Understand the relationship
The reasoning behind the result
Matching rotation fixes the circular radius
Torbit=2π√(r³/μ)=Trotation
A synchronous orbit repeats one orbital revolution in the same time as the body rotates once relative to inertial space. For an ideal circular test-particle orbit, solving the period law gives the radius from the centre. Subtract the separately entered body radius to express a height above the reference surface.
The inverse starts with a circular radius and finds the body rotation period that would match it. This is a mathematical matching condition; it does not predict how the body actually rotates. A day of 86,400 seconds is only a unit here, not an automatic sidereal-period assumption.
Synchronous does not automatically mean stationary
For a satellite to stay above one longitude in this idealized model, its orbit must be circular, equatorial and prograde as well as synchronous. Inclined or eccentric synchronous motion changes the satellite's apparent position during a day. This workflow assumes the circular equatorial prograde geometry for its longitude comparisons.
A formal radius at or inside the entered surface fails the exterior point-mass interpretation. The status remains visible instead of presenting a negative height as a usable orbit. Even an exterior result does not assess an atmosphere, terrain, tides, radiation, stability, collision avoidance or stationkeeping.
Drift is the difference between two angular rates
ωrelative=360°/Torbit−360°/Trotation
At a smaller circular radius, the satellite completes its orbit faster than the prograde rotating surface and drifts east in this reference convention. At a larger radius it falls behind and drifts west. Exactly matching the period makes the ideal relative rate zero.
Multiplying the rate by elapsed time gives unwrapped longitude change, which may exceed one revolution. The wrapped longitude is an equivalent angle within ±180° from the initially aligned reference. Wrapping does not discard the separate unwrapped total, and neither output is a full ground-track prediction.
Follow the numbers
A period-matching unit reference
- Let μ=1 m³/s² and Trotation=2π s. The matching radius is ∛[1×(2π/2π)²]=1 m.
- With an entered reference body radius of 0.1 m, the formal exterior height is 1−0.1=0.9 m. Circular speed is √(1/1)=1 m/s.
- At radius 2 m, orbital period is 2π√8 s. After one body rotation, relative longitude has changed by 360(1/√8−1)=−232.720779°.
- That unwrapped westward change is equivalent to +127.279221° within ±180°. Both figures refer to the same final direction; the first retains how far the relative angle accumulated.
The radius that matches the clock has no ideal relative drift. A larger prograde orbit falls behind the rotating reference.
Quick guide
How to use this calculator
- Choose a radius solve, an inverse rotation-period solve or a candidate comparison.
- Enter central gravity, reference body radius and units. A sidereal rotation period is measured relative to inertial space.
- For drift, enter candidate radii and elapsed time. Positive longitude drift is eastward relative to the assumed prograde rotating surface.
- Check the exterior-radius status. Period matching alone does not establish a stationary sky position or a viable orbit.
Calculation method
Calculation and interpretation
Distinguish a period-matching radius from the extra conditions required for a satellite to remain fixed over a rotating body.
rsync=∛[μ(Trotation/2π)²]; altitude=rsync−Rbody; drift rate=360(1/Torbit−1/Trotation); Δlongitude=drift rate×elapsed time.
Worked example
A period-matching unit reference
The radius that matches the clock has no ideal relative drift. A larger prograde orbit falls behind the rotating reference.
rsync=∛[μ(Trotation/2π)²]; altitude=rsync−Rbody; drift rate=360(1/Torbit−1/Trotation); Δlongitude=drift rate×elapsed time.
Supported inputs
Precision and limits
Exterior point-mass approximation
The body is spherical and the satellite has negligible mass. At or inside the entered surface, results are formal point-mass arithmetic only and are explicitly flagged.
Restricted drift geometry
Drift comparisons assume equatorial prograde circular orbits and constant body rotation. They do not support inclined or eccentric ground tracks, precession, orientation epochs or perturbations.
No operational or geographic claim
No longitude, current position, orbit allocation, radio coverage, stationkeeping requirement or viable mission is inferred.
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