# Orbit Simulator - Kepler's Laws, Orbital Elements & 3-Body Problem

> Drag a satellite's velocity to get circular, elliptic or hyperbolic orbits with live orbital elements, Kepler's equal areas, N-body demos and Δv tools.

- URL: https://qilingtools.com/en/math/orbit
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Place a satellite anywhere, drag the yellow arrow to give it a velocity, and see whether it follows a circle, an ellipse, a parabola or a hyperbola. The semi-major axis, eccentricity, periapsis, apoapsis and period are computed live from the state vectors, and equal-time sectors make Kepler's second law visible. The integrator is a symplectic leapfrog with steps that shrink as √(r³/μ), so energy drift stays tiny; there are also classic N-body scenes (figure-eight, butterfly, Pythagorean) and a calculator for orbital and escape speeds, synchronous orbits and Hohmann transfers.

## How to use

1. Pick a preset (ISS, geostationary, Molniya, Moon, Halley's Comet, escape velocity…) or a different central body
2. Press play; drag the arrow tip to change the velocity, drag the satellite to move it, or type an altitude, speed and flight-path angle
3. Read the orbit type, orbital elements, energy and sector areas, then try the N-body and calculator tabs

## Features

- Drag-to-set initial state with the full predicted conic
- Orbital elements a, e, rₚ, rₐ, T, h and ε from state vectors
- Kepler's second law: equal-time sectors with measured areas
- Symplectic leapfrog with adaptive substeps and an energy-drift readout
- N-body presets: figure-eight, butterfly, binary star, Lagrange triangle, Pythagorean
- Circular and escape speeds, periods, synchronous orbits and Hohmann Δv for the planets

## FAQ

### Why does a slightly faster satellite end up on an ellipse instead of a circle?

At distance r an orbit is circular only if the speed is exactly √(μ/r) and perpendicular to the radius. Any faster and that point becomes the periapsis, so the satellite climbs onto an ellipse. At √(2μ/r), the escape speed and √2 times the circular speed, the specific orbital energy is zero and the path is a parabola. Faster still gives a hyperbola and the satellite never returns. Slower turns the point into the apoapsis, and if the periapsis falls below the surface the satellite crashes.

### Why does Kepler's second law hold, and how are the sector areas measured?

Gravity points at the centre, so it exerts no torque and the angular momentum h = r × v is conserved. The rate at which the radius vector sweeps area is exactly dA/dt = h/2, so equal times sweep equal areas. The tool cuts the path into twelfths of the period, measures each sector with the shoelace formula and divides by h·Δt/2. The ratios should come out very close to 1, even though sectors near periapsis are short and wide and those near apoapsis long and thin.

### Why not just use Euler's method? What is a symplectic integrator?

Euler's method adds a little energy every step, so the satellite spirals outward and after a few dozen orbits the result is meaningless. Leapfrog (velocity Verlet) is symplectic: it preserves phase-space volume, so the energy error oscillates within a small band instead of growing and orbits stay closed over long runs. Here the step also shrinks as √(r³/μ), which keeps high-eccentricity orbits such as Halley's accurate at perihelion. The relative energy drift is shown on screen.

### Is the three-body problem really unsolvable? What is the figure-eight orbit?

There is no general closed-form solution, and most starting conditions give chaotic motion in which tiny differences grow exponentially. The Pythagorean problem, with masses 3, 4 and 5 released from rest, is the textbook example and ends by ejecting one star. Special periodic solutions do exist, though: Lagrange's equilateral triangle, and the figure-eight found by Moore in 1993 and proved by Chenciner and Montgomery in 2000, where three equal masses chase each other around one figure-eight curve. Šuvakov and Dmitrašinović later found more, such as the butterfly.

### How are the synchronous orbit altitude and the Hohmann Δv calculated?

Set the orbital period equal to the body's rotation period (one sidereal day, 23 h 56 min 4 s, for Earth) and solve T = 2π√(r³/μ) for r = (μT²/4π²)^(1/3). Subtracting the radius gives about 35,786 km for Earth; the tool uses the mean radius, so it differs slightly. A Hohmann transfer uses an ellipse tangent to both circles: burn Δv₁ to enter it and Δv₂ at apoapsis to circularise. LEO to GEO takes about 3.9 km/s in total and roughly 5.3 hours.

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