Kepler’s Third Law of Planetary Motion

Kepler’s Third Law of Planetary Motion

Kepler’s Third Law: Law of Harmonics or Law of periods

The square of the orbital period of a planet is proportional to the cube of mean distance from the sun. We also say that, the semi – major axis of the ellipse, half the sum of smallest and greatest distance from the sun.

Planets farther away from the sun have larger orbits, so even if they moved at the same speed as the closer in planets, you did expect them to take longer to move around the sun. And they do. But they also move at a lower rate of speed, which further lengthens the time is takes to revolve around the sun. What Kepler discovered to be exact, is that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi major axis of its orbit (r).

By Kepler’s third law can be derived from newton’s inverse square law of gravitation force acts as centripetal force for the planet.

Therefore,

$$\frac{m{{v}^{2}}}{r}\,=\,\frac{GmM}{{{r}^{2}}}$$,

Where,

m = mass of planet

M = Mass of sun

v² = (GM/r)

Where,

v = rω

v² = r²ω²

v²r = GM

r³ω² = GM

$${{\omega }^{2}}\,=\,\frac{GM}{{{r}^{3}}}$$,

$$\frac{{{(2\pi )}^{2}}}{{{T}^{2}}}\,=\,\frac{GM}{{{r}^{3}}}$$,

r³ (2π)² = (GM)T²

T² α r³

The planets move slower when they’re farther from the Sun, and whatever influence makes planets go around the Sun weakens with distance.  It was left to the great British scientist and world-historical genius, Isaac Newton, to explain this relationship with his own law of gravity, a law upon which all of modern physical science is based.

Kepler’s three laws appear to hold not just in our own solar system, but in all circumstances where one body moves under the influence of another’s gravity.  They are general universal relationships.