Using the force constant in equations

[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.

 

[marcus]

Yuk, this is hard work. and because the project is eccentric if not bizarre it is mostly in social isolation (but thanks to those who have dropped in so far!)

I am checking to see about these natural units----like ordinary Planck but with |F|=1
instead of the more usual |G|=1

First, they do seem to work better than conventional Planck and this is confirmed by what I notice in Quantum Gravity articles. Increasingly I notice a kappa ("gravitational constant") which is 8piG. And which can be set to equal one to further simplify the equations. The Newtonian G is yielding a little---not yet a secondary constant but not as predominant.

the moment one sets
|F|= |c|=|hbar|=|k|=|e|=1
one has a fairly universal set of units and it is interesting to see what some familiar quantities come out to be.

In part I am just interested in rough sizes, in part I want to know basic constants like Hubble parameter, proton mass, cosmological constant, in these terms because oddly enough I've found it is sometimes actually convenient to work with data that way

rough sizes:

pound E8
year E50
handbreadth E33
pace (32 inch) E34
halfmile E37
lightyear E50
food Calorie E-5
lab calorie E-8
quartervolt E-28
green photon energy 10E-28
average Earth surface temp E-29
2/3 mph E-9
67 mph E-7
cold air speed of sound E-6
one "gee" acceleration E-50
weight of 50 kg sack of cement E-40
power of 160 watt bulb E-49

some constants (approx.):

reciprocal proton mass 2.6E18
electron mass 2.1E-22
Hubble time 1.6E60
Lambda 0.85 E-120
rho-Lambda 0.85 E-120
rho-crit (critical density) 1.16 E-120
more exact Earth year 1.1676 E50
more exact lightyear 1.1676 E50
avg Earth orbit speed E-4
earth mass 1.38 E33
earth radius 7.86 E40
sun mass 4.6 E38
solar surface temp 2.0E-28
CMB temperature 9.6E-32
earth surface pressure 1.4E-106
earth surface gravity 0.88E-50
fuel energy released by one O₂ 17E-28
density of water 1.225 E8/E99

the time scale is important enough to treat separately:
1/222 of a minute E42
4.5 minutes E45
(to have a named power-of-ten for a convenient time interval, imagine counting out loud rapidly, at the rate of 222 counts a minute, each count is E42 natural time units. A thousand counts is 4 and 1/2 minutes. It just happens that one year is roughly E8 counts, or E50 natural.)

Named powers of ten can help assimilate and remember quantities expressed in natural units. The way I remember Earth surface air pressure is to think of the weight of a sack of cement (E-40) on a sq. handwidth area (E66) which gives me an idea of the pressure E-106, and it is 1.4 of those.
I remember the density of water as 1.225 E8/E99, that is somewhat more than one "pound" per
pint-size cubic handbreadth volume, which simplifies to 1.225E-91.
Richard suggested that a distaste for extreme numbers be called "googlephobia"---isn't google
somebody's name for 10¹⁰⁰?
I think of the Earth radius as 7860 halfmiles (a halfmile being E37)
instead of 7.86E40. another way of coping with googlephobia, or of
bridging between humanscale and natural.

 

[marcus]

special numbers

some special numbers go with these units, most are pure math numbers and would be factors in the equation no matter what system of units, but here they sometimes jump out a little more clearly.

80/pi this tells the evap time of a BH. cube the mass and multiply by 80/pi

pi²/15 tells the per-volume radiant energy density at some temp. quart the temp (raise to fourth) and multiply by pi²/15

pi²/60 tells the brightness at some temp (power radiated per unit area). quart the temp (raise to fourth) and multiply by pi²/60

3zeta(4)/zeta(3) = 2.701 tells the average photon energy at some temp.
just multiply the temperature by 2.701. Since sun temp is 2E-28, the average sunlight photon has energy 5.402E-28---anyway that's the idea.

1 tells the bekenhawking temperature of a BH. just take 1 over the mass.

