Using the force constant in equations

[nightcleaner]

Hi all.

Just want to add my vote in the Marcus popularity poll. I think this work is excellent, Marcus, and am very interested in seeing it in table form. Natural units are, in my opinion, likely to overtake and replace all our human based systems. It is very exciting for me to see them demonstrated so clearly.

Be well,

nc

 

[marcus]

Removable gear, as in

1. Wires: like nuts on wires in many sizes
2. little pieces of soft metal that can be hammered into small holes
3. Camming devices (such as 'friends'): these are very nice for putting in cracks
and come in many sizes

Of course, all this stuff is heinously expensive.

since you like mountaineering I want to try to calculate the dry air lapse rate. the rate that air gets colder with altitude.

you mentioned these cabins, one with a "freezer door", that hikers can use in the NewZealand mountains I guess to be more comfortable than in a tent. I guess there are some places where the height is great enough to be quite cold.

there is this beautiful temperature gradient which is the threshold for convection. if it doesn't cool off faster than this then the air can be stable, but if it cools off with height more abruptly than this then convection starts and wind, and mixing etc etc, which after a while redistributes the heat so that it cools off at only this rate, and then convection stops.

homeostasis is nice. Oh, Earth surface gravity is about 0.88E-50 natural.

\text{limiting temperature gradient} = \frac{\text{weight of air molecule}}{\frac{7}{2}k}

\text{limiting temperature gradient} = \frac{0.88E-50 \times 29 \div 2.6E18}{\frac{7}{2}k}

k is one, like the other constants. so the calculation is trivial and
comes to 28 of our "fahrenhalf degrees for each halfmile of altitude.

Well, in natural units it is 2.8E-68, but you know how these things go,
E37 natural length units is a half mile and E-32 on the temp scale is a fahrenhalf step so that is how it is coming out to be what I said.

whenever the wind blows you know that somewhere on Earth there must be convection which means that somewhere the gradient must have temporarily exceeded this 28 halfdegrees per halfmile. it is called the lapse rate. Kea knows this but I am saying it so as to be very explicit

BTW they used to sell a toy which was just a cube---a box---with a switch on the outside. You flip the switch and you hear the box whirr and it even shakes a little

and then a door flies open and a little hand sticks out

and flips off the switch

 

[marcus]

and then of course the little arm pulls back in and the door slams shut

and everything is the way it was before



hello nightcleaner that story of the box was partly for you
homeostasis is a good thing

 

[RingoKid]

I remember those boxes !

they were coffins and you had to put a coin on the switch then a withered hand would reach out and snatch it...

 

[marcus]

ringo, that could well be it. I am going on hearsay from someone I thought reliable, and partly on my imperfect memory. In my version there is no coin--the being in the box simply wants the switch restored to the off position.

Another piton or point of reference. Remembering the density of water. At ordinary swimming pool conditions it is around 1.225 pounds per pint----where a pound is E8 natural mass units and a pint is E99 volume units----so that translates to 1.225E-91. somehow it seems easier to remember 1.225 pounds per pint (which encode the powers of ten)

We had better rescue a baez-in-space story from a stalled thread in the Atoms Molecules forum. Kea got me in the frame of mind to tell this:

John Baez is exploring the galaxy in a small spacecraft and finds himself in low circular orbit around a glistening planet. His wristwatch tells time in natural units and after traveling one radian (1/2pi of full circle) he looks at his watch and sees it has taken

T_radian = 7E45 time units

What is the density of the planet?

\text{reciprocal density} = \frac{8 \pi G}{6} {T_{radian}}^2

remember that in the Force system 8 pi G = 1

\text{reciprocal density} = \frac{1}{6} (7\times 10^{45})^2

\text{reciprocal density} = \frac{49}{6} \times 10^{90}

\text{reciprocal density} = \frac{49}{60} \times 10^{91}

\text{planet's density} = \frac{60}{49} \times 10^{-91}

and 60/49 is about 1.2 so this planet is the density of water. in fact it is one large lucent drop. in the story Baez abandons the spacecraft and goes swimming. I suppose his orbit was low and slow enough to allow this.

The density of the Earth's moon is close to that of jade---the green semiprecious stone. I imagine that radian time for a planet made of jade would be just under 4E45

the radian time of a round planet depends only on the density and not on the size----big or small, the same radian time.

for the Earth I rather think the radian time is 3E45, but I haven't checked.

 

[nightcleaner]

and then of course the little arm pulls back in and the door slams shut

and everything is the way it was before



hello nightcleaner that story of the box was partly for you
homeostasis is a good thing

thanks, at last something I can understand. For some people, it takes falling off a roof...

