Suppose we are given a huge collection of astronomical bodies, such as the stars
ID: 3881087 • Letter: S
Question
Suppose we are given a huge collection of astronomical bodies, such as the stars clouds of dust, and dark matter in a galaxy. The problem is to simulate the evolu tion of the galaxy. Each body has a mass and an initial position and velocity. Gravity causes the bodies to accelerate and to move. The motion of an n-body system is simu- lated by stepping through discrete instants of time. At each time step, we calcu- late the forces on every body and then update their velocities and positions. A simulation of the gravitational n-body problem thus has the following structure: initialize bodies: for time start to finish by Dr] calculate forces move bodies The value of Dr (delta time) is the length of a time step. From Newtonian physics, the magnitude of the gravitational force between two bodies i and j is G m, m The m's are the masses of the two bodies, and r is the distance between them. Gravity is an extremely weak force, so the value of the gravitational constant G is the very small number 6.67.10-1 For simplicity, we will assume the bodies are all on the same spatial plane, and hence that we have only a two-dimensional problem. Let the positions of two bodies be (p, x, p,"y) and (p, x, p, y). Then the Euclidean distance between them is given by If two bodies come close to each other (a near collision), then r will be very small. This will lead to numerical instability in calculating forces, but we ignore the problem here The direction of the force on body i exerted by body j is given by a unit vec- tor that points from i to j. The direction of the force on body j due to body i is given by a unit vector that points in the opposite direction. Thus, the magnitudes of the forces of each body on the other are the same, but the directions are oppo- site. The total force on a body is the sum of the forces exerted by every other body. (Vector sums order.) are commutative, so the force vectors can be added in anyExplanation / Answer
import java.text.*;
public final class nbody {
private static final NumberFormat nf = new DecimalFormat("#0.000000000");
public static void main(String[] args) {
int n = Integer.parseInt(args[0]);
NBodySystem bodies = new NBodySystem();
System.out.println(nf.format(bodies.energy()) );
for (int i=0; i<n; ++i) {
bodies.advance(0.01);
}
System.out.println(nf.format(bodies.energy()) );
}
}
final class NBodySystem {
private Body[] bodies;
public NBodySystem(){
bodies = new Body[]{
Body.sun(),
Body.jupiter(),
Body.saturn(),
Body.uranus(),
Body.neptune()
};
double px = 0.0;
double py = 0.0;
double pz = 0.0;
for(int i=0; i < bodies.length; ++i) {
px += bodies[i].vx * bodies[i].mass;
py += bodies[i].vy * bodies[i].mass;
pz += bodies[i].vz * bodies[i].mass;
}
bodies[0].offsetMomentum(px,py,pz);
}
public void advance(double dt) {
double dx, dy, dz, distance, mag;
for(int i=0; i < bodies.length; ++i) {
for(int j=i+1; j < bodies.length; ++j) {
dx = bodies[i].x - bodies[j].x;
dy = bodies[i].y - bodies[j].y;
dz = bodies[i].z - bodies[j].z;
distance = Math.sqrt(dx*dx + dy*dy + dz*dz);
mag = dt / (distance * distance * distance);
bodies[i].vx -= dx * bodies[j].mass * mag;
bodies[i].vy -= dy * bodies[j].mass * mag;
bodies[i].vz -= dz * bodies[j].mass * mag;
bodies[j].vx += dx * bodies[i].mass * mag;
bodies[j].vy += dy * bodies[i].mass * mag;
bodies[j].vz += dz * bodies[i].mass * mag;
}
}
for(int i=0; i < bodies.length; ++i) {
bodies[i].x += dt * bodies[i].vx;
bodies[i].y += dt * bodies[i].vy;
bodies[i].z += dt * bodies[i].vz;
}
}
public double energy(){
double dx, dy, dz, distance;
double e = 0.0;
for (int i=0; i < bodies.length; ++i) {
e += 0.5 * bodies[i].mass *
( bodies[i].vx * bodies[i].vx
+ bodies[i].vy * bodies[i].vy
+ bodies[i].vz * bodies[i].vz );
for (int j=i+1; j < bodies.length; ++j) {
dx = bodies[i].x - bodies[j].x;
dy = bodies[i].y - bodies[j].y;
dz = bodies[i].z - bodies[j].z;
distance = Math.sqrt(dx*dx + dy*dy + dz*dz);
e -= (bodies[i].mass * bodies[j].mass) / distance;
}
}
return e;
}
}
final class Body {
static final double PI = 3.141592653589793;
static final double SOLAR_MASS = 4 * PI * PI;
static final double DAYS_PER_YEAR = 365.24;
public double x, y, z, vx, vy, vz, mass;
public Body(){}
static Body jupiter(){
Body p = new Body();
p.x = 4.84143144246472090e+00;
p.y = -1.16032004402742839e+00;
p.z = -1.03622044471123109e-01;
p.vx = 1.66007664274403694e-03 * DAYS_PER_YEAR;
p.vy = 7.69901118419740425e-03 * DAYS_PER_YEAR;
p.vz = -6.90460016972063023e-05 * DAYS_PER_YEAR;
p.mass = 9.54791938424326609e-04 * SOLAR_MASS;
return p;
}
static Body saturn(){
Body p = new Body();
p.x = 8.34336671824457987e+00;
p.y = 4.12479856412430479e+00;
p.z = -4.03523417114321381e-01;
p.vx = -2.76742510726862411e-03 * DAYS_PER_YEAR;
p.vy = 4.99852801234917238e-03 * DAYS_PER_YEAR;
p.vz = 2.30417297573763929e-05 * DAYS_PER_YEAR;
p.mass = 2.85885980666130812e-04 * SOLAR_MASS;
return p;
}
static Body uranus(){
Body p = new Body();
p.x = 1.28943695621391310e+01;
p.y = -1.51111514016986312e+01;
p.z = -2.23307578892655734e-01;
p.vx = 2.96460137564761618e-03 * DAYS_PER_YEAR;
p.vy = 2.37847173959480950e-03 * DAYS_PER_YEAR;
p.vz = -2.96589568540237556e-05 * DAYS_PER_YEAR;
p.mass = 4.36624404335156298e-05 * SOLAR_MASS;
return p;
}
static Body neptune(){
Body p = new Body();
p.x = 1.53796971148509165e+01;
p.y = -2.59193146099879641e+01;
p.z = 1.79258772950371181e-01;
p.vx = 2.68067772490389322e-03 * DAYS_PER_YEAR;
p.vy = 1.62824170038242295e-03 * DAYS_PER_YEAR;
p.vz = -9.51592254519715870e-05 * DAYS_PER_YEAR;
p.mass = 5.15138902046611451e-05 * SOLAR_MASS;
return p;
}
static Body sun(){
Body p = new Body();
p.mass = SOLAR_MASS;
return p;
}
Body offsetMomentum(double px, double py, double pz){
vx = -px / SOLAR_MASS;
vy = -py / SOLAR_MASS;
vz = -pz / SOLAR_MASS;
return this;
}
}
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