Illustrative tidal distance scales and planetary synodic periods

Different tidal assumptions produce different distance scales

The factor 2 estimate balances the leading radial tidal acceleration against surface self-gravity, without rotational acceleration or deformation. It is not a universal breakup radius for a crystalline moon. Material strength, spin and shape affect disruption.

For a homogeneous strengthless fluid in synchronous circular motion, the familiar approximate coefficient 2.44 gives a larger distance scale. Literature also quotes about 2.46; this worksheet fixes 2.44 for its illustrative calculation. The synchronously rotating rigid-sphere estimate instead uses a cube-root coefficient based on 3, not 2.

Both distances are from the primary center, not altitude. Use the same density units for primary and satellite and positive inputs. Neither calculation predicts a real ring boundary or automatically determines whether a particular moon breaks up.

Nonrotating tidal-balance estimate (km)

Radius B2 is in km and B3/B4 is a dimensionless density ratio. This simplified estimate intentionally omits rotation and material strength.

=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0),"Invalid inputs",ROUND(B2 * ((2 * B3 / B4) ^ (1 / 3)), 1)))

Mean synodic period for same-direction orbital motion

Use positive sidereal periods in the same time unit. The reciprocal difference gives the relative-angle beat period for uniform prograde motions; real launch windows require orbital geometry and mission constraints. Equal periods give zero relative mean angular rate and no finite beat period. Equality alone does not prove a Trojan configuration or stable resonance. Opposite-direction motions need a sum of reciprocal periods instead.

=IF(COUNT(B5:B6)<>2,"Invalid inputs",IF(OR(B5<=0,B6<=0),"Invalid inputs",IF(B5 = B6, "No finite beat period", ROUND(1 / ABS((1 / B5) - (1 / B6)), 2))))

Fluid Roche approximation with coefficient 2.44

Compare this result with the nonrotating force-balance estimate; the different coefficients reflect different assumptions.

=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0),"Invalid inputs",ROUND(2.44 * B2 * ((B3 / B4) ^ (1 / 3)), 1)))

Further reading

• Microsoft Support — ABS function
• Hesselbrock and Minton — rigid and fluid Roche models (Astronomical Journal, 2019)
• US Naval Observatory — astronomical glossary: synodic period