Ch 13: Gravitation

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 13: Gravitation

Problem 30b

Chapter 13, Problem 30b

In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term 'hot Jupiter'). The orbit was just 1/9 the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). How fast (in km/s) is this planet moving?

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First, understand that the problem involves calculating the orbital speed of a planet. The orbital speed can be found using the formula: \( v = \frac{2\pi r}{T} \), where \( v \) is the orbital speed, \( r \) is the radius of the orbit, and \( T \) is the orbital period.

Next, determine the radius \( r \) of the planet's orbit. The problem states that the orbit is \( \frac{1}{9} \) the distance of Mercury from the Sun. The average distance of Mercury from the Sun is approximately 57.9 million kilometers. Therefore, \( r = \frac{1}{9} \times 57.9 \text{ million km} \).

Convert the orbital period \( T \) from days to seconds, since the speed will be calculated in km/s. There are 86400 seconds in a day, so \( T = 3.09 \times 86400 \text{ seconds} \).

Substitute the values of \( r \) and \( T \) into the orbital speed formula: \( v = \frac{2\pi \times (\frac{1}{9} \times 57.9 \times 10^6)}{3.09 \times 86400} \).

Simplify the expression to find the orbital speed \( v \) in km/s. This will give you the speed at which the planet is moving in its orbit around the star HD 179949.

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Key Concepts

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

Orbital Mechanics

Orbital mechanics, or celestial mechanics, is the study of the motions of celestial objects under the influence of gravitational forces. It involves understanding how planets, moons, and other bodies move in their orbits. For this problem, knowing the orbital period and distance allows us to calculate the orbital speed using the formula for circular motion.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or rotation along a circular path. In this context, the planet's orbit is assumed to be circular, which simplifies calculations. The speed of an object in circular motion is given by the circumference of the orbit divided by the orbital period.

Gravitational Forces

Gravitational forces are the attractive forces between two masses. In the context of planetary motion, these forces keep planets in orbit around stars. Understanding gravitational forces helps explain why planets maintain their orbits and how their speeds are influenced by their proximity to the star they orbit.

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