Are planetary orbits actually perfect ellipses?

They are extremely close to ellipses over short times, but mutual planetary perturbations cause slow precession and small deviations. Mercury's perihelion advance includes a measurable general-relativistic contribution.

Why is area swept per time constant?

Angular momentum about the Sun is conserved for central gravity; the areal rate is L/(2m), a constant of motion.

Can I use Kepler's third law on a satellite around Earth?

Yes, with the Earth's mass as the central body: T² = (4π²/GM) a³ for circular orbits is a common form. For low Earth orbit, the orbit is still nearly elliptical with Earth at a focus.

What breaks if I treat a comet as a circular orbit?

Many comets are highly eccentric; assuming a circle mis-estimates period, energy, and heliocentric speed. Use the actual eccentricity from observations.

Kepler's Laws

Elliptical orbits with equal-area sweeps visualized.

About this model

Kepler's three laws summarize planetary motion in the Sun's gravitational field: (1) orbits are ellipses with the Sun at one focus, (2) the radius vector sweeps equal areas in equal times—conservation of angular momentum, and (3) the square of the orbital period scales with the cube of the semi-major axis, T² ∝ a³ for a given central mass. Newton later showed these follow from F = GMm/r² and F = ma. This simulator lets learners vary eccentricity, semi-major axis, and primary mass to connect geometry to period and speed, often visualizing an ellipse with perihelion and aphelion speeds obeying vis-viva energy relations. Idealizations include a single dominant point mass, negligible perturbations from other planets, and non-relativistic speeds. Students should exit understanding why circular orbits are a special case (e = 0), why area sweep rate is constant even for eccentric paths, and how the third law enables weighing the Sun from planetary periods.

Who it's for: High school and early undergraduate astronomy and mechanics courses linking observational astronomy to Newtonian gravitation.

Key terms

Kepler's laws
Ellipse
Semi-major axis
Eccentricity
Areal velocity
Orbital period
Vis-viva equation
Inverse-square law

How it works

Kepler I: bound orbits are ellipses with the central mass at one focus. Kepler II: the radius vector sweeps equal areas in equal times — here, colored wedges use equal Δt and should have nearly equal area (numerical spread shown). Kepler III: for this two-body model, T² ∝ a³ with constant GM; adjust a and GM and watch T²/a³ track the inverse of GM scaling.

Key equations

M = E − e sin E → planet (a(cos E − e), b sin E)

T = 2π √(a³ / GM), n = 2π / T

dA/dt = L / (2m) = const.

Frequently asked questions

Are planetary orbits actually perfect ellipses?: They are extremely close to ellipses over short times, but mutual planetary perturbations cause slow precession and small deviations. Mercury's perihelion advance includes a measurable general-relativistic contribution.
Why is area swept per time constant?: Angular momentum about the Sun is conserved for central gravity; the areal rate is L/(2m), a constant of motion.
Can I use Kepler's third law on a satellite around Earth?: Yes, with the Earth's mass as the central body: T² = (4π²/GM) a³ for circular orbits is a common form. For low Earth orbit, the orbit is still nearly elliptical with Earth at a focus.
What breaks if I treat a comet as a circular orbit?: Many comets are highly eccentric; assuming a circle mis-estimates period, energy, and heliocentric speed. Use the actual eccentricity from observations.

Other simulators in this category — or see all 27.

View category →

School

Escape Velocity

Launch Simulator

Launch from different planets and see trajectory results.

NewUniversity / research

Gravitational Lensing

Launch Simulator

Massive objects bending light. Visual distortion effects.

NewUniversity / research

Lagrange Points L1–L5

Launch Simulator

CRTBP effective potential; L1–L5; Coriolis test particle.

NewSchool

Earth–Moon Tides

Launch Simulator

Equilibrium tide bulges; orbit speed; ~12.4 h spacing note.

NewSchool

Binary Star (circular)

Launch Simulator

COM orbits; r₁,r₂; Kepler T² ∝ a³/(M₁+M₂).

NewUniversity / research

Roche Limit

Launch Simulator

Fluid d ≈ 2.456 R_p (ρ_p/ρ_s)^(1/3); vs orbit distance.

Kepler's Laws

About this model

How it works

Key equations

Frequently asked questions

More from Gravity & Orbits

Escape Velocity

Gravitational Lensing

Lagrange Points L1–L5

Earth–Moon Tides

Binary Star (circular)

Roche Limit

Kepler's Laws

About this model

How it works

Key equations

Frequently asked questions