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The Earth - Sun Distance

Or, how far are the planets from the Sun?

To solve this problem, it was necessary in the History of Science for the following to have occurred:

Tycho Brahe

Johannes Kepler

Tycho Brahe (1546 - 1601) who was born in Skane, Denmark [ now in Sweden ] left his homeland with his books, planetary data, and instruments after falling out with King Christian IV and settled in Prague in 1599 as the Imperial Mathematician in the court of Emperor Rudolph II. In Prague in 1600 until his death in 1601 he hired Johannes Kepler (1571-1630), Austrian mathematician and astronomer, as an assistant to continue the calculation of the planetary orbits. Kepler knew that Tycho's unprecedented data were accurate to within 1-2 arcminutes and in fact was never off as much as 8 arcminutes. In fact Tycho is credited with having established the most accurate astronomical data of his time! Kepler published the First and Second of his three laws of planetary orbits in his Astronomia Nova (1609) and his Third Law in Harmonice Mundi (1618). Kepler's Laws describe how the planets move, not why. Therefore, Kepler's Laws are empirical, not physical laws of planetary science. It took the mathematical genius of Isaac Newton (1643-1727) in Philosophiae Naturalis Principia Mathematica commonly known as the Principia (1687) to correctly describe in differential and integral mathematics the correct physical explanation of gravity and hence of planetary orbits. 

Now, Kepler knew from his calculations of Tycho's data that Mars has an orbital period of 1.88 Earth-Years and from Kepler's 3rd Law ( Harmony Law ), we can derive Mars's relative distance from the Sun:

That is, Mars is 1.524 times the distance from the Sun as is Earth's distance from the Sun.

The other "relative distances" of the planets which can be derived from Tycho's data and Kepler's 3rd Law are:

However, the actual absolute value of 1.0 AU, the Earth-Sun distance, at this point is still unknown!

This answer was determined by Giovanni Cassini in 1672 by using the mathematics of Parallax and came within 93.3% accuracy of the modern accepted AU value. A 7% error in 1672!

Cassini first determined the Parallax distance of Earth-Mars where

Hence, the Earth-Sun distance becomes:

The consequence of this determination of 1.0 AU in 1672 will be the first approximation of the constancy of the speed of light in 1676 by Danish astronomer Olaf Roëmer, assistant to Cassini at the Royal Observatory, Paris!! 

The historical record in determining AU:

Aristarchus of Samos ( ca. 310 BC - 230 BC ) ≈ sun's distance is 19 times further away than the moon or about 7,300,000 km ( 4.5 million miles )^∗

Tycho Brahe ≈ 8,000,000 km ( 5 million miles )

Johannes Kepler ≈ 24,000,000 km ( 15 million miles )

Giovanni Cassini ≈ 140,000,000 km ( 87 million miles )

Captain James Cook 1769 Tahiti voyage to view the Venus Transit ≈ 153,000,000 km ( 95 million miles )

modern accepted ≈ 150,000,000 km ( 93 million miles )

^∗Aristarchus used correct Euclid geometry but his instruments for observation were inadequate - true value is about 390 times further away

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