 

[marcus]

electron mass is 21E-23, or is 2.1E-22 better?

another piton, the electron mass is 2.1E-22 natural.

so we can calculate the colors a hydrogen atom will glow

another piton: photon energy E-27 is green

if you remember that then you can figure out later that
energy 0.75E-27 is red, and 1.25E-27 is blue, the visible spectrum is a narrow band around green, only some 20 percent higher and 20 percent lower. so one benchmark kind of locates it.

the thing about the colors is nice. if you calculate the ionization energy of the hydrogen atom then the jumps between energy levels are like this

\text{jump from level 3 to level 2} = (1/2^2 - 1/3^2) E_{ionization}

\text{jump from level 3 to level 2} = (1/4 - 1/9) E_{ionization}

\text{jump from level 4 to level 2} = (1/4 - 1/16) E_{ionization}

I am simplifying, but this is roughly the story: this is how much energy is released if it jumps down from any higher level to some lower level (this relates to Richard too, regretably). And so if we can once calculate what the ionization energy is then maybe we can find two levels where the transition calculates out to release exactly the right amount for a green photon!
Or a red, or a blue.

You see how great it would be if we could calculated the hydrogen atom ionization energy?!

But it is a cinch because

E_{ionization} = \frac{1}{2}\times (\frac{1}{137})^2\times \text{rest energy of electron}

in natural units the rest energy of something is the same number as its mass (because c has value 1, so E and mc^2 are the same number) so the electron rest energy is what I said earlier: 2.1E-22 energy units.

E_{ionization} = \frac{1}{2}\times (\frac{1}{137})^2\times 2.1 \times 10^{-22} = 5.6\times 10^{-27}

but this is great! All we have to do is find transition "m to n" where the (1/n² - 1/m²) number is small enough to cancel off the 5.6 and we will have green! So let's experiment:

(1/4 - 1/16) \times 5.6 = 1.05

that's it!

a jump from level 4 down to level 2, in the H atom, will release a photon of energy 1.05E-27.

that will make a very pretty green.

if we pick a higher level to jump down from we might get a turquoise or a blue or a violet. Let's try:
(1/4 - 1/25) \times 5.6 = 1.176

(1/4 - 1/36) \times 5.6 = 1.244

whoa. it looks like jumping from 6 down to 2 releases so much energy it is up at the violet limit of what is visible, beyond that could be UV.

my guess is that the jump from 5 down to 2, whose resulting photon has energy 1.18E-27, is blue.

you can check for yourself if the H atom will also make red. If it does it would be a jump from level 3 to level 2.
There are also all the transitions from higher levels down to level 1, but I haven't calculated what they yield because i know it is always up in the UV where only a bee can see. Bees can see UV too high-energy for you and me, so they would recognize "colors" from hydrogen atom transitions that don't have any aesthetic meaning for us. well enough said

 

[marcus]

Hi all.

...Natural units are, in my opinion, likely to overtake and replace all our human based systems.

Listen. it might actually be right

(but that is not the immediate aim, we are climbing this particular face to see what we can learn from doing that)

 

[marcus]

I think it is about assimilation.
you have to know that a photon with E-27 energy is green
that is a kind of landmark or piton into the wall

then someone comes and says well the mass of an electron is 2.1E-22
and that means the energy nature has invested in the electron's existence is also 2.1E-22 (in natural energy units)

OK then you belay and you progress sideways by the ratio 1/137
and you come to the ionization energy of hydrogen-----5.6E-27.

then you see the E-27 in that number----hey that's green.

so then it turns out that although the ionization of hydrogen involves morethanvisible energy---involves a UV photon or equivalent---some of the smaller transitions within that atom involve visible energy, and indeed jumping from 4 down to 2 actually makes very close to E-27, so you do get green light

so we keep driving reference points into the originally blank wall.
am I the only one who is learning by this? some people have said maybe not.
of course selfAdjoint already knows the reference points but in another system.
Like, in conventional units H ionization is 13.6 electron volts and you get the eevees of the colors by scaling down the same way.

 

[marcus]

I just happened to see a 1998 webpage that mentions the LHC and says that it will give the pair of colliding protons an collision energy of 5.4 TeV

http://www.nupecc.org/nupecc/report97/report97_nnc/node16.html

Is this right? I don't know what to expect from the LHC when it begins to operate up to spec, say in 2007.

but anyway, suppose it is right. and each proton has a kinetic energy of about 2.7 TeV, when they meet. What is that in natural?

by serendipity it turns out right around 10^-15

I know that because it turns out that if you were to use a quarter of a volt, to measure voltage, then your 'electron quartervolt' would come out
10^-28 of the natural energy unit

so we can take 2.7 x 10¹² eV and multiply by 4 and get
around 11 x 10¹² "eQ", which is 11 x 10^-16 of the natural energy unit. Given that it is several years off and we don't know what energy LHC will actually attain, I am willing to call that 10^-15

Now we have this "avogadro-like" number 2.6E18 which is how many proton rest masses make a natural mass unit-----or how many proton rest energies make a natural energy unit (same ratio).
And that tells us that to go from proton rest energy to LHC energy is a factor of 2.6E5. Have I made some mistake with powers of ten? It looks like what the accelerator does is increase the energy of a proton by 260,000 (a quarter of a million) and then smack two together head-on.

the speed that the proton must be going is, as you might expect, so close to 1 that, if I try to find the speed naively, my calculator cannot calculate it---it just says exactly 1.
I would have to proceed indirectly or else get a calculator with more than 12 digit accuracy because

\beta ^2 = 1 - (\frac{1}{260,000})^2

Does anyone know if this figure of 5.4 TeV agrees with current expectations? Or, if not, what the current target energy is at LHC?

