PHL 329U Dr. Leon
MWF
3-4pm
Philip Kromer & Jason Avent
Size of the Moon: History and Mystery
The moon illusion is an old and compelling mystery that both highlights philosophical issues of perception, measurement, and belief, and motivates interesting astronomical questions with connections to basic geometry and algebra.
To engage the students we would like to illustrate the moon illusion first-hand. In the absence of an overnight field trip, it is difficult to directly demonstrate the moon illusion to the class. Instead, we would ask students to observe the moon on their own over several nights and at different times.
For the in-class engagement, we present several folk descriptions of the moon – images and illustrations showing the horizon moon as we perceive it:
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We start by showing students the first two pictures – a painting by Daumier, and the same painting modified to show a moon with a realistic angular extent. Most people feel that the first painting (with a very large moon) better matches their perception of the size of the moon. In the remaining illustrations it is obvious that an exaggerated moon is depicted, but it makes the point that people find the horizon moon to be much larger than the zenith moon.
Next, we show real images of the moon:
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In these images the angular extent of the moon is much less than expected. This disconnect between what we perceive, what we see, what a camera sees, and what we measure engages and motivates the students.
To further motivate the issue we briefly mention the history of the moon illusion: that it is quite old – going back to the birth of astronomy – and that scientists even today disagree on the cause of the illusion. There are, however, some very strong and reasonable arguments that serve to explain the illusion, and the students will be asked to judge how well these fit the data.
Now we ask students to think of reasons why the moon looks bigger at the horizon. As students respond, we write their suggestions on the board. Reasons may include:
Because it IS bigger
Atmospheric Refraction
Because You’re Closer
Because nearby objects make it appear bigger
Optical Illusion
The perceived Dome of the Sky makes us think it’s farther at the horizon
Psycho-Optical Effects
The atmospheric refraction hypothesis should be briefly discussed (the air is denser and warmer near the earth, which changes its optical properties).
Once a reasonable list has been generated, we ask whether anyone can find reasonable objections to any of the hypotheses. Students should be able to argue that the first reason is not true for a variety of reasons. (It implies the moon is changing size or distance, which would have measurable astronomical implications. Also, our horizon moon is Hawaii’s zenith moon; and our zenith moon is France’s horizon moon.)
Next we show the students the following photographs:
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Now we return to the list of hypotheses for the moon illusion, and ask our students whether they want to rule out any of the hypotheses. Since the moon is not actually changing size, only the last four hypotheses – the ones concerning our perception – still stand. As teachers we need to hammer home that the illusion is purely perceptual: the actual angular size of the moon is roughly constant; only our perception of that size changes.
Next we explore two of the remaining hypotheses: the optical illusion and sky dome arguments.
We show students two relevant optical illusions, but first make the following points:
We are going to show some pictures and ask questions about those pictures. You may have seen these before
However, don’t give the answer that you might think we want, or an answer you may know from seeing an explanation – just answer honestly based on your perception.
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With each illusion, we ask students to indicate which object looks bigger, and take a vote. Then we have a student come and measure the object (more on this below). We also discuss the various ways in which the given optical illusion could induce the moon illusion (perspective vanishing lines are like train tracks in Ponzo Illusion; nearby objects are like circles in Ebbinghaus illusion).
Next we discuss the effect of the perceived sky dome. The “Vault of the Heavens” seems to be a flattened parabola, rather than the circular orbit the moon actually travels. Obviously, a cloud far on the horizon is further away than one overhead. This mechanism in the brain is good at helping estimate the relative sizes of clouds, but for the moon there is a very small difference in distance. Ironically, the moon is actually closest to the observer when at its zenith, even though it appears largest when on the horizon.
First we have students discuss the shape of the sky dome. Is a cloud on the horizon or a cloud on the zenith closer? Can we calculate how much closer or farther it is? Is there any other reason (lack of distance cues) why we might perceive a shallow sky dome? Then we present the perceived sky dome:
We then ask what effect that has on the perceived size of the moon. It’s interesting to note that the sky dome model can be used to argue for a smaller or larger horizon moon (If our brain thinks the moon is farther, does that affect our perception of the angular or linear size? How do those interplay? How much processing is going on? These are questions without well-understood answers.)
