Tidal Force Calculator | Tidal Acceleration, Differential Gravity & Roche Limit
Compute tidal acceleration, differential gravitational force, tidal deformation, and the Roche limit for any planet-moon or star-planet system. Includes data presets for Earth-Moon, Sun-Earth, Jupiter-Io, and Saturn-Titan, plus custom mass and distance inputs.
What Is the Tidal Force Calculator | Tidal Acceleration, Differential Gravity & Roche Limit?
Tidal forces arise because gravity varies with distance: the near side of a body experiences stronger gravitational pull than the far side. This differential produces a stretching effect along the line joining two bodies. The tidal acceleration scales as 1/d³, falling off much faster than gravity itself (1/d²). The Roche limit is the critical distance at which tidal forces overcome a secondary body's self-gravity, causing it to break apart — explaining Saturn's rings and some comet disruptions.
Formula
a_tidal = 2G·M·R/d³ · F_diff = 2G·M·m·R/d³ · d_R(rigid) = R·(2M_p/M_s)^(1/3) · d_R(fluid) = 2.44·R_s·(M_p/M_s)^(1/3)
How to Use
- 1
Choose a preset from the top buttons to fill in all fields for a known system.
- 2
Or enter the perturber mass M (kg) — the body causing the tidal force.
- 3
Enter the primary body radius R (m) — the body experiencing the tidal force.
- 4
Enter the separation distance d (m) between the two bodies' centres.
- 5
Enter a test mass m (kg) to compute the differential tidal force on that mass.
- 6
Fill in the Roche limit section: primary mass, secondary mass, and secondary radius.
- 7
Click "Calculate Tidal Force" to see tidal acceleration, force, Roche limits, and stability status.
Select a preset system (Earth-Moon, Sun-Earth, Jupiter-Io, Saturn-Titan) or enter custom values for perturber mass, primary radius, separation distance, and test mass. The Roche limit section requires the primary and secondary masses and the secondary radius.
Example Calculation
Example 1 — Moon on Earth: M = 7.342×10²² kg, R = 6.371×10⁶ m, d = 3.844×10⁸ m, m = 1 kg. a_tidal = 2×6.674×10⁻¹¹×7.342×10²²×6.371×10⁶/(3.844×10⁸)³ ≈ 1.1×10⁻⁶ m/s². Example 2 — Roche limit Earth-Moon: d_R(fluid) = 2.44×1.737×10⁶×(5.972×10²⁴/7.342×10²²)^(1/3) ≈ 18 400 km. Moon is at 384 400 km — well outside.
Understanding Tidal Force | Tidal Acceleration, Differential Gravity & Roche Limit
Tidal accelerations in the solar system
| System | Perturber | Primary R (km) | Distance (km) | a_tidal (m/s²) |
|---|---|---|---|---|
| Moon on Earth | Moon (7.34×10²² kg) | 6 371 | 384 400 | 1.1×10⁻⁶ |
| Sun on Earth | Sun (1.99×10³⁰ kg) | 6 371 | 149 600 000 | 5.0×10⁻⁷ |
| Jupiter on Io | Jupiter (1.90×10²⁷ kg) | 1 822 | 421 700 | 4.9×10⁻³ |
| Saturn on Titan | Saturn (5.68×10²⁶ kg) | 2 576 | 1 221 870 | 2.6×10⁻⁴ |
Roche limits for solar system bodies
| Primary | Secondary | Rigid Roche (km) | Fluid Roche (km) | Actual distance (km) | Status |
|---|---|---|---|---|---|
| Earth | Moon | ~9 500 | ~18 400 | 384 400 | Safe |
| Jupiter | Io | ~35 000 | ~78 000 | 421 700 | Safe (tidal volcanism) |
| Saturn | Ring particles | ~55 000 | ~120 000 | ~80 000 | Inside — rings form |
| Sun | Earth | ~556 000 | ~1 230 000 | 149 600 000 | Far outside |
Physical effects of tidal forces
- ›Ocean tides: The Moon's tidal acceleration on Earth (~1.1×10⁻⁶ m/s²) slightly deforms the ocean, creating two tidal bulges. The Sun contributes about 46% as much, causing spring and neap tides.
- ›Tidal volcanism on Io: Jupiter's tidal force on Io (a_tidal ≈ 5×10⁻³ m/s²) flexes Io's interior thousands of times per orbit, generating enough heat for intense sulfuric volcanism — the most volcanically active body in the solar system.
- ›Tidal locking: When the tidal torque acts over geological time, it synchronizes a moon's rotation with its orbital period (1:1 resonance). The Moon is tidally locked — we always see the same face.
- ›Saturn's rings: Saturn's rings exist within the fluid Roche limit. Any moon-sized body that drifts inside is torn apart by tidal forces before it can accrete; the resulting debris forms the ring system.
Frequently Asked Questions
Why does the tidal acceleration fall off as 1/d³ instead of 1/d²?
The tidal acceleration is the gradient of gravity — the rate at which gravity changes with distance. Taking the derivative of g = GM/d² gives dg/dd = −2GM/d³. The differential across the primary's radius R is a_tidal ≈ 2GM·R/d³, hence the 1/d³ dependence.
What is the difference between the rigid and fluid Roche limits?
The rigid Roche limit (~1.26·R) assumes the secondary is a rigid solid whose structural strength counters tidal forces. The fluid Roche limit (~2.44·R) applies to liquid or rubble-pile bodies that cannot exert tensile strength. Most real moons fall between the two. Saturn's rings lie within the fluid Roche limit.
Is Io inside the Roche limit?
No — Io is well outside Jupiter's fluid Roche limit (~78 000 km) at its orbital distance of ~421 700 km. But it is so close that tidal flexing from Jupiter and orbital resonance with Europa and Ganymede generates enormous internal heat, powering its extreme volcanism.
Can tidal forces be felt by humans?
The tidal acceleration across a human body (~1.8 m tall) due to Earth's gravity at the surface is about 2×9.8×1.8/(6.371×10⁶) ≈ 5×10⁻⁶ m/s² — utterly imperceptible. Tidal effects are only significant across planetary distances.
Why does the Moon cause bigger tides than the Sun despite being much less massive?
Tidal acceleration scales as M/d³, not M/d². Although the Sun is 27 million times more massive than the Moon, it is ~389 times farther away. The Moon's tidal effect is (1/389)³ × 27×10⁶ ≈ 2.2 times larger. The Sun contributes about 46% of the Moon's tidal force, combining at new/full moon (spring tides) and offsetting at quarter moon (neap tides).
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