If you had to write beta, the speed, out as a decimal number it would be
11 nines followed by a three

beta = 0.999999999993

 

[marcus]

the hot tub of Baez

When he goes on extended travels John Baez always remembers to turn off the heater at the hot tub, to conserve energy.

The hot tub contains 10²⁹ molecules of water
and the heat capacity, per molecule, is 9k, where k is Boltzmann constant.

(this could be seen as typical, in a lot of materials it is 3k per atom, and the water molecule having 3 atoms, 9k is not totally unexpected.)

During Baez absence the water temperature has subsided to ambient
outdoor temperature 10^-29 (corresponds to 49 Fahrenheit).
To be even tolerable, it must be raised to 1.11 x 10^-29

We now see that to heat up the tub, on his return, the noted Category Theorist must expend one natural unit of energy

This is because the total heat capacity of the water is 9 x 10²⁹ k, so the energy to raise its temperature by 0.11 x 10^-29 is
9 x 10²⁹ x 0.11 x 10^-29 = 1

---------
Now some people may harbor the unworthy suspicion that I have arranged for these numbers to be artificially nice and tractable, but indeed not, for I calculated the number of water molecules assuming metric
(which is as good as saying random) dimensions. I assumed a cylinder of water one meter deep and one meter radius so that it would have a mass of 3142 kilograms. that mass of water just turns out to comprise 10²⁹ molecules. this example shows that one natural unit of energy is quite a lot---enough to heat a hot tub.

 

[marcus]

I thought last night that what I am doing in this thread might be called "natural units with named powers of 10"
and I might argue that one needs a few named power-of-ten to help assimilate natural units (not to get indigestion, or boggled)

Like 10⁸ mass units is a "pound" and that is a helpful handle on the mass unit. As I mentioned earlier this pound is 434 grams. This pound is a named power of ten multiple of the unit mass.

And it helps when it comes to hot tubs. Baez hot tub is metric, meter radius and meter deep, so the mass of water is 3142 kilograms----divide by 0.434 and you get 7240 pounds-----7.24 x 10¹¹ natural mass units of water.

So far not particularly pretty numbers, nothing memorable. I just chose the hot tub dimensions at random as about the right size for a hot tub.

But then we have our avogadro-like number 2.6E18 which says that one natural mass unit comprises 2.6 quintillion protons. And remember that it is 18 proton masses to one water molecule.

So take the modest unprepossessing number 7.24E11 and multiply by 2.6E18 and divide by 18 and one gets 1.046E29. My goodness when one thinks that warm water is considerably less dense than the chilly metric ideal one might even get the notion there are exactly E29 molecules in Baez tub!

I once stayed at the house of someone who lives up the slope overlooking Boulder, more or less off the grid, and I remember her deck. it had a long refractor on a massive tripod and I was able to watch the satellites of Jupiter several consequtive nights. I remember splitting wood with wedges and sledge. There was all that wood under the raised deck. You watched planets and split wood.

 

[marcus]

Phobos pointed us today to NASA and CNN pages about the deep impact mission to comet Tempel 1.

One aspect of this mission is that a copper block with mass 8.6E10 natural mass units will collide with the comet at a speed of 3.4E-5 natural speed units.

the problem is always to interpret and named powers-of-ten help

remember that E8 mass units is a pound. so the mass is simply 860 pounds!

in much of the Earth's atmosphere the speed of sound is around E-6, that is a millionth of the speed of light. So think of 34E-6 as Mach 34!

this is actually rather close to escape velocity from the Earth's surface which I believe I recall is Mach 37, but that doesn't matter. Another reference point is that E-9 (a billionth of c) is 2/3 mph. So E-7 is 67 mph and E-6 (a "Mach" in the cold above the clouds) is 670 mph. The Earth's orbit speed is E-4 (Mach 100)

Anyway the natural speed scale is not too hard to assimilate

E-9   2/3 mph
E-8   6.7 mph
E-7   67 mph
E-6   cold air sound, "Mach"
E-5   communication satellite in stationary orbit
E-4   Earth orbit speed

so as part of studying Tempel 1, the comet, an 860 pound block of copper, equipped with cameras, will collide at Mach 34 with the comet
HOW MUCH ENERGY WILL BE RELEASED

Well it works out to 50 natural units. Recall that one unit is enough to heat a hot tub.

