Tidal Effects in the Solar System

A Discreet but Powerful Force

In space, the balance between gravity and inertia shapes movements. Among gravitational interactions, tidal forces stand out for their subtlety and reach. They act on any extended body, generating differential deformations because gravity decreases with distance. These forces modify planetary rotations, trigger internal heating, or stabilize orbits. Their importance is such that no planetary system can be correctly modeled without taking them into account.

Why Variable Gravity Deforms Extended Bodies

Imagine a spherical celestial body (like Earth or Io) subject to the gravitational attraction of another massive body, such as the Moon or Jupiter. The Newtonian gravitational force is locally expressed by: \[ F = \frac{GMm}{r^2} \] This force depends on the distance \( r \) between the centers of mass. However, an extended body presents a significant difference in distance between its parts close to and far from the attracting body. This gravity gradient induces a differential force between the hemisphere facing the body and the opposite one.

This difference in force causes a stretching of the affected body: it adopts a slightly ellipsoidal shape, with the main axis oriented towards the attracting object. This phenomenon is purely gravitational and is proportional to the radius of the affected body, making it stronger for large moons close to massive planets.

The modified shape is not perfectly aligned with the external object if the body is rotating: this creates a tidal torque, which acts to dissipate mechanical energy as heat and modify the rotation. This mechanism is the origin of many rotational locks and the slowing down of Earth.

In summary, tidal forces are the geophysical expression of a fundamental fact: gravity is not uniform over an extended object, which naturally generates tensions and internal reorganizations.

An Invisible but Essential Mechanism

Far from being anecdotal, tidal effects deeply structure the evolution of celestial bodies. From the synchronization of natural satellites to the habitability of moons, they are at the heart of planetary dynamics. Their understanding is essential for modeling orbits, predicting geological activities, or evaluating the biological potential of an oceanic world. In the study of exoplanets as well as icy moons, tides are an invisible but decisive key.

Gravitational Gradient

A tidal force arises from the variation of the gravitational field over an extended body. One side of the body is closer to the attracting object (usually a planet or a star), the other farther away. The difference in the intensity of the gravitational force generates an internal tension in the body, resulting in an elastic or viscous deformation, depending on its composition.

The Newtonian approximation expresses the intensity of the tidal force by the second derivative of the gravitational potential: \[ a_\text{tide} \approx \frac{2GM R}{d^3} \] where \( G \) is the gravitational constant, \( M \) the mass of the attracting body, \( R \) the radius of the affected body, and \( d \) their distance. The term in \( 1/d^3 \) shows that tidal effects decrease very rapidly with distance, explaining their power in tight satellite systems like Io-Jupiter or Enceladus-Saturn.

Tidal Gradient on the Inner Moons of Jupiter and Saturn

This table presents the differential accelerations (gravitational gradient) exerted by the giant planets on their nearby moons. The stronger the gradient, the more significant the tidal effects.

Calculations based on: Jupiter Mass \( M_J = 1.898 \times 10^{27} \) kg, Saturn Mass \( M_S = 5.683 \times 10^{26} \) kg, \( G = 6.674 \times 10^{-11} \ \mathrm{m^3 \cdot kg^{-1} \cdot s^{-2}} \). Sources: NASA NSSDC, JPL Solar System Dynamics.

Dynamic Consequences: Rotation and Orbit

Tidal forces tend to align the rotation axis of the affected body with the direction towards the source object. This creates a torque that slows down the rotation of the body, dissipating energy as heat. This internal friction causes long-term rotational locks (e.g., the Moon rotates at the same speed as it orbits the Earth).

On Earth, this dissipation slows the Earth's rotation (increasing the length of the day by about 2.3 milliseconds per century) and transfers angular momentum to the Moon, which slowly moves away at a speed measured by lunar retro-reflectors (≈3.8 cm/year). This process of orbital evolution is general: it also affects exoplanets close to their host star (e.g., "hot" planets like 55 Cancri e).

Geological Effects: Io, Titan, and Europa

Tidal forces do more than shape orbits. In inner moons, they are responsible for viscous dissipation heating: the interior constantly deforms under the effect of periodic stresses. This internal heating can reach several tens of milliwatts per m²:

• Io: the most volcanic moon in the Solar System. Its slightly eccentric orbit (maintained by resonance with Europa and Ganymede) causes rhythmic stretching that liquefies the rocky interior. Eruptions cover its surface with sulfur and molten silicates.
• Europa: its icy crust rests on a 100 km deep saltwater ocean. Tidal energy keeps the water liquid, making Europa a major target in the search for extraterrestrial life.
• Enceladus: its polar geysers, observed by Cassini, release plumes of water vapor and particles, feeding Saturn's E ring.
• Titan: although farther away, it shows shape variations consistent with an internal ocean maintained by tidal effects and gravitational pressure.

Table of Tidal Effects in the Solar System

Sources: NASA Solar System Exploration, arXiv:2206.01297, PSJ 2021