Kepler’s Laws of Planetary Motion

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Kepler's Laws of Motion

The Kepler’s Law of planetary motion typically revolves around the periods, orbits, and areas with regard to planet and sun. The principle thus states that the planets revolve around the sun in elliptical orbits with the sun at one focus in a sun-centred solar system. 

Owing to this evaluation of the motions of the planets, Kepler introduced a series of laws, now popularly known as Kepler’s three laws. The three laws by Kepler described the behaviour of planets depending on their paths through space. The first two Kepler's laws of motion were published in 1609 in The New Astronomy. Their discovery turned out to be an earnest approach to the development of modern science.


Kepler's First Law

Referred to as the law of ellipses, this law describes that planets are orbiting around the sun in a path explained as an ellipse. This Kepler's first law is also popularly known as the law of orbits. The law suggests that the orbit of any planet is an ellipse along with the Sun with Sun at one of the two focal points of an ellipse. We are already acquainted with the fact that planets move around the Sun in a circular orbit. However, as per Kepler, it is beyond doubt that planets revolve around the Sun, but not in a circular orbit. However, it revolves around an ellipse. In an ellipse, we have two foci. Sun is situated at one of the focus points of the ellipse.


Kepler's Second Law

Known as the law of equal areas- this law defines the speed at which any given planet will revolve while orbiting the sun. The speed at which any planet revolves through space changes constantly. A planet revolves the fastest when it is closest to the sun and slowest when it is farthest from the sun. Still, if an imaginary line is drawn from the centre of the planet to the centre of the sun that line would blow off the similar area in equal periods. For example, if an imaginary line is drawn from the earth to the sun, then the area swept aside by the line every 31-day month will be the same. This is depicted in the diagram below.


Diagrammatic Representation Of Kepler's Second Law

As can be seen from the diagram below, the areas developed when the earth is closest to the sun can be estimated as a broad but short triangle. On the other hand, the areas developed when the earth is farthest from the sun can be estimated as a slender but long triangle. These areas are of similar size. Seeing that the base of these triangles are shortest when the earth is farthest from the sun. It is observed that the earth would have to be in motion more slowly for the sake of this imaginary area to be the same size as when the earth is closest to the sun.

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Diagrammatic Representation Of Kepler's Second Law


Kepler’s Third Law

Referred to as the law of harmonies, Kepler’s third law sets a comparison for the orbital period and radius of orbit of a planet against other planets. Distinct from Kepler's 1st and 2nd laws that explain the motion attribute of a single planet, the 3rd law compares the motion characteristics of different planets. The comparison established is that the ratio of the squares of the periods to the cubes of their average distances from the sun is similar for every one of the planets.


Solved Examples for Kepler’s Laws of Motion

Problem 1

a. Which scientist is attributed to the gathering of the data relevant to support the planet's elliptical motion?

b. Which scientist is credited with the correct description of the data?

c. Which scientist is acknowledged with the long and complex task of evaluating the data?

Answer

  1. Tycho Brahe collected the data. 

  2. Johannes Isaac Newton appropriately described the data. 

  3. Kepler evaluated the data.


Problem 2

The average orbital distance of Mars is 1.52 times to that of the average orbital distance of the Earth. Having known that the Earth orbits around the sun in about 365 days, use Kepler’s third law (law of harmonies) to anticipate the time that Mars would require to orbit the sun.

Answer

Given that, the ratio of Rmars / Rearth is 1.52

= Tmars will be 684 days

Problem 3

Employ the graphing competencies of your TI calculator to trace out T2 vs. R3 (T2 must be plotted about the vertical axis) and to identify the equation of the line. Express the equation in slope-intercept form below.

Answer

T2 = [3.03 ×10-16] × R3 - 4.62 × 10+9

Given the unpredictability in the y-intercept value, it can be estimated as 0.

Hence, T2 = (3.03 × 10-16) × R3

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FAQ (Frequently Asked Questions)

1. What are Kepler’s Three Laws?

Ans. Kepler’s 3 laws of planetary motion can be explained as follows:

  1. The Law of Ellipses: By this postulate, the path of the planets along the sun is elliptical, with the nucleus (centre) of the sun being situated at one focus.

  2. The Law of Harmonies: The proportion of the squares of the periods of any two planets is equivalent to the proportion of the cubes of their mean distances from the sun.

  3. The Law of Equal Areas: An imaginary segment drawn from the nucleus of the sun to the nucleus of the planet will eradicate equal areas at equal intervals of time.

2. How to manually draw an Ellipse?

Ans. You can easily construct an ellipse using a pencil, string, sheet of paper, piece of cardboard and two tacks and wrap the string tightly around the tacks. Tack the paper sheet to the cardboard with two tacks. Then tie the string into a loop and around the two tacks. Take your pencil and tug the string until the two tacks and pencil form a triangle. Then start to draw a path with the pencil. The resulting shape is an ellipse.

An ellipse is a unique curve in which the sum of the total distances from each point on the curve to two other points is constant. For proper understanding, refer to the illustrative image below. 

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