= Solution
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. \b[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 $2\pi r\,dr$ square arcseconds, so
$$
F(r_m)=\int_0^{r_m}2\pi r\,a(1-kr)\,dr
=\boxed{2\pi a\left(\frac{r_m^2}{2}-\frac{kr_m^3}{3}\right)}.
$$
This 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 \b[16 <photons> at the boundary]. With pixel area $\Delta\Omega=1\,\mathrm{arcsec}^2$,
$$
h\nu=3.9756\times10^{-19}\,\mathrm J,\qquad
N(r)=\frac{At\Delta\Omega\,a}{h\nu}(1-kr)=32.6991(1-kr).
$$
Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec and
$$
\boxed{E_m=AtF(r_m)=2.8100\times10^{-15}\,\mathrm J}.
$$
Integrating to the model's outer radius $R=1/k=20$ arcsec instead gives
$$
\boxed{F_{\rm total}=\frac{\pi a}{3k^2}=5.4454\times10^{-17}\,\mathrm{W\,m^{-2}}},
\qquad E_{\rm total}=5.4454\times10^{-15}\,\mathrm J.
$$
The pixel cut retains \b[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.
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2014/paper-1-isophotal-flux.png]
{title=Original photon-counting profile and fraction of total flux inside the imposed detection radius}
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.
Back to article page
