Using the force constant in equations

[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.

 

[marcus]

thinking about minus 32 (winter night in upper Wisconsin)
what is it on the natural scale

I've been using 1.000 E-29 as a reference point: it works out to 49 Fahrenheit, and bumping the last digit up or down is like half a Fahrenheit degree

so 1.002 E-29 is 50 Fahr.
1.004 E-29 is 51 Fahr.
1.040 E-29 is twenty Fahrenheit steps up from 49, so room temperature 69 F.

but your minus 32 is 81 Fahrenheit steps DOWN from that reference point. so i double that: 162
and have to subtract

1.000
0.162
0.838

so in these very absolute terms---one a scale where 0 is absolute zero and 1 is more or less the temperature people estimate for the big bang, some kind of universal top temperature, the temperatures we directly experience are in a narrow range around E-29

winter night in Midwest 0.838 E-29
thaw 0.966 E-29
49 F benchmark 1.000 E-29
room temp 1.040 E-29
body temp 1.100 E-29
hot tub (104 F) 1.110 E-29

surface of sun 2 E-28

in a way you experience the temperature of the sun's surface through the color of the light (like when you look into a furnace or kiln you see the temperature somewhat by the color, and by the brightness that the bricks glow) so I enlarged the range to include that

but it really is a pretty narrow range around E-29, in universe terms, that we actually feel.

 

[marcus]

back in post #59 Richard made a case for caution or restraint in using named power of ten. there were a couple of points, one was the anthropomorphism when one's main desire might be to describe the universe in its own (non-human) terms----appreciate its intrinsic proportions if that is possible.

well having named powers of ten interferes with that, clearly, but it still does leave it as an option
if you initially learn things in terms of a handbreadth (which is E33 natural units of length, i.e. 8.1 cm) or a pace (E34 natural, 81 cm) then if you have a need or desire to transcend the humanscale references you can convert to natural units by changing the exponents in your data to include the factor of E33 or E34.
but it is a point. there are reasons to favor the purely natural units route, with no crutches or anthropomorphic bridges.

anyway my height is 22 hands
(or it is 22 E33)
and my mass is (I am sorry to say) 220 pounds
or it is 220 E8.

but these pounds are just 434 gram ones, and besides it the fault of the Seasonal Holidays, i swear it! And I rounded up, I'm not really 220 (even in these small pounds!) and besides, it's coming off.

well, in the end one just has to discover by trying out which works and feels better.
To say that I am 22 hands and 220 pounds,
or to say that I am 22 E33 height units and 220 E8 mass units.

(assuming one wants to gain some acquaintance with these natural units in the first place)

above all (following the line of Richard's comment) remember that named powers of ten are OPTIONAL means of assimilating the natural system and that even if one does use them as a help, one should sometimes make a point of NOT using them, so as not to get too habituated, and adhering to the pure, or raw, natural units.

 

[marcus]

my main dissatisfaction with this system of natural units may seem frivolous or idiosyncratic to some people (but the topic only interests a few people anyway so maybe I shouldn't worry)

what bothers me is that Earth atmospheric pressure is 1.4 E-106
natural pressure units

and the "solar constant" or brightness of sunlight is 5.7 E-117
natural energy units per unit area per unit time.

this solar constant thing is what you sometimes see written as 1370 watts per square meter. or 1380----something like that. and watt means joules per second, so that is 1370 joules per square meter per second.
it is the rate at which it warms your back up when you walk out of the woods into a clearing.

I have decided to take drastic measures.

there will be a named power of ten which is a considerable force.
sort of like the weight of a 50 kilo sack of cement.
You may well object. there will be this force which is 10^-40
of the Big Force, the natural unit.
this named decimal fraction E-40 of the natural force unit will be
more than most people can lift easily and it will be called a BAG of force.
what a gross name.

Not too long ago (for me to remember it vividly) we had to lay some concrete and I was carrying 50 pound and 70 pound bags of mix (the stuff called "quik-crete" that has the sand and gravel already in)
and a Bag of force is more force and therefore even worse

but anyway, air pressure on Earth surface is 1.4 Bags per sq. hand.

You take a handbreadth 8.1 cm, and you make a square, and that is how much the air presses on it: 1.4 Bags of force.

the conversion is easy because Bag = E-40
and hand is E33, so sq hand is E66 (natural units of area, think of graph paper)
and Bag per sq. hand is E-40/E66
which is E-106

so "1.4 bags per sq hand" is definitely and unmistakably 1.4E-106 natural force unit per area unit.

the google-phobe in me groans with relief, ahhhh, that's better.