\text{kinetic energy} = \frac{1}{2}\times 8.6 \times 10^{10} \times (3.4\times 10^{-5})^2

you can check, it comes out to 50

estimates of the resulting diameter of crater that will be put in the face of Tempel 1 range from 10 meters to 150 meters

 

[marcus]

...
I once stayed at the house of someone who lives up the slope overlooking Boulder, more or less off the grid, and I remember her deck. it had a long refractor on a massive tripod and I was able to watch the satellites of Jupiter several consequtive nights. I remember splitting wood with wedges and sledge. There was all that wood under the raised deck. You watched planets and split wood.

I remember when I visited that lady in Colorado I split a lot of logs. She cooks and heats the house with a couple of very efficient stoves and of course that takes small wood and the logs down below the deck (with the telescope) are big wood.

There was no path through the trees to the house because everytime you walked back the road, or brought stuff from the car to the house, she liked you to wend a different way thru the trees. So no path got worn. you picked a random path. There was lichen and pine-needles on the ground and rock sticking out in places, but no bare dirt. no mud ever.

I am trying to picture how much firewood releases one natural energy unit of energy when it burns. My first guess was 50 pounds, but now I am guessing 70 pounds. You remember that for us a pound is just a name for a certain power of ten of the mass unit---it means E8 mass units. So 70 pounds is 7E9 mass units. We multiply by our avogadrolike number 2.6E18----to find the combined molecular weight: typically that's in (CH₂O) chunks which weigh 30 apiece. Each of those yield about 17 eQ when it burns. (16 or 17 eQ is typical of the energy released when one molecule of oxygen is consumed in burning most common fuels)
The eQ is a named power of ten, that really means 10^-28 of the energy unit.

So we do everything in terms of the natural units and multiply the mass 7E9 by avogadro 2.6E18 and divide by 30 and multiply by 17E-28. don't worry it will give one energy unit.

\text{heat from 70 pounds dry wood} = 7\times 10^9 \times 2.6\times 10^{18} \div 30 \times 17 \times 10^{-28} = 1 \text{ energy unit}

 

[marcus]

Whether it's diesel fuel or wheat,
the Oh-Two used will tell the heat.
On the average, each Oh-Two
releases 17 eee-kyoo.

It turned out to be convenient to define a Quartervolt as a unit of voltage which is about one quarter of a conventional volt. that gives us a microscopic unit of energy "electron-quartervolt" which is just exactly
10^-28 of the natural energy unit. So the "eekyoo" becomes one of our Named Powers-of-Ten.

the eekyoo is the amount of energy it takes to make an electron hop a barrier of a quarter of a volt.

eekyoo is a good measure of energy released in chemical reactions. One carbon atom, by burning to form one CO₂ releases about 17 eekyoo. Funny thing is that two H₂ molecules also release about the same when they burn to form two H₂O. To a first approximation it depends on the oxygen used---and that is across the board from hydrocarbons to carbohydrates: a broad spectrum of common fuels and even foods. So eekyoo is a good size for keeping track of that

it is also a good measure of energy carried by photons of light. A green photon delivers right about 10 eekyoo.

 

[marcus]

Pretty soon I will make a list of a few things to remember to help interpret natural units---landmarks, like that E-9 c is 2/3 mph.
Named powers of ten, like eQ is E-28 energy unit
and a 10 eQ photon is green. Anyone who wants to understand and use a system of units should keep their own list of stuff it helps them to remember and update it and extend it from time to time.

But it is tempting to explore some more. It is cold here and we have a gas heater that is not really adequate to the house (commonplace with old houses in N. Calif.) So we wear several layers indoors and expect to be a bit uncomfortable that is how we know it is january.

what is the heating value of a pint of CH₄?
If you would be happier to say cubic meter then multiply by 1880. I will
do it for pint because that is a named power of ten meaning E99 volume units.

It is really just a matter of how many molecules CH₄ are in a pint, because each molecule uses TWO Oh-Two when it burn, each releasing 17 eQ, and so each methane is worth 34 eQ.

Well for our purposes standard temperature is E-29 and standard pressure is 1.4E-106 so the usual Gas Law PV = NkT says that

1.4 \times 10^{-106} \times 10^{99} = N \times 10^{-29}

N = 1.4 \times 10^{22}

I know the order of magnitude is right because I know when I take a deep breath it is slightly less than a gallon and contains E23 air molecules, so here's a pint and it's roughly consistent with that.

Now each molecule releases 34E-28 energy unit and multiplying by the number of molecules N gives 47E-6 energy unit. Looks like a piddling amount. What volume to heat a hot tub? 21 cubic steps----or, dividing by 1.88 to make it sound more familiar, 11 cubic meters.

I am trying a 1000 pint thing called a cubic step. a step is an 81 cm (32 inch) length that is ten times the handbreadth.

this is one of these cases that almost seems to cook the goose of these units, because atmospheric pressure is so off-putting: 1.4E-106 in the natural unit of pressure. for some reason I can accept that a pint volume is E99. I have a handbreadth that is E33, and a square hand---a sort of palm area---is E66 area units. So E99 even though a big number seems OK for a handy practicalsize volume. But having normal air pressure come out to 1.4E-106 could dampen someone's enthusiasm right away.
Maybe a way to assimilate it will turn up.

