As described above, we expect a perfect circle if the optical axis of the camera is aimed at the North Pole of the Celestial Sphere. 3 This cone defines a circle of points on the sensor with Polaris very near its center. The collection of stars that make an angle of θ (the declination 𝛿=𝜋/2− 𝜃) with the north pole of the celestial sphere (close to Polaris) produces a right circular cone of light rays that passes through the pinhole to produce a cone of light inside the camera. So, for this discussion, I will assume a pinhole that a light ray from any star passes through the before striking the sensor. Here I will display exact calculations of star trails that can serve as benchmarks for approximations and rules of thumb for photographers.Ī perfect, distortion-free, rectilinear lens would project an image identical to that found in a pinhole camera. It has been prompted by the approximate equations 1 and inaccurate calculators 2 that have been posted online. The rate of growth of trails for stars with various declinations and positions on the sensor is the subject of this study. 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. With a slight tilt away from Polaris, the star trails become ellipses and more extreme tilts produce parabolic or hyperbolic paths. 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. Do stars move in circles around Polaris in your camera? As the lawyers say, it depends.

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