The star Rho ^1 Cancri is 57 light-years from the earth and...

The star Rho $^{1}$ Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho I Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho "Cancri?

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Key Concepts

Newton’s Law of Universal Gravitation

This principle states that two bodies with mass attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. It is fundamental in orbital calculations because the gravitational force acting on a planet provides the centripetal force required to keep it in orbit.

Circular Orbital Motion and Centripetal Force

In a circular orbit, the gravitational force acting on an orbiting body is exactly balanced by the centripetal force needed to keep it moving along its circular path. By equating these forces, one can derive key parameters such as orbital speed and orbital period, which describe the motion of planets and satellites.

Kepler’s Third Law of Planetary Motion

Kepler's Third Law relates the orbital period of a planet to the size of its orbit, typically stating that the square of the period is proportional to the cube of the orbital radius (with adjustments for the mass of the central object). This law is integral in determining how changes in orbital distance and central mass affect the duration of an orbit.

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