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How do point particles create atoms with size?

This article is archived for use with my students from Ask Ethan: If Matter Is Made Of Point Particles, Why Does Everything Have A Size?

Forbes, Stars With a Bang, by Ethan Siegel 9/16/17

Proton Structure Brookhaven

Proton Structure Brookhaven National Laboratory

The big idea of atomic theory is that, at some smallest, fundamental level, the matter that makes up everything can be divided no further. Those ultimate building blocks would be literally ἄ-τομος, or un-cuttable.

As we’ve gone down to progressively smaller scales, we’ve found that molecules are made of atoms, which are made of protons, neutrons, and electrons, and that protons and neutrons can be further split into quark and gluons. Yet even though quarks, gluons, electrons, and more appear to be truly point-like, all the matter made out of them has a real, finite size. Why is that? That’s what Brian Cobb wants to know:

Many sources state that quarks are point particles… so one would think that objects composed of them — in this instance, neutrons — would also be points. Is my logic flawed? Or would they be bound to each other in such a way that they would cause the resulting neutron to have angular size?

Let’s take a journey down to the smallest scales, and find out what’s truly going on.

Magdalena Kowalska Nuclear Scale to quarks

Magdalena Kowalska / CERN / ISOLDE team

If we take a look at matter, things behave similar to how we expect they should, in the macroscopic world, down to about the size of molecules: nanometer (10-9meter) scales. On smaller scales than that, the quantum rules that govern individual particles start to become important.

Single atoms, with electrons orbiting a nucleus, come in at about the size of an Angstrom: 10-10 meters. The atomic nucleus itself, made up of protons and neutrons, is 100,000 times smaller than the atoms in which they are found: a scale of 10-15 meters. Within each individual proton or neutron, quarks and gluons reside.

While molecules, atoms, and nuclei all have sizes associated with them, the fundamental particles they’re made out of — quarks, gluons, and electrons — are truly point-like.

Standard-Model Quarks Leptons Bosons

E. Siegel / Beyond The Galaxy

The way we determine whether something is point-like or not is simply to collide whatever we can with it at the highest possible energies, and to look for evidence that there’s a composite structure inside.

In the quantum world, particles don’t just have a physical size, they also have a wavelength associated with them, determined by their energy. Higher energy means smaller wavelength, which means we can probe smaller and more intricate structures. X-rays are high-enough in energy to probe the structure of atoms, with images from X-ray diffraction and crystallography shedding light on what molecules look like and how individual bonds look.

Electron density map of protein

Imperial College London

At even higher energies, we can get even better resolution. Particle accelerators could not only blast atomic nuclei apart, but deep inelastic scattering revealed the internal structure of the proton and neutron: the quarks and gluons lying within.

It’s possible that, at some point down the road, we’ll find that some of the particles we presently think are fundamental are actually made of smaller entities themselves. At the present point, however, thanks to the energies reached by the LHC, we know that if quarks, gluons, or electrons aren’t fundamental, their structures must be smaller than 10-18 to 10-19 meters. To the best of our knowledge, they’re truly points.

quark-gluon plasma

Brookhaven National Laboratory

So how, then, are the things made out of them larger than points? It’s the interplay of (up to) three things: Forces, Particle properties, and Energy.

The quarks that we know don’t just have an electric charge, but also (like the gluons) have a color charge. While the electric charge can be positive or negative, and while like charges repel while opposites attract, the force arising from the color charges — the strong nuclear force — is always attractive. And it works, believe it or not, much like a spring does.

Warning: Analogy ahead!

Caution analogies

Here we go:

Quarks and Gluons

How did the Proton Get Its Spin? Brookhaven National Laboratory

Above: The internal structure of a proton, with quarks, gluons, and quark spin shown. The nuclear force acts like a spring, with negligible force when unstretched but large, attractive forces when stretched to large distances

When two color-charged objects are close together, the force between them drops away to zero, like a coiled spring that isn’t stretched at all.

When quarks are close together, the electrical force takes over, which often leads to a mutual repulsion.

But when the color-charged objects are far apart, the strong force gets stronger. Like a stretched spring, it works to pull the quarks back together.

