Tidal Force Calculator
Find the tidal acceleration and force that the Moon, the Sun or any other body produces across a planet, and see why the Moon wins by about 2.2 to 1.
🌙 What is Tidal Force?
Tidal force is the stretching effect that arises because gravity is not the same everywhere across an extended body. The Moon pulls harder on the side of Earth that faces it than on Earth's center, and harder on the center than on the far side. Subtract the pull on the center and what remains is a stretching along the line to the Moon and a slight squeezing at right angles to it. That leftover, called the tide-generating acceleration, is about 1.1 millionths of a meter per second squared at Earth's surface, yet it is what raises the ocean tides.
It is easy to confuse tidal force with the total gravitational force, so the distinction deserves emphasis. The total pull of the Moon on Earth is about 3.3e-5 m/s², roughly 30 times larger than the tidal part. Most of that pull simply keeps Earth and Moon in orbit around their common center of mass. Only the difference across Earth's radius deforms the planet, and that difference falls off as the cube of distance, not the square. This is why the nearby Moon outdoes the far more massive Sun for tides, and why a Moon that is only 12 percent closer at perigee than at apogee produces tides almost 50 percent stronger.
Tidal force matters well beyond the seashore. Oceanographers use it to predict tidal ranges and to understand spring and neap tides. Astronomers use it to explain the tidal locking of moons, the heating of Jupiter's moon Io, and the disruption of comets and satellites that pass inside the Roche limit. Geophysicists use it to compute the twice-daily Earth tides in solid rock, and engineers who handle precise gravimeters and long baseline instruments correct for it.
This calculator evaluates the standard first-order formula for any perturbing mass, distance and body radius. It shows the on-axis and perpendicular components, the force on a test mass you choose, and it always compares the Moon with the Sun using standard values so you can see the roughly 2.2 to 1 ratio. Two charts show how steeply the effect drops with distance and how much it varies between the Moon's perigee and apogee.
📐 Formula
The formula comes from a first-order Taylor expansion of Newtonian gravity: the difference between GM/(d − R)² and GM/d² is approximately 2GMR/d³ when R is much smaller than d. It is accurate to a few percent for Earth and the Moon and improves as d/R grows. For bodies that are very close together the exact difference should be used instead.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — The Moon on Earth and an 80 kg Person
M = 7.342e22 kg, d = 384,400 km, R = 6371 km, m = 80 kg
Example 2 — The Sun on Earth and the Moon to Sun Ratio
M = 1.989e30 kg, d = 149.6 million km, R = 6371 km, m = 1 kg
Example 3 — Moon at Perigee Versus Apogee
M = 7.342e22 kg, d = 356,500 km (perigee), R = 6371 km
Example 4 — Jupiter Stretching Its Moon Io
M = 1.898e27 kg (Jupiter), d = 421,700 km, R = 1821.6 km (Io), m = 1 kg
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for tidal force?
The tide-generating acceleration at the point of a body of radius R nearest to a perturbing mass M at distance d is a = 2GMR/d³, where G = 6.6743e-11 m³/(kg s²). Multiply by an object's mass m to get the tidal force F = m a. At right angles to the line to the body, the acceleration is half as large and points inward.
Why is the Moon's tidal effect greater than the Sun's?
Tidal acceleration depends on M/d³. The Sun has about 27 million times the Moon's mass but is about 389 times farther away, and 389 cubed is about 59 million. The distance wins, so the Moon's tidal effect is about 2.18 times the Sun's.
What is the difference between gravity and tidal force?
Gravity is the full pull of a body, GM/d². Tidal force is the difference between the pull on one side of an extended object and the pull on its center, approximately 2GMR/d³. For the Moon acting on Earth, gravity is about 3.3e-5 m/s² while the tidal part is only about 1.1e-6 m/s².
How much stronger are tides at perigee than at apogee?
The Moon's distance varies from about 356,500 km at perigee to 406,700 km at apogee. Because tidal acceleration goes as 1/d³, perigee tides are (406,700/356,500)³ = 1.485 times as strong, about 48 percent more, as at apogee. Compared with the mean, perigee is about 25 percent higher and apogee about 16 percent lower.
What is the tidal acceleration of the Moon on Earth in g?
At the mean distance of 384,400 km, the Moon's tidal acceleration across Earth's 6371 km radius is 1.0993e-6 m/s², which is 1.121e-7 g. The Sun's is 5.052e-7 m/s², or 5.15e-8 g. Both are tiny compared with Earth's own gravity, yet they move whole oceans.
Why are there two tidal bulges, one on each side of Earth?
On the near side the Moon pulls the water more than it pulls Earth's center, so water bulges toward it. On the far side the Moon pulls the water less than it pulls the center, so the water is left behind and bulges away. Both effects come from the same difference in gravity, and both have magnitude close to 2GMR/d³.
Does the mass of the object matter for tidal acceleration?
Not for the acceleration. Tidal acceleration depends on the perturbing mass M, the distance d and the radius R of the body being stretched. The mass m of a test object only scales the force on it, F = m a. A 1 kg test mass feels 1.0993e-6 N from the Moon, an 80 kg person 8.794e-5 N.
What are spring tides and neap tides?
At new and full Moon the Sun and Moon line up and their tidal accelerations add, giving the largest range (spring tides). At the quarter phases they act at right angles and partly cancel, giving the smallest range (neap tides). With a Moon-to-Sun ratio of about 2.2, the spring-to-neap range ratio is roughly (2.2 + 1)/(2.2 - 1), about 2.7.
Why does tidal force scale as 1/d³ instead of 1/d²?
Gravity falls as 1/d². The tidal force is the difference in gravity across a size R, which is the derivative of GM/d² with respect to d times R, giving 2GMR/d³. One extra power of distance comes from taking that derivative, so the tidal effect falls off faster than gravity itself.
Can this calculator work for other planets and moons?
Yes. Choose the custom option, enter the perturbing mass in kilograms, the distance in kilometers and the radius of the body in km. For example, Jupiter (1.898e27 kg) at 421,700 km acting on Io (radius 1821.6 km) gives about 6.15e-3 m/s², thousands of times the Moon's tidal effect on Earth.
Do tidal forces act on the solid Earth too?
Yes. The solid Earth deforms by up to about 30 cm twice a day, called Earth tides, which is much smaller than the ocean tide because rock is stiff. The same tidal acceleration drives both, but the ocean responds more freely.