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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

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How science works – examples

Science is a process used to approach claims. We approach claims skeptically: That doesn’t mean that that we don’t believe anything. Rather, it means we don’t accept a claim unless we are given compelling evidence. Skepticism is a provisional approach to claims.

skeptic no amount of belief

In 1976 during the Viking missions, NASA scientists found a pattern of chemical reactions that indicated some form of bacterial life may be living in the martian soil.

In the late 1990s, studies of a Martian meteorite provided evidence that microscopic, bacteria-like life on Mars may have existed. Did simple forms of life once lived on Mars? Does bacterial life live in the Martian soil today?

If this interests you, look up Viking lander biological experiments, and the meteorite Allan Hills 84001 (ALH84001) 

mars-life-new-look-old-data_51551_990x742

Many people in Scotland reported a creature swimming in Loch Ness (a large freshwater lake in the Scottish Highlands.) A few blurry photographs have been taken of an object in the water. Newspapers named this supposed creature “the Loch Ness Monster”. Are there unknown, large sea monsters living in this lake?

If this interests you look up Loch Ness “monster”

Hoaxed_photo_of_the_Loch_Ness_monster

In the 1970’s doctors created an oral pill, Loniten, to control high blood pressure. It works by dilating the blood vessels, so blood can flow better. One of the side effects that patients reported was excess body hair growth. Could this be the first drug to regrow more hair? If this interests you look up the discovery of Minoxidil.

Male pattern baldness minoxidilMinoxidil molcule

Charles Darwin (1809 –1882) was an English naturalist. He discovered evidence that today’s animals are modified versions of animals that lived in the past; he discovered that many forms of life have descended over time from common ancestors. Has life on Earth evolved from earlier forms of life? If this interests you look up the discovery of evolution by natural selection.

Charles Darwin quote

How can we tell which of these claims are true?  We use the scientific method to investigate such claims. 

 

Learning Objectives

2016 Massachusetts Science and Technology/Engineering Standards
Students will be able to:
* plan and conduct an investigation, including deciding on the types, amount, and accuracy of data needed to produce reliable measurements, and consider limitations on the precision of the data
* apply scientific reasoning, theory, and/or models to link evidence to the claims and assess the extent to which the reasoning and data support the explanation or conclusion;
* respectfully provide and/or receive critiques on scientific arguments by probing reasoning and evidence and challenging ideas and conclusions, and determining what additional information is required to solve contradictions
* evaluate the validity and reliability of and/or synthesize multiple claims, methods, and/or designs that appear in scientific and technical texts or media, verifying the data when possible.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Implementation: Curriculum, Instruction, Teacher Development, and Assessment
“Through discussion and reflection, students can come to realize that scientific inquiry embodies a set of values. These values include respect for the importance of logical thinking, precision, open-mindedness, objectivity, skepticism, and a requirement for transparent research procedures and honest reporting of findings.”

Next Generation Science Standards: Science & Engineering Practices
● Ask questions that arise from careful observation of phenomena, or unexpected results, to clarify and/or seek additional information.
● Ask questions that arise from examining models or a theory, to clarify and/or seek additional information and relationships.
● Ask questions to determine relationships, including quantitative relationships, between independent and dependent variables.
● Ask questions to clarify and refine a model, an explanation, or an engineering problem.
● Evaluate a question to determine if it is testable and relevant.
● Ask questions that can be investigated within the scope of the school laboratory, research facilities, or field (e.g., outdoor environment) with available resources and, when appropriate, frame a hypothesis based on a model or theory.
● Ask and/or evaluate questions that challenge the premise(s) of an argument, the interpretation of a data set, or the suitability of the design

MA 2016 Science and technology

Appendix I Science and Engineering Practices Progression Matrix

Science and engineering practices include the skills necessary to engage in scientific inquiry and engineering design. It is necessary to teach these so students develop an understanding and facility with the practices in appropriate contexts. The Framework for K-12 Science Education (NRC, 2012) identifies eight essential science and engineering practices:

1. Asking questions (for science) and defining problems (for engineering).
2. Developing and using models.
3. Planning and carrying out investigations.
4. Analyzing and interpreting data.
5. Using mathematics and computational thinking.
6. Constructing explanations (for science) and designing solutions (for engineering).
7. Engaging in argument from evidence.
8. Obtaining, evaluating, and communicating information.

