Kepler's Laws
Learning Objectives
- Differentiate Johannes Kepler's observational geometry from Newton's causal physics.
- Understand Kepler's Three Laws of Planetary Motion regarding ellipses, sweeping areas, and periods.
- Analyze how Kepler's Third Law allows engineers to precisely design satellite constellation architectures.
Newton proved why orbits exist via gravity. But decades earlier, Johannes Kepler had already figured out exactly how they behave geometrically by staring directly at the sky.
1.3.5 The Pre-Telescope Astronomer
In the early 1600s, before telescopes were widely used and long before Newton hypothesized gravity, Johannes Kepler painstakingly analyzed decades of naked-eye astronomical data collected by Tycho Brahe. From this raw data, Kepler deduced three absolute laws that dictate the geometry of all orbital bodies.
I. The Law of Ellipses
Orbits are not perfect circles. They are ellipses, with the primary gravitational body sitting off-center at one of the two mathematical foci. This means every orbit inherently has a closest point (Periapsis) and a furthest point (Apoapsis).
II. The Law of Equal Areas
An orbiting body will sweep out equal areas in equal amounts of time. Practically, this means a satellite moves much faster when it swoops down close to the planet (Periapsis) and much slower when it swings far out into deep space (Apoapsis).
III. The Law of Periods
The square of a satellite's orbital period (how long it takes to complete one lap) is directly proportional to the cube of its semi-major axis (its altitude). $T^2 \propto a^3$
1.3.6 Modern Engineering Context: GPS
Kepler's Third Law ($T^2 \propto a^3$) is one of the most practically useful equations in modern aerospace architecture. It dictates that an orbit's altitude perfectly locks in its time period, and vice-versa.
You cannot have a satellite orbiting at $400\text{ km}$ that takes 5 hours to circle the Earth. If it is exactly $400\text{ km}$ high, the laws of physics dictate it must lap the Earth every $\sim 92$ minutes.
FIGURE 3.2: CONSTELLATION TIMING
When designing the Global Positioning System (GPS), engineers wanted exactly 24 satellites to pass over specific points on Earth exactly twice per day. Because they wanted a 12-hour period ($T$), Kepler's third law dictated exactly where they had to put them: $20,200\text{ km}$ ($a$).