Copyright © 2001 by Leonard Cottrell, www.planck.com. All rights reserved.
This essay takes a look at the Planck quantities by constructing a model in which the quantities are scaled by powers of 105 to make them less extreme, and therefore easier to visualize, assimilate, and remember. Along the way we can truncate experimental uncertainty so as to get versions with exact metric equivalents which are equal to the natural quantities to within the accuracy of current measurement.
It's possible that a number of users of the world wide web already have some familiarity with measures such as the Planck mile, Planck gallon, Planck ounce (mass), Planck ton (force), and Planck minute. These are traditional-sized measures which happen to be power-of-ten multiples of Planck units. On the one hand, except for powers of ten, they arise naturally from light and gravity. On the other, they are close enough to traditional counterparts to have the same subjective feel.
Several human-scale systems of units can be constructed using power of ten multiples of the Planck quantities. One can truncate uncertainties so that the units have exact metric equivalents. In choosing the set of basic units, there is a special convenience to using multiples of the natural Planck quantities which are not merely powers of ten but are, in fact, powers of 105. All the exponents are then multiples of 5 and are easier to remember.
This means that when the basic units are combined to make units of other types, they always turn out to be in a power of 105 relation to the corresponding Planck unit. For example, the unit of energy tonpace (about 5 food Calories, illustrated by pushing for one pace with ton force) is 10-5 of the Planck energy.
The values of fundamental constants typically turn out to be powers of 105 as well. Planck's h-bar, for example, is 10-40 tonpaceminute, and the speed of light is 1010 pace/minute.
To be able to handle electric quantities and temperature we need two more human-scale Planck units:
One can think of the system constructed from these as a human-scale version of nature's units. The basic advantages are threefold: (1) the human-scale units are perceptual — they tend to be in the range of direct sensation and perception — (2) all physical constants have the same values, except for powers of ten, that they do in terms of the natural units, and (3) the values of certain fundamental constants (such as G and the Stefan-Boltzmann sigma) are identical to those they have in the natural system. The fundamental physical constants are listed in a table at the bottom of this page.
The Planck quantities are normally defined in terms of c, h-bar, and G—the speed of light in empty space, Planck's constant, and the gravitation constant. Here are the essential elements of the human-scale units in those terms:
1. The speed of light c is replaced by 10-10 c, a scaled-down version of the speed of light (actually quite slow) which will be a pace per minute as soon as we have defined those units. For convenience it will be nicknamed dime speed. Dime comes from decima, Latin for tenth, and is a reminder of what fraction of the speed of light we are using.
2. The Planck force c4/G is replaced by the ton-sized force 10-40 c4/G. This ensures that G will have value unity (G= 1 dime4/ton) and makes the ton, so defined, a force of some 2700 pounds or about 12 100 newtons. Obviously the ton is much closer to human-scale than the Planck force, which is immensely strong. (It is the basic tension in string theory, an attempt at a unified understanding of the nature of matter and its forces.)
3. The Planck power c5/G is replaced by the tondime power, 10-50 c5/G. The tondime level, about half a horsepower or 360 watts, is the power delivered by pushing at dime speed with ton force, or raising a ton weight one pace per minute.
4. The Planck frequency ZP is that which h-bar associates with the Planck power. Indeed h-bar times the square of this frequency is Planck power, namely c5/G.
5. The Planck time tP is the reciprocal of Planck frequency. In the scale model it is replaced by the minute-sized interval 1045 tP.
6. The Planck length lP is ctP. In the model it's replaced by the five-foot pace-sized length 1035 lP. A pace is the distance traveled at dime speed in one minute.
7. The Planck temperature TP is replaced by a roughly hundred-degree-sized temperature gradation 10-30 TP. When we need a name for it, we'll refer to this hundred-degree-sized step as a grade. It will serve as an analog of the Planck temperature, similarly to how the others work.