 

[marcus]

this means we have another handle on the Great Force that we started this thread with, in post #1

this is the force which is the central coefficient in General Relativity.

it is 10⁴⁰ Bags

I sincerely hope it doesn't mind being called that.

this force is the coefficient in the Einstein equation that tells how matter curves spacetime----it's our model of how gravity works, and how the shape of the universe evolves, very basic.

And I have this idea of a Bag of force which in metric terms is 480 Newtons, or the weight in Earth gravity of right around 50 kilo.
and this Universal Force Constant, it turns out, is just
ten-to-the-forty Bags.

it is what curves space, and space doesn't bend easily (by our human standards) so that is why the force is so big (by our standards)

it tells us, so to speak, the "stiffness" of space


it also tells us the maximum force with which two distinct things can attract each other while still remaining two different things
(any more attraction and they collapse the space between them and merge)
you can calculate it by studying two black holes rushing into each other, the attraction they experience just before they merge.
so it is a notion of maximal force

but it is also a notion of stiffness-----given a curvature, multiplying that curvature by the force produces an energy density-----telling what concentration of energy would be needed to produce that much curvature.

It takes, by our standards, a lot of concentration of energy to produce some curvature, because, by our standards, the force you multiply by is big.

so that is what ten-to-the-forty Bags (of cement) is about.

 

[marcus]

just a little reminder about Lambda----the cosmological constant is another one of these things that is so extremely small, one often hears that it is vaguely somewhere around 10^-120
but despite miniscule size, it is important to the past and future evolution of the universe.
It is the very slight intrinsic curvature the world has which is not (as far as we can tell) due to any matter or energy we can measure

or perhaps this curvature is due to a postulated transparent and invisible "dark energy" that is evenly distributed thoughout space, in the usual way that energy curves the universe.

I want to know what that Lambda is, expressed in these natural units.

just to flavor the discussion, today Tanmay Vachaspati of Case Western Reserve posted a new idea of how the universe came into existence.

Islands in the Lambda Sea
http://arxiv.org/astro-ph/0501396

Tanmay has published some 40 papers, some co-authored with such people as Mark Trodden, Lawrence Krauss, and Alex Vilenkin.

We live in a time when new ideas of Cosmogony are very common, popping up all the time, and they all involve some intersection of General Relativity and Quantum Theory. General Relativity is our prevaling theory of spacetime and how gravity works so all Cosmologies arise in that framework, naturally enough, but they all seem to have some Quantum element too: a "quantum fluctuation" in some field, or a "quantum bounce".

the vision of Tanmay and his co-author, is at least tranquil and may in fact be beautiful in a sense: there is nothing but an empty expanding universe with the very same Lambda we observe today---everywhere. and then
a little fluctuation, it does not have to be a scalar field, even a photon will do----and as Tanmay tells the story all this that we see comes from that and then (his equations tell us) gradually over trillions of years, it all fades away again and goes back to an expanding emptiness, with nothing in it but Lambda, once again.

and there can be several islands that appear and eventually fade out, in this Lambda sea---and this same Lambda is in and around them all, in the expanding emptiness that surrounds and separates the islands

it has a similar visual format to some other pictures we've seen (like eternal inflation) but it doesn't need an "inflaton" scalar field (a kind of mythical beast like the unicorn which other stories invoke but has never been seen) and it is different enough to be Vachaspati own idea---well read it and see if you think so too.

Now Trodden and Krauss and especially Vilenkin are extremely reputable people and they have each written papers with this guy. What it means to me is that we are going to see more and more of these ideas, including very imaginative and possibly beautiful ones. And there will be no way to choose for a long time which one is the best fit to Real Things. We are going to keep on seeing them and some will look pretty sexy and we will not know which is right for a long time.

So I am just going to calculate the value of Lambda in natural units, because at least there is that. It is an unexplained thing that seems to be there in the world and also in every one of these theories, which a theory must eventually explain (if it is at all adequate) why Lambda is the number that it is.

Baez was asking about what is Lambda in Planck terms just a while ago, here is a PF thread related to that
https://www.physicsforums.com/showthread.php?t=50108

In this thread our natural units are variants of Planck units (for better or worse) so it doesn't come out exactly the same as in conventional Planck.
Lambda comes out to be 8.46 x 10^-121 in natural units of curvature----and as rough idea of the size, that is pretty close to 10^-120 which is what quite a few people seem to have been saying
all along.

