(Under construction!  Diagrams to follow.)

I taught second-semester Trig at Good Counsel in 1999 to a small class of nine seniors, and we spent a little time on a project which applied some trigonometry to astronomy.  With the right class it could be done in the first semester, since the trig involved is nothing more difficult than solving right triangles (although if they’ve had logarithms they can also compute the absolute magnitude of the stars from their distance and visual magnitude).

I regret that I didn’t get photos!  But I’ll describe the project here and give some useful information, as well as insights gained by my experience.  If I do it again in the future, as I hope to, I’ll know to do some things differently.

The idea is to build a cubical frame, about 2 feet on each side, in which beads are hung that represent the sun (in the center) and the nearby stars in their actual locations in space.  Since we found a bag of beads of assorted sizes at an art shop, we were able to use the size of the bead to indicate the star’s absolute magnitude (brightness irrespective of distance), and we used fluorescent paint on most of them to indicate the star’s color.

The great thing was to find out where the star should hang in 3-D.  With advice from an astronomer from Cornell University, we went to the online Hipparcos catalog and searched it to produce a table with information on the stars within 10 parsecs.  (You’ll have to study your astronomy to do this project -- a parsec is about 3.26 light years, if I remember right.)  Since the catalog gives distances as parallaxes, the actual search was for stars whose parallax is greater than or equal to 100 milliseconds, sorted by parallax.  From the parallax you can compute the star’s distance in parsecs -- find the formula in any introductory astronomy text.

Also on the table was each star’s right ascension and declination, which are like its longitude and latitude, respectively, in the sky as seen from earth.  Knowing this and its distance gives us all the information we need in principle to find its position in three dimensional space, so the rest is merely trigonometry.

Our frame for the map used two square plexiglas sheets on top and bottom, on which we drew polar coordinates with an indelible marker.  Keeping with the standard usage in astronomy, the radii of the coordinate system were at intervals of 15 degrees and they were marked in “hours” -- 0h, 1h, 2h, 3h, ... up to 23h.  A star’s right ascension is given as 3h 14m 34s, for example, where an hour is 15 degrees, a minute is one sixtieth of an hour, and a second is one sixtieth of a minute.  (Don’t confuse these minutes and seconds with the usual DMS system!)

Right ascension is the easy coordinate; the star’s right ascension gets plotted directly onto the polar grid.  The other two dimensions needed are the distance from the center of the  polar coordinate planes on the plexiglas sheets, and the length of thread between the upper plexiglas sheet and the bead that represents the star.  (Actually we used monofilament nylon fishing line, which is clear and cheap, rather than thread.)  These dimensions are easily computed from the declination (the angle between the star and the celestial equator) and the distance from the star to the earth, as illustrated in the diagram below.

One big problem we encountered is that plexiglas is flexible.  Therefore, as we attached more and more stars between the two sheets, pulling each one tightly into place, the top and bottom sheets both bowed inward, becoming quite noticeably concave.  The worst part of this is that the stars hung earlier, which had originally been tight, became loose and flopped around.  A better design is needed to prevent this problem.

And yet plexiglas was an excellent material in another way.  For attaching the threads to the top and bottom plates, we simply melted a hole in the plexiglas with a woodburning pencil (like a small soldering iron, with a tip perfectly shaped for this operation).  The plexiglas melted readily and hardened again quickly, leaving a little crater that could be broken off with fingernails or a spatula.  Then the nylon line could be threaded through the hole and secured with a knot tied at exactly the right place (which was another challenge!).

Regarding attaching the beads to the nylon line: the best way is to enclose the bead in a loop of nylon.  But note that this means the nylon has to fit through the bead twice!  When you sort your beads into different sizes, test several of the smallest beads to make sure they will all take two lines through them; if they won’t, discard that size.  We were disappointed to find the smallest beads couldn’t be used after we had planned that stars with magnitude over a certain number would take that size.  (A larger magnitude means a dimmer star; consult an astronomy text to find out how to calculate absolute magnitude from visual magnitude.)

You also may want to color the beads, based on the star’s spectral class:  again, consult an astronomy text, but briefly class O = blue, B = blue, A = blue, F = blue to white, G = white to yellow, K = orange to red, and M = red.  Most stars are red.

My plan for doing this in the future would be as follows:

Prepare for the project by getting permission to have students use the Hipparcos catalog.  Instruct the students in enough observational astronomy for them to understand and do the project.  (If possible, work with the science department on this.)  Also, depending on how much you want to do for the students, you might want to find and acquire fluorescent paints in the colors needed and beads from an art-supply store.  The rest of the materials needed will be standard hardware.  If you think the students are pretty good at building things you could leave construction of the frame to them; otherwise you’d do well to have it put together in advance.  Note that plexiglas is emphatically not a construction material; it’s too flexible.  The frame will need to have at least two and preferably three sides of some solid material, such as plywood.  The finished product will be much more attractive if the inner surfaces of the solid sides are painted black or dark blue, to resemble the night sky.

Have the students access the Hipparcos catalog to get the information they will need for their own stars.  There are about 180 known stars within 10 parsecs of earth (they’re still being discovered!); you could go out farther depending on how much work you want each student to do and how big your class is, etc.  Divide up the stars among the students, and secretly assign each star to several students, not telling the students who else has each of their stars.

Once the students have the information for their stars downloaded and printed in a useful report, have each student do the calculations for his or her stars and present the information on each star in a table.  The table should show the star’s identifying number (probably just the line number 1-180 from your own “master” table); the star’s right ascension; the star’s distance from the origin of the polar coordinate system (2-dimensional) on the plexiglas; the star’s distance from the top plane of the 3-d map; the size of the bead they are to use (based on its absolute magnitude, and you will need to decide in advance on a table of cutoff-points for each available bead size), and the color to paint the star.  When everyone is done, have them check their work with the other people who did the same stars.

Construction:  To avoid the plexiglas bowing problem, I wonder whether the design might temporarily mount the lower plate an inch closer than its final position, then when all the stars are tied in, lower it the final inch and fix it in place so everything tightens up at once?  In any case, after the frame is constructed, students will need to:  1) choose the bead of the appropriate size; 2) cut a piece of nylon and fix the bead onto it at the right place (inexactly, leaving a few inches at each end to tie knots); 3) paint the bead and let it dry, keeping it labelled so they can find it later; 4) carefully melt holes in the plexiglas plates, top and bottom, in the correct positions to accept the nylon; 5) thread the correct end of the line through the top hole, and tie a knot to affix the bead in exactly the right position; 6) thread the other end of the line through the bottom hole, and if the construction design allows the lower plate to be moved out later, tie a knot in the line exactly 1 inch beyond the bottom plate; 7) put a numeric label on the bottom hole, oriented so it will be easy to read when the 3-d map is finally displayed.

Note:  you will save yourself and your students much agony if you find a way to do all the stars in order from one side of the map to the other!  If you instead take the easy way and work from the center out, as we did, you will have to reach between dozens of nylon lines to attach some of the later stars, thereby pulling some of them loose again and causing headaches.  One thought would be to have the students plot their right-ascension, radius points on a big paper grid (like the plexiglas sheets), numbering each point, and then use the paper to plan the order in which to hang the stars.

Questions, corrections, comments:  Send me e-mail at  markthom@flash.net