Understand the relationship
The reasoning behind the result
Color is an ordered magnitude difference
C=mA−mB
The first band is subtracted from neither anything implicit nor a universal visual reference: the order is exactly first minus second. Swapping the bands reverses the sign. Negative colors are valid.
A color is not a temperature by itself. Spectrum, reddening, passband response and calibration affect it. This calculator does not assign a stellar class, infer temperature or choose a color–temperature relation.
Equal magnitudes need not mean equal physical densities
fA/fB=(f0A/f0B)10^(−0.4C)
The magnitude difference fixes a ratio of densities normalized to their own band zero points. If the zero densities differ, their physical density ratio has an additional f0A/f0B factor. Equal magnitudes then need not mean equal densities in physical units.
The optional zero-density ratio is accepted only as an external same-unit, same-density-type comparison. It does not turn a per-frequency quantity into a per-wavelength quantity or integrate flux across the two bands.
Entered color excess is a separate subtraction
C0=Cobs−E
Color excess is defined here as observed minus intrinsic color for this exact band pair. Removing the entered E yields the intrinsic-color scenario; adding E reverses it. E(B−V) must not be reused for another color without an applicable external law.
A negative entered excess can describe a signed measurement or scenario, but it is not automatically evidence of negative dust attenuation. No dust model, extinction coefficient or intrinsic source color is inferred.
Uncertainty propagation needs independence
σC²=σA²+σB²; σderived²=σknown²+σE²
For independent magnitude errors, variances add when subtracting the two bands. Independent uncertainty in a supplied excess adds again to the derived color's variance. The calculation reports the standard uncertainty of that linear arithmetic.
Shared calibration errors or an excess estimated from the same observed color introduce covariance, so this independent-input propagation does not apply. It is not a confidence interval or evidence that systematic calibration uncertainty is absent.
Follow the numbers
Separate normalized and physical density ratios
- Let mA=mB=10. Their color is C=0, giving a normalized density ratio of 10⁰=1.
- If the externally established zero densities have f0A/f0B=2, the physical density ratio is 2×1=2.
- With independent uncertainties of 0.03 and 0.04 mag, the measured color uncertainty is √(0.03²+0.04²)=0.05 mag.
A zero color identifies equal normalized densities; the band zero points determine the physical comparison.
Quick guide
How to use this calculator
- Name the first and second bands including their magnitude conventions.
- Enter a measured pair or a known observed/intrinsic color, retaining the first-minus-second order.
- Supply only an externally justified color excess for that same band pair.
- If adding uncertainties or a physical density ratio, verify the independence and compatible zero-density assumptions first.
Calculation method
Calculation and interpretation
Keep band order, calibration, color excess and any claimed flux ratio explicit.
C=mA−mB; E=Cobs−C0; (fA/f0A)/(fB/f0B)=10^(−0.4C); fA/fB=(f0A/f0B)10^(−0.4C).
Worked example
Separate normalized and physical density ratios
A zero color identifies equal normalized densities; the band zero points determine the physical comparison.
C=mA−mB; E=Cobs−C0; (fA/f0A)/(fB/f0B)=10^(−0.4C); fA/fB=(f0A/f0B)10^(−0.4C).
Supported inputs
Precision and limits
No inferred source properties
No temperature, spectral class, metallicity, dust law or intrinsic color is selected from the result.
Explicit uncertainty and calibration assumptions
The optional uncertainties require independent inputs. Physical density ratios require compatible same-unit reference densities for the named bands.
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