Stellar Magnitudes: Flux Ratios and the Combined Brightness of Stars

How do magnitudes describe brightness?

The numbers are easy to misread because their direction is opposite to an ordinary brightness score. A magnitude-2 object is brighter than a magnitude-5 object in the same band. A negative magnitude is a valid bright-source value; it does not mean negative light.

This guide uses invented numerical examples, not current measurements of named stars. The purpose is to explain the conversion, the addition of light and the information needed for a meaningful comparison.

State the band and zero point before using the formula

Write the ordinary logarithmic definition as m = −2.5 log₁₀(F/F₀), where F₀ is the reference flux defining the zero point. The logarithm acts on a dimensionless ratio. At F = F₀, the magnitude is zero, but the flux is plainly not zero.

For two measurements with the same reference, subtracting their definitions cancels F₀. This gives the magnitude-difference equation above. If the filters or zero-point systems differ, that cancellation cannot simply be assumed. A colour index uses two bands and needs their defined calibration context.

The Jodrell Bank teaching notes explain the logarithmic flux relation and distinguish apparent brightness from emitted luminosity. The worked examples here remain within a common band and zero point so that the arithmetic has an unambiguous meaning.

Convert positive and negative magnitude differences

The direction of the ratio matters. If m₂ − m₁ = 2, then F₂/F₁ is approximately 0.158489. The second source supplies about 15.85 percent of the first source's flux; equivalently, the first is about 6.3096 times as bright in that band.

For Δm = −2.5, the ratio is ten: the second source is brighter. Reversing the order of the two sources changes the sign of Δm and replaces the flux ratio with its reciprocal. Checking this symmetry is a quick way to catch a sign mistake.

A magnitude difference of one corresponds to approximately 2.511886 in the brighter-to-fainter direction. Use the full power-of-ten expression for calculations. Repeatedly multiplying a rounded value such as 2.5 does not reproduce the defined five-magnitude factor of exactly one hundred.

Add fluxes, then convert the total back to magnitude

When independent source fluxes add at the detector, their combined flux is Ftotal = F₁ + F₂ + … . Their magnitude numbers do not add. Substituting each flux into the logarithmic definition gives mcombined = −2.5 log₁₀[Σ 10^(−0.4mᵢ)] for a shared zero point.

For two sources, set Δm = m₂ − m₁. Factoring out the first source's flux gives mcombined = m₁ − 2.5 log₁₀[1 + 10^(−0.4Δm)]. The quantity subtracted is the brightening relative to the first source.

These expressions assume that the component fluxes being combined are the fluxes actually received under comparable conditions. An occultation can hide part of a source, and a saturated detector does not record a simple proportional count. Such effects cannot be repaired merely by applying the logarithm to unsuitable inputs.

Two equal stars brighten by about 0.75 magnitude

Two sources of magnitude 5 each have twice the flux of one source. Their combined magnitude is 5 − 2.5 log₁₀(2), approximately 4.247425. Adding 5 + 5 to obtain magnitude 10 would predict a fainter object, the opposite of adding light.

For N equal sources of magnitude m, the general result is m − 2.5 log₁₀(N). Ten equal sources brighten by 2.5 magnitudes; one hundred brighten by five. These are statements about additive flux, not about whether a telescope can resolve the individual sources.

The table deliberately holds the magnitude of every component at five. If the components are unequal, counting them is not enough. Their individual flux contributions must be used, which leads to the next set of examples.

Combine sources with different magnitudes

For magnitudes 2 and 3, the second source has 10^(−0.4) times the first source's flux, approximately 0.398107. The total is therefore about 1.398107 times the first flux, giving a combined magnitude of approximately 1.636149.

Add a magnitude-4 source and the combined result becomes approximately 1.519560. The extra light makes the magnitude smaller again. This illustrates a useful check: adding a positive flux must make the combined source brighter than the brightest component alone.

The examples 0 + 5 and 20 + 25 have the same five-magnitude gap, so they produce the same brightening of approximately 0.010803. Shifting every component magnitude by a common constant shifts the combined magnitude by that constant while leaving all relative flux ratios unchanged.

How much does a faint companion contribute?

Here the primary is the brighter component, so the horizontal magnitude gap is nonnegative. At zero gap the two sources are equal and the brightening is largest, about 0.752575 magnitude. As the companion becomes fainter, both curves approach zero.

The upper ratio is the companion's flux divided by the primary's flux, not its share of the combined light. For a ratio r, the share of the total is r/(1+r). At equal brightness, r = 1 but the companion contributes one half of the combined flux.

A detector example: compare net count rates

The Sheffield relative-photometry notes describe estimating background and measuring source counts. The figures above are invented to illustrate the arithmetic. Assume a linear detector, comparable throughput and the same filter; the listed background estimates refer to the respective source apertures.

Subtracting background gives 12,000 target counts and 30,000 comparison counts. Because the exposures differ, divide by their durations: 200 and 1,000 counts per second. The target-to-comparison flux ratio is then 0.2, not the raw net-count ratio of 0.4.

The target is −2.5 log₁₀(0.2), approximately 1.747425 magnitudes fainter. If the comparison has magnitude 8 in the matching system, the target has approximately 9.747425. These conclusions rely on the stated detector and calibration assumptions, not on the count totals alone.

Use a stable calculation and useful sanity checks

For very large magnitude numbers, directly evaluating each tiny flux can underflow in numerical software. Choose the smallest input magnitude b and calculate mcombined = b − 2.5 log₁₀[Σ 10^(−0.4(mᵢ − b))]. The brightest component then contributes exactly one to the scaled sum.

For N positive component fluxes, the combined magnitude must lie between b − 2.5 log₁₀(N) and b. The lower end occurs when all N magnitudes equal b. With more than one finite positive component, the combined magnitude is strictly smaller than b; a very faint companion may make only a tiny difference.

Input order must not matter. Combining two groups in flux space must agree with combining every component at once. A common magnitude offset must shift the answer by the same offset. These checks test the structure of the calculation, not just one numerical fixture.

Do not confuse decimal precision with measurement quality

Differentiating the magnitude definition gives the small-error relation σm ≈ (2.5/ln 10)(σF/F) ≈ 1.085736(σF/F). A five-percent flux uncertainty therefore corresponds to roughly 0.0543 magnitude in this local linear approximation.

The approximation assumes a sufficiently small fractional uncertainty. Near zero net flux, the logarithm and its error behaviour need more careful treatment. A zero or negative background-subtracted estimate has no finite real magnitude under the ordinary logarithmic formula used here; it should not be repaired by silently taking its absolute value.

The tables show six decimals to make the model reproducible. They are not measurements accurate to six decimal places. A real result needs its uncertainty, filter, zero-point system and observing context recorded alongside the magnitude.

Apparent magnitude, absolute magnitude and visibility

Apparent magnitude describes received flux. Absolute magnitude refers to the magnitude an object would have at a standard distance of ten parsecs under the defined convention. A bright-looking object need not emit more total energy than a fainter-looking object, because distance and attenuation also matter.

An integrated magnitude also differs from surface brightness. Two extended objects can have the same total flux spread over different angular areas. In a uniform illustrative model, spreading the same flux over ten times the area reduces flux per unit area by ten, a difference of 2.5 magnitudes per matching area unit.

Geometry and conditions still affect an observation. Airmass, background light, angular extent and instrument response are separate from the conversion rule. Neither this chart nor an astrological natural-strength score determines whether a source can be seen on a particular night.

Frequently asked questions