Scientific inquiry and engineering design are dynamic and complex processes. Each requires engaging in a range of science and engineering practices to analyze and understand the natural and designed world. They are not defined by a linear, step-by-step approach. While students may learn and engage in distinct practices through their education, they should have periodic opportunities at each grade level to experience the holistic and dynamic processes represented below and described in the subsequent two pages… http://www.doe.mass.edu/frameworks/scitech/2016-04.pdf

Origin of the moon

There are a number of models describing how Earth’s moon originated.

Deep Space Climate Observatory DSCOVR transit Earth Dark side of moon

A NASA camera aboard the Deep Space Climate Observatory (DSCOVR) satellite captured a unique view of the moon as it moved in front of the sunlit side of Earth last month. The series of test images shows the fully illuminated “dark side” of the moon that is never visible from Earth. The images were captured by NASA’s Earth Polychromatic Imaging Camera (EPIC), a four megapixel CCD camera and telescope on the DSCOVR satellite orbiting 1 million miles from Earth. From its position between the sun and Earth, DSCOVR conducts its primary mission of real-time solar wind monitoring for the National Oceanic and Atmospheric Administration (NOAA). https://www.nasa.gov/feature/goddard/from-a-million-miles-away-nasa-camera-shows-moon-crossing-face-of-earth

These models are together called the giant-impact hypothesis. It proposes that a Mars-sized body, called Theia, impacted Earth, creating a large debris ring around Earth, which then accreted to form the Moon. This collision also resulted in the 23.5° tilted axis of the earth, thus causing the seasons. (adapted loosely from Wikipedia)

Giant Impact Moon Theia

Giant Impact Moon Theia another
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What Made the Moon? New Ideas Try to Rescue a Troubled Theory

Quanta Magazine, Rebecca Boyle, 8/2/17

http://www.quantamagazine.org/what-made-the-moon-new-ideas-try-to-rescue-a-troubled-theory-20170802/

In the past five years, a bombardment of studies has exposed a problem: The canonical giant impact hypothesis rests on assumptions that do not match the evidence. If Theia hit Earth and later formed the moon, the moon should be made of Theia-type material. But the moon does not look like Theia — or like Mars, for that matter. Down to its atoms, it looks almost exactly like Earth.

Confronted with this discrepancy, lunar researchers have sought new ideas for understanding how the moon came to be. The most obvious solution may also be the simplest, though it creates other challenges with understanding the early solar system:

  • Perhaps Theia did form the moon, but Theia was made of material that was almost identical to Earth.
  • The second possibility is that the impact process thoroughly mixed everything, homogenizing disparate clumps and liquids the way pancake batter comes together. This could have taken place in an extraordinarily high-energy impact, or a series of impacts that produced a series of moons that later combined.
  • The third explanation challenges what we know about planets. It’s possible that the Earth and moon we have today underwent strange metamorphoses and wild orbital dances that dramatically changed their rotations and their futures.

Theia Four Ways To Make The Moon

Lucy Reading-Ikkanda/Quanta Magazine

Bad News for Theia

To understand what may have happened on Earth’s most momentous day, it helps to understand the solar system’s youth. Four and a half billion years ago, the sun was surrounded by a hot, doughnut-shaped cloud of debris. Star-forged elements swirled around our newborn sun, cooling and, after eons, combining — in a process we don’t fully understand — into clumps, then planetesimals, then increasingly larger planets. These rocky bodies violently, frequently collided and vaporized one another anew. It was in this unspeakably brutal, billiard-ball hellscape that the Earth and the moon were forged.

solar system forming GIF 2

To get to the moon we have now, with its size, spin and the rate at which it is receding from Earth, our best computer models say that whatever collided with Earth must have been the size of Mars. Anything bigger or much smaller would produce a system with a much greater angular momentum than we see. A bigger projectile would also throw too much iron into Earth’s orbit, creating a more iron-rich moon than the one we have today.

