Kepler’s Laws Of Planetary Motion Part 2 – The Law of Areas

Reading Time – 9 Minutes, Difficulty Level – Challenging

The second in a series of three posts about Kepler’s Laws of Planetary Motion. Here we look at the Kepler’s second law or, ‘The Law of Areas’ – A line that connects a planet to the Sun sweeps out equal areas in equal times.

The first section here is the accessible part, which you should be able to understand with some concentration, you don’t need to read through the mathematics!

Johannes Kepler’s Second Law of Planetary Motion, also known as the “law of equal areas,” is a fundamental principle that helps us understand the movement of planets in our solar system. This law describes how the speed of a planet changes as it moves through its elliptical orbit around the sun, and how it sweeps out equal areas in equal times.

Kepler’s Second Law In a nutshell

Kepler’s second law states that a planet will travel faster when it is closer to the sun and slower when it is farther away. Specifically, the planet’s speed is such that the line connecting the planet to the sun sweeps out equal areas in equal times. In other words, as a planet moves closer to the sun, it travels faster and covers a larger area, while when it moves farther away, it travels slower and covers a smaller area.

Kepler’s Second Law in Detail

This law is particularly important because it explains why planets do not move at a constant speed throughout their orbits. Instead, they speed up and slow down as they move closer to and farther away from the sun. This concept is critical for understanding how planets move and how they interact with each other.

Kepler’s Second Law also explains why planets have “perihelion” and “aphelion” points. The perihelion point is the point in a planet’s orbit where it is closest to the sun, while the aphelion point is the point where it is farthest away. Because of the way planets move in their elliptical orbits, they spend more time near the aphelion point, where they travel more slowly, than near the perihelion point, where they travel faster.

Seasonality

One of the most important implications of Kepler’s Second Law is that it helps us understand why planets experience seasons. As a planet moves through its elliptical orbit, its distance from the sun changes, causing changes in the amount of sunlight it receives. This is particularly true for Earth, which experiences more extreme seasons because its axis is tilted relative to its orbit around the sun.

Kepler’s Second Law was also critical in the development of Isaac Newton’s law of gravitation, which describes how objects with mass attract each other. By observing the motion of planets and understanding how their speed changes as they move through their orbits, Kepler was able to make significant contributions to our understanding of gravity and the laws that govern the movement of celestial bodies.

Kepler’s Second Law of Planetary Motion is a fundamental principle that helps us understand the movement of planets in our solar system. By describing how the speed of a planet changes as it moves through its elliptical orbit and how it sweeps out equal areas in equal times, this law explains why planets do not move at a constant speed and helps us understand how they interact with each other. Kepler’s Second Law has had a profound impact on our understanding of gravity and the laws that govern the movement of celestial bodies.

Kepler’s Second Law Mathematics

The mathematics of Kepler’s second law is extremely challenging for the non-scientist, but I will include it here for you to look through anyway. Understanding this section absolutely isn’t required to gain a high level overview of the second law.

Kepler’s second law can be derived mathematically as follows:

Consider a planet of mass m moving in an elliptical orbit around the sun of mass M. Let r be the distance between the planet and the sun, and let v be the velocity of the planet.

By the conservation of angular momentum, we have:

mvr=constant

This equation means that the product of the planet’s mass, velocity, and distance from the sun remains constant. In other words, if the planet’s distance from the sun decreases, its velocity increases in such a way that the product of these three quantities remains the same. Similarly, if the planet’s distance from the sun increases, its velocity decreases accordingly. This conservation of angular momentum is a fundamental principle of physics, and it plays an important role in the motion of planets and other celestial bodies.

Differentiating both sides with respect to time t, we get:

Since the mass of the planet is constant, dm/dt = 0, so the above equation becomes:

This is a vector equation, so we can write it in terms of the components of r and v. Let θ be the angle between r and v, so that:

Differentiating both sides with respect to time, we get:

Since r and v are perpendicular at the point where the angle θ is a right angle, we have:

Therefore:

This is known as the area law, since it implies that the area swept out by the planet in a small time interval dt is given by:

Thus, the area swept out by the planet in a complete orbit is given by:

where Δθ is the change in the angle θ over a complete orbit. Since the time taken for a complete orbit is T, we have:

Δθ = 2π

Therefore, the area swept out by the planet in a complete orbit is:

This is a constant, since r is a function of time and varies periodically. Therefore, the area swept out by the planet in equal time intervals is the same, which is Kepler’s second law.

Further Reading

Kepler’s Laws of Plantery Motion Part 1 – The Law of Orbits
Kepler’s Laws of Planetary Motion Part 3 – The Law of Periods