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

Objective

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

Key Concepts

• Elliptical orbits have two special points:

  • Perihelion – the point of closest approach to the Sun.
  • Aphelion – the point of greatest distance from the Sun.

• Gravitational potential energy of a mass \$m\$ at distance \$r\$ from the Sun (mass \$M_{\odot}\$) is

  \$U = -\frac{G M_{\odot} m}{r}\$

  where \$G\$ is the universal gravitational constant.

• Kinetic energy of the moving object is

  \$K = \frac12 m v^{2}\$

  with \$v\$ the orbital speed.

• The total mechanical energy \$E\$ of the orbiting object is the sum of kinetic and potential energy:

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

  For a given orbit \$E\$ is constant (conserved).

Why Speed Increases Near the Sun

When the object moves from aphelion to perihelion the distance \$r\$ decreases. Because the potential energy term \$-\frac{G M_{\odot} m}{r}\$ becomes more negative, the total energy \$E\$ must stay the same. To compensate, the kinetic energy \$K\$ must increase, which means the speed \$v\$ must increase.

Mathematically, setting the total energy at aphelion (\$ra\$, \$va\$) equal to that at perihelion (\$rp\$, \$vp\$):

\$\$\frac12 m va^{2} - \frac{G M{\odot} m}{r_a}

= \frac12 m vp^{2} - \frac{G M{\odot} m}{r_p}\$\$

Rearranging gives the relationship between speeds:

\$vp^{2} = va^{2} + 2 G M{\odot}\!\left(\frac{1}{rp}-\frac{1}{r_a}\right)\$

Since \$rp < ra\$, the term in parentheses is positive, so \$vp > va\$.

Numerical Example (Earth’s Orbit)

Even a modest change in distance produces a noticeable change in speed, illustrating the principle.

Key Points to Remember

1. The total mechanical energy of an object in a closed orbit is constant.
2. As the object approaches the Sun, \$r\$ decreases, making the gravitational potential energy more negative.
3. To keep the total energy constant, the kinetic energy must increase, so the orbital speed increases.
4. This explains why planets (and other bodies) move fastest at perihelion and slowest at aphelion.