Photophys.com: The Science of Photography

Do stars move in circles around Polaris in your camera? As the lawyers say, it depends. If you have the usual rectilinear lens (anything other than a fisheye lens); and your camera is not pointed directly at Polaris, the answer is no. With a slight tilt away from Polaris, the star trails become ellipses and more extreme tilts produce parabolic or hyperbolic paths. Of course, all the star trails are beautiful, so why does one care about mathematical descriptions of their shapes? The answer is that photographers need to calculate maximum exposure times that will still avoid noticeable star trails resulting from motion. The rate of growth of trails for stars with various declinations and positions on the sensor is the subject of this study. It has been prompted by the approximate equations¹ and inaccurate calculators² that have been posted online. Here I will display exact calculations of star trails that can serve as bench marks for approximations and rules of thumb for photographers.  For the complete article see Luminous-Landscape.  The article can also be found Star trails - exact.