Early geochemical studies of troctolite 76536 and other rocks bolstered this story. They showed that lunar rocks would have originated in a lunar magma ocean, the likes of which could only be generated by a giant impact. The troctolite would have bobbed in a molten sea like an iceberg floating off Antarctica. On the basis of these physical constraints, scientists have argued that the moon was made from the remnants of Theia. But there is a problem.

Back to the early solar system. As rocky worlds collided and vaporized, their contents mixed, eventually settling into distinct regions. Closer to the sun, where it was hotter, lighter elements would be likelier to heat up and escape, leaving an excess of heavy isotopes (variants of elements with additional neutrons). Farther from the sun, rocks were able to keep more of their water, and lighter isotopes persisted. Because of this, a scientist can examine an object’s isotopic mix to identify where in the solar system it came from, like accented speech giving away a person’s homeland.

isotopes-of-carbon

These differences are so pronounced that they’re used to classify planets and meteorite types. Mars is so chemically distinct from Earth, for instance, that its meteorites can be identified simply by measuring ratios of three different oxygen isotopes.

In 2001, using advanced mass spectrometry techniques, Swiss researchers remeasured troctolite 76536 and 30 other lunar samples. They found that its oxygen isotopes were indistinguishable from those on Earth. Geochemists have since studied titanium, tungsten, chromium, rubidium, potassium and other obscure metals from Earth and the moon, and everything looks pretty much the same.

This is bad news for Theia. If Mars is so obviously different from Earth, Theia — and thus, the moon — ought to be different, too. If they’re the same, that means the moon must have formed from melted bits of Earth. The Apollo rocks are then in direct conflict with what the physics insist must be true.

“The canonical model is in serious crisis,” said Sarah Stewart, a planetary scientist at the University of California, Davis. “It has not been killed yet, but its current status is that it doesn’t work.”

Stewart has been trying to reconcile the physical constraints of the problem — the need for an impactor of a certain size, going a certain speed — with the new geochemical evidence. In 2012, she and Matija Ćuk, now at the SETI Institute, proposed a new physical model for the moon’s formation. They argued that the early Earth was a whirling dervish, rotating through one day every two to three hours, when Theia collided with it. The collision would produce a disk around the Earth, much like the rings of Saturn — but it would only persist for about 24 hours. Ultimately, this disk would cool and solidify to form the moon.

Supercomputers are not powerful enough to model this process completely, but they showed that a projectile slamming into such a fast-spinning world could shear away enough of Earth, obliterate enough of Theia and scramble enough of both to build a moon and Earth with similar isotopic ratios. Think of smacking a wet lump of clay on a fast-spinning potter’s wheel.

For the fast-spinning-Earth explanation to be right, however, something else would have to come along to slow down Earth’s rotation rate to what it is now. In their 2012 work, Stewart and Ćuk argued that under certain orbital-resonance interactions, Earth could have transferred angular momentum to the sun. Later, Jack Wisdom of the Massachusetts Institute of Technology suggested several alternate scenarios for draining angular momentum away from the Earth-moon system.

Angular momentum ice skater

But none of the explanations was entirely satisfactory. The 2012 models still couldn’t explain the moon’s orbit or the moon’s chemistry, Stewart said. Then last year, Simon Lock, a graduate student at Harvard University and Stewart’s student at the time, came up with an updated model that proposes a previously unrecognized planetary structure.

In this story, every bit of Earth and Theia vaporized and formed a bloated, swollen cloud shaped like a thick bagel. The cloud spun so quickly that it reached a point called the co-rotation limit. At that outer edge of the cloud, vaporized rock circled so fast that the cloud took on a new structure, with a fat disk circling an inner region. Crucially, the disk was not separated from the central region the way Saturn’s rings are — nor the way previous models of giant-impact moon formation were, either.

