Question

now consider a second, identical satellite on the same orbital path but trailing one sixth of a period behind the first satellite. how far from the centre of the planet is the centre of mass position for the planet and both satellites?

Answer 1

The center of mass position for the planet and both satellites, considering a second satellite trailing one-sixth of a period behind the first on the same orbital path, remains at the planet's center.

Step-by-step explanation:

In this scenario, the two identical satellites are in the same orbital path, with one trailing one-sixth of a period behind the other. For the center of mass (CM) position calculation, we consider the masses and distances involved.

The center of mass formula is CM = (m₁r₁ + m₂r₂) / (m₁ + m₂), where m₁, m₂ are the masses and r₁, r₂ are the distances of the respective objects from the center.

Given the two satellites have the same mass and are identical, their contribution to the CM is symmetrical. As they have the same distance from the planet's center, their combined effect on the CM remains at the center of the planet.

The trailing of one satellite by one-sixth of a period does not impact the CM since the configuration involves equal masses and symmetrical positions. Thus, the CM remains fixed at the center of the planet.

This conclusion holds due to the equal masses and identical distances of the satellites from the planet's center, ensuring the CM position remains unchanged at the planet's center, irrespective of the orbital phase difference between the satellites.