Based on the magnitude of the color charges and the strength of the strong force, along with the electric charges of each of the quarks, that’s how we arrive at the size of the proton and the neutron: where the strong and electromagnetic forces roughly balance.

Quarks and protons

APS/Alan Stonebraker

The three valence quarks of a proton contribute to its spin, but so do the gluons, sea quarks and antiquarks, and orbital angular momentum as well. The electrostatic repulsion and the attractive strong nuclear force, in tandem, are what give the proton its size.

On slightly larger scales, the strong force holds protons and neutrons together in an atomic nucleus, overcoming the electrostatic repulsion between the individual protons. This nuclear force is a residual effect of the strong nuclear force, which only works over very short distances.

Because individual protons and neutrons themselves are color-neutral, the exchange is mediated by virtual, unstable particles known as pions, which explains why nuclei beyond a certain size become unstable; it’s too difficult for pions to be exchanged across larger distances. Only in the case of neutron stars does the addition of gravitational binding energy suppress the nucleus’ tendency to rearrange itself into a more stable configuration.

Nuclear Force GIF

Wikimedia Commons user Manishearth

And on the scale of the atom itself, the key is that the lowest-energy configuration of any electron bound to a nucleus isn’t a zero-energy state, but is actually a relatively high-energy one compared to the electron’s rest mass.

This quantum configuration means that the electron itself needs to zip around at very high speeds inside the atom; even though the nucleus and the electron are oppositely charged, the electron won’t simply hit the nucleus and remain at the center.

Instead, the electron exists in a cloud-like configuration, zipping and swirling around the nucleus (and passing through it) at a distance that’s almost a million times as great as the size of the nucleus itself.

Wavefunctions of the electron of a hydrogen atom PoorLeno Wikipedia

The energy levels and electron wavefunctions that correspond to different states within a hydrogen atom, although the configurations are extremely similar for all atoms. The energy levels are quantized in multiples of Planck’s constant, but the sizes of the orbitals and atoms are determined by the ground-state energy and the electron’s mass.

There are some fun caveats that allow us to explore how these sizes change in extreme conditions. In extremely massive planets, the atoms themselves begin to get compressed due to large gravitational forces, meaning you can pack more of them into a small space.

Jupiter, for example, has three times the mass of Saturn, but is only about 20% larger in size. If you replace an electron in a hydrogen atom with a muon, an unstable electron-like particle that has the same charge but 206 times the mass, the muonic hydrogen atom will be only 1/206th the size of normal hydrogen.

And a Uranium atom is actually larger in size than the individual protons-and-neutrons would be if you packed them together, due to the long-range nature of the electrostatic repulsion of the protons, compared to the short-range nature of the strong force.

Planet's axes are tilted at different angles axis

Image credit: Calvin Hamilton.

The planets of the Solar System, shown to the scale of their physical sizes, show a Saturn that’s almost as large as Jupiter. However, Jupiter is 3 times as massive, indicating that its atoms are substantially compressed due to gravitational pressure.

By having different forces at play of different strengths, you can build a proton, neutron, or other hadron of finite size out of point-like quarks. By combining protons and neutrons, you can build nuclei of larger sizes than their individual components, bound together, would give you. And by binding electrons to the nucleus, you can build a much larger structure, all owing to the fact that the zero-point energy of an electron bound to an atom is much greater than zero.

In order to get a Universe filled with structures that take up a finite amount of space and have a non-zero size, you don’t need anything more than zero-dimensional, point-like building blocks. Forces, energy, and the quantum properties inherent to particles themselves are more than enough to do the job.


Ethan Siegel is the founder and primary writer of Starts With A Bang!


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Radar was developed secretly for military use by several nations, before and during World War II.The term was coined in 1940 by the United States Navy as an acronym for RAdio Detection And Ranging. It entered English and other languages as a common noun, losing all capitalization.

Radar uses radio waves to determine the range, angle, or velocity of objects.

EM waves can be of many different wavelengths.
Longer wavelengths we perceive as orange and red
Shorter wavelengths are towards the blue end of the spectrum

Fields are at right-angles to each other

They travel through vacuum (empty space) at the speed of light

c  =  speed of light
c  =  3 x 108 m/s       =   186,282 miles/second

So all parts of the EM spectrum – radio, light, Wi-Fi, X-rays,
are all made of exactly the same thing! The only thing different among them? wavelength and frequency!


