All About Kepler’s Third Law – The Law of Periods

Kepler’s theories extended Copernicus’ model further. They were the most important addition to the advancement of Copernicus’ model. Kepler also accepted Copernicus’ theory of planetary orbit eccentricity as zero. It was discovered that the planets’ orbits follow a curved course, their centre point is the Sun, and their speed in the primary orbit is constant.

The Copernicus model was improved by Kepler’s law, which stated that the planets’ orbits are an ellipse with cycles. This leads to Kepler’s first law, “the orbit of a planet is an ellipse with the Sun at one of the two foci.”The speed of the planets’ field remains constant as well. The planets’ linear and angular momentum were not found to be constant during their orbits.

Kepler’s Scientific Discoveries

Johannes Kepler, a German astronomer, discovered that circular orbits were impractical. As a result, he investigated celestial objects and their orbits, developing laws to show that the orbits were elliptical rather than circular. When Kepler was studying Mars’ orbital motions, he discovered that they were elliptical or oval-like.

The data also demonstrated that other planets orbiting farther from the Sun have elliptical orbits. In a letter, he explained his discovery to another astronomer, David Fabricius. He penned his newfound discovery on October 11, 1605, and the majority of his works were published between 1605 and 1607.

Kepler’s Third Law

• The square of a planet’s time period is proportional to the cube of its semi-major axis, according to this law.
• If the Earth rotates around the Sun, the square of the time it takes to complete one rotation around the Sun is proportional to the cube of the semi-major axis.
• Because it is based on the planets’ time periods, Kepler’s third law is known as the law of periods.
• The assumption that the path of the planet is circular is used to derive Kepler’s third law.

Formula of Kepler’s Third Law

The formula of Kepler’s third law is given below.

Here, ‘m’ is the planet’s mass.

M is the Sun’s mass.

Newton’s Law of Gravitation states:

F = GMmr2

Fc = mv2r

where, 

Fc = The centripetal force aids the planet’s eccentric orbit around the Sun.

F = Fc

GMmr2 = mv2r 

where r is the distance between two particles 

GMr = v² (1)

v = 2πrT

Squaring both sides of the equation above,

v² = 4π2r2T2

Placing the value of (1)

GMr = 4π2r2T2

T2 = (4π2r3GM4 π² r³/GM) 

where (4π2GM) = constant

T² = r³

The radius of the circle is the same as the semi-major axis in an ellipse.

Kepler’s third law is in contrast to Kepler’s first and second laws. Kepler’s first and second laws help establish that when a planet is closer to the Sun, it travels faster. Kepler’s third law states that the farther a planet is from the Sun, the slower its orbital speed. The law of coherence is Kepler’s third law, which compares a planet’s orbital period and radius of orbit to those of other planets.

Using the equations for Newton’s law of gravitation and the laws of motion, Kepler explained the third law, adding that the smaller the planet’s orbit around the Sun, the longer it would take to complete one revolution. Kepler’s formula related to this is described here:

P2= 4π2[G(M1+M2)]a3

Which denotes the formula of Kepler’s third law, where the masses of the two circling objects, M1and M2are shown in solar masses.

How to Use Kepler’s Third Law

Information about how to use Kepler’s third law and what kind of benefits can be obtained from it are presented here:

• One thing to keep in mind concerning Kepler’s third law is that it does not mention an object’s mass. However, it follows Newton’s gravitational laws, which are a more generalised version of Kepler’s equation.

• The mass of the bodies involved in the specified system can be determined using this generalised variant of the equation of Kepler’s third law. Given the masses 1 (M1) and 2 (M2), the masses of the two bodies, the M1 star’s description is often so much larger than M2 that the orbiting body’s mass might have been ignored. 

• The exact masses and average densities found for Mars, Jupiter, Saturn, Uranus, and Neptune were obtained using Kepler’s third law, as well as the successful application of the harmonic law to compute the mass of the planets in our solar system.

• Kepler’s third law has recently permitted a reliable determination of Venus’ mass and average density thanks to the recent placement of artificial satellites around the planet. The harmonic law is used to calculate the overall mass of the Pluto-Charon system.

Conclusion

Kepler’s third rule has aided astronomers in obtaining measurements of comets’ highly irregular orbits around the Sun. But it is not limited to the solar system. Many discoveries pertaining to everyday life have been made. The mass of stars in binary systems has been calculated using Kepler’s third law in conjunction with the second law, which has been useful in understanding the structure and evolution of stars.

At the same time, it is found here that although Kepler had no knowledge of gravitation when he devised his three laws, they were essential in Isaac Newton’s development of his theory of universal gravitation. This explains the unknown force of Kepler’s third law. Kepler’s theories were crucial in gaining a better grasp of the dynamics of our solar system, as well as serving as a springboard for newer models that more properly approximate our planetary orbits.