8. The elementary (electron) charge is replaced by 1020 e, referred to as a score because of that word's associations with the number twenty. The Planck mass mP is replaced by 1015 mP. This mass is a quadrillion times Planck mass and as a reminder of this we call it a quad. These will serve as analogs of the electon charge and Planck mass.
These human-scale versions of the Planck quantities then form a consistent system of units which mimic the natural units. As a shorthand designation we can call them "TPM units", referring to three of the main ones in the set: ton, pace and minute. In all cases the power-of-ten scaling is made easier to remember by the fact that it only involves powers of 105 — the exponents, in other words, are multiples of five.
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In the scale model (or if you prefer, in TPM units) the gravitational constant is unitary. It has the same value (namely one) in our humanscale units as it does in the natural system. The corresponding thing also happens with the Stefan-Boltzmann constant, which relates temperature to the brightness of thermal radiation. As we see further on, Coulomb's constant, basic to electricity and magnetism, also has the same value (approximately 1/137) that it does in terms of the natural units — except for a factor of 1010.
G = 1 dime4/ton.
This means that if circular orbit speed is one dime—one pace per minute—at some distance from a central body then the body would weigh a ton in its own gravity at that distance. A planet weighing a ton in its own surface gravity would have a circular orbit speed at its surface equal to one pace per minute. The weight of a central body at any distance is proportional to the fourth power of the circular orbit speed at that distance and one of the many roles G plays is to serve as the proportionality in this case.
Dime and ton stand in the same relation as Planck speed and Planck force.
In symbols, 10-10 c and 10-40 FP (dime and ton) stand in the same algebraic relation to each other as c and FP=c4/G (Planck speed and Planck force).
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Mass has twin aspects of gravitational attractiveness and inertia, but it is conventionally equated exclusively with inertia. By way of illustration, in the metric system the kilogram is the unit of inertia. In metric terms, to impart unit acceleration (1 m/s2) to an object with unit inertia (1 kg) requires unit force (1 newton). In one proposed version of the metric system a prior force standard is established electrically (by watt balance) and the kilogram unit of inertia is derived from the units of force and acceleration. In that version the kilogram would be defined as a newton second2/meter.
The metric unit mass has no perceptible gravitational attractiveness. If you put two kilogram blocks side by side their attraction for each other cannot be detected. Partly for educational reasons it would be useful to have a unit of mass which is LARGE enough that its attraction for another unit could be detected and yet SMALL enough that one could see the acceleration effects of forces applied to it. And in our human-scale model (or TPM system of units) the unit of mass does, in fact, turn out to have both perceptible inertia and gravitational attractiveness.
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In TPM terms the unit of inertia is the tonminute/dime. It is 1015 times the Planck mass. To have a nametag for it, since it is a quadrillion Planck masses we can call it a quad. In conventional terms a quad is about 22 billion grams. It is a not-unusual mass for ocean-going freighters — a freighter with quad mass needs a tonminute shove to get it up to dime speed. Anyone who has spent much time aboard ocean-going vessels or watching them being maneuvered in harbor has probably acquired some notion of the inertia involved.
On the other hand this unit of mass also has a humanly perceptible gravitational attraction. A pair of compact unit masses, 100 paces apart, would attract each other with a gentle force which you could feel with your fingers. The attraction between unit masses at that separation would, in fact, be 10-4 ton: a tenthousandth of a ton is roughly a quarter pound.
A seven-year-old human being is apt to have mass around a millionth of a quad, namely 22 kilograms, and as adults our masses are apt to be a few millionths. A useful auxilliary scale of mass is in terms of a Planck ounce (about 22 grams). An ounce is one million times the Planck mass and there are a billion ounces to a quad.
To judge how well this mass fits into our scale model of the Planck quantities notice that it makes the gravitational constant G unitary:
G = 1 ton pace2/quad2.
That is why two spherical quad masses, centers 100 paces apart, attract each other with a force which is 10-4 of a ton. It shows that force, mass and distance are related in human-scale units in exactly the same way they are in Planck units, where one has:
G = 1 FP lP2/mP2.
