The Attractiveness of Laotzu's Dream Planet

Laotzu dreamed he was drifting unhurriedly a few feet above the surface of a grassy planet. He was in orbit and skimspeed on this planet was slow as sleepwalk — ten paces a minute. Dream-air, which doesn't interfere with motion, let Laotzu smell the heat in the grass.

After a little over two hours he began recognizing places and realized that he had made a circuit of the planet. He looked at his watch, which told time in natural minutes (ten percent shorter than ordinary Earth minutes). It had taken 40 pi minutes to come full circle around this planet, so it must take twenty minutes to go the length of the planet's radius. This 20 minute radian arctime or "skimtime" is typical of planets with about the same density as the Earth's moon. All planets with the same density have the same skimtime.

Laotzu thought the planet radius must be 200 paces (two city blocks) because that's how far you go in 20 minutes when you travel at ten paces a minute.

The cube of a planet's radius over the square of its skimtime is an "attractiveness" quantity that summarizes a lot of facts concerning orbits of all sizes and shapes around the planet. In this case the radius cubed is 8 million cubic paces and the skimtime squared is 400 square minutes. So for this planet the attractiveness is 8 million pace3/400 minute2, or in other words 20 thousand pace3/minute2.

The Emperor's finger and trice units used in the Forbidden City are compatible with paces and minutes in the curious sense that the attractiveness units are identical: 1 pace3/minute2 is exactly the same as 1 finger3/trice2

Attractiveness, not mass, of central bodies is what NASA uses for navigation and so you can find the attractiveness of the sun listed at this NASA website of astrophysical constants In old metric units it is:

1.32712440018 × 1020old meter3/old second2.

Laotzu just happens to use paces (of a hundred fingers) and minutes (of a thousand trice) instead of NASA's meters and seconds. When Laotzu told Confucius his dream, the other sage asked him all sorts of questions about orbit speeds and sizes around the planet and Laotzu was always able to answer from the planet's attractivness. To find the speed of a circular orbit at any distance from center, just divide the attractiveness by the distance—it gives the speed squared.

Confucius asked what the circular orbit speed would be 800 paces from center. Laotzu divided 20 thousand pace3/minute2 by 800 paces and got 25 pace2/minute2. That is the square of the orbit speed so the speed would have to be 5 pace/minute.

Confucius asked what the speed would be 5 miles from center, which is 5000 paces. Laotzu divided 20 thousand pace3/minute2 by 5000 paces and got 4 pace2/minute2, which meant that the speed would have to be 2 pace/minute.

The attractiveness quantity is a single quantity that works for all distances from the planet. Johannes Kepler discovered the attractiveness invariant for the sun. He found a cubic distance over square time quantity that was the same no matter which of the planets he used to calculate it. With a little care, the quantity works with non-circular orbits as well.

The Emperor was amused to hear the two sages discussing a planet which existed only in Laotzu's dream and he asked about the acceleration of gravity at the planet's surface. Laotzu did not happen to know what it was but he knew that to calculate the falling acceleration at ANY distance all he needed to do was divide the attractiveness by the square of the distance. The surface is 200 paces from center, which squared is 40 thousand square paces. So the sage divided 20 thousand pace3/minute2 by 40 thousand pace2 and got half a pace/minute2. His dream planet had the very gentle gravity of half a pace per minute per minute.

*

Copyright © 1999, 2002 by Leonard Cottrell. All rights reserved.
New Metric Fables: Table of Contents

Some units issues are still unresolved. ORIGINALLY THE STORY WAS TOLD exclusively in ton-pace-minute units like this:

Laotzu dreamed he was drifting unhurriedly a few feet above the surface of a grassy planet. He was in orbit and skimspeed on this planet was slow as sleepwalk — ten paces a minute. Dream-air, which doesn't interfere with motion, let Laotzu smell the heat in the grass.

After a little over two hours he began recognizing places and realized that he had made a circuit of the planet. He looked at his watch, which told time in natural minutes (ten percent shorter than ordinary Earth minutes). It had taken 40 pi minutes to come full circle around this planet, so it must take twenty minutes to go the length of the planet's radius. This 20 minute radian arctime or "skimtime" is typical of planets with about the same density as the Earth's moon. All planets with the same density have the same skimtime.

Laotzu thought the planet radius must be 200 paces (two city blocks) because that's how far you go in 20 minutes when you travel at ten paces a minute.

The cube of a planet's radius over the square of its skimtime is an "attractiveness" quantity that summarizes a lot of facts concerning orbits of all sizes and shapes around the planet. In this case the radius cubed is 8 million cubic paces and the skimtime squared is 400 square minutes. So for this planet the attractiveness is 8 million pace3/400 minute2, or in other words 20 thousand pace3/minute2.

Hopefully the reader is familiar with this type of quantity. The NASA website on astrophysical constants lists it for the sun, in old metric units:

1.32712440018 × 1020old meter3/old second2.

Laotzu just happens to use paces and minutes instead of NASA's meters and seconds. When Laotzu told Confucius his dream, the other sage asked him all sorts of questions about orbit speeds and sizes around the planet and Laotzu was always able to answer from the planet's attractivness. To find the speed of a circular orbit at any distance from center, just divide the attractiveness by the distance—it gives the speed squared.

Confucius asked what the circular orbit speed would be 800 paces from center. Laotzu divided 20 thousand pace3/minute2 by 800 paces and got 25 pace2/minute2. That is the square of the orbit speed so the speed would have to be 5 pace/minute.

Confucius asked what the speed would be 5 miles from center, which is 5000 paces. Laotzu divided 20 thousand pace3/minute2 by 5000 paces and got 4 pace2/minute2, which meant that the speed would have to be 2 pace/minute.

The attractiveness quantity is a single quantity that works for all distances from the planet. Johannes Kepler discovered the attractiveness invariant for the sun. He found a cubic distance over square time quantity that was the same no matter which of the planets he used to calculate it. With a little care, the quantity works with non-circular orbits as well.

The Emperor was amused to hear the two sages discussing a planet which existed only in Laotzu's dream and he asked about the acceleration of gravity at the planet's surface. Laotzu did not happen to know what it was but he knew that to calculate the falling acceleration at ANY distance all he needed to do was divide the attractiveness by the square of the distance. The surface is 200 paces from center, which squared is 40 thousand square paces. So the sage divided 20 thousand pace3/minute2 by 40 thousand pace2 and got half a pace/minute2. His dream planet had the very gentle gravity of half a pace per minute per minute.

*

Copyright © 1999, 2002 by Leonard Cottrell. All rights reserved.
New Metric Fables: Table of Contents