I have always liked those comparisons that try to illustrate the very tiny by resizing it to more imaginable objects. For instance, one says: if an orange were as big as the Earth, then the atoms of that orange would be a big as grapes. Another says: if an atom were as big as the galaxy, then the Planck Length would be the size of a tree.
The question I have with these is: How accurate are these comparisons? Can I trust them to provide any real sense of the scale involved? If I imagine an Earth made of grapes, am I also imagining an orange and its atoms?
So, I did a little math.
It turns out the one about the Earth and the grapes is pretty much on the money.
But the one about the galaxy and the tree missed the mark by several orders of magnitude.
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Let’s start with the orange.
Except, what size orange? Oranges vary in size, but the average diameter is supposedly 2.5 inches (seems a little small for diameter, but whatever). That gives it a radius of 1.25 inches or 0.03175 meters.
Magnifying that average orange to the size of the Earth blows it two-hundred million times. Which makes the carbon atoms in the orange about 2.7 inches in diameter.
Which is maybe a little big for a grape, but pretty close.
If we start with a 5-inch orange (twice as big), blow it up only one-hundred million times, then the carbon atoms are half as small: 1.35 inches in diameter, and those are definitely grape-like.
So, this one gets the thumbs up. Orange as big as the Earth has grape-sized carbon atoms (especially if it’s a large orange).
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Compared to an amoebae!
To expand a carbon atom to the size of the Milky Way galaxy requires magnifying it by a factor of 2.9×10³⁰ — 30 orders of magnitude!
(That’s much more than the mere 2×10⁸ used to expand an average orange to the size of the Earth.)
But the Planck Length, believed to be the smallest possible distance, is so small that even that much magnification only gives us a value of 47.53 micro-meters (1.87 thousandths of an inch) — which is a lot smaller than a tree.
It is, in fact, about the size of an amoeba, which is the comparison I heard Jim Baggott make during a talk at the Royal Institute earlier this year.
So, the one I heard about the tree is wrong, but change the tree to an amoeba, and it’s right.
Now think about that for a moment: The Planck Length is to an atom as an amoeba is to the Milky Way galaxy.
Amoebae are tiny just on our human scale, vanishingly tiny on the scale of the Earth, not really noticeable on the scale of the Solar system… on the scale of the whole galaxy?
The Planck Length is really, really, really tiny.
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There’s a group of other comparisons that expand atoms to something building-sized in order to show how very small the nucleus is compared to the whole atom.
This gets complicated because, for one thing, the size of an atom is one of those “it depends on what you mean” things. (I’ve been using the van der Waals radius.)
A further complication is that, other than the nucleus, an atom is a cloud of electrons in various possible orbitals, so it doesn’t have fixed (let alone hard) boundaries.
Finally, even settling on how to judge its size, different types of atoms have different sizes, so it depends on which atom we pick. (A potassium atom is 2.5 times larger than a hydrogen atom.)
The cake icing is that the size of the atomic nucleus also varies with atom type. The point is to compare the tiny nucleus with the overall size of the atom, so we need to be very specific.
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Hydrogen atom. Sort of.
It’s easiest to deal with a hydrogen atom, since it has just the one proton.
It turns out the radius of a hydrogen atom is 1,307 times larger than the radius of the single proton that is its nucleus.
That might not sound like much, but remember that the volume of a sphere increases with the cube of the radius. That seemingly paltry 1,307 increases the volume by a factor of 2.2×10⁹ (which is a lot of empty space for the electrons to zip around in).
Let’s magnify a hydrogen atom by a factor of 4.156×10¹¹ (for reasons that will become immediately apparent).
That means our hydrogen atom now has a radius of 45.72 meters and, therefore, a diameter of 91.44 meters. That distance is known to fans of American Football as 100 yards — the length of a football field.
So try to imagine a fuzzy sphere just big enough to enclose a football field, keeping in mind it’s a sphere, so it goes up 50 yards and down in the ground 50 yards. The single-proton nucleus would be in the center, on the ground on the 50-yard line.
Speaking of which, that single proton would have a diameter of 69.94 millimeters — about 2.75 inches. Pretty much the size of an average orange.