1/4pi tells the Schw. radius of a BH. just take that times the mass.

1/4pi tells the area of the BH. take that times the square of the mass.

3 tells the critical density of the universe. just multiply 3 by the square of the Hubble parameter

6 tells the density of a round planet. divide 6 by the square of the radian time in low orbit.

9 or thereabouts is the heat capacity of a molecule of water

29 is the molecular weight of air. It is handy to know.
(atomic and molecular weights generally are)

Oh, they tell us that the density of the universe is at or very close to the critical value. So 3 also tells the actual density of the universe.

1/137 (more exactly 1/137.036...) is the coulomb constant. it tells the force between two charges separated by a distance. just multiply the charges by 1/137 and divide by the square of the distance.

1/137 also tells the force between parallel currents (measured on a test segment with length equal half the separation). just multiply the currents by 1/137

(1/137)² tells the energy needed to ionize a hydrogen atom. multiply the rest energy of an electron (2.1E-22) by it and you get a quantity of energy called the Hartree----which is twice the ionization energy (so you still need to divide by two)

in each case i am assuming that the calculation is done in natural units terms, so that I don't have to specify the units each time I say something.

there's lots more but maybe this is enough for now

 

[marcus]

finally a way to remember the solar constant

Back in post #71 I listed some rough sizes, including these force and power benchmarks.

weight of 50 kg sack of cement E-40
power of a 160 watt lightbulb E-49

I am thinking of the force E-40 as a "sack" force benchmark
and imagine a 50 kg weight on a pulley descending at speed E-9
(which is 2/3 mph, or a billionth of the speed of light)
and as it descends it does work, like turning a spindle, maybe even
generating electricity.

The power output of that descending weight is E-49. To see that,
you just have to multiply the force E-40 by the speed E-9
and you get the power. of course if you are generating electricity there
will be some loss because of inefficiencies.
but basically this force exerted at that speed delivers that much power.
and I'm going to call that level of power a BULB of power.

this is a drastic solution to the problem of remembering the brightness of sunlight. the solar constant at this distance from the sun---the power per unit area delivered by direct unattenuated sunlight----is 5.7 BULBS PER SQUARE PACE.

In natural unit terms, a pace (81 cm) is E34 and a square pace is E68 and a bulb of power is E-49. So a bulb of power spread over a square pace is
E-49/E68 = E-117
I am saying that the brightness of sunlight is 5.7 times that.
It is like about SIX of those 160 watt litebulbs set in a pace-wide square.

In natural units, 5.7E-117 is what the handbook value of the solar constant actually turns out to be. but I don't find that so easy to remember. So I visualize it as 5.7 bulbs per sq. pace.

A pace is just one of my steps----around 32 inches----so I can easily pace out a square that size on the flagstones in the garden. It is an easy area for me to visualize. and the litebulbs are easy to visualize. so I have a visual handle on this 5.7E-117
===================

In the "Force" system of natural units, the unit of power is of course E49 bulbs (because bulb was defined as E-49) and it is the power delivered by the unit Force pushing at the speed of light.

this is a lot of power and if you count as fast as you probably can outloud, say 222 counts a minute, then WITH EVERY COUNT UNIT POWER DELIVERS ENOUGH ENERGY TO CREATE 2000 SUNS.

We discussed this, it is enough power to create a galaxy in something on the order of 100 days. or if you wanted to produce such a power by annihilating stars and converting their whole mass into energy then you would have to annihilate about 2000 stars like the sun with every count.

As with conventional Planck units, these natural units are fundamentally Big Bang-scale. the temperature, the density, the pressure, ...and so on...are mostly at the level of big bang conditions. I guess that could be seen as reassuring. You can be sure ahead of time that you are not going to encounter any temperature less than zero or greater than one. the physical scales tend to be bounded between zero and one----like with speed too.