 Press_release_cartoon_website_complete_rast

A synestia would be made of a bagel-like mass of vaporized rock surrounding a rocky planet.

Simon Lock and Sarah Stewart

Conditions in this structure are indescribably hellish; there is no surface, but instead clouds of molten rock, with every region of the cloud forming molten-rock raindrops. The moon grew inside this vapor, Lock said, before the vapor eventually cooled and left in its wake the Earth-moon system.

Given the structure’s unusual characteristics, Lock and Stewart thought it deserved a new name. They tried several versions before coining synestia, which uses the Greek prefix syn-, meaning together, and the goddess Hestia, who represents the home, hearth and architecture. The word means “connected structure,” Stewart said.

“These bodies aren’t what you think they are. They don’t look like what you thought they did,” she said.

Synestia artists impression

Mike Zeng for Quanta Magazine

In May, Lock and Stewart published a paper on the physics of synestias; their paper arguing for a synestia lunar origin is still in review. They presented the work at planetary science conferences in the winter and spring and say their fellow researchers were intrigued but hardly sold on the idea. That may be because synestias are still just an idea; unlike ringed planets, which are common in our solar system, and protoplanetary disks, which are common across the universe, no one has ever seen one.

“But this is certainly an interesting pathway that could explain the features of our moon and get us over this hump that we’re in, where we have this model that doesn’t seem to work,” Lock said.

Let a Dozen Moons Bloom

Among natural satellites in the solar system, Earth’s moon may be most striking for its solitude. Mercury and Venus lack natural satellites, in part because of their nearness to the sun, whose gravitational interactions would make their moons’ orbits unstable. Mars has tiny Phobos and Deimos, which some argue are captured asteroids and others argue formed from Martian impacts. And the gas giants are chockablock with moons, some rocky, some watery, some both.

In contrast to these moons, Earth’s satellite also stands out for its size and the physical burden it carries. The moon is about 1 percent the mass of Earth, while the combined mass of the outer planets’ satellites is less than one-tenth of 1 percent of their parents. Even more important, the moon contains 80 percent of the angular momentum of the Earth-moon system. That is to say, the moon is responsible for 80 percent of the motion of the system as a whole. For the outer planets, this value is less than 1 percent.

The moon may not have carried all this weight the whole time, however. The face of the moon bears witness to its lifelong bombardment; why should we assume that just one rock was responsible for carving it out of Earth? It’s possible that multiple impacts made the moon, said Raluca Rufu, a planetary scientist at the Weizmann Institute of Science in Rehovot, Israel.

In a paper published last winter, she argued that Earth’s moon is not the original moon. It is instead a compendium of creation by a thousand cuts — or at the very least, a dozen, according to her simulations. Projectiles coming in from multiple angles and at multiple speeds would hit Earth and form disks, which coalesce into “moonlets,” essentially crumbs that are smaller than Earth’s current moon. Interactions between moonlets of different ages cause them to merge, eventually forming the moon we know today.

Planetary scientists were receptive when her paper was published last year; Robin Canup, a lunar scientist at the Southwest Research Institute and a dean of moon-formation theories, said it was worth considering. More testing remains, however. Rufu is not sure whether the moonlets would have been locked in their orbital positions, similar to how Earth’s moon constantly faces the same direction; if so, she is not sure how they could have merged. “That’s what we are trying to figure out next,” Rufu said.

Meanwhile, others have turned to another explanation for the similarity of Earth and the moon, one that might have a very simple answer. From synestias to moonlets, new physical models — and new physics — may be moot. It’s possible that the moon looks just like Earth because Theia did, too.

All the Same Stuff

The moon is not the only Earth-like thing in the solar system. Rocks like troctolite 76536 share an oxygen isotope ratio with Earth rocks as well as a group of asteroids called enstatite chondrites. These asteroids’ oxygen isotope composition is very similar to Earth’s, said Myriam Telus, a cosmochemist who studies meteorites at the Carnegie Institution in Washington, D.C. “One of the arguments is that they formed in hotter regions of the disk, which would be closer to the sun,” she said. They probably formed near where Earth did.

