Kepler's Third Law (Orbital Period) Calculator
Kepler's Third Law says the square of an orbital period T is proportional to the cube of the semi-major axis a, with proportionality 4π²/(G·M). Enter a central mass (Sun, Earth or custom) plus a semi-major axis to get the orbital period T for a planet, moon or spacecraft — or flip the mode to recover a from a known period.
Enter a positive mass and a positive semi-major axis or period.
Result
—
—
Other units
- Minutes
- —
- Hours
- —
- Days
- —
- Years (Julian)
- —
- Mean orbital speed
- —
Formula
T = 2π · √(a³ / (G · M))
Uses CODATA 2018 value G = 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻² and 1 AU = 1.49598 × 10¹¹ m. Assumes the two-body limit with the orbiting body much less massive than the primary — a good approximation for planets, moons and spacecraft. Use the semi-major axis a for elliptical orbits; for circular orbits a is just the orbital radius. Real periods differ slightly due to perturbations from other bodies and relativistic corrections.
Formula
T = 2π · √(a³ / (G · M)); invert to a = ∛(G · M · T² / (4π²))
- · Uses the CODATA 2018 value G = 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻².
- · 1 AU (astronomical unit) = 1.49597870700 × 10¹¹ m — the IAU 2012 defined value, roughly the mean Earth–Sun distance.
- · Assumes two-body motion with the orbiting body much less massive than the primary (planets, moons and satellites all qualify).
- · Elliptical orbits use the semi-major axis a, not the instantaneous distance. For a circular orbit a is just the orbital radius.
- · Presets: Earth a = 1 AU; Mars a ≈ 1.524 AU; Moon a ≈ 384 399 km around Earth; ISS-like LEO at 400 km altitude; GEO at 35 786 km altitude.
- · The result also includes the mean orbital speed v = √(G·M / a) — equal to the actual speed for a circular orbit.
Frequently asked
Why does Earth come out to almost exactly 365.25 days?
When a = 1 AU = 1.49598 × 10¹¹ m and M = M☉ = 1.989 × 10³⁰ kg, T = 2π·√(a³/(G·M)) ≈ 3.156 × 10⁷ s ≈ 365.25 days. That's exactly the definition of a Julian year — astronomers chose 365.25 days as a clean time unit (it's what powers the "light-year"). NASA's quoted sidereal year of 365.256 days differs because (a) Earth itself adds a tiny bit of mass making the effective G·(M+m) slightly larger, and (b) other planets perturb the orbit. The tool's idealised two-body answer of ~365.25 days agrees with NASA to four significant figures.
Why is a geostationary satellite period 23 h 56 min, not 24 h?
A geostationary satellite must keep step with Earth's rotation relative to the distant stars, and that's a sidereal day of 23 h 56 min 4 s — not the 24 h solar day. The extra ~4 minutes in a solar day come from Earth also orbiting the Sun by ~1° per day, so the planet must rotate that extra degree before the Sun is overhead again. Plug 23.9345 h and M = M⊕ into this tool and you get a ≈ 42 164 km — i.e. 35 786 km altitude, the ITU-standard GEO height.
What if it's a binary star or a satellite that isn't much lighter than its primary?
The full two-body form is T² = 4π²·a³ / (G·(M + m)), i.e. you use the total mass. When m/M < 0.001 the error is well below 0.05 %; for the Earth–Moon system m/M ≈ 0.0123 so the idealised version is off by ~0.6 %. For planet–Sun systems the approximation is rock solid. Binary stars and near-equal pairs like Pluto–Charon need both bodies counted, and you also need to remember that each body orbits the common centre of mass rather than the other body.
Related tools
Ohm's Law Calculator (V / I / R / P)
Enter any two of voltage, current, resistance, or power — the calculator solves for the other two using V = IR and P = VI.
Speed, Distance & Time Calculator
Enter any two of distance, time and speed to get the third — with km/h, mph, m/s, km, miles, hours and minutes supported.
