The magnitude scale

Builds on: Sections 4.1, O1.1 and S3
Reading time: 40 min
Exercises: 4
Olympiad problems: 1

The Sun has magnitude −26.74 and the full Moon −12.74. Fourteen magnitudes separate them, so the Sun gives about 400 000 times as much light as the full Moon, even though both look much the same size in the sky. The scale that makes this so goes back more than two thousand years, to Greek astronomers who called the brightest stars "of the first magnitude" and the faintest they could see "of the sixth". In this section we turn their scale into exact numbers, use it in both directions, and see why the magnitude of two stars seen together is not the sum of their magnitudes.

4.2.1Pogson's rule

Before you readMake a guess first

The full Moon is 14 magnitudes fainter than the Sun. Roughly how many times less light does it give?

The eye judges brightness by ratios. A star that gives twice the light of another looks brighter by the same step whether the two are bright or faint. In 1856 Norman Pogson made the old scale exact by declaring that a difference of 5 magnitudes is a factor of exactly 100 in flux. One magnitude is then a factor of $100^{1 / 5} = 2.512$ , and a difference in magnitude is a ratio of fluxes on a logarithmic scale, with the minus sign because brighter stars have smaller magnitudes.

ResultPogson's rule(4.2)

\begin{aligned} m_{1} - m_{2} & = - 2.5 \log_{10} \frac{F_{1}}{F_{2}} \\ \frac{F_{1}}{F_{2}} & = 10^{- 0.4 (m_{1} - m_{2})} \end{aligned}

$m_{1}, m_{2}$	the magnitudes of two objects, measured in the same band
$F_{1}, F_{2}$	their fluxes in that band

The rule gives only differences. A zero point, fixed by agreement for each band, turns a ratio of fluxes into a magnitude. Vega, for example, has magnitude very close to 0 in the V band. The two forms are the same rule read in opposite directions, and Section S3 gives the logarithms behind them.

Quick checkThe faintest star against the brightest

The faintest stars the naked eye can see have magnitude 6, and the brightest star in the night sky has magnitude −1.5. What is the ratio of their fluxes, faint to bright?

Figure 4.1From the Sun to the faintest galaxiesinteractive

First objectSecond object

Difference in magnitude
Ratio of fluxes

The figure shows the magnitude scale from −28 at the top to +31 at the bottom, with the magnitudes of some familiar objects, brightest at the top. The two lists choose two objects, the heavy bar joins them on the scale, and the table gives the difference in magnitude and the ratio of their fluxes by (4.2). Choosing gives the factor of about 400 000 above, a factor of about 950, and a factor of about

5 \times 10^{22}

4.2.2Adding the light of two stars

Two stars too close together to be told apart look like one star. Their light adds, so their fluxes add, but their magnitudes do not, because a magnitude is a logarithm. We turn each magnitude into a flux, add the fluxes, and turn the sum back into a magnitude.

ResultMagnitude of two stars together(4.3)

m = - 2.5 \log_{10} (10^{- 0.4 m_{1}} + 10^{- 0.4 m_{2}})

$m$	the combined magnitude of two stars seen as one
$m_{1}, m_{2}$	the magnitudes of the two stars

The combined magnitude is always a little smaller, that is brighter, than the brighter of the two. Two equal stars together are

2.5 \log_{10} 2 = 0.75

magnitudes brighter than either alone.

Example 4.3A binary of magnitudes 1.0 and 2.0

The two stars of a binary have magnitudes 1.0 and 2.0. What is their combined magnitude when they are seen as one star?

Solution

By (4.3),

\begin{aligned} m & = - 2.5 \log_{10} (10^{- 0.4} + 10^{- 0.8}) \\ = - 2.5 \log_{10} (0.398 + 0.158) = 0.64 . \end{aligned}

AnswerThe pair has magnitude 0.64, brighter than either star alone and far from 3.0, the sum of the two magnitudes. This is a problem from IOAA 2010, theory problem S1, which asked how bright the pair can appear at most, when neither star hides the other.

Example 4.4A nova brightensfinish it

A star of magnitude 15.0 flares up as a nova and reaches magnitude 5.0. By what factor did its flux grow?

The start

The magnitude fell by $15.0 - 5.0 = 10.0$ , which is two steps of 5 magnitudes.

Your stepNow turn this difference into a ratio of fluxes.

times

Common errorsSection 4.2

A larger magnitude means a fainter object. A star of magnitude 3 is fainter than one of magnitude 1, by a factor of ${2.512}^{2} = 6.3$ .
The magnitudes of two stars seen together cannot be added. Their fluxes add, and the combined magnitude comes from the sum of the fluxes.
A difference of 1 magnitude is a factor of 2.512 in flux, so a difference of 3 magnitudes is a factor of ${2.512}^{3} = 15.9$ . Each magnitude multiplies the ratio again by 2.512.

ExercisesSection 4.24 questions

4.2.1
How many times more flux does a star of magnitude 2 give than a star of magnitude 7?
times
4.2.2
Two stars of magnitude 0.00 are seen together as one. What is their combined magnitude?
4.2.3
A star has twice the flux of a star of magnitude 4.25. What is its magnitude?
4.2.4
A cluster holds 100 equal stars of magnitude 10. What is the magnitude of the whole cluster, seen as one patch of light?

Olympiad problemsPast IOAA problems, restated1 problem

2013Splitting the light of a binary10 points

Adapted from IOAA 2013 (Volos), theory problem S8. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

A binary star cannot be split in a telescope and is seen from Earth as one star of magnitude 5.00. One of its stars is twice as bright as the other. What are the magnitudes of the two stars?

I solved it without the solution

AnswerThe brighter star has magnitude 5.44 and the fainter 6.19.