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The Planck energy is what is delivered by Planck force pushing for a distance of one Planck length. It was a typical energy for primordial photons to have in the Big Bang and it's pretty large by human standards. One equivalent of pushing with Planck force (1040 tons) for one Planck length (10-35 pace) would be to lift a ton weight 100 miles into the air. Multiplying Planck force and Planck length together (1040 × 10-35) tells us what Planck energy is in terms of human-scale units and evidently it works out to 105 tonpaces.
The human-scale unit of energy is a simple analog of the Planck one — a tonpace is enough energy to raise a ton weight by one pace. It turns out to be 4.7 food Calories. It's amazing how much energy there is in food. Imagine eating something like a fragment of a cookie with 4.7 Calories and then visualize cranking a ton weight up one pace (five feet) with some sort of winch. Four-point-seven Calories is a lot of work. Since I weigh 1/14 of a ton I can experience that same amount of work by climbing a flight of stairs which rises 14 paces. Seven stories — no matter how you look at it, it's still a lot of work.
I guess nobody will object if I say that a tonpace of energy (a little under 5 food Calories) is human scale. We know what a tonpace feels like to eat as food and to perform as work. By contrast, the metric unit, the joule, is a bit harder to relate to significant everyday experience. A tonpace unit is about 20 thousand joules — imagine dividing 4.7 Calories of cookie up into 20 thousand little joule specks (crumbling the crumbs!) or dividing the seven-story stairclimb into 20 thousand tiny little joule efforts.
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In terms of human-scale energy and time units, tonpace and minute, Planck's h-bar constant is:
h-bar = 10-40 tonpace minute.
In terms of nature's units it is simply the product of unit energy and unit time:
h-bar = 1 FP lP tP
To verify this, just put in what the natural quantities actually are:
= 1 (1040ton)×(10-35pace)×(10-45 minute.)
This means that in our human-scale model, energy and time are related in way that is power-of-ten similar to the way they are related in the natural system of units. In the case of the gravitational constant G, force, distance and time were related in human-scale units exactly as in nature's. Here we must do some power-of-ten scaling and the result is mere similarity.
What does this value of h-bar (10-40 tonpace minute) mean?
It means that for a quantum bundle of light to carry one tonpace of energy (4.7 Calories, it would be quite a lot for one to carry!) it would have to be vibrating 1040 radians per minute. h-bar is what you multiply a photon's frequency by to get its energy and this checks in our case: Multiply h-bar (10-40 tonpace minute) by 1040per minute and you get one tonpace of energy.
A radian is 1/2pi cycles, it's just one of those trivial details that needs getting used to. When you work with h-bar you need to reckon vibration in radians per unit time instead of in cycles. It's like deciding to classify circles by the size of the radius instead of by the size of the circumference — either way will work but you need to be consistent about which.
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It just happens that middle D on the piano, or the violin's D string, has a frequency which is 105 per minute. On the other hand the Planck frequency is 1045 per minute. (That's because it is defined as passing through one radian of phase in each Planck interval of time.) So as a result, the Planck frequency is 1040 higher than middle D.
Middle D provides a useful alternative way to think about frequencies. For example, pure green light from a prism is typically vibrating two trillion times higher than middle D. At least for people like myself who often participate in music, it can sometimes be easier to think of the frequency-range in the colors of light as multiples of middle D than in terms of radians per minute. Typical red, green, and violet frequencies are 1.5, and 2, and 2.5 trillion (1012) times middle D. But on the whole it's usually simpler to stick to the basic unit and think of the frequencies of these colors as 1.5, 2, and 2.5 × 1017 per minute.
