Orbital Period Calculator

Kepler's Laws of Planetary Motion

Kepler's Third Law

The square of the orbital period is proportional to the cube of the semi-major axis:

T² = (4π²/GM) × a³

Where:

• T = orbital period
• G = gravitational constant (6.674 × 10⁻¹¹ m³/kg/s²)
• M = mass of central body
• a = semi-major axis

Orbital Velocity

For circular orbits:

v = √(GM/r)

For elliptical orbits (at any point):

v = √(GM(2/r - 1/a))

Kepler's First Law

Planets orbit in ellipses with the central body at one focus. The eccentricity (e) describes the shape:

• e = 0: Perfect circle
• 0 < e < 1: Ellipse
• e = 1: Parabola (escape orbit)
• e > 1: Hyperbola (unbound orbit)

Kepler's Second Law

A line drawn from the central body to the orbiting object sweeps equal areas in equal times.

Examples of Orbital Systems

System	Semi-major Axis	Period	Central Mass	Eccentricity
Earth around Sun	1.00 AU	365.25 days	1.00 M☉	0.017
Moon around Earth	384,400 km	27.3 days	1.00 M⊕	0.055
ISS around Earth	~408 km altitude	~93 minutes	1.00 M⊕	~0.001
Jupiter around Sun	5.20 AU	11.9 years	1.00 M☉	0.049
Proxima Cen b	0.05 AU	11.2 days	0.12 M☉	~0.11

Applications

• Satellite Design: Determine orbital parameters for communications and Earth observation satellites
• Exoplanet Research: Calculate planet masses and orbital characteristics from transit data
• Mission Planning: Design spacecraft trajectories and encounter timing
• Binary Stars: Determine stellar masses from orbital motion