Understand the relationship
The reasoning behind the result
Magnitude compares a density with a stated zero point
m=−2.5 log₁₀(f/f0)
Both f and f0 must use the same spectral-density definition and unit. Their ratio is dimensionless. A factor of ten less density adds 2.5 magnitudes, while a zero magnitude simply means the density equals its reference.
The AB option deliberately uses the common 3,631-Jy reference. The also common formula with an exact additive constant of 48.60 in cgs implies about 3,630.780548 Jy instead; the two differ by roughly 0.0000656 mag. This workbench does not silently mix them.
ST is constant per wavelength, AB per frequency
mST=−2.5 log₁₀(fλ in erg/s/cm²/Å)−21.10
The ST convention uses a zero density of 10^(−21.10/2.5), about 3.630780548×10⁻⁹ erg/s/cm²/Å. A constant density per wavelength is not a constant density per frequency.
An entered custom zero point is interpreted as the density for magnitude zero in the exact selected unit and band. This supports an externally supplied calibration, but it is not a universal Vega conversion. A Vega-based broad-band reference depends on the passband and adopted spectrum.
The spectral Jacobian includes wavelength squared
fλ=fν |dν/dλ|=fν c/λ²
Frequency is ν=c/λ. Equal energy in a small spectral interval means fν dν and fλ dλ have matching magnitudes, producing the c/λ² conversion. The wavelength must first be in metres when using SI c.
Per-ångström and per-nanometre units require their own length factors: one erg/s/cm²/Å is 10⁷ W/m²/m, while one W/m²/nm is 10⁹ W/m²/m. For compatible band-averaged densities the appropriate pivot wavelength is needed; a guessed central wavelength does not reproduce arbitrary passband integration.
A signed measurement can exist without a logarithmic magnitude
Background subtraction can yield zero or negative estimated flux. Those are valid signed measurements to retain, but the ordinary logarithm has no finite magnitude for them. The tool labels that state and does not clamp the flux, invent a pseudocount or claim a nondetection limit.
Asinh magnitudes and statistical upper limits require additional conventions or uncertainty data and are not computed here. Large finite magnitudes are calculated in log space; if the inverse density cannot be represented in the chosen unit, an explicit numerical-range error is returned.
Follow the numbers
AB magnitude 20 in microjansky
- Use the declared AB zero point f0=3,631 Jy=3.631×10⁹ µJy.
- For m=20, the density ratio is 10^(−0.4×20)=10⁻⁸.
- The density is 3.631×10⁹×10⁻⁸=36.31 µJy, equivalent to 3.631×10⁻³¹ W/m²/Hz.
The unit changes the density number, while the physical density and magnitude remain the same.
Quick guide
How to use this calculator
- Select forward or inverse conversion, and identify the band and system.
- Choose per-frequency or per-wavelength density and its exact unit.
- When crossing spectral forms, supply the monochromatic wavelength or the appropriate filter pivot wavelength.
- Inspect the actual zero density and the signed measured value. A nonpositive measurement is not replaced by a limit.
Calculation method
Calculation and interpretation
Choose the spectral-density and zero-point convention before taking a logarithm.
m=−2.5 log₁₀(f/f0); f=f0·10^(−0.4m); fλ(per metre)=fν·c/λ²; 1 Jy=10⁻²⁶ W/m²/Hz.
Worked example
AB magnitude 20 in microjansky
The unit changes the density number, while the physical density and magnitude remain the same.
m=−2.5 log₁₀(f/f0); f=f0·10^(−0.4m); fλ(per metre)=fν·c/λ²; 1 Jy=10⁻²⁶ W/m²/Hz.
Supported inputs
Precision and limits
No spectrum integration
This converts compatible spectral densities. It does not integrate a source spectrum through a response curve or infer a passband's pivot wavelength.
Calibration remains explicit
Catalog-specific offsets, extinction, detector response, color transformations and uncertainty limits are not automatically applied.
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