Kepler’s First Law: The Law of Ellipses

The first law states that the path of each planet around the Sun is an ellipse with the Sun located at one of the two foci. The shape of the orbit is determined by its eccentricity ($e$): an eccentricity of zero denotes a circular orbit, while an eccentricity greater than zero specifies an elliptical orbit.

Kepler's Second Law: The Law of Equal Areas

The second law states that the imaginary line connecting a planet to the Sun sweeps out equal areas in equal intervals of time. Consequently, planets travel faster when they are closer to the Sun (perihelion) and slower when they are farther away (aphelion).

Kepler's Third Law: The Law of Harmonies

The third law establishes a mathematical relationship between a planet's orbital period ($T$) and its mean distance from the Sun ($a$): the square of the orbital period is directly proportional to the cube of the semi-major axis of its orbit:

$$T^2 \propto a^3 \quad \text{or} \quad \frac{T^2}{a^3} = \text{constant}$$

For instance, Mercury completes an orbit in 88 days, whereas Saturn requires approximately 29.5 years. Later, Isaac Newton proved this law derives directly from universal gravitation, independent of planetary mass.

Applications and Limitations

Applications: Used for calculating orbits of planets, comets, and asteroids, designing spacecraft trajectories, and detecting exoplanets.

Limitations: Assume isolated two-body systems without external gravitational perturbations from other celestial bodies, and omit relativistic effects.