Apparent Magnitude & Brightness Ratio Calculator

Background.

"Sirius is magnitude −1.46 and Vega is magnitude 0.03" is a statement most people can read but few can feel. How much brighter is that, actually? This calculator answers exactly that: give it two apparent magnitudes and it tells you the ratio of the light they deliver, or give it a brightness ratio and it tells you how many magnitudes that is. It also combines two magnitudes into the single value a photometer would report if it could not separate them.

The scale is logarithmic and it runs backwards, which is why the arithmetic is not obvious. Norman Pogson formalised the inherited Greek ranking in 1856 by fixing five magnitudes as exactly a hundredfold difference in brightness. That makes one magnitude a factor of 100^(1/5) = 2.5118864315 — usually quoted as 2.512 — and it forces the minus sign in the definition, m = −2.5 log₁₀(flux) + constant, so that brighter objects keep the small numbers they had inherited. ESA's education material states the same property directly: "the difference between each step on the scale is equal to a decrease in brightness of 2.512 and (2.512)⁵ = 100".

The practical formulas follow immediately. The ratio of two fluxes is F₁/F₂ = 10^(0.4(m₂ − m₁)), and the difference of two magnitudes is Δm = 2.5 log₁₀(F₁/F₂). Notice that no zero point appears in either: because both are built from a difference or a ratio, whatever constant defines the photometric system cancels exactly. That is what lets this page work in the V band, in Gaia's G band, in bolometric magnitudes or in anything else — provided both magnitudes come from the same band. Mix a V magnitude with a bolometric one and the cancellation is invalid, the answer is silently wrong, and no arithmetic can detect it. The result note says so on every calculation.

The combined-magnitude output answers a different and genuinely useful question: what does a close pair look like when a photometer cannot resolve them? Fluxes add, magnitudes do not, so you have to convert down to flux, add, and convert back: m_total = −2.5 log₁₀(10^(−0.4m₁) + 10^(−0.4m₂)). Two identical stars blend to 2.5 log₁₀(2) = 0.753 magnitudes brighter than either alone — not one magnitude, and certainly not the sum of the two. This assumes the fluxes simply add, which holds for incoherent sources in the same band, and it models no extinction.

One limit worth naming before you use the number: apparent magnitude describes what reaches you, not what the object emits. Two stars with the same apparent magnitude can differ by orders of magnitude in real output if one is far more distant, and interstellar dust dims things further along the way. If you want intrinsic brightness, you need the distance modulus and an absolute magnitude — that is the stellar magnitude calculator — or a luminosity in watts, which is the luminosity calculator. This page compares what you see, and only what you see.

What is apparent magnitude calculator?

Apparent magnitude, m, is the logarithmic measure of how bright a celestial object appears from Earth. It combines the object's real output, its distance, and any dimming along the way into one number, and it is what a telescope actually measures. The scale is inverted — the brightest objects have the most negative values — because it descends from a Greek ranking in which the brightest stars were "of the first magnitude" and the faintest visible ones "of the sixth".

Pogson's 1856 convention fixed that inherited ranking to an exact ratio: five magnitudes is exactly a factor of 100 in flux, so one magnitude is 100^(1/5) = 2.5118864315. From this everything on this page follows. The flux ratio of two objects is F₁/F₂ = 10^(0.4(m₂ − m₁)); the magnitude difference for a known flux ratio is Δm = 2.5 log₁₀(F₁/F₂); and the blended magnitude of an unresolved pair is m_total = −2.5 log₁₀(10^(−0.4m₁) + 10^(−0.4m₂)).

Because all three are ratios or differences, the photometric zero point cancels and none of them needs one. The IAU did fix exact zero points in 2015 (Resolution B2) for the bolometric scale specifically — m_bol = 0 at an irradiance of 2.518021002 × 10⁻⁸ W m⁻² — but those are needed only to convert a magnitude into a physical flux, which this page deliberately does not do. What it needs from the IAU definition is just the −2.5 log₁₀ form.

What apparent magnitude is not: it is not absolute magnitude (which is the apparent magnitude at 10 parsecs), it is not luminosity (total power in watts), and it is not comparable across photometric bands. All three of those distinctions cause real errors, and all three are handled by other pages here rather than fudged into this one.

How to use this calculator.

1. 01Pick your direction: two magnitudes to a brightness ratio, or a brightness ratio to a magnitude difference.
2. 02Enter the magnitude of the reference object as object 1. Both modes need it.
3. 03Enter the second magnitude, or the flux ratio, depending on your chosen mode.
4. 04Check that both magnitudes come from the same photometric band. This is the one error the calculator cannot catch for you.
5. 05Read the brightness ratio, the percentage, and — if you are dealing with an unresolved pair — the combined magnitude.

The formula.

Start from the definition m = −2.5 log₁₀(F) + C, where C is the photometric zero point of whatever system you are in. Subtract two of them and C vanishes: m₂ − m₁ = −2.5 log₁₀(F₂/F₁) = 2.5 log₁₀(F₁/F₂). Exponentiating gives F₁/F₂ = 10^(0.4(m₂ − m₁)), which is the first formula on this page, and rearranging gives Δm = 2.5 log₁₀(F₁/F₂), which is the second. The disappearance of C is not a convenience — it is the reason this page is band-agnostic, and it is also precisely why the two magnitudes must come from the same band, since two different C values do not cancel.

The sign is the part that trips people. Δm is defined here as m₂ − m₁, so a positive Δm means object 2 has the larger magnitude, which on an inverted scale means object 2 is the fainter one and object 1 is brighter. Sirius at −1.46 and Vega at 0.03 give Δm = +1.49 and a ratio of 3.94, meaning Sirius delivers 3.94 times Vega's flux. Swap the inputs and Δm becomes −1.49 and the ratio becomes 1/3.94 = 0.2535 — the calculator does exactly that, and a test asserts it.

The combined magnitude cannot be done in magnitude space at all, because magnitudes are logarithms and light adds linearly. Each magnitude is converted to a relative flux with 10^(−0.4m), the two fluxes are added, and the sum is converted back with −2.5 log₁₀. For two identical sources the answer is m − 2.5 log₁₀(2) = m − 0.7525749892: doubling the light is three quarters of a magnitude, not one. For Sirius and Vega the blend is −1.7053219872, brighter than Sirius alone but by only a quarter of a magnitude, because Vega contributes about a fifth of the combined light.

Rounding stage: no intermediate rounding. All arithmetic runs at 40 significant digits in a private decimal context, and rounding happens once at the return boundary — magnitudes to ten decimal places, ratios and percentages to twelve significant digits (a ratio can legitimately be 10⁻²⁰ or 10²⁰, which decimal-place rounding would destroy). The brighter/fainter/equal decision is made on the unrounded magnitude difference, so the exactly-equal case is deterministic.

Invalid-domain behaviour: a flux ratio of zero or below raises a field error, since the logarithm has no value there and negative brightness has no physical reading. Magnitudes outside −60 to +70 raise an error, as does a flux ratio outside 10⁻⁴⁰ to 10⁴⁰, because beyond those the derived quantities exceed what a double can hold; for scale, the Sun is about −26.8 and the faintest objects Hubble has detected are around +31, so the supported range is far wider than astronomy uses.

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.