Kepler’s Law of Planetary Motion

In the 17th century, Johannes Kepler introduced three important laws that changed the way we understand how planets move around the Sun. These laws were a huge step forward in astronomy because they gave scientists a clear, mathematical way to describe planetary orbits. Before Kepler, most people believed that planets moved in perfect circles. But Kepler showed that their paths are actually elliptical—like stretched-out circles—with the Sun sitting at one of the focal points. He also discovered that a planet doesn’t move at a constant speed; it travels faster when it’s closer to the Sun and slower when it’s farther away. Finally, he found a simple relationship between a planet’s distance from the Sun and how long it takes to complete one orbit. These discoveries not only supported the idea that the Sun is at the center of the solar system (as Copernicus suggested) but also helped pave the way for Isaac Newton’s theory of gravity. Even today, Kepler’s laws are essential tools in both physics and astronomy.

1.0Theories About Planetary Motion

1. The Geocentric Model: In ancient times, Greek astronomers like Ptolemy believed that Earth was the center of the universe. According to this geocentric model, the Sun, Moon, stars, and planets all revolved around Earth. This idea made sense to people at the time because it matched what they saw in the sky and aligned with many religious beliefs. As a result, it remained widely accepted for centuries.
2. Aryabhatta’s Contribution: Aryabhatta, a 5th-century Indian astronomer and mathematician, proposed that the Earth rotates on its axis—explaining the apparent movement of stars. Though he followed the geocentric model, his ideas challenged traditional views and influenced future astronomical thought.
3. The Heliocentric Model: In the 16th century, Nicolaus Copernicus introduced a revolutionary idea: that the Sun—not the Earth—is at the center of the solar system. In this heliocentric model, the planets, including Earth, revolve around the Sun. This concept corrected many errors in earlier models and became the starting point for modern astronomy.
4. Tycho Brahe’s Contribution: Tycho Brahe, a Danish astronomer, was known for his incredibly detailed and accurate observations of the planets—long before telescopes were invented. Although he proposed a compromise model where the planets orbited the Sun, and the Sun orbited Earth, his data was groundbreaking. It later played a crucial role in helping Johannes Kepler discover the true nature of planetary motion.
5. Johannes Kepler’s Contribution: Kepler's three laws showed that planets orbit the Sun in ellipses, move faster when nearer to it, and have orbital periods linked to their distance. His work paved the way for Newton’s law of gravity.

2.0Kepler’s First Law (Law of Orbits)

Each planet revolves around the sun in an elliptical orbit with the sun situated at one of the two foci.

PA=2a =major axis, BC=2b=minor axis

• Perihelion: The point in a planet's orbit closest to the Sun.
• Aphelion: The point in a planet's orbit farthest from the Sun.

Ellipse : Ellipse is the set of all points such that the sum of the distance from the foci (plural of focus) and any point on the ellipse is constant.

General equation of ellipse, $\frac{x^2}{a^2}+\frac{x^2}{b^2}=1$

Note: Kepler's Problem: The derivation of Kepler’s three laws of planetary motion from Newton law of gravitation is called Kepler’s problem.

Proof of Kepler's First Law 

The planetary motion takes place under the action of the gravitational force exerted by the sun,

$\vec{F}=-\frac{G{M}_S{m}_p}{r^3}\vec{r}$

$M_S=$Mass of the Sun. This force is radial and central and the negative sign indicates that $\vec{F}$ is oppositely directed to $\vec{r}$

Torque exerted on the planet P about the sun is

$\vec{\tau }=\vec{r}\times \vec{F}=\vec{r}\times (-\frac{G{M}_S{m}_p}{r^3}\vec{r}=\vec{0}(\because \vec{r}\times \vec{r}=\vec{0})$

$\vec{\tau }=\frac{\overrightarrow{dL}}{dt}$=Rate of change of angular momentum

$\frac{\overrightarrow{dL}}{dt}=0$ or $\vec{L}$=Constant

• This shows that the planet's angular momentum about the Sun remains constant in both magnitude and direction.
• The direction of $\vec{L}=\vec{r}\times \vec{P}$ is fixed , $\vec{r}$ and $\vec{v}$ lie in a plane normal to $\vec{L}$.

3.0Kepler’s Second Law ( Law of Areas)

• A line connecting any planet to the Sun sweeps equal areas in equal time, keeping the areal velocity constant.

