Colour, temperature and bolometric correction

Builds on: Sections 4.3, 4.4 and 4.6
Reading time: 30 min
Exercises: 4
Olympiad problems: 2

Antares, the heart of Scorpius, is so red that its name means "rival of Mars", while Spica in Virgo shines blue-white. A camera with a blue filter and a yellow-green filter records the difference as two magnitudes for each star. Through the blue filter Antares is 1.87 magnitudes fainter than through the yellow-green one, while Spica is 0.24 magnitudes brighter. By Section 4.6, a star's colour is set by its temperature, so these two numbers are already a thermometer. In this section we define colour with filters, use it to find temperatures and the dust in front of a star, and turn the light seen through one filter into the light given out at all wavelengths.

4.7.1Colour index

Before you readMake a guess first

A star's magnitude through a blue filter, B, is 1.5 more than its magnitude through a yellow-green filter, V. Is the star hotter or cooler than the Sun?

Photometry measures magnitudes through standard filters (Section 5.8). The most used are the U, B and V filters of the Johnson system, centred near 365 nm in the near ultraviolet, 440 nm in the blue and 550 nm in the yellow-green. A magnitude measured through a filter is named after it, so $B$ is the blue magnitude and $V$ the visual one. The difference of two such magnitudes is a colour index, such as $B - V$ .

By (4.2), $B - V = - 2.5 \log_{10} (F_{B} / F_{V})$ plus a constant, so the colour index measures the ratio of the blue to the yellow-green light. The constant is fixed so that Vega, a white star at about 9600 K, has colour indices close to 0. A hot blue star gives relatively more blