe: 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 (p_x, p_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.

Simulating the universe: 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 (p_x, p_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.

• Step 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).

• Step 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 (p_x, p_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 (p_x + Δt v_x, p_y + Δt v_y).

• Step 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 (about 40 frames per second looks good). You will use several methods from the StdDraw library; click here to see a summary of what this library contains.

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. (As the checklist explains, you may need to use java and javac in place of java-introcs and javac-introcs depending on your machine's setup.) To compile your program from the command line, type:

% javac-introcs 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-introcs NBody 157788000.0 25000.0 < planets.txt

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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

Getting started. Before you begin coding, do the following:

• Get familiar with 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.

• Get familiar with 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 should already be working for you. 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). The Stdlib cheatsheet and your book's appendix have a summary of the APIs (but they are missing StdDraw.picture(x, y, filename)).

• Use the java-introcs and javac-introcs commands. If you did not review these already in class, this is explained on the checklist. See also the basic test on the Command-line cheatsheet.

• Download the data files. To test your program, you will need the image and music files, and the formatted planet data text files. We also provide you with the readme.txt template, as well as additional interesting test universes and their associated files. To get these files, you can either download them as a zip archive nbody.zip or you can copy the folder itself via anonymous FTP from ftp://ftp.cs.princeton.edu/pub/cs126/nbody. The checklist has more details on both methods.

Submission. Submit NBody.java and a completed readme.txt file.

Extra credit. Submit an alternate universe (in our input format) along with the necessary image files. Describe it in your readme.txt. 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.