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Nicholas Copernicus (1473-1543)

(Latinized of Polish Mikolai Kopernik). He is credited with introducing the modern heliocentric model. We do not know what was his reasoning. Perhaps, he just wanted to light the world from the center!

Limitations:

Assumed that planets go on circles around the Sun. In order to fit observations, he had to include equants and epicycles.
Parallax was a sticking point. Copernicus had to move stars further away.

Big success:

Natural explanation for the retrograde motion of planets.
Predictive power: a new planet should rotate slower than existing ones.

Tycho Brahe (1546-1601)

He was a famous scientists and a heavy drinker (from which he finally died), which means that the two are not mutually exclusive. He

repeated his measurements, i.e. made sure that his results are reproducible.
estimated the errors of his measurements; no one did it before him!

Remember, Aristarchus calculated that

$\begin{displaymath}R_\oplus \approx 3 \times R_{\rm m}. \end{displaymath}$

Tycho Brahe would write this as

$\begin{displaymath}R_\oplus = (3 \pm 1) \times R_{\rm m}. \end{displaymath}$

A modern value

$\begin{displaymath}R_\oplus = (3.672 \pm 0.001) \times R_{\rm m}. \end{displaymath}$

Brahe was convinced that

Ptolemy model was wrong, since it did not fit observations.
Copernicus model was wrong, since he could not detect the parallax.

If parallax is not observed, it could only be because of two reasons:

stars are too far away.
the Earth does not move.

Brahe believed the stars were near because he detected their size (This however was an optical illusion.). Thus, he concluded that the Earth does not move. This is a perfect example of correct scientific reasoning that leads to an incorrect conclusion.

However, Brahe is not considered a great scientist because he made such a great mistake!

What Brahe did:

$\bullet$ In 1572 Brahe observed a ``new'' star, i.e. he saw a star where no star was before. Now we know that it was a supernova. This proved that the heavens were not immutable, eternally unchangeable. This was like a revelation to many people!

$\bullet$ He also proved that comets flew beyond the orbit of Mars. They thus flew through the celestial spheres, which thus cannot be solid and impenetrable (as Aristotle believed). That meant that the Earthly and celestial realms were not distinct, but might obey the same laws and be made of the same substances.

$\bullet$ He quarreled with the king of Denmark, moved to Prague, and there took an assistant named Johannes Kepler. About his is the next page of our story.

Johannes Kepler (1571-1630)

After Tycho died at one of the royal banquets (apparently from overdrinking), Kepler inherited his huge collection of observational data.

He was faced with a serious problem: none of then existing models could fit the observational data to within Tycho's stated errors. He believed that Tycho calculated his errors correctly, so he embarked on developing a world model that was in agreement with observations.

And then he had an inspiration!.. Not a circle but an ellipse. A single ellipse with the Sun in its focus was able to fit all the data, instead of hundreds of epicycles in the Ptolemaic model.

Kepler formulated three laws of planetary motion that are still called Kepler laws:

Planets orbit the Sun in an ellipse, with the Sun at one focus.
The line from the Sun to the planet sweeps out an equal are in an equal time. Thus, planets move faster when they are nearer the Sun.
The square of the period of the orbit is equal to the cube of the semimajor axis of the ellipse.

Mathematically, this means:

P² = R³.

Very important note: This is only true if the period P is measured in years, and the semimajor axis R is measured in astronomical units. One astronomical unit is equal to the length of the semimajor axis of the Earth orbit.

If we measure P in, say, days, P=365.25 days, and we measure R in miles, R=93 million miles, we can calculate:

$\begin{displaymath}P^2 = 365.25^2 = 133,000 = 1.33\times10^5, \end{displaymath}$

and

$\begin{displaymath}R^3 = 93,000,000^3 = 8.04\times10^{23}. \end{displaymath}$

They are not equal at all!!!

Scientists would rather write the Kepler's third law as:

$\begin{displaymath}\left(P\over 1{\rm\,year}\right)^2 = \left(R\over 1{\rm\,AU}\right)^3. \end{displaymath}$

Let's check: if P=365.25 days and R=93 million miles, then we get

$\begin{displaymath}\left(365.25{\rm\,days}\over 1{\rm\,year}\right)^2 = 1^2 = 1. \end{displaymath}$

and

$\begin{displaymath}\left(93{\rm\,million~miles}\over 1{\rm\,AU}\right)^3 = 1^3 = 1. \end{displaymath}$

And, finally,

1 = 1.

It works!