The inner and outer diameters of Saturn's B Ring are 184,000 and \(235,000 \mathrm{km},\) respectively. If the average thickness of the ring is 10 meters and the average density is 150 kilograms per cubic meter \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\), what is the mass of Saturn's B Ring?

Saturn's B Ring is one of the most prominent and luminous rings of Saturn, characterized by its considerable span. To understand its size, let's dive into its dimensions.

The inner diameter of the B Ring is given as 184,000 kilometers, while the outer diameter is 235,000 kilometers. Converting these diameters to radii allows us to work with more manageable numbers:

• Inner Radius: 92,000 kilometers
• Outer Radius: 117,500 kilometers

These radii provide the needed measurements to calculate the ring's surface area, which is an essential step in determining the volume of the ring.

With a substantial thickness of 10 meters, these dimensions help us grasp the size and scale of the B Ring more effectively, preparing us for further calculations.

Determining the volume of Saturn's B Ring involves a few straightforward mathematical steps. It's all about finding the volume using the ring's surface area and thickness.

First, we calculate the area of the ring's face using the formula for the area of a ring: \( A = \pi (R_{outer}^2 - R_{inner}^2) \), where:

• \( R_{outer} = 117,500 \text{ km} \)
• \( R_{inner} = 92,000 \text{ km} \)

By substituting the values, we get: \[ A = \pi ( (117500 \times 10^3)^2 - (92000 \times 10^3)^2 ) \] \[ A = \pi (1.380625 \times 10^{10} - 0.8464 \times 10^{10}) \] \[ A = \pi \times 5.34225 \times 10^9 \text{ km}^2 \] Converting the area from square kilometers to square meters simplifies upcoming calculations: \[ 1 \text{ km}^2 = 10^6 \text{ m}^2 \] \[ A = 5.34225 \times 10^{15} \text{ m}^2 \] Using the area and multiplying by the thickness of the ring (10 meters), we find the volume: \[ \text{Volume} = A \times \text{thickness} = 5.34225 \times 10^{15} \text{ m}^2 \times 10 \text{ m} = 5.34225 \times 10^{16} \text{ m}^3 \]

To determine the mass of the B Ring, we move from the known volume to mass by factoring in the ring's density. The given density of Saturn’s B Ring is 150 kilograms per cubic meter (\text{kg/m}^3).

The formula for mass calculation is: \[ \text{Mass} = \text{Density} \times \text{Volume} \] By substituting the values: \[ \text{Mass} = 150 \text{ kg/m}^3 \times 5.34225 \times 10^{16} \text{ m}^3 \] We get: \[ \text{Mass} = 8.013375 \times 10^{18} \text{ kg} \] This massive figure represents the mass of Saturn’s B Ring. Understanding density and how it relates to mass and volume is crucial in astrophysical calculations. It allows us to comprehend the physical properties and scale of celestial objects like Saturn's rings, providing insight into their composition and structure.