# Past Models of Our Universe

By

Dr Lim Zhi Han

●

Published 2022-07-30

Humans have been mapping out the stars for a very long time. Several ancient architecture, artifacts and drawings were designed with reference to the positions of celestial objects. These objects could be made to record and study the seasons (to optimize agriculture activities), for navigation, or even for religious and political purposes. A few examples are listed below.

Greek philosopher Plato (424-348 BC) and later his student Aristotle (384-322 BC) advocated a model where the Earth is at the center of the Universe, while the Moon, Sun and planets orbit in concentric circles around Earth. Although now we know that this model is wrong, what the ancient Greeks did was still highly commendable. They tried to make sense of the periodic motion of planets and Sun along the ecliptic with a physical model. The act of doing so is aligned with modern scientific thinking and processes.

It should also be noted that not all Greek philosophers believed in the geocentric model. The Pythagorean school of thought believed that the Earth, the planets and the Sun are orbiting around a "central fire". Aristarchus (310-230 BC) proposed the first heliocentric model where the Earth orbits the Sun. These models were generally sidelined in favour for the more influential Aristotlean school of thought.

However as no stellar parallax had been observed in ancient times, it was logical to conclude that Earth is static.

There were unsurprisingly some elements of mysticism in the early models. It was believed that circles and spheres are geometrically perfect, and thus the motion of heavenly bodies must be in perfect circles.

While the geocentric model generally accounts for the observed motions of celestial bodies, it could not explain an anomalous behaviour of planetary movement known as retrograde motion. Planets generally drift from east to west along the ecliptic over the year. But sometimes they appear to move backwards for weeks or months, before continuing back in the usual direction.

Claudius Ptolemy (AD 100-170), Roman/Greek/Egyptian mathematician and geographer made an ingenious twist to the geocentric model to allow the model to fit observational data better. To the original circular orbit of a planet which he called the deferent, he added a smaller circle called the epicycle. The center of the epicycle is on the (circumference of the) deferent, and the planet is on the (circumference of the) epicycle.

While the epicycle moves clockwise along the deferent, the planet moves clockwise along the epicycle. When the (linear) direction of motion in the epicycle is opposite to that in the deferent, retrograde motion occurs.

This circle within circle model allowed Ptolemy to have another set of independent parameters: radius and period associated with the epicycle. In more modern terms, Ptolemy decomposed a complicated periodic motion (derived from astronomical observations) into two simple circular periodic motion. Does this sound like something you have learned before?

Speaking of sound, to visualise with our ears Ptolemy's trick, run the following in Wolfram|Alpha .

play sin(880 pi t) for 10 s

play sin(880 pi t)+0.1{*}sin(440 pi t) for 10 s

The first sound \sin(880\pi t) you hear is analogous to the original geocentric model with 1 circle. The second sound \sin(880\pi t)+0.1\sin(440\pi t) is analogous to Ptolemy's geocentric model with an epicycle. Hear for yourself to note the difference.

Nicolaus Copernicus (1473-1543) published On the Revolutions of the Heavenly Spheres in 1543, where he re-proposed the heliocentric (Sun-centered). Earth was treated as just another planet doing circular orbit around the Sun. He arranged the planets in order of distance from the Sun using the period of orbit he calculated. Earth was the third rock from the Sun, after Mecury and Venus, before Mars, Jupiter and Saturn. He also used geometry and observational data to estimate the relative distance from the Sun for different planets.

The elegance of the heliocentric model comes from its ability to demonstrate retrograde motion of planets in a simple way. Retrograde motion of a planet can be observed around the time when it is at it's closest distance to Earth.

Unfortunately, despite the elegance, Copernicus' model cannot account for the observational data very well when it comes down to numbers. To match the numerical data, he had to add epicycles to the heliocentric orbits! This is one of the main reasons why many were skeptical of Corpernicus' heliocentric model..

Tycho Brahe (1546-1601) was the most respected astronomer of his time. He took pride to make systematic and careful astronomical observations. One of his early works was the sighting of a supernova in the constellation of Cassiopeia in 1572. Tycho Brahe used this as evidence to challenge the Aristotelian view that the heavens is unchanging,

He was offered an island by the Danish king, on which he built an observatory and alchemy research centre. There, he constructed several custom built (non-telescope) astronomical instruments capable of measuring positions of celestial objects to 40'' (arcseconds!), which far surpassed his contemporaries. The bulk of his observational data was compiled compiled by Johannes Kepler in the Rudolphine Tables published in 1627, several years after his death.

Brahe knew about Copernicus' heliocentric model and appreciated its elegance but could not accept a moving Earth. He could not measure stellar parallax, even with his accurate instruments. He maintained a static Earth centered model where the Moon and Sun orbit the Earth while the other planets orbit the Sun.

Tycho fell out of favour with the new Danish king and moved to Prague to begin work as Imperial Mathematician.

In 1600, Prague, Brahe employed Johannes Kepler, a German mathematician, to work on mathematical modelling to fit his observational data of planetary motion. One year later, Brahe passed away and Kepler was appointed as Brahe's successor and continued to work on the project. Instead of using Brahe's geo-heliocentric model, Kepler adopted a modified version of Copernicus' heliocentric model. Kepler found that the data will fit very well in a heliocentric model if the planetary orbit follows an elliptical path instead of a circular one.

In 1609, he published two laws of planetary motion and in 1619, he published the third law. The laws are:

1. Law of ellipses: The orbit of each planet is an ellipse, with the Sun located at one of its foci.
2. Law of equal areas: A line drawn between the Sun and the planet sweeps out equal areas in equal times as the planet orbits the Sun.
3. Harmonic Law: The square of the time taken for a planet to complete one revolution about the Sun (relative to the stars) is directly proportional to the cube of the semimajor axis of the planet's orbit.

Mathematically, the equation of an ellipse in polar coordinates (r,\theta) is given by

r=\frac{a(1-\epsilon^{2})}{1+\epsilon\cos\theta}

where a is the semi-major axis which is the longest line from the center of the ellipse to its circumference, and \epsilon is the eccentricity, which characterises how elongated the ellipse is.

The second law can be expressed mathematically by

\frac{dA}{dt}=\text{constant}

It gives information of how fast the planet is moving at different parts of the orbit.

It was later known that the second law is equivalent to the law of conservation of angular momentum. Interested reader may refer to the appendix for details.

The third law relates the sizes and periods of orbits for different planets.

\left(\frac{T_{1}}{T_{2}}\right)^{2}\propto\left(\frac{a_{1}}{a_{2}}\right)^{3}

It is interesting to note that Kepler published the first and second law in 1609 using only data from Mars. The third law was published much later in 1619 after a long attempt to find patterns that could represent ''universal music'' , an old philosophical concept that movements of celestial bodies follow some beautiful proportions, like the proportions that give rise to musical notes.