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 (Cambridge AO1‑AO3):

Analyse and interpret planetary data – orbital distance, orbital period, density, mean surface temperature and surface‑gravity (uniform gravitational field strength).

Use the data to calculate orbital speed, surface gravity, planetary mass and escape velocity, and to explain the structure and physical differences between the planets.

1. Overview of the Solar System

• Central star: The Sun – a G2 V main‑sequence star.
• Major planets (in order from the Sun): Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune.
• Dwarf planets & minor bodies: Pluto, Eris, Ceres (asteroid belt), plus typical comets, asteroids and Kuiper‑belt objects.
• Other components: Natural satellites, rings, the Kuiper belt, the Oort cloud.

1.1 Earth’s rotation and axial tilt

• Rotation period ≈ 24 h → day/night cycle.
• Axial tilt ≈ 23.5° to the orbital plane.
• The tilt causes the Sun’s apparent altitude to vary during the year, producing the four seasons.

1.2 Moon’s orbit and phases

• Sidereal orbital period ≈ 27.3 days; synodic (full phase cycle) ≈ 29.5 days.
• Phases arise because we see different portions of the Moon’s sun‑lit hemisphere as it orbits Earth.
• Lunar eclipses occur when the Earth lies between the Sun and the Moon (full Moon) and the three bodies are aligned.

2. Key Concepts, Formulae and Constants

Constants (to 3 s.f.):

\(G = 6.674\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\)  \(1\ \text{AU}=1.496\times10^{8}\ \text{km}\)  \(1\ \text{yr}=3.156\times10^{7}\ \text{s}\)

3. Planetary Data (selected planets)

4. Worked Examples

4.1 Verifying Kepler’s 3^rd Law

Calculate \(\displaystyle \frac{T^{2}}{a^{3}}\) for each planet (using the values in the table). The result should be ≈ 1 when \(T\) is in years and \(a\) in AU.

The near‑constant ratio confirms Kepler’s law for the Solar System.

4.2 Deriving Mass from Density and Surface Gravity (Earth)

1. Given: \(\rho = 5515\ \text{kg m}^{-3}\), \(g = 9.8\ \text{m s}^{-2}\), \(R = 6.371\times10^{6}\ \text{m}\).
2. Mass from density:

   \[

   M = \rho\frac{4}{3}\pi R^{3}

   = 5515\;\frac{4}{3}\pi (6.371\times10^{6})^{3}

   \approx 5.97\times10^{24}\ \text{kg}.

   \]

3. Check surface gravity:

   \[

   g = \frac{GM}{R^{2}}

   = \frac{(6.674\times10^{-11})(5.97\times10^{24})}{(6.371\times10^{6})^{2}}

   \approx 9.8\ \text{m s}^{-2}.

   \]

4. The calculated \(g\) matches the tabulated value, confirming internal consistency.

4.3 Mass and Radius of Mars from \(\rho\) and \(g\)

Because both \(\rho\) and \(g\) are given, we can solve for the radius first.

\[

g = \frac{G\rho\frac{4}{3}\pi R^{3}}{R^{2}} = \frac{4\pi G\rho}{3}\,R

\qquad\Longrightarrow\qquad

R = \frac{3g}{4\pi G\rho}.

\]

\[

R = \frac{3(3.7)}{4\pi(6.674\times10^{-11})(3930)}

\approx 3.39\times10^{6}\ \text{m}=3\,390\ \text{km}.

\]

Now the mass:

\[

M = \rho\frac{4}{3}\pi R^{3}

= 3930\;\frac{4}{3}\pi (3.39\times10^{6})^{3}

\approx 6.4\times10^{23}\ \text{kg}.

\]

4.4 Average Orbital Speed of Earth

\[

v = \frac{2\pi a}{T}

= \frac{2\pi(1.00\ \text{AU})}{1.00\ \text{yr}}

= \frac{2\pi(1.496\times10^{8}\ \text{km})}{3.156\times10^{7}\ \text{s}}

\approx 29.8\ \text{km s}^{-1}.

