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

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)

[Image 2]

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 r_min and r_max:

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

The semi-latus rectum p is the harmonic mean between r_min and r_max:

The eccentricity ε is the coefficinent of variation between r_min and r_max:

The area A of an ellipse is given by:

For the special case of a circle, ε = 0, r = p = r_min = r_max = 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.

[Image 3]

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

Images

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

Previous articles of the series

Rectlinear motion

Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
Projectile motion as a plane motion -> missile/bullet motion as a plane motion
Smooth Circular motion -> smooth circular motion theory
Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

Force and Newton's first law -> force, 1st law
Mass and Newton's second law -> mass, 2nd law
Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
Contact forces and friction -> contact force, friction
Dynamics of Circular motion -> circular motion dynamics, applications
Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
Contact force and friction examples -> exercises in force and friction
Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
Internal Energy and Work -> Internal Energy, Internal Work

Momentum and Impulse

Conservation of Momentum -> Momentum, Conservation of Momentum
Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
Collision Examples -> Various Elastic and Inelastic Collision Examples
Impulse -> Impulse with Example
Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

Angular motion basics -> Angular position, velocity and acceleration
Rotation with constant angular acceleration -> Constant angular acceleration, Example
Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
Parallel Axis Theorem -> Parallel axis theorem with example
Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
Angular Momentum -> Angular Momentum and its conservation
Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
Exercises around Angular motion -> Angular motion examples

Equilibrium and Elasticity

Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
Force Couple System -> Force Couple System, Example
Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples

Gravity

Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
Gravitational Fields -> Gravitational Field Mathematics and Visualization
Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites

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!