Measuring the distance to the stars
Nobody can stretch a tape measure to a star, but the Earth’s orbit provides a baseline 186 million miles wide. Over six months, a nearby star seems to shift slightly against far more distant stars. Half of that shift is the star’s parallax, and simple geometry turns it into a distance.
The formulas
Distance (parsecs) = 1 ÷ parallax (arcseconds)
With the parallax in milliarcseconds, as catalogs list it, the distance is 1000 ÷ parallax. One parsec equals 3.2616 light-years or about 30.9 trillion km.
Once the distance is known, the absolute magnitude M tells you how bright the star really is:
M = m − 5 log₁₀(d ÷ 10)
where m is the apparent magnitude and d the distance in parsecs. The luminosity compared with the Sun follows from the Sun’s absolute visual magnitude of 4.83:
L ÷ L☉ = 10^(0.4 × (4.83 − M))
Worked example: Sirius
Sirius, the brightest star in the night sky, has a parallax of 379.21 milliarcseconds and an apparent magnitude of −1.46:
- Distance: 1000 ÷ 379.21 = 2.637 parsecs, or 8.60 light-years (about 81 trillion km).
- Distance modulus m − M: −2.89, negative because Sirius is closer than 10 parsecs.
- Absolute magnitude: −1.46 + 2.89 = 1.43.
- Visual luminosity: about 22.8 times the Sun.
So Sirius looks so bright partly because it is close and partly because it really is a more powerful star than the Sun.
The magnitude scale
Magnitudes run backward: the smaller the number, the brighter the star. A difference of 5 magnitudes is exactly a factor of 100 in brightness, so 1 magnitude is a factor of about 2.512. The faintest stars visible to the naked eye under a dark sky are around magnitude 6.
Limits of the method
From the ground, the atmosphere blurs images and parallaxes were reliable only for a few thousand nearby stars. The Hipparcos satellite, launched in 1989, measured about 118,000 stars, and Gaia, launched in 2013, has measured parallaxes for more than a billion. For the brightest stars, Gaia reaches a precision of a few hundredths of a milliarcsecond.
The calculator gives visual luminosity, not total energy output, and ignores interstellar dust. Dust dims and reddens distant stars, which makes them look fainter than they are and leads to an underestimated luminosity. For very small parallaxes with large error bars, simply inverting the parallax also becomes biased, and astronomers use statistical methods instead.