**The Earth - Sun Distance**

**( Or, how far are the planets from the Sun? )**

**"**My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble mind**"** - Albert Einstein ( 1879 - 1955 )

- Tycho Brache's accumulated historic planetary data, especially that of Mars;
- The mathematics of Parallax utilized by Giovanni Cassini in 1672 at the Royal Observatory, Paris;
- Kepler's Equations of Planetary Motion.

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

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:

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

Relative Distances derived from Tycho's Data and Kepler's 3rd Law | |||
---|---|---|---|

Planet | Period | Mean Relative Distance from Sun | |

Sun | --- | 0.000 AU, by definition | |

Mercury | 0.241 | 0.387 AU | |

Venus | 0.616 | 0.723 AU | |

Earth | 1.000 | 1.000 AU, by definition | |

Mars | 1.880 | 1.524 AU | |

Jupiter | 11.900 | 5.204 AU | |

Saturn | 29.500 | 9.539 AU | |

Uranus | 84.000 | 19.191 AU | |

Neptune | 165.0 | 30.071 AU | |

Pluto | 248.0 | 39.457 AU |

*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 *universal 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 Tahiti voyage to view the June 3, 1769 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 - the true value is about 390 times further away!