 

[marcus]

Since air pressure being 1.4E-106 is a hard one to get used to, I should do some more exercises with it
Like how many helium atoms are needed to lift a 200 pound person,
and what volume is that under ordinary conditions of temperature (E-29) and pressure (1.4E-106)

in natural units the mass to be lifted is 2E10
I know that each air molecule(wt. 29) replaced by a helium atom (wt. 4) reduces the molecular weight in the balloon by wt. 25
which in natural mass units is 25/(2.6E18)

So I have to solve N 25/(2.6E18) = 2E10
N = 2.08E27

then the PV = NkT Gas Law is where the pressure comes in


1.4 \times 10^{-106} \times V = 2.08 \times 10^{27} \times 10^{-29}

V = 1.5 \times 10^{104}

so it is 1.5E5 pints-----150 cubic steps, divide 150 by 1.88 if you want the size of the balloon in cubic meters
(pint was too small for visualizing larger volumes like this so I needed to resort to cubic step----a step is about 2 and 1/2 feet----30 inches----half the regulation 5 foot pace that miles are based on----cubic step is 0.532 cubic meter. regret having to add one more tool of imagination to the kit, but seems needed)

 

[marcus]

reference quantities update

reference quantities for the "Force" system of units

the system is defined by making hbar and c the units of ang.mom. and speed and making the unit force be the coefficient in the Einst. eqn.

the charge unit is the elementary charge e,
and unit heat capacity be the Boltzmann k, which just says that kT and T are the same number.

speeds:
E-9  2/3 mph
E-7  67 mph
E-6   sound in cold air
E-4  Earth orbit

masses:
E8  "pound" (434 gram)

2E-22 electron
1/(2.6E18) proton
1.1E31 europa
1.7E31 moon
1.38E33 earth

energies:

E-28  "eQ" (electron "quartervolt")
10E-28  green photon (8 eQ for red and 12eQ for blue)
17E-28  yield from one O[sub]2[/sub] in combustion or metabolism
(of a number of common hydrocarbon and carbohydrate fuel/food)
1 unit  70 pounds of firewood, heats the hot tub
50  impact mission collision with comet Tempel 1

lengths:
E33  handbreadth, 8.1026 cm.
E27 "microhand" the (angular) wavelength of green light
E34  one step (half a conventional pace)
E37 conventional halfmile
7.86E40  average radius of earth, 7860 "halfmiles"

areas:
E66  sq hand

volumes:
E99  cubic hand, "pint", 532 cc (1880 to the cubic meter)
E102 cubic step (for visualizing larger volumes) 0.532 cubic meter


temperatures:

E-32 half a Fahrenheit step (above absolute zero)
9.6E-32 cosmic microwave background
E-29  approx. Earth surface average, 49 Fahrenheit
2E-28  solar surface

forces:

E-43  half a Newton
1/137   idealized: force betw. pair of electrons sep by unit dist.
(to make this more realistic, divide by the square of a larger separation,
like E27, the wavelength of green light.)
(1/137)E-54  force betw. pair of unit charge separated by E27

power:
E-52   one sixth of a watt

acceleration:
0.88E-50   Earth surface gravity

current:
E-24  about 0.6 amp 

pressure:
1.4E-106  typical air pressure at Earth surface

 

[marcus]

The force between two parallel currents, measured on a segment of one of them whose length is half the separation, is 1/137 F times the sizes of the two currents.

the picture is two long straight wires running parallel and each carrying steady direct current: if the two are in the same direction the force is an attractive, and otherwise the opposite. It is standard Freshman physics and the aim is to see how it goes with these natural units.

Down near the bottom of the code box in the previous post I noted that E-24 of the current unit is about 0.6 amp, which gives an idea of what sizes we might want to have the currents be. Let's make them both E-21, and going in the same direction. admittedly that is a lot of current

then the attractive force between the wires is

is 1/137 F E-42

we have a named power of ten of the force unit which is close to half a metric Newton----it is E-43 F
this force---the attraction of one wire for the other measured on a segment which is half-separation long----is 10/137 of those.

this is a primitive example of calculating magnetic effects with these units.
for better or worse, you multiply the fine structure constant (actually 1/137.036..., but 1/137 for short)
by the sizes of the two currents.
maybe that's bad, maybe good, maybe it seems awkward, maybe neat, but that's how it goes.

 

[marcus]

Whether it's to burn or eat,
the Oh-Two count will tell the heat.
On the average, each Oh-Two
releases 17 eekyoo.

how much fuel energy is in a 20 pound tank of propane?
I don't know how it is at the present but it used to be that if you
had a camper you owned your propane tank and you could
drive into a filling station (where they sold propane too)
and buy a few pounds of liquid propane.
they might weigh the tank before and after filling it, to
tell how much to charge.

