Change the shape
The star occupies one focus rather than the geometric centre, except for a circular orbit. With size and mass unchanged, changing the eccentricity preserves the period.
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One orbit.
A thousand ways to understand.
A small planet around a fixed star. Change the shape, size and gravitation to see what happens.
Motion follows the calculated time: the planet speeds up near the star and slows down as it moves away.
Star · focus Planet Centre of the ellipse Sectors to compare
Elapsed time…
—Current distance to the star
—Current speed
The gold and turquoise sectors have the same area, even when their shapes differ.
One complete orbit
—Nearest · periapsis
—Farthest · apoapsis
—Try, then compare
Three small experiments.
Change the speed
Equal time intervals correspond to equal swept areas. Near the star, the planet travels a longer arc: its speed is greater. Look at the two sectors, then start the animation.
For a circle, distance and speed remain constant.
Change the orbital period
Doubling the semi-major axis multiplies the period by about 2.83. Doubling the star’s mass, for an orbit of the same size, multiplies the period by about 0.707.
Buttons are disabled if doubling would exceed the simulator’s range.
The equations behind the motion.
P = 2π √(a³ / μ)P is the period, a the semi-major axis in metres and μ the star’s gravitational parameter. The planet’s mass is neglected here.
rmin = a(1 − e)
rmax = a(1 + e)Eccentricity changes the nearest and farthest distances at a fixed semi-major axis.
M = E − e sin(E)
M = 2π t / PKepler’s equation relates time to position. The animation advances in time, producing the changing speed.
v² = μ(2/r − 1/a)The vis-viva equation gives the speed at distance r. In the physical equations, distances are in metres and time in seconds.
Model, units and sources
Isolated Newtonian model: a planet of negligible mass, a fixed point star at one focus and a flat elliptical orbit. Relativity, other planets, collisions and the star’s actual radius are ignored. Changing a setting constructs a different orbit: no thrust or actual change of mass during flight is simulated.
The semi-major axis a controls size, e controls shape and the star’s mass controls gravitation. The displayed phase is the fraction of time elapsed since periapsis, not the fraction of angle travelled. The time slider allows the entire experiment without animation. Animation starts only when you request it and pauses when you leave the tab.
Playback speed is expressed in simulated days per real second. It stays the same when physical parameters change, making their effect visible. The “Adjust” button selects a playback of about 20 seconds per orbit; it changes only the animation.
Constants: 1 AU = 149,597,870,700 m; one day = 86,400 s; one displayed year = 365.25 days. The reference solar parameter is μ☉ = 1.32712440041279419 × 10²⁰ m³/s², multiplied by the selected number of solar masses.
The examples use only rounded values of a and e from J2000 elements published by JPL. “Earth” is a rounded reference for the Earth–Moon system. No date, actual inclination or perturbation calculations are performed. To prepare an observation for a particular date, use the planner.
Each sector covers 1/12 of the period: one begins at periapsis, the other at apoapsis. Displayed areas are calculated in the model; the chart uses the same scale on both axes. The 1 AU circle is a distance reference, not Earth’s exact path. The arrow shows the direction of motion; its length does not encode speed.
The CSV contains 181 evenly spaced times over one orbit, with parameters, positions, distances and speeds. The final row closes the orbit. Data are calculated on your device; no account or saved settings are required.
References accessed on 19 October 2026: NASA Science · Kepler’s laws ; NASA/JPL · Kepler’s equation and planetary elements ; NASA/JPL · constants and units.
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