Physics - Classical Mechanics - Kepler's Laws of Planetary Motion

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Introduction

Hey it's a me again @drifter1!

In this article we will continue with Physics, and more specifically the branch of "Classical Mechanics". Today's topic are Kepler's Laws of Planetary Motion, extending the articles around the topic of Gravity even more into space!

So, without further ado, let's get straight into it!


How Kepler's Laws came to be

The Story of Kepler

German mathematician Johannes Kepler lived in Graz, Austria during the early 17th century. Do to religuous and political difficulties of the era he was banished from Graz. Due to the more accurate astronomical observations of his time, famous astronomer Tycho Brache was impressed by young Kepler. Thus, young Kepler moved from Graz to Brahe's home in Praque to work as an assistant of Brahe. Because Brahe mistrusted Kepler, he didn't show all his volunimous planetary data, fearing that young Kepler might surpass him. [Ref1]

The orbit of planet Mars didn't fit into the universal model set by Aristotle and Ptolemy, and so Brahe set Kepler the task of understanding the orbit of Mars. Brahe hoped that the difficulty of this task would occupy Kepler while he could work on perfecting his own theory of the solar system, where the earth was considered the center of the solar system. Based on this geocentric model, all other planets orbit around the Sun, while the Sun orbits around the earth. Kepler beliveved into the Copernican model of the solar system, known as heliocentric, which correctly placed the Sun at its center. The problem with the Copernican system was that it incorrectly assumed that the orbits of the planets are circular. [Ref1]

After a lot of struggling, Kepler finally realized that the orbits of the planets are not circles, but elongated or flattened circles, called ellipses. The orbit of planet mars was the most elliptical among the planets for which Brahe had extensive data. In a twist, Brahe unwittingly gave Kepler the very part of his data that would enable Kepler to formulate the correct theory for the solar system, banishing Brahe's own theory. [Ref1]

Elliptical Orbit

Since the orbits of the planets are ellipses, each orbit has three basic properties:

  • An ellipse is defined by two points, called focus, which together are called foci.
    • The sum of the distances to the foci from any point on the ellipse is always constant (similar to how the distance to the center of a circle is always constant and equal to its radius)
  • An ellipse is described by its eccentricity (amount of flatting)
    • The flatter an ellpse the more eccentric it is
    • The eccentricity is a value between zero (a circle) and one (a flat line called parabola)
  • An ellipse has two axes:
    • the longest axis is called major axis
    • the shortest axis is called minor axis
    • the half of the major axis is termed a semi major axis

Using such elliptical orbits, the motion of the planets, and comets as well, can be described very accurately. The orbit of each planet is thus defined by the foci, eccentricity and two axes.

Kepler's Laws

Kepler's Laws describe how the planets orbit around the Sun, and more specifically:

  • How planets move in elliptical orbits with the Sun at one focus (First Law)
  • How a planet covers the same area of space in the same amount of time, no matter where it is in its orbit (Second Law)
  • How the orbital period of a planet is proportional to the size of its orbit (Third Law)


Kepler's First Law (The Law of Ellipses)

Each planet's orbit about the Sun is an ellipse, with the Sun's center always located at one focus of the orbital ellipse. The planet follows the ellipse, making the distance the planet has to the Sun change constantly.

An ellipse can be described in many ways, but all are more general equation for conic sections.

Mathematically, in polar coordinates (r, θ), an ellipse can be represented by the formula:

  • r: distance from the Sun to the planet
  • p: semi-latus rectum of the conic section, which is half of the chord parallel to the directix passing though a focus
  • ε: eccentricity of the ellipse
  • θ: the angle of the planet's current position from its closest approach (perihelion)



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At θ = 0°, perihelion, the distance is minimum:

At θ = 180°, aphelion, the distance is maximum:

At θ = 90° and θ = 270° the distance equals p.

The semi-major axis a of the ellipse, is the arithmetic mean between rmin and rmax:

The semi-minor axis b of the ellipse, is the geometric mean between rmin and rmax:

The semi-latus rectum p is the harmonic mean between rmin and rmax:

The eccentricity ε is the coefficinent of variation between rmin and rmax:

The area A of an ellipse is given by:



For the special case of a circle, ε = 0, r = p = rmin = rmax = a = b giving A = πr2

Eccentricity of the Solar System's planets

  • Mercury - ε = 0.206
  • Venus - ε = 0.0068
  • Earth - ε = 0.0167
  • Mars - ε = 0.0934
  • Jupiter - ε = 0.0485
  • Saturn - ε = 0.0556
  • Uranus - ε = 0.0472
  • Neptune - ε = 0.0086
The ellipses of all but Mercury and Mars are basically circles.


Kepler's Second Law (The Law of Equal Areas)

Drawing an imaginary line from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. In other words, the planets do not move in constant speed along their orbits, but the speed various based on the distance to the sun. The closer the planet is to the sun the faster it moves, the further away the slower it moves. The nearest approach is termed perihelion, whilst the greatest sepeartion a aphelion.



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The orbital radius and angular velocity of the planet vary during its elliptical orbit.

During infinitesimal time dt, the planet sweeps out a small triangle with base line r and height r dθ, with area:



and so a planet has the following constant areal velocity:


Kepler's Third Law (The Law of Harmonies)

The period of a planet's orbit increases rapidly with the radius of its orbit. More specifically the law states that:

The squares of the orbital periods of the planets are directly proportional
to the cubes of the semi major axes of their orbits



[Image 4]

Combining Kepler's Third Law with Newton's Law of Gravitation, the following equation comes out:

Using this law its easier to compare the orbits of the various planets with each other. For example in the case of the Earth and Mars:

  • Earth: Period = 3.156 x 10 7 secs, Average Distance = 1.4957 x 1011 m and T2/a3 = 2.977 x 10-19 s2/m3
  • Mars: Period = 5.930 x 10 7 secs, Average Distance = 2.2780 x 1011 m and T2/a3 = 2.975 x 10-19 s2/m3
A similar ratio can also be computed for the other planets.


RESOURCES:

References

  1. https://solarsystem.nasa.gov/resources/310/orbits-and-keplers-laws/
  2. https://www.physicsclassroom.com/class/circles/Lesson-4/Kepler-s-Three-Laws
  3. https://cnx.org/contents/8sj3SsYT@12/Kepler-s-Laws-of-Planetary-Motion
  4. http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
  5. https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

Images

Mathematical equations used in this article, where made using quicklatex.


Previous articles of the series

Rectlinear motion

Plane motion

Newton's laws and Applications

Work and Energy

Momentum and Impulse

Angular Motion

Equilibrium and Elasticity

Gravity


Final words | Next up

And this is actually it for today's post!

Next time we will get into Kepler's Laws of Planetary Motion...

See ya!

Keep on drifting!
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