Interactively simulate gravitational slingshot maneuvers. Adjust planet mass, velocity, and launch angle to see real-time velocity changes before and after flyby.
Newton's law of gravitation (force on spacecraft):
$$\vec{F}= -\frac{G M_p m}{r^2}\hat{r}$$Equations of motion in 2D (relative position $\vec{r}= (x-x_p,\,y-y_p)$):
$$\ddot{x}= -\frac{G M_p (x-x_p)}{r^3}, \qquad \ddot{y}= -\frac{G M_p (y-y_p)}{r^3}$$Numerical integration: 4th-order Runge–Kutta (RK4). Planet moves at constant velocity.
Maximum velocity gain from an ideal trailing-side flyby:
$$\Delta v_{\max}\approx 2\,V_p$$$V_p$: planet orbital speed. Achieved when velocity vector is deflected 180° in the planet frame.
Voyager 1 & 2 (1977): Multiple gravity assists at Jupiter and Saturn boosted both probes to solar escape velocity. Voyager 1 is now over 24 billion km from Earth — the most distant human-made object.
Cassini (1997): Four gravity assists (Venus ×2, Earth ×1, Jupiter ×1) delivered the Saturn probe with far less propellant than a direct trajectory would require.
MESSENGER (2004): Six gravity assists including three at Mercury progressively shed orbital energy for Mercury orbit insertion — the first spacecraft ever to achieve it.
Here are some points where beginners often get tripped up when mastering this simulator. A major misconception is the idea that the closer you get to a planet, the more you can accelerate. While gravity does get stronger as you approach, real missions are limited by the "planet's atmosphere" and the "Roche limit" (the distance at which a satellite would be torn apart). For example, a Jupiter flyby at an altitude of 100,000 km is safe, but lowering it to 10,000 km risks the probe being destroyed by intense radiation belts. Even if it's possible in the simulation, a "safety margin" is essential for a real spacecraft.
Next, tips for setting initial conditions. The "approach velocity" and "approach angle" relative to the planet determine everything. Here, "approach angle" refers to the spacecraft's relative direction of approach against the planet's velocity vector. To maximize the effect in the simulator, the basic principle is to approach from almost directly behind the planet's direction of travel, in a nearly straight line. For instance, if you set the planet's velocity to $V_p=13\,\text{km/s}$ and the approach velocity to $V_\infty=10\,\text{km/s}$, passing at the optimal angle does not yield a theoretical maximum velocity increment of $\Delta v \approx 2V_\infty \approx 20\,\text{km/s}$. That's only in the planet's reference frame; note that the increment seen from the solar system is much smaller (on the order of a few km/s). Always be mindful of the reference frame when adjusting parameters.
Finally, regarding the simplification of the "three-body problem". This tool uses a simple two-body (planet-spacecraft) model, approximating the Sun's gravitational influence by having the planet move in a straight line at constant velocity. Therefore, it's not suitable for predicting long-term trajectories in the real solar system. In practice, you need "n-body simulations" that simultaneously consider the gravity of multiple celestial bodies. Think of this tool purely as a first step to understanding the physical essence of the flyby phenomenon.
The simulator is based on the governing equations of Gravity Slingshot (Gravity Assist) Orbital Simulator. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Engineering Design: The concepts behind Gravity Slingshot (Gravity Assist) Orbital Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Cassini-style Saturn flyby: planet mass = 5.68e26 kg, incoming velocity = 26 km/s, approach angle = 50°. Spacecraft reaches closest approach at 120,000 km altitude. Speed before flyby = 26.0 km/s, speed after = 34.2 km/s. RK4 integration tracks hyperbolic orbit; velocity gain of 8.2 km/s redirects probe toward outer system without fuel burn.