By the way, the Planck minute is about 90 percent of the conventional one. To have a definite number for conversion purposes, there are 66.78, or almost 67 Planck minutes in an hour. This is the slightly shorter minute one must use in order for our scale model "ton-pace-minute" units to relate well with the Planck ones. Personally I don't find it objectionable to count on 67 minutes in an hour — it can be a refreshing change — and I hope having a few extra minutes in an hour is something that readers will not find too difficult to adjust to. It is a minute which is subjectively about the same as a conventional one and which facilitates visualizing Planck quantities and working numerical examples. I do not propose it as a minute to use for keeping appointments (one would always be arriving slightly early!)
In practice, different piano tuners and different orchestra leaders use different frequency standards, so that middle D is not the same everywhere. But the variations in pitch tend to be inaudible or hardly noticeable. Although there is no one right standard, if you assume 66.78 minutes to an hour and use the figure of 105 per minute (radians, of course), you do get a recognizable middle D frequency.
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There is a standard relation between frequency and temperature given by the ratio of two fundamental constants, Boltzmann's k and h-bar. In the natural Planck units this ratio is:
k/h-bar = Planck frequency/Planck temperature = ZP/ TP.
The relation has to do with the tell-tale thermal glow at any given temperature. The thermal glow at some temperature has a characteristic (you could say a keynote) frequency which is proportional to the temperature and which distinguishes that temperatures from others. In practice, the way temperature is actually measured over a wide range is to analyze a sample of glow to determine this characteristic frequency (the temperature's "color"). From the extremely low temperature of the microwave background in space to the surface-temperatures of stars, in a host of cases in between, temperatures are measured (in effect) by assuming a value for k/h-bar and analyzing frequency.
The k/h-bar ratio is how temperature is related to characteristic frequency in nature. In our human- scale "ton-pace-minute" units, the grade temperature step (mentioned earlier in the introduction) is used and the Planck temperature is 1030 grade. Since the Planck frequency is 1045 perminute, the same fundamental ratio comes out to be:
k/h-bar = (1045 perminute)/(1030 grade) = 1015 perminute/grade.
This is one of the nature's fundamental proportions. You multiply a temperature by this ratio to find what keynote frequency it corresponds to. This means, for instance, that a temperature of 1 grade corresponds to a frequency of 1015 per minute. Other ways to say that are "quadrillion per minute" or 1010 above middle D.
The grade is serving as our human-scale analog of the very high Planck temperature. In fact the grade is a giant step of absolute temperature which consists of a hundred ordinary-sized Planckian degrees. (Planckian degrees are about forty percent larger than kelvin. Where a metric-speaker would say that ice melts at 273 kelvin we say in terms of our scale model that it melts at 193 degrees — and this is the same as 1.93 grade.) If someone wants a non-absolute scale relative to some arbitrary substitute zero they can arrange it in either Planckian or metric terms, but for physical laws to work we need to use the real zero.
Room temperature is 2.07 grade. This means it has a characteristic frequency which is 2.07 quadrillion per minute. If you are in room temperature surroundings, you could find that frequency by analyzing the invisible infrared shining from the walls. It would be the key frequency in a distinctive mound-shaped distribution curve which you could determine from the light itself.
Normal body temperature is 2.19 grade. This means it has a characteristic frequency which is 2.19 quadrillion per minute. You could find that frequency by analyzing the heatshine coming out of someone's mouth.
Frequency and temperature are related in our scale model of the Planck units in a way that is similar to, and parallels, the way they are related in the natural units. When cosmologists tell us that the primordial temperature, the temperature of the universe when it came into existence, was Planck temperature, they are talking about 1 TP, the power-of-ten analog of the temperature we are calling 1 grade.
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In the natural Planck units, temperature and brightness (that is radiant flux or power per unit area) are related by a simple fourth-power (quartic) formula. The relevant fundamental physical constant is called sigma (the Stefan-Boltzmann constant.) The value of sigma in the natural units is just pi2/60.