So football field-sized hydrogen atom — big electron cloud — with an orange-sized proton in the center. Think about that next time you see a football field.
The single electron in the cloud, as far as we know, has no size or structure, so even at this size it’s an invisible mote.
One test puts an upper bound of 10⁻²² (meters) on its radius. If we round our magnification factor up to a nice even 10¹², that upper limit is still only 10⁻¹⁰ meters — roughly the size of a small atom.
So electrons, even if they have any size at all, are really tiny.
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A potassium atom, 2.5 times the size of a hydrogen atom, has an atomic number of 19, so at the same magnification it’s much bigger than a football field.
Alternately, to make it football field-sized, we magnify it only 1.66×10¹¹, which makes the protons in the nucleus only 28 millimeters in diameter — about one inch.
So there’s another image: A potassium atom the size of a football field with 39 one-inch nucleons in the center. Maybe a bit smaller than a soccer ball?
§ §
So there it is.
Orange to the Earth, atoms to a football field or, skipping lots of ground, an entire galaxy.
But you should do your own math (or at least check mine).
To help you get started, below are the numbers I used.
They’re pretty much all in metric; deal with it.
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A Planck Length (PL) is: 1.616255×10⁻³⁵ meters.
So, in just one meter (about three feet; 39 inches), there are:
61,871,424,991,724,696,907,356,821,788,641,000 PL
That’s a 61 followed by 33 more digits.
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A proton has a charge radius of 8.414×10⁻¹⁴ meters (841.4 atto-meters).
A neutron, being roughly the same thing, is the same size. You could line up 11,884,953,648,680 of them in a meter — almost 12-trillion.
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A single atom of carbon has a (van der Waals) radius of 1.70×10⁻¹⁰ meters (170 pico-meters).
A single atom of hydrogen has a radius of 1.10×10⁻¹⁰ meters (110 pico-meters).
A single atom of sulfur has a radius of 1.05×10⁻¹⁰ meters (105 pm, which is just about one angstrom).
On the larger side, single atom of potassium has a radius of 2.75×10⁻¹⁰ meters (275 pm).
The biggest is francium, which has a radius of 3.48×10⁻¹⁰ meters (348 pm).
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The Earth has an average radius of 6.371×10⁶ meters (6,371 kilometers).
That’s 250,826.8 inches, so the Earth is about a half-million inches in diameter.
And its radius in Planck Lengths: 3.9418×10⁴¹.
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The Milky Way Galaxy has a radius about 52,850 Light Years (105,700 LY in diameter).
Which means our galaxy is 6.18×10⁵⁵ Planck Lengths across.
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The Visible Universe (VU) has a diameter estimated at about 9.3×10¹⁰ Light Years (93 billion LY).
Which means the Visible Universe is 5.4437×10⁶¹ Planck Lengths across.
§ §
And finally,…
The speed of light is: 299,792,458 meters / second.
Or, if you prefer, 11,802,859,050.7 inches / second.
A year is 365.25 days long. A day is 24 hours × 60 minutes (per hour) × 60 seconds (per minute), so a day is 86,400 seconds long, and a year is 31,557,600 seconds long.
A light-year (how far light travels in one year) is: 299,792,458 meters / second × 31,557,600 seconds / year = 9,460,730,472,580,800 meters / year.
Let’s call a light-year (LY): 9.45425×10¹⁵ meters. That’s almost nine-and-a-half quadrillion meters. Probably easier to think of it as over nine-trillion kilometers.
(It’s 372,469,904,778,553,354 inches. Or 5,878,623,554,478 miles. That is, almost six-trillion miles.)
And there are 5.8535×10⁵⁰ Planck Lengths in a Light Year.
§ §
Sooooo many numbers! 😁
Perhaps now you see why Mandelbrot zooms with factors of 10¹⁰⁰ (and considerably beyond) impress me so much. The scale is jaw-dropping.
Stay metric, my friends!
∇
Logos con carne 
August 8th, 2019 at 1:58 pm
One take away is that, since atoms are in the one- to three-angstrom range, which is 1–3 × 10-10, magnifying an atom by a factor of 1010 gives you a one- to three-meter atom.
