Understand the relationship
The reasoning behind the result
Assign background to the same effective area
B=A b; S=C−A b
C is the measured sum within the source aperture. The local background b is signal per pixel-area unit, so multiplying by the aperture's effective area A gives the background assigned to that sum. Subtracting it leaves signed net signal S.
Fractional overlap, masks and image boundaries can make effective area differ from πr² or an integer pixel count. Use the area associated with the actual summation weights. This tool does not inspect an image or choose an aperture.
A sky region supplies a mean, not a second source sum
b=Csky/Asky
A background-region sum must be divided by that region's own effective area before applying it to the source aperture. Subtracting the entire sky sum would be wrong when the two areas differ.
The simple sum/area option represents a mean over the measured region. If a robust or clipped estimator was used, enter its already determined background mean instead. Contamination, gradients, masking and estimation bias remain measurement concerns outside this arithmetic.
Exposure normalization precedes instrumental magnitude
R=S/t; minstr=−2.5 log₁₀(R / 1 unit s⁻¹)
The rate uses the entered exposure time once. Its numerical logarithm has an explicit reference of one selected signal unit per second and an additive constant of zero. Switching from ADU to electrons without applying a real gain changes that instrumental convention.
Zero and negative residuals remain visible as signed measurements, but have no finite ordinary logarithmic instrumental magnitude. They do not become zero-flux detections, infinite magnitudes or automatically established upper limits.
Background estimation uncertainty is not pixel scatter
σS²=σC²+(Aσb)²
The optional propagation assumes the aperture sum and estimated background mean are independent, and treats the entered area and exposure as exact. σC is the uncertainty of the aperture sum; σb is the uncertainty of the estimated mean background, not the scatter of individual background pixels.
For positive signal, the first-order magnitude uncertainty is (2.5/ln10)σS/S. It is a linear approximation that becomes unreliable at low signal-to-noise. Signed S/σS is reported as a measurement ratio without a detection threshold or Gaussian significance claim. No Poisson, read-noise, gain or correlated-pixel model is inferred.
Follow the numbers
Reconcile 12,000 measured ADU
- With C=12,000 ADU, A=100 pixel² and b=20 ADU/pixel², the assigned background is B=2,000 ADU.
- The net signal is S=10,000 ADU. Over t=50 seconds the rate is 200 ADU/s and instrumental magnitude is −2.5 log10(200)=−5.752574989.
- If independent σC=100 ADU and σb=0.3 ADU/pixel² are supplied, σS=√(100²+30²)=104.4030651 ADU.
The measured aperture sum, assigned background and net source estimate reconcile without discarding a signed residual.
Quick guide
How to use this calculator
- Enter the integrated aperture sum and its effective unmasked area.
- Supply a local background mean or obtain it from a measured region's sum divided by its matching area.
- Enter exposure time once. Choose the actual signal unit; no ADU-to-electron gain is assumed.
- If uncertainties are available, distinguish uncertainty of the background mean from scatter among individual background pixels.
Calculation method
Calculation and interpretation
Separate the light measured in an aperture from the background assigned to that same effective area.
B=A·b; S=C−B; R=S/t; σS=√(σC²+A²σb²); minstr=−2.5 log₁₀(R / 1 selected unit/s), only if R>0.
Worked example
Reconcile 12,000 measured ADU
The measured aperture sum, assigned background and net source estimate reconcile without discarding a signed residual.
B=A·b; S=C−B; R=S/t; σS=√(σC²+A²σb²); minstr=−2.5 log₁₀(R / 1 selected unit/s), only if R>0.
Supported inputs
Precision and limits
No image processing or automatic noise model
No aperture optimization, pixel masking, deblending, saturation test, sky clipping or detector gain is performed. Signal units and effective areas must come from the actual measurement.
Conditional uncertainty arithmetic
Supplied errors must be independent and must not double-count background uncertainty. Low-signal magnitude errors are not reliable symmetric confidence intervals.
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