Kepler’s Laws

Planetary motion has been observed since ancient times, and by the 16th century, extensive data had been collected. Johannes Kepler studied these observations and formulated three fundamental laws describing planetary motion, known as Kepler’s Laws. These laws describe the elliptical nature of planetary orbits, their varying speeds, and the relationship between their orbital periods and distances from the Sun.

Types of Motion and Orbital Shapes:

1. Bounded Motion (E < 0): The total energy of the moving object is negative, meaning it remains within a closed orbit (circle or ellipse).

• Circular orbits have eccentricity (e) = 0.
• Elliptical orbits have eccentricity (0 < e < 1).

2. Unbounded Motion (E > 0): The total energy is positive, meaning the object escapes its orbit.

• Parabolic paths have eccentricity (e = 1).
• Hyperbolic paths have eccentricity (e > 1).

A planet moves around the Sun in an elliptical orbit, with the Sun located at one of its foci.

• An ellipse is a closed curve where the sum of the distances from any point on the curve to two fixed points (called foci) remains constant.
• The Sun is not at the centre of a planet’s orbit but is positioned at one of the two foci of the ellipse.
• Since an ellipse is not a perfect circle, the distance between the planet and the Sun continuously changes during its orbit.
• When the planet is closest to the Sun, this point is called the perihelion (147 million km).
• When the planet is farthest from the Sun, this point is called the aphelion (152 million km).

This law disproved the earlier belief that planets move in perfect circles. It provided a more accurate description of planetary motion.

The line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

• As a planet moves along its elliptical orbit, it does not travel at a constant speed.
• When the planet is closer to the Sun (at perihelion), it moves faster.
• When the planet is farther from the Sun (at aphelion), it moves slower.
• However, in equal time intervals, the area covered by the imaginary line connecting the planet and the Sun remains the same.

For example, if a planet moves from point A to point B in a certain time and from point C to point D in the same time, then the areas ASB and CSD will be equal, even though the distances AB and CD may be different.

The orbit of a planet moving around the Sun.

This law explains why planets speed up and slow down in their orbits. It follows from the conservation of angular momentum in physics.

The square of a planet’s orbital period (T²) is directly proportional to the cube of its average distance (r³) from the Sun.

T² ∝ r³

or

`T^2/r^3` = constant = K

where,

• T = Time taken by the planet to complete one revolution around the Sun (orbital period)
• r = Average distance of the planet from the Sun
• K = Constant value for all planets in the solar system

This law shows a mathematical relationship between the time a planet takes to orbit the Sun and its distance from the Sun. It implies that planets farther from the Sun take longer to complete one orbit compared to planets closer to the Sun.

For example, Mercury, which is closest to the Sun, has the shortest orbital period, while Neptune, the farthest, takes the longest.

This law helped astronomers understand the structure of the solar system. It was later used by Newton to develop his law of universal gravitation, providing a deeper explanation for planetary motion.

Kepler formulated these laws based on observational data, but he did not know why planets obeyed them. Later, Isaac Newton provided an explanation using his law of universal gravitation, which states:

`F = G(m_1m_2)/(r^2)`​​

where,

• F = Gravitational force between two objects
• G = Universal gravitational constant
• m₁, m₂ = Masses of the two objects
• r = Distance between them

Newton showed that the gravitational force exerted by the Sun on the planets is responsible for their elliptical orbits, variable speeds, and orbital periods, perfectly aligning with Kepler’s laws.