Some of these rocks came together to form Earth; others would have combined to form Theia. The enstatite chondrites are the detritus, remnant rocks that never combined and grew large enough to form mantles, cores and fully fledged planets.

In January, Nicolas Dauphas, a geophysicist at the University of Chicago, argued that a majority of the rocks that became Earth were enstatite-type meteorites. He argued that anything formed in the same region would be made from them, too. Planet-building was taking place using the same premixed materials that we now find in both the moon and Earth; they look the same because they are the same. “The giant impactor that formed the moon probably had an isotopic composition similar to that of the Earth,” Dauphas wrote.

David Stevenson, a planetary scientist at the California Institute of Technology who has studied lunar origins since the Theia hypothesis was first presented in 1974, said he considers this paper the most important contribution to the debate in the past year, saying it addresses an issue geochemists have grappled with for decades.

“He has put together a story which is quantitative; it’s a clever story, about how to look at the various elements that go into the Earth,” Stevenson said. “From that, he can back out a story of the particular sequence of Earth’s formation, and in that sequence, the enstatite chondrites play an important role.”

Not everyone is convinced, however. There are still questions about the isotopic ratio of elements like tungsten, Stewart points out. Tungsten-182 is a daughter of hafnium-182, so the ratio of tungsten to hafnium acts as a clock, setting the age of a particular rock. If one rock has more tungsten-182 than another, you can safely say the tungsten-filled rock formed earlier. But the most precise measurements available show that Earth’s and moon’s tungsten-halfnium ratios are the same. “It would take special coincidences for the two bodies to end up with matching compositions,” Dauphas concedes.

Chrondrite asteroid

Jean Lachat/ University of Chicago (Dauphas portrait); Nicolas Dauphas (enstatite chondrite)

Clues on Other Worlds

Understanding the moon — our constant companion, our silvery sister, target of dreamers and explorers since time immemorial — is a worthy cause on its own. But its origin story, and the story of rocks like troctolite 76536, may be just one chapter in a much bigger epic.

“I see it as a window into a more general question: What happened when terrestrial planets formed?” Stevenson said. “Everybody is coming up short at present.”

Understanding synestias might help answer that; Lock and Stewart argue that synestias would have formed apace in the early solar system as protoplanets whacked into each other and melted. Many rocky bodies might have started out as puffy vapor halos, so figuring out how synestias evolve could help scientists figure out how the moon and other terrestrial worlds evolved.

More samples from the moon and Earth would help, too, especially from each mantle, because geochemists would have more data to sift through. They would be able to tell whether oxygen stored deep within Earth is the same throughout, or if three common oxygen isotopes preferentially hang out in different areas.

“When we say that Earth and the moon are very close to being identical in the three oxygen isotopes, we are making an assumption that we actually know what the Earth is, and we actually know what the moon is,” Stevenson points out.

New tweaks to solar system origin theories, which are often based on complex computer simulations, are also illuminating where planets were born and where they migrated. Scientists increasingly suggest we can’t count on Mars to tell this story, because it may have formed in a different area of the solar system than Earth, the enstatites and Theia. Stevenson said Mars should no longer be used as a barometer for rocky planets.

Ultimately, lunar scientists agree that the best answers may be found on Venus, the planet most like Earth. It may have had a moon in its youth, and lost it; it may be very similar to Earth, or not. “If we can get a lump of rock from Venus, we can answer this question [of the moon’s origins] very simply. But sadly, that is not on anyone’s priority list right now,” Lock said.

Absent samples from Venus, and without laboratories that can test the unfathomable pressures and temperatures at the heart of giant impacts, lunar scientists will have to keep devising new models — and revising the moon’s origin story.
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Fair use. This website is educational. Materials within it are being used in accord with the Fair Use doctrine, as defined by United States law.

