Focal Length and Image Scale

This journal is the Week 4 companion document in the Austronomic foundational series, which approaches amateur astronomy as a systems-oriented engineering and scientific discipline. It establishes focal length as the geometric parameter governing the angular scale of the image formed at the telescope focal plane, and derives the two governing relationships from first principles. The plate scale equation s = 206,265 × p / f quantifies the angular extent of a single detector pixel in arcseconds, where p is the pixel pitch in micrometres and f is the effective focal length in millimetres. The field-of-view equation FOV = 3438 × d_s / f gives the angular span of the full sensor along a given axis in arcminutes. Both relationships are derived from the thin lens equation under the small-angle approximation and are instrument-independent: they apply to any telescope-camera combination for which the focal length and pixel size are known. The journal treats focal reducers and Barlow lenses as effective-focal-length multipliers within the same framework. A visual-astronomy analogue is developed through the eyepiece field stop to connect the photographic field-of-view concept to the more familiar visual experience. The methodology is illustrated through worked examples spanning eight telescope-camera configurations with focal lengths from 490 mm to 2350 mm and image scales from 1.22 to 0.25 arcseconds per pixel. An experimental programme for empirically confirming any predicted plate scale by astrometric plate solving is defined with a 5 percent tolerance criterion and a structured discrepancy analysis framework. The procedure is demonstrated using Omega Centauri as a reference field from a southern-hemisphere observing site. This journal is produced as part of a 12-week educational series. Week 4 addresses the geometric mapping link in the complete photon-to-pixel imaging chain, following Week 3 (aperture and resolution) and preceding Week 5 (focal ratio and exposure efficiency).