Kepler's Law Calculator: Orbital Period, Velocity & Perihelion

Calculate planetary orbital periods, perihelion, aphelion, and orbital velocities using Kepler's three laws of planetary motion.

Quick 1-Click Celestial Orbits

Orbital Geometry & Stellar Mass Parameters

1 AU = 1.496 × 10⁸ km
0 = Circular, >0 = Elliptical
1 M☉ = 1.989 × 10³⁰ kg
Orbital Period (T)
1.88years
687 Earth days
Perihelion Distance
1.382AU
Max Speed: 26.5 km/s
Aphelion Distance
1.666AU
Min Speed: 22.0 km/s
Mean Orbital Speed
24.1km/s
Semi-minor axis b = 1.517 AU

Keplerian Elliptical Orbit Diagram

Sun at primary focus (F1) showing perihelion, aphelion, and orbital eccentric deviation.

Sun (F1)Perihelion (1.38 AU)Aphelion (1.67 AU)Orbiting Body

Orbital Velocity vs True Anomaly

Velocity (km/s) variation across a full 360° orbital revolution (Vis-Viva profile).

Solar System Benchmark Orbital Data

Observed astronomical parameters verified against Kepler's Third Law.

Celestial BodySemi-Major Axis (AU)Eccentricity (e)Period (Years)Mean Speed (km/s)
Mercury0.387 AU0.20560.241 yr47.36 km/s
Venus0.723 AU0.00680.615 yr35.02 km/s
Earth1.000 AU0.01671 yr29.78 km/s
Mars1.524 AU0.09341.881 yr24.07 km/s
Jupiter5.204 AU0.048511.862 yr13.07 km/s
Saturn9.582 AU0.055529.457 yr9.69 km/s
Halley's Comet17.834 AU0.967175.32 yr7.05 km/s

Worked Example: Mars Orbit (a = 1.524 AU, e = 0.0934, M = 1.0 M☉)

Calculating the complete orbital characteristics of Mars around the Sun:

Semi-Major Axis (a)
1.524 AU
Orbital Eccentricity (e)
0.0934 (dimensionless)
Central Star Mass (M)
1.0 Solar Mass (M☉)
Gravitational Model
Newtonian Keplerian Two-Body Mechanics
Orbital Mechanics Formulations:
•1. Kepler's 3rd Law Period: T = √(a³ / M) = √(1.524³ / 1.0) = √3.5396 = 1.8814 Earth years (687.2 days).
•2. Semi-Minor Axis: b = a × √(1 - e²) = 1.524 × √(1 - 0.0934²) = 1.524 × 0.9956 = 1.5173 AU.
•3. Perihelion Distance: q = a(1 - e) = 1.524 × (1 - 0.0934) = 1.3817 AU (closest approach to Sun).
•4. Aphelion Distance: Q = a(1 + e) = 1.524 × (1 + 0.0934) = 1.6663 AU (farthest point from Sun).
•5. Perihelion Velocity: v_peri = 29.78 × √(1/a) × √((1+e)/(1-e)) = 29.78 × 0.810 × 1.098 = 26.50 km/s.
•6. Aphelion Velocity: v_aph = 29.78 × √(1/a) × √((1-e)/(1+e)) = 29.78 × 0.810 × 0.911 = 21.97 km/s.

💡 Astrophysics Insight: Mars orbits the Sun in 1.88 Earth years. Because of its 0.093 eccentricity, its orbital velocity swings from 26.5 km/s at perihelion down to 22.0 km/s at aphelion, illustrating Kepler's 2nd law in action.

Frequently Asked Questions

Authoritative Celestial Mechanics & Orbital Standards

NASA Planetary Fact Sheet - Solar System Orbital Data

Official NASA astronomical data for planetary semi-major axes, eccentricities, orbital periods, and velocities.

MIT OpenCourseWare - Celestial Mechanics & Keplerian Orbits

University lecture series on Newton's derivation of Kepler's laws, angular momentum conservation, and vis-viva kinematics.

IAU Working Group on Cartographic Coordinates & Rotational Elements

International Astronomical Union reference standards for celestial ephemerides and orbital parameters.

Cambridge University Press - Orbital Mechanics for Engineering Students

Standard aerospace engineering textbook on orbital maneuvering, transfer orbits, and two-body dynamics.

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Quick Tips

  • 1Use consistent units throughout calculations
  • 2Consider significant figures in results
  • 3Document experimental conditions
  • 4Verify results with control calculations