Well propane is C3H8
so one molecule takes 5 oxygen molecules to burn
and by this naive rhyming rule that means that APPROXIMATELY
one propane will give you 17 x 5 = 85 eekyoo

an eekyoo is just a tiny named power of ten fraction of the natural energy unit-----it's E-28.

so if we want to know the energy in 20 pound of propane we use that avogadrolike number and say that a pound has 2.6E26 total molecular weight, multiply by 20, divide by weight 44 of one propane, multiply by 85 eekyoo

that comes to 1.00E28 eQ, and multiply E-28 to get 1 natural energy unit.

Darn! wasnt expecting that close! looks like 20 pounds of propane is roughly equivalent to 70 pounds of firewood (if I did the firewood right) and that either one would be enough to heat that hot tub (up from ambient outdoors temperature to something comfortable)

if you would like to check this using CRC handbook values for propane, you just need to know that one of our pounds is 434 gram.
and the natural energy unit comes to 390 MJ.

 

[marcus]

doing a little editing on that list of benchmarks

speeds:
E-9  2/3 mph
E-7  67 mph
E-6   sound in cold air
E-4  Earth orbit

masses:
E8  "pound" (434 gram)
2E-22 electron
1/(2.6E18) proton
1.1E31 europa, 1.7E31 moon
1.38E33 earth, 4.6E38  sun

energies:

E-28 is an "eQ" (electron "quartervolt")
10E-28 is energy of a green photon (8 eQ for red and 12eQ for blue)
17E-28 is yield from one O[sub]2[/sub] in combustion or metabolism
(of a number of common hydrocarbon and carbohydrate fuel/food)
E-8 unit is about one lab calorie (approx. 4 Joules)
E-5 is a food calorie
1 unit:  70 pounds of firewood, 20 pounds propane, heats the tub
50 units:  Tempel 1 mission collision impact

lengths:
E33 is a handbreadth, 8.1026 cm.
E27 is  one "microhand" the (angular) wavelength of green light
E34 is one step (half a conventional pace)
E37: conventional halfmile
7.86E40  average radius of earth, 7860 "halfmiles"

area:
E66:  sq hand

volumes:
E99:  cubic hand, "pint", 532 cc (1880 to the cubic meter)
E102 is a cubic step (for visualizing larger volumes) 0.532 cubic meter

temperatures:
E-32 half a Fahrenheit step (above absolute zero)
9.6E-32 cosmic microwave background
E-29  approx. Earth surface average, 49 Fahrenheit
2E-28  solar surface

forces:
E-43  half a Newton
1/137   idealized: force betw. pair of electrons sep by unit dist.
(to make this more realistic, divide by the square of a larger separation,
like E27, the wavelength of green light.)
(1/137)E-54  force betw. pair of unit charge separated by E27

power:
E-52   one sixth of a watt

acceleration:
0.88E-50   Earth surface gravity

current:
E-24  about 0.6 amp 

pressure:
1.4E-106  typical air pressure at Earth surface

some of these figures, e.g. the mass of the sun, are ones we calculated in earlier posts of this thread.
I have read that the Tempel 1 impact is supposed to be the equivalent of some 4 and 1/2 tons TNT. I calculated 50 natural energy units so according to that, a tonTNT is around 11 energy units. wonder if that's right? We could check it, sounds about right order magnitude. It would make a ton TNT equiv to about 770 pounds of firewood which might seem off until you recall that the firewood gets to burn. Maybe i will be able to check this directly.
BTW the natural energy unit metric equiv is 390 MJ.

Hmmm, this site
http://www.sengpielaudio.com/calculator-energyunits.htm
says a ton TNT is 4.18 GJ
That agrees with what I just estimated because it says a ton TNT is 10.7 energy units-----about 11.

 

[marcus]

the energy of Pizza

you are about to scarf up a delicious 400 Calorie slice of gooey pizza, loaded with toppings and you pause to ask what is the food energy of this slice of pizza, expressed (of course) in natural units.

As it happens you can immediately reply, "Why 400 x 10^-5, to be sure!"

Because 10^-5 of the natural energy unit IS approximately equal to one food Calorie. (in terms of lab calories, that is a kilocalorie)

so pizza = 4E-3 E

If someone were in need of more precision one could say that E-8 of the natural units is 3.902 joules----while a calorie is a little bigger, more like 4.184 joules. but one is never too precise about the Calories in pizza.

