Kepler Orbital Period
Compute orbital period T = 2π√(a³/GM) from semi-major axis (m) and central mass (kg).
Kepler's laws: T² = a³ (AU and years)
Kepler formulated three laws of planetary motion between 1609 and 1619, based on Tycho Brahe's observations. 1st law: planets orbit in ellipses with the Sun at one focus (not a circle, as previously believed). 2nd law: the radius vector sweeps equal areas in equal times — a direct consequence of angular momentum conservation, so planets move faster at perihelion than aphelion. 3rd law: T² = k·a³; expressed in astronomical units (AU) and years for the solar system, the constant becomes exactly 1, giving T² = a³. The generalized form is T² = (4π²/GM)·a³, valid for any central body of mass M. Example: Mercury orbits with a = 0.387 AU, so T² = 0.058 → T = 0.24 years (about 88 days), matching observation. Modern exoplanet missions (TESS, James Webb) detect distant worlds by the transit method — dimming as the planet crosses the host star.
Applications
Space mission design (Hohmann transfer orbits, gravitational slingshots used by Voyager 1 and 2 to leave the solar system), GPS constellation (satellites orbit at ~20,200 km with T = 12 h, a = 26,560 km), geostationary satellites for TV and communications (T = 23 h 56 min sidereal day, a = 42,164 km from Earth's center), determining stellar masses in binary systems, and confirming exoplanet detection.
FAQ
Why does T² = a³ work without units in the solar system? Choosing AU for distance and years for period makes the constant 4π²/GM equal to 1 when M is one solar mass. For other systems (Jupiter's moons, exoplanets), you need the full equation with G and M.
Do the laws apply to artificial satellites? Yes — they apply to anything orbiting a central mass under gravity, using Earth's mass instead of the Sun's. A satellite at a = 7,000 km from Earth's center has T ≈ 97 min.
How did Newton explain Kepler's laws? Newton derived all three from his law of universal gravitation (F = GMm/r²) plus F = ma — turning Kepler's empirical descriptions into mathematical consequences of a single physical law.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.