Jupiter is an oblate planet with an average radius of 69,900 \(\mathrm{km},\) compared to Earth's average radius of \(6,370 \mathrm{km}\) a. Given that volume is proportional to the cube of the radius, how many Earth volumes could fit inside Jupiter? b. Jupiter is 318 times as massive as Earth. Show that Jupiter's average density is about one-fourth that of Earth's.

One of the first steps in comparing planetary measurements is understanding how their volumes relate. To determine the volume of a sphere, we use the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( V \) stands for the volume and \( r \) represents the radius of the sphere. Since the volume is proportional to the cube of the radius, if you know the radius, you can easily calculate the volume.

For Jupiter and Earth, we start by noting their average radii. Jupiter's average radius is 69,900 km and Earth's is 6,370 km. To find how many Earth volumes can fit inside Jupiter, calculate the ratio of their radii and then cube that number:

- Ratio of radii: \( \frac{69,900}{6,370} = 10.97 \) - Ratio of volumes: \( 10.97^3 \approx 1,320 \)

Hence, approximately 1,320 Earth-sized spheres could fit inside Jupiter. This massive difference highlights the gigantic scale of Jupiter compared to Earth.

After understanding the volume ratio, it’s crucial to look at the density ratio to get a full picture of the planetary measurements. Density is defined as mass per unit volume, expressed mathematically as: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) We know that Jupiter is 318 times as massive as Earth and, from our previous calculations, that its volume is 1,320 times greater. To find the density ratio, you divide Jupiter’s mass ratio by its volume ratio:

- Ratio of masses: 318 - Ratio of volumes: 1,320 - Density ratio: \( \frac{318}{1,320} \approx 0.24 \)

Thus, Jupiter's average density is approximately one-fourth (or 0.24 times) that of Earth's. This lower density is mainly because Jupiter is a gas giant, composed mostly of hydrogen and helium, unlike Earth, which has a rocky composition.

Now that we've explored volume and density, let's compare Jupiter and Earth overall. The two planets differ significantly in composition, size, and characteristics:

- **Volume**: Jupiter is about 1,320 times the volume of Earth, making it the largest planet in our solar system.
- **Mass**: Jupiter is 318 times as massive as Earth, yet it’s not as dense.

From these comparisons, we can deduce that if you were to somehow combine Earths to match Jupiter’s mass and volume, you would need 1,320 Earth-sized spheres.

It’s also fascinating to consider how these differences affect each planet's gravity, magnetic field, and overall structure. Jupiter's immense size and mass contribute to its powerful magnetic field and thick atmospheric composition, while Earth supports diverse ecosystems with its higher density and suitable environmental conditions. Understanding these comparisons helps drive home the immense scale and diversity of our solar system.