Apparent Magnitude & Brightness Ratio Calculator

Background.

Star brightness is still graded on a scale handed down from Greek star catalogues, where the most prominent stars were "of the first magnitude" and the faintest ones the eye could make out were "of the sixth". Two quirks survive from that origin. The numbers run backwards, so a smaller magnitude means a brighter object. And the steps multiply rather than add: going from sixth magnitude to first is a hundredfold gain in light, not a fivefold one.

Norman Pogson made that hundredfold figure exact in 1856. Five magnitudes became a flux ratio of precisely 100, so a single magnitude is the fifth root of 100, about 2.5119 — the "2.512" quoted by textbooks and by ESA's education material. Everything here follows from that one choice. Enter two apparent magnitudes and you get how many times more light the brighter object sends you; enter a brightness ratio and you get the magnitude gap it corresponds to. Either way you also get the single magnitude the two would read as if an instrument could not separate them.

The compression of the scale is what surprises people. Sirius at −1.46 and Vega at 0.03 sit only a magnitude and a half apart in the catalogue, yet Sirius delivers almost four times as much light. Ten magnitudes is a factor of 10,000, and the 25 magnitudes between the naked-eye limit near sixth magnitude and the faintest objects Hubble has detected, around +31, is a factor of ten billion.

Keep in mind what is being compared: light arriving at Earth, measured through one particular filter. A star can look faint because it is distant or dimmed by dust rather than feeble, and two magnitudes can only be compared when they come from the same photometric band.

What is apparent magnitude calculator?

Apparent magnitude is a logarithmic measure of the light an object delivers to an observer, defined as m = −2.5 log₁₀(F ÷ F₀), where F is the flux received and F₀ is the reference flux of the photometric system. Each system measures through a defined filter — Johnson V is centred in the yellow-green near 550 nanometres, Gaia's G band is much broader — so every magnitude belongs to a band. For bolometric magnitudes, which count all wavelengths, IAU Resolution B2 of 2015 fixed the reference exactly: m_bol = 0 at 2.518021002 × 10⁻⁸ W m⁻². Apparent magnitude differs from absolute magnitude, which is the value the object would have at a distance of 10 parsecs, and from luminosity, the total power it emits in watts.

How to use this calculator.

1. 01Under "What do you want to find?" choose Brightness ratio when you have two magnitudes, or Magnitude difference when you know how many times brighter one object is.
2. 02Enter the reference object as object 1. SIMBAD lists V magnitudes for named stars and the Gaia archive lists G magnitudes; brighter objects have smaller or negative numbers.
3. 03Enter object 2's magnitude from the same band, or in the other mode the ratio F₁ / F₂, which must be above zero; a ratio below 1 means object 1 is the fainter one.
4. 04Use the combined magnitude only when the two would be blended into one measurement, such as a close double star that a small telescope cannot split.

The formula.

Write each magnitude in its defining form, m = −2.5 log₁₀(F) + C, where C is the zero-point constant of the band. Subtracting one from the other removes C: m₂ − m₁ = 2.5 log₁₀(F₁ ÷ F₂). Solving for the flux gives F₁ ÷ F₂ = 10^(0.4 (m₂ − m₁)), and the reverse direction is Δm = 2.5 log₁₀(F₁ ÷ F₂). The factor 0.4 is simply 1 ÷ 2.5, and 10^0.4 is Pogson's 2.5119. Because C cancels only when both magnitudes share it, magnitudes from different bands cannot be fed into the same ratio.

The difference is taken as m₂ − m₁, so a positive Δm means object 1 is brighter and the ratio exceeds 1; swap the inputs and the ratio becomes its reciprocal. The extra-flux percentage is (F₁ ÷ F₂ − 1) × 100.

Combining two sources has to happen in flux, because light adds while logarithms do not. Each magnitude is turned into a relative flux, 10^(−0.4 m), the two are added, and the total goes back through −2.5 log₁₀: m_total = −2.5 log₁₀(10^(−0.4 m₁) + 10^(−0.4 m₂)). This assumes the sources are incoherent and in the same band, and it applies no correction for extinction.

Magnitudes are accepted from −60 to +70 and ratios from 10⁻⁴⁰ to 10⁴⁰, far beyond anything observed; in ratio-to-magnitude mode the implied second magnitude must also land in that window.

A worked example.

Frequently asked questions.

References& sources.

1. [1]European Space Agency, Science & Technology education resource, "Stellar Distances" (retrieved 2026-07-29). States the defining property of the Pogson scale: "a magnitude 1 star is 100 times brighter than a magnitude 6 star" because "the difference between each step on the scale is equal to a decrease in brightness of 2.512 and (2.512)⁵ = 100". Also defines apparent versus absolute magnitude. Independent of the IAU documents below; open access.
2. [2]Mamajek, E. E., et al. (2015), "IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales", arXiv:1510.06262 (retrieved 2026-07-29). Fixes the −2.5 log₁₀ form of the magnitude definition with exact zero points (m_bol = −2.5 log₁₀(f/f₀), f₀ = 2.518021002 × 10⁻⁸ W m⁻²). This page needs only the functional form, since every quantity it returns is a ratio in which the zero point cancels. Open-access preprint of an adopted IAU resolution.
3. [3]SIMBAD astronomical database (CDS, Strasbourg), objects "Sirius" (alf CMa, V = −1.46) and "Vega" (alf Lyr, V = 0.03), both from reference 2002yCat.2237....0D with quality flag C and no quoted uncertainty. Retrieved 2026-07-29. The two inputs to the worked example. Independent data centre; open access.
4. [4]Pogson, N. R. (1856), "Magnitudes of Thirty-six of the Minor Planets for the First Day of each Month of the Year 1857", Monthly Notices of the Royal Astronomical Society 17, 12. The paper that proposed fixing a light ratio of 100^(1/5) per magnitude, giving the modern m = −2.5 log I scale. Bibliographic reference only — the 1856 volume is an archival scan and was not machine-readable at the time of writing; the scale's defining property is verified here against the ESA citation above rather than against the original.