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Chapter 7: Problem 35

Jupiter has a mass equal to 318 times Earth's mass, an orbital radius of \(5.2 \mathrm{AU}\), and an orbital velocity of \(13.1 \mathrm{km} / \mathrm{s}\). Earth's orbital velocity is \(29.8 \mathrm{km} / \mathrm{s}\). What is the ratio of Jupiter's orbital angular momentum to that of Earth?

Short Answer

Expert verified
The ratio of Jupiter’s orbital angular momentum to that of Earth is approximately 733.56.

Step by step solution

01

Understanding Orbital Angular Momentum

Orbital angular momentum for a planet can be calculated using the formula: \[ L = m \times v \times r \] where \( L \) is angular momentum, \( m \) is mass, \( v \) is orbital velocity, and \( r \) is the orbital radius.
02

Calculate Earth's Orbital Angular Momentum

Use the given data for Earth: Earth’s mass \( m_E \), Earth’s orbital velocity \( v_E = 29.8 \, \mathrm{km/s} \), Earth’s orbital radius \( r_E = 1 \mathrm{AU} \). Thus, \[ L_E = m_E \times v_E \times r_E = m_E \times 29.8 \, \mathrm{km/s} \times 1 \, \mathrm{AU} \]
03

Calculate Jupiter's Orbital Angular Momentum

Use the given data for Jupiter: Jupiter’s mass is \( m_J = 318 \times m_E \), Jupiter’s orbital velocity \( v_J = 13.1 \, \mathrm{km/s} \), Jupiter’s orbital radius \( r_J = 5.2 \, \mathrm{AU} \). Thus, \[ L_J = m_J \times v_J \times r_J = 318 \times m_E \times 13.1 \, \mathrm{km/s} \times 5.2 \, \mathrm{AU} \]
04

Find the Ratio of Orbital Angular Momenta

To find the ratio \( \frac{L_J}{L_E} \), we set up the ratio: \[ \frac{L_J}{L_E} = \frac{318 \times m_E \times 13.1 \, \mathrm{km/s} \times 5.2 \, \mathrm{AU}}{m_E \times 29.8 \, \mathrm{km/s} \times 1 \, \mathrm{AU}} \] Cancel out \( m_E \) and simplify: \[ \frac{L_J}{L_E} = \frac{318 \times 13.1 \times 5.2}{29.8} \]
05

Simplify the Ratio

Calculate the numerical value: \[ \frac{L_J}{L_E} = \frac{318 \times 13.1 \times 5.2}{29.8} \approx \frac{21863.6}{29.8} \approx 733.56 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Jupiter
Jupiter is the largest planet in our solar system and has a mass that is 318 times greater than that of Earth. This gas giant plays a significant role in shaping the orbits of other objects in the solar system due to its immense gravitational influence. Understanding Jupiter's properties, such as its mass and velocity, is essential when studying orbital mechanics and comparing it to other planets like Earth.
Earth
Earth, our home planet, has a relatively moderate mass compared to Jupiter. It orbits the Sun at an average distance of 1 Astronomical Unit (AU) and has an orbital velocity of 29.8 km/s. By comparing Earth’s orbital properties with those of Jupiter, we can better understand the concept of orbital angular momentum and the factors influencing it.
orbital mechanics
Orbital mechanics is the study of the motions of celestial bodies influenced by gravitational forces. Key concepts include:
  • Orbital radius: The average distance between a celestial body and the object it orbits (such as Earth orbiting the Sun).
  • Orbital velocity: The speed at which a celestial body travels along its orbit.
  • Gravitational force: The force that attracts two bodies towards each other, dependent on their masses and the distance between them.
Laws like Kepler’s Laws of Planetary Motion and Newton's Law of Universal Gravitation are fundamental to understanding how objects move in space.
mass and velocity
Mass and velocity are two critical factors in determining a planet's orbital angular momentum. Angular momentum, denoted as L, is calculated using:

\[ L = m \times v \times r \]

where:
  • \( m \): mass of the planet
  • \( v \): orbital velocity
  • \( r \): orbital radius
For example, Jupiter's greater mass and orbital velocity significantly affect its angular momentum compared to Earth. When combined, these factors illustrate why Jupiter's angular momentum is so much larger than Earth's.
calculating ratios
Calculating the ratio of angular momentum between two celestial bodies involves comparing their respective values. For Jupiter and Earth, this entails:
  • Observing the given values of mass, velocity, and orbital radius
  • Substituting these values into the angular momentum formula for each planet
  • Finding the ratio by dividing Jupiter's angular momentum by Earth's
In the specific example provided, this ratio calculation showed that Jupiter's angular momentum is approximately 733.56 times that of Earth's. This illustrates the immense difference in their orbital mechanics.

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Most popular questions from this chapter

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