Luminosity Calculator

The luminosity calculator finds a star's luminosity, absolute magnitude, and apparent magnitude.

Luminosity3.828e+26 W1 times the Sun — L = 4πR²σT⁴, so temperature matters to the FOURTH power and radius only to the second

Peak wavelength502.0395 nmWien's law — visible

Absolute bolometric magnitude4.74the magnitude scale runs BACKWARDS — smaller is brighter, and five magnitudes is exactly a factor of 100

At twice the temperature16× the luminositythe fourth power, which is why a small hot star outshines a large cool one

At twice the radius4× the luminosityonly the square

Apparent magnitude4.74brighter than magnitude 6, so visible to the naked eye from a dark site

Distance modulus0m − M = 5log₁₀(d/10pc), which is zero at exactly 10 parsecs by definition

Flux at that distance3.199e-10 W/m²against 1361 W/m² of sunlight at Earth

Light travel time32.6156 years

Radiation pressure at that distance1.067e-18 PaI/c on a black surface, and DOUBLE that at 2.134e-18 Pa on a perfect mirror, because reflection reverses the photon momentum instead of just absorbing it

Transit depth10,092.17 ppm1.00922% of the star's light — (R_p/R_*)², a ratio of AREAS, which is why the depth does not depend on distance at all

Drop in magnitude during transit0.011013easily detectable from the ground

Planet radius69,890 km10.97 Earth radii, or 0.9786 Jupiter radii

Fraction of the stellar disc covered1.00922%the same number as the depth, because a transit measures exactly that

Drake equation estimate78 civilisationscommunicating civilisations in the galaxy right now, on the factors entered

What the Drake equation is fororganising ignorancethe last four factors are unmeasured and the answer spans twenty orders of magnitude across plausible inputs. Its value is in naming the unknowns, not in the number it produces

Lifetime dominatesN is LINEAR in Lif civilisations last ten times longer, ten times as many exist at once — the one factor we know least about is the one the answer is most sensitive to

The formula

L = 4πR²σT⁴; m − M = 5log₁₀(d/10pc); N = R*f_pn_ef_lf_if_cL

Temperature to the fourth power

A star's luminosity is 4πR²σT⁴. Radius enters as a square but temperature as a fourth power, so temperature dominates: doubling it multiplies the output by sixteen, while doubling the radius only quadruples it. This is why a small, hot blue star can outshine a far larger cool red one, and why the Hertzsprung–Russell diagram separates stars so cleanly.

The magnitude scale runs backwards

Smaller magnitudes are brighter, a convention inherited from Hipparchus ranking stars first, second and third class by eye. It was later made precise so that exactly five magnitudes is a factor of 100 in flux, making each magnitude a factor of about 2.512. Absolute magnitude is what a star would show at 10 parsecs, so the distance modulus m − M is zero there by definition.

The Drake equation organises ignorance

It is a chain of seven factors, of which the first three are now reasonably constrained by exoplanet surveys and the last four are essentially unknown. Across plausible inputs the answer spans twenty orders of magnitude, so it does not predict anything. Its value is in making the unknowns explicit — and in showing that N is linear in civilisation lifetime, meaning the factor we understand least is the one the result depends on most.