That means if everything is expressed in Planck terms you can find the radiant flux, or energy per unit time and area, from a blackbody surface just by raising the temperature to power four and multiplying by pi2/60. This tells how brightly hot things glow. Pi squared is about ten, so multiplying by pi2/60 is about like dividing by six. This is a lot messier in the metric context, where the units are unnaturally sized, but that observation applies pretty generally.
Now what I've been emphasizing here is that the quantities in this scale model of the Planck units are related in analogous, power-of-ten-similar ways. So we should check to see how the human-scale units of power, area, and temperature are related.
The "ton-pace-minute" unit of power is a tondime, which is the same as one tonpace per minute. (Dime is just a nickname for the unit speed of one pace per minute, which is exactly a tenbillionth of the speed of light.) One tondime is the level of power delivered by something pushing with ton force at dime speed — picture a slow freight elevator raising a ton of freight.
A tondime is approximately half a horsepower or, if you think lightbulbs, 360 watts.
We should try to see what sigma is in terms of grade temperature and tondimes-per-square-pace brightness.
Here it turns out that our scale model quantities are related in exactly the same way as the natural Planck ones.
sigma = (pi2/60) tondime/pace2 per grade4.
For instance suppose you go outside and find it is a brisk fall day and the temperature is 2.00 grade (that means 283 kelvin or 50 Fahrenheit.) If you want to know how brightly the flagstones on the terrace are glowing with thermal radiance, all you need to do is raise 2 to power four, getting 16, and divide by six. That gives the thermal brightness in tondimes per square pace of area. A physicist asked to to the same calculation in metric terms might well need to consult a handbook and use his calculator. The sizes of metric units were established circa 1790 and they are far from convenient for physical calculation.
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Except for possibly a factor of 1010, the same factor that was used to scale down the speed of light, our quantities of force, distance, and charge have the same relation to each other in the human-scale model that they do in the natural, or Planck, system of units.
There is a famous number alpha which physicists often write simply as 1/137 (although the experimentally measured value is more like 1/137.036) and which is called the fine structure constant. Arnold Sommerfeld, who was Heisenberg's teacher, named it in 1916. In the natural Planck units, alpha happens to be the value of the Coulomb constant (basic to E&M.) In human-scale terms the Coulomb constant has the value alpha×1010.
There are actually two attractive alternative units of charge to use with our TPM units in building a scale model of the Planck quantities. One is the score of charge, is 1020 times the elementary charge, which is the charge on one electron. The score (called so because of associations with the number twenty) is larger than the metric unit, being about 16 coulombs. It turns out to fit in with the other units well, but is a bit on the large side for convenience. The other alternative is a convenient small fraction, namely 10-5 of a score. This is 1015 times the elementary charge. It's the charge of quadrillion electrons and it's a quantity you can feel by its effects of attraction and repulsion.
Score is the real unit of charge in this context but 1015 e is such a convenient size that I would very much like to have a way to refer to it. Maybe just calling it 10-5 score will work.
In the natural Planck units, Coulomb's constant is:
1/137 FP lP2/e2
On the other hand in TPM units the Coulomb constant turns out to be:
1/137 × 1010 ton pace2/score2
which means that a pair of 10-5 score charges, a pace apart, repel each other with a force which is 1/137 ton. To check this you can multiply the Coulomb constant by the product of the two charges — the 10-10 cancels the 1010 — and divide by the square of the one-pace distance, which doesn't have any numerical effect. The force comes out to a few pounds, something one could feel by hand.
Thinking of the units as a model, the ton plays the part of Planck force; the pace serves in place of Planck length, and 10-5 score plays the part of the charge on the electron.
Using the smaller charge quantity Coulomb's constant can be written:
1/137 ton pace2/(10-5 score)2
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The Planck unit of current (e/tP) is large in metric terms, about 3 × 1024 amperes. On the other hand, calculations using the natural unit, such as finding the force between currents in long parallel wires, tend to be simple.
One way to quantify the attraction or repulsion between indefinitely long straight current-carrying wires is to ask about the force experienced by a halfspread test segment along one of the wires. This is a test segment whose length is one half of the separation between the wires. The force on such a test segment is purely a function of how much current is flowing in each wire.
