Why spreading the same galaxy light makes it harder to measure

Project a physical radius onto the sky

The detector centre is the origin, with right and up positive. The source centre is $(x_s,y_s)$ , in arcseconds. Physical half-light radius $r_{1/2}$ encloses half the intrinsic total light. Angular-diameter distance converts it to an angular radius.

A circular Gaussian has a smooth tail. Its width parameter differs from its half-light radius: integrating to the latter includes exactly half the total flux.

\theta_{1/2}=\frac{r_{1/2}}{D_A},\qquad \sigma_{\rm src}=\frac{\theta_{1/2}}{\sqrt{2\ln2}}

Put radius and distance in the same length unit, then convert radians to arcseconds. At redshift ten in the adopted cosmology, 0.3 kiloparsecs subtends about 0.0721 arcseconds.

Reading: Hogg (1999) · Sandbox Physics

Let blur and pixels act on that same profile

The example also approximates diffraction blur as a Gaussian, with its width specified from each filter’s pivot wavelength and an aperture of six and a half metres. Gaussian variances add under convolution. A real JWST PSF has more structure.

\mathrm{FWHM}_{\rm proxy}=1.028\frac{\lambda_{\rm pivot}}{D}

Use the same length unit for wavelength and aperture, obtaining radians before conversion to arcseconds. This specifies a diffraction scale and assigns its width to the Gaussian proxy.

\sigma^2=\sigma_{\rm src}^2+\sigma_{\rm proxy}^2,\qquad \sigma_{\rm proxy}=\frac{\mathrm{FWHM}_{\rm proxy}}{2\sqrt{2\ln2}}

The full width at half maximum and Gaussian width use the same angular unit. The total width includes the intrinsic source and the specified optical proxy.

w_{ij}=\int_{\mathrm{pixel}\,ij}\frac{\exp\!\left[-\frac{(x-x_s)^2+(y-y_s)^2}{2\sigma^2}\right]}{2\pi\sigma^2}\,dx\,dy

A weight is the fraction of total light received inside one pixel area. Sampling the Gaussian height at its centre is a different operation.

Reading: Sandbox Physics · Photutils

Retain the light lost at the field edge

Thirty-two pixels per side at 0.03 arcseconds each form a square field 0.96 arcseconds across. Gaussian tails can extend beyond it. Moving or broadening a source reduces the total profile weight inside the field.

Keep that weight. Renormalizing a clipped profile to unity changes how the specified source flux enters the image. The original fractions identify how much light arrives and how much lies outside.

\sum_{ij\in\mathrm{field}}w_{ij}\leq1

Equality is the limit of a field containing all light. A finite Gaussian field is normally below one, although extremely faint tails can fall below floating-point resolution.

Reading: Sandbox Physics · Sandbox Physics

Estimate one total flux from those pixels

Let $F$ denote total input flux and $I_p$ the measured pixel value. Index $p$ lists the two-dimensional pixels in a single sequence. Each contains the profile-weighted signal and independent Gaussian noise with zero mean and equal standard deviation.

Supply the true position and profile, leaving only total flux unknown. Minimize the squared differences between the model and pixels. At the minimum, a small flux change has zero first-order effect. The result is a linear weighted estimate.

\chi^2(F)=\sum_p\frac{(I_p-Fw_p)^2}{\sigma_{\rm pix}^2},\qquad\frac{d\chi^2}{dF}=-\frac{2}{\sigma_{\rm pix}^2}\sum_p w_p(I_p-Fw_p)=0

Each term is a squared pixel residual in units of the common noise. Setting the derivative to zero gives the total-flux estimate below.

I_p=Fw_p+n_p,\qquad \widehat F=\frac{\sum_p w_pI_p}{\sum_p w_p^2},\qquad \sigma_F=\frac{\sigma_{\rm pix}}{\sqrt{\sum_p w_p^2}}

The specified profile weights each pixel. The examples use 0.15 nanojanskys of noise per pixel; total flux and its uncertainty are in nanojanskys.

\mathbb E[\widehat F]=F,\qquad\mathrm{Var}(\widehat F)=\frac{\sigma_{\rm pix}^2\sum_p w_p^2}{(\sum_p w_p^2)^2}

Independent noise contributes no cross-covariance terms. A correct known template can estimate the total flux without bias after field loss, with less information and a larger uncertainty.

Reading: Sandbox Physics · Photutils

Compare compact and diffuse sources at the same input flux

Fix redshift at ten, absolute UV magnitude at minus nineteen and a half, and retain the same screen and spectrum. Both F150W sources have 18.67 nanojanskys of input total flux. Increasing physical half-light radius from 0.3 to 1.2 kiloparsecs spreads light into shallow outer pixels and sends roughly a tenth beyond the field.

The compact source has total-flux uncertainty about 1.16 nanojanskys; the diffuse source about 4.39. Each noisy image uses its own saved draw, with actual signal-to-noise about 15.3 and 2.79. The uncertainty isolates the size effect, while a particular measurement also fluctuates around the true flux.

Compact and diffuse F150W sources at the same input total flux, with separate ideal expectations and actual saved noisy pixels — Both columns share total input flux and one fixed intensity scale. Ideal expectations occupy the upper row; each saved simulated noise realization is below. Half-light radii are 0.3 and 1.2 kiloparsecs. These are controlled Gaussian sources, not observed JWST galaxy photographs.

