Kepler's Laws: The Harmony of Planetary Motion

Introducing Johannes Kepler

Johannes Kepler, born on December 27, 1571, in Germany, stands as one of the most influential figures in the history of astronomy. His groundbreaking work unveiled the mechanics of planetary motion, revolutionizing our understanding of the solar system. Despite facing religious and political turbulence, which led to his banishment from Graz in 1600, Kepler’s perseverance brought him to the court of Tycho Brahe, a brilliant Danish astronomer known for his precise, pre-telescopic observations.

Tycho Brahe tasked Kepler with solving the "Mars problem," which referred to Mars' erratic motion that couldn’t be explained by the geocentric models of Aristotle and Ptolemy, or even by Copernicus' heliocentric theory, which assumed circular orbits. Kepler’s groundbreaking realization that planets follow elliptical orbits, rather than perfect circles, which challenged centuries of astronomical thought and laid the foundation for the celestial mechanics we know today.

Image from: NASA

Vocabubbleary: geocentric theories are those that assume that Earth is the centre of the structure of the solar system, as well as the universe.

Quick Recap of Eclipses

An ellipse is a geometric shape defined by two fixed points, called foci. The key property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is always constant. The shape of an ellipse can vary in how "stretched" or "flattened" it appears, and this is quantified by its eccentricity. 

Image from: Hyperphysics

Eccentricity measures the degree of flattening of the ellipse; a perfect circle has an eccentricity of zero, while a highly elongated ellipse has an eccentricity approaching one. The eccentricity is calculated as the ratio of the distance between the foci to the length of the major axis. The longest axis of the ellipse is called the major axis, while the shortest axis is the minor axis. Half of the major axis is referred to as the semi-major axis. Understanding these properties is essential for describing the elliptical orbits of planets and other celestial objects.

1st Law: The Law of Ellipses

Planets orbit the Sun in elliptical paths, with the Sun at one focus.

Image from: Fleet Science Centre

Kepler’s insight overturned the long-held belief in circular orbits, solving the Mars problem. He demonstrated that while some orbits appear circular, these are special cases where the foci coincide. For most planets, the Sun’s position at one focus explains the varying planet-to-Sun distance throughout its orbit.

2nd Law: The Law of Equal Areas

A line connecting a planet to the Sun sweeps out equal areas in equal time intervals.

Kepler's Second Law explains that the elliptical shape of the orbit results from the inverse square law of gravity. This means that a planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion), due to the change in gravitational force as the planet’s distance from the Sun varies. 

Image from: Britannica

This law also relates to the conservation of angular momentum. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time. The formula denoting this is L = mvr, where less radius (distance between the bodies) results in higher velocity.

3rd Law: The Law of Periods

The square of a planet’s orbital period (T²) is proportional to the cube of the semi-major axis of its orbit (r³).

Image from: Jimmy Newland

This mathematical relationship connects a planet’s distance from the Sun to its orbital duration. For example, Mercury, the closest planet, takes 88 days to orbit the Sun, while Saturn requires over 10,000 days. 

Image from: Hyperphysics

Newton later generalized this law to include gravitational interactions, making it applicable to systems beyond our solar system. A way to understand this is that Kepler only determined a constant k related period and radius in a relationship of T=kr3/2, but Newton used his law of universal gravitation to derive the constant. This allowed for calculations beyond just celestial bodies orbiting the Sun, but a generalized formula for any gravitational systems.

Image from: Soul Food

Fun activity: try deriving this using these formulas of centripetal acceleration and the universal law of gravitation!

Legacy of Kepler's Laws

Kepler’s laws provided the framework for Newton’s formulation of the laws of motion and universal gravitation, bridging observational astronomy and physical science. Today, these laws remain vital in space exploration, satellite mechanics, and exoplanet studies. From NASA’s Kepler Space Telescope, which discovered thousands of exoplanets, to calculations of dark matter's effects on galaxies, Kepler’s legacy continues to expand our cosmic horizons.

Image from: NASA

☕ Hubble Tea of the post! ☕

A new video has been compiled using footage from Chandra (in red) and Palomar (in blue) data which reveals details about the Guitar Nebula, named for its resemblance to the instrument in glowing hydrogen light. The nebula surrounds the pulsar PSR B2224+65, which is a rapidly rotating neutron star moving through space. As the pulsar travels, it creates bubbles of particles that form the guitar-like shape and emits a filament of X-rays about two light-years long, visible in Chandra’s observations. Red in the image corresponds to X-ray data from Chandra, showing the filament of high-energy matter and antimatter particles emitted from the pulsar. Blue corresponds to optical data from Palomar, which highlights the hydrogen gas and structures forming the Guitar Nebula.

The pulsar's fast rotation and strong magnetic fields accelerate particles, producing matter-antimatter pairs (electrons and positrons) by converting energy into mass. These particles spiral along magnetic field lines, creating X-rays. As the pulsar collides with denser gas regions, energetic particles escape, forming the filament stretching away from the pulsar.

Image from: NASA Missions

That wraps our post today on the Kepler’s laws which give us insight into the relationships and interactions of celestial bodies and systems in our grand universe. Without these laws, our understanding of the cosmos, and our place within it, would be vastly diminished. Hope you enjoyed this read and keep an eye out for more. As always, enjoy reading while sipping on space tea! 🔭✨

Follow us on our socials!

Instagram: hubbleteabynabira

Medium: Nabira Rashid