Tidal Force Calculator

Background.

A tide is not gravity — it is the difference in gravity across an extended body. The Moon pulls harder on the ocean nearest it than on Earth's centre, and harder on Earth's centre than on the ocean on the far side, and it is those differences, not the pull itself, that raise the tides. This calculator computes that differential acceleration for any perturbing mass, separation and body radius, in four forms at once: the standard linearised expression, the exact near-side and far-side values, and the transverse component that squeezes rather than stretches.

The distinction from ordinary gravitational attraction is worth being precise about, because it produces a result most people find backwards. Total attraction falls off as the square of distance; a tide falls off as the cube. The Sun's total pull on Earth is about 178 times the Moon's, yet the Sun's tide is only about 46 % of the Moon's — a ratio of roughly 2.18 in the Moon's favour. That inversion happens entirely because of the extra power of distance, and it is why the Moon rather than the Sun governs the tides.

Sign convention matters here more than in most calculations, so it is stated beside the results rather than left implicit. Every number returned is a positive magnitude, but the directions differ. The radial components point away from the body's centre on both the near side and the far side — the tidal field stretches the body along the line joining it to the perturber. That is why there are two tidal bulges and two high tides a day, not one. The transverse component points inward: at right angles to that line the tidal field squeezes, with exactly half the magnitude of the linearised radial term. Stretch along one axis and squeeze along the other two, in that two-to-one ratio, is the complete description.

The linearised expression 2GMr/d³ that appears in every textbook is the first term of an expansion in r/d, and it is exact only in the limit where the body is vanishingly small compared with the separation. This page reports the exact near-side value alongside it and tells you the percentage difference. For the Earth–Moon system, where r/d is 0.0166, the linearised form is 2.54 % low. For the Earth–Sun system, where r/d is 4.3 × 10⁻⁵, it is low by 0.0064 %. The error grows quickly as bodies approach each other, and the page raises a warning once it passes 5 %, which happens at r/d ≈ 0.032. Past that point the near and far bulges also stop being the same size, so quoting a single tidal figure becomes misleading.

One scope limit belongs where the answer is. This calculator gives the tidal FORCING, and nothing more. Actual tides at a coastline depend on ocean depth, basin resonance, coastline geometry, friction and the solid Earth's own elastic response — effects that can shift high tide by hours and change its height by an order of magnitude between two places on the same shore. A tide table is the product of numerical models fitted to a century of gauge records, not of this equation. What this equation gives you is the driver those models respond to.

What is tidal force calculator?

Tidal force is the differential gravitational force across an extended body: the difference between the pull on its near side, its centre, and its far side. Because it is a difference rather than a total, the uniform part of the field cancels, and what is left is a stretching along the line to the perturber and a compression at right angles to it.

For a body of radius r at distance d from a mass M, the leading term is a = 2GMr/d³. The cube in the denominator is the defining feature, and it comes from differentiating the inverse-square law: d/dr of GM/r² is −2GM/r³. This is why a nearby small body can dominate a distant large one. It is also why tidal effects become extreme near compact objects, where d can be tiny — the stretching an infalling body experiences near a black hole, informally called spaghettification, is this same expression evaluated at very small d.

Tidal forces do a great deal of work in the solar system beyond raising ocean tides. They lock moons into synchronous rotation, so that one face permanently points at the planet. They heat interiors: Io, squeezed by Jupiter on an eccentric orbit, is the most volcanically active body known, and Europa and Enceladus owe their subsurface oceans to the same mechanism. Push far enough and tides disrupt a body entirely, which is the subject of the Roche Limit calculator.

How to use this calculator.

1. 01Choose which component you want as the headline: the linearised radial term, the exact near or far side, or the transverse squeeze.
2. 02Enter the perturbing body's mass in kilograms. The Moon is pre-filled; for the Sun use 1.98841 × 10³⁰ kg.
3. 03Enter the centre-to-centre separation in metres. It must be larger than the body radius, or the near-side expression is singular.
4. 04Enter the radius of the body being stretched — for Earth, its mean radius; for a person, about half their height.
5. 05Enter a test mass if you want the answer as a force in newtons rather than an acceleration.
6. 06Check the linearisation error before quoting the simple 2GMr/d³ figure. Above 5 % it is flagged, and you should use the exact values instead.
7. 07Remember this is the forcing only: it is not a tide height and not a tide time.

