I recently made a mini spectrometer using a CD. I have used that to measure the spectrum of the Sun, moon and various artificial light sources.
My question is how can I use it or modify it to measure the spectrum of stars
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You'll need to amplify the light coming in. There are two classes of devices which do this: cameras and telescopes. You can understand them by analogy to your eye. A telescope works by having a very wide "pupil", letting more light in, but then using lenses to focus this light onto a smaller "retina" (which is in fact your pupil). A camera is slightly different: in these cases the pupil and lens configuration is already set up for you, but you control exposure time, which is how much time the "retina" collects information before we finalize the signal (in eyes, as an action potential along the ocular nerve to the brain; in real cameras, as an amount of light-sensitive chemical reaction happening in the film inside).
You will probably want a mixture of these: You will probably want to use a telescope to amplify the light of the star and exclude other ones, and you will probably want to use a sheet of light-sensitive paper or an actual controllable camera with a long exposure time to see where the light comes out.
If you have access to a proper darkroom where you can expose film onto photographic paper and then develop and fix that paper, your device could simply be augmented by a pinhole camera made out of a Pringles can with photographic paper inserted directly. You'd of course have to calibrate it with some lasers if the paper was not color-sensitive.
Otherwise, you'll really want a digital camera that can have a long exposure time, mounted directly on the spectrometer. Just keep doubling the time until you can get results from dim objects, and you should be able to find something.
I've built several CD based spectrometers myself. The best resolution is obtained with a CD-RW because the data lines are much closer together than on a ordinary CD.
But evaluating the light from stars is seriously difficult because so very little light reaches the spectrometer because stars are so far away.
At the very least you'll need a good quality spectroscope (no chromatic aberration) on high magnification and with a tracking mechanism (to keep the star in sight over prolonged periods of time), some way of mounting the spectroscope on the eyepiece and a decent digital camera set to high light sensitivity and long exposure times.