Solar System Simulator — Planetary Orbits & Kepler's Laws

The most famous and practical of Kepler's Laws is the Third Law, which relates a planet's orbital period (its "year") to the semi-major axis of its elliptical orbit (essentially its average distance from the Sun).

Where $T$ is the orbital period (e.g., in Earth years), and $a$ is the semi-major axis (often in Astronomical Units, AU, where 1 AU is the Earth-Sun distance). The constant is the same for all objects orbiting the same central body (like our Sun).

When we include Newton's Law of Universal Gravitation, we can derive the exact constant, leading to a more powerful form of the law.

Here, $G$ is the gravitational constant, and $M_{\odot}$ and $m$ are the masses of the Sun and the planet. Since the Sun's mass is enormous, $M_{\odot}+ m \approx M_{\odot}$. This equation is the foundation for calculating orbits of planets, asteroids, and human-made satellites.

Satellite Trajectory Design: Kepler's Laws are the ABCs of placing satellites in correct orbits. Engineers use them to calculate the precise speed and altitude needed for a communications satellite to remain in geostationary orbit, appearing fixed over one point on Earth.

Space Mission Planning: Every interplanetary mission, like the Mars rovers, relies on these laws. Mission planners calculate efficient transfer orbits (called Hohmann transfers) based on Kepler's Third Law to determine launch windows and flight duration with minimal fuel.

Exoplanet Discovery: Astronomers apply these laws in reverse to discover planets around other stars. By measuring the periodic wobble or dimming of a star (caused by an orbiting planet), they can use Kepler's Third Law to estimate the planet's orbital distance and mass.

Aerospace CAE & Simulation: Before launch, CAE software uses orbital mechanics to simulate structural loads on a spacecraft during launch, predict thermal cycles it will experience in orbit, and model the extreme aerodynamic heating it must withstand during atmospheric re-entry.

There are a few key points you should be aware of when starting to use this simulator. First, don't assume the orbits are perfect circles. While the tool displays them as clear circles for simplicity, actual planetary orbits are ellipses, with the Sun located at one "focus," not the center. For example, Mars's orbital eccentricity is about 0.09, meaning its distance from the Sun varies by roughly 20% between perihelion and aphelion. This "deviation" is one reason why launch windows for spacecraft are so narrow.

Second, understand that the simulation ignores the gravitational influence of other planets. This simulation is fundamentally based on the two-body problem of "the Sun and one planet." However, in reality, the gravity of massive planets like Jupiter perturbs the orbits of others. Practical trajectory calculations must handle this "n-body problem," making the computations significantly more complex.

Third, watch out for pitfalls with the "simulation speed" setting. If you set the speed to maximum to observe long-term motion, computational errors can accumulate, causing the orbit to drift slightly over time. This stems from the numerical integration method used (like Euler's method). In practice, higher-precision methods like Runge-Kutta are employed. You can sometimes observe this error effect by zooming in fully and watching Mercury's motion at ultra-high speed in the tool—give it a try.