This leads us into to an examination of philosophical issues concerning the moon illusion. We lead students in a discussion of questions such as
Why do we call it an Illusion?
Why is a photograph convincing?
Why is a measurement with a ruler convincing?
Is Human Perception ever superior to instrumentation?
Is Perception valid in Scientific Inquiry?
How do we deal with Inconsistent Experiences?
Students should confront issues of perception, measurement, and belief in this discussion.
There are several ways in which this lesson could develop from here. We describe one that connects to mathematics, and another that connects to observational astronomy.
Determining the angular size of the moon requires students to draw on knowledge from Astronomy, Physics, and Mathematics. We would ask students to use astronomical data to find the “expected” angular size of the moon, and explore how that size would change under various modifying conditions. Based on the level of the class, we could provide data for the linear size and distance of the moon, or ask them to draw on previously determined values from their work with Kepler’s and Newton’s law.
In either case, students calculate the angular extent of the moon, and estimate how many “moons” would fit in a circle going from horizon to horizon (that is, what portion of 180º the moon subtends). They can compare that value with the photographs from before and with values from literature.
In the moon illusion, researchers find that the moon appears about 2.5 times larger near the horizon. What does that say about our perception of the distance and size of the moon? [It will seem to either be closer, or bigger.]
If the moon actually were 2.5 times bigger in angular size, but stayed the same diameter, what would its distance need to be? Would it be closer or farther? [Students calculate to find the answer] If it moved that much close, what would change about our solar system and our life on earth? [The moon’s period would increase/a month would be shorter – by how much? Tides would be much stronger and more frequent.]
Now instead what if the moon actually were 2.5 times bigger in angular size, but stayed the same distance, what would its new diameter need to be? Would it be bigger or smaller? [Students calculate to find the answer] If it grew that much, what would change about our solar system and our life on earth? [The moon’s mass would increase by the change in diameter cubed – mention scaling! This would also change tides. Would it change moon’s orbit? NO! (Kepler/Newton eqns: no mass)]
(The last two exercises are themselves an Explore/Explain cycle: students calculate to find the change in size or distance, then consider ways in which this affects the universe. They then explain those effects, and perhaps explore again ways to estimate their magnitude.)
Finally, students should briefly compare their confidence in their result for angular size, as compared to one found through direct (say, photographic) measurement.
Another exploration would have students measure and experience the moon illusion directly. The class could brainstorm for methods to do so, or the teacher could suggest the following procedure:
Students make a tube out of a sheet of notebook paper during these observations of the moon. First, it will be large enough to block out the horizon and they will determine if the tube helps diminish the illusion, or if the tube becomes a long tunnel to the horizon. They will then constrict the aperture of the tube and when it matches the size of the moon, they will apply tape. Because all of the tubes are about 11 inches long, all of the students should have roughly equal apertures at the end of their tubes.
The students are asked to go outside to look at the full moon as it rises at sunset. They note the size of the moon. These students will repeat the measurement at another time when the moon is near its zenith, allowing the students to determine whether the moon’s size is indeed the same. This should include making a new “tube” measurement and using the earlier.
The students are also asked to turn away from the moon, bend over, and look at the moon through their legs. Some people find that the moon illusion disappears as their perspective and perception of the horizon is flipped. Because the horizon is always on the bottom of the visual field, the brain cannot find it in an upside-down world, and hence cannot make assumptions about the distance to the moon.
Students will compare their results the day of the lesson to discover the value of tools in taking measurements. Students will also discuss questions including
Is the “tube” measurement a convincing demonstration of the moon illusion?
How do these observations relate to the various hypotheses discussed earlier? Should we modify, discard, or decide in favor of any of them?
Why do different observers’ measurements differ? Is there more than just instrumental error at play?
As evaluation, we would have students repeat the classroom exercises for various other satellites and planets. We could ask questions such as
Is there also a Sun illusion, or does it just happen to the moon? (There is a sun illusion, and it works in just the same way. We aren’t as aware of it because we can’t look directly at the sun except at sunset or sunrise).
What is the angular size of the sun? How does it compare to the size of the moon? (Coincidentally, they are quite close and this explains the perfect fit of a total solar eclipse).
What is the angular size of a star? Of a planet? Are they similar?
Given the distances to each planet from the earth, rank the planets in order of angular size. Can you explain why they were discovered in the order they were? (Brightness matters for this question).
In class, students explored the relation between the angular measure, distance and size of the moon. One extension question asks how the angular measure changes as a function of the distance. The moon is actually receding from the earth at a rate of 3.8 cm/yr. Using their calculator, students can plot the percent change in distance and angular extent after 1,000, 2,000, …, 5,000, 10,000 20,000, etc. years. (What was happening on Earth at each of those times? Middle Ages, Romans, Ice Age, etc.). They will find that the angular extent grows as the square root of the change in distance. This can be connected to the derivative in an advanced class, or treated as an application of interpolation in an intermediate class. Students should also estimate how long it would take the moon to recede a “noticeable” amount. (Even a 5% change in angle would take an astronomically long time).
We had in mind an upper-level Physics or Astronomy class while preparing this lesson, but the lesson plan we have outlined above addresses TEKs from several different courses.
(c) Knowledge and skills.
(c2) Scientific processes. The student uses scientific methods during field and laboratory investigations. The student is expected to:
(B) Collect data and make measurements with precision;
(Students use a homemade paper tube from notebook paper to measure the moon. These apertures are measured and averaged to give the class average of the best size of hole at the end of the tube.)
(c3) Scientific processes. The student uses critical thinking and scientific problem solving skills to make informed decisions. The student is expected to:
(A) Analyze, review, and critique scientific explanations, including hypotheses and theories, as to their strengths and weaknesses using scientific evidence and information;
(Many theories- some lead down an incorrect road and students must choose.)
(C) Evaluate the impact of research on scientific thought, society, and the environment;
(Senses are not to be trusted without instruments- the moon illusion is one of the few experiences actually inconsistent with the reality and requires more than a cursory explanation.)
(E) Research and describe the history of astronomy and contributions of scientists.
(We noted the historical figures such as Aristotle and Ptolemy who offered explanations false ones; as to the cause of the moon illusion. i.e. the moon appears larger due to distortion by the atmosphere- it would actually appear smaller if this effect was significant.)
(b) Introduction.
(b1) In Integrated Physics and Chemistry, students conduct field and laboratory investigations, use scientific methods during investigations, and make informed decisions using critical-thinking and scientific problem-solving.
(Observational explorations; defending, assessing hypotheses)
(b2) Science is a way of learning about the natural world. Students should know how science has built a vast body of changing and increasing knowledge described by physical, mathematical, and conceptual models, and also should know that science may not answer all questions.
(The questions here are in many ways still open, and the applicability of various models and hypotheses is a matter that the student debate and defend)
(c) Knowledge and skills.
(c2) Scientific processes. The student uses scientific methods during field and laboratory investigations. The student is expected to:
(A) plan and implement investigative procedures including asking questions, formulating testable hypotheses, and selecting equipment and technology;
(B) collect data and make measurements with precision;
(C) organize, analyze, evaluate, make inferences, and predict trends from data; and
(D) communicate valid conclusions.
(c3) Scientific processes. The student uses critical thinking and scientific problem solving to make informed decisions. The student is expected to:
(A) analyze, review, and critique scientific explanations, including hypotheses and theories, as to their strengths and weaknesses using scientific evidence and information;
(b) Introduction.
(c1) In Physics, students conduct field and laboratory investigations, use scientific methods during investigations, and make informed decisions using critical thinking and scientific problem solving. Students study a variety of topics that include: laws of motion; changes within physical systems and conservation of energy and momentum; force; thermodynamics; characteristics and behavior of waves; and quantum physics. This course provides students with a conceptual framework, factual knowledge, and analytical and scientific skills.
(c2) Science is a way of learning about the natural world. Students should know how science has built a vast body of changing and increasing knowledge described by physical, mathematical, and conceptual models, and also should know that science may not answer all questions.
