Why Planets Fly in Ellipses: A New Proof ✎ express
A thrown stone arcs through the air, but a planet around the Sun follows a closed curve. Why an ellipse, and not a circle or a spiral? Kepler's first law provides the answer, but its usual proof requires heavy computation. This work offers a shorter path — through a single simple ratio of velocities.
The authors note: if you take the planet's velocity and project it onto two directions (one along the line from the Sun to the planet, and one onto a fixed axis) the ratio of these projections remains constant throughout the motion. This follows directly from Newton's laws, without integrating the equations. And the constancy of such a ratio is a well-known geometric hallmark of an ellipse (or, more precisely, of any conic section).
The result is a bridge between physics and geometry: the inverse-square law by itself guarantees that the orbit is an ellipse. The proof is short and intuitive: you can sketch it on a single chalkboard.
This approach reminds us that planetary motion is shaped not only by force but also by pure geometry. The ellipse is not a random shape but a direct consequence of how attraction works.
If gravity fell off not as the square of the distance but in some other way, planetary orbits would not be ellipses: they could be open curves or even escape to infinity.