Our eyes can only see a tiny amount of the EM spectrum.
There are longer and shorter waves as well.

Gamma rays Spectrum Properties NASA

Is  used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain.

A radar system consists of:

transmitter producing electromagnetic radio waves

a receiving antenna (often the same antenna is used for transmitting and receiving)

a receiver and processor to determine properties of the object(s)

Radio waves from the transmitter reflect off the object and return to the receiver

This gives info about the object’s location and speed.


air and terrestrial traffic control

radar astronomy

air-defence systems / antimissile systems


marine radars to locate landmarks and other ships

Commercial marine radar antenna

aircraft anticollision systems

radar by Marshall Brain

outer space surveillance and rendezvous systems

meteorological (weather) precipitation monitoring

Weather radar

flight control systems

guided missile target locating systems

ground-penetrating radar for geological observation

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

6.MS-PS4-1. Use diagrams of a simple wave to explain that (a) a wave has a repeating pattern with a specific amplitude, frequency, and wavelength, and (b) the amplitude of a wave is related to the energy of the wave.

HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium.

HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. Clarification Statements:
• Emphasis is on qualitative information and descriptions.
• Examples of technological devices could include solar cells capturing light and
converting it to electricity, medical imaging, and communications technology.

Massachusetts Science and Technology/Engineering Curriculum Framework (2006)

6. Electromagnetic Radiation Central Concept: Oscillating electric or magnetic fields can generate electromagnetic waves over a wide spectrum. 6.1 Recognize that electromagnetic waves are transverse waves and travel at the speed of light through a vacuum. 6.2 Describe the electromagnetic spectrum in terms of frequency and wavelength, and identify the locations of radio waves, microwaves, infrared radiation, visible light (red, orange, yellow, green, blue, indigo, and violet), ultraviolet rays, x-rays, and gamma rays on the spectrum.

Can we stop a hurricane

Can we stop a hurricane?

Can tropical cyclones be stopped?


Can Science Halt Hurricanes? Tropical cyclones are nature’s most powerful storms. Can they be stopped?


Engineers could stop hurricanes with the ‘sunglasses effect’ — but it’d require a huge sacrifice


What would be need to stop a hurricane?


Offshore wind farms could tame hurricanes before they reach land, Stanford-led study says


Hurricane Research Division NOAA: Tropical Cyclone Modification and Myths



“Taming Hurricanes with Arrays of Offshore Wind Turbines,” appears online on Feb. 26 in Nature Climate Change


The Science and History of the Sea

The Science and History of the Sea

Session 1: TBA at the USS Constitution Museum. Museum staff led.

Introductory movie (10 minutes)

  • Design your own frigate based on the templates of Constitution’s ship designer Joshua Humphreys: Students will produce drawings.
  • Made in America – what materials were used to create the USS Constitution? Students will create a list of 5 materials from the New England region.
  • Which of these woods is the hardest? Through dropping balls into difference woods, we can study the difference in how the ball bounces back. The kinetic energy of the rebounding ball is related to the amount of energy absorbed by the wood. Students will review with the teacher the difference between kinetic energy and potential energy.
  • Test your ship against other frigates in this hands-on challenge. Choose between three different types of ships for the ultimate test of size, speed and power: Students use this interactive computer simulation.
  • What’s so great about copper? Learn about the metals used in construction
  • Build a ship: Assemble 2D pieces into a 3D model – how quickly can they accurately complete the task?
  • Construction and launch: View this video, and then explain how a ship is safely launched from a drydock into the ocean.  Students will demonstrate that they understand the procedure by writing a step-by-step paragraph explaining the sequence.
  • How can a ship sail against the wind? Through a hands on experiment, see how changing the angle of the sail affects the motion of the boat: Students should be able to explain in complete sentences how the same wind can make a ship move forwards or backwards.
  • On the 2nd story of the museum, operate a working block-and-tackle system. This uses a classic simple machine. It is a system of two or more pulleys with a rope or cable threaded between them, usually used to lift or pull heavy loads. Back in the school building, we’ll review each of the classic simple machines.

