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Within the same photometric band, the logarithmic magnitude scale converts a flux ratio into a brightness difference.
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Formula
m1 and m2 in magnitudes; F1 and F2 in the same flux unit and band
| Symbol | Meaning | Unit |
|---|---|---|
| m1, m2 | Apparent magnitudes of two sources on the same photometric scale | mag |
| F1, F2 | Measured fluxes for sources 1 and 2 with the same band definition | W/m2 or compatible flux unit |
m_1-m_2=-2.5\log_{10}\!\left(\frac{F_1}{F_2}\right)The relation follows from m=−2.5 log10(F/F0); subtracting cancels the same reference flux F0. Both measurements must use the same passband and calibration convention. A smaller magnitude means a brighter source. Five magnitudes correspond to a flux factor of 100; differences in extinction or spectral band must be accounted for first.
Both fluxes must be measured in the same photometric band and magnitude convention.
Subtract the magnitude definitions; the shared reference flux cancels.
Flux ratio
The same passband is still required.
Given F1/F2=100, find m1−m2.
−2.5log10(100)=−5 mag; source 1 is brighter.
A source is 2.5 mag brighter. What is its flux factor?
Fbright/Faint=10^(0.4·2.5)=10.
Reversing the sign and treating larger magnitude as brighter.
m1−m2 is negative when F1 exceeds F2.
Directly comparing fluxes from different filters.
Passbands and photometric zero points must match.
Formula cluster
Magnitudes encode flux logarithmically; distance and extinction affect the observed value.
If F1/F2=100 in the same filter, m1−m2=−2.5 log10(100)=−5 mag: source 1 is five magnitudes brighter.
Compare stellar brightnesses from compatible photometric measurements.
Here is how to work through a typical Magnitude difference from a flux ratio (m₁ − m₂ = −2,5 log₁₀(F₁/F₂)) task step by step:
Task
Given F1/F2=100, find m1−m2.
Solution path
−2.5log10(100)=−5 mag; source 1 is brighter.
Task
A source is 2.5 mag brighter. What is its flux factor?
Solution path
Fbright/Faint=10^(0.4·2.5)=10.
The equation describes a specific relation among astronomical quantities; it supports a reliable conclusion only when measurements match the variables’ definitions. Start by identifying each quantity, unit and reference frame. The relation m1-m2=-2.5\log_{10}\!\left(\frac{F1}{F2}\right) organizes those values mathematically, while the physical model determines whether it applies to the situation. A number alone does not establish that. Record assumptions, signs and any approximations. This keeps separate what follows directly from calculation and what requires interpretation. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
The relevant limits are stated in the subject background: The relation follows from m=−2.5 log10(F/F0); subtracting cancels the same reference flux F0. Both measurements must use the same passband and calibration convention. A smaller magnitude means a brighter source. Five magnitudes correspond to a flux factor of 100; differences in extinction or spectral band must be accounted for first. Read them before substitution because ideal geometry, small angles, constant properties or a particular cause of motion may be part of the derivation. Measurement uncertainty is also distinct from model validity: a precise measurement does not make an unsuitable approximation correct. Check that quantities use consistent conventions. If the assumptions fail, use a more suitable model or explicitly report the result as an estimate rather than an exact measurement. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Use the documented worked example as a calculation check: If F1/F2=100 in the same filter, m1−m2=−2.5 log10(100)=−5 mag: source 1 is five magnitudes brighter. Carry units through the arithmetic rather than handling numbers alone. Do they combine into the unit of the requested quantity? A dimensional mismatch often reveals a conversion error or swapped variables. If possible, substitute the result back into a rearranged original relation. Finally compare its order of magnitude with familiar astronomical scales; plausibility is not proof, but it is a useful error check. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Read the exponents and placement of quantities in m1-m2=-2.5\log_{10}\!\left(\frac{F1}{F2}\right): direct proportionality behaves differently from an inverse or power relation. Mentally double one input at a time and infer how the requested quantity changes. In observational data, a small relative uncertainty can matter greatly when a relation uses a small angle, a difference or an inverse quantity. Distinguish this sensitivity from measurement precision: the equation describes error propagation, not how well an instrument measures. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
State the numerical value with its unit, the input measurements and the main model assumption. If using an approximation, label the result as an estimate and say which effects were ignored. Avoid conclusions beyond the equation: a geometric or photometric quantity is not automatically a complete description of a star or galaxy. When measurement errors are known, do not report more precision than the data support. This concise record makes both calculation and interpretation traceable and reproducible. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Quanta exam set
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Which formula describes Magnitude difference from a flux ratio?
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How do you rearrange Magnitude difference from a flux ratio for Flux ratio?
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Which common mistake happens with Magnitude difference from a flux ratio?
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