What to do

Watch the yellow fan the Sun–planet line sweeps out.
Raise the eccentricity e — the orbit stretches; the planet races through perihelion and crawls at aphelion.
Predict first: at higher e, where is the planet fastest?

What to observe

The swept fan covers the SAME area per unit time everywhere — that is Kepler's second law, dA/dt = L/2m = constant.
Near the Sun the planet must move fast to sweep the same area as it does slowly far away: vₚrₚ = vₐrₐ.
Changing a changes the year (T = √a³) but changing e does NOT — period depends only on the semi-major axis.

What it means

The law of areas is just angular-momentum conservation, and it holds for ANY central force — only T²∝a³ is special to gravity's inverse square.

Why this matters

Equal areas in equal times is conservation of angular momentum wearing a geometric disguise. Gravity acts along the line joining the two bodies, so it exerts no torque about the Sun, which means L cannot change — and constant L is precisely constant areal speed.
That is why a planet races through perihelion and dawdles at aphelion: the same area swept in the same time means a short radius must be paired with fast motion.
Because the argument uses only that the force is central, it holds for ANY central force, not just an inverse-square one. Kepler's second law is therefore more general than his first and third, which do need the inverse square.

🪐 Kepler's Laws — equal areas in equal times