On the 2nd story of the museum, operate a working block-and-tackle system. This uses a classic simple machine. It is a system of two or more pulleys with a rope or cable threaded between them, usually used to lift or pull heavy loads.

pulley simple machine


Session 2: TBA at the USS Constitution Museum. Museum staff led.

Details TBA.


Session 3: USS Constitution Visitor Center, Building 5 (teacher led)

10 minute orientation video

Can you locate where our school is on the 3D Boston Naval Shipyard model?

As students tour the visitor center, they practice ELA reading and writing skills (listed below) by briefly summarizing something they learn from each of these sections: They are encouraged to create drawings/tracings as they see fit to help illustrate their text.

  • Describe how ropes are made from string in the ropewalk
  • From wood & sail to steel & steam
  • Preparing for new technology
  • The shipyard in the Civil War
  • Ships and shipbuilding
  • The Navy Yard 1890-1974
  • Chain Forge and Foundary
  • The Navy Yard during World Wars I and II
  • Shipyard workers 1890 to 1974
  • The shipyard during the Cold War era 1945-1974


Session 4: Teaching math using the USS Constitition

Teaching math: Lessons from the USS Constitution

This teaching supplement contains math lessons organized in grade-level order. However, because many of the math skills used in these lessons are taught in multiple grades, both grade-level and lesson content are listed below.

Pre K–K 
Estimating Numbers of Objects

Grade 1
Estimating and Comparing Numbers of Objects

Grade 2
Estimating and Comparing Length, Width and Perimeter

Grade 3
Computing Time and Creating a Schedule

Grade 4
Drawing Conclusions from Data Sets

Grade 5
Creating and Interpreting Graphs from Tables

Grade 6
Range, Mean, Median and Mode and Stem-and-Leaf Plots

Grade 7
Converting Between Systems of Measurement

Grade 8
Calculating Volume

Algebra I (Grade 9–10)
Describing Distance and Velocity Graphs

Algebra I (Grade 9–10)
Writing Linear Equations

Algebra II (Grade 9–12)
Using Projectile Motion to Explore Maximums and Zeros

Precalculus & Advanced Math (Grade 10–12)
Using Parabolic Equations & Vectors to Describe the Path of Projectile Motion


Learning Standards

MA 2006 Science Curriculum Framework

2. Engineering Design. Central Concept: Engineering design requires creative thinking and consideration of a variety of ideas to solve practical problems. Identify tools and simple machines used for a specific purpose, e.g., ramp, wheel, pulley, lever.

Massachusetts Science and Technology/Engineering Curriculum Framework

HS-ETS4-5(MA). Explain how a machine converts energy, through mechanical means, to do work. Collect and analyze data to determine the efficiency of simple and complex machines.

Benchmarks, American Association for the Advancement of Science

In the 1700s, most manufacturing was still done in homes or small shops, using small, handmade machines that were powered by muscle, wind, or moving water. 10J/E1** (BSL)

In the 1800s, new machinery and steam engines to drive them made it possible to manufacture goods in factories, using fuels as a source of energy. In the factory system, workers, materials, and energy could be brought together efficiently. 10J/M1*

The invention of the steam engine was at the center of the Industrial Revolution. It converted the chemical energy stored in wood and coal into motion energy. The steam engine was widely used to solve the urgent problem of pumping water out of coal mines. As improved by James Watt, Scottish inventor and mechanical engineer, it was soon used to move coal; drive manufacturing machinery; and power locomotives, ships, and even the first automobiles. 10J/M2*

The Industrial Revolution developed in Great Britain because that country made practical use of science, had access by sea to world resources and markets, and had people who were willing to work in factories. 10J/H1*

The Industrial Revolution increased the productivity of each worker, but it also increased child labor and unhealthy working conditions, and it gradually destroyed the craft tradition. The economic imbalances of the Industrial Revolution led to a growing conflict between factory owners and workers and contributed to the main political ideologies of the 20th century. 10J/H2

Today, changes in technology continue to affect patterns of work and bring with them economic and social consequences. 10J/H3*

Massachusetts History and Social Science Curriculum Frameworks

5.11 Explain the importance of maritime commerce in the development of the economy of colonial Massachusetts, using historical societies and museums as needed. (H, E)

5.32 Describe the causes of the war of 1812 and how events during the war contributed to a sense of American nationalism. A. British restrictions on trade and impressment.  B. Major battles and events of the war, including the role of the USS Constitution, the burning of the Capitol and the White House, and the Battle of New Orleans.