 

[marcus]

seemed like a nice bit of serendipity that the natural unit of energy turned out to be roughly 100 thousand food Calories

makes it easy to do an order-of-magnitude check on that heating-the-tub calculation. the assumed volume of water was a one meter deep and meter radius cylinder----pi cubic meters---3142 liters---each of them takes about one kilocalorie (a food Calorie) to raise one Celsius step.
the temperature increase was assumed (in natural units terms) to be from a chilly ambient 1.00E-29 to a nice hot 1.11E-29. In metric terms this is a rise of 31 celsius (in conventional fahrenheit going from 49F to 104F, a comfortable hottub setting)
Multiplying 3142 by 31 shows that heating the tub takes essentially 100 thousand Calories---short by less than 3 percent.

so we got the right answer by the somewhat unfamilar route of assuming
that water's heat capacity was 9k per molecule (3k per atom if you like) and thinking of this volume of water as 10²⁹ molecules.

 

[marcus]

it is spring, buds are out on the peach tree (always gets loaded with fruit, which i look forward to in August)
as i walked down the flagstone steps through the garden the sun warmed my back---the air is chilly but suddenly there is more light and warmth.

I want to know the power density of sunlight in natural units

the easiest thing to remember is that the solar surface temperature is 2E-28, and that the angular size of the sun's radius seen from Earth is 1/215 radians.

if the sun's temp were significantly less (or more) than 2E-28 it would look reddish (or bluish) and the number 215 just indicates how big the sun looks in the sky, about the same size as the full moon. So these are direct visual things that I am always aware of someplace in my mind.

and they tell me the solar constant easily enough---it's just that it will turn out to be a really small number because the natural unit of power is so huge. the rate energy is delivered by sunlight, per unit area, will turn out to be a real small number by universe standards. well it warms my back so I am not going to quarrel, it is how it is.

I did this before but today is the first springlike day of 2005 and it won't hurt to repeat

\text{sunlight power per unit area} = (\frac{1}{215})^2 \times \frac{\pi^2}{60}\times (2\times 10^{-28})^4 \text{power unit per area unit}

that wouldn't be so small if we used and a more human scale unit of power----like E-52 of the natural one, about one sixth of a watt---and a larger area, like a "square handbreadth", the hand being some 8.1 centimeters. This palmsized patch of area is E66 times the natural unit. If we changed to humanscale units----multiplying by E66 and dividing by E-52---we'd simply get 57 practical power units (roughly 10 watts) on a palmsized area and that would be that.

as it works out in natural, the solar constant is 5.7 E-117
changing to humanscale units means multiplying by (E66 x E52 =) E118, so that's where the 57 comes from

I just went out in the sunlight in the garden again to think it over---could I stand to have sunlight be such a small number as 5.7E-117. A lot of people wouldn't understand that----why you'd care.

the natural unit of power could make our galaxy in a few minutes---supply the mass-energy comprising all its stars---so I guess its all right

 

[marcus]

in papers on quantum gravity including semiclassical stuff going back to 1970s (hawking temp, unruh radiation, black holeology) it has become common, even typical, to use units in which the basic constants hbar, c, Boltzmann k, and either G or 8piG, have unit value. So in effect either conventional Planck units or a variant system is gaining currency in a cluster of related research areas.
Assuming humans eventually find a workable quantum theory of spacetime, and the rest of physics is carried over and rebuilt on that foundation, a system of natural units (like the one I am experimenting with in this thread) will be embedded in our picture of nature

what I want to do is to scrape away the dirt and see how the intrinsic scales look and what it's like to use them.
the first thing that becomes obvious is that they are unfamiliar sizes, which has led me to use "named powers of ten" as handles on them

the natural time unit is tiny so I think of E42 time units and call it a "count", and it turns out that for practical purposes there are 222 counts to an ordinary minute.
the natural length unit likewise so I think of E33 of them as a handbreadth, and E37 as a halfmile. A "halfmile" turns out to be 810.26 meters which is pretty close to half an ordinary mile, so it works reasonably well.

these are just some ways to keep from getting lost in a wilderness of extremes.

Since Saturn's moon Titan is on people's minds these days, I'm going to see how it's physical characteristics would look in terms based on natural units

 

[marcus]

NASA-JPL has online data for Titan. they list GM instead of M which makes sense for astronomers and for accurate navigation purposes. I will translate, and fill in some from other sources:

GM(halfmile³/count²) 1233
Radius (halfmiles) 3180
Surface temp (degrees) 337
Distance from Saturn (halfmiles) 1,508,000
Orbit period (counts) 5.1 million
Surface pressure (natural) 2.2E-106

If you had to, you could probably convert these to raw natural units---halfmile means E37 length units, count means E42 time units, a degree is E-32 on the natural temp scale. I haven't figured out what to do about pressure---earth normal is 1.4E-106 and this estimate for Titan is 60 percent higher. BTW that should be revised using Huygens data, if anyone hears better figures for surface temp and pressure please tell me.
Also the composition of the atmosphere---so far all I hear is that it is mostly nitrogen (but with plenty of hydrocarbons giving the smoggy look)

http://ssd.jpl.nasa.gov/sat_props.html
http://ssd.jpl.nasa.gov/sat_elem.html

 

[marcus]

I'm feeling discouraged about this system of natural units. Lots of nice features but normal air pressure 1.4E-106 and power density of sunlight
5.7E-117 are so outlandishly small, and hard to remember.

