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 ${ }^{1}$ 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?

Key Concepts

Kepler's Third Law

Kepler's Third Law states that the square of a planet’s orbital period is proportional to the cube of its orbital radius, with the proportionality constant depending on the mass of the central object. This law is instrumental in determining the orbital period of a planet around a star, especially when using comparative units such as those relative to the Earth-Sun system.

Astronomical Scaling and Units

In astrophysics, using scaled or normalized units, such as expressing orbital radii in terms of astronomical units (the average Earth-Sun distance), simplifies calculations and comparisons. This concept allows one to relate measured distances and orbital parameters of extrasolar planets to those in our own solar system, facilitating the determination of orbital speeds and periods.

Newton's Law of Gravitation

Newton's Law of Gravitation describes the force between two masses as directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This concept is central to understanding how stars like Rho ¹ Cancri exert a gravitational pull on orbiting planets, determining their orbital speeds and paths.

Circular Orbital Motion

For objects in circular orbits, the gravitational force provides the necessary centripetal force needed to keep the object moving in a circle. By equating the gravitational force to the centripetal force, one can derive the orbital speed formula and analyze how changes in mass or orbital radius affect the motion.

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