Kepler’s law of planetary motion

Introduction to Kepler’s Laws

The law revolves around stating that the motion is always relative. Being the energy of the particle under motion as a base, the motions are divided into two categories:

•       Bounded Motion
•       Unbounded Motion

If a system lies constricted in a specific region of space is called Bounded motion. It usually occurs when the particle’s energy is less than or equal to the total potential barrier at the infinite separation.

For eccentricity 0≤ e <1 and energy E < 0

 A circular orbit always has eccentricity e = 0, and an elliptical orbit always has eccentricity e < 1.

According to unbounded motion, the particle in motion has total positive energy (E > 0) and has a single extreme point. The total energy is always equivalent to the particle’s potential energy; that is, the particle’s kinetic energy falls and becomes zero.

For eccentricity e ≥ 1, E > 0

This States that the body has unbounded motion. A Parabolic orbit always has eccentricity e = 1, and the Hyperbolic path always has eccentricity e>1.

 KEPLER’S LAW OF PLANETARY MOTION

Johannes Kepler coined the law of Kepler motion. Kepler’s law of planetary motion enlightens us about how planets revolve around the sun. He described the following about the planets and their motion around the sun:

•       How planets move in elliptical orbits with the sun being the center of focus.
•       A planet covers an equal amount of area in the same amount of time, irrespective of where it is lying in orbit.
•       How a planet’s time revolves or completes one orbit is proportional to the size of its orbit.

KEPLER’S FIRST LAW- THE LAW OF ORBITS

The first law of Kepler reveals and demonstrates that all the planets in the solar system revolve around the sun in a fixed ellipses orbit, having the sun as a focus. Kepler’s first law also revealed that Perihelion is the point in the orbit where the planets are closest to the sun, and Aphelion is the point in the orbit where the planets are farthest from the sun. The characteristics of the ellipses are duly responsible that the sum distance of the plant from any two foci on-orbit remains constant. Also, here is the fact that the elliptical orbit of the planets is undoubtedly responsible for the falling and changing seasons.

CHARACTERISTICS OF ELLIPSES

Let us dig insights into the characteristics of ellipses. The three essential properties of ellipses are:

1.     An ellipse is the sum of two defined points, each is known as a focus, and together they are known as foci. The sum of the distances from the foci to any point on the ellipse is always constant.
2.     The second property of an ellipse is that the flattening is termed eccentricity. The eccentricity of the ellipse is more when it is flat. Every ellipse has an eccentric value ranging between zero, a circle, and one.
3.     The third property of an ellipse is that the longest axis of the ellipse is termed as the central axis, while the shortest axis is recalled as the minor axis. If we divide it into half, the central axis is recalled as the semi-major axis. Understanding that period of time, the orbits of the planets are elliptical planets, Johannes Kepler took on three laws of planetary motion. These laws also accurately described the motion of comets in the solar system.

KEPLER’S SECOND LAW-THE LAW OF EQUAL AREAS

Kepler’s second law states that an imaginary line drawn from planets to the sun joining them together sweeps space into equal areas at equal intervals as planets start to orbit around the sun. The kinetic energy of the elliptical planetary orbits won’t be constant. It keeps on changing. More kinetic energy is near the perihelion, and less energy is near the aphelion. This states that there is more speed at perihelion and less speed at aphelion. We can keep a check on the period and radius of the planet orbits through the following table:

KEPLER’S THIRD LAW-THE LAW OF PERIODS

Kepler’s third law states that the square of the orbital period of the planets around the sun are proportional to the cube of the semi-major axes of the given orbit. This law reveals that as the radius of the orbit increases, the period to complete one orbit increases. That is why Mercury requires only 88 days to orbit around the Sun, while the earth takes 365 days, and Saturn takes 10,759 days to complete one orbit.

Using Newton’s law of gravitation, we can derive the formula for Kepler’s third law:

p2 = 4π2 /[G(M1+ M2)] × a3

here p is period, and a is the semi-major axis.

Let’s understand these laws by some numerical examples.

Ex. 1) A satellite of a planet, has an orbital radius of 2.25×109 m. Its orbital period is 25.65 days. Another satellite of this planet has an orbital radius of 1.65×109 m. Use Kepler’s third law of planetary motion and predict the orbital period of the second satellite in days.

Ans: Given parameter 𝑟A = 2.25×109 m

 𝑇A = 25.65 𝑑𝑎𝑦𝑠 

𝑟B = 1.65×109 m

 𝑇B =? 

From Kepler’s third law 

(TATB)2 = (rArB)3

TB = TA2rB3rA31/2

After solving, we will get

TB  = 16.11 days.

Ex. 2 Distance between the earth and the sun is 149.5 x 106 km and  Earth’s revolution time period is 1 year. Calculate the value of  T2 / r3.

Ans. Given the data in question

Distance between the earth and the sun r =  149.5 x 106 km

Earth’s revolution time period T =  1 year

So T2r3 = (1 year)2(149.5 x 106 km)3 

T2r3 = 2.99 10-25years2/km3

CONCLUSION

From the above article, we learned about Kepler’s law of planetary motion. The basic idea behind the three laws is: How planets move in elliptical orbits with the sun being the Center of focus to them, how a planet covers an equal amount of area in the same amount of time, irrespective of where it is lying in orbit. How a planet’s time period revolves or completes one orbit is proportional to the size of its orbit.