National Council for the Social Studies: National Curriculum Standards for Social Studies

Time, Continuity and Change: Through the study of the past and its legacy, learners examine the institutions, values, and beliefs of people in the past, acquire skills in historical inquiry and interpretation, and gain an understanding of how important historical events and developments have shaped the modern world. This theme appears in courses in history, as well as in other social studies courses for which knowledge of the past is important.

A study of the War of 1812 enables students to understand the roots of our modern nation. It was this time period and struggle that propelled us from a struggling young collection of states to a unified player on the world stage. Out of the conflict the nation gained a number of symbols including USS Constitution. The victories she brought home lifted the morale of the entire nation and endure in our nation’s memory today. – USS Constitution Museum, National Education Standards

Common Core ELA: Reading Instructional Texts

Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings

Common Core ELA Writing

Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims.

Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing.

Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

External links

The USS Constitution Museum, located in the Charlestown Navy Yard, which is part of the Boston National Historical Park

What kinds of radiation cause cancer

An infographic on the intersection of physics and health:

For most people the biggest cancer risk from radiation hovers in the sky above us giving us all warmth and light. There is no cancer risk from Wi-Fi or microwaves.

Wear sunscreen, but use WiFi without fear. (Image: Spazturtle/SMS (CC))

What is radiation, and where does it come from? nuclear chemistry

What is cancer? How is caused?  Cancer

Microwaves, Radio Waves, and Other Types of Radiofrequency Radiation: American Cancer Society



Soundly Proving the Curvature of the Earth at Lake Pontchartrain

Excerpted from an article by Mick West

A classic experiment to demonstrate the curvature of a body of water is to place markers (like flags) a fixed distance above the water in a straight line, and then view them along that line in a telescope. If the water surface is flat then the markers will appear also in a straight line. If the surface of the water is curved (as it is here on Earth) then the markers in the middle will appear higher than the markers at the ends.

Here’s a highly exaggerated diagram of the effect by Alfred Russel Wallace in 1870, superimposed over an actual photograph.

Lake Pontchartrain power lines demonstrating the curvature Metabunk

This is a difficult experiment to do as you need a few miles for the curvature to be apparent. You also need the markers to be quite high above the surface of the water, as temperature differences between the water and the air tend to create significant refraction effects close to the water.

However Youtuber Soundly has found a spot where there’s a very long line of markers permanently fixed at constant heights above the water line, clearly demonstrating the curve. It’s a line of power transmission towers at Lake Pontchartrain, near New Orleans, Louisiana.

The line of power lines is straight, and they are all the same size, and the same height above the water. They are also very tall, and form a straight line nearly 16 miles long. Far better than any experiment one could set up on a canal or a lake. You just need to get into a position where you can see along the line of towers, and then use a powerful zoom lense to look along the line to make any curve apparent

One can see quite clearly in the video and photos that there’s a curve. Soundly has gone to great lengths to provide multiple videos and photos of the curve from multiple perspectives. They all show the same thing: a curve.

Lake Pontchartrain curve around Earth

One objection you might make is that the towers could be curving to the right. However the same curve is apparent from both sides, so it can only be curving over the horizon.



People have asked why the curve is so apparent in one direction, but not in the other. The answer is compressed perspective. Here’s a physical example:


Compressed perspective on a car

That’s my car, the roof of which is slightly curved both front to back and left to right. I’ve put some equal sized chess pawns on it in two straight lines. If we step back a bit and zoom in we get:

Compressed perspective on a car II

Notice a very distinct curve from the white pieces, but the “horizon” seems to barely curve at all.

