Pyxis· 03 · Kepler
Newtonian gravity · two-body problem
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Kepler's laws connect orbit shape, central mass, and period. Pick a system — the ISS, Halley's Comet, S2 around Sagittarius A* — and watch the math reveal perihelion, aphelion, and how long an orbit takes.

01 Pick a system central body · orbit
Orbital scenarios
Earth orbiting Sun
Our home world. The reference orbit.
Semi-major axis 1.000 AU
Eccentricity 0.0167
Orbit shape · central body marked at focus
02 Orbital period T = 2π√(a³/GM)
One orbit takes
1.000 yr
Earth completes one orbit of the Sun every 365.25 days.
Mean orbital speed: 29.78 km/s — about 30 times faster than a bullet.
Period (seconds)
3.156 × 10⁷ s
Semi-major axis
1.496 × 10¹¹ m
Mean speed
29.78 km/s
Orbit count / yr
1.000
03 Orbital characteristics vis-viva · conservation laws
Perihelion (closest)
0.983 AU
Closest approach to the central body. Earth reaches this point in early January each year.
r_peri = a(1 - e)
Aphelion (farthest)
1.017 AU
Farthest distance from the central body. Earth reaches this in early July.
r_apo = a(1 + e)
Velocity at perihelion
30.29 km/s
Fastest orbital speed — reached at closest approach (Kepler's 2nd law: equal areas in equal times).
v² = GM(2/r - 1/a)
Velocity at aphelion
29.29 km/s
Slowest orbital speed — reached at farthest distance.
v_apo × r_apo = v_peri × r_peri
Escape velocity at perihelion
42.13 km/s
The speed required to leave the system entirely from this distance. Higher than orbital velocity by a factor of √2.
v_esc = √(2GM/r)
Specific orbital energy
-4.43 × 10⁸ J/kg
Total energy per kilogram. Negative means bound. Zero means escape. Positive means hyperbolic flyby.
ε = -GM/(2a)
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03 · Kepler · orbital mechanics