Final answer:
The Roche limit for an Earth-orbiting body with the same density as Earth would be greater than the Earth's radius but less than the geosynchronous orbit distance, likely around 6,000 km given the structural strength of silicate bodies and Earth's density.
Step-by-step explanation:
The question asks for the Roche limit of an Earth-like body orbiting Earth. The Roche limit is the minimum distance at which a celestial body, due to tidal forces, will disintegrate due to gravitational forces exceeding its own structural cohesion.
Considering the density information provided, we conclude that Earth's density is about 5.51 g/cm³ (based on the given mass and volume), making it the densest of the planets in our solar system.
The Roche limit can be calculated using a formula that involves the densities of the two bodies and the radius of the parent body.
Given that the celestial body has the same density as Earth, and considering the strength of materials that make up Earth-like celestial objects as mentioned in the reference material, the Roche limit would indeed be close to the Earth's radius; however, it would need to be a bit further away to prevent tidal disintegration.
Hence, the Roche limit will be greater than the Earth's radius but less than the geosynchronous orbit distance. Considering this information, we're led to the conclusion that option b) 6,000 km, roughly Earth's radius, is the likely Roche limit in this scenario.
Objects within this limit could disintegrate due to Earth's tidal forces before their gravity can pull them into a spherical shape.