Solar System Simulation

Physics, Mathematics, and Numerical Methods

Adam Macháček

High School Presentation • 2026

Press → or click to continue

What's on the Agenda?

1Gravitational Physics

2Kepler's Laws

3Numerical Methods

43D Simulation

🎯 Presentation Goal

Understand how we can simulate planetary motion in the Solar System using physical laws and programming.

💡 Why is it important?

NASA uses the same principles for mission planning, satellites for navigation, and scientists for asteroid prediction.

Newton's Law of Universal Gravitation

Every two objects in the universe attract each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them.

$$\vec{F} = G \frac{m_1 \cdot m_2}{r^2} \hat{r}$$

Newton's Law of Universal Gravitation (1687)

Variables

G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant)
m₁, m₂ = masses of bodies [kg]
r = distance between bodies [m]
F = gravitational force [N]

🌍 Example: Earth-Sun

Sun Mass: 2×10³⁰ kg

Distance: 150 million km

Force: 3.54 × 10²² N

Kepler's Laws of Planetary Motion

1st Law (Ellipses)

Planets orbit in ellipses with the Sun at one focus.

2nd Law (Areas)

A line joining planet and Sun sweeps equal areas in equal times.

3rd Law (Periods)

$$T^2 = \frac{4\pi^2}{GM} a^3$$

The square of the orbital period is proportional to the cube of the semi-major axis.

Johannes Kepler (1571–1630) derived these laws from Tycho Brahe's observations

Equations of Motion

From Force to Motion

Combining Newton's Law of Gravitation and Newton's Second Law (F = ma) gives us:

$$\vec{a} = -\frac{GM}{|\vec{r}|^3} \vec{r}$$

Gravitational Acceleration

This is a 2nd order differential equation that we must solve numerically.

$$\frac{d^2\vec{r}}{dt^2} = -\frac{GM}{|\vec{r}|^3} \vec{r}$$

Decomposition into Components

// Distance from Sun
r = √(x² + y²)

// Acceleration in x direction
aₓ = -GM × x / r³

// Acceleration in y direction
aᵧ = -GM × y / r³

// Velocity update
vₓ = vₓ + aₓ × Δt
vᵧ = vᵧ + aᵧ × Δt

// Position update
x = x + vₓ × Δt
y = y + vᵧ × Δt
                

Numerical Integration Methods

Differential equations usually do not have analytical solutions. We must solve them numerically — in small time steps.

Method	Accuracy	Energy Conservation	Complexity
Euler	O(Δt)	Poor ❌	Simple
Velocity Verlet	O(Δt²)	Excellent ✓	Medium
Runge-Kutta 4	O(Δt⁴)	Good	Higher
Leapfrog	O(Δt²)	Excellent ✓	Simple

For orbital mechanics, we use symplectic integrators (Verlet, Leapfrog), which conserve energy.

Velocity Verlet Algorithm

Algorithm Steps

$$\vec{v}_{n+\frac{1}{2}} = \vec{v}_n + \frac{\Delta t}{2} \vec{a}_n$$

1. Half velocity step

$$\vec{r}_{n+1} = \vec{r}_n + \Delta t \cdot \vec{v}_{n+\frac{1}{2}}$$

2. Full position step

$$\vec{a}_{n+1} = f(\vec{r}_{n+1})$$

3. Calculate new acceleration

$$\vec{v}_{n+1} = \vec{v}_{n+\frac{1}{2}} + \frac{\Delta t}{2} \vec{a}_{n+1}$$

4. Complete velocity step

Code Implementation

function verletStep(planet, dt) {
  // 1. Half velocity step
  planet.vx += 0.5 * planet.ax * dt;
  planet.vy += 0.5 * planet.ay * dt;

  // 2. Full position step
  planet.x += planet.vx * dt;
  planet.y += planet.vy * dt;

  // 3. New acceleration
  const r = Math.sqrt(
    planet.x**2 + planet.y**2
  );
  const a = -GM / (r**3);
  planet.ax = a * planet.x;
  planet.ay = a * planet.y;

  // 4. Complete velocity
  planet.vx += 0.5 * planet.ax * dt;
  planet.vy += 0.5 * planet.ay * dt;
}
                

Speed

🪐 Simulation

Time:Year 0

Method:Verlet

Planets:8

Interactive 3D Simulation

Planet Name

Description goes here.