August 8th, 2019 at 7:41 pm
Nice analysis Wyrd!
One of the things these kinds of analyses always impress on me is how empty the universe is. Atoms are mostly empty space, which means most macroscopic objects are mostly space. The solar system, when considered to scale, is mostly empty. And the orbits of the planets are barely noticeable in the distances between stars.
This gets even more profound when the point like nature of all elementary particles are considered (at least point like after the “wave function collapse”). It’s like there is no matter, just relations between points with various intrinsic properties.
August 8th, 2019 at 8:50 pm
Thanks!
“And the orbits of the planets are barely noticeable in the distances between stars.”
And pretty empty between galaxies, too. Yeah, almost entirely empty space.
“It’s like there is no matter, just relations between points with various intrinsic properties.”
Yeah, at that level, it’s easy to see the matter-energy equivalence. What we call matter, “particles,” are just vibrations in a fermion field.
(As an aside, I’m not a “relationist” — one who sees the relations between things as more primary than the things. I think relationships emerge from the things. Which are real and fundamental.)
((As a further aside, I hadn’t fully appreciated until I read Through Two Doors At Once that wavefunction collapse is another example of “spooky” entanglement. The entire wavefunction collapses everywhere instantly.))
August 9th, 2019 at 7:36 am
“As an aside, I’m not a “relationist” ”
It is easy to see where relationists are coming from when contemplating that matter reduces to the excitation of fields, particularly when you remember that a “field” is itself a relational concept. Still, there’s the old question, “What breathes fire into the equations?” What’s the difference between the entities in equations that predict reality and those that don’t?
“The entire wavefunction collapses everywhere instantly.”
When I first started reading about quantum mechanics, I gravitated toward the De Broglie–Bohm interpretation, the pilot-wave one. It seemed like the most common sense one, until I came to understand this point, which seemed to make pilot-wave just as bizarre as the other interpretations.
August 9th, 2019 at 11:58 am
“It is easy to see where relationists are coming from…”
For me it’s actually not easy. I’m in the same boat here as I am with the idea of time being emergent. I think it’s putting the cart in front of the horse.
There are things, like materialism or computationalism, that I believe are probably false, but I can absolutely see they could be correct. I can’t fault anyone for holding those views even if I don’t.
But relationships having any sort of primary ontology makes no sense to me. I see a relation, by definition, as an operator between two things. Which seems to me to clearly require the two things have the primary ontology, to be fundamental. The relation is emergent.
(Maybe it’s me. Maybe I just don’t get it. As I mentioned recently, we all have intellectual blind spots… ideas our minds just can’t see.)
“…particularly when you remember that a “field” is itself a relational concept.”
How so? (I would have said a “field” (as used in physics) is something with a value at every point in space.)
“What’s the difference between the entities in equations that predict reality and those that don’t?”
An interesting way to put it… is there an analogy in writing with fact versus fiction? The same words, same techniques, same phrasing,… but one connects with reality, the other doesn’t.
It relates to the discussion about platonism, the reality of abstractions or fantasies.
For instance, I could tell a story about taking a walk this morning (which I did, in fact, do). That story could be accurate or utter fantasy or any mix. Equations, likewise, have a Yin-Yang of fact and fantasy. In all cases, the syntax is the same, and even a lot of the semantics. And in all cases, logically consistent.
In philosophy, I’m not always a fan of “logical worlds” arguments — it really annoys me how much Chalmers depends on zombie arguments — but it is kind of a vexing question you asked. The only difference seems to be that one story happens to match the facts while another doesn’t. Somehow there should be more to it. Seems too accidental.
“I gravitated toward the De Broglie–Bohm interpretation, the pilot-wave one.”
I kinda liked that one, too (and as an alternate theory, still see attractions — it’s a realist theory, and I do lean towards realism). But it seems we’re stuck with spooky “magic” one way or another.
Given the “crisis” in high-energy physics right now, I’m really starting to wonder if a full understanding of reality will be forever beyond us.
