Kepler's Law of Planetary Motion Class 11 Physics Notes, Definition, Working Principle, Formula & Real-Life Applications

The figure depicts an illustration of the first Kepler's law of planetary motion. It says that "All the planets move around the sun in ellipitcal orbits with sun at one of the focus not at centre of orbit." The planets' orbits around the sun are found to be roughly circular or considerably less ecentric.

Please note that Gravitation is an important chapter for both school (CBSE) and entrance examinations such as NEET and JEE exams.

Kepler's second law is a different way to express the concept of conservation of momentum. The picture in figure (a) serves as an illustration of it. Because of the conservation of angular momentum, we know that an elliptical orbit plane will travel more quickly the closer it is to the sun. As a result, a planet on an elliptical orbit experiences constant changes in its angular speed. Perihelion is the term used to describe the point at which a planet is closest to the sun. Aphelion is the term used to describe the point of maximum separation. The planet moves at its fastest speed when it is close to perihelion and its slowest speed when it is close to aphelion, according to the principle of angular momentum conservation.

According to second Kepler's law of planetary motion, "The line joining the sun and planet sweeps out equal areas in equal time or the rate of sweeping area by the position vector of the planet with respect to sun remains constant." The figure (b) illustrates this. Law of conservation of angular momentum is used to validate the preceding statement of Kepler's second law. Examine the planet orbiting the sun at a general point C at speed  to confirm this. Let r be the planet's current distance from the sun. If θ is the angle between the planet's location vector $\stackrel{\to }{r}$ and velocity vector (v), then the planet's angular momentum at this point is

$L=mvrsin\theta$ (1)

In elemental time, the planet will cover a small distance $CD=dl$  and will travel to another adjacent point D as shown in figure (a), thus the distance CD=vdt. In this duration dt, the position vector $\stackrel{\to }{r}$ sweep out an area which is equal to that of triangle SCD, which is calculated as

Area of triangle SCD is

$$\mathrm{dA}=\frac{1}{2}\times r\times vdtsin(\pi -\theta )=\frac{1}{2}rvsin\theta \cdot dt$$ Thus the rate of sweeping area by the position vector $\stackrel{\to }{r}$ is

$$\frac{dA}{dt}=\frac{1}{2}rvsin\theta$$ Now from equation (1)

$$\frac{dA}{dt}=\frac{L}{2m}=\mathrm{constant}$$ (2)

The Kepler's second law formula shown in the equation (2) validates the statement of Kepler' second law of planetary motion.

Kepler's Third Law discusses the planets' revolution period. Kepler's III law states that "The time period of revolution of a planet in its orbit around the sun is directly proportional to the cube of semimajor axis of the elliptical path around the sun." If ''T" denotes the revolution period and 'a' denotes the semi-major axis of the planet's path, then:

$$T^2\propto a^3$$ When a circular orbit's major and minor axes are equal, it is a particular instance of an ellipse. The revolution speed of a planet in a circular orbit of radius r around the sun must is expressed as $$v=\sqrt{\frac{G{M}_s}{r}}$$ where $M_s$ stands for the sun's mass. As you may remember, this speed is unaffected by the planet's mass. In this case, the revolution time period can be expressed as:

$$T=\frac{2\pi r}{v} \text{or} T=\frac{2\pi r}{\sqrt{\frac{G{M}_s}{r}}}$$ By squaring the previous equation, we obtain: 

$$T^2=\frac{4{\pi }^2}{G{M}_s}{r}^3\left(1\right)$$ The assertion of Kepler's third law for circular orbits is confirmed by equation (1). In a similar manner, we can confirm it for elliptical orbits. We begin with the relationship we previously established for the rate of sweeping area by the planet's position vector with respect to the sun, which is:

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

1. First Kepler’s Law of Planetary Motion (The Law of Orbits)

Comets and planets do not go around the Sun in perfect circles. Instead, they follow stretched paths called ellipses. This is important for scientists because it helps them know exactly where a comet or planet will be at a certain time. The same idea is used when planning how a satellite should travel in space. Since the path is not a perfect circle, planners can choose the best path depending on how far or close the satellite needs to be from Earth. This helps make better use of energy and time during space missions.

2. Second Kepler’s Law of Planetary Motion (The Law of Equal Areas)

When a planet or a spacecraft is closer to the Sun, it moves faster. When it is farther away, it moves slower. This is helpful for planning trips in space. If a spacecraft is near a planet, it can go faster using less fuel. Space engineers use this idea to decide the best time for a spacecraft to speed up or slow down. This saves fuel and helps the spacecraft reach its goal more easily.

3. Third Kepler’s Law of Planetary Motion(The Law of Periods)

There’s a connection between how far a planet is and how long it takes to go around the Sun. If something takes a long time to orbit, it is probably far away. Scientists use this idea to measure distances in space. They also use it to find out how heavy a planet or a star is. By watching how moons or planets move around it, they can guess the size of the thing in the center; even if they cannot see it clearly.

1. Kepler First Law – Works only when there are only two bodies involved

2. Kepler Second Law – Does not cover everything

3. Kepler Third Law – It shows a pattern but not the reason

Limits in all three Kepler's law planetary motion

The following points highlight the limitations of Kepler's law of plantary motion: