Vol. Science · Astronomy · Stellar brightness

Magnitude
the backwards scale

Apparent magnitude m is how bright something looks from here. Absolute magnitude M is how bright it would look from a standard distance of 10 parsecs. The difference is the distance modulus, m − M = 5 log₁₀(d / 10 pc). At 10 pc, m = M. The scale runs backwards: a smaller number is brighter, and the brightest objects go negative. Five magnitudes is a factor of 100 in flux, because 10^0.4·5 = 100. Slide the distance and M and watch m move. Distance in parsecs is the reciprocal of parallax in arcseconds, the rule on the parallax desk. The nearest stars are the neighborhood that yardstick measures, and the stars desk is what their letters mean.

Classroom schematic. Visual (V) band: Johnson V for the star ticks, and the visual magnitudes printed on the NASA fact sheets. Extinction is ignored. Dust between us and a star would make m fainter than this formula. The naked-eye lines are approximate ranges for an average person, not a forecast for a city or a night. Why a city sky is brighter is the light-pollution desk. Not a research catalog, not a finder chart, and not a sky survey.

Interactive · The scale runs backwards

Smaller numbers are brighter · negatives included · tap a tick or a button

Cited ticks · approximate ones say so

The sliders need JavaScript. The cited ticks are still in the sources below: Sun −26.74, full Moon −12.74 (mean opposition), Venus −4.8 at brightest, Sirius −1.46, Vega 0.03, Polaris 2.02 (variable), an inner-city limit near 4.0, and an ideal dark-sky limit near 6.0 to 6.5.

Interactive · Five magnitudes, a factor of 100

Δm → flux ratio · both formulas · and the inverse

A positive Δm is that many magnitudes fainter

Δm

Ratio

Interactive · Distance modulus

m − M = 5 log10(d / 10 pc) · extinction left out

Distance and absolute magnitude · each one moves m

Distance

Interactive · Move the Sun

Absolute magnitude fixed at the fact sheet’s +4.83 · slide the distance

Where we put the Sun · not a catalog of real suns

Distance

Why the numbers run backwards

Brighter is smaller

Hipparchus sorted stars into six classes, 1 the brightest and 6 the faintest. Pogson kept that order and made it a logarithm: a gap of five magnitudes is a factor of 100 in flux. A smaller m is the brighter object. Past the bright end the numbers go negative. The Sun at −26.74 is not “minus bright.” It is so much flux that the old classes ran out of room.

Absolute magnitude uses the same backwards scale, after the star is moved to 10 pc in the formula. A star can look faint because it is far, and still have a small or negative M. Extinction is a third dimming, and this page leaves it out.

Literacy · words on the scale

Apparent magnitude m: How bright the object looks from the observer. Visual magnitude on this page is the V band, not a bolometric sum over every wavelength.
Absolute magnitude M: The apparent magnitude the object would have at 10 parsecs, with extinction left out. NASA’s Marshall activity sheet calls that standard distance 10 parsecs, about 32.6 light-years. The Sun’s fact-sheet value is +4.83.
Distance modulus: m − M = 5 log₁₀(d / 10 pc). Flux falls as 1/d², and magnitudes are a base-10 log of flux, so the distance shows up as that factor of 5. At d = 10 pc the log is zero and m = M. Dust is not in the formula.
Flux ratio: 10^0.4·Δm, which is the same number as 100^Δm/5. Δm = 5 gives 100. Δm = 1 gives about 2.512. The inverse is Δm = 2.5 log₁₀(ratio).
Parallax tie-in: d in parsecs is 1 divided by the annual parallax p in arcseconds. At 10 pc, p = 0.1″. The parallax desk draws that angle. Very small distances make a huge angle; that is not a stellar parallax.
Naked-eye limit: The faintest star an average person can see, as a magnitude. NASA’s Imagine the Universe says that under ideal conditions that is about 6.0 to 6.5. Sky & Telescope’s account of Bortle class 9 puts an inner-city sky at about 4.0 or brighter. Both are approximate. The light-pollution desk is why the city number is brighter.