August 9th, 2019 at 2:06 pm
“How so? (I would have said a “field” (as used in physics) is something with a value at every point in space.)”
That’s actually the definition I had in mind. It doesn’t seem like an overall relational description to you? Where is the “thing” in it?
I’m not definitively in the relational camp. Ultimately there may be brute facts that all relations are built on. But I’m not confident we know what those are yet. Or how we could know whether we’d ever found the final and fundamental ones, or just ones we hadn’t managed to break into further relations yet.
“The only difference seems to be that one story happens to match the facts while another doesn’t. Somehow there should be more to it.”
A mathematical platonist would argue that the difference is that one fits into the equation we think of as “the universe” while the other one doesn’t. I can see the appeal of that view, although I find mathematical empiricism more likely (at least at the moment).
“I’m really starting to wonder if a full understanding of reality will be forever beyond us.”
I’m not sure if we’ll ever have a full understanding. We’re part of reality, which means as patterns in the framework, we may be unequipped to comprehend the framework itself, or its boundaries or foundations. We may be like characters in a video game attempting to understand the computer, but with only the in-game concepts to work with. Although I hope we never stop trying.
August 9th, 2019 at 4:40 pm
“Where is the ‘thing’ in it?”
I would say the field is the ‘thing.’
“But I’m not confident we know what those are yet.”
I agree. If an electron is a disturbance in the electron field, how does that square with the idea it has no size or structure? I liked string theory at first because it had a good answer to that problem. Point “particles” is weird.
I think something has to be fundamental. There is a fabric.
“I can see the appeal of that view, although I find mathematical empiricism more likely (at least at the moment).”
I’m beginning to really see the appeal of anti-realism; that there’s just no fact in the matter (at least in some situations). It may be entirely in how one looks at it.
I have a hard time thinking some parts of math, even of thought, are not somehow discovered. Kronecker: “God made the integers, all else is the work of man.” Maybe it’s a blend of what we discover and what we add to that.
Per the fact/fiction analogy, one can start with fact and “embellish” it.
“We may be like characters in a video game attempting to understand the computer, but with only the in-game concepts to work with.”
Exactly. And, possibly exactly that. 🙂
I keep thinking there are so many aspects of our existence that make a lot of sense from a design point of view. IOW, if I were designing a universe simulation, I might very well make similar design choices.
For example, DNA. If I wanted a population of beings, all slightly different within the species, I might very well (A) have the design template data be part of the organism and (B) randomly combine that data from two existing beings to create a new one.
Star system planet formation. It depends on initial conditions and chaos, so it’s possible to get lots of different kinds of star systems using a basic set of rules. Exactly the sort of thing you’d do to avoid having to create them manually.
Even the disconnect between QFT and GR makes me wonder: Different rules for reality at different scales?
I’m increasingly annoyed that every creation and existence story is utterly preposterous. Anti-real, indeed. 😦
August 9th, 2019 at 5:45 pm
“I would say the field is the ‘thing.’”
I can see that, but it also feels like a field is an infinite collection of scalar values, all things in their own right. But they’re also relations between the field and space.
“I think something has to be fundamental. There is a fabric.”
Like the edge of space discussion, it seems like any answer will be absurd. Either we’ll have relations all the way down (absurd) or brute facts that simply can’t be reduced any further, whereupon the question becomes, why those facts?
“I’m beginning to really see the appeal of anti-realism”
I’m a scientific instrumentalist, but for me, that’s not a desirable thing but a cautionary epistemic strategy. I often do think there is a fact of the matter, and I care what it is, even when I’m forced to admit we don’t know it.
Along those lines, I do think there’s a fact of the matter with platonism. It either is or isn’t true. I currently don’t think it is. But I see no way to conclusively resolve it.
“Different rules for reality at different scales?”
But why switch rules? Why should it matter whether we’re working at 10^-35 meters or 10^100 meters? And it’s not clear that size is the actual delineation.
“I’m increasingly annoyed that every creation and existence story is utterly preposterous. Anti-real, indeed.”
Reality is absurd.
August 9th, 2019 at 9:02 pm
“I can see that, but it also feels like a field is an infinite collection of scalar values, all things in their own right.”