 

[marcus]

An explorer once visited three planets and went into low orbit around each in order to take pictures with his digital camera to put on his website. He finds each planet more delightful than the one before it and, while skimming around the third, he breaks down and calls you on the cellphone.
Hello, says the explorer, on each planet it took a different length of time to travel one radian of the low orbit. In natural time units it took
7E45
4E45
and 3E45
on planets A, B, and C respectively.
what are the densities of the three planets?

the formula for the density is D = 6/T²
you just divide 6 by the square of the radian time, so the three densities are
6/(49E90) = 1.224E-91
6/(16E90) = 3.75E-91
and 6/(9E90) = 6.66E-91

the first, you tell the explorer, is virtually the same the density of water.
planet B, on the other hand, is slightly over 3 times the density of water and is therefore comparable to many of the solar system's satellites including the Earth's moon
planet C, however, is 5.4 times the density of water, quite close to Earth itself, which is 5.5!
Indeed, says the explorer, that is within experimental error. I believe I am just passing over Sausalito.

BTW E45 natural time units was listed a couple of posts back as lasting 4.5 minutes, so the radiantime for low orbit in the earth-like case, namely 3E45, becomes 3 x 4.5 = 13.5 minutes. that is for a hedgetop skimming orbit neglecting air--- not practical, of course, but raising it above the atmosphere does not make the orbit all that much slower. so it is a pretty good estimate. (multiply 13.5 minutes by 2pi to get the period)

 

[marcus]

Another time the explorer cruises by the nightsides of each of 3 planets in order to gauge the infrared heat brightness from each. He wants to know how warm or cold they are. He finds that the heatglow brightness of the three is as follows:

1.4E-117
1.9E-117
and 2.5E-117

the question is, what is the night-time surface temperature on each planet?

Simply put, you just multiply each number by 6 and take the fourth root (press square root twice) although officially what you multiply each number by is 60/pi².
However, pi-square is almost the same as ten, and 60/10 is six, so it's almost the same either way.

So let's multiply each planet's heatglow by 60/pi²

8.511 E-117
11.55 E-117
15.20 E-117

and press the squareroot button twice to get the temps

0.960 E-29
1.037 E-29
1.110 E-29

The first is below freezing, the second is room temperature, and the third is the perfect temperature for a hot tub!

To help with interpreting these temperatures, remember all those we usually experience are close to E-29 and 1.000E-29 is our basic reference 49 Fahrenheit. Going up from 1.000 to 1.110 is equivalent to going up 110 halfFahrenheit steps, that is 55 F-degrees, which if you add it to 49 gets you 104 Fahrenheit.
On the other hand, going from 1.000 up to 1.037 is equivalent to 37 of those steps which is 18-some Fahrenheit-degrees. Adding that which to 49 gets you 67 Fahrenheit, and what could be a more comfortable than that?

So the moral is: go for the planet that glows 1.9 E-117 in the infrared.

 

[marcus]

two more benchmarks
the temperature in the energyproducing core of the sun is 5E-25

(remember that solar surface temp is 2E-28
and avg Earth surface temp is E-29
the way to remember solar surface is that E-28 is an eQ
and green photons have energy 10eQ, so solar surface is going to be
around E-28, and the temp there happens to be 2E-28.
so from core out to surface, temp goes down by factor of 2500)

and middle D on piano is (1/2)E-39, have to go, back later
that means the D in the soprano/alto range, on the fourth line
of the treble clef, is E-39
the natural unit frequency is E39 times higher than that note
the sopranos in our chorus sing---maybe I could get it in falsetto.

THE CAT ENGINES OF ORNISH

I suspect Kea of liking cats, so I have devised a story about the wicked space pirates of Planet Ornish and how they propel their giant Bagel-shaped troopships. I hope this will scandalize Kea.

 

[marcus]

------THE CAT ENGINE OF THE SHIPS OF ORNISH----

The ships of Ornish are driven by Cat Motors
which consume cats as fuel by converting each cat entirely into energy.

The captain of a ten-million-pound troopship wishes to achieve a speed of E-3 (one thousandth of the speed of light, about 300 km/second) in order to depart a plundered system. Once in the clear he will enter warp and rendezvous with the rest of the pirate band. The initial change of velocity is accomplished by the ship's efficient photon drive.