Sources · public NASA, ESA, SIMBAD, IAU

Cited, not invented

NASA NSSDCA — Sun Fact Sheet — Sun column, visual magnitude V(1,0) −26.74. Absolute magnitude +4.83. The Earth column on the V(1,0) row is −3.86 and is not used here.
NASA NSSDCA — Moon Fact Sheet — Mean values at opposition: distance 378,000 km, apparent visual magnitude −12.74. Marked approximate for a given full Moon.
NASA NSSDCA — Venus Fact Sheet — Maximum visual magnitude −4.8. One decimal, so the tick is approximate.
Ducati, J. 2002 — Catalogue of Stellar Photometry in Johnson’s 11-color system (VizieR II/237, 2002yCat.2237....0D), as listed by SIMBAD: * alf CMa (Sirius) V = −1.46, * alf Lyr (Vega) V = 0.03, * alf UMi (Polaris) V = 2.02. SIMBAD types Polaris cC*, a classical Cepheid, so 2.02 is marked approximate.
NASA Night Sky Network — Check your sky quality with Orion — A smaller magnitude is brighter. Most visible stars run from 1st to 6th magnitude. A magnitude 1 star appears 100 times brighter than a magnitude 6 star. Sirius is given as −1.46. Venus can shine brighter than −4.
NASA Marshall — Star Light, Star Bright — Five magnitudes, a 100:1 flux ratio. Absolute magnitude is brightness from 10 parsecs, about 32.6 light-years. The Sun numbers in that student sheet are a coarser rounding. This page uses the fact sheet’s −26.74 and +4.83.
NASA GSFC Radio JOVE glossary — Absolute magnitude: the apparent magnitude at 10 parsecs (32.6 LY). 1 parsec = 3.26 light-years.
NASA GSFC Imagine the Universe — Ask an Astrophysicist (archived night-sky answers) — Under ideal conditions the average person should see stars as faint as magnitude 6.0 to 6.5. Pogson’s scale. The same archive says Vega defines magnitude 0.0, which is the historical zero. The catalog V used here is 0.03.
ESA Science & Technology — Stellar distances — d(parsec) = 1 / p(arcsecond). 1 parsec = 3.26 light-years. Sirius is the worked example at parallax 0.37921″. The Sirius-distance button uses that parallax, then this page’s Ducati V, and prints the absolute magnitude the formula returns. It is arithmetic, not a second catalog.
ESA Science & Technology — Teacher’s notes, stellar distances — Absolute magnitude is the apparent magnitude at 10 parsecs. M = m − 5 log(D/10), with D in parsecs. The logarithm is base 10. Their printed Sirius magnitude is −1.44. This page keeps Ducati’s −1.46.
IAU 2012 Resolution B2 — The astronomical unit is exactly 149597870700 m. The light-year switch uses 1 pc ≈ 3.2616 ly: four-decimal rounding of the parsec (648000/π astronomical units, one arcsecond of parallax) divided by light-travel in a Julian year, 365.25 × 86400 s at 299792458 m/s. The unrounded quotient is 3.261563777…. NASA and ESA round the same length to 3.26 ly. The slider uses 3.2616 so a typed light-year matches the sentence.
Sky & Telescope — How dark is your night sky? — Bortle class 9, an inner-city sky, has a limiting magnitude of 4.0 or less. Approximate. The idea of a naked-eye limit under a bright sky is the light-pollution desk.

Classroom picture of a backwards scale. Δm = 5 gives a flux ratio of 100. At 10 pc, m = M, so the Sun’s +4.83 is the apparent magnitude at that distance. At 1 AU the same formula gives −26.742, which rounds to the fact sheet’s −26.74. Extinction is ignored. Visual band. Not a catalog. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

Yardstick

Magnitude desk

A classroom picture. Visual band. Extinction is ignored. Not a catalog and not a forecast.

The number line

The number line
Direction	Brighter is toward smaller numbers, including negatives. Fainter is toward larger numbers.
Ticks	Each button is a cited value. Approximate or variable ones say approx. The ideal dark-sky tick is a short range, 6.0 to 6.5.
Tap	Tap the line near a tick, or use the buttons. Left and Right move through the buttons when one of them has focus.

Flux ratio

Flux ratio
Δm	A magnitude difference. Positive means that many magnitudes fainter. The ratio is 10^(0.4·Δm), also written 100^(Δm/5). Δm = 5 gives 100.
Ratio field	The inverse. Type a flux ratio greater than zero and Δm becomes 2.5 log10 of that ratio.
Sirius vs Vega	Δm is Vega’s V minus Sirius’s V, 0.03 − (−1.46) = 1.49. Sirius is the brighter one.

Distance modulus

Distance modulus
Distance	Parsecs, light-years, or AU. 1 pc ≈ 3.2616 ly on this desk. The slider runs from a millionth of a parsec to 1,000 pc. The number field accepts a wider span.
M	Absolute magnitude. The Sun preset is the fact sheet’s +4.83.
10 pc	m equals M. The modulus is zero.
Parallax	p in arcseconds is 1/d with d in parsecs. At solar-system distances that angle is huge. The parallax desk is the stellar case.
Sirius	Distance from ESA’s 0.37921″. M is computed so the apparent magnitude comes back as Ducati’s −1.46. Arithmetic, not a catalog absolute magnitude.

Move the Sun

Move the Sun
Fixed M	+4.83, the Sun Fact Sheet. Only the distance moves.
1 AU	Our sky. The formula gives about −26.742, which rounds to the sheet’s −26.74.
10 pc	m = +4.83. Fainter than the inner-city line near 4, brighter than the ideal line near 6.
m = 4 and m = 6	The distances where this Sun would sit on those two naked-eye lines. Approximate comparison, not a forecast.

Keyboard and touch

Keyboard and touch
Sliders	Arrow keys nudge the focused slider. The number field is where a cited figure is typed exactly.
Buttons	Enter or Space selects. Left and Right step through that group. The pressed button is aria-pressed.
Touch	Buttons, the question mark, Close, and the fields are at least 44 pixels on a side.
This dialog	The question mark opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Magnitude
the backwards scale

Brighter is smaller

Cited, not invented

Parallax

Nearest stars

Stars

Light pollution

Solar map

Local sky

Magnitude desk

The number line

Flux ratio

Distance modulus

Move the Sun

Keyboard and touch

Magnitudethe backwards scale

Brighter is smaller

Cited, not invented

Parallax

Nearest stars

Stars

Light pollution

Solar map

Local sky

Magnitude
the backwards scale