§107. Limitations on Exclusive Rights: Fair Use

Notwithstanding the provisions of section 106, the fair use of a copyrighted work, including such use by reproduction in copies or phone records or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use, the factors to be considered shall include:

the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes; the nature of the copyrighted work;
the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and the effect of the use upon the potential market for or value of the copyrighted work. (added pub. l 94-553, Title I, 101, Oct 19, 1976, 90 Stat 2546)

 

 

 

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.

c

20170722-105907-h6wr6

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:

c

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.

I.

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:

θ=1.220.4μmD=2m24kmθ=1.220.4μmD=2m24km
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)

 

 

Crystals in metals

States of matter: Why do metals have the properties that they have?
(Section under construction)

types of Metals

Solid / Liquid / Gas

Metal is a type of solid

Metal is usually an imperfect crystal

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At any temperature above absolute zero, atoms vibrate, so even in solids the atoms are always somewhat in motion

Iron atoms, like many other metals, take on this shape

Body-Centered Cubic (BCC) Structure: there are 8 atoms at the 8 corners, and one atom in the centre of the unit cell.  This structure is then repeated over and over.

BCC Body centered cubic crystal Iron

“The structure of iron atoms isn’t continuous throughout the entire paper clip. When a metal cools and is transitioning from liquid to solid, its atoms come together to form tiny grains, or crystals. Even though the crystalline structure does not continue from crystal to crystal, the crystals are bound to one another. In this diagram, each square represents an individual atom.”

Crystals form grains PBS

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atoms held together with metallic bonds

(add pics here)

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Defects break the bonds

“When a metal crystal forms, the atoms try to assemble themselves into a regular pattern. But sometimes there isn’t an atom available to fill in a space, and sometimes a growing layer is halted by other growing layers. There are many imperfections within each crystal, and these flaws produce weak points in the bonds between atoms. It is at these points, called slip planes, that layers of atoms are prone to move relative to adjacent layers if an outside force is applied. Adding other elements to a metal can counteract the effects of the imperfections and make the metal harder and stronger. Carbon, for example, is added to iron to make steel, and tin is added to copper to make bronze.”

Atoms can slip into a new position

metal atoms move PBS

Slipping

metal atoms slip PBS

Metal atoms can bend

metal atoms bend PBS

Heat can loosen the fixed positions of metal atoms

metal atoms heated PBS

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PBS NOVA: Building on Ground Zero – The Structure of Metals

PBS NOVA: Interactive Structure of Metals

PBS NOVA: Engineering Ground Zero

Learning Standards

Massachusetts Science and Technology/Engineering Curriculum Framework

High School Chemistry
HS-PS2-6. Communicate scientific and technical information about the molecular-level structures of polymers, ionic compounds, acids and bases, and metals to justify why these are useful in the functioning of designed materials.*

PS1.A Structure of matter. That matter is composed of atoms and molecules can be used to explain the properties of substances, diversity of materials, how mixtures will interact,
states of matter, phase changes, and conservation of matter. States of matter can be modeled in terms of spatial arrangement, movement, and strength of interactions between particles.

PS2.B Types of interactions.  Electrical forces between electrons and the nucleus of
atoms explain chemical patterns. Intermolecular forces determine atomic composition, molecular geometry and polarity, and, therefore, structure and properties of substances.

 

MCAS Open Response questions

Content Objectives: SWBAT construct answers to open-response questions on the physics MCAS.

2015, High School Intro Physics: sample open response question

2011 sample open response questions

2012 sample open response questions

2013 sample open response questions

2014 sample open response questions

2015 sample open response questions

2016 sample open response questions

  • Learning Standards:
  • For answering open-response questions – ELA Core Curriculum
  • CCRA.R.1 – Read closely to determine what the text says explicitly and to make logical inferences from it; cite
  • specific textual evidence when writing or speaking to support conclusions drawn from the text.
  • For answering problems involving equations: Massachusetts Curriculum Framework for Mathematics
  • Functions: Connections to Expressions, Equations, Modeling, and Coordinates.
  • Determining an output value for a particular input involves evaluating an expression; finding inputs that yield a
  • given output involves solving an equation.