Tidal Acceleration Calculator

Enter calculation inputs

Example data table

The example below uses Moon-on-Earth values with average Earth-Moon distance. Output is shown in µm/s² using the exact geometry model.

Angle (deg)	Radial (µm/s²)	Tangential (µm/s²)	Total (µm/s²)
0	1.128095	0.000000	1.128095
30	0.696473	-0.733730	1.011650
60	-0.149655	-0.717305	0.732751
90	-0.549837	0.013670	0.550007
120	-0.125726	0.711393	0.722418
150	0.678714	0.696113	0.972228
180	1.073370	1.955719e-16	1.073370

Formula used

1) Exact point-to-attractor distance

s = √(d² + R² − 2dR cosθ)

Here, d is center distance, R is body radius, and θ is the surface angle from the sub-attractor point.

2) Differential gravity vector components

gₓ = GM[(d − R cosθ)/s³ − 1/d²]

gᵧ = GM[−R sinθ/s³]

These subtract the attractor's gravity at the body's center from the gravity at the selected surface point.

3) Local radial and tangential components

a_rad = gₓ cosθ + gᵧ sinθ

a_tan = −gₓ sinθ + gᵧ cosθ

a_total = √(a_rad² + a_tan²)

Radial values describe stretching or compression. Tangential values describe sideways driving along the surface.

4) Classical approximation

a_axis ≈ 2GMR/d³

a_side ≈ GMR/d³

These are helpful quick checks when the body radius is much smaller than the center distance.

How to use this calculator

Enter the attracting body's mass and choose the mass unit.

Enter the center-to-center distance between both bodies.

Enter the radius of the body experiencing tidal distortion.

Set the surface angle. Use 0° for near side, 90° for side point, and 180° for far side.

Choose an output acceleration unit and graph step size.

Press the calculate button to display the result section above the form.

Review the graph to compare radial, tangential, and total tidal acceleration over angle.

Use the CSV or PDF buttons to export the summary table after calculation.

Frequently asked questions

1) What does tidal acceleration mean?

Tidal acceleration is the difference in gravitational pull across a body's size. One side feels a slightly different force than the center or opposite side, which creates stretching, compression, and surface-driving effects.

2) Why is the surface angle important?

The tide is not uniform over the surface. Angle changes both distance and direction relative to the attractor, so radial and tangential components vary strongly from the near side to the far side.

3) What is the difference between exact and approximate results?

The exact model uses full geometry and works better when the body's radius is not tiny compared with separation. The approximation is faster and useful for quick checks when radius is much smaller than distance.

4) Can I use this for planets, moons, or stars?

Yes. The calculator works for any two-body setup as long as you enter a consistent mass, center distance, and body radius. Unit selectors make it easier to switch between astronomical and engineering scales.

5) What does a positive radial value mean?

A positive radial value means the local differential pull points outward from the body's center along the surface normal. This is associated with tidal stretching near the sub-attractor and anti-attractor regions.

6) What does the tangential component represent?

The tangential component measures sideways differential acceleration along the surface. It helps explain why material can be driven away from or toward certain latitudes relative to the attractor direction.

7) Why is the total not just radial plus tangential?

Radial and tangential parts are perpendicular vector components. The total differential acceleration is their vector magnitude, found from the square root of the sum of squared components.

8) When does the simple approximation become less reliable?

It becomes less reliable when the body is large compared with the separation, such as very close orbits, compact systems, or situations where exact geometry meaningfully changes the local distance and direction.