Similarly in the front-back direction, where there’s an even greater curve:

Compressed perspective on a car III

There’s a lot more discussion with photos here Soundly Proving the Curvature of the Earth at Lake Pontchartrain



Lord Of The Rings Optics challenge

A great physics problem for senior year students:

In J. R. R. Tolkien’s The Lord of the Rings (volume 2, p. 32), Legolas the Elf claims to be able to accurately count horsemen and discern their hair color (yellow) 5 leagues away on a bright, sunny day.

“Riders!” cried Aragorn, springing to his feet. “Many riders on swift steeds are coming towards us!”
“Yes,” said Legolas,”there are one hundred and five. Yellow is their hair, and bright are their spears. Their leader is very tall.”
Aragorn smiled. “Keen are the eyes of the Elves,” he said.
“Nay! The riders are little more than five leagues distant,” said Legolas.”

Make appropriate estimates and argue that Legolas must have very strange-looking eyes, have some means of non-visual perception, or have made a lucky guess. (1 league ~ 3.0 mi.)

On land, the league is most commonly defined as three miles, though the length of a mile could vary from place to place and depending on the era.
At sea, a league is three nautical miles (3.452 miles; 5.556 kilometres).

Several solutions are possible, depending on the estimating assumptions

Eye focusing rays of light figure_10_24_labeled

When parallel light waves strike a concave lens the waves striking the lens surface at a right angle goes straight through but light waves striking the surface at other angles diverge. In contrast, light waves striking a convex lens converge at a single point called a focal point. The distance from the long axis of the lens to the focal point is the focal length. Both the cornea and the lens of the eye have convex surfaces and help to focus light rays onto the retina. The cornea provides for most of the refraction but the curvature of the lens can be adjusted to adjust for near and far vision.


By Chad Orzel is an Associate Professor in the Department of Physics and Astronomy at Union College in Schenectady, NY

The limiting factor here is the wave nature of light– light passing through any aperture will interfere with itself, and produce a pattern of bright and dark spots.
So even an infinitesimally small point source of light will appear slightly spread out, and two closely spaced point sources will begin to run into one another.
The usual standard for determining whether two nearby sources can be distinguished from one another is the Rayleigh criterion:

Rayleigh Criterion circular aperature

sine of the angular separation between two objects = 1.22 x ratio of the light wavelength to the diameter of the (circular) aperture, through which the light passes.
To get better resolution, you need either a smaller wavelength or a larger aperture.

Legolas says that the riders are “little more than five leagues distant.”
A league is something like three miles, which would be around 5000 meters, so let’s call it 25,000 meters from Legolas to the Riders.
Visible light has an average wavelength of around 500 nm, which is a little more green than the blond hair of the Riders, but close enough for our purposes.

The sine of a small angle can be approximated by the angle itself.

The angle = the size of the separation between objects divided by the distance from the objects to the viewer.

Putting it all together, Legolas’s pupils would need to be 0.015 m in diameter.
That’s a centimeter and a half, which is reasonable, provided he’s an anime character. I don’t think Tolkien’s Elves are described as having eyes the size of teacups, though.

We made some simplifying assumptions to get that answer, but relaxing them only makes things worse. Putting the Riders farther away, and using yellower light would require Legolas’s eyes to be even bigger. And the details he claims to see are almost certainly on scales smaller than one meter, which would bump things up even more.

Any mathematical objections to these assumptions? Sean Barrett writes:

“The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer.”

Technically this is not quite right; the separation divided by the distance is not the angle itself, but rather the tangent of the angle. (SOHCAHTOA: sin = opposite/hypoteneuse; tangent = opposite/adjacent.)

Because the cos of a very small angle is very nearly 1, however, the tangent is just as nearly equal the angle as is the sine. But that doesn’t mean you can just skip that step. And there’s really not much need to even mention the angle; with such a very tiny angle, clearly the hypoteneuse and the adjacent side have essentially the same length (the distance to either separated point is also essentially 25K meters), and so you can correctly say that the sine itself is in this case approximated by the separation divided by the distance, and never mention the angle at all.

(You could break out a calculator to be on the safe side, but if you’re going to do that you need to know the actual formulation to compute the angle, not compute it as opposite/adjacent! But, yes, both angle (in radians) and the sine are also 1/25000 to about 10 sig figs.)

