Johannes Kepler deduced that planets have an elliptical orbit based on his extensive observations and analysis of the motion of Mars. He made critical contributions to the field of astronomy and planetary motion by formulating his three laws of planetary motion, which revolutionized our understanding of the solar system.

Here's how Kepler determined the elliptical shape of planetary orbits:

1. Tycho Brahe's Observations: Kepler had access to the exceptionally precise observational data collected by his mentor, the Danish astronomer Tycho Brahe. Brahe's meticulous measurements of planetary positions and meticulous records provided Kepler with a wealth of high-quality data.

2. Heliocentric Model: Kepler adopted Copernicus's heliocentric model, which placed the Sun at the center of the solar system with planets orbiting around it. This model contradicted the prevailing geocentric model of the time.

3. Mars' Orbit: Kepler focused his efforts on understanding the orbit of Mars, which was known for its irregularities. He realized that the planet's motion deviated from the circular paths predicted by earlier astronomers.

4. Elliptical Path: Through painstaking calculations and analysis, Kepler discovered that Mars' orbit is not circular but elliptical, or egg-shaped. He realized that the Sun was not at the center of the circle, but at one of the two foci of the ellipse.

5. Kepler's First Law: Kepler's first law of planetary motion, also known as the "Law of Ellipses," states that the orbit of a planet around the Sun is an ellipse, with the Sun at one focus.

6. Area Law: Kepler also discovered that a line connecting a planet to the Sun sweeps out equal areas in equal intervals of time. This is known as "Kepler's Second Law" or the "Law of Equal Areas."

7. Third Law (Law of Harmonies): Kepler further revealed the relationship between the orbital periods and distances of planets from the Sun. This relationship, known as "Kepler's Third Law" or the "Law of Harmonies," quantifies the proportional relationship between the square of the orbital period of a planet and the cube of its average distance from the Sun.

Kepler's laws revolutionized astronomy by providing a mathematical and physical framework for understanding planetary motion. They laid the foundation for Isaac Newton's theories of gravitation, which further advanced our comprehension of celestial mechanics and the dynamics of the universe.