In Planets, Moons and Small Bodies you sorted the planets into small rocky ones near the Sun and giants farther out. One number tells the two families apart and reveals what each planet is made of, even ones no spacecraft has landed on: its average density.
Why it works
Density is mass divided by volume, $\rho = \frac{M}{V}$. We can measure both from far away. A planet's mass comes from the orbits of its moons or of passing spacecraft (see weighing with orbits), and its radius from its angular size and distance. A planet is close to a sphere, so $V = \frac{4}{3}\pi r^3$ (volume of spheres). Then we compare with known materials: water is $1$ g/cm³, rock about $3$, iron about $8$, and hydrogen and helium far less. A planet's average density tells the mix.
Key idea
Average density $= \dfrac{M}{\frac{4}{3}\pi r^3}$. Rocky (terrestrial) planets come out at about $4$ to $5.5$ g/cm³: rock with iron cores. The giants come out at about $0.7$ to $1.6$ g/cm³: mostly hydrogen and helium (Jupiter, Saturn) or ices (Uranus, Neptune) around small dense cores.
| Planet | Mass (Earths) | Radius (km) | Density (g/cm³) | Family |
|---|---|---|---|---|
| Mercury | 0.055 | 2,440 | 5.43 | rocky |
| Venus | 0.815 | 6,052 | 5.24 | rocky |
| Earth | 1 | 6,371 | 5.51 | rocky |
| Mars | 0.107 | 3,390 | 3.93 | rocky |
| Jupiter | 318 | 69,911 | 1.33 | gas giant |
| Saturn | 95 | 58,232 | 0.69 | gas giant |
| Uranus | 14.5 | 25,362 | 1.27 | ice giant |
| Neptune | 17.1 | 24,622 | 1.64 | ice giant |
Worked examples
Worked example: Earth's density
Earth's mass is $5.97 \times 10^{24}$ kg and its radius $6371$ km. Find its average density.
- Radius in meters: $6.371 \times 10^{6}$ m.
- $V = \frac{4}{3}\pi (6.371 \times 10^{6})^3 = 1.083 \times 10^{21}\ \text{m}^3$ (scientific notation keeps this manageable).
- $\rho = \frac{5.97 \times 10^{24}}{1.083 \times 10^{21}} = 5510$ kg/m³ $= 5.51$ g/cm³.
- Surface rocks are only about $2.7$ to $3$ g/cm³, so the inside must be denser: Earth has an iron core.
Answer: $5.51$ g/cm³.
Worked example: A planet that floats
Saturn's mass is $5.68 \times 10^{26}$ kg and its radius $58{,}232$ km. Find its density.
- $V = \frac{4}{3}\pi (5.8232 \times 10^{7})^3 = 8.27 \times 10^{23}\ \text{m}^3$.
- $\rho = \frac{5.68 \times 10^{26}}{8.27 \times 10^{23}} = 687$ kg/m³ $= 0.69$ g/cm³.
- Less than water: in a big enough ocean, Saturn would float.
Answer: $0.69$ g/cm³.
Common mistake
It is tempting to think the biggest planets must be the densest. The opposite is true: Jupiter is 318 times Earth's mass but about 1300 times its volume, so it is only a quarter as dense. Density compares mass with volume; size alone tells nothing. Check: Earth, a small planet, is the densest of all.
Earth's density is a little higher than it would be uncompressed, because its own weight squeezes the deep rock and iron. The Moon, at $3.34$ g/cm³, has only a small iron core.