II. Another solution

Using the Rayleigh Criterion. In order for two things, x distance apart, to be discernible as separate, at an angular distance θ, to an instrument with a circular aperture with diameter a:

θ > arcsin(1.22 λ/a)

5 leagues is approximately 24000 m.
Sssume that each horse is ~2 m apart from each other
So arctan (1/12000) ≅ θ.
We can use the small-angle approximation (sin(θ) ≅ tan(θ) ≅ θ when θ is small)
So we get 1/12000 ≅ 1.22 λ/a

Yellow light has wavelengths between 570 and 590 nm, so we’ll use 580.

a ≅ 1.22 * (580E-9 m)* 12000 ≅ .0085 m.

8 mm is about as far as a human pupil will dilate, so for Legolas to have pupils this big in broad daylight must be pretty odd-looking.
Edit: The book is Six Ideas that Shaped Physics: Unit Q, by Thomas Moore

III. Great discussion on the Physics StackExchange

Could Legolas actually see that far? Physics StackExchange discussion

Here, Kyle Oman writes:

For a human-like eye, which has a maximum pupil diameter of about mm and choosing the shortest wavelength in the visible spectrum of about 390 nm, the angular resolution works out to about 5.3×105  (radians, of course).

At a distance of 24 km, this corresponds to a linear resolution (θd, where is the distance) of about 1.2m1. So counting mounted riders seems plausible since they are probably separated by one to a few times this resolution.

Comparing their heights which are on the order of the resolution would be more difficult, but might still be possible with dithering.

Does Legolas perhaps wiggle his head around a lot while he’s counting? Dithering only helps when the image sampling (in this case, by elven photoreceptors) is worse than the resolution of the optics. Human eyes apparently have an equivalent pixel spacing of something like a few tenths of an arcminute, while the diffraction limited resolution is about a tenth of an arcminute, so dithering or some other technique would be necessary to take full advantage of the optics.

An interferometer has an angular resolution equal to a telescope with a diameter equal to the separation between the two most widely separated detectors. Legolas has two detectors (eyeballs) separated by about 10 times the diameter of his pupils75 mm or so at most. This would give him a linear resolution of about 15cm at a distance of 24 km, probably sufficient to compare the heights of mounted riders.

However, interferometry is a bit more complicated than that. With only two detectors and a single fixed separation, only features with angular separations equal to the resolution are resolved, and direction is important as well.

If Legolas’ eyes are oriented horizontally, he won’t be able to resolve structure in the vertical direction using interferometric techniques. So he’d at the very least need to tilt his head sideways, and probably also jiggle it around a lot (including some rotation) again to get decent sampling of different baseline orientations. Still, it seems like with a sufficiently sophisticated processor (elf brain?) he could achieve the reported observation.

Luboš Motl points out some other possible difficulties with interferometry in his answer, primarily that the combination of a polychromatic source and a detector spacing many times larger than the observed wavelength lead to no correlation in the phase of the light entering the two detectors. While true, Legolas may be able to get around this if his eyes (specifically the photoreceptors) are sufficiently sophisticated so as to act as a simultaneous high-resolution imaging spectrometer or integral field spectrograph and interferometer. This way he could pick out signals of a given wavelength and use them in his interferometric processing.

A couple of the other answers and comments mention the potential difficulty drawing a sight line to a point 24 km away due to the curvature of the Earth. As has been pointed out, Legolas just needs to have an advantage in elevation of about 90 meters (the radial distance from a circle 6400 km in radius to a tangent 24 km along the circumference; Middle-Earth is apparently about Earth-sized, or may be Earth in the past, though I can’t really nail this down with a canonical source after a quick search). He doesn’t need to be on a mountaintop or anything, so it seems reasonable to just assume that the geography allows a line of sight.

Finally a bit about “clean air”. In astronomy (if you haven’t guessed my field yet, now you know…) we refer to distortions caused by the atmosphere as “seeing”.

Seeing is often measured in arcseconds (3600 arcse60 arcmi13600 arcsec = 60arcmin = 1∘), referring to the limit imposed on angular resolution by atmospheric distortions.

