I'm president of the University of Chicago's astronomy club, the Ryerson Astronomical Society (RAS for short). You can visit our website here. Carl Sagan used to be a member! Below are several emails I've written to our club email listhost with some information about moon maps and positional astronomy that I thought was worth preserving online. They're unaltered, so please excuse me if anything seems poorly written or out of context.

Third, here's something interesting we could do next year: make rectified moon maps. A little background: in 1960 Gerard Kuiper edited a collection of moon photographs called the Photographic Lunar Atlas. The point of this atlas was to provide good quality coverage of the Moon's near side at several sun angles. The photographs come from five different observatories, including Yerkes and Lick. (Unfortunately, the Atlas is now really rare: not very because it's a collection of photographs stored unbound in a big box. [The box is maroon-coincidence, since Kuiper had it published by the University of Chicago press??] Actually, the Air Force version of the Photographic Lunar Atlas is bound, but that's rare for other reasons.)

The PLA was followed by three supplements: the Orthographic Atlas of the Moon, the moon with a grid system imposed; the Rectified Lunar Atlas; and the Consolidated Lunar Atlas, a collection of the best pictures of the Moon. You can find the Consolidated Lunar Atlas online and it's well worth looking at. Anyway, let me talk about the Rectified Lunar Atlas. We view the Moon from Earth essentially in orthographic projection. As a consequence, the features of the Moon (particularly near its limb) are distorted. (How can we quantitatively understand this distortion? Have you heard of Tissot indicatrices? If you haven't, you've probably seen them somewhere. http://en.wikipedia.org/wiki/Tissot%27s_Indicatrix Anyway, as I just realized, the moon is covered with Tissot indicatrices!! Impact craters are almost always circular unless the impactor came in at a very shallow angle/was really, really big and came in at a somewhat less shallow angle, and they're spotted across the moon's surface, so they make for built-in Tissot indicatrices. And they work just fine... I calculated roughly with a "flux through square" idea how much a crater at the latitude of Plato should be distorted (ignoring longitude), shrunk to an ellipse with a minor/major axis ratio of 0.6, and that basically agrees with measurements I made from CLA images.) So we can see the distortions, and they really ruin our view of the moon.

But we can correct that by making rectified moon maps, projecting the orthographic images we have of the moon back onto a sphere. (If this is confusing, notice that we almost always do the opposite: like, we have to take aerial photos and orthorectify them to make orthophotos that are useful as maps.) For the Rectified Lunar Atlas, this projection was quite literal: Ewen Whittaker projected photos of the moon on a large white sphere and photographed the sphere to un-distort the moon and see it as it really was for the very first time! There's a technical paper giving more details which I unfortunately can't find at the moment--ask me later if you're interested. William Hartmann was the first to see the multi-ring structure of the Orientale Basin in these rectified images and his insight into the importance of huge impacts on the moon revolutionized lunar science.

I think it would be cool to make some rectified moon maps. We have this excellent moon mosaic thanks to Dean-http://dwarmstr.blogspot.com/2009/02/another-moon-mosaic.html-and software can reproject maps so we don't have to make the whole white sphere apparatus. (The white sphere apparatus may still be at Yerkes, which I should ask about.) So how about we wait for a favorable libration, take pictures of the moon, and rectify them to see clearly the structure of the Orientale Basin? I don't know how to do the rectification in software yet: I tried "draping over globe" in ArcGlobe but I couldn't get it to work. It should be possible to do it in Adobe Illustrator, but I'd rather use ArcInfo... is anyone an Arc guru?

We need to raise sundial awareness on campus. After last Monday meeting, Dean proposed we chalk a sundial at the very center of the quads. It's supposed to be warm next week-we could do it early in the morning Wednesday or Thursday, weather permitting. Tell me this Monday if you want to help out. Here are the technical details.

An analemmatic sundial is the clear choice. You can read about the analemmatic sundial here-http://pass.maths.org.uk/issue11/features/sundials/index.html-or here: http://dls-website.com/documents/Analemmatic_Sundials.htm

Why an analemmatic sundial? We can't set up a gnomon at the center of the quads. But the analemmatic sundial is an azimuth dial, so we can use a vertical style: the shadow cast by a person. There is a dial like this right outside the visitor center at Fermilab. We also want a sundial automatically corrected for equation of time. It's possible to correct an equatorial dial just by rotating it (think about it) but it's not possible to correct a more complicated, "derived" dial this simply. Fortunately since this sundial is chalk, it won't survive very long, so we only have to worry about correcting for the equation of time once.

So construction steps: 1) Trammel an ellipse with major axis 5 feet (I haven't actually measured), minor axis 5 feet times sin(latitude). 2) Plot hour points on the ellipse; x = 5 feet times sin(hour angle + equation of time), y = 5 feet times sin(latitude)cos(hour angle + equation of time), and the origin is the center of the ellipse 3) Plot sunrise and sunset and erase the ellipse past those points 4) Draw "stand here" footprints at 5 feet times tan(declination of sun)cos(latitude) from the center, towards the south (I think).