The formula.

Start from Newton's law. The gravitational acceleration a perturbing mass M produces at distance x is GM/x². A body of radius r centred at distance d has its near face at d − r and its far face at d + r, so the near face feels GM/(d − r)² and the centre feels GM/d². The difference between them is the near-side tidal acceleration, GM[1/(d − r)² − 1/d²], and it is directed towards the perturber. On the far side the same subtraction gives GM[1/d² − 1/(d + r)²], directed away from the perturber. Both point away from the body's centre, which is why there are two bulges.

Expanding either expression for small r/d gives 2GMr/d³, the familiar textbook form, which is the same on both sides at first order. The inverse cube is the whole story: differentiating an inverse-square field with respect to distance necessarily raises the power by one.

The transverse component comes from geometry rather than magnitude. At 90° from the line of centres the perturber's pull is essentially the same size as at the body's centre but points in a slightly different direction, and the component of that difference is directed inward with magnitude GMr/d³ — exactly half the radial term. The two-to-one ratio of stretch to squeeze is a signature of any inverse-square field and is a direct consequence of the tidal tensor being trace-free in vacuum.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at 40 significant digits in a dedicated high-precision decimal context. Rounding happens once, at the return boundary, to 12 significant digits rather than a fixed decimal place — tidal accelerations are of order 10⁻⁷ m/s² and the gradient of order 10⁻¹³ s⁻², both of which a fixed decimal place would round to a flat zero. The linearisation error is rounded once and that same rounded value drives both the displayed percentage and the 5 % warning, so the note and the number on screen can never disagree.

SIGNIFICANT FIGURES. The arithmetic is exact to twelve figures, but the inputs are not: a mass entered in kilograms carries the 22-parts-per-million uncertainty of the gravitational constant G, and real separations vary continuously — the Moon's distance swings between roughly 363 000 and 406 000 km over a month, changing the lunar tide by about 40 %. Three significant figures is a generous ceiling for any real application.

SIGN CONVENTION AND INVALID DOMAIN. All outputs are positive magnitudes; the directions are given in words beside the result. Every input must be strictly positive, and the separation must exceed the body's radius: at d = r the term 1/(d − r)² is a genuine singularity, and d < r would place the perturbing mass inside the body, where none of these expressions apply. That case raises a labelled error naming the minimum valid separation.

APPROXIMATION REGIME. Newtonian, point-mass perturber, rigid body, no ocean response, no Love numbers, no resonance, no coastline. Near a black hole or a neutron star the Newtonian expression also stops being adequate and a general-relativistic treatment is required.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST, CODATA 2022 recommended value of the Newtonian constant of gravitation: G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻², relative standard uncertainty 2.2 × 10⁻⁵. Independent, open access. Retrieved 2026-07-29.
2. [2]NASA JPL Solar System Dynamics, 'Planetary Satellite Physical Parameters': Earth's Moon, GM = 4902.800 ± 0.001 km³/s² and mean radius 1737.4 km, from ephemeris DE440. Source of the default perturbing mass. Independent, open access. Retrieved 2026-07-29.
3. [3]NASA JPL Solar System Dynamics, 'Astrodynamic Constants': astronomical unit 149 597 870 700 m; heliocentric gravitational constant GM_Sun = 1.327 124 400 412 794 19 × 10²⁰ m³ s⁻² (DE440). Source of the solar comparison figures. Independent, open access. Retrieved 2026-07-29.
4. [4]NASA JPL Solar System Dynamics, 'Planetary Physical Parameters': Earth volumetric mean radius 6371.0084 ± 0.0001 km. Source of the default body radius. Independent, open access. Retrieved 2026-07-29.
5. [5]NOAA, Center for Operational Oceanographic Products and Services, 'Our Restless Tides', chapter 4: states that the tide-producing force varies 'in inverse proportion to the third power of the distance' between the bodies. Chapter 3 of the same document states that 'the tide producing force of the moon is approximately 2.5 times that of the sun' — a figure this page does NOT adopt; see the FAQ on the Moon-to-Sun ratio for the recorded disagreement. Independent, open access. Retrieved 2026-07-29.
6. [6]NASA Science, 'Moon Facts' (page updated 12 February 2026): 'The Moon is an average of 238,855 miles (384,400 kilometers) away.' Source of the default separation. Independent, open access. Retrieved 2026-07-29.