What are Kepler's three laws of planetary motion?

1. All planets move in elliptical orbits with the Sun at one focus. 2. A line joining a planet and the Sun sweeps out equal areas in equal time intervals (planets move faster when closer to the Sun). 3. The square of the orbital period is proportional to the cube of the semi-major axis (T² ∝ a³).

What is the vis-viva equation?

The vis-viva equation v² = GM(2/r − 1/a) gives the orbital speed at any point in an elliptical orbit. Here r is the current distance from the central body, a is the semi-major axis, G is the gravitational constant, and M is the central body's mass. It shows that speed increases as distance decreases.

What determines if an orbit is circular or elliptical?

The eccentricity (e) determines the orbit shape. When e = 0, the orbit is a perfect circle. As e increases toward 1, the orbit becomes more elongated. Earth's orbit has e = 0.017 (nearly circular), while Comet Halley has e = 0.967 (highly elongated). At e = 1, the trajectory becomes parabolic (escape trajectory).

Orbital Mechanics Simulator

Explore Kepler's laws and orbital dynamics. Adjust semi-major axis and eccentricity to see how orbits change shape, and observe how speed varies with distance from the central body.

Orbit Parameters

Presets

Semi-major axis = 1.000 AU

1.0

Eccentricity = 0.017

0.017

Central Mass = 1.0 M⊙

1.0

Show

Velocity vector Kepler sweep areas Gravitational field Orbit trace

Properties

Semi-major axis	1.000 AU
Eccentricity	0.017
Period	--
Perihelion	--
Aphelion	--
Distance	--
Speed	--
Escape v	--

The Physics Behind It

Orbital Formulas

Vis-viva: v² = GM(2/r − 1/a)
Kepler's 3rd law: T² = 4π²a³/(GM)
Orbit equation: r(θ) = a(1−e²)/(1+e cosθ)
Escape velocity: vₒ = √(2GM/r)

Key Concepts

Kepler's 1st law: orbits are ellipses with the central body at one focus
Kepler's 2nd law: a line from planet to star sweeps equal areas in equal times — planets move faster near perihelion
Kepler's 3rd law: the square of the period is proportional to the cube of the semi-major axis
Eccentricity determines orbit shape: e=0 is circular, e near 1 is highly elongated

Orbital Mechanics Simulator

Orbit Parameters

Properties

The Physics Behind It

Orbital Formulas

Key Concepts

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