How many standard 10 pound cats must be converted?

---answer---
"pound" is just a handle on E8 mass units
so clearly the mass of each cat is one billion mass units (10 pounds is 10 E8 units).

so since c = 1 each cat yields one billion natural energy units.
(the mass is the same number as the energy in natural units)

The ship mass is 10 million pounds (E15), so the desired momentum change is E15 x E-3 = E12 momentum units. This requires discharging a photon pulse with E12 energy units, which consumes 1000 cats.

 

[marcus]

---------FAST FOOD EXERCISE------
According to Defense Analyst Daniel Pinkwater, the Earth is in danger of invasion by the Fat Men: a space-faring race which plunders other planets for their fast food.

Their planet, planet Ornish, is deficient in basic resources like french fries, mayonaise, potato pancakes, Colonel Sanders fried chicken, and sour cream, which has forced them into a life of nomadic piracy.

The Fat Men spacewar uniforms consist of loud plaid sports-jackets, green dacron slacks, loafers, and eyeglasses with heavy black plastic frames. They travel by the millions in a fleet of troopships shaped like enormous bagels.

Professor Pinkwater fears that, before long, hordes of Fat Men will descend on Earth and ravage our fast food outlets.

The defense analyst has calculated that one raid by the Fat Men could deplete the Earth of a million pounds of its mayonaise. How much energy does this represent?

---answer---

If you just go into the kitchen and look on a jar of Best Foods Real Mayonaise it will say that a 13 gram serving has 90 food Calories.
One of our (E8 natural mass unit) pounds is 434 grams so it contains almost exactly 3000 Calories.

Also to a reasonably close approximation, the natural unit of energy is 100 thousand food Calories. So a million pounds of mayonaise, with its 3 billion Calories, represents 30 thousand natural energy units.

 

[marcus]

Baez posted some facts about Titan on SPR today including at what altitude the atmosphere was coldest----in natural units terms it was
surface temp 3.33E-30
lowest temp 2.48E-30 (at altitude 6.2E38)

let's look at the indicated gradient 0.85E-30/6.2E38 =1.4E-69
this was for a mainly nitrogen atmosphere which was as much as 7 percent methane at some places, but for simplicity I want to calculate as if it is "dry" nitrogen with no methane "vapor" and see what the theoretical lapse rate would be-----this can be compared with the measured gradient.

http://www.esa.int/SPECIALS/Cassini-Huygens/SEMMF2HHZTD_0.html

-------GRAVITY ON TITAN-----
Titan surface gravity is 0.12E-50.
(I write it that way because I think of acceleration E-50 as about one "gee")
-----TEMPERATURE GRADIENT ON TITAN-----
in natural units terms the lapse-rate (rate of change in temp with altitude) is just 2/7* x wt of molecule

the weight of a nitrogen molecule = 0.12E-50 x 28/(2.6E18) = 1.29E-68
and 2/7 of that is 0.37E-68 = 3.7E-69

So the theoretical lapse-rate (with no methane assumed) is 3.7E-69
but what was actually observed was 1.4E-69

The presence of a few percent methane could explain why the observed gradient was less.

----notes----
That ESA link says Titan radius is 0.4 earth's, and Titan mass is 1/45 of earth. this means surface gravity is around 13.8 percent of earth

As a crutch I remember an order of magnitude "gee" is E-50 natural acceleration units-----official Earth surface gravity is 0.88E-50.
So* titan is 0.12E-50

Wind is the result of convection which happens when the temperature gradient (cooling off with altitude) exceeds the threshold gradient called "lapse-rate".

The lapse rate in moist air is less than the dry-air lapse rate, because having moisture (able to condense to form clouds and give up its energy) serves as a "fuel" for convection---a reservoir of energy which makes convection easier and able to happen with more gradual temperature gradients.

