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 a 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).
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.
So the one I heard about the tree is wrong, but change the tree to an amoebae, and it’s right.
Now think about that for a moment: The Planck Length is to an atom as an amoebae 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.
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.
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.
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.
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?
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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.
A Planck Length (PL) is: 1.616255×10-35 meters.
So, in just one meter (about three feet; 39 inches), there are:
That’s a 61 followed by 33 more digits.
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.
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).
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.
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.
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.
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.
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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!