Magnitude scales and measured colours

A smaller magnitude means a larger calibrated flux

Astronomical magnitude is a logarithmic way to report brightness. Within a specified system and passband, a simple calibrated definition is m = −2.5 log₁₀(F/F₀), where F is the measured positive source flux and F₀ is the reference flux assigned magnitude zero.

The ratio inside the logarithm is dimensionless. F and F₀ must use compatible units and the same passband and response convention. Raw counts from different exposures cannot be substituted without the corresponding calibration.

The minus sign reverses the ordering: increasing the flux decreases the magnitude. A negative magnitude is therefore allowed; it means the source is brighter than that zero-point reference in the specified system. It does not describe negative energy or negative flux.

This simple logarithm is undefined for zero or negative flux. Background subtraction can yield a noisy negative flux estimate, but it does not describe a physically negative source. Low-signal measurements need an appropriate flux likelihood, limit or explicitly different reporting convention.

Five magnitudes correspond exactly to a factor of one hundred

For two positive source fluxes in the same calibrated band and system, subtracting magnitudes gives m₁ − m₂ = −2.5 log₁₀(F₁/F₂). Their common F₀ cancels. This is a statement about a shared reference, not permission to mix unrelated systems.

Equivalently, F₁/F₂ = 10^[−0.4(m₁ − m₂)]. A source five magnitudes brighter has one hundred times the flux. A difference of one magnitude corresponds to a flux factor 10^0.4 ≈ 2.511886, rather than exactly 2.5.

The original table uses one arbitrary positive reference flux F₀. Multiplying its flux by ten lowers the magnitude by 2.5; dividing it by ten raises the magnitude by 2.5. The same rule applies to a ratio whether the absolute physical flux is large or small.

Magnitude precision and flux precision have different numerical units. For small errors around a positive, well-measured F, differentiating the logarithm gives the first-order relation |δm| ≈ (2.5/ln 10)|δF/F|. This local approximation is not a valid description of every low-signal or asymmetric uncertainty.

An unresolved pair is combined in flux

Flux contributions from unresolved incoherent sources add before the magnitude is calculated. Magnitudes themselves do not add, because a logarithm does not turn a sum into a sum of logarithms.

Suppose an original hypothetical pair contains two components each of magnitude 10 in the same band and system. Each has the same flux F. Together they have 2F, giving a magnitude shift of −2.5 log₁₀2 ≈ −0.752575 relative to either component.

The unresolved pair therefore has magnitude about 9.247425, not 20 and not the arithmetic mean 10. The result assumes both component signals are included and measured with a compatible calibration; it does not identify whether a real apparent pair is physically bound.

For unequal components, convert each magnitude to flux relative to one common zero point, add those relative fluxes and convert the sum back. A faint companion can make a smaller change than the equal-component example, while still affecting a physical inference that assumes one star.

A colour index compares two separately calibrated passbands

A colour index such as B − V subtracts the magnitude measured in one defined band from that in another. The band names stand for response functions, and each magnitude has its own system and zero point. The index is not simply a wavelength or an uncalibrated ratio of two pixel counts.

Stellar spectral shape influences colour, but so can atmospheric absorption lines, surface gravity, chemical composition and material along the line of sight. A colour-to-temperature calibration has a domain and assumptions. An observed colour alone need not fix one unique effective temperature.

Consider a stipulated star with intrinsic B = 6.0 and V = 5.5, so its intrinsic B − V is 0.5. Let the supplied band extinctions be A_B = 0.6 and A_V = 0.4 magnitudes. Its observed values become 6.6 and 5.9, and observed B − V becomes 0.7.

Here the colour excess E(B − V) is 0.7 − 0.5 = 0.2 magnitudes, also A_B − A_V. These are original supplied values, not a universal dust law. Inferring the intrinsic colour or total extinction for a real star requires further evidence. Broad stellar passbands should not be described as universally narrow wavelength channels.

Absolute magnitude supplies a distance convention

Apparent magnitude describes the source flux received at the observer within the specified system. Absolute magnitude M in a band describes the apparent magnitude the source would have at a distance of 10 parsecs with intervening extinction removed, using the same band and magnitude conventions.

For the nearby transparent inverse-square geometry, with extinction A in that band added separately, the distance modulus is m − M = 5 log₁₀(d/10 pc) + A. Both d and 10 pc are lengths, so their ratio is dimensionless. This is not a complete cosmological photometry formula.

For an original example with m = 12, d = 100 pc and stipulated A = 1 magnitude in the same band, the geometric term is 5. Thus M = 12 − 5 − 1 = 6. Omitting the supplied extinction would instead give 7 and attribute too little intrinsic band brightness to the source.

Absolute magnitude remains band dependent. A bolometric magnitude and its correction from a band magnitude require a defined bolometric zero point and a spectral estimate or calibration. Keep those conventions visible when using magnitudes to compare total luminosities rather than only selected-band light.

Worked examples

Try the reasoning

1. In one shared passband, source A has one hundredth the flux of source B. What is m_A − m_B?

   Show the answer and reasoning

   +5 magnitudes.

   −2.5 log₁₀(0.01) = +5. The smaller flux has the larger magnitude.

2. A and B have magnitudes 16 and 11 in the same system and band. What is F_A/F_B?

   Show the answer and reasoning

   0.01.

   Their difference is +5, so the flux ratio is 10^(−0.4 × 5) = 10⁻².

3. For m = 12, d = 100 pc and band extinction A = 1, what is M in the stipulated nearby model?

   Show the answer and reasoning

   6.

   The distance term is 5 log₁₀(100/10) = 5, and M = 12 − 5 − 1.

4. Does one observed B − V value identify a unique stellar temperature without other assumptions?

   Show the answer and reasoning

   No.

   Extinction, composition, gravity, the photometric system and the calibration domain can also affect that colour.

What to carry forward

• Magnitude differences encode positive same-band flux ratios with a reversed sign.
• Add unresolved light in flux before converting the total to a magnitude.
• Colour and absolute magnitude are defined measurements with band, zero-point, extinction and calibration conventions.