As a way of interpreting the observed gradient of 1.4E-69, think of E-32 on the temp scale as half a Fahrenheit degree and E37 on the distance scale is half a mile
so this is saying 1.4 E-32/E37 which is 1.4 fahrenhalfsteps per halfmile.* it is a very slight gradient, because of the weak gravity, by Earth standards. so convection happens easily and there is a lot of wind action, and the convection shifts heat upwards and evens out the gradient.

 

[marcus]

Numbers that mean home

I think Quantum Gravity research is probably going to succeed and get a quantum theory of spacetime (and of its shape, which we call gravity)

and I expect physics will be rebuilt on this new spacetime (just like since 1680 it has been built mostly on Newton's absolute flat spacetime and the 1905 minor variation of that)

and today as usual I looked at a QG paper and as usual it was using nonmetric QG units where c and hbar are set equal one (or something else convenient) and G likewise---so it is some variant of the Planck units (which we've had around since 1899 or so). As physics gets rebuilt on a quantum spacetime basis, I expect people will gradually get used to gauging things in natural units: scales intrinsic to quantum spacetime.

So after reading I went out in the garden, bright with insects flying around and some stuff opening, a few leaves coming out to get the sun, and I thought: 5.7E-117 spells life and it is a longterm number. It's the brightness of sunlight expressed in natural units----as long as sunlight is this level, as long as there is a rich chemistry, as long as the rivers run...

some crows were talking, a redtail hawk cried, some other birds were calling too
well I am going for a walk up the hill with my wife, for exercise etc.
be back later

these longlasting numbers that spell earth, spell home, most are not too interesting

E-29 is global avg. surface temp
E-50 is a "gee"
E-91 is the density of water
E-106 is a "bar"----standard air pressure

but 5.7E-117 is the brightness of sunlight, not too strong or too weak, that is driving all this life, the crows cawing, trees budding, wind stirring the branches, and all that,
so if I happened to be riding around with some aliens in their saucer I think I would would ask to be put off at a planet with steady 5.7E-117 sunlight.
it is the number that more than any other characterizes home

btw in natural units what I gave for gee and bar etc were only approximate
the official gee is more like 0.9E-50
and the normal density of water is more like 1.2E-91
and the usual sealevel pressure norm is 1.4E-106
but I don't care too much about a few percent. I think I could be adapt
(as long as the chemistry was good) somewhere with
gravity E-50 and pressure E-106. But I think I might get depressed,
or sick of the place, if illumination wasnt right.

 

[marcus]

Freshman physics on the round number planet.
I'm thinking of a planet with the same general chemistry as on earth, air has the same composition, soil and vegetation similar etc. but where some basic features like the surface gravity are round numbers.

E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year

(Note that on Earth the official gee is not exactly E-50 but more like 0.9E-50,
the usual sealevel pressure norm more like 1.4E-106,
the planet mass 1.4E33,
although estimates vary the average temperature is close to E-29,
and the year is 1.167E50 instead of exactly E50.
So this round number planet is not an exact match to Terra but akin to it.)

One thing we will not compromise about is the brightness of sunlight. It is 5.7E-117 expressed in natural units. And as to some things like the proton mass and the density of water we have no choice:
1.2E-91 is the density of water at standard temperature and pressure (E-29 and E-106).
2.6E18 protons make one mass unit, so if we need a figure for the mass of a water or air molecule we can say 18/(2.6E18) and (29/2.6E18).

I'm thinking of a series of exercises---a kind of natural units Yoga---which is finding out about this round number planet: the speed of sound, the threshold of convection in the air, the minimal orbit time, the planet's radius and density, the escape velocity, the rate pressure falls with altitude, the equilibrium temperature in direct sunlight. these might be good exercises to do for several reasons.

 

[marcus]

summary of inputs
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

so to begin, let's calculate the planet radius.

surface gravity = E-50 = GM/R² = (1/8pi)E33/R²

R² = (1/8pi)E83

R = 6.308E40 = 6.3E40 rounded.

to humanize this recall E37 is half a mile. the radius is 6308 halfmiles.

