Kepler’s second law. A line from the Sun to the comet sweeps out equal areas in equal times. Near the Sun the line is short, so the comet has to move quickly to sweep the same area; far from the Sun the line is long and the comet moves slowly.

Why? The only force on the comet is the Sun’s gravity, which points along the line from the Sun, so there is no torque about the Sun and the angular momentum L = mrv_⊥ is constant. The area swept per second is dA/dt = L/2m, which is constant.

dA/dt = ½ r v⊥ = L/2m = constant

Energy. The total energy of the comet, E = KE + PE = −GMm/2a, is constant. Near the Sun the PE is more negative and the KE is bigger. The comet has a mass of 100 billion tonnes (1014 kg).

The graph. The area swept grows steadily with time, whatever the shape of the orbit (the straight line). The distance travelled along the orbit does not: it speeds up and slows down.

The orbit has a semi-major axis of 1.8 AU, and its period comes from Kepler’s third law, T² = a³. The areas are measured by adding up the thin triangles of the shaded sectors, so they are checked independently of the formula.