Orbital Period Calculator

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Orbital Period Calculator

Calculate the orbital period of any object orbiting a central body using Kepler's Third Law of Planetary Motion.

Semi-Major Axis (a)

km m AU light-years

Distance from the orbiting body to the center of mass of the central body

Mass of Central Body (M)

kg Solar masses Earth masses

Mass of the body being orbited (e.g. Sun = 1.989 × 10³⁰ kg)

Orbital Eccentricity (e) (optional, default = 0)

Eccentricity does not affect the period — only the semi-major axis matters (Kepler's 3rd Law)

Output Unit

Seconds (s) Minutes Hours Days Years

Calculate Orbital Period

function orbCalc() { const resultDiv = document.getElementById('orb-result'); const errorDiv = document.getElementById('orb-error'); resultDiv.style.display = 'none'; errorDiv.style.display = 'none';

// --- Read inputs --- const aRaw = parseFloat(document.getElementById('orb-semi-major-axis').value); const mRaw = parseFloat(document.getElementById('orb-central-mass').value); const eccRaw = document.getElementById('orb-eccentricity').value.trim(); const axisUnit = document.getElementById('orb-axis-unit').value; const massUnit = document.getElementById('orb-mass-unit').value; const outUnit = document.getElementById('orb-output-unit').value;

// --- Validate --- function showError(msg) { errorDiv.textContent = msg; errorDiv.style.display = 'block'; }

if (isNaN(aRaw) || document.getElementById('orb-semi-major-axis').value.trim() === '') { return showError('Please enter a valid semi-major axis.'); } if (aRaw = 1) { return showError('Eccentricity must be between 0 (inclusive) and 1 (exclusive) for a bound orbit.'); } }

// --- Constants --- const G = 6.674e-11; // m³ kg⁻¹ s⁻²

// --- Convert semi-major axis to metres --- let a_m; if (axisUnit === 'm') a_m = aRaw; else if (axisUnit === 'km') a_m = aRaw * 1e3; else if (axisUnit === 'au') a_m = aRaw * 1.495978707e11; else if (axisUnit === 'ly') a_m = aRaw * 9.4607304725808e15;

// --- Convert mass to kg --- let M_kg; if (massUnit === 'kg') M_kg = mRaw; else if (massUnit === 'solar') M_kg = mRaw * 1.989e30; else if (massUnit === 'earth') M_kg = mRaw * 5.972e24;

// --- Kepler's Third Law: T = 2π √(a³ / GM) --- const T_s = 2 * Math.PI * Math.sqrt(Math.pow(a_m, 3) / (G * M_kg));

if (!isFinite(T_s) || isNaN(T_s)) { return showError('Calculation resulted in an invalid number. Please check your inputs.'); }

// --- Convert period to desired output unit --- let T_out, unitLabel; if (outUnit === 's') { T_out = T_s; unitLabel = 'seconds'; } else if (outUnit === 'min') { T_out = T_s / 60; unitLabel = 'minutes'; } else if (outUnit === 'hr') { T_out = T_s / 3600; unitLabel = 'hours'; } else if (outUnit === 'day') { T_out = T_s / 86400; unitLabel = 'days'; } else if (outUnit === 'yr') { T_out = T_s / 31557600; unitLabel = 'years'; } // Julian year

// --- Also compute a human-readable breakdown --- function breakdown(seconds) { const yrs = Math.floor(seconds / 31557600); const rem1 = seconds % 31557600; const days = Math.floor(rem1 / 86400); const rem2 = rem1 % 86400; const hrs = Math.floor(rem2 / 3600); const rem3 = rem2 % 3600; const mins = Math.floor(rem3 / 60); const secs = Math.round(rem3 % 60); let parts = []; if (yrs > 0) parts.push(yrs + ' yr' + (yrs !== 1 ? 's' : '')); if (days > 0) parts.push(days + ' day' + (days !== 1 ? 's' : '')); if (hrs > 0) parts.push(hrs + ' hr' + (hrs !== 1 ? 's' : '')); if (mins > 0) parts.push(mins + ' min' + (mins !== 1 ? 's' : '')); if (secs > 0 || parts.length === 0) parts.push(secs + ' sec' + (secs !== 1 ? 's' : '')); return parts.join(', '); }

// --- Format number nicely --- function fmt(n) { if (n >= 1e6 || (n 0)) return n.toExponential(6); return parseFloat(n.toPrecision(8)).toLocaleString('en-US', {maximumFractionDigits: 6}); }

// --- Orbital velocity at periapsis / apoapsis (bonus info) --- const ecc = eccRaw !== '' ? parseFloat(eccRaw) : 0; const mu = G * M_kg; // standard gravitational parameter // vis-viva at semi-major axis (mean orbital speed approximation for circular) const v_mean = 2 * Math.PI * a_m / T_s; // m/s (exact for circular, approximate for elliptical)

resultDiv.innerHTML = '' + 'T = ' + fmt(T_out) + ' ' + unitLabel + '' + 'Breakdown: ' + breakdown(T_s) + '' + 'Period in seconds: ' + fmt(T_s) + ' s' + 'Mean orbital speed: ' + fmt(v_mean / 1000) + ' km/s' + 'Semi-major axis used: ' + fmt(a_m / 1.495978707e11) + ' AU (' + fmt(a_m) + ' m)';

resultDiv.style.display = 'block'; }

#### Formula

Kepler's Third Law (Newton's form):

T = 2π √( a³ / GM )

T — Orbital period (seconds)
a — Semi-major axis of the orbit (metres)
G — Gravitational constant = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²
M — Mass of the central body (kg)

Note: The orbital period depends only on the semi-major axis and the central mass — not on eccentricity. Two orbits with the same semi-major axis but different eccentricities have identical periods (Kepler's Third Law).

Mean orbital speed (circular approximation): v = 2πa / T

#### Assumptions & References

Reference: Newton, I. (1687). Principia Mathematica; Kepler, J. (1619). Harmonices Mundi.

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