The Hill Sphere is the region of gravitational influence of a celestial body, where its gravity is dominant over the gravity of other nearby celestial bodies.

The size of a Hill Sphere depends on the mass of the central body, the mass of the orbiting body, and the distance between the two bodies.

The formula for calculating the radius of a Hill Sphere is $R_H = (1 + m/3M)^{1/3} a$, where $R_H$ is the radius of the Hill Sphere, $m$ is the mass of the orbiting body, $M$ is the mass of the central body, and $a$ is the distance between the two bodies.

The Hill Sphere is significant in celestial mechanics because it determines the stability of orbits, affects the tidal forces, and influences the Roche Limit.

Within the Hill Sphere, the region where the gravitational influence of the central body is dominant is called the Inner Hill Sphere.

Beyond the Hill Sphere, the region where the gravitational influence of the central body becomes negligible is called the Outer Hill Sphere.

Within the Hill Sphere, the Roche Limit is the distance from the central body where the tidal forces are strong enough to disrupt a celestial body.

Within the Hill Sphere, the Roche Limit is significant because it affects the tidal forces and determines the stability of orbits.

The Hill Sphere and the Lagrange Points are located at different distances from the central body.

The Sun has the largest Hill Sphere in our solar system due to its immense mass.

In the context of exoplanets, the Hill Sphere is significant because it determines the habitable zone, affects the tidal forces, and influences the Roche Limit.

The Hill Sphere affects the stability of orbits around a celestial body by determining the orbital period, influencing the orbital eccentricity, and affecting the orbital inclination.

The Hill Sphere and the gravitational sphere of influence overlap, but they are not the same.

In the context of spacecraft missions, the Hill Sphere is significant because it determines the trajectory of the spacecraft, affects the communication with the spacecraft, and influences the power requirements of the spacecraft.

The Hill Sphere affects the formation of planetary systems by determining the number of planets in a system, influencing the size of the planets, and affecting the orbital spacing of the planets.