The asteroid Ida (mass 4.2 ×10^16 kg ) is attended by a tiny asteroidal moon, Dactyl, which orbi

step 1 Hi, everybody. So neglecting the mass of the moon, what is the ductal orbital period in hours? S

The asteroid Ida (mass 4.2 ×10^16 kg ) is attended by a tiny...

The asteroid Ida (mass $4.2 \times 10^{16} \mathrm{kg}$ ) is attended by a tiny asteroidal moon, Dactyl, which orbits Ida at an average distance of $90 \mathrm{km} .$ Neglecting the mass of the tiny moon, what is Dactyl's orbital period in hours?

Key Concepts

Kepler's Third Law

Kepler's Third Law relates the orbital period of a body to the size of its orbit around a central mass. In the context of circular orbits under gravity, it can be mathematically expressed using the gravitational constant and the mass of the central object, allowing us to calculate the period of orbit from the orbital radius.

Two-Body Problem Simplification

When one object's mass is negligibly small compared to the other, the orbital calculations can be simplified by ignoring the smaller mass. This allows the central mass's gravitational influence to dictate the dynamics of the orbiting body, which simplifies the application of Kepler's Third Law and other orbital mechanics formulas.

Newton's Law of Universal Gravitation

This law states that every two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is fundamental in understanding how a central body, such as an asteroid, exerts an attractive force on an orbiting object.

Centripetal Force in Circular Orbits

For an object moving in a circular orbit, the gravitational force acts as the centripetal force that keeps the object in its circular path. By equating the gravitational force to the necessary centripetal force, one can derive the relationship needed to calculate the orbital period or speed of the orbiting body.

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