A comparison might be the real number line. The set of real numbers is a ‘thing’ and each real number, as a magnitude along that line, is also a ‘thing.’ There is a hierarchy of things there, since the set contains all the members.
“But they’re also relations between the field and space.”
I wondered if that’s the relation you had in mind — a relation between location and field value? I don’t see that as a relation because it’s fixed — it is never the case that a given point in the field has some other location. The value changes, but the location is the location. To me a tautology isn’t a relation.
But I’m not sure this is a case with a correct answer.
“…or brute facts that simply can’t be reduced any further, whereupon the question becomes, why those facts?”
I’m fine with that and see it as ultimately necessary. It’s not turtles all the way down, something is true, and it may well be a set of fundamental facts. (Such as I believe time is, for instance. Time just is. We can study it very closely, but we’ll never account for it other than “it just is.”)
Or maybe not and we’ll find it’s some law that just is, but ultimately there is something that just is. (I, as a realist, believe.)
“I often do think there is a fact of the matter, and I care what it is, even when I’m forced to admit we don’t know it.”
Agree. When it comes to physical reality, I do think there is an answer. It’s more on philosophical matters I think there may be no correct answer.
“Along those lines, I do think there’s a fact of the matter with platonism.”
I can see Platonism having some fact of the matter, but platonism seems just a point of view to me. What sort of facts could resolve it, do you think?
“But why switch rules?”
We’re way into speculation, but I can speculate. The quantum rules, as we know, are extremely computationally intense. If reality is a computer, its lowest level is weird and complex mathematically.
But the macro world works according to much simpler rules. So why use complex computational resources if simpler ones will do most of the time. Why not only use the complex rules when the situation requires it.
Stars work because of weak force interactions, so quantum rules are required. For most of the stuff in our world, they aren’t.
And when clever scientists start peering at the exact workings, then quantum rules are needed again. 😀
“Reality is absurd.”
And then you die. 😮
August 10th, 2019 at 9:26 am
“I can see Platonism having some fact of the matter, but platonism seems just a point of view to me. What sort of facts could resolve it, do you think?”
I don’t know if there’s anything observable that could resolve it. But facts that might would include convergence on structure, function, or properties for objects which share no underlying commonalities, where essentially the only thing they share is the form. The problem is that everything appears to at least have the laws of physics in common, so as a practical matter, eliminating confounding explanations doesn’t seem possible.
“And when clever scientists start peering at the exact workings, then quantum rules are needed again.”
Of course, at that point all that really needs to be taken care of are the conscious experience of the scientists. But then, that’s all that ever really needs to be taken care of.
August 10th, 2019 at 1:34 pm
“The problem is that everything appears to at least have the laws of physics in common, so as a practical matter, eliminating confounding explanations doesn’t seem possible.”
Exactly. Seems impossible to untangle!
September 6th, 2019 at 12:10 pm
There’s a good article by Jennifer Ouellette in ars technica, Physics not “broken” after all? We’re close to resolving proton radius puzzle.
It talks about a new experiment designed to test the charge radius of the proton. In my post I used the best recent value, 0.841 femto-meters. The new measurement puts the value at 0.833 femto-meters.
Or, if you prefer, 833 atto-meters. 😀
September 6th, 2019 at 12:14 pm
The important part is that scientists were thinking there might be new physics involving the muon, because muonic hydrogen (with a muon instead of an electron) was used to test the proton size and seemed to deliver different results than when a plain old electron was used. The new experiment confirms that muons don’t involve new physics.
September 11th, 2019 at 4:24 pm
Natalie Wolchover has a good article about the Proton size in Quanta: Physicists Finally Nail the Proton’s Size, and Hope Dies
Back in 2016 she wrote an article about how using muons instead of electrons created a puzzle that maybe suggested new physics: New Measurement Deepens Proton Puzzle.
But, so sadly, there doesn’t appear to be any new physics.
Move along. Nothing to see here…
August 25th, 2022 at 9:04 am
Kind of a cool video about the size of atoms:
February 15th, 2023 at 8:10 am
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