What Is Kepler's 3rd Law of Planatery Motion? Kepler's 3rd Law Explained!

In previous blogs, I discussed Kepler's first and second laws of planetary motion. In this blog, I will be discussing Kepler's third law of planetary motion.

Statement: "The ratio of the square of the orbital period of objects to the cube of the semi-major axis is the same for all objects orbiting the same prime celestial body.

Kepler's 3rd Law of Planatery Motion

T₁²/R₁³= T₂²/R₂³ = T₃²/R₃³ or vice versa

R₁³/T₁²= R₂³/T₂²= R₃³/T₃²

Different Elements of Ellipse

For example, if you take the sun as being a prime celestial body orbited by objects that are planets like Mercury, Venus, Earth, and Mars.

^Planets	^{Average distance from the Sun during a complete revolution (In Astronomical Unit)}	^{Time Taken for a complete revolution} ^{(In Earth days)}
^Mercury	^0.387	^87.969
^Venus	^0.723	^244.700
^Earth	¹	^365.256
^Mars	^1.523	^686.979

0.387³/87.960² =0.723³/244.7² = 1³/365.25² = 1.523³/686.979² = 7.49*10^-6AU³/days²

This was used by Newton in his equivalence relationship between centripetal force and gravitational force, which was published in the year 1687 almost sixty years after Kepler proposed his three laws.

Deducing Kepler's 3rd law by using Newton's equivalence relationship between centripetal force and gravitational force.

Centipetal force and Gravitational Force

centripetal force = Gravitational force

ma = GMm/ R²

mv/ T = GMm/ R²

mv²/R = GMm/ R² (by multiplying and dividing LHS by v)

m (R w)²/R = GMm/ R² (by v = Rw)

mR(2pi/T)² = GMm/ R²

T² = 4pi²R³/GM

T²proportional to R³

In conclusion, Johannes Kepler's Third Law of Planetary Motion has left an indelible mark on our understanding of celestial mechanics. By establishing a precise relationship between a planet's orbital period and its distance from the sun, Kepler provided a crucial bridge between the observations of Tycho Brahe and the groundbreaking insights of Sir Isaac Newton. This law, often referred to as the Law of Harmonies, laid the groundwork for Newton's Universal Law of Gravitation and paved the way for the modern understanding of how celestial bodies move in space. Kepler's Third Law remains a testament to the power of empirical observation and mathematical rigor in advancing our knowledge of the cosmos.

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