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from Google AI:

The Roche limit is the distance from a celestial body at which the tidal forces from a more massive primary body overcome the smaller body's self-gravity, causing it to disintegrate. This disintegration forms a ring of debris, a common way planetary rings form, and the limit's exact distance depends on the relative densities and structural integrity of the two bodies.

What happens at the Roche limit?

- Gravitational Overwhelm: The larger body's gravitational pull creates tidal forces that are stronger on the near side of the smaller body than on its far side. 

 

- Disintegration: When the smaller body (like a moon or comet) comes within the Roche limit, these tidal forces stretch and tear it apart. 

 

- Formation of Rings: The resulting debris then spreads out to form a ring around the primary body.

Factors influencing the Roche limit

The Roche limit is not a fixed distance but varies depending on:

- Density of the primary body: A more massive or denser primary body has stronger tidal forces and a larger Roche limit. 

 

- Density and size of the satellite: A smaller, less dense satellite will be torn apart at a greater distance than a larger, more dense one. 

 

-Structural strength of the satellite: A rigid object like a solid ball will have a closer Roche limit than a "fluid" or deformable object, which will stretch and break apart more easily.

Examples

- Saturn's rings: Many astronomers believe that Saturn's rings are made from the debris of a moon that crossed inside Saturn's Roche limit. 

 

- Phobos' future: Phobos, a moon of Mars, is on a path to eventually cross into Mars' Roche limit, which could lead to the formation of a ring around Mars

More:

The Roche limit is the distance within which a celestial body will disintegrate due to tidal forces, while the event horizon is a boundary around a black hole from which nothing, not even light, can escape. You cannot directly calculate a single "Roche limit event horizon" because they are distinct astrophysical concepts. However, you can calculate the Roche limit for an object approaching a black hole using the formula d = R(2ρM/ρm)¹/³ (or d = R(2MM/Mm)¹/³) for rigid or fluid bodies, respectively, and the Schwarzschild radius for the event horizon using the formula rs = 2GM/c², noting that the Roche limit is usually much larger than the event horizon. 

Calculating the Roche Limit

The Roche limit depends on the radius and densities of the two bodies involved. For a rigid satellite. 

d = R(2 * ρM/ρm)¹/³ 

• d: The Roche limit (distance of disintegration)
• R: The radius of the larger body
• ρM: The density of the larger body
• ρm: The density of the smaller body (satellite)
• For a fluid satellite:

d ≈ 2.423R(ρM/ρm)¹/³ 

• This accounts for the satellite's ability to deform.

Calculating the Event Horizon (Schwarzschild Radius)

The event horizon for a non-rotating black hole is defined by the Schwarzschild radius: 

• rs = 2GM/c²:
  • rs: The Schwarzschild radius (event horizon radius)
  • G: The gravitational constant
  • M: The mass of the black hole
  • c: The speed of light

Key Differences

• Mechanism: 

  The Roche limit is caused by differential tidal forces stretching an object apart. The event horizon is a point of no return due to extreme gravitational warping of spacetime. 
• Relationship to Black Holes: 

  The event horizon is a specific property of the black hole itself, while the Roche limit is a distance defined by the interaction between a black hole and another object. 
• Typical Ratios: 

  For a sufficiently massive black hole, its Roche limit for a stellar-mass object can be well outside its event horizon, meaning you can get close to a black hole without immediate spaghettification, though you would still eventually cross the event horizon

 

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