Density Calculator (mass / volume)
Compute density from mass and volume (ρ = m / V), or solve for the missing variable. Built-in reference table for 19 common substances.
Projectile Motion Calculator
Enter launch speed, angle and height to compute projectile range, peak height and flight time (no air resistance). Pick from Earth, Moon, Mars and more.
Wind Chill Calculator
Compute the wind chill (feels-like temperature) from air temperature and wind speed using the 2001 Environment Canada / US NWS formula, with frostbite risk levels.
Dew Point Calculator
Compute dew point from air temperature and relative humidity using the Magnus formula — handy for HVAC, photography and weather analysis.
Kinetic Energy Calculator (KE = ½ m v²)
Compute kinetic energy KE = ½ m v² with mixed units (kg / g / lb and m/s / km/h / mph) and see the result in joules, kilojoules, food calories, foot-pounds and watt-hours.
Half-Life & Exponential Decay Calculator
Enter any three of initial amount, remaining amount, elapsed time and half-life to solve for the fourth — useful for radioactive decay, drug pharmacokinetics and radiometric dating.
Resistor Color Code Calculator (4 / 5 band)
Pick the colour bands and instantly read the resistance and tolerance — 4-band and 5-band notations supported, with Ω / kΩ / MΩ formatting and a closest E12 / E24 preferred-value check.
GPS Distance Calculator (Haversine)
Enter two latitude/longitude pairs to compute the great-circle distance using the haversine formula (km, miles, nautical miles), with bearing and midpoint.
Solution Dilution Calculator (C₁V₁ = C₂V₂)
Solve any one of C₁, V₁, C₂, V₂ from the dilution equation C₁V₁ = C₂V₂ — a daily lab essential for chemistry, biology and pharmacy work.
Decibel (dB) Sum Calculator
Two 80 dB sound sources do not equal 160 dB. Enter multiple dB values to compute the combined SPL, and subtract background noise to recover the signal alone.
Resistor Parallel / Series Calculator
Enter up to 8 resistor values to see the series total (R₁ + R₂ + …) and the parallel total (1 / Σ(1/Rᵢ)) at the same time.
Wavelength ↔ Frequency Calculator
Convert between electromagnetic wavelength and frequency via c = λf, with the matching spectrum band (radio / microwave / visible / X-ray / γ) and photon energy.
Tank Volume Calculator
Compute the capacity of vertical or horizontal cylindrical, rectangular and spherical tanks, including partial-fill volumes at a given liquid level.
Pendulum Period Calculator (T = 2π√L/g)
Enter the pendulum length and local gravity to get the period, frequency and angular frequency, with Earth/Moon/Mars/Jupiter presets — and reverse-solve for the length needed to hit a target period.
Heat Index Calculator
Enter air temperature and relative humidity to get the apparent temperature (NOAA Rothfusz heat index) and the corresponding heat-stress risk band.
Vehicle Stopping Distance Calculator
Enter speed, reaction time and road friction to estimate reaction, braking and total stopping distance.
Snell's Law Refraction Calculator
Enter the refractive indices of two media and an angle of incidence — get the refraction angle and critical angle from Snell's law (n₁ sin θ₁ = n₂ sin θ₂).
Capacitor Energy Calculator
Enter capacitance (F, mF, µF, nF, pF) and voltage to compute the stored energy (E = ½CV²) and charge (Q = CV) on a capacitor.
Boiling Point at Altitude Calculator
Enter altitude to compute the boiling point of water (°C / °F) and local air pressure using the ICAO standard atmosphere and the Antoine equation — useful for hiking, cooking and high-altitude baking.
Specific Heat (Q = mcΔT) Calculator
Solve Q = m × c × ΔT for any one of heat energy, mass, specific heat capacity or temperature change — with presets for water, aluminium, iron, copper, glass, air and more.
pH and Hydrogen Ion Concentration Calculator
Convert between pH, pOH, hydrogen-ion concentration [H⁺] and hydroxide concentration [OH⁻] — with acid / neutral / alkaline classification.