Fact goes here.

Solar System Explorer — Click a Planet!

Galaxy Simulation (Particle System)

⚫ Black Hole Mode

Particles are pulled towards a central point. They gain a reddish hue as they approach the "event horizon".

Force

Size

👆 Click and drag to spawn particles!

Particle Physics Sandbox

Real World Applications

🚀 Space Missions

NASA uses similar simulations to plan trajectories for probes like Voyager, New Horizons, or Mars missions.

Gravity assists save fuel using a "slingshot" around planets.

🛰️ GPS Satellites

32 GPS satellites must know their exact position. The same gravitational calculations are used.

Accuracy: ±3 meters on Earth

☄️ Earth Protection

Tracking asteroids and calculating collision risks. The DART mission successfully altered an asteroid's orbit!

~2,300 potentially hazardous asteroids

🎮 Also in Games!

Games like Kerbal Space Program use realistic orbital mechanics. Try planning space missions!

Summary

📚 What We Learned

Newton's law of gravitation describes the attraction of bodies
Kepler's laws describe the elliptical orbits of planets
We solve differential equations numerically
Velocity Verlet conserves system energy
3D visualization helps understand dynamics

🔧 Technologies Used

Three.js – 3D graphics in the browser
WebGL – hardware acceleration
GLSL Shaders – effects and lighting
JavaScript – physics calculations
MathJax – beautiful equations

Thank you for your attention!

Adam Macháček • 2026

Universe Sandbox

☀️ Pick an object, click canvas to place, then drag to set velocity!

The Three-Body Problem

Why it's unsolvable

For two bodies, Newton's laws yield a clean closed-form orbit. Add a third body and the system becomes chaotic — tiny differences in initial conditions explode exponentially. Henri Poincaré proved in 1887 that no general analytical solution exists. The trajectories are non-integrable.

Chaos in practice

Real systems — binary stars with a planet, the Sun-Earth-Moon system, spacecraft trajectories — must be solved numerically. Long-term prediction breaks down. Even the solar system is chaotic on ~100 million year timescales.

Lyapunov exponent > 0 No closed-form solution Sensitive initial conditions

Special solutions & workarounds

Lagrange points (1772): stable equilibria for restricted 3-body. Figure-8 orbit (Chenciner & Montgomery, 2000): one exact periodic solution exists. Modern approach: symplectic integrators + long double precision for spacecraft. Machine learning is now being explored for trajectory prediction.

LIVE 3-BODY SIMULATION

🎮 Cosmic Toys

☀ SOLAR MUNCH Score: 0

☀

Solar Munch

Move your sun with the mouse.
Eat smaller suns, avoid bigger ones!

☄ GRAVITY SLINGSHOT

drag to launch •

🎨 PLANET PAINTER click planet to paint

✕

Solar System Simulation

Solar System Simulation

What's on the Agenda?

🎯 Presentation Goal

💡 Why is it important?

Newton's Law of Universal Gravitation

Variables

🌍 Example: Earth-Sun

Kepler's Laws of Planetary Motion

1st Law (Ellipses)

2nd Law (Areas)

3rd Law (Periods)

Equations of Motion

From Force to Motion

Decomposition into Components

Numerical Integration Methods

Velocity Verlet Algorithm

Algorithm Steps

Code Implementation

🪐 Simulation

Interactive 3D Simulation

Planet Name

Solar System Explorer — Click a Planet!

Galaxy Simulation (Particle System)

⚫ Black Hole Mode

Particle Physics Sandbox

Real World Applications

🚀 Space Missions

🛰️ GPS Satellites

☄️ Earth Protection

🎮 Also in Games!

Summary

📚 What We Learned

🔧 Technologies Used

Object Settings

Universe Sandbox

The Three-Body Problem

Why it's unsolvable

Chaos in practice

Special solutions & workarounds

Bonus: Blender Solar System Render!

Video not found

Solar System Simulation