There is a set of units built into nature—based on proportions intrinsic to light and gravity—called the natural units or Planck units. Human-scale versions of the natural units are fun to play around with and convenient to use for some purposes.
In the version described here, natural units have been scaled by powers of ten to give them handy sizes. The natural units of time, length, and mass are things you can find on the web—e.g. just google with "fundamental constants" and select "universal". You will get a short list of universal fundamental physical constants with the accepted values of the Planck units, as well as things like c, G, h-bar, on which the natural units themselves are based.
It just happens that when you scale the mite-sized natural mass unit up by a billion, to make it human-size, you get a mass which is (within a fraction of a percent) equal to 48 conventional pounds, about what a classical talent weighed. The natural length unit, scaled up by a power of ten, is (within half a percent of) an ordinary mile. The time unit, likewise scaled by a power of ten to give it a familiar size, comes out to be (within two tenths of a percent) 1/1600 of a day, or nine tenths of a conventional minute. This is just how things turn out, and it indicates the real, but not impossible, hurdles involved in getting used to human-scale natural units. It means getting accustomed to using a reduced minute in certain technical situations—for definiteness the minute in these stories will be 54 seconds, ten percent less than the conventional one—and to describing masses and lengths with convenient fractions of the talent and mile.
Together with the minute, the talent and mile specify a unit of force which is somewhat more than the weight of a kilogram in normal sealevel gravity—more precisely about 2.7 pounds. There is a traditional unit of this size used in parts of eastern Europe and the mediterranean which is called the ocque or oke—the spelling here is simplified to oc. An oc is the force needed to accelerate talent mass by one mile a minute per minute. The ocmile unit of energy, about 5 food Calories worth, is that delivered by a mile-long oc push. These are essentially Planck units of force and energy scaled by powers of ten to make them a practical human size.
Technically the Talent Mile units used in the stories are defined by specifying exact round values for the main physical constants:
the speed of light, c = E7 miles per minute (E7 stands for 107, ten million.)
Planck's hbar = E-40 ocmile minute
Boltzmann's k = E-25 ocmile per grade.
It turns out that with these definitions the gravitation constant G takes an approximately power-of-ten value.
G = 1.00E-15 cubicmile/squareminute per talent.
Reform is not urged here. You are urged to continue using ordinary pounds and miles, or metric units if you prefer, and the customary hours and minutes of the clockface. But for certain kinds of physics calculations it can be fun to switch over to "talent mile units"— units connected to nature and to the proportions built into light and gravity, in terms of which the most important natural constants take on power-of-ten values.
Three units (time, length, mass) are defined and then used to specify others, particularly some smaller subdivisions.
This makes the mile come out to be 1618.88 meters (a fraction of a percent longer than the conventional one) and the talent come out 21.73 kilograms. The pace, one thousandth of a mile, is used as a convenient subdivision. If we need to describe volume, one thousandth of a cubic pace is gallon-sized and will be called so.
For some peculiar reason the stories seem to require a name for the speed which is one hundredth of the unit speed in our system. One hundredth of a mile per minute will be called a cent of speed. One cent of speed is 2/3 mph, or ten paces in a minute. Another usage peculiar to the stories is to call the 1000 oc (2700 pound) force a ton. I have several times tried to get rid of these terms and simply describe speed in miles per minute and force in ocs, but so far have not found a graceful way to do it.
I also divide the talent mass into 1000 ounces. In one story there is a 100 ounce rabbit and a 10 ounce frog. This ounce is roughly 3/4 of a conventional avoirdupois ounce. It seems to work fairly well and so far I have not needed any other subdivision of the talent. Within the walls of the Forbidden City the Emperor has established the mass of a white Samoyed dog as the standard. The mass of the sun is calculatated for the Emperor as a multiple of the mass of this dog. Of course the dog's mass is one talent. Perhaps Talent should be the dog's name, but this is not mentioned in the story.
The width of the Emperor's little finger at the first knuckle is exactly one hundredth of a pace, or one hundredthousandth of a mile. People in the Forbitdden City believe that this small distance was, in ancient times, called the bone. In the fables the mile is so divided.
The plain Talent Mile units do not have all this extra terminology—the units are simply based on minute, mile, talent, and the oc force that arises from them. The unit of power is that which supplies unit energy (ocmile) in unit time (minute). One ocmile per minute works out to be about 360 conventional watts or half a horsepower: a good nickname for it would be a pony.
Again for storytelling reasons, particularly where some Chinese sages were involved, I needed a name for a tenth of a pony and I called that amount of power one chi. A chi of power, about 36 conventional watts, is exemplified by raising a ten oc weight at cent speed—ten ocs at one hundredth of a mile a minute. Fortunately these extra complications occur rarely, in only one or two of the fables.
In classical Chinese the word chi, spoken on fourth tone, means dynamic principle or vital breath. The written character is radical #84 "breath" with the symbol for a living plant inside it.