(Yes, we did do this, and it worked out very well! Future ideas: projecting map on analemmatic sundial. Keep an eye out for my Sundial Science Simplified page, to come.)

WEEKS 2 AND 3 -- SUNDIALS (Tote and Aaron)

The way I saw it is that Tote's talk could be about telling time with sundials--like an exploration of the motions of the sun and different types of time, then an exploration of the classical dials (where the gnomon points towards the pole, so that the angle the sun sweeps out as it moves along the gnomon is simply the hour angle. If the plate's in the plane of the equator, then you're done. What if it's horizontal? Vertical? At some other inclination (slope)? At some declination (turning-away from N/S direction)? What would a horizontal dial at the equator look like?) You could talk about body dials, like using your hand or length of shadow as a sundial. You could debunk the boy scout hour-hand-toward-sun trick. You can discuss Babylonian and Italic hours, unequal hours too. Did you ever read The Name of the Rose? Do you know about the liturgy of the hours? When's Nones, when's Sext? You could even finish up with the equation of time. Is Sext at noon? (The difference between apparent and mean time).

Then Aaron's talk could be about the date and sundials. You could talk about how curves traced out by the tip of a shadow are related to conics. As the sun changes declination, what do the traces on an equatorial dial look like? Why are the traces at our latitude hyperbolae? How can this be used to determine the date? From the date you can look at lines on a dial to see how many hours of sunlight there will be in a day. http://www.quns.cam.ac.uk/Queens/Images/Sundial2.jpg Since you know date, you also know the correction of equation of time and can combine this information to make a sundial that reads mean time. The "analemma"-http://scienceblogs.com/startswithabang/upload/2009/08/why_our_analemma_looks_like_a/analemma-1.gif Finally, you could finish off with the extremely cool analemmatic dial, based on the orthographic projection. There's one of these outside the visitor center at Fermilab. If you do talk about this, please explain how to use it properly-the gnomon (you at the Fermilab dial) only moves up and down along the minor axis of the ellipse. I've always done this wrong!

The most thorough sundial book I've read is Rene Rohr's "Sundials: History, Theory, and Practice." I can't say it's perfect, but it does cover your two talks very well, particularly chapters 3-6. There's good information on types of dials online. Look at the NASS (North American Sundial Society) website.

WEEK 4 -- MOONDIALS AND NOCTURNALS (Tad)

On the night of the full moon, an ordinary sundial should tell time properly at night (the moon's directly opposite the sun, you just add 12 hours). On the night of the new moon, no moondial can tell time properly at night, since the moon is not up. In between things get complicated. Actually, even on these dates things are complicated, since the moon's motion in its orbit is continuous. I'm interested in hearing about the motion of the moon and the "equation of moon" you have to add to make a functioning moondial (only during the portion of the moon's phases when it's bright enough to cast a shadow). Rohr has a very short chapter on moondials. Sorry I don't know any more.

If you know the date, you know the Sun's position in the ecliptic. If you know the sidereal time, you know where the ecliptic is in the sky. So if you know both, you know where the sun is in the sky and the local apparent time. In the northern latitudes we're lucky enough to have a clock in the sky that reads sidereal time, with the pole at the center of the face and, for example, the pointer stars in the Big Dipper as the hour hand. The nocturnal is an instrument that uses that hour hand and the date to tell the time. http://brunelleschi.imss.fi.it/museum/esim.asp?c=500044

WEEK 5 -- ASTROLABES (Lisa)

A good way to figure out answers to tough geometry problems is to draw a picture. The ancients wanted to solve tough geometry problems in the sky--given latitude and the altitude and coordinates of a celestial object, they wanted to find the local time. But the sky has to be treated as a sphere, and they didn't have a fully developed theory of spherical trigonometry. Even after they developed the theory, the calculations proved prohibitive. So they drew a picture. But how do you draw a picture of a sphere? You can draw a picture *on* a sphere (the idea behind the armillary sphere) or you can draw a picture on a plane that preserves some essential properties of the sphere (hence the "planispheric" astrolabe).

The stereographic projection wasn't inevitable. The Greeks probably stumbled upon conics while trying to solve the problem of doubling the cube, and the development of their theory led naturally to the stereographic projection (since the vertex of a single-napped cone can be considered as the origin of the projection, and slicing planes to be planes of projection. An orthographic projection would work too, and actually a later instrument uses an orthographic projection onto the solstitial colure.

Anyway, the stereographic projection was used, with the south celestial pole as the aspect and the celestial equator as the plane of projection. Usually the astrolabe cuts off at the Tropic of Capricorn so as to avoid being infinitely large. In your lecture, you can talk about angle and circle preservation properties of the stereographic projection. I drew a picture for my last astrolabe talk that's up in the office if you want to look at that.