If the currents are running in the same direction, the force is an attraction, while if their flows are opposed it is a repulsion, and what one basically does is multiply them and divide by 137. (The number 1/137 called the "fine structure constant" is basic to electricity and magnetism.) I am going to spell this out in more detail and see how the analogous calculation goes using our scale model "ton-pace-minute" quantities.
If everything is expressed in natural units then to find the force you just multiply the sizes of the two currents and divide by 137. Those are the actual steps in the calculation using Planck units. To keep things clerically in order you should first divide each current by the speed of light. Magnetic force is a relativistic effect and in calculating it both speeds and currents ought to be divided by c. But that doesn't effect the numbers because the speed of light is unitary. Including clerical detail, the whole story of what you are doing is this: Divide both currents by the speed of light, multiply them together, and multiply by the Coulomb constant.
We would like something comparably simple to be the case of "ton-pace-minute" units. It won't be quite so simple because in that case, instead of being just equal to one, the speed of light is 1010 pace/minute. So dividing the current by it will make a difference of some powers of ten.
Consider an example of two parallel currents each of which is 105 score/minute. We want to find the force on a half-spread test segment. Dividing each by the speed of light and multiplying them together we get 10-10 score2/pace2. The Coulomb constant is 1/137 × 1010 ton pace2/score2. So when you multiply by the Coulomb constant the powers of ten cancel and you simply get that the force on the segment is 1/137 ton.
The relation between current and force is nearly as simple in our scale model units as it is in nature's. The extra thing you have to remember to do is divide by a couple of factors of 1010, to allow for the fact that the ton-pace-minute speed of light is 1010 instead of one. Other than that you just multiply the two currents together with the Coulomb constant.
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As I already indicated, the Planck minutes we're using here are exactly 66.78 to the hour — which makes them about 90 percent of a conventional minute. The 66.78 figure is chosen so as to get agreement out to four decimal places with the experimentally measured Planck quantities and, on the other hand, to have some definite number to use in converting units. The pace is the distance traveled in one minute at 10-10 of the speed of light in vacuum, and it's approximately 1.616 meters. A Planck mile of a thousand paces is a useful multiple which, at 1616 meters turns out to be within half a percent of the US statute mile (1609 meters.)
A cubic pace is 4221 liters and therefore a Planck gallon defined as a thousandth of that volume turns out to be 4.221 liters. This puts it half way between the US and British gallons — a compromise gallon size.
A billionth of quad mass is 21.766 grams. This makes it slightly over ¾ of the conventional ounce mass (which is 28.35 grams) and qualifies it as a Planck ounce. A thousand ounces is typical for a seven-yearold youngster. Most of us have masses which are several thousand ounces. My own, for instance, happens to be about 4000.
The ton unit is a force of about 2700 pounds or 12100 newtons. For definiteness, it can be treated as 1.234 metric ton-force.
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Because the Planck minute and pace have exact metric equivalents the conventional metric norm for the acceleration of gravity can be translated with arbitrarily high precision. To five-place precision it translates as 17634 dimes per minute. In this metric idea of normal gravity, our quad unit mass weighs 17 634 tons. I'd like to round this off but am reluctant to do so since the metric norm is given officially as 9.80665 meters per second2 and I don't want to risk losing convertibility.
Somewhat less formally, normal human body temperature is 2.19 grade and the conventional norm for atmospheric pressure translates into ton-pace-minute units as 21.9 tons per square pace. At these conditions of pressure and temperature, which are certainly normal in some sense, the unit volume of a perfect gas contains 1026 molecules.
This is a straightforward application of the PV=NkT law which describes the behavior of a perfect gas by the latter's definition.
