Focal Ratio and Exposure Efficiency

This journal is the Week 5 companion document in the Austronomic foundational series, which approaches amateur astronomy as a systems-oriented engineering and scientific discipline. It establishes focal ratio as the optical parameter governing focal-plane irradiance and exposure efficiency for extended astronomical sources, and develops the governing relationships from first principles. The focal ratio equation N = f / D defines focal ratio as the ratio of focal length to aperture diameter. For extended sources, the focal-plane irradiance relationship E = (pi/4) * L / N^2 shows that irradiance depends on source surface brightness and the inverse square of focal ratio. The corresponding relative integration-time relationship t2 / t1 = (N2 / N1)^2 quantifies the exposure penalty or advantage between two configurations under matched imaging conditions. These relationships are developed in instrument-independent form and apply to any telescope-camera combination for which focal length and aperture diameter are known. The journal distinguishes clearly between extended-source and point-source imaging regimes. For diffuse targets such as nebulae and galaxies, focal ratio governs focal-plane irradiance and therefore exposure efficiency. For unresolved point sources, aperture governs the total collected signal, while the point spread function governs how that signal is distributed across the detector. The methodology is illustrated through worked examples spanning the Austronomic telescope-camera configurations, with focal ratios from f/5.44 to f/10.0 and plate scales from 1.38 to 0.25 arcseconds per pixel. An experimental programme for empirically confirming the predicted focal-plane irradiance ratio is defined through matched-region measurements on the Eta Carinae Nebula (NGC 3372), using sky-subtracted nebular surface brightness in calibrated image stacks. The procedure is framed with a 10 percent tolerance criterion and a structured discrepancy analysis that includes calibration quality, transparency variation, reducer spacing, focus shift, and tilt. This journal is produced as part of a 12-week educational series. Week 5 addresses the exposure-efficiency link in the complete photon-to-pixel imaging chain, following Week 4 (focal length and image scale) and preceding Week 6 (atmospheric seeing).