[NOTE: Since writing this post I realized that I had the orientation of the dipper all wrong. I realized this when double checking the location I used for Dubhe, the alpha star and the upper of the pointer stars. In the photo where I traced out the Big Dipper constellation I realized that Dubhe was actually the bottom star in the orientation I was looking at the model. I’ve added some photos to correct this and am now going over things to see where I went wrong with some of the calculations because some stars are clearly in the wrong place. Back to the drawing board.]
When we look at the stars we often feel that we can just reach up and pluck them out of the sky, like grapes hanging on a vine. Or, if we are drawing out the constellations, we see the stars as points on a giant one-dimensional game of connect the dots. The truth is that the stars in constellations dot all points of our three-dimensional space. Some are closer to us while others lie at incredible distances away from us.
To help illustrate this concept, and just to have a little fun in my down time, I decided to build a scale 3D model of the Big Dipper. First I had to completely relearn my basic trigonometry and thankfully a couple of my co-workers were more than happy to sit down and walk me through everything I had forgotten since my one physics course in university.
After that I had to find the distances to the stars and their positions in the sky with Declination and Right Ascension. Having access to a copy of Starry Night definitely helped on this front. The last thing to do was to figure out how to convert spherical coordinates into 3D Cartesian coordinates.
Here again one of my co-workers came to the rescue and supplied with three basic formulas for the conversion. Using x = r Sin q Cos f ; y = r Sin q Sin f and z = Cos q (where q was Declination in degrees, f was Right Ascension in degrees and r was distance from the Earth in Light Years) I sat down with pencil, paper and calculator and set to work.
Once I had all my Cartesian coordinates I plotted out all my x and y values on a 2D grid using 1 cm as 10 Light Years (LY). Using the grid as I guide, I then poked holes into a cardboard box. For my z co-ordinates I simply used pipe cleaners to hold small balls of plasticine at the right heights above the box.
With everything in place I took a look at my 3D model. It turns out I had to turn the box upside down to see the Dipper when I was hopping to keep it upright on the table. I also think the stars are mirrored from how they appear in the sky at this time of year, but I am still very happy with the end result as it was great fun getting to use math outside of the class room and seeing all the stars in 3D space.
The Bigger Dipper probably isn’t the best example of stars at different distances forming a constellation, as they are all within 40 LY of each other, but it still fun to see and look at it from different angles that we don’t see from Earth. Next I think I’ll try and correct the errors in my calculations and make a larger model. After that, Orion here I come.