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| ton | 12 100 newtons | |
| pace | 1.616 meters | |
| minute | 66.78 to the hour | |
| grade | 141.7 kelvin | |
| score (charge) | 16.02 coulombs | |
| tonpace/score | 1221 conv. volts | |
| score/minute | 0.2972 amperes | |
| tondime, tonpace/minute | 362.9 watts |
I am in doubt as to whether names are needed for the unit voltage (tonpace/score) and unit current (score/minute), and, if they are, what they should be. Because the voltage unit is about a thousand familiar-sized voltage units, what comes to mind is to call it a grand. The current unit, about 0.3 conventional amperes, stumps me. Maybe since it is smaller than an ampere the thing to do is call it ambit. This sounds diminutive, and the Latin word ambitus had the idea of a going around, or circuit.
To faciliate the use of natural units, a set of precise counterparts to the Planck quanitities has been constructed by removing experimental uncertainties. These are described in more detail in Natural Units for Scientists and Engineers. The precise counterparts are equal to the natural quantities either identically or else to the accuracy with which the latter have so far been measured. They are essentially the same as the natural units except that uncertainties have been removed, giving them precise metric equivalents to facilitate their use as units. For comparison's sake, here's what the precise counterparts to the Planck quantities look like in the SI and in the ton-pace-minute systems:
| Planck speed | 299792458 meter/second | 1010 dimes |
| Planck frequency | 371/200 × 1043 persecond | 1045 perminute |
| Planck time | 200/371×10-43 second | 10-45 minute |
| Planck length | 1.61613... ×10-35 meter | 10-35 pace |
| Planck voltage | 1.221...×1028conv. volts | 1025 grand |
| Planck current | 2.972...×1024 amperes | 1025 ambit |
| Planck resistance | 4108.3... ohms | 1 grand/ambit |
| Planck power | 3.629... ×1052 watts | 1050 tondime |
| Elementary charge | 1.602... ×10-19 coulomb | 10-20 score |
| Planck energy | 1.956... ×109 joules | 105 tonpace |
| Planck force | 12.104... ×1043 newtons | 1040 ton |
| Planck acceleration | 5.561...×1051 meter second-2 | 1055 dime/minute |
| Planck mass | 21.766... ×10-9 kilogram | 10-15 quad |
| Planck momentum | 6.525... kilogram meter/second | 10-5 ton minute |
| Planck temperature | 1.417... ×1032 kelvin | 1030 grade |
Two valuable auxiliary units are the practical Planck volt (1.221 conventional volts) which is a thousandth of the principal voltage unit in this system and the practical Planck degree (1.417 kelvin) which is a hundredth of a grade. The former auxilliary unit helps define a familiar-sized eevee (1.221 of the conventional.) The principal energy unit, tonpace, is thus equal to 1023 eevee.
Here are a few fundamental constants in ton-pace-minute units. In a couple of cases (written 1.000) the values are accurate to whatever accuracy the natural quantity is measured. In others (like c, h-bar, and e) they are exact. I've included one example of a conventional constant as well: the standard earth gravity that corresponds to the conventional metric figure:
| speed of light in vacuum c | 1010 pace/minute |
| Planck's h-bar | 10-40 tonpace minute |
| gravitational constant G | 1.000 pace3minute-2 quad -1=1.000 ton pace2 quad-2 |
| elementary charge e | 10-20 score |
| Boltzmann's k | 10-25 tonpace/grade (equiv. 10-2 eevee/grade) |
| (Angular) frequency-to-temperature k/h-bar | 1015 perminute/grade |
| Stefan-Boltzmann sigma | (pi2/60) tondime pace-2 grade-4 |
| Coulomb's constant | 1/137×1010 ton pace2score-2 |
| Planck voltage VP | 1.000×1025 grand (equiv. 1028 volt) |
| Avogadro mass | 0.7629... ounce (equiv. 0.7629...×10-9 quad) |
| Avogadro number | 1025 per mole |
| molar gas constant R | 1 tonpace/grade-mole |
| standard Earth gravity | 17634 dime/minute |
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