SNW #39: Kepler´s Laws of Planetary Motion

 In this article I would like to focus on Kepler´s three laws of planetary motion. They are the backbone to all of astrodynamics and space travel. They have been formed centuries ago, but they still apply to this day.

Johannes Kepler (1571-1630), a German astronomer, stands as the monumental figure in the history of science, whose work fundamentally shaped humanity´s understanding of celestial motion.

Before him, the prevailing belief, even within the then-revolutionaty Copernican heliocentric model (which correctly placed the Sun at the center of the solar system), was that planetary orbits were perfect circles. His research first struggled, as he could not understand how the orbits were precise circles (Mars´ orbit was especially problematic). He then came to a brilliant realisation - the planetary orbits were not perfect circles, but rather elongated or flattened circles know as ellipses.

KEPLER´s FIRST LAW:

This law is known as the law of ellipses. It states that “each planet´s orbit around the Sun is an ellipse. The Sun´s center is always located at one focus of the orbital ellipse”. This law corrects the common misconception that orbits are circular, when in fact, they´re not.

An ellipse can be visualised as a elongated circle. Unlike a circle, which has a single center, an ellipse is defined by two internal points, each called a “focus” (plural: foci).

A critical implication of this law is that because the central body (such as the Sun) is positioned at one of the foci, and not at the geometric center of the elipse, the distance between the orbiting object and the central body is constantly changing throughout its orbit.

KEPLER´s SECOND LAW:

This law is known as the law of equal areas. It states that "the imaginary line joining a planet and the Sun sweeps equal areas of space during equal time intervals as the planet orbits".

For an orbiting object to sweep out equal areas in equal amounts of time, its speed must necessarily vary along its elliptical path. When the object is closer to the central body, it moves faster, and conversely, when it is farther away, it moves slower. The point where the orbiting object is closest to the central body is called the “periapsis” (or “perihelion” for orbits around the Sun), while the point of greatest separation is called “apoapsis” (or “apohelion” for orbits around the Sun). Consequently, a planet moves faster at perihelion and the slowest at aphelion.

This law provides a vivid illustration of how the central body's gravitational pull dynamically influences an orbiting object's speed. The stronger gravitational attraction experienced when closer to the central body causes faster movement, while the weaker attraction at greater distances results in slower movement.

KEPLER´s THIRD LAW:

This law is known as the law of periods and states that “the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit". It establishes a precise mathematical relationship between the size of an orbit and the time it takes to complete that orbit. This relationship is commonly expressed mathematically as T squared ∝ a cubed, where 'T' is the orbital period (the time required for one full orbit) and 'a' is the semi-major axis (the average distance from the central body).

A key implication of this law is that larger orbits have significantly longer periods, while smaller orbits have shorter periods. This law provides a direct, quantitative tool for predicting orbital periods based on orbital size, and vice versa. This predictive capability is absolutely foundational for designing and executing satellite missions, determining how long a spacecraft will take to reach its destination, and precisely placing satellites in specific orbits.

THEIR IMPACT ON SPACE TRAVEL: 

While Kepler's laws were initially formulated to describe the motion of planets around the Sun, their profound power lies in their universality. These laws apply to any two celestial bodies orbiting each other under the influence of gravity. This means that the same elegant mathematical framework that describes the grand dance of planets can be directly applied to human-made objects in space: artificial satellites orbiting Earth, probes embarking on journeys to other planets, and even orbital debris.

Kepler's First Law, by defining all bound orbits as elliptical, serves as the fundamental blueprint for designing spacecraft trajectories. Whether a satellite is placed in Earth orbit or a probe is dispatched to Mars, its path (or segments of its path) will inherently be an ellipse, with the central body (Earth or Sun) positioned at one of its foci. This elliptical nature of orbits is not just a descriptive fact; it is the fundamental geometric foundation that enables all space travel. Every maneuver, from achieving initial orbit to navigating across the solar system, involves carefully shaping and transitioning between these elliptical paths.   

A prime application of this principle is in the design of transfer orbits. To move a spacecraft from one orbit to another (e.g., from a low Earth orbit to a higher one), engineers design specific elliptical paths. The Hohmann transfer orbit is a classic and highly fuel-efficient example: it is an elliptical path that precisely touches the initial and final circular orbits at its periapsis and apoapsis, respectively. This maneuver typically requires two brief engine burns to initiate and complete the transfer.   

Furthermore, after launch, rockets frequently place payloads into a temporary "parking orbit" around Earth. This is a stable elliptical (or near-circular) orbit used to await the optimal moment for a subsequent engine burn, which will then send the spacecraft to its final destination, whether a higher operational orbit or an interplanetary trajectory. The precise shape and characteristics of this parking orbit are governed by Kepler's First Law.