Given the orbital elements of the earth, sun, and moon, calculate the location and appearance of the moon in the sky in relation to the background stars. Appearance includes both the phase of the moon and the angle of the crescent in relation to the background stars.
The moon is the only naked eye astronomical body that's not a point or circle, so the only one we can ask questions like what angle it is turned.
Given the location of the observer on earth, we can also ask about the orientation of the crescent in relation to the horizon. Near the equator, one can observe the moon shaped like a boat.
It also might be interesting to try to compute taking into account that the speed of light is finite. General relativity: gravitational lensing, gravitational redshift.
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