Astronomy Calculator — Kepler’s Laws, Orbital Period

Compute orbital period, escape velocity and gravitational acceleration using Kepler’s and Newton’s laws.

About this calculator

An astronomy calculator handles orbital mechanics (Kepler's third law), gravitational acceleration, escape velocity, and stellar magnitude/luminosity calculations. Essential for astrophysics coursework, satellite trajectory planning, and amateur astronomy.

Kepler's third law (T² ∝ a³) governs every orbit in the solar system and beyond. The constant of proportionality involves the mass of the central body — measuring a planet's orbital period and distance gives its star's mass.

Formula

T^2 = \frac{4\pi^2}{GM} a^3 \;;\; v_{esc} = \sqrt{\frac{2GM}{r}}

  • T — Orbital period
  • a — Semi-major axis (orbit radius for circular)
  • G — 6.674 × 10⁻¹¹ N·m²/kg²
  • M — Mass of central body
  • v_esc — Escape velocity from radius r

Worked example

Inputs: Escape velocity from Earth's surface

  1. M_Earth = 5.972 × 10²⁴ kg
  2. r = 6.371 × 10⁶ m (Earth radius)
  3. v_esc = √(2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / 6.371 × 10⁶)
  4. v_esc = √(1.251 × 10⁸)
  5. v_esc = 11.18 km/s — must reach this to leave Earth's gravity well

Frequently asked questions

Why are all the planets in the same orbital plane?

They formed from a single rotating disk of gas and dust (the solar nebula). Conservation of angular momentum kept everything in roughly the same plane during accretion. Small inclinations (Mercury 7°, Pluto 17°) come from later gravitational interactions and small initial irregularities. The exoplanets observed by Kepler and TESS also typically orbit in coplanar configurations, confirming the model.

How is the apparent magnitude scale defined?

Apparent magnitude m measures how bright a star appears from Earth. The scale is logarithmic and inverted: lower magnitude = brighter. A difference of 5 magnitudes corresponds to exactly 100× brightness ratio (each magnitude = 100^(1/5) ≈ 2.512 ratio). Sun: m = −26.7; full Moon: −12.6; Venus at brightest: −4.4; faintest naked-eye stars: +6; faintest telescope objects: +30.

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