F150W quantity	Compact	Diffuse
Input total / nJy	18.67	18.67
Fraction inside field	about 1	0.901
Total-flux uncertainty / nJy	1.16	4.39
One actual signal / noise	15.3	2.79

Reading: Sandbox Physics

Understand the size dependence of the uncertainty

Independent noise variances add when pixels are combined. The standard deviation of an equal-weight sum grows as the square root of pixel count. A broader profile brings more noisy pixels into the estimate. For fine square pixels and a large field, the weight-square sum has a useful continuous limit.

\sum_p w_p^2\simeq\frac{a^2}{4\pi\sigma^2},\qquad\sigma_F\simeq\sigma_{\rm pix}\frac{\sqrt{4\pi}\,\sigma}{a}

Pixel angular pitch

a

must be much smaller than the profile width, and the field sufficiently large. In this limit, doubling profile width approximately doubles total-flux uncertainty.

Reading: Sandbox Physics

Give a known-position recovery probability

Under these assumptions the linear flux estimate remains Gaussian. A fixed threshold turns it into a recovery probability. The standard-normal cumulative distribution is the probability that a zero-mean Gaussian with unit standard deviation lies below its argument.

\Pr(\widehat F>k\sigma_F\mid F)=\Phi\!\left(\frac{F}{\sigma_F}-k\right)

Position and profile are fixed here; only noise varies. Input flux relative to its uncertainty determines this single-threshold curve.

Reading: Sandbox Physics · Photutils

Include source finding in the experiment

A blind finder first decides where targets might be. It searches many positions, estimates a background and deals with overlapping neighbours. A noise peak may be selected, or a diffuse source partly removed with the background. A single known-template Gaussian probability omits those operations.

To measure a blind pipeline’s completeness, inject sources of specified brightness and shape into the corresponding images. Keep their positions for evaluation, but withhold them from the finder. Run the complete pipeline, then match its detections to the planted sources. The recovery fraction now includes that finding and measurement procedure.

The independent model checks use four geometries with 1024 acquisitions each, comparing known-template means, variances and threshold fractions with Gaussian predictions. Those statistical tests use 0.05 nanojanskys per pixel; the five saved imaging examples use 0.15. The linear formula specifies the scaling with noise.

This small analytic result helps design the larger experiment: choose the source properties, identify the image conditions and decide which step must face an unknown position. Selection probability then has a specific experimental meaning when it enters a population count.

Reading: Sandbox Physics · Photutils

Try the observation

Keep total light fixed and change its distribution

Hold redshift, UV magnitude and screen fixed while broadening half-light radius. Predict total flux and its uncertainty separately.
Move the centre toward the edge and compare enclosed fraction with total-flux uncertainty.
Switch measured pixels and ideal expectation, identifying the saved noise and the supplied position/profile assumptions.

The example integrates a specified Gaussian over finite pixels and estimates its total flux under fixed independent Gaussian noise.

Exercises and explained answers

At fixed total flux and per-pixel noise, double Gaussian angular width in the fine-pixel, large-field limit. What happens to mean signal-to-noise?

Total-flux uncertainty approximately doubles and mean signal-to-noise halves. If the source is already much narrower than the optical blur, shrinking its intrinsic radius cannot make the total width arbitrarily small.

Divide a clipped template by its enclosed weight before fitting. Does its fitted normalization still represent the original infinite-field total?

The new unit-normalized template describes the light inside the field. Its fitted amplitude has a different meaning. Renormalization supplies no new information about the missing light; continuing to call both amplitudes the same total confuses two quantities.

A pipeline receives every injected position before making its measurements. Which probability is it testing?

It tests recovery and photometry near supplied positions. A blind experiment must also make the pipeline discover sources and include background, neighbours, false positives and matching rules.

Use the preceding unattenuated flat-frequency example. At fixed physical radius and intrinsic UV output, how does flux density per solid angle scale with redshift?

Total frequency flux contains $(1+z)/D_L^2$ and angular area contains $1/D_A^2$ . Their ratio gives $(1+z)^{-3}$ . Integrating the whole spectrum gives the familiar fourth-power bolometric dimming. The pixel model already includes the flux and angular conversions; an additional dimming factor would count the effect twice.

Primary sources and calculation notes

Hogg (1999), Distance measures in cosmology
Luminosity-distance section: bolometric flux, angular/luminosity distance and spectral-density transformations. The numerical examples use the explicitly adopted radiation-free model.
Photutils, PSF photometry
Finding sources, fixing supplied positions, fitting pixel-integrated profiles and grouping blends. Used to explain why source finding and a known-position flux estimate are different operations.
Sandbox Physics, finite physical-model checks
Actual 58 finite checks: matched Astropy cosmology, scalar/adaptive photon integrals, finite Gaussian pixels and four 1024-acquisition white-noise tests. Those statistical tests use 0.05 nJy per pixel; the saved five imaging examples use 0.15 nJy.
Sandbox Physics, specified synthetic imaging examples
Original synthetic examples with exact parameters, distances, native-band totals and saved noisy measurements. No observed galaxy catalog, foreign source ZIP, executable Engine or measured cosmic LF is included.