\]

Typical speeds: Mercury ≈ 47 km s⁻¹, Neptune ≈ 5.4 km s⁻¹ – inner planets move faster because they are nearer to the Sun.

4.5 Escape Velocity of Mars (example)

\[

v_{e}= \sqrt{2gR}

= \sqrt{2(3.7\ \text{m s}^{-2})(3.39\times10^{6}\ \text{m})}

\approx 5.0\ \text{km s}^{-1}.

\]

Compare with Earth’s \(v_{e}\) ≈ 11.2 km s⁻¹ – the lower value on Mars explains why rockets need less thrust to leave the planet.

5. Interpretation & Discussion (AO2 – AO3)

1. Kepler’s 3^rd law: The constant \(\frac{T^{2}}{a^{3}}\) shows that all planets experience the same central inverse‑square force from the Sun.
2. Mass, radius and surface gravity: Using \(\rho\) and \(g\) we can deduce each planet’s size and mass. Gas giants have low densities but huge radii, giving high \(g\); terrestrial planets are dense and smaller.
3. Why Venus is hotter than Mercury: Venus’s thick CO₂ atmosphere produces a strong greenhouse effect, trapping infrared radiation and raising the surface temperature to ≈ 735 K despite its greater distance from the Sun.
4. Density trends: Terrestrial planets (ρ ≈ 3900–5500 kg m⁻³) are rocky/metallic. Gas giants (ρ ≈ 600–1600 kg m⁻³) consist mainly of hydrogen, helium and ices, often with a solid core.
5. Weight of a 70 kg person: The smallest surface gravity is on Mars (3.7 m s⁻²).

   \[

   W = mg = 70\times3.7 \approx 260\ \text{N}\;(≈26\ \text{kgf}).

   \]

6. Escape velocity and space missions: Lower escape velocities (e.g., Mars) mean less propellant is required for launch, influencing mission design.

6. Alignment with IGCSE Assessment Objectives

• AO1 – Knowledge: Students recall definitions, formulae and constants; identify the eight major planets and their key physical properties.
• AO2 – Application: Perform multi‑step calculations (mass, radius, orbital speed, escape velocity) using the provided data and formulae.
• AO3 – Analysis & Evaluation: Interpret tables and graphs, assess the consistency of data (e.g., checking \(g\) from \(\rho\) and \(R\)), and explain trends such as the temperature difference between Venus and Mercury.

7. Practical Skills Expected in the IGCSE

• Conversion between units (AU ↔ km, yr ↔ s, kg ↔ 10²⁴ kg).
• Use of scientific notation and correct significant figures.
• Plotting and interpreting graphs (e.g., \(T^{2}\) vs. \(a^{3}\), bar chart of surface gravities).
• Carrying out multi‑step calculations with clear working shown.
• Evaluating tabulated data for internal consistency.
• Communicating results concisely, including a brief conclusion for each question.

8. Suggested Classroom Activities

1. Graph Kepler’s law: Plot \(T^{2}\) (yr²) against \(a^{3}\) (AU³) for the eight planets. Draw the best‑fit straight line, determine its slope and comment on the constant of proportionality.
2. Bar chart of surface gravity: Create a vertical bar chart comparing \(g\) values. Identify the planet with the strongest and weakest gravity.
3. Atmosphere research: For each planet, list the main atmospheric components and thickness, then relate these to the observed mean surface temperature.
4. Orbital‑speed ranking: Calculate \(v\) for all planets using \(v=2\pi a/T\) and arrange them from fastest to slowest.
5. Escape‑velocity comparison: Using the masses derived in 4.2–4.3, compute \(v_{e}\) for Earth and Mars. Discuss the implications for launching spacecraft.
6. Data‑consistency check: Choose any planet, use the given \(\rho\) and \(g\) to calculate its radius, then compare with the tabulated radius. Discuss possible sources of discrepancy (e.g., non‑uniform density).