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?

Ch 13: Gravitation

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 13: Gravitation

Problem 31

Chapter 13, Problem 31

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?

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To find the orbital speed of the planet, we can use the formula for orbital speed: \( v = \sqrt{\frac{GM}{r}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the star, and \( r \) is the orbital radius of the planet.

First, calculate the mass of Rho1 Cancri. Given that its mass is 0.85 times the mass of the sun, we have \( M = 0.85 \times M_{\text{sun}} \).

Next, determine the orbital radius \( r \) of the planet. It is given as 0.11 times the radius of Earth's orbit (1 astronomical unit, AU). Therefore, \( r = 0.11 \times 1 \text{ AU} \).

Substitute the values of \( G \), \( M \), and \( r \) into the orbital speed formula to find the speed \( v \).

To find the orbital period \( T \), use Kepler's third law: \( T^2 = \frac{4\pi^2r^3}{GM} \). Solve for \( T \) by substituting the known values of \( G \), \( M \), and \( r \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Force and Orbital Motion

Gravitational force is the attractive force between two masses, such as a star and a planet. It provides the necessary centripetal force to keep a planet in orbit. The balance between gravitational force and the planet's inertia determines the orbital speed and period, following Kepler's laws of planetary motion.

Kepler's Third Law

Kepler's Third Law states that the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit. This law helps relate the orbital period to the distance from the star, allowing us to calculate the period if the orbital radius and the star's mass are known.

Circular Orbital Speed

The orbital speed of a planet in a circular orbit can be calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of the star, and r is the orbital radius. This formula derives from equating gravitational force to the centripetal force required for circular motion.

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