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How do shepherd moons prevent a planetary ring from spreading out?

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Images obtained by Voyager 2. — Uranus Two views of the southern hemisphere of Uranus, produced from images obtained by Voyager 2 on Jan. 17, 1986. In colours visible to the unaided human eye, Uranus is a bland, nearly featureless sphere (left). In a colour-enhanced view processed to bring out low-contrast details, Uranus shows the banded cloud structure common to the four giant planets (right). From the polar perspective of Voyager at the time, the bands appear concentric around the planet's rotational axis, which is pointing nearly toward the Sun. Small ring-shaped features in the right image are artifacts arising from dust in the spacecraft's camera.

Shepherd moons prevent planetary rings from spreading by exerting gravitational forces that maintain the rings' narrow width. Because collisions between ring particles naturally cause a ring to spread radially, these moons act as dynamic stabilizers through a process called shepherding.1

A shepherd moon establishes a stable resonance with ring particles, where their orbital periods are related by a ratio of small whole numbers. As the moon and particles interact gravitationally, they exchange angular momentum. This interaction causes energy to be dissipated through collisions among the ring particles, effectively repelling the moon and the ring from one another:

Inner shepherd moons transmit angular momentum to the ring, pushing it outward.
Outer shepherd moons receive angular momentum from the ring, pushing it inward.

By flanking a ring on both sides, a pair of shepherd moons creates a gravitational boundary that prevents the ring from spreading beyond the radius of resonance, thereby keeping it confined to a narrow band. At Uranus, the moons Cordelia and Ophelia serve this function for the Epsilon ring.1