Free · Physics models
Newtonian Orbital and Surface-Grazing Periods
Find two-body orbital period or the ideal circular period immediately outside a spherical body of known mean density.
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
For a mean body density of 1 g/cm³, the surface-grazing circular period is about 3.301 h.
T = 2π√(a³/[G(m1+m2)]); surface-grazing T = √(3π/[Gρ]), with G = 6.67430×10⁻¹¹ SI.
Worked example
Enter these known values and leave the other values blank.
- Central body density
- 1 g/cm³
- Orbital period
- 3.301 hrs
Assumptions and limitations
- The density relation is a hypothetical circular orbit immediately outside a spherical body. The radius cancels after substituting its mass from mean density. It is not the period of every orbit around that body.
- For the separate two-body relation, a is the semimajor axis of the relative orbit, and both masses contribute to the gravitational parameter.
- The Newtonian model omits drag, nonspherical gravity, third-body forces and relativistic effects. Positive density, periods, axis and both masses are retained.
- G is the 2022 CODATA central value 6.67430×10⁻¹¹ m³/(kg·s²); its uncertainty is not propagated. Stellar and planetary mass unit labels are illustrative conversion scales, rather than precision measured gravitational parameters.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
- Reviewed hash-pinned model correction: The density relation is a hypothetical circular orbit immediately outside a spherical body. The radius cancels after substituting its mass from mean density. It is not the period of every orbit around that body. For the separate two-body relation, a is the semimajor axis of the relative orbit, and both masses contribute to the gravitational parameter. The Newtonian model omits drag, nonspherical gravity, third-body forces and relativistic effects. Positive density, periods, axis and both masses are retained. G is the 2022 CODATA central value 6.67430×10⁻¹¹ m³/(kg·s²); its uncertainty is not propagated. Stellar and planetary mass unit labels are illustrative conversion scales, rather than precision measured gravitational parameters. Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Why does density alone give a period?
For an ideal orbit immediately outside a sphere, substituting mass from its mean density cancels radius.
Which semimajor axis belongs in the binary calculation?
Use the semimajor axis of the relative separation orbit, rather than one body’s individual orbit about the center of mass.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
- NIST: Newtonian gravitational constant
- NIST CODATA physical constants
- Calculation definition and unit reference
Use current CODATA G = 6.67430e-11 and mathematical pi for the Newtonian two-body and surface-grazing circular limits.