Show me the sky!

Picture yourself being invited to an exquisite dinner when suddenly your right-hand neighbor turns to you with genuine curiosity and asks:

"By the way, Charles [if you are called Charles], do you happen to know the minimum distance between us and Jupiter?"

Your heart skips a beat as you realize you don't have an encyclopedia - or mobile phone - at hand! But fear not, for I am here to rescue you with a captivating mathematical trick that will effortlessly impress your conversation partner.

In the late 18th century, two brilliant German scientists, Johann Daniel Titius and Johann Elert Bode, stumbled upon a fascinating observation: the average distance of planets from the Sun - expressed in millions of kilometers - seemed to follow a mathematical pattern. Allow me to unveil this pattern to you:

d = 60 + 11.25 x 2ⁿ

Here, the variable "n" represents the rank of the planet in relation to the Sun. For instance, let's consider Venus, the second planet of the Solar system. Applying the formula, we find its distance to the Sun to be 60 + 11.25 x 2^2 = 105 million kilometers. This remarkable relationship worked astonishingly well for the five known planets at that time, although it required a few adjustments. For Mercury, "n" should be set to -∞ (rather than 1!). Interestingly, when the law was formulated, the fifth rank was vacant. However, the subsequent discovery of Ceres, an asteroid situated between Mars (rank 4) and Jupiter (rank 6), supported the credibility of this law.

In the table above, D1 represents the average measured distance (in million kilometers) of each planet from the Sun. D2, on the other hand, illustrates the corresponding distance calculated using the Titius-Bode law. To calculate the distance between Earth and a specific planet, you can use the formula |D2 - 150|, where 150 is the distance (in million kilometers) between Earth and the Sun.

We observe an excellent correlation between the measured average distances (D1) and the calculated distances (D2), except for Neptune. This law, formulated over two centuries ago, is no longer considered an empirical law but rather an astronomical curiosity.

Therefore, armed with the knowledge mentioned above, you can confidently respond to your dinner table neighbor that the minimum distance between Jupiter and Earth is approximately | 780 - 150 | = 630 million kilometers!