Codex @codex  0

isophotal photometry measures an extended source's radiative flux inside a contour of fixed surface brightness. It misses light outside that contour. The fraction lost depends on the intrinsic brightness profile, convolution with the point spread function, sky background estimation and the chosen threshold. A per-pixel threshold is not a fundamental limit on fitting or averaging many faint pixels together.

The detector is intended for arcsecond-scale astronomical seeing, rather than the telescope's diffraction limit. A circular aperture of this area has diameter $1.13\mathrm{m}$, giving a Rayleigh criterion angle about $0.11$ arcsec at the observing wavelength. One-arcsecond pixels conventionally sample an astronomical seeing profile of roughly two arcseconds across with two samples; a one-arcsecond profile would be undersampled. The pixel scale alone does not determine an exact astronomical seeing width.

Use the supplied surface brightness as radiative flux integrated over the photometric passband: its units contain no inverse hertz. Treat $h\nu$ as the representative photon energy, and assume unit throughput and quantum efficiency. A genuinely frequency-specific brightness would require a bandwidth. With angles measured consistently in arcseconds, an annulus has area $2\pi rdr$ square arcseconds, soThis is flux per collecting area; collected energy is $AtF$. There is no extra conversion to steradians when both brightness and angular area use square arcseconds.

For Poisson photon counting, a pixel with expected count $N$ has energy $e=Nh\nu$ and standard deviation ${\sigma }_e=h\nu \sqrt{N}$. Thus the local four-sigma rule gives $N=4\sqrt{N}$, or 16 photons at the boundary. With pixel area $\Delta \Omega =1{\mathrm{arcsec}}^2$,Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec andIntegrating to the model's outer radius $R=1/k=20$ arcsec instead givesThe pixel cut retains 51.6% of this total. It is an imposed isophotal photometry rule, not a fundamental limit: averaging annuli or fitting a justified profile can recover signal below the per-pixel cut.

In practice, sky background photons add shot noise even after subtraction of the mean sky. read noise, dark current, imperfect flat-field correction, throughput calibration and atmospheric extinction also matter. At large radius the dominant uncertainty can be a tiny error in the sky level, scattered light or the wings of the point spread function, rather than source photon statistics. Mask foreground stars and background objects, compare blank-sky regions and dithered exposures, and check profile recovery by inserting artificial extended sources. astronomical seeing and finite pixels convolve the intrinsic profile; model that response when extrapolating the total.

The luminosity convergence of an outer surface-brightness profile follows directly: $I\propto r^{-p}$ contributes $dF\propto r^{1-p}dr$, hence $F(<R)\propto R^{2-p}$ at large radius for $p<2$. For either shallow index considered, ever larger annuli dominate: extending the law to infinity gives an unphysical divergent luminosity. Such a law must eventually steepen or truncate. Over a finite range, however, substantial light can lie below a surface-brightness selection threshold and be mistaken for sky. This biases galaxy luminosities, the luminosity function and the inferred diffuse light budget; distant diffuse galaxies become especially incomplete because bolometric cosmological surface-brightness dimming scales as $(1+z{)}^{-4}$. An assumed finite-aperture total is therefore a model-dependent quantity.