The gravity constant G is, among many other things, a relation linking distance and square speed to mass. With two-decimal-place accuracy, here is how it looks in the metric system:
G = 6.67 ×10-11 meter3/second2 per kilogram.
This is too cumbersome for the fables. Their lightness depends on the arithmetic doing itself. Using the corresponding format in our units the same thing is:
G = 1.00 ×10-15
mile3/minute2 per talent,
and calling 1015 a quadrillion lets us rephrase that:
G = 1.00 mile3/minute2 per quadrillion
talents.
Looking at G in this form tells us about satellites circling a body with quadrillion talent mass—the orbit radius multiplied by the square of the speed is 1.00 mile3/minute2. So if the speed is a tenth of a mile per minute (and the square is therefore 1/100 square speed units) the radius must be 100 miles. Conversely if some character in a story happens to visit a small planet where the radius mulitiplied by the squared speed of ground-level orbit is 1.00 mile3/minute2 then he or she knows that the planet's mass is a quadrillion talents.
A bit more algebraic tinkering, using the fact that oc force equals talent mile/minute2 (unit mass times unit acceleration) gets us:
G = 1.00 mile4/minute4 per quadrillion
ocs, and using the ton to stand for a thousand oc force:
G = 1.00 mile4/minute4 per trillion tons.
In this format, G relates the fourth power of circular orbit speed at some distance from a body to what a copy of that body would weigh placed at the same distance. If someone in a story notices a small satellite circling the planet at mile-a-minute speed then he or she can immediately tell that what the planet would weigh in its own gravity at the same distance—a trillion tons. A planet's weight in its own gravity is proportional the the fourth power of circular orbit speed. So if orbit speed is a hundredth of a mile a minute—a cent in other words—the weight force must be (1/100)4 of 1012 tons, which is 1012-8 or tenthousand tons. Each fourthpower or quartic cent corresponds to tenthousand tons of weight:
G=1.00 cent4 per tenthousand tons.
Putting G in this format makes it seem highly outlandish compared with the standard metric version that many people are familiar with (and which was included at the top of the paragraph.) But it is the same natural constant, cast in a different role—the ratio of quartic speed to force. One squares the speed twice in succession to get the fourth power—the "quartic" as opposed to "cubic" speed—and each quartic cent of speed is worth ten thousand tons. So if the orbit speed is 3 cents, the weight is 81 times tenthousand tons.
The oc and ton units of force are essentially power-of-ten fraction of the universe's natural unit of force FPlanck. Among other things this is the central constant in the 1916 Relativity equation describing gravity as a proportion between energy density and curvature. So a decimal cousin of our units is basic to how gravity works.
FP = 1.00 × 1043 oc = 1.00 × 1040 ton.
There is a curious analogy to the preceeding, where we squared an orbit speed twice and got a force. If we treat the speed of light the same way we get Planck force. Recalling that G makes each quartic mile-a-minute worth 1012 (a trillion) tons and that the speed of light is 107 miles a minute, raise that to the fourth (1028), multiply by 1012 and in fact you do get 1040 tons, the universe's natural unit.
The universe has a natural unit of power—that delivered by Planck force pushing at the speed of light—and our pony unit is essentially a decimal fraction of that natural power. As was just mentioned, the force is 1043 oc and the speed is 107 mile a minute—putting those together gives us 1050 ocmile a minute, which is 1050 ponies. If you prefer, think of it as 360E50 watts, either way it is a large power. Since the universe has a built-in connection between power and (squared) frequency, this natural power gives us a handle on the intrinsic frequency unit in nature.
Planck's hbar appears in so many roles that instead of trying to suggest their variety I will just describe one that is at the same time familiar and never mentioned in conventional physics texts.
Your two hands palms up make a roughly square platform with an area that is about a hundredth of a square pace. In direct vertical sunlight such an area receives one chi of light-power. The figure of a chi (36 watts) is for vertical sunlight on a hundredth of a square pace at this distance from the sun. Chi, I should remind you, is an idiosyncratic extra unit I keep around for storytelling purposes—the primary power unit in Talent-Mile system is equivalent to 360 watts or half a conventional horsepower.
On a square pace the power of direct sunlight is this, or in other words 10 ponies. We can relate this power to how rapidly quanta of sunlight arrive (1024 per minute) and the average rate they vibrate (1017 per minute). In fact if we multiply those two rates togther we get a square frequency (1041 per minute per minute) that is proportional to the power.
Among other things hbar is the ratio connecting square frequency to power and one equivalent way to write it is:
h-bar = 10-40 pony minute2
The upshot is that each square frequency unit is worth 10-40 pony. So if an area is getting a rush of vibration which is 1041 per square minute, then the power that rush represents is 1041-40=10 ponies.
Planck frequency, 1045 per minute, is connected to the natural unit of power in the fashion just described for sunlight. The square of the natural frequency unit, 1090 per square mintue, describes a rush of vibration with a power equal to 1090-40=1050 ponies, the natural unit.
This preface is already too long, so let's move on to the fables.
Copyright 2002 Leonard Cottrell. All rights reserved.
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