Ideal Gas Law (PV = nRT) Calculator
Pick the unknown (P, V, n or T), enter the other three and PV = nRT is solved instantly — works in Pa / kPa / atm / bar / mmHg / psi, m³ / L / mL, mol / mmol / kmol and K / °C / °F.
Coulomb's Law Calculator
Enter two point charges and the separation distance to compute the electrostatic force between them via F = kₑ·q₁·q₂/r² (attractive or repulsive).
Newton's Law of Universal Gravitation Calculator
Enter two masses and the distance between them to compute the gravitational attraction via F = G·m₁·m₂/r².
Thin Lens Equation Calculator
Given any two of the three quantities (object distance u, image distance v, focal length f), solve for the remaining one and the lateral magnification.
Distance to Horizon Calculator
Enter the observer eye height above the surface to compute the distance to the geometric and refraction-corrected horizon.
Hooke's Law (Spring Force) Calculator
Given any two of spring constant k, displacement x or restoring force F, solve for the third and the elastic potential energy U = ½kx².
Buoyancy Force (Archimedes' Principle) Calculator
Enter fluid density, submerged volume and gravitational acceleration to compute the buoyant force F = ρ V g, plus whether the object floats, sinks or stays neutral.
Voltage Divider Calculator
Enter the input voltage and two series resistor values to find the divider output voltage, current and power dissipated in each resistor.
Escape Velocity Calculator
Enter the mass and radius of a celestial body (or pick Earth, Moon, Mars and other presets) to compute the minimum surface launch speed v = √(2GM/r) needed to escape its gravity.
Newton's Law of Cooling Calculator
Enter initial temperature, ambient temperature, cooling constant and elapsed time to estimate an object's temperature with T(t) = T∞ + (T₀ − T∞)·e^(−kt), plus half-cooling time and time constant.
Centripetal Force Calculator
Enter mass and radius, then linear speed, angular speed or period, and instantly read off centripetal force F = m·v²/r, centripetal acceleration, tangential speed, angular speed, period and frequency.
LC Resonant Frequency Calculator
Enter inductance L (H) and capacitance C (F) to compute the resonant frequency f = 1 / (2π√(LC)) of an LC tank circuit, plus its period and angular frequency.
Speed of Sound in Air Calculator
Enter the air temperature and compute the speed of sound in dry air using v = 331.3 × √(1 + T/273.15) — output in m/s, km/h, mph, ft/s plus the "count-the-seconds" thunder distance rule.
Reynolds Number Calculator
Enter fluid density, velocity, characteristic length and viscosity to compute the Reynolds number and classify the flow as laminar, transitional or turbulent.
Spring Potential Energy Calculator
Enter the spring constant k and displacement x to instantly compute the elastic potential energy stored in a Hookean spring via U = ½·k·x², with conversions to kJ, kcal, ft·lbf and Wh for easy comparison.
Beer-Lambert Law Absorbance Calculator
Use A = ε·c·ℓ to compute absorbance, transmittance and concentration — any three of the four inputs determines the fourth.
Drag Force Calculator
Enter fluid density, velocity, drag coefficient Cd and frontal area and compute the drag force from Fd = ½·ρ·v²·Cd·A — handy for cycling, automotive, ballistic and skydiving scenarios.
Hydrostatic Pressure Calculator
Compute the hydrostatic pressure P = ρ·g·h from fluid density, depth and gravity, and convert it to kPa, bar, psi, atm, mmHg and metres of water — useful for diving, aquariums and piping.
Buffer pH (Henderson–Hasselbalch) Calculator
Enter pKa together with the conjugate-base [A⁻] and weak-acid [HA] concentrations and apply the Henderson–Hasselbalch equation to compute buffer pH, the buffer ratio and the effective buffering range.