But I will try to keep in practice with the system by finding out the speed of sound on Titan.

Earlier we found a typical speed of sound at Earth surface using the fact that a commonplace temperature for air at Earth surface is E-29, and also the average molecular weight of air is 29

Until they tell us different i assume Titan atmospher is mainly nitrogen N2
this means that in natural mass unit the average molecule mass is
28/(2.6E18)

In any system of units a standard speed of sound formula (for a biatomic gas like nitrogen) is

\text{speed of sound} = \sqrt{\frac{\frac{7}{5}kT}{\text{mass of molecule}}}

So if the Titan surface temp really is 337---which in natural terms is 337E-32, or 3.37E-30----then

\text{speed of sound on Titan} = \sqrt{\frac{\frac{7}{5}\times 3.37 \times 10^{-30}}{28/(2.6E18)}}

\text{speed of sound} = \sqrt{\frac{7}{5}\times 3.37\times 10^{-30}\times 2.6 \times 10^{18} \div 28}

\text{speed of sound} = \sqrt{4.38 \times 10^{-13}}

\text{speed of sound} = 6.6 \times 10^{-7} c

it comes out 0.66 millionths, and that is millionths of the speed of light because c is the natural speed unit

when we did it for typical conditions at the Earth surface it came to 1.12 millionths. not dramatically different: Titan's somewhat more than half Earth's

 

[marcus]

Let's find the escape velocity from Titan surface
here's some of the data posted earlier:

GM(halfmile³/count²) 1233
Radius (halfmiles) 3180
Surface temp (degrees) 337
Distance from Saturn (halfmiles) 1,508,000
Orbit period (counts) 5.1 million


\text{escape speed} = \sqrt{\frac{2GM}{R}}

\text{escape speed} = \sqrt{\frac{2466}{3180}}

that comes to 0.88 halfmile per count. Argh. what I want is a
fraction of the speed of light. Yes I believe it is 0.88E-5 c
8.8 millionths of the speed of light. which is right. that is what escape speed from Titan surface is.

I am still learning how to cope with these units. If we use some named powers of ten, some crutches, like "halfmile" (E37) and "count"(E42)
then I have to know that the speed of light is 100,000 halfmiles per count.

maybe instead of crutches the better word is "bridges"---they help relate the quantum gravity scale with the everyday scale. the universal with the practical.

 

[nightcleaner]

Hi Marcus

I have been following this thread with interest. Your expositions are very clear and understandable, and make good reading. I am sorry to hear you are feeling discouraged, but, in a slightly jealous sort of way, am also glad for you to be able to feel the ridiculous-ly small touch of spring sunshine on your shoulder. As a strict Northerner, of course, I have to look on your talk of spring with some disdain. It is warmer here, too, only fifteen below zero tonight where last night at this time it was below -thirtytwo. Never the less, I am really happy for you in your garden with thoughts of peaches to come, and thank you for sharing that warm moment.

Now about this natural unit exposition. In my mind, the reason for natural units is to take a step toward removing anthropomorphisms from measurement, and hence calculations. So I am a little wary of your named powers of ten, halfmile and count and palm and pint and so on. Well as bridges they are fine, but one of the goals is to remove conversion factors from the formulae. I am concerned that the conversion factors will sneak back in by way of your bridges.

Btw, I wonder, if there is a name for the fear of very large numbers, googleplexophobia or something.

Which brings me to the reason for my interruption of this beautiful thread. The use of natural units makes calculations easier in part because the dimensional analysis isn't cluttered up by conversions. The other part, aside from the calculations, is the lifting of reason out of the strictly human domain, and in so doing, the lifting of human thought above locally restrictive surfaces.

I imagine the real value of natural units will become apparent when working with the extremes where general relativity and quantum mechanics are thought to meet. Perhaps your analysis might extend to Black Holes and other singularities?

Thank you in any case for this interesting reading material.

Richard

 

[marcus]

...last night at this time it was below -thirtytwo...

Now about this natural unit exposition. In my mind, the reason for natural units is to take a step toward removing anthropomorphisms from measurement, and hence calculations. So I am a little wary of your named powers of ten, halfmile and count and palm and pint and so on. Well as bridges they are fine, but one of the goals is to remove conversion factors from the formulae. I am concerned that the conversion factors will sneak back in by way of your bridges.

Btw, I wonder, if there is a name for the fear of very large numbers, googleplexophobia or something.

...

thanks for these comments, which are astute and thoughtful: the hazards of bridges, the need for wariness

aversion to large numbers----like ambient air pressure being 1.4E-106---
is in part reasonable because based on one's limited memory resources.

if you know that all the numbers you need to remember are between one and twenty then it can be somewhat reassuring-----but if some of them are like 106 then there is a feeling of memory resources being stretched too thin.
I don't want to over-emphasize this, though, because it may be possible to accommodate it, with a little practice.