The best seeing, achieved from mountaintops in perfect conditions, is about arcsec,
or in radians 4.8×106 . This is about the same angular resolution as Legolas’ amazing interferometric eyes.

I’m not sure what seeing would be like horizontally across a distance of 24 km. On the one hand there is a lot more air than looking up vertically; the atmosphere is thicker than 24 km but its density drops rapidly with altitude. On the other hand the relatively uniform density and temperature at fixed altitude would cause less variation in refractive index than in the vertical direction, which might improve seeing.

If I had to guess, I’d say that for very still air at uniform temperature he might get seeing as good as 1 arcsec, but with more realistic conditions with the Sun shining, mirage-like effects probably take over limiting the resolution that Legolas can achieve.


IV. Also on StackExchange, the famous Luboš Motl writes:

Let’s first substitute the numbers to see what is the required diameter of the pupil according to the simple formula:

I’ve substituted the minimal (violet…) wavelength because that color allowed me a better resolution i.e. smaller θθ. The height of the knights is two meters.
Unless I made a mistake, the diameter DD is required to be 0.58 centimeters. That’s completely sensible because the maximally opened human pupil is 4-9 millimeter in diameter.
Just like the video says, the diffraction formula therefore marginally allows to observe not only the presence of the knights – to count them – but marginally their first “internal detailed” properties, perhaps that the pants are darker than the shirt. However, to see whether the leader is 160 cm or 180 cm is clearly impossible because it would require the resolution to be better by another order of magnitude. Just like the video says, it isn’t possible with the visible light and human eyes. One would either need a 10 times greater eye and pupil; or some ultraviolet light with 10 times higher frequency.
It doesn’t help one to make the pupils narrower because the resolution allowed by the diffraction formula would get worse. The significantly more blurrier images are no helpful as additions to the sharpest image. We know that in the real world of humans, too. If someone’s vision is much sharper than the vision of someone else, the second person is pretty much useless in refining the information about some hard-to-see objects.

The atmospheric effects are likely to worsen the resolution relatively to the simple expectation above. Even if we have the cleanest air – it’s not just about the clean air; we need the uniform air with a constant temperature, and so on, and it is never so uniform and static – it still distorts the propagation of light and implies some additional deterioration. All these considerations are of course completely academic for me who could reasonably ponder whether I see people sharply enough from 24 meters to count them. 😉

Even if the atmosphere worsens the resolution by a factor of 5 or so, the knights may still induce the minimal “blurry dots” at the retina, and as long as the distance between knights is greater than the distance from the (worsened) resolution, like 10 meters, one will be able to count them.

In general, the photoreceptor cells are indeed dense enough so that they don’t really worsen the estimated resolution. They’re dense enough so that the eye fully exploits the limits imposed by the diffraction formula, I think. Evolution has probably worked up to the limit because it’s not so hard for Nature to make the retinas dense and Nature would be wasting an opportunity not to give the mammals the sharpest vision they can get.

Concerning the tricks to improve the resolution or to circumvent the diffraction limit, there aren’t almost any. The long-term observations don’t help unless one could observe the location of the dots with the precision better than the distance of the photoreceptor cells. Mammals’ organs just can’t be this static. Image processing using many unavoidably blurry images at fluctuating locations just cannot produce a sharp image.

The trick from the Very Large Array doesn’t work, either. It’s because the Very Large Array only helps for radio (i.e. long) waves so that the individual elements in the array measure the phase of the wave and the information about the relative phase is used to sharpen the information about the source. The phase of the visible light – unless it’s coming from lasers, and even in that case, it is questionable – is completely uncorrelated in the two eyes because the light is not monochromatic and the distance between the two eyes is vastly greater than the average wavelength. So the two eyes only have the virtue of doubling the overall intensity; and to give us the 3D stereo vision. The latter is clearly irrelevant at the distance of 24 kilometers, too. The angle at which the two eyes are looking to see the 24 km distant object are measurably different from the parallel directions. But once the muscles adapt into this slightly non-parallel angles, what the two eyes see from the 24 km distance is indistinguishable.


V. Analyzed in “How Far Can Legolas See?” by minutephysics (Henry Reich)