Switch between distance, mass, or μ inputs for precise periods every time. Get seconds to years, plus downloadable CSV and PDF summaries in one.
These examples show typical inputs and expected magnitudes. You can paste them into the calculator using the “Load example” button.
| Scenario | Mode | Preset | a or r | Unit | μ (if used) | μ unit | Expected T |
|---|---|---|---|---|---|---|---|
| Low Earth orbit (approx.) | a + μ | Earth | 6771 | km | (blank) | km³/s² | ~ 92 minutes |
| Geostationary orbit | r (circular) + μ | Earth | 42164 | km | (blank) | km³/s² | ~ 23.93 hours |
| Earth around the Sun | a + μ | Sun | 1 | AU | (blank) | m³/s² | ~ 1 year |
For a two-body orbit, Kepler’s third law gives the sidereal period:
T = 2π √(a³ / μ)
When the orbit is circular, a = r and the circular speed is:
v = √(μ / r)
To convert an observed synodic period to a sidereal period:
1/Psyn = | 1/Psid − 1/Pref |
The sign depends on whether the object’s motion is faster or slower than the reference.
Use exports to document results and compare scenarios.
The sidereal period is the true time a body needs to complete one orbit relative to distant stars, not relative to the Sun. For satellites, it sets repeatability of ground tracks and timing for station keeping. For planets, it defines orbital cadence used in ephemerides and long-term simulations.
Kepler’s third law shows a powerful scaling: period grows with the 3/2 power of size. Doubling the semi-major axis increases the period by about 2.828×. This is why high-altitude orbits quickly become slow and why compact systems can have rapid orbital cycles.
Use the semi-major axis for elliptical orbits because it represents the energy-equivalent size of the trajectory. In a circular orbit, the semi-major axis equals the orbital radius. A common pitfall is using periapsis altitude instead of distance from the central body’s center.
Gravitational parameter μ combines the gravitational constant and the central mass into one value that is measured very precisely for major bodies. For Earth, μ ≈ 3.986×1014 m³/s² (or 398600.4418 km³/s²). Using μ often reduces uncertainty compared with entering mass directly.
Mass-based calculation is valuable for custom systems: asteroids, binary stars, or laboratory analogs. The calculator converts mass to μ using G, then applies Kepler’s law. In many spacecraft problems, the satellite mass is negligible compared with the central mass, so μ ≈ GM is adequate.
A synodic period is an observed cycle relative to a moving reference, such as the time between conjunctions or phases. Converting synodic to sidereal uses reciprocals of periods; the correct sign depends on whether the object moves faster or slower than the reference motion.
Low Earth orbit at about 6771 km from Earth’s center yields a period near 92 minutes. A geostationary radius near 42164 km yields about 23.93 hours, matching Earth’s rotation relative to stars. Earth’s orbit around the Sun at 1 AU produces roughly one year.
Non-spherical gravity, atmospheric drag, third-body perturbations, and thrusting maneuvers all shift the instantaneous period. For precision operations, analysts use numerical propagation, but Kepler-based sidereal estimates remain the fastest way to sanity-check inputs and compare designs.
No. A solar day is measured relative to the Sun. A sidereal period is measured relative to distant stars, so it excludes the extra rotation needed to bring the Sun back to the same position.
Use μ when available because it is typically known with high precision from tracking data. Use mass when modeling a custom body or when μ is not known for your system.
You can, but convert correctly: r = (planet radius + altitude). The calculator expects distance from the central body’s center for r and semi-major axis for a.
For an ideal two-body orbit, the period depends only on the semi-major axis, not eccentricity. Eccentricity changes speed along the orbit, but the full orbital time remains set by a.
Mean motion is the average angular rate of orbital progress. It is useful for quick timing estimates, resonance checks, and comparing orbits because it is directly linked to the sidereal period.
Drag, oblateness, third-body forces, and maneuvers can shift the real period. Also ensure you used the correct distance reference and consistent units when entering inputs.
Use it when you observe repeating geometry relative to another moving body, such as phase cycles, conjunction cycles, or periodic alignments. The conversion estimates the underlying sidereal period that produces the observed repetition.
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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.