Solving problems with the astrolabe: there's an Arabic text that gives 1000 uses for the astrolabe. This seems a little excessive. The prototypical problem is going from altitude of a star and date to time (or date and time to altitude, etc.) Suppose you sight down the alidade and see some star of mid-northern declination at 20 degrees altitude in the E. (The qualifications ensure that this star is not near the meridian, i.e. its motion is mostly vertical. You can see it's good to measure altitude when it's changing a lot-it's like why you don't take the cosine of small angles.) Then you move the rete to position this star on the 20 degree almucantar in the E. OK, then you know the hour angle of a star and the sidereal time (the pointer off the first point of Aries). Let's say you know the date. Then you know the position of the sun on the ecliptic and hence the hour angle of the sun, or local apparent time.

References: http://www.astrolabes.org/personal.htm James Morrison sells inexpensive paper astrolabes. He also wrote a book called "The Astrolabe" which is really good--unfortunately our library doesn't have a copy. I got a copy through interlibrary loan but have to return it soon. "The Planispheric Astrolabe for the Renaissance Faire Performer"-http://members.cox.net/hapnueby/astrolabe/contents.html

WEEK 6 -- ARMILLARY SPHERES AND CELESTIAL GLOBES (Annie)

"Armilla" means "ring" in Latin. Armillary spheres are skeletal celestial spheres with rings for the equator, tropics, ecliptic, horizon, etc. Some of the rings are moveable and it's a simple way to find approximate answers to positional astronomy problems. Celestial globes are globes with star positions marked (from "outside" the celestial sphere) occasionally with rings to find approximate answers to rising/setting/culminating time problems.

What can you talk about? The armillary sphere is connected to measurements of the heavens. Ptolemy determined the moment of the equinox and measured positions of heavenly bodies with an armillary sphere. http://www.hps.cam.ac.uk/starry/armillobser.html It might also be interesting to reconsider the celestial sphere in the context of this instrument--like give an intro-to-positional-astronomy talk with a historical flavor. How did declination and right ascension get their names? An armillary sphere can also be used as a sundial (sort of). For the celestial globe, it might be interesting to consider the history of the constellations, the zodiac, stuff like that. There are a bunch of good star lore books. There's an armillary sphere on campus--do you know where?

WEEK 7 -- QUADRANTS AND SEXTANTS (Jake)

The equatorial system of celestial coordinates is based on terrestrial longitude and latitude. There's a good reason for this--the main motion of the skies is caused by the rotation of the earth, and making the axis of the world the axis of the heavens means that only one celestial coordinate will change with time. It's also suggestive of the fundamental idea of celestial navigation: that each celestial position has a geographical position "below" it, a ground point (a geographical position with that celestial position at the zenith). If you see the celestial north pole at your zenith, you're at the north pole.

Suppose you find the altitude of the sun at noon and you know the sun's declination. Then you know your latitude. (Why?) Suppose you have an accurate chronometer set to GMT and you compare it with the time of noon at your position. Then you know your longitude. (Why?) This is useful, and, better yet, it's simple. But we want to find our position from the sight of any heavenly body at any time, not just the sun at noon). We go back to the geographical position idea: we find the altitude of a heavenly body at some time. Then, using trigonometry, we're able to find how far we are from the ground point (it's like a flagpole: if the top of the flagpole is right above you, you know you're right under the flagpole. If it's 45 degrees down from the zenith, or up from the horizon, you know you're as far away from the flagpole as it it is high. Actually, you're on a circle around the flagpole). So you're on a circle around this ground point, and you know the radius. You take a sight of another body, the circles intersect in two places, from dead reckoning you know your position well enough to choose one for certain and you get a "fix" on your position. Sight reduction tables are books of solved spherical trigonometry problems so you can find your distance from the ground position. There are some very clever simplifications here--look up the intercept method. I recommend the American Practical Navigator, which you can download for free online, for more information on celestial navigation and navigation in general.

Now, how do you actually take these altitude measurements? On land, you can get degree accuracy with fairly simple instruments. With enormous quadrants Tycho was able to get, I think, minute accuracy. But minutes mean miles! Accuracy really matters for getting a fix. It's the difference between making it safely to home port and running aground on the Lizard. There are two problems: 1) Measure down from zenith or up from horizon? You can't get a plumb on a ship, so the choice is made for you. You need to correct for the curvature of the earth, though, and the height of your eye. 2) Looking at two things at once. The cross-staff and the quadrant require you to look at the horizon and the object you're sighting at once. The sextant lets you bring the horizon and the object into coincidence http://en.wikipedia.org/wiki/File:Using_sextant_swing.gif

Finally, what objects do you take sights from? You need to see the horizon when you sight, so your objects must be visible at twilight (sextants have a little telescope, which helps) or be the sun/moon. The sun has lots of advantages-it's the main celestial object navigators sight (unless you count GPS satellites). I recommend the book Celestial Navigation by Frances Wright.

Will Vaughan. Last revision September 7, 2010.