 

[marcus]

summary of inputs
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

Let's calculate the speed of sound at the planet's surface.
the speed of sound formula for biatomic gas at temp T
speed² = (7/5) kT/(mass of molecule)

In our case the average mass of the air molecule is 29/(2.6E18) and temp is E-29

speed² = (7/5) E-29 (2.6E18)/29 = 1.255E-12
speed = 1.12E-6
just over a millionth of the speed of light

BTW I added the planet's orbit speed to the list of easy numbers. It matches the Earth's orbit speed almost exactly and will allow us to calculate the mass of the planet's sun.

 

[marcus]

summary of inputs
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

I want to use just these inputs and demonstrate how various things follow from them.

mass of the sun, in natural units the mass M is related to the year period P and the orbit speed v by
M/4 = P v³ = E50 (E-4)v³ = E50 E-12 = E38
M = 4E38

remember the planet mass is E33, so this is 400,000 times that.

(rather like how Earth and sun masses are related, you can always
try comparing between what we get for the round number planet and the handbook data for Earth and it usually won't be far off. in this case Jupiter is a bit less than 1/1000 of sun and Earth is roughly 1/300 of Jupiter and 400,000 wouldn't be a bad guess)

 

[marcus]

Over in general astronomy there is Little Orbits thread
https://www.physicsforums.com/showthread.php?p=447657#post447657
which among other things dealt with calculating the circular orbit period of two identical gold balls separated by 3 times the radius, center to center.
A toy gravitating system consisting of two equal balls with a gap between them equal to their common radius.
That should be a snap for us with these units. In interpreting the orbit time we are going to calculate remember that the time interval E45 is about 4.5 minutes. The density of gold is 24E-91, which is to say 24 pounds per pint (24 E8/E99). I had to look it up in the CRC handbook.

If T is the radian time (that is P/2pi) then
T² = 3³/8E-91
T= 5.8E45 = about 26 minutes
Multiply by 2pi to get 164 minutes for the period.

For those who like more symbols let S be the ratio of the separation to the radius, which in this case is 3 (sep is thrice radius) and then
T² = 3 S³/density
all we needed to do was put 24E-91 in for the density.

As another illustration, if the separation is 4 radii, so that the gap is equal to the diameter of one of the balls, then
T² = 3 4³/24E-91 = 4³/8E-91 = 8E91

 

[marcus]

just now out in the garden was impressed by the vivid colors
blue sky, green leaves, red camelia (we have this big tree-almost, which is covered)
and recalled that I knew the energies and frequencies and wavelengths of these colors
green energy is E-27
which means the frequency is also E-27 (radians per natural unit of time)
and the angular wavelength is E27 (natural length units)
so it is so easy:
you know this one number, for the energy of a green photon, and it works for the frequency and wavelength too.

and the extemes of the visible tend to be about 25 percent higher and lower than green, which is right in the middle
so red energy is 0.8E-27, which makes its frequency the same 0.8E-27
and its wavelength 1.25E27 so that is the camelias
(we have white and pink ones too but I am just thinking of the very red ones)

and the blue of the sky, its photon energy is 1.25E-27 also with frequency 1.25E-27, and of course the wavelength is shorter than green, namely the reciprocal number 0.8E27

So I am remembering last night, our chorus rehearsal, where I was in the bass section sitting right next to the sopranos and right at my elbow was Dacia or Dasha whose high range is wonderful and she was singing plenty of frequency E-39
which is the D on the fourthline of the trebleclef
and we of course match each other singing by whole numbers and precise fractions because that is what 4-part harmony is about
and maybe it doesn't matter but her D which is E-39 natural
is related to the green in sunlight and outside in the garden, which is E-27

by merely a trillion, by merely a factor of E12

when the sopranos sing it is a transfusion of light
or sometimes fire

 

[marcus]

round# planet inputs
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

I want to use just these inputs and demonstrate how various things follow from them.

