Analyse and interpret planetary data about orbital distance, orbital duration, density, surface temperature and uniform gravitational field strength at the planet's surface

6.1.2 The Solar System

Objective: Analyse and interpret planetary data about orbital distance, orbital duration, density, surface temperature and uniform gravitational field strength at the planet's surface.

Key Concepts

• Orbital distance – average distance from the Sun (semi‑major axis), usually expressed in astronomical units (AU) or kilometres.
• Orbital duration – time taken to complete one revolution around the Sun (orbital period), expressed in Earth years or days.
• Density – mass per unit volume, \$\rho = \frac{m}{V}\$, expressed in \$\text{kg·m}^{-3}\$.
• Surface temperature – equilibrium temperature determined by solar radiation and atmospheric effects, expressed in Kelvin (K) or degrees Celsius (°C).
• Uniform gravitational field strength at the surface – surface gravity \$g\$, calculated from \$g = \frac{GM}{R^{2}}\$, where \$G\$ is the universal gravitational constant, \$M\$ the planet’s mass and \$R\$ its radius.

Planetary Data (selected planets)

Interpreting the Data

Use the table to answer the following questions. Show all working, using the appropriate formulas.

1. Verify Kepler’s third law for the planets listed. The law states that \$T^{2} \propto a^{3}\$, where \$T\$ is the orbital period (in Earth years) and \$a\$ the orbital distance (in AU). Calculate \$\frac{T^{2}}{a^{3}}\$ for each planet and comment on the constancy of the ratio.
2. Calculate the mass of each planet using the surface gravity formula rearranged as \$M = \frac{gR^{2}}{G}\$. You will need the planet’s radius \$R\$, which can be obtained from the relation \$V = \frac{4}{3}\pi R^{3}\$ and the given density \$\rho\$ (since \$M = \rho V\$). Provide a step‑by‑step example for Earth.
3. Discuss why \cdot enus, despite being closer to the Sun than Earth, has a higher mean surface temperature. Include the role of atmospheric composition and the greenhouse effect.
4. Compare the densities of the terrestrial planets (Mercury, Venus, Earth, Mars) with those of the gas giants (Jupiter, Saturn, Uranus, Neptune). What does this tell you about their internal structure?
5. Determine which planet would experience the weakest weight for a 70 kg person and calculate that weight. Use \$W = mg\$ where \$g\$ is the surface gravity from the table.

Worked Example: Surface Gravity of Earth

Given:

• Mean density of Earth, \$\rho = 5515\ \text{kg·m}^{-3}\$
• Surface gravity, \$g = 9.8\ \text{m·s}^{-2}\$ (to be verified)

Step 1 – Find Earth’s radius using density and known mass (or use known radius \$R = 6.371\times10^{6}\ \text{m}\$ for verification).

Step 2 – Compute mass from \$M = \rho \times \frac{4}{3}\pi R^{3}\$:

\$M = 5515 \times \frac{4}{3}\pi (6.371\times10^{6})^{3} \approx 5.97\times10^{24}\ \text{kg}\$

Step 3 – Use \$g = \frac{GM}{R^{2}}\$ (with \$G = 6.674\times10^{-11}\ \text{N·m}^{2}\text{kg}^{-2}\$):

\$g = \frac{(6.674\times10^{-11})(5.97\times10^{24})}{(6.371\times10^{6})^{2}} \approx 9.8\ \text{m·s}^{-2}\$

The calculated value matches the tabulated value, confirming the data.

Suggested Activities

• Plot \$T^{2}\$ against \$a^{3}\$ on graph paper to visualise Kepler’s law.
• Create a bar chart (hand‑drawn) comparing surface gravities of the planets.
• Research the composition of each planet’s atmosphere and write a short paragraph linking it to surface temperature.