If you’re a fan of math and physics, you might have come across the YouTube channel 3blue1brown, run by Grant Sanderson. Grant creates engaging videos that delve into mathematical concepts and their applications. In one of his videos, he explores a fascinating topic: why planets orbit in ellipses, inspired by a lost lecture from the legendary physicist Richard Feynman.
The journey begins with a simple geometric construction. Imagine drawing a circle and selecting a point inside it, not at the center. This point is termed “eccentric.” From this eccentric point, draw lines to the circle’s edge and rotate each line 90 degrees at its midpoint. Surprisingly, an ellipse emerges. This construction, while seemingly arbitrary, holds the key to understanding why planets orbit the sun in ellipses.
Richard Feynman, a Nobel Prize-winning physicist, was known for his ability to make complex topics accessible. His lectures at Caltech are renowned, but not all were published. One such lecture, “The Motion of Planets Around the Sun,” was lost for years until it was rediscovered and reconstructed by Judith Goodstein and her husband David. This lecture provides a unique perspective on why celestial bodies follow elliptical paths.
At the heart of this phenomenon is the inverse square law, which states that the gravitational force between two objects is inversely proportional to the square of the distance between them. This law explains why orbits are elliptical. While gravitational interactions between celestial bodies mean no orbit is a perfect ellipse, they are remarkably close.
Feynman’s approach to explaining this didn’t rely on complex mathematics. Instead, he aimed for an “elementary demonstration,” requiring minimal prior knowledge but a good deal of focus. His method involved understanding the properties of ellipses and how they relate to orbital mechanics.
An ellipse can be constructed using two thumbtacks and a piece of string. By fixing the string’s ends with thumbtacks and tracing a curve while keeping the string taut, you create an ellipse. The defining property of an ellipse is that the sum of the distances from any point on the curve to the two thumbtacks (foci) is constant.
Returning to the initial construction, the lines drawn from the eccentric point to the circle’s circumference, when rotated, form the ellipse’s tangent lines. This connection between geometry and physics is crucial. The tangency direction of these lines corresponds to the velocity of an orbiting object.
Kepler’s second law states that a planet sweeps out equal areas in equal times as it orbits the sun. This law, derived from the conservation of angular momentum, doesn’t assume elliptical orbits but is consistent with them. It highlights the relationship between a planet’s speed and its distance from the sun.
Feynman demonstrated that if you plot the velocity vectors of an orbiting object, their tips trace a perfect circle. This circle, with an off-center point, is key to understanding the elliptical nature of orbits. The velocity vectors’ constant change in length and direction aligns with the inverse square law, leading to a regular polygon that approaches a circle as the slices become finer.
By rotating the velocity circle and its vectors, an ellipse emerges, mirroring the orbit’s shape. This clever geometric insight ties the velocity diagram to the actual path of the orbiting object, confirming that orbits are indeed elliptical.
Feynman’s exploration of elliptical orbits showcases the beauty of physics and geometry. His ability to simplify complex ideas into intuitive concepts is a testament to his genius. This journey through his lost lecture not only deepens our understanding of celestial mechanics but also highlights the elegance of the universe’s design.
Engage in a hands-on activity by constructing an ellipse using a string and two thumbtacks. Fix the thumbtacks on a board, loop a string around them, and trace the ellipse by keeping the string taut with a pencil. Reflect on how this simple construction relates to the elliptical orbits of planets.
Conduct a simulation or use software to visualize the inverse square law in action. Observe how varying the distance between two objects affects the gravitational force. Discuss how this principle underpins the elliptical nature of planetary orbits.
Create a velocity circle diagram for a hypothetical planet orbiting a star. Plot the velocity vectors and observe how their tips form a circle. Analyze how this geometric representation connects to the planet’s elliptical orbit.
Work in groups to derive Kepler’s second law from the conservation of angular momentum. Use diagrams to illustrate how a planet sweeps out equal areas in equal times and discuss its implications for elliptical orbits.
Watch Grant Sanderson’s video on Feynman’s lost lecture and participate in a seminar-style discussion. Focus on Feynman’s approach to explaining elliptical orbits and how his insights can be applied to modern physics education.
Math – The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics). – In advanced math courses, students explore complex functions and their applications in real-world scenarios.
Physics – The natural science that involves the study of matter, its motion and behavior through space and time, and the related entities of energy and force. – Physics provides the foundational principles that explain how the universe operates, from the smallest particles to the largest galaxies.
Ellipse – A regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant. – The orbits of planets in our solar system are not perfect circles but ellipses, as described by Kepler’s First Law.
Orbits – The gravitationally curved trajectory of an object, such as the path of a planet around a star or a satellite around a planet. – The study of orbits is crucial for understanding satellite deployment and interplanetary travel.
Eccentric – A measure of how much an orbit deviates from being circular, with a value ranging from 0 (a perfect circle) to 1 (a parabolic trajectory). – The eccentric orbit of Halley’s Comet brings it close to the Sun and then sends it far into the outer solar system.
Gravity – A natural phenomenon by which all things with mass or energy are brought toward one another, including planets, stars, galaxies, and even light. – Gravity is the force that keeps the planets in orbit around the Sun and governs the motion of celestial bodies.
Momentum – The quantity of motion of a moving body, measured as a product of its mass and velocity. – In physics, the conservation of momentum is a fundamental principle that explains the behavior of colliding objects.
Geometry – The branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. – Understanding geometry is essential for solving complex problems in physics, such as calculating the angles of incidence and reflection.
Velocity – The speed of something in a given direction, a vector quantity that denotes the rate of change of position. – In physics, velocity is a crucial factor in determining the kinetic energy of an object in motion.
Kepler – Referring to Johannes Kepler, a key figure in the 17th-century scientific revolution, known for his laws of planetary motion. – Kepler’s laws provided one of the first accurate models of the solar system, describing how planets move in elliptical orbits.
