Star magnitude calculator

What is the formula for apparent magnitude difference?

m₁ − m₂ = −2.5 × log₁₀(F₁ ÷ F₂), where F is the received flux (brightness). A difference of 5 magnitudes is exactly a factor of 100 in brightness, so one magnitude is a factor of 100^(1/5) = 2.512. A star of magnitude 1 is therefore 100 times brighter than one of magnitude 6.

Why do brighter stars have smaller magnitudes?

The scale is historical and runs backward: the larger the magnitude number, the fainter the object. OpenStax Astronomy puts it as the system that goes backward. Negative numbers are for the brightest objects.

What are apparent and absolute magnitude?

Apparent magnitude m is how bright a star looks from Earth. Absolute magnitude M is the apparent magnitude it would have at a standard distance of 10 parsecs (32.6 light-years), so M compares the stars’ real output without the effect of distance.

How does the distance modulus work?

m − M = 5 × log₁₀(d ÷ 10 pc), with d in parsecs. A star that appears 5 magnitudes fainter than its absolute magnitude is at 100 pc: m − M = 5 gives 100 pc, or 326 light-years. At 1,000 pc the modulus is 10. One parsec is 3.262 light-years.

What is the absolute magnitude of the Sun?

It depends on the band. The IAU's 2015 resolution on the bolometric scale sets M_bol = 0 at 3.0128 × 10²⁸ W, so the nominal solar luminosity 3.828 × 10²⁶ W corresponds to M_bol = 4.74. In visual light (the V band) a 2018 study by Willmer gives 4.81; some textbooks still use 4.83. Pick the one that matches the data you are comparing.

How faint a star can my telescope show?

A telescope collects light in proportion to its area, so a lens of diameter D gathers (D ÷ d)² times what a pupil of diameter d does, which is a gain of 5 × log₁₀(D ÷ d) magnitudes. With a naked-eye limit of 6.0 and a 6 mm pupil that is 2.1 + 5 × log₁₀(D in mm). A 4 inch (101.6 mm) telescope reaches about 12.1 and an 8 inch 13.6.

Is that limiting magnitude exact?

No, it is a rule of thumb for a dark sky and a young, dark-adapted eye. A 7 mm pupil instead of 6 mm lowers the 4 inch result to 11.8; a bright sky, a hazy night, an older eye or too much magnification lowers it more. The rules you meet online differ by half a magnitude or so because they assume different eyes.

The magnitude scale in one paragraph

The scale is defined by a ratio: 5 magnitudes is 100 times in brightness, so the ratio for one magnitude is 2.512. A difference of 1 gives a factor of 2.512, 5 gives 100 and 10 gives 10,000. The naked-eye limit under a dark sky is about magnitude 6 (OpenStax Astronomy, figure 17.2).

Worked example: distance from two magnitudes

A star has an apparent magnitude of 10 and an absolute magnitude of 5. The distance modulus is 5, so d = 10 pc × 10^(5÷5) = 100 pc, about 326 light-years. Because it is 10 times farther than the standard 10 pc, the light we receive is 100 times weaker than it would be at 10 pc, which agrees with the 5-magnitude difference. For luminosity: an absolute bolometric magnitude of 0 means 78.7 times the Sun’s luminosity (3.01 × 10²⁸ W); −5 means 7,870 times. The Sun itself at 4.74 is 3.83 × 10²⁶ W.

Limits of this method

The distance modulus assumes no dust between you and the star. Interstellar dust dims starlight, so a star looks fainter than its distance alone would explain and the simple formula returns a distance that is too large; astronomers add an extinction term for that. Magnitudes are also measured through filters: visual (V), blue (B) and others, and the bolometric scale covers all wavelengths. Mixing a V-band magnitude with a bolometric solar value gives a wrong luminosity, which is why the calculator asks for the band. Last, the telescope result is not a measurement but a scaling of the naked-eye limit by the light-collecting area.

Common mistakes

Reading the scale in the wrong direction is the first one: magnitude 2 is brighter than magnitude 4. The second is to use the distance in light-years in the modulus formula; it needs parsecs (1 pc = 3.26 ly). A third is using the base-e logarithm; the formulas here use base 10. And for telescopes people compare aperture in inches against a formula that wants millimeters.

Sources: OpenStax Astronomy 2e, section 17.1 “The Brightness of Stars” (magnitude scale, 5 magnitudes = 100:1, naked-eye limit near +6); IAU 2015 Resolution B2 on bolometric magnitude zero points (arXiv:1510.06262: L₀ = 3.0128 × 10²⁸ W, M_bol of the nominal Sun 4.74, parsec = 648,000/π au); C. Willmer, “The absolute magnitude of the Sun in several filters,” ApJS 236, 47 (2018), arXiv:1804.07788 (M_V = 4.81); distance modulus m − M = 5 log₁₀ d − 5 as given on the astro.wku.edu labs page “Magnitudes and distance” and the Swinburne Astronomy Online glossary. The telescope limit is derived here from collecting area, not taken from a table. Related: blackbody radiation calculator, redshift and Hubble’s law calculator, Schwarzschild radius calculator, orbital period calculator.