Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Verified step by step guidance

1. Step 1: Recall the formula for density, which is defined as mass divided by volume. The formula is given by \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume.
2. Step 2: Use the given information to express the mass of Titania relative to Earth's mass. Since Titania's mass is 1/1700 of Earth's mass, write it as \( m_{\text{Titania}} = \frac{1}{1700} m_{\text{Earth}} \).
3. Step 3: Calculate the volume of Titania using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. Given that Titania's radius is 1/8 of Earth's radius, express it as \( r_{\text{Titania}} = \frac{1}{8} r_{\text{Earth}} \) and substitute this into the volume formula to get \( V_{\text{Titania}} = \frac{4}{3} \pi \left(\frac{1}{8} r_{\text{Earth}}\right)^3 \).
4. Step 4: Simplify the expression for Titania's volume by calculating \( \left(\frac{1}{8} r_{\text{Earth}}\right)^3 \) and substituting back into the volume formula.
5. Step 5: Substitute the expressions for Titania's mass and volume into the density formula to find the average density of Titania. Simplify the expression to find the ratio of Titania's density to Earth's density.

Key Concepts

Density

Density is defined as mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is a fundamental property of materials that helps determine their composition and behavior under various conditions. In this context, calculating the density of Titania involves using its mass and volume to understand its material composition.

Volume of a Sphere

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius. This formula is essential for determining the volume of Titania, given its radius is provided. Understanding how to apply this formula allows for the calculation of Titania's volume, which is necessary for finding its density.

Comparative Planetology

Comparative planetology is the study of planets and moons by comparing their physical and chemical properties. This concept is relevant in understanding Titania's density in relation to other celestial bodies, such as Earth and ice-rich bodies. It provides context for interpreting Titania's composition and the implications of its lower density compared to rocky materials.