Question

Find the following center-of-mass information about objects in the Solar System. You can look up the necessary data on the Internet or in the tables in Chapter 12 of this book. Assume spherically symmetrical mass distributions for all objects under consideration. a) Determine the distance from the center of mass of the Earth-Moon system to the geometric center of Earth. b) Determine the distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun.

Short answer

Question: Calculate the distance from the center of mass to the geometric center of Earth in the Earth-Moon system and the distance from the center of mass to the geometric center of the Sun in the Sun-Jupiter system. Answer: The distance from the center of mass to the geometric center of Earth in the Earth-Moon system is 4,676 meters. The distance from the center of mass to the geometric center of the Sun in the Sun-Jupiter system is 741,500 meters.

Step-by-step solution

(Data Collection for Earth-Moon System)

Look up the necessary data for the Earth and Moon masses and their distance apart. Earth mass (M_E) is 5.97 × 10^24 kg, Moon mass (M_M) is 7.34 × 10^22 kg, and their distance apart (r_EM) is 3.84 × 10^8 meters.

(Calculate Center of Mass for Earth-Moon System)

Use the center of mass formula: Center of mass = (M_E * r1 + M_M * r2) / (M_E + M_M). In this case, we will set the position of Earth at the origin (r1 = 0). Thus, r2 = r_EM. Center of mass = (M_E * 0 + M_M * r_EM) / (M_E + M_M) Center of mass = (7.34 × 10^22 kg * 3.84 × 10^8 meters) / (5.97 × 10^24 kg + 7.34 × 10^22 kg) Center of mass = 4,676 meters The center of mass is 4,676 meters from the geometric center of Earth.

(Data Collection for Sun-Jupiter System)

Look up the necessary data for the Sun and Jupiter masses and their distance apart. Sun mass (M_S) is 1.989 × 10^30 kg, Jupiter mass (M_J) is 1.898 × 10^27 kg, and their distance apart (r_SJ) is 7.78 × 10^11 meters.

(Calculate Center of Mass for Sun-Jupiter System)

Use the center of mass formula: Center of mass = (M_S * r1 + M_J * r2) / (M_S + M_J). In this case, we will set the position of Sun at the origin (r1 = 0). Thus, r2 = r_SJ. Center of mass = (M_S * 0 + M_J * r_SJ) / (M_S + M_J) Center of mass = (1.898 × 10^27 kg * 7.78 × 10^11 meters) / (1.989 × 10^30 kg + 1.898 × 10^27 kg) Center of mass = 741,500 meters The center of mass is 741,500 meters from the geometric center of the Sun.

Key concepts

Celestial Mechanics

Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of celestial bodies. This field uses the principles of physics and mathematics to explain the orbits of planets, moons, and other objects as they move through space. At the heart of celestial mechanics is the concept of the center of mass, often called the barycenter. This point is the balance point of an entire system where all the mass could be concentrated without changing the gravitational interactions within the system.

Understanding the concept of the center of mass is critical in predicting the motion of bodies in space. For example, when two bodies orbit each other, they do so around their shared center of mass. In a system like the Earth-Moon system, this results in both bodies orbiting a point in space that is not at the center of either body, but rather at a point that reflects their respective masses and the distance between them.

Earth-Moon System

The Earth-Moon system is a prime example of two celestial bodies interacting with each other, with the Moon orbiting the Earth. However, contrary to popular belief, the Moon does not orbit precisely around the Earth's center. Instead, both Earth and the Moon orbit around their common center of mass. This point is located inside the Earth, but not at its geometric center, due to the relative masses of the Earth and the Moon.

To find the center of mass in the Earth-Moon system, as demonstrated in the exercise, we consider the masses of the two bodies and the distance between them. The location of the center of mass tells us about the nature of their gravitational dance. The Earth experiences a slight 'wobble' as a result, contributing to the complex motion we observe from our planetary perspective. This concept is essential for understanding not just how the Moon moves, but also for deep space missions that must navigate the gravitational interplay between Earth and its natural satellite.

Sun-Jupiter System

The Sun-Jupiter system, with its significant contrast in mass between the Sun and Jupiter, still obeys the same laws of celestial mechanics. The center of mass for this system lies within the Sun, yet it's not at the Sun's center. Jupiter's massive size compared to other planets means it has a considerable influence on the location of the barycenter of the system.

The exercise calculation reveals the surprising fact that even as massive as the Sun is, the center of mass from which it orbits in relationship with Jupiter is quite a distance from its geometric center. This point illustrates the immense mass of Jupiter and its effect on the solar gravitational field. The study of the Sun-Jupiter system, and how both bodies influence each other's motion, is a vivid example of the dynamic nature of celestial mechanics, with far-reaching implications for understanding the movement and stability of our solar system as a whole.