(E-120 is in the same ballpark as 0.846 E-120, so good enough for some purposes, and easier to remember)

 

[marcus]

the universal force constant, which in some sense this thread is about (using it as a natural unit of force)
how far do you imagine that force would have to push to do a certain large amount of work------like enough work to bring the sun into existence?

imagine the Force harnessed, like a tractor, and it is pulling a cable that unwinds off a spindle and turns an electric generator. But all this is way epic scale. And the energy generated by the Force is used to make matter---by E = mc²----and becomes the energy invested in the existence of particles. And as those particles accumulate they coallesce to form the Sun.

Energy is equal to the product of force and distance-----a certain force pushing for a certain distance defines a quantity of work.
how far does the force have to push in order to deliver a quantity of energy equal to all the energy bound up in the sun?

I think the answer is around 23 miles. I have to check, but I'm reasonably sure of it.

 

[marcus]

Baez was asking about what is Lambda in Planck terms just a while ago, here is a PF thread related to that
https://www.physicsforums.com/showthread.php?t=50108

In this thread our natural units are variants of Planck units (for better or worse) so it doesn't come out exactly the same as in conventional Planck.
Lambda comes out to be 8.46 x 10^-121 in natural units of curvature----and as rough idea of the size, that is pretty close to 10^-120 which is what quite a few people seem to have been saying
all along.

(E-120 is in the same ballpark as 0.846 E-120, so good enough for some purposes, and easier to remember)

In that "Planck Lambda" thread I just checked that in these units the cosmological constant and the dark energy density have the same numerical value, which is convenient. You only have to remember one number. And the number is essentially just E-120 which is comparatively easy to remember. I tried visualizing the dark energy density as E-10 calorie in a pace-size box.

It is also one tenth calorie in a half-mile cube.

figure it this way my fat old Webster's says a traditional regulation "pace" is 30 inches (a "Roman" pace is two steps, twice that, but this is our kind) so "pacing off" some distance is counting steps and 1000 paces is HALF A MILE. Now in these natural units a pace of 81 centimeters or 32 inches is E34.

a half mile is E37

so the halfmile cube volume is E37 x E37 x E37 = E111

and the dark energy density, being E-120, can be written
E-9/E111

so in a halfmile cube the amount of dark energy is E-9, one billionth of a natural unit of energy.

We already played around with this natural unit of energy and discovered that it is 100 million calories (roughly, close enough anyway) and a calorie is E-8 of the natural unit.

so E-9 is one tenth of a calorie, which is the amount of darkness in this halfmile cube


Well, since we all eat calories (actually we measure food in food Calories which are kilocalories, but yeah we sort of know calories) we have a direct handle on the "observed" dark energy density.
========

I don't believe dark energy exists. It could, maybe it will be proven, but I am skeptical and I can't accept it based on evidence to date. What I have more confidence about is Lambda, the cosmological constant, which is a CURVATURE.

that curvature might or might not be produced by some uniformly distributed form of energy called dark energy. maybe it is not the effect of any stuff, maybe it is just an intrinsic curvature that the world has, for some other reason. I am skeptical about postulating an impalpable energy to attribute the curvature to.

curvature is the reciprocal of area.

let's look at Lambda directly. What is the area that is the reciprocal of E-120 natural units of curvature?

 

[marcus]

well obviously the area (reciprocal to Lamda, one over the cosmological constant) is E120 natural area units

and that would be the area of a square which is E60 length units on a side.

we were using paces and halfmiles to visualize, before.
a pace is E34 and a halfmile is E37
so the length of the side of this square is E23 halfmiles.
and that is still mindboggling, so let's try light traveltime

in the natural system c = 1 so the distance E60 (the side of the square to be visualized) is exactly the distance light travels in E60 time units.

all we had about the natural time unit is that E42 is a "count" and there are 222 of them to a minute. this makes around 117 million to the year.

So, hey this is not bad, a year is 1.17 E50 time units
a lightyear is 1.17 E50 length units.
for order of magnitude we can just call a lightyear E50.

so the side of this square is E60 natural
which we can interpret (order of magnitude) as E10 lightyears.

OK I know, it is still boggling huge. A square 10 billion lightyears on a side.
the cosmological constant is the curvature which is ONE OVER this vast area. It is a tiny tiny curvature. We (or at least I) do not ever grasp it, we can only make a more or less gallant, and more or less awkward, attempt.

