Understand the relationship
The reasoning behind the result
The period belongs to a whole orbit
T²=4π²a³/[G(M+m)]
In isolated Newtonian two-body motion, the relative separation traces an ellipse. Its semimajor axis a fixes the orbital energy, and the sum of the masses fixes the gravitational parameter μ. For the same μ and a, the period T is independent of eccentricity, although the speed is not uniform around an eccentric orbit.
A circle's radius equals a, but an eccentric orbit's current radius generally does not. Replacing a with a current separation, an altitude, or one component's barycentric semimajor axis answers a different question and produces the wrong period. The two barycentric semimajor axes add to the relative a.
The inverse keeps the same units and physical meaning
a=∛[μ(T/2π)²]; μ=4π²a³/T²
To infer orbital size, convert time to seconds and μ to m³/s², form μ(T/2π)² and take its cube root. The result is a length. To infer μ from size and period, the length cubed divided by time squared has the required m³/s² units.
Distance entries may be metres, kilometres or exact astronomical units. Day entries use 86,400 seconds and do not imply a body's sidereal rotation period. Each row retains converted period and axis values so the inverse can be checked without another calculator.
Total mass is determined; individual masses are not
Mtotal=M+m=μ/G
A measured period and relative semimajor axis constrain the sum of the masses. They cannot, by themselves, identify either component's share. A primary-only approximation is appropriate only when the omitted mass is negligible for the intended accuracy; it is not automatically valid for binary stars or comparable masses.
The implementation uses the measured gravitational constant G=6.67430×10⁻¹¹ m³/(kg·s²). A directly supplied μ does not acquire a new uncertainty from this conversion when calculating periods or axes. The equivalent mass display is a conversion, not a precision claim or an uncertainty estimate.
Scaling provides a useful independent check
T₂/T₁=(a₂/a₁)^(3/2) at fixed μ
At unchanged gravity, multiplying a by four multiplies the period by eight. At unchanged period, multiplying total mass by eight doubles the inferred semimajor axis. At unchanged size, a larger total mass shortens the period by the inverse square root of the mass ratio.
The live curve uses the first row's μ. If mass-inverse rows imply different μ values, their markers need not lie on that reference curve; the full ledger keeps those differences visible rather than silently averaging incompatible measurements.
Follow the numbers
Recovering a common μ from two orbital scales
- For an ideal reference with a=1 m and T=2π s, μ=4π²×1³/(2π)²=1 m³/s².
- For a=4 m and T=16π s, μ=4π²×64/(16π)²=1 m³/s² again. The axis increased by four and the period by eight.
- Each case implies the same total mass: 1/(6.67430×10⁻¹¹)=1.498284464×10¹⁰ kg. This is the sum of the two masses, not either component alone.
Matching μ demonstrates the expected two-body scale relationship. These ideal small-scale inputs are a mathematical benchmark, not a claimed observed system.
Quick guide
How to use this calculator
- Select the missing quantity and whether known gravity is supplied as μ or as two masses.
- Choose distance and time units before entering named scenarios. Use the relative semimajor axis measured between components.
- The mass inverse needs independently established orbital size and period. An angular orbit also needs a distance scale and appropriate deprojection before it can be entered here.
- Read both the requested result and the reconstructed gravity, period and size in the ledger. Total system mass alone does not separate the components.
Calculation method
Calculation and interpretation
Recover the missing orbital scale from two independently known quantities while distinguishing total system mass from either component mass.
T=2π√(a³/μ); a=∛(μ(T/2π)²); μ=4π²a³/T²; Mtotal=μ/G; μ=G(M+m).
Worked example
Recovering a common μ from two orbital scales
Matching μ demonstrates the expected two-body scale relationship. These ideal small-scale inputs are a mathematical benchmark, not a claimed observed system.
T=2π√(a³/μ); a=∛(μ(T/2π)²); μ=4π²a³/T²; Mtotal=μ/G; μ=G(M+m).
Supported inputs
Precision and limits
Isolated two-body approximation
Perturbations, general relativity, extended mass distributions, orbital decay and time-variable gravity are excluded. Only positive size, period and gravity are supported.
True relative semimajor axis required
This tool does not deproject visual binary orbits, fit astrometric data, infer an unknown distance or turn an instantaneous separation into a semimajor axis.
No component or uncertainty inference
A total mass does not identify component masses. Measurement uncertainty is not propagated; rounded output is not a statement of observational precision.
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