AC02 · Counting starlight

Crowded-star Photometry

Can a neighboring star make a steady star look brighter? Yes. A circular aperture also collects light spread from nearby stars. Fitting the shapes of both stars can separate their light, but the uncertainty grows as their images overlap.

Drag the amber star into the cyan aperture and watch its light enter the measurement. Compare a circular aperture with a two-star fit to the same noisy detector pixels.

Aperture flux, corrected for lost light $8.32\times10^3\,\mathrm e^{-}$

Target flux from the two-star fit $6.76\times10^3\,\mathrm e^{-}$

Fit uncertainty, one standard deviation $104\,\mathrm e^{-}$

Neighbor light inside the aperture $1.55\times10^3\,\mathrm e^{-}$

Whose light did you collect?

Aperture radius

2.8

r\;[\mathrm{pixel}]

Star separation

3.2

d\;[\mathrm{pixel}]

Blur width

1.2

\sigma\;[\mathrm{pixel}]

Neighbor / mean target brightness

0.8

F_{\mathrm n}/\overline F_{\mathrm t}

Sky electrons per pixel

90

B\;[\mathrm e^{-}]

Drag the amber star or the cyan aperture. Each exposure measures the same pixels in two ways.

Background error and reading

Extra sky subtraction per pixel

0

\Delta B\;[\mathrm e^{-}]

The fit knows the source positions, the Gaussian profile and the true pixel variances. It also fits a constant sky. Real images require their own calibration.

Learn photometry and extract a real TESS light curve

Model assumptions

This detector is simulated. Both stars have a known circular Gaussian point-spread function and known positions. Each pixel integrates that profile, then receives independent Poisson photoelectrons and Gaussian read noise. The aperture uses fractional pixel areas and an exact-model aperture correction; the fit solves both fluxes and a constant sky jointly, with true model variances supplied as weights. These favorable assumptions omit PSF errors, saturation, cosmic rays, flat-field errors and interpixel correlations. The cyan star varies sinusoidally for the demonstration; time is accelerated. The separate TESS tutorial uses measured, already background-subtracted SPOC pixels.

Follow one star through the measurements

White: injected target flux. Cyan: the two-star fit. Amber: aperture flux with the known target aperture correction. All values are photoelectrons per simulated exposure.

Repeated simulated exposures

F\;[\mathrm e^{-}]

What the uncertainty includes

The quoted fit error comes from the weighted linear fit to the two pixel-integrated Gaussian profiles and a constant background. Photon and read noise are included. Source positions, profile width and weights are supplied by the generating model. Uncertainty in those inputs would increase the error in a real observation. Aperture leakage is an ideal contribution from the neighbor, corrected by the same target enclosed fraction as the aperture flux.

The detector uses one false-color scale for acquired simulated pixel counts. For display only, the known input sky is removed, negative remaining values use the darkest display color, and an asinh stretch runs from dark blue to white with a ceiling one thousand electrons above the sky. Cyan and amber mark the supplied aperture and star positions. Stored pixels and both measurements retain the original counts.

\widehat{\boldsymbol f}=(A^{\mathsf T}WA)^{-1}A^{\mathsf T}W\boldsymbol y,\qquad\operatorname{Cov}(\widehat{\boldsymbol f})=(A^{\mathsf T}WA)^{-1}

Start with the essentials

Focus question: How does the same detector image give two different brightness estimates?
One-sentence intuition: An aperture adds pixels inside a boundary; a two-star fit uses their spatial pattern to divide the light.

Core mathematical model

Light in each detector pixel

y_i=F_{\mathrm t}P_{i\mathrm t}+F_{\mathrm n}P_{i\mathrm n}+B+\epsilon_i

The target and neighbor contribute their total photoelectrons times the fraction falling in pixel i. The constant sky adds photoelectrons to every pixel; noise fluctuates around this mean.

Aperture correction

\widehat F_{\mathrm{ap}}=\frac{\sum_i w_i(y_i-\widehat B)}{\sum_i w_iP_{i\mathrm t}}

A pixel weight is the fraction of its area covered by the aperture. The denominator is the target profile fraction captured with those weights. It corrects target light loss; neighbor light remains in the numerator.

Noise of a pixel

\operatorname{Var}(y_i)=F_{\mathrm t}P_{i\mathrm t}+F_{\mathrm n}P_{i\mathrm n}+B+\sigma_{\mathrm{read}}^2

Independent Poisson electrons have variance equal to their expected count. The readout contributes an independent variance. The simulation supplies this true variance to the fit.

Common difficulties

More pixels do not always improve the measurement

Typical misconceptionThe widest aperture is best because it catches more target light.

Better mental modelThe outer pixels contain progressively less target light, but still add background noise and neighbor light. Try changing only the radius and compare the corrected aperture flux with the injected white curve.

Run the experiment

01
Move the neighbor away
Select A star on its own, then drag the amber star toward the cyan aperture.
What to observe: The aperture estimate rises as neighbor light enters. The two-star fit still follows the injected target flux.
02
Blur the images together
Choose Two images merge and reduce Star separation. Compare the fit uncertainty with the isolated preset.
What to observe: The two profiles become harder to distinguish. The fit can remain unbiased while its uncertainty increases.
03
Give the sky the wrong value
Choose Subtract too much sky. Open Background error and vary only Extra sky subtraction per pixel.
What to observe: The aperture estimate falls because every covered pixel loses extra electrons. The fit solves its own constant background from the image.

Crowded-star Photometry

Follow one star through the measurements

Separating two overlapping pools of light

Core mathematical model

Light in each detector pixel

Aperture correction

Noise of a pixel

Common difficulties

More pixels do not always improve the measurement

Run the experiment

Move the neighbor away

Blur the images together

Give the sky the wrong value

Related experiments and articles

Measuring the sky

Measuring starlight in crowded fields