Discover the Orbital Period of a Planet Using Newton's Law of Gravitation

[lnl]

Question:

Use Newton's version of Kepler's third law to answer the following questions. (Hint: The calculations for this problem are so simple that you will not need a calculator.) Imagine another solar system, with a star of the same mass as the Sun. Suppose there is a planet in that solar system with a mass twice that of Earth orbiting at a distance of 1 AU from the star. What is the orbital period of this planet? Explain.

Any thoughts?

 

[tiny-tim]

Welcome to PF!

Use Newton's version of Kepler's third law to answer the following questions. (Hint: The calculations for this problem are so simple that you will not need a calculator.) Imagine another solar system, with a star of the same mass as the Sun. Suppose there is a planet in that solar system with a mass twice that of Earth orbiting at a distance of 1 AU from the star. What is the orbital period of this planet? Explain.

Hi lnl! Welcome to PF!

Show us what you've tried, and where you're stuck, and then we'll know how to help.

 

[baba89]

I would approach this question by first understanding the principles behind Newton's Law of Gravitation and Kepler's Third Law. Newton's Law states that the gravitational force between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them. Kepler's Third Law states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis.

In this scenario, we are given a star with the same mass as the Sun and a planet with twice the mass of Earth orbiting at a distance of 1 AU (the distance between Earth and the Sun). Using Newton's Law, we can calculate the force of gravity between the star and the planet. Since the mass of the star is the same as the Sun, the force of gravity will also be the same as the force of gravity between the Sun and Earth.

Next, we can use Kepler's Third Law to calculate the orbital period of the planet. Since the planet's semi-major axis (the distance from the planet to the star) is 1 AU, we can plug in the values into the equation and solve for the orbital period. Since the planet's mass is twice that of Earth, the calculation would be (1 AU)^3 = (2 Earth mass)^3, which simplifies to 1 = 2^3. Therefore, the orbital period of the planet would be the same as Earth's, approximately 365 days.

In summary, by using Newton's Law of Gravitation and Kepler's Third Law, we can determine the orbital period of a planet in a given solar system. In this particular scenario, the planet's orbital period is the same as Earth's due to its distance from the star and its mass. This calculation showcases the power and accuracy of these laws in predicting the behavior of celestial bodies.