How the Planckhead reckoned a planet's density

One time a planckhead and a meterhead were in low orbit over the surface of a planet they had never seen before. They were riding in a Mini Minor which had been especially modified for space travel. The planckhead was in the front seat where he could get a good look at the ground and the meterhead was in the back seat reading a comic book.

After they had been cruising the planet for almost two hours (40pi minutes to be exact) the planckhead spotted a drive-in theater that he had seen when they first entered orbit. It was right next to a miniature golf lot and it was playing Boondock Saints, a hard-drinking Irish-American movie in which the brothers kill gangsters.

"Dig this, we have just now completed a circuit of the entire orb," said the planckhead glancing at the Mini's dashboard clock, "it took 40pi minutes."

"In that case," said the meterhead, turning a page of his comic book, "it takes 20 minutes to go the length of the planet's radius."

The planckhead, still staring avidly out of the window on the driver's side, squared 20 minutes, getting 400, and multiplied by 4pi/3 with a result that came to something like 1600. Multiplying by 4pi/3 is rather like multiplying by 4, and in fact the answer he got was 1676.

"Hey note this, man, I've found the density of this whole hard-drinking Irish-American planet. You will not believe this, man, a talent of this planet averages 1676 cubic pinkies in volume."

***footnote****

Don't try this in metric unless you actually ENJOY messy arithmetic. You still have to square the radian time in low orbit and you multiply by the same 4pi/3 but then you still have to multiply by the hard-drinking Irish-American number 6.673×10-11.

But it works easily in Talent-Mile units with any planet including Earth, which is a 15 minute rather than a 20 minute planet. In the lowest earth orbits it takes 15 (tech) minutes to go the length of the earth's radius over the ground. So square 15 and multiply by 4pi/3 and you get nearly 1000. And indeed the average density of the earth, with all its compact rock and iron, is such that a talent occupies 1000 cubic pinks—which, and consider this well Sahib, is one natural gallon.