Know that an object in an elliptical orbit travels faster when closer to the Sun and explain this using the conservation of energy

6.1.2 The Solar System

Learning objectives

By the end of this section students will be able to:

• Identify the Sun, the eight named planets (in order), the four IAU‑recognised dwarf planets, the main asteroid belt, typical moons, comets and other small bodies.
• Explain how Earth’s rotation gives day and night, and how its revolution together with the 23.5° axial tilt produces the seasons.
• Describe the Moon’s orbit and explain the origin of its phases.
• Calculate orbital speed using both v = 2πr / T and v = √(GM/r).
• Use the conservation of mechanical energy to explain why a body in an elliptical orbit moves faster when it is nearer the Sun.

Solar‑System overview

Earth’s rotation and revolution

• Rotation: one complete turn about its axis ≈ 24 h → day and night.
• Axial tilt: ≈ 23.5° to the orbital plane.
• Revolution: one orbit around the Sun ≈ 365.25 days. The combination of tilt and orbital motion produces the seasons.

The Moon

• Mean orbital radius ≈ 3.84 × 10⁵ km.
• Sidereal period ≈ 27.3 days; synodic (phase) period ≈ 29.5 days.
• Phases arise because the illuminated half of the Moon is seen from different angles as it moves around Earth.

Orbital‑speed formulas

For a (near‑circular) orbit of radius r and period T the average speed v can be obtained in two equivalent ways:

1. Geometric definition – distance travelled in one complete orbit divided by the period:
   

   \$v = \frac{2\pi r}{T}\$

2. Force balance – equating centripetal force to the Sun’s gravitational attraction:
   

   \$v = \sqrt{\frac{GM}{r}}\$

   where G = 6.674 × 10⁻¹¹ N m² kg⁻² and M is the mass of the central body (the Sun for planets, Earth for the Moon).

Why an object moves faster near the Sun – conservation of mechanical energy

The total mechanical energy of a body of mass m in the Sun’s gravitational field is the sum of kinetic and gravitational potential energy:

\$E = K + U = \frac12 m v^{2} - \frac{G M_{\odot} m}{r}\$

• Kinetic energy \$K = \tfrac12 m v^{2}\$
• Gravitational potential energy \$U = -\dfrac{G M_{\odot} m}{r}\$ (negative because the force is attractive)

In a closed (elliptical) orbit \$E\$ is constant. The two extreme points are:

• Perihelion – closest approach (\$r{p}\$, speed \$v{p}\$).
• Aphelion – farthest distance (\$r{a}\$, speed \$v{a}\$).

Setting the total energy equal at perihelion and aphelion gives

\$\frac12 m v{a}^{2} - \frac{G M{\odot} m}{r{a}} \;=\; \frac12 m v{p}^{2} - \frac{G M{\odot} m}{r{p}}\$

and, after rearranging,

\$v{p}^{2} = v{a}^{2} + 2 G M{\odot}\!\left(\frac{1}{r{p}}-\frac{1}{r_{a}}\right).\$

Because \$r{p}{a}\$, the term in parentheses is positive, so \$v{p}>v{a}\$. The body therefore travels fastest at perihelion and slowest at aphelion.

Numerical example – Earth’s orbit

Using \$v = 2\pi r/T\$ with \$T = 1\$ yr (≈ 3.156 × 10⁷ s) reproduces the same speeds, confirming that the increase in kinetic energy at perihelion exactly compensates the more negative potential energy.

Key points to remember

1. The Solar System consists of the Sun, eight planets, four recognised dwarf planets (Pluto, Eris, Haumea, Makemake) plus Ceres, the main asteroid belt, numerous moons, comets and other small bodies.
2. Earth’s 24‑h rotation creates day and night; its 23.5° axial tilt together with its 365‑day revolution produces the seasons.
3. The Moon orbits Earth in ~27 days; the changing view of its illuminated half gives the lunar phases.
4. Average orbital speed can be found with \$v = 2\pi r/T\$ or \$v = \sqrt{GM/r}\$.
5. In an elliptical orbit the total mechanical energy \$E = \tfrac12 m v^{2} - GM_{\odot}m/r\$ remains constant.
6. When the body moves toward the Sun (\$r\$ decreases) the potential energy becomes more negative; to keep \$E\$ constant the kinetic energy must increase, so the speed rises.
7. Consequently, planets (and any orbiting object) travel fastest at perihelion and slowest at aphelion.