When the sun and its planets had just been made the people and animals were disappointed by one detail. The contractors had left a black hole where Jupiter should have been, as a place-holder. The Jovian moons, which some animals with their sharp eyes could see, were going around the black hole (which had the same mass as the planet that should have been there) instead of a brilliant visible thing.
So the people and animals made a great outcry unto the Lord Apollo Maker of the Universe and called on him to get himself over to the solar system and correct the oversight.
There was a glorious arpeggio of plucked lyre strings and Apollo appeared. He examined the contract and agreed there had been an oversight. Instead of a black hole, he had specified a gas giant planet with a mass of 1026 talents. Unfortunately the contractors had already been paid off and had vanished back into the untold from whence they came—it was too late to get them to fix the problem.
However, said the god, there was a band of gypsies traveling through that part of the galaxy with a bag of planets of various sizes, which nobody knew how they came by, and which they were willing to trade for black holes. Being compact and indistinguishable from universe to universe, black holes are used as currency in many places gypsies travel.
Whether or not the gypsies could help you, said the god, depended on whether the gypsies happened to have a planet of exactly the same mass as your black hole. If so, then it was legal for them to swap. The Jovian moons could continue orbiting the swapped-in planet just as they had been orbiting the swapped-out black hole. The moons wouldn't know the difference and everything would be all right.
Among the animals there was a parrot who knew how to deduce the mass of a central body from things in orbit. The parrot was suspicious of contractors and wanted to check to see what the mass of the black hole actually was. The parrot observed that one of the moons was in million mile radius circular orbit (around the invisible thing at the center). He and a falcon with very sharp eyesight had been watching that moon and they had determined that its radian arctime (full circuit time divided by 2 pi) was 3160 minutes. The parrot observed that the orbit radius was 106 miles, and he did a little arithmetic, cubing the radius (1018) and squaring the time (107). Finally he divided the cubed radius by the squared time and said that the attractiveness of the hole must indeed be 1011 miles3/minute2.
Because of the G ratio, which was universal, each unit of attractiveness is worth a quadrillion talents, which is to say 1015. By multiplying 1011 and 1015, the parrot and the falcon could tell that the mass of the black hole must indeed be 1026 talents. So the parrot told the other animals that if the gypsies had a planet with the very same mass they could swap it in and everything would be fine.
Apollo felt responsible for the error so he made it his business to consult the gypsies. He found them sitting around the campfire with their bag of planets telling metric-system stories. They laughed scornfully at the god and said they didn't rate planets by inertia they rated them by the size of the black hole you could swap them for. Show us the diameter of the black hole, they said, and we'll see if we have a planet to swap for it. Apollo was not versed in these matters (he considered himself a musician and only created universes as a sideline) but fortunately he had brought the parrot along.
Just divide the attractivess by the square of the speed of light, said the parrot, and that will give the halfradius. 1011 mile3/minute2 divided by 1014 mile2/minute2 (which the parrot had learned from Ole Roemer was the square of the speed of light) is 10-3 or one thousandth of a mile. It is the distance of one pace. The halfradius of the black hole we want to swap is just one pace, said the parrot, and the hole's diameter is four paces. Ask them, it urged impatiently, if they have a planet to match! It turned out that the gypsies did.
The next evening the people and animals were waiting to see and they watched Jupiter rise in the heavens, very bright, with its little moons twinkling around it which some of the animals could see with their sharp eyes.
note: G=1.00 mile3/minute2 per quadrillion
talents.
1/G=1.00 quadrillion talents per mile3/minute2.
The speed of light is 107 mile/minute, so its square is
1014 mile2/minute2.
Copyright © 2002 Leonard Cottrell. All rights reserved.
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