 

[marcus]

yesterday i took a bird feather out to an undeveloped canyon where there is this old peach tree which still blossoms
I tried to reach the blossoms to tickle them, the way a bee would,except that from a bee's point of view it might still be too cold ( sometimes there arent enough bees at this time of year)
the tree is between 50 and 100 years old and is half dead and has fallen over so the trunk is horizontal, out over a sharp dropoff, but the part that is still alive is still blossoming copiously every year. I guess the canyon used to be farm. but it is all overgrown now with brush and coyote bush and eucalyptus and bay. you should watch out for the poison oak.

I want to think about the natural unit of power---the rate of delivering energy, like the calories per unit time.
A "count" is about as fast as you can count outloud, say counting repeatedly up to 20. and a count is E42
So in one count, the natural unit power delivers E42 units of energy.

(each one can heat JB's hot tub, so E42 a huge amount of energy)

remember that one natural unit is E8 calories. so one can say that in one count it delivers E50 calories, if you like.

So as you count rapidly, with each number you say, the power brings E50 calories.

but how to visualize that much energy. For instance, how does it compare with mass-energy invested in the existence of the sun? Every particle of matter has some energy bound up in its very existence which is released if that particle experiences annihilation. By our standards it is quite a lot, even a small amount of mass (by our standards) when annihilated releases a large (by our standards) energy. If the sun went out of existence with a flash. What.

As you count, with each number you say, the power brings you the energy that would be needed to create how many suns?

I think it is 2000 suns, but i will have to check.

well that is about right, you are counting as fast as you can (222 counts a minute) and everytime you say a number
the natural power gives you 2000 suns

it gives you the energy that would be set free if 2000 suns suddenly went out of existence (not just the energy which they would produce in their lifetimes which is only a very small fraction of their total mass-energy)

so in not too long a time (maybe a couple hundred days) this power could deliver a galaxy-equivalent of energy.

I had better check that the number 2000 is right, or find out what it is more precisely just to be sure of not being too far off

-----
I did check it and it seems all right.
this time it actually seems simpler if you do NOT use calories or any of that, but stick with natural units. We already worked out that the mass of the sun is 4.6E38, and because c = 1 that means the ENERGY of the sun is the same number 4.6E38 units of energy. And at every count you get E42 units of energy. So how many suns-worth is that?

Easy just divide E42 by 4.6E38.

I get 2170 suns-worth. So about 2000.

in a minute (222 counts) you get about half a million suns.
That is enough about that for a while!

 

[marcus]

today's best estimate of the Hubble parameter is 71 km/second per Mpc. This is the reciprocal of a time, called the "Hubble time" which you sometimes see listed as 13.77 billion years, or some such. It is not the estimated age of the universe, although the figures are close, but just one over the Hubble parameter.

in natural units, a year of 365.25 days is 1.1676 E50
I am keeping unnecessary precision to round off later.
It seems handy that a year comes out close to E50 time units
and it makes a lightyear also come out roughly E50 length units.

this figure they have for the Hubble, of 71, translates to
a Hubble time of 1.6080 E60 natural time units. (keeping spurious precision again for later rounding)

if you divide that by the length of a year you do get 13.77 billion years, so it checks.

I'm thinking that the reciprocal of 1.6E60 might actually be a very handy figure for the Hubble. I will use it to calculate the critical density---the density ( including dark energy) we theoretically must have in order for the universe to be spatially flat.
I actually think that by the Friedmann equation it must be

3/(1.6E60)²------you just do 3H² (other stuff is one)

it comes out 1.16 E-120

By WMAP and other means they think they observe that as the actual density of the universe, and that dark energy is 73 percent of it. The 0.73 is another WMAP number. well what a nice surprise , 73 percent of 1.16E-120 is the figure we got for dark energy density earlier.
0.85 E-120
you may be disappointed that this is all so trivial, please excuse, it's just me getting used to these units. I hadnt realized that a year was E50 and the Hubble time was of order E60 and the critical density was around E-120 (rho crit a little more, and rho Lambda a little less)

A couple of years back when Baez opined, on SPR, that Planck units would be better if 8piG = 1 they didnt bother to see how it would work out, they argued abstractly, and voices rallied to the flag of ancient custom. I think eventually he was persuaded to drop it. More important matters to discuss.

Still many's the Loop Quantum Gravity paper I see that has a kappa in it standing for 8pi G_N, and that kappa is not unlikely to be referred to as "the gravitational constant" at some point and set equal to one by a change of units.