The Celestial Sphere - 200 Brightest Stars - Star Projector
by sphynx, published
This thing allows you to print out the Celestial Sphere with small pentagonal holes drilled for each star. It has a hole in the South Pole, so if you put a bright point-source light such as an ultra-bright LED inside it, it acts as a simple star projector.
What is the Celestial Sphere?
According to Wikipedia:
"In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the observer. All objects in the observer's sky can be thought of as projected upon the inside surface of the celestial sphere, as if it were the underside of a dome or a hemispherical screen. The celestial sphere is a practical tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant."
If you look up into the night sky, you can imagine that the stars are projected onto the inside of a sphere that surrounds the earth. This is the Celestial Sphere. As the Earth spins around it's North/South axis every 24 hours, so the stars appear to rotate around the North and South Poles. Another way of looking at this, is that you can imagine the Earth being stationary and the Celestial Sphere rotating around the Earth once every 24 hours.
Print out the file 01CelestialSphere.stl.
This is quite a long print because I have made the walls of the sphere 5mm thick in order to make it opaque. You can probably take this down to 2 or 3mm if you want.
My Cura settings are in settings01.jpg and settings02.jpg.
This is quite a technical print. There are three problems:
1) The sphere has to be large enough to adequately separate all of the stars. For example, if you make the sphere much smaller than the dimensions I have used, the stars in Orion's Belt all munge together far too much. If that's OK with you - go ahead - make it smaller.
2) The holes for the stars have to be large enough to be resolved by your printer. For me, I can't make the stars smaller than about 1mm radius or they loose all definition.
3) Printing hollow spheres can be difficult. As the curvature of the sphere flattens towards the horizontal, the print can fail. This is why I have made the walls so thick.
The science bit
The positions of the stars are recorded in star almanacs by right ascension (ra) and declination (dec). I have used the data from Wolfram Alpha (see the CelestialSphere.nb Mathematica notebook).
The ra measures how far a star is in its "orbit" around the Earth's North Pole. It is given in decimal hours. This is because as the Earth rotates over a 24 hour period, the positions of the stars sweep 360 degrees around the Poles. So the rotation of the stars from some defined starting point can be expressed both as a time in hours and as an angle in degrees. A corollary of this is that if a telescope wishes to track a star, then it must rotate at a rate of 360/24 = 15 degrees per hour in the same plane of rotation as the stars. There are 24 facets on the printed sphere, so each facet is 1 hour or 15 degrees.
Declination is the distance of the star above the plane of the ecliptic. The ecliptic is the plane in which the Earth and all other planets in the solar system orbit around the Sun.
You can convert ra and dec to rotations about the Y and Z axis as follows:
zrot = ra * 360/24
yrot = 90 - dec
Given this information, modelling the Celestial Sphere is very easy:
Draw a sphere radius r. // The Celestial Sphere
For each star
Draw a cylinder at the origin with axis along the Z axis
Rotate the cylinder first by yrot then by zrot
Project the cylinder out from the origin until it pierces the sphere.
The tinkering bit
You need the OpenSCAD files:
1) stars.scad - the 500 brightest stars in Cartesian and Polar coordinates.
2) 01CelestialSpherePolar.scad - the Celestial Sphere with a hole in the South Pole for a LED.
If you want to tinker with the OpenSCAD files (and I hope you do!) please be aware that OpenSCAD takes an age to render all the stars. For 200 pentagonal stars, I had to leave it chundering away for a several hours on my 2011 MacBook Air. Ideally, I would like to print 500 stars, but life is too short. Also, I was going to make the stars the shape of Koch Snowflakes (http://www.thingiverse.com/thing:35246), but that way lies madness...
The file 01CelestialSphere.nb is a Mathematica notebook that shows how to get the star coordinates and format them for OpenSCAD.
As a future development of this project, I want to connect up the stars to show the constellations. This will involve drawing geodesic acs on the surface of the sphere between stars. There is an easy way to do this in OpenSCAD, but I haven't finished working out the maths yet!
Another development I am working on is to do a simplified sphere (or even cylinder) that only has the Zodiac constellations.
Watch this space (pun intended)!