In astronomy, Kepler’s laws of planetary motion are three scientific laws that explain the movement of planets around the Sun.

Kepler’s First Law - The Law of Orbits

Kepler’s Second Law: The law of equal areas.

Kepler’s Third Law: The law of periods.

Table of Contents:

• Introduction to Kepler’s Laws
• Kepler’s First Law - The Law of Orbits
• Kepler’s Second Law - The Law of Equal Areas
• Kepler’s Third Law - The Law of Periods

Introduction to Kepler’s Laws

Motion is always relative. Based on the energy of the particle under motion, the motions are classified into two types:

Motion is always relative, and the energy of the particle in motion determines the type of motion. There are two types of motion:

• Bounded Motion
• Unbounded Motion

The particle in bounded motion has a negative total energy (E < 0), and has two or more extreme points where the total energy is equal to the potential energy of the particle, resulting in the kinetic energy of the particle becoming zero.

If the eccentricity of an orbit is between 0 and 1 (0 ≤ e < 1), then a body has bounded motion if its energy (E) is less than 0. A circular orbit has an eccentricity of 0, and an elliptical orbit has an eccentricity of less than 1 (e < 1).

The particle in unbounded motion has a total energy of E > 0, and its potential energy is equal to its total energy at a single extreme point where the kinetic energy of the particle is zero.

For eccentricity e ≥ 1, if E > 0 then the body has unbounded motion. A Parabolic orbit has eccentricity e = 1, and a Hyperbolic path has eccentricity e > 1.

Also Read:

Gravitational Potential Energy

Gravitational Field Intensity

Kepler’s laws of planetary motion can be stated as follows:

Kepler’s First Law - The Law of Orbits

According to Kepler’s first law, all the planets revolve around the sun in elliptical orbits having the sun at one of the foci. The point at which the planet is close to the sun is known as perihelion, and the point at which the planet is farther from the sun is known as aphelion.

The characteristic of an ellipse is that the sum of the distances of any planet from two foci is constant. This is the reason why the elliptical orbit of a planet is responsible for the occurrence of seasons.

Kepler’s First Law - The Law of Orbits

Kepler’s Second Law - The Law of Equal Areas

Kepler’s second law states that “the radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time”.

As the orbit is not circular, the planet’s kinetic energy is not constant in its path. It has more kinetic energy near the perihelion (at a distance of r_min from the sun), and less kinetic energy near the aphelion (at a distance of r_max from the sun) which implies that the planet has more speed at the perihelion and less speed (v_min) at the aphelion.

r_min + r_max = 2a * (Length of Major Axis of an Ellipse) . . . . . . . (1)

The law of conservation of angular momentum states that the total angular momentum of a system will remain constant over time. This can be verified mathematically, as at any point in time the angular momentum, L, can be expressed as $L=m{r}^2\omega$, where m is the mass of the system, r is the distance of the mass from the axis of rotation, and ω is the angular velocity.

The length of the arc covered in a small area $\mathrm{\Delta }A$ described in a small time interval $\mathrm{\Delta }t$ with a covered angle $\mathrm{\Delta }\theta$ and a radius of curvature r is equal to $r\mathrm{\Delta }\theta$.

$\Delta A=1/2[r\cdot (r\cdot \Delta \theta )]=1/2r^2\cdot \Delta \theta$

Therefore, $$\frac{\Delta A}{\Delta t} = \frac{1}{2r^2}\frac{\Delta \theta}{dt}$$

Taking limits on both sides as, $$\lim_{\Delta t \to 0}$$, we get;

$$(\lim_{\Delta t \rightarrow 0}\frac{\Delta A}{\Delta t} = \lim_{\Delta t \rightarrow 0}\frac{1}{2}r^2\frac{\Delta \theta}{\Delta t})$$

$$(\frac{dA}{dt} = \frac{1}{2}r^2\omega)$$

$$\frac{dA}{dt}=\frac{L}{2m}$$

Now, according to conservation of angular momentum, L is a constant.

Thus, $$\frac{dA}{dt} = constant$$

The area swept in equal intervals of time remains constant.

Kepler’s second law states that the areal velocity of a planet revolving around the sun in an elliptical orbit remains constant, implying that the angular momentum of a planet is also constant. As a result of this, all planetary motions are planar motions, which is a direct consequence of central force.

Check: Acceleration due to Gravity

Kepler’s Third Law - The Law of Periods

“According to Kepler’s law of periods, the square of the time period of revolution of a planet around the sun in an elliptical orbit is proportional to the cube of its semi-major axis.”

$T^2\propto a^3$

The shorter the orbit of the planet around the sun, the shorter the time taken to complete one revolution. Using Newton’s law of gravitation and laws of motion, Kepler’s third law can be expressed in a more general form.

$$P2 = \frac{4\pi^2}{G(M_1 + M_2)} \times a^3$$

Where M₁ and M₂ are the masses of the two orbiting objects, both in Solar masses.

Frequently Asked Questions on Kepler’s Law

What does Kepler’s first law explain?

Kepler’s First Law explains that planets move in elliptical orbits, with the Sun at one of the two foci.

According to Kepler’s first law, all the planets revolve around the Sun in elliptical orbits with the Sun as one of the foci.

What does Kepler’s Second Law explain?

According to Kepler’s Second Law, the speed of planets in space is constantly changing. This law explains that when planets are closer to the Sun, they will move faster.

Kepler’s Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

What is Kepler’s third law?

Kepler’s third law, also known as the law of periods, states that the square of the orbital period is proportional to the cube of its mean distance (R).

What Causes the Orbits of the Planets to be Elliptical?

If the velocity of the planet changes, it is highly unlikely that the orbit will remain circular; instead, it will become elliptical.