For a paraboloid mirror, the angular expansion of an image due to coma, depends on the telescope diameter D and focal length /by the following relation:
tan 9 (in radians)
where Q is the angle of the incident ray to the optical axis. The angular expansion of the image due to astigmatism is:
For the case near the optical axis tan 8 — 8 in radians and so in this case these relationships simplify to:
It is more convenient in practical terms to express the angular distance off-axis in arcminutes and the aberrations in arcseconds, when the first result becomes
This is in close agreement with Bell.9 Since | is linear in 8, whilst a is quadratic, it follows that, on moving off-axis, coma is always the first aberration to appear and that, for the small 8 values with which we are concerned, astigmatism is generally negligible compared with coma for all except very extreme focal ratios. Their ratio is a/^ = 8F8/3 which, for example, only reaches unity at f/6 rather more than 3.5° off-axis and is < 0.1 at this f-ratio out to 8 = 21.5'. A Newtonian showing astigmatic star images is, therefore, either grossly misaligned - to the point that the reflection of the diagonal in the main mirror will be wildly eccentric - or has a badly distorted optical figure.
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