Real-time visualisation of the tidal bulges that the differential gravity of the Moon and Sun raises on the Earth. Change the orbital radius and the masses to compute the Roche limit and spring and neap tides.
Parameters
Primary mass M (×10²⁴ kg)
×10²⁴kg
Secondary mass m (×10²² kg)
×10²²kg
Orbital radius d (×10³ km)
Mm
Primary radius R (km)
km
Secondary density ρ_m (kg/m³)
kg/m³
While paused, moving a slider updates the result immediately.
Results
Tidal bulges and differential-gravity vectors (real time)
Tidal acceleration vs orbital radius
Theory & Key Formulas
$$a_{\rm tidal}= \frac{2GmR}{d^3}$$
Roche limit (rigid-body approximation; ρ_M=M/(4πR³/3) is the mean density of the primary):
Tidal-locking time estimate (until the secondary rotates synchronously; order-of-magnitude guide assuming an initial spin period of 10 h, Q/k₂=3000 and I=0.4mR_s²):
$$t_{\rm lock}\approx \frac{\omega_0 d^6 I Q}{3GM^2 k_2 R_s^5}$$
M: primary mass, m: secondary mass (the body raising the tide), R: primary radius (the deformed body), ρ_m, R_s: density and radius of the secondary, d: orbital distance
About the Tidal Forces Simulator
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Isn't tidal force just the Moon's gravity pulling on the sea? What does this simulator actually show?
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Roughly speaking, what matters is the difference in gravity. The sea nearest the Moon and the sea on the far side of the Earth feel different pulls, right? That difference is the force that deforms the Earth into an ellipsoid – the tidal force. In this simulator you can change the Moon's mass and distance with the sliders above and watch in real time how the tidal acceleration $a_{\rm tidal}$ changes.
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Wait, the far side of the Earth bulges too? Not just the side being pulled? And what is the Roche limit?
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That's right. The near side is pulled strongly, the Earth's centre normally, and the far side least of all. The result is that the whole Earth is stretched, so bulges form on both sides. The Roche limit is the closest distance at which a satellite can survive before the primary's tidal force tears it apart. Saturn's rings, for example, are said to be the debris of a former moon that strayed inside the Roche limit and was shattered. Raise the secondary's density in the simulator (rock 3,340 → iron 8,000 kg/m³) and the Roche limit moves closer to the primary; low-density bodies such as ice (917 kg/m³) start to break up farther out. With the Earth defaults it is about 9,500 km.
🙋
I see! So if I make the Sun's mass huge and use it as the primary, would the Earth cross its Roche limit? And can spring and neap tides be reproduced?
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The Sun is massive, but it is so far from the Earth that its tidal force is only about half the Moon's, so no shattering. Spring and neap tides are decided by whether the solar and lunar tidal forces add up or partly cancel. Switch between "Spring tide (new/full moon)" and "Neap tide (half moon)" in the simulator and you see at a glance how the Sun's tide (computed internally with fixed values) combines with the Moon's.
Frequently Asked Questions
Inside the Roche limit the tidal force exceeds the satellite's self-gravity, so the satellite is destroyed and forms a ring around the primary. In the simulator, when the orbital radius drops below the Roche limit the check card turns to "Danger" and the Roche-limit ring is drawn outside the satellite's orbit (the break-up itself is not drawn).
Spring tides occur at full and new moon, when the Moon and Sun line up with the Earth and the tidal bulges are largest. Neap tides occur at first and last quarter, when the Moon and Sun are at right angles and the bulges are smallest. In the simulator the "Spring tide (new/full moon)" and "Neap tide (half moon)" buttons switch the Sun's direction, and the relative tidal range changes between 1.00× and 0.60×.
The tidal force is proportional to the secondary's mass m and inversely proportional to the cube of the distance d. A larger mass increases the tidal bulge, but the rigid-body Roche limit depends only on the density ratio of primary and secondary, so changing the mass does not move it. Move the mass slider in the simulator and watch the change in real time.
The simulator visualises the basic tidal mechanism driven by the differential gravity of the Moon and Sun, but real tides are also shaped by seabed topography, coastlines and the Earth's rotation, so it is meant for qualitative understanding. It is not suited to quantitative prediction.
