Calculate tidal acceleration from mass, distance, and object radius, with preset Moon, Sun, Jupiter, Saturn, and custom scenarios.

Estimates radial center-to-edge tidal acceleration, not force in newtons. Uses the small-object approximation r ≪ d; preset distances and masses are approximate.

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Tidal Acceleration (Tidal Force per Unit Mass) Formula

The following formula is used to estimate the tidal acceleration (radial differential gravitational acceleration from its center to its edge) caused by a celestial body. This approximation assumes the object’s radius is much smaller than the separation distance (r ≪ d).

aₜ = (2 G M r) / (d³)

Variables:

  • at is the tidal acceleration (m/s²), i.e., tidal force per unit mass
  • G is the universal gravitational constant (6.67430 × 10-11 m3 kg-1 s-2)
  • M is the mass of the celestial body (kilograms)
  • r is the object’s radius or radial center-to-edge length (meters); the near-to-far separation is 2r
  • d is the distance from the center of the celestial body to the center of the object (meters)

To calculate the tidal acceleration, multiply twice the universal gravitational constant by the mass of the celestial body and the radius of the object, then divide by the cube of the center-to-center distance. If you need an estimated tidal force on a specific mass m at the object’s edge, use Ftidal = m·at.

What is Tidal Force?

Tidal force is the secondary effect of the force of gravity and is responsible for the tides on Earth. It arises because the gravitational force exerted by one body on another is not constant across it; the nearest side is attracted more strongly than the farthest side. This difference in attraction creates a stretching effect on the body, which can cause it to become elongated along the axis pointing towards the celestial body causing the tidal effect. This effect is most clearly observed in the Earth-Moon system but occurs in other systems as well, such as the tidal forces between a planet and its satellites.

How to Calculate Tidal Force?

The following steps outline how to calculate the tidal acceleration (tidal force per unit mass).

  1. First, determine the mass of the celestial body (M) in kilograms.
  2. Next, determine the center-to-center distance from the celestial body to the object (d) in meters.
  3. Next, determine the radius of the object (r) in meters.
  4. Next, gather the formula from above: at = (2 G M r) / d3.
  5. Finally, calculate the tidal acceleration (at) in m/s².
  6. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Mass of the celestial body (M) = 7.342 × 1022 kg (Moon’s mass)

Distance from the celestial body (d) = 384,400,000 m (average distance from Earth to Moon)

Radius of the object experiencing the tidal effect (r) = 6,371,000 m (Earth’s radius)