Understand the relationship
The reasoning behind the result
Instrumental magnitude starts from a normalized signal
mi=−2.5 log₁₀[(S/t)/(1 selected signal unit/s)]
S is a positive net signal after background subtraction and t is exposure time. Dividing once by exposure produces a rate. The logarithm uses an explicit numerical reference of one selected signal unit per second, with zero additive constant.
Instrumental magnitudes are accepted without reconstructing a count rate, because external software may use another additive constant. That constant must be common to the reference and target measurements, or the supplied zero point must use the same convention. ADU and electron conventions are not interchangeable without the actual gain.
A reference star connects the instrument and catalog scales
Zi=mcat,i−mi; mtarget=mtarget,instr+Z
Each reference star supplies an individual zero-point estimate. With one reference this is the familiar differential formula mtarget=mcat,ref+mtarget,instr−mref,instr. The shared instrumental offset cancels.
The same zero point is meaningful only for compatible passband, instrument response and observing conditions. Normalizing different exposure lengths removes exposure scaling; it does not correct airmass, transparency, color terms, nonlinearity or changing calibration between frames.
An ensemble retains every reference's contribution
Z=ΣwiZi; equal wi=1/n; inverse-variance wi∝1/σZi²
Equal weighting treats each entered reference zero point equally. When independent uncertainties are supplied, inverse-variance weighting gives more influence to a better determined zero point. Every weight and residual remains in the output ledger, and no apparent outlier is silently removed.
Reference scatter is the weighted root-mean-square deviation around the fitted zero point. It is descriptive disagreement, not automatically the uncertainty of the mean or proof that the calibration is accurate. Systematic catalog offsets and shared measurement errors do not vanish by adding references.
Independent uncertainty propagation is conditional
σZi²=σcat,i²+σmi²; σZ²=Σwi²σZi²; σtarget²=σtarget,instr²+σZ²
The optional formulas require independent reference errors and a target measurement independent of the zero-point fit. For count or rate input, magnitude uncertainty is approximated to first order as (2.5/ln10)σS/S, with exposure treated as exact.
This linearized magnitude uncertainty becomes unreliable when signal uncertainty is large relative to positive net signal. Correlated backgrounds, shared zero-point systematics and transformations require additional covariance modeling. If no uncertainties are supplied, none are fabricated from the number of decimal places or from reference scatter.
Check stars test the applied calibration without fitting it
Check residual=mcheck,instr+Z−mcheck,catalog
An independent check star has a known catalog magnitude but is excluded from the reference ensemble. Its residual compares the applied calibration with that separate measurement. The sign is calibrated minus catalog, so a positive residual means the calibrated value is numerically fainter.
A check residual does not by itself prove variability, identify a bad reference or certify photometric accuracy. No pass/fail threshold, automatic check-star selection, transformation coefficient or reporting submission is generated. All work stays in the current browser calculation.
Follow the numbers
Calibrate from two unequally precise references
- Reference A gives Z=12−(−8)=20 with σZ=√(0.03²+0.04²)=0.05 mag. Reference B gives Z=13−(−7.2)=20.2 with σZ=0.10 mag.
- Inverse variances are 400 and 100, giving normalized weights 0.8 and 0.2. The zero point is 0.8×20+0.2×20.2=20.04 mag.
- A target instrumental magnitude of −6 becomes 14.04 mag. With independent target uncertainty 0.02 mag, combined standard uncertainty is √[(0.8×0.05)²+(0.2×0.10)²+0.02²]=0.048989795 mag.
The fitted zero point, reference disagreement and propagated target uncertainty answer separate questions.
Quick guide
How to use this calculator
- Choose instrumental-only, known-zero, zero-point derivation or complete reference-star calibration.
- Select integrated net signal, rate or precomputed instrumental magnitudes. Keep the same signal convention and band for every record.
- Enter each reference star's catalog magnitude and measurement; targets and independent check stars have separate records.
- Inspect the individual reference zero points and residuals. Check stars never influence the fitted zero point, and no row is automatically clipped.
Calculation method
Calculation and interpretation
Make the reference-star calibration and every target's result inspectable from measured signal to magnitude.
mi=−2.5 log₁₀(S/t); Zi=mcat,i−mi; Z=ΣwiZi; mtarget=mtarget,instr+Z; Σwi=1.
Worked example
Calibrate from two unequally precise references
The fitted zero point, reference disagreement and propagated target uncertainty answer separate questions.
mi=−2.5 log₁₀(S/t); Zi=mcat,i−mi; Z=ΣwiZi; mtarget=mtarget,instr+Z; Σwi=1.
Supported inputs
Precision and limits
No automatic standard-system transformation
Same-frame differential calibration does not automatically remove color terms, extinction differences, detector nonlinearity or catalog-system offsets. No standard band transformation or observation submission is performed.
Independent uncertainty model only
Shared calibration errors and covariance are excluded. Low-signal logarithmic errors are first-order approximations; nonpositive net signals must remain outside this positive-signal calibration and can be inspected in the aperture workbench.
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