Real-World Applications
Ocean tide prediction: Applied to the Earth–Moon–Sun system, this physical model is the basis for predicting the times and heights of high and low water (spring and neap tides). It is indispensable for navigation and the design of coastal facilities.
Evolution and structure of celestial bodies: The intense volcanism of Jupiter's moon Io is powered by tidal heating from the strong tidal forces of Jupiter and the other moons. The fact that Saturn's rings lie inside the Roche limit is strong evidence for this theory.
Exoplanet research: Hot Jupiters orbiting extremely close to their stars are deformed into ellipsoids by strong tidal forces, and the degree of deformation is used to infer their internal structure, such as whether they have a core.
Spacecraft trajectory design: When exploring the surroundings of giant planets such as Jupiter and Saturn, the stress on a spacecraft grows in regions of strong tidal force. Assessing tidal forces is important for designing safe orbits.
Common Misconceptions and Pitfalls
When you start using this simulator, there are a few points where newcomers to CAE often stumble. First of all, be very aware that tidal force is not gravity itself. It is true that increasing the secondary's mass in the tool strengthens the tidal force, but the effect of distance is far more dramatic. Doubling the Moon's mass doubles the tidal force, but bringing the Moon from 10 to 5 Earth radii multiplies it eightfold, because the tidal force falls with the cube of distance. When exploring parameters, pushing the sliders to extremes is the way to get a feel for the changes.
Second, the Roche-limit formula is an approximation that assumes a rigid body. Real bodies are deformable fluids or piles of fragmented material. So the numbers from the simulator are only a guide. A highly viscous body, for example, can approach the primary a little closer without being destroyed, while a low-density, loosely bound body starts to come apart farther out. For practical applications a more elaborate model that accounts for the body's material properties is needed.
Finally, a pitfall when reproducing the Earth's tides: the simulator computes a static, instantaneous force, whereas the real ocean takes time to respond and is strongly influenced by seabed topography and coastlines. That is why the computed time of high water differs from the actual one. What you learn from the tool is the nature of the driving force; actual tide prediction combines this with the equations of ocean motion in numerical models.
Set the primary mass M in the range 1–20×10²⁴ kg; the Earth is 5.97×10²⁴ kg
Enter the secondary mass m and the orbital radius d; for the Moon, 7.34×10²² kg and 384×10³ km
Set the primary radius R and the secondary density ρ_m; the tidal acceleration and the Roche limit are recomputed immediately
Check the satellite's stability and the time scale of spin synchronisation from the Roche-limit check (Safe/Danger) and the tidal-locking time estimate (Myr)
Worked Example
With the default Earth (M=5.97×10²⁴ kg, R=6,371 km), Moon (m=7.34×10²² kg, ρ_m=3,340 kg/m³) and orbital radius 384×10³ km, the tidal acceleration is 1.10×10⁻⁶ m/s² and the Sun's contribution about half of that, 5.05×10⁻⁷ m/s². The rigid-body Roche limit is 9.5×10³ km; the Moon is about 40 times farther out, so the check reads "Safe", and the tidal-locking time estimate is about 41 Myr. Setting the secondary density to that of ice, 917 kg/m³, gives a Roche limit of 1.46×10⁴ km; iron at 8,000 kg/m³ gives 7.1×10³ km. Moving the orbital radius to 800×10³ km lowers the tidal acceleration to 1.22×10⁻⁷ m/s² and lengthens the locking time to about 3,400 Myr.
Practical Notes
The Roche limit depends strongly on the satellite's density. A dense iron satellite (8,000 kg/m³) can approach (8000/917)^(1/3)≈2.1 times closer than an icy one (917 kg/m³): 7.1×10³ km versus 1.46×10⁴ km in this tool
The tidal-locking time scales with d⁶, so satellites close to their planet synchronise orders of magnitude faster. The tool's value is an order-of-magnitude guide assuming an initial spin of 10 h and Q/k₂=3000; the real body's rigidity and viscosity move it by about an order of magnitude. The Moon and Mars's Phobos are already tidally locked
Real satellite break-up does not happen at the instant the Roche limit is crossed but progressively over weeks, so treat the check as a guide