• The radius vector from the Sun to a planet covers equal areas in equal time, keeping the areal velocity steady.
• If a planet takes the same amount of time to travel from position A to B as it does from C to D, then according to Kepler's second law, the areas swept out — ASB and CSD — must be equal. Since the planet travels a greater distance from C to D (closer to the Sun) than from A to B (farther from the Sun) in the same time, this means its linear velocity is higher when it is nearer to the Sun and lower when it is farther away.

$Area=\frac{1}{2}(base)(height)$

$dA=\frac{1}{2}(rd\theta )(r)=\frac{1}{2}{r}^{2}d\theta$

So, areal velocity (area per unit time)

$\frac{dA}{dt}=\frac{1}{2}{r}^2\frac{d\theta }{dt}=\frac{1}{2}{r}^2\omega =\frac{1}{2}rv$

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

Areal velocity,

$L=I\omega =m{r}^2\omega =mvr$

Here $I=m{r}^2$ (instantaneous moment of inertia of planet about sun)

Angular momentum of planet about the Sun $L=mvr$

As the force of gravitation on the planet by the Sun passes through the Sun itself, the Torque of gravitational pull is zero about the Sun.

Angular momentum of planet about the Sun is constant

$L=mvr(Constant)$

$\frac{L}{m}=vr\Rightarrow \frac{L}{2m}=\frac{1}{2}vr\Rightarrow \frac{1}{2}vr=\frac{dA}{dt}$

$r\times v=constant$

$v_a{r}_a=v_p{r}_p\Rightarrow \frac{v_a}{v_p}=\frac{r_p}{r_a}=\frac{r_{min}}{r_{max}}$

Conclusion

At minimum distance from the Sun, velocity is maximum. At maximum distance from the Sun, velocity is minimum.

4.0Kepler’s Third Law (Law of Periods)

The quadret of a planet's orbital period is directly proportional to the cube of its orbit's semi-major axis.

$T^2\propto a^3$

a :  mean distance from the Sun.

$a=\frac{r_{max}+r_{min}}{2}$

• The third law is derived by assuming the planet's circular orbit, with gravity supplying the centripetal force for its motion. 

$\frac{G{M}_S{m}_p}{R^2}=m_p({\omega }^{2}R)$

$\frac{G{M}_S}{R^3}={\omega }^2$

$\frac{G{M}_S}{R^3}=(\frac{2\pi }{T}{)}^2$

$\frac{G{M}_S}{R^3}=\frac{4{\pi }^2}{T^2}\Rightarrow T^2=\frac{4{\pi }^2}{G{M}_S}{R}^3$

Illustration-1. A small satellite is moving in an elliptical orbit around the Earth, as shown in the diagram. Let L represent the magnitude of its angular momentum about the center of the Earth, and K denote its kinetic energy. Given that the satellite is at two different points in its orbit, labeled as Position 1 (closer to Earth) and Position 2 (farther from Earth), compare the kinetic energies at these two positions: K₁ (at Position 1) and K₂ (at Position 2).

Solution: The torque of gravitational pull is zero about the sun. So angular momentum of the planet about the sun is constant.

$L_1=L_2\Rightarrow m{v}_1{r}_1=m{v}_2{r}_2\Rightarrow v_1{r}_1=v_2{r}_2\Rightarrow r_1>r_2$

$v_1=\frac{v_2{r}_2}{r_1}$

$v_2>v_1$

$\frac{1}{2}m{v}_2^2>\frac{1}{2}m{v}_1^2\Rightarrow K_2>K_1$

Illustration:2. A planet twists around the Sun in an elliptical orbit. The areal velocity of the planet is given as 4.0 × 10¹⁶ m²/s, and the minimum distance between the planet and the Sun (perihelion) is 2 × 10¹² m. What is the maximum speed of the planet (in km/s) as it passes through its closest point to the Sun?

Solution:

$\frac{dA}{dt}=\frac{r^2\omega }{2}$ is constant

$\frac{dA}{dt}=\frac{r_{max}^2{\omega }_{min}}{2}=\frac{r_{min}^2{\omega }_{max}}{2}\Rightarrow {\omega }_{max}=\frac{2\frac{dA}{dt}}{r_{min}^2}$

$v_{max}={\omega }_{max}{r}_{min}=\frac{2dA/dt}{r_{min}}=\frac{2\times 4\times 10^{16}}{2\times 10^{12}}=40km/sec$