(3) A system is a collection of cycles, structures, and processes that interact. Students should understand a whole in terms of its components and how these components relate to each other and to the whole. All systems have basic properties that can be described in terms of space, time, energy, and matter. Change and constancy occur in systems and can be observed and measured as patterns. These patterns help to predict what will happen next and can change over time.
(4) Investigations are used to learn about the natural world. Students should understand that certain types of questions can be answered by investigations, and that methods, models, and conclusions built from these investigations change as new observations are made. Models of objects and events are tools for understanding the natural world and can show how systems work. They have limitations and based on new discoveries are constantly being modified to more closely reflect the natural world.
(c) Knowledge and skills.
(3) Scientific processes. The student uses critical thinking and scientific problem solving to make informed decisions. The student is expected to:
(A) analyze, review, and critique scientific explanations, including hypotheses and theories, as to their strengths and weaknesses using scientific evidence and information;
(B) express laws symbolically and employ mathematical procedures including vector addition and right-triangle geometry to solve physical problems;
(6) Science concepts. The student knows forces in nature. The student is expected to:
(A) identify the influence of mass and distance on gravitational forces;
(a) Basic understandings.
(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; shapes and figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them.
(3) Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures.
(4) The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems.
(b) Geometric structure: knowledge and skills and performance descriptions.
(2) The student analyzes geometric relationships in order to make and verify conjectures.
(e) Congruence and the geometry of size: knowledge and skills and performance descriptions.
(1) The student extends measurement concepts to find area, perimeter, and volume in problem situations.
(B) The student finds areas of sectors and arc lengths of circles using proportional reasoning.
We consulted these books while preparing this lesson:
Ross HE and Plug C. The mystery of the moon illusion: Exploring Size Perception. Oxford University Press, 2002. LOC: QP 495 R67 2002; ISBN: 0 19 850862 X.
Minnaert, M. Light & Colour in the Outdoors. Springer, 1974. LOC: QC 975 M552 1954.
Hershenson, M, ed. The Moon illusion. L. Erlbaum Associates, 1989. LOC: QP 495 M66 1989; ISBN: 0805801219.
These webpages further explain the moon illusion:
Waeber, M. "Moon
Illusion,"
Archimedes Lab
A simple, clear
presentation.
http://www.archimedes-lab.org/moon_illusion/moon.html
Borghuis, Bart.
"The
Moon Illusion,"
Bart's World
An overview of the rival
hypotheses.
http://retina.anatomy.upenn.edu/~bart/scriptie.html
Simanek, Donald.
"The
Moon Illusion, An Unsolved Mystery"
Another
overview of the rival
hypotheses.
http://www.lhup.edu/~dsimanek/3d/moonillu.htm
MacCready, Don.
"The
Moon Illusion Explained."
The most in-depth, though I think he overstates the superiority
of his theory over the others.
http://facstaff.uww.edu/mccreadd
Adler Planetarium.
"Facts
about the moon,"
Learning Astronomy.
Reference figures for the size, mass,
distance, etc. of the moon.
http://www.adlerplanetarium.org/learn/moon/facts.ssi
Finally,
here are the sources for the images in our presentation:
“Daumier with
realistic and artistic Moons:”
http://www.michaelbach.de/ot/sze_moon/
Movie
posters:
Underworld http://horror.about.com/library/blunderimgb.htm
VoyageToTheMoon
http://www.cinemacom.com/vintage-shockers.html
Honeymooners
http://www.tvcrazy.net/tvclassics/americantv/honey.htm
E.T http://www.idave.com/ETbikemoon.jpg
Blue Moon
http://www.philsp.com/data/images/blue_moon.jpg
PontiacMoon
http://www.sd455.com/moviepontiacmoon.jpg
“Small Moon
over
Brasilia”
http://www.trekearth.com/gallery/South_America/Brazil/photo57006.htm
“Moon Over White Sands”
“Moon At Horizon”
“Timelapsed Moon
over Seattle”
http://antwrp.gsfc.nasa.gov/apod/ap031011.html
“Timelapsed Moon
over Seattle:” Photographer’s
website
http://www.shaystephens.com/galleryFineart.asp
“Moon at Horizon
and Zenith:”
http://www.stargazing.net/david/moon/moonrisesize.html