Say that here is the round number planet and a can judge force on the soles of my feet. I go out in the garden and just stand there feeling the sunlight and the force of my weight on the ground, and watching what is going on.

On each foot I feel a force of E-40 natural.

How many nucleons are in my body?

 

[marcus]

these natural units are based on the universal force constant which is the force that appears in the Einst. eqn. relating energy density in space to curvature of space

the force of my own weight is something I know. I have to cope with this force when I get out of bed in the morning, and it helps me compact the trash so I can get a bit more in the can on pickup day.
this force is
2E-40 of the natural constant force

so how many baryons am I made of? how many protons and neutrons?
well gee is E-50

so my mass is 2E10 natural mass units (that is force/gee)

and baryons are 2.6E18 to each mass unit it says in the list of 3 hard numbers.

So I am made of 5.2E28 baryons.

 

[marcus]

round# planet inputs
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

I want to use just these inputs and demonstrate how various things follow from them.

right outside the front gate there are tall eucalyptus, and a lot of the time we hear the wind. So I often think of the lapse rate----the temperature gradient necessary for convection. there is a deadend barrier here and beyond it the ravine with the creek. for much of the year we hear the creek making a quiet roaring and that also is driven by convection, since it lifts water to make rain uphill from us. so these constant voices tell me that somewhere the gradient has become steeper than the threshold for convection.

If I am on the round number planet, what is that threshold gradient?
It is going to be some temperature drop per pace or per mile, in human terms, some number of degrees cooling off per measure of height. but energy per distance is force, so it is basically a certain force which we will measure in relation to the universal force constant.

the formula is simply the WEIGHT of an average air molecule, divided by (7/2)k. but k = 1, so we just have to divide by 7/2.
the average air molecule mass is 29/(2.6E18) in natural terms.
And gee on this planet is E-50

threshhold cooling rate for convection = (2/7)(29/2.6)E-68 = 3.2E-68

so this is the number which I want to hear in the wind, if I live on the round number planet.

to humanize it, a halfmile is E37 length units, and a halfFahrenheit step is E-32 of the natural unit. So a halfdegree per halfmile drop in temperature is E-32/E37 = E-69. so what the calculation showed is that on the round number planet, which is prettymuch like earth, a drop of 32E-69, which is
32 of those temp steps per halfmile (16 Fahrenheit if you have to think F) is a kind of limit on the rate the air can cool with height. If it cools off faster then convection, and mixing, will set in. Interestingly, Richard advised against humanizing numbers too often. maybe he would say to keep the number that is implicit in the wind a simple 3.2E-68. And implicit in the clouds of moisture lofted by the rising air.

 

[marcus]

Some of the eucalyptus are 4E35 tall. that is 40E34, or 40 paces, in more familiar terms 100 feet. They tower.

we are all familiar with the drop in pressure as you gain altitude. what is the scale of this. it is exponential, so there must be some distance D such that the pressure falls off with height h as exp(h/D). that is, if you want the pressure to be less by 5 percent you should ascend by 5 percent of D.

we are on the round number planet, this pressure drop distance is part of getting acquainted with the planet. what is it?

the average temperature in the air column we are looking at might as well be E-29 since that is typical of the planet

the weight of an airmolecule in this planet's 'gee' gravity E-50 is
gee times 29/(2.6E18), which is 11.15E-68

the pressure drop scale D = temp/air molecule weight = E-29/(11.15E-68) = 9.0E37, do you remember E37 as a halfmile? the distance D is 9 of those (or 4.5 miles if you prefer miles to halfmiles a whole lot)

so D = 9 halfmiles. that means if you want the pressure to decline by 1/9 of what it is now, you should go up the mountain 1 halfmile (one ninth of D).

What physics facts about round number planet might you want to know that you think we might calculate from the given data?
Suppose we go scuba diving on this planet. how deep do you go to get an increased pressure of one atmosphere? (it's the limit of a suction well-pump that we learned about in middleschool)