Star brightness and magnitude

The backwards scale

Around 130 BC the Greek astronomer Hipparchus sorted the visible stars into six ranks. The brightest he called first magnitude, and the dimmest a naked eye can catch, sixth. The scale stuck. Modern astronomy is therefore saddled with a system in which brighter means a smaller number. Once telescopes and detectors made brightness measurable the ranks were pinned down, and apparent magnitude m became the measure of how bright an object looks from Earth. Objects brighter than first magnitude simply continue through zero into negative numbers.

The eye judges brightness on a squashed, roughly logarithmic scale, so Hipparchus's equal-feeling steps are really equal ratios. The modern definition makes that exact. Five magnitude steps correspond to a brightness ratio of exactly 100, so one magnitude step is a ratio of 2.51, the fifth root of 100. Brightness here means the intensity arriving at your detector. The unaided eye's judgement is subjective, and tying the scale to measured ratios is what removed the subjectivity.

Luminosity and radiant flux

Underneath the magnitude language sit two physical quantities worth separating. A star's luminosity L is the total power it radiates, in watts. What reaches a detector a distance d away is the received flux, the power per square metre, strictly a flux density or irradiance; some boards' papers name it radiant flux intensity. Geometry alone dilutes it:

the star's whole output spread over a sphere of radius d. Measure F, and the inverse square gives the distance the moment anything tells you L. Objects whose luminosity is known independently are called standard candles, and the usual examples are certain pulsating stars and, further out, type Ia supernovae. Spot one in a distant galaxy, compare its known L with its measured F, and the galaxy's distance follows. CIE frames the whole distance ladder in exactly these terms.

Three ways to say how far

Astronomy keeps three units of distance, each built for its own scale. The astronomical unit (AU) is the mean Earth-Sun distance, 1.50 × 10¹¹ m. The light year (ly) is the distance light travels in a year, 9.46 × 10¹⁵ m. The third is subtler and matters most for this unit: watch a nearby star from opposite ends of the Earth's orbit and it appears to shift slightly against the distant background, an effect called parallax. The smaller the shift, the further the star.

One parsec (pc) is the distance at which one AU subtends an angle of one arcsecond, a 3600th of a degree. The name is the definition compressed, parallax of one arcsecond. Work the triangle through and 1 pc = 3.08 × 10¹⁶ m, which is 3.26 light years. The unit justifies itself twice over. Real parallax measurements arrive in arcseconds, and the magnitude equation below is built around the parsec.

The parallax angle p of a star is the angle that one AU subtends at it, which is half the total sideways shift the star makes over six months. Because the parsec was defined to make that triangle come out at one, the distance follows with no constant to carry. OCR sets the relation as an equation, and no other board does:

p in seconds of arc, d in parsecs, and those units are not a footnote to the equation, they are the equation. Written in radians and metres the same physics needs the awkward constant that the parsec was invented to absorb, so p = 1/d is simply false in any other pair of units. Quote them every time you quote it.

Put numbers through it and the scale of the sky comes out. A star with a parallax of 0.10 arcseconds lies at 1/0.10 = 10 pc; halve the parallax to 0.050 arcseconds and the star is twice as far, at 20 pc. Proxima Centauri, the nearest star, holds the record at 0.77 arcseconds, which puts it at 1.3 pc, or 4.2 light years. Nothing lies closer, so every stellar parallax ever measured is below one arcsecond, and they shrink fast: the method runs out of precision within a few thousand parsecs, and the standard candles of the next lessons take the distance ladder on from there.

Absolute magnitude: the level playing field

Apparent magnitude mixes up two different things, how much light a star pours out and how far away it happens to sit. A feeble nearby star can outshine a giant across the galaxy. To compare stars fairly, imagine picking every one up and placing it at the same distance, 10 parsecs. The apparent magnitude a star would have at 10 pc is its absolute magnitude M, a like-for-like measure of the star's own output. The two magnitudes are linked by the star's actual distance d, in parsecs:

The logic sits in the signs. A star nearer than 10 pc has d/10 below 1, the log goes negative, and m comes out smaller than M. Moving a star closer than its standard shelf makes it look brighter. Further than 10 pc, the log is positive and the star looks dimmer than its absolute magnitude. The difference m − M is called the distance modulus, because knowing it gives d.