The star Rho1 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 Rho1 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 Rho1 Cancri?

Verified step by step guidance

1. Identify the given values: mass of Rho1 Cancri (M) is 0.85 times the mass of the Sun, and the orbital radius (r) of the planet is 0.11 times the Earth's orbital radius around the Sun.
2. Use Newton's version of Kepler's third law to find the orbital speed (v) of the planet. The formula to use is v = \sqrt{\frac{GM}{r}}, where G is the gravitational constant.
3. Substitute the values into the formula, noting that M = 0.85M_{sun} and r = 0.11r_{Earth}. Use the known values of G, M_{sun}, and r_{Earth} to calculate v.
4. To find the orbital period (T) of the planet, use the formula T = \frac{2\pi r}{v}. This formula relates the orbital period to the orbital radius and the orbital speed.
5. Substitute the values of r and v calculated in the previous steps into the formula to find T.

Key Concepts

Gravitational Force and Orbital Motion

The gravitational force between two masses governs the motion of celestial bodies. For a planet in orbit, this force provides the necessary centripetal acceleration to maintain a circular path. The gravitational force can be calculated using Newton's law of universal gravitation, which states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between their centers.

Orbital Speed

Orbital speed is the velocity at which a planet travels along its circular orbit. It can be derived from the balance between gravitational force and the required centripetal force for circular motion. The formula for orbital speed (v) is given by v = √(GM/r), where G is the gravitational constant, M is the mass of the star, and r is the orbital radius of the planet.

Orbital Period

The orbital period is the time it takes for a planet to complete one full orbit around a star. Kepler's third law relates the orbital period (T) to the semi-major axis (r) of the orbit, stating that T² is proportional to r³ for orbits around the same central mass. The formula T = 2π√(r³/GM) allows us to calculate the period based on the radius of the orbit and the mass of the star.