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 pace^{3}/400
minute^{2}, or in other words 20 thousand
pace^{3}/minute^{2}.

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 pace^{3}/minute^{2} is exactly the same
as 1 finger^{3}/trice^{2}

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 × 10^{20}old meter^{3}/old
second^{2}.

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 pace^{3}/minute^{2} by 800
paces and got 25 pace^{2}/minute^{2}. 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 pace^{3}/minute^{2} by 5000
paces and got 4 pace^{2}/minute^{2}, 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 pace^{3}/minute^{2} by 40 thousand
pace^{2} and got half a pace/minute^{2}. 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 pace^{3}/400
minute^{2}, or in other words 20 thousand
pace^{3}/minute^{2}.

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 × 10^{20}old meter^{3}/old
second^{2}.

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 pace^{3}/minute^{2} by 800
paces and got 25 pace^{2}/minute^{2}. 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 pace^{3}/minute^{2} by 5000
paces and got 4 pace^{2}/minute^{2}, 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 pace^{3}/minute^{2} by 40 thousand
pace^{2} and got half a pace/minute^{2}. 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