Kepler’s Laws of Planetary Motion
In the early 17th century, Johannes Kepler formulated three laws that describe the motion of planets around the Sun.
These laws, derived from meticulous astronomical observations, laid the groundwork for our understanding of orbital dynamics.
Kepler’s First Law: Elliptical Orbits
Planets move in elliptical orbits with the Sun at one focus.
- An ellipse is a flattened circle characterized by two foci.
- The Sun occupies one of these foci, not the center.
This means that the distance between a planet and the Sun varies as the planet orbits.
ExampleThe orbit of Earth is nearly circular, but comets like Halley’s Comet have highly elongated elliptical orbits.
TipRemember, the Sun is not at the center of the ellipse, it’s at one of the foci.
Kepler’s Second Law: Equal Areas in Equal Times
A line joining a planet and the Sun sweeps out equal areas in equal time intervals.
This law implies that a planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion).
Example- A planet sweeps out equal areas in equal time intervals as it orbits the Sun, in accordance with Kepler's second law.
- This means that if a planet takes 30 days to travel from point A to point B near the Sun, the area swept out will be the same as the area swept out in 30 days when the planet is farther from the Sun, moving from point C to point D.
- However, the planet moves faster when closer to the Sun and slower when farther away.
- This law is a consequence of the conservation of angular momentum.
- As a planet moves closer to the Sun, it speeds up to maintain its angular momentum.
Kepler’s Third Law: Orbital Period and Radius
The square of the orbital period $T$ of a planet is proportional to the cube of the semi-major axis $r$ of its orbit.
- Mathematically, this is expressed as: $$T^2 \propto r^3$$
- For any planet orbiting the Sun, the ratio $\frac{T^2}{r^3}$ is constant.
