Apparent Magnitude

What magnitude measures

Apparent magnitude m expresses how bright a star, planet or other object appears from Earth. The scale runs backwards — the smaller the number, the brighter the object, and the very brightest objects have negative magnitudes. It is also logarithmic, which matches the way the eye responds to light: equal steps in magnitude look like roughly equal steps in brightness.

The system goes back to the Greek astronomer Hipparchus in the 2nd century BC, who sorted the naked-eye stars into six classes: the brightest were of the first magnitude, the faintest visible ones of the sixth. In 1856 the English astronomer Norman Pogson put the scale on a precise footing by defining a difference of 5 magnitudes as exactly a factor of 100 in brightness.

The Pogson formula

One magnitude therefore corresponds to a brightness ratio equal to the fifth root of 100, about 2.512. For two objects with measured fluxes F₁ and F₂:

m₁ − m₂ = −2.5 × log₁₀ (F₁ / F₂)F₁ / F₂ = 10^(−0.4 × (m₁ − m₂))

Example: Venus at magnitude −4.6 and Sirius at −1.46 differ by 3.14 magnitudes, so Venus is 10^(0.4 × 3.14) ≈ 18 times brighter than the brightest star. Sirius, in turn, is almost 1,000 times brighter than a star at the naked-eye limit of magnitude 6. Magnitudes are additive, not brightnesses: two stars of magnitude 1 together shine at magnitude 0.25, not 2.

Typical apparent magnitudes

Absolute magnitude

Apparent magnitude mixes two things: how luminous an object really is and how far away it lies. Absolute magnitude M removes the effect of distance by asking how bright a star would appear from a standard distance of 10 parsecs (32.6 light-years):

M = m − 5 × log₁₀ (d / 10)       (d in parsecs)m − M = 5 × log₁₀ d − 5          (distance modulus)

Sirius, at m = −1.46 and 2.64 parsecs, has M ≈ +1.4. The Sun, overwhelming in our sky at −26.7, would shine at only M ≈ +4.8 from 10 parsecs — a modest naked-eye star. The quantity m − M is called the distance modulus. Combined with a parallax measurement, which gives the distance directly, it tells astronomers how luminous a star really is.

Magnitudes of the planets

Planets shine by reflected sunlight, so their brightness depends on their distance from the Sun (r) and from Earth (Δ), both in astronomical units, and on their phase angle i — the Sun–planet–Earth angle. Astronomical Algorithms gives formulae of the form:

m = H + 5 × log₁₀ (r × Δ) + f(i)

H is the planet's magnitude if it were 1 AU from both the Sun and Earth and fully lit, and f(i) is an empirical phase correction. Saturn needs an extra term for the tilt of its rings, which can change its brightness by almost a full magnitude. These effects explain why Mars ranges from about +1.8 near conjunction, on the far side of the Sun, to −2.9 at a close opposition.

Limiting magnitude and light pollution

The faintest star you can see — the limiting magnitude — depends on the darkness of the sky, the altitude of the star, the transparency of the air and how well your eyes are dark-adapted. From a truly dark site the naked-eye limit is about 6.5, which reveals roughly 9,000 stars over the whole celestial sphere, about half of them above the horizon at any time. Under suburban skies the limit drops to 4 or 5; in a city centre it can be 3 or brighter, leaving only a few dozen stars.

Each magnitude step roughly triples the number of stars that become visible, so a small improvement in sky darkness makes a big difference. Binoculars of 10×50 reach about magnitude 9 to 10, and a 200 mm telescope about magnitude 14.

The Mag slider on the AstroAlgos sky disk sets the limiting magnitude of the map, from 2 to 6. At 2 you see only the brightest stars, close to what a city observer sees; at 6 you get the full naked-eye sky of a dark site. Matching the slider to your own sky makes it much easier to recognise the constellations when you step outside.

Explore it on AstroAlgos

• Sky disk — adjust the limiting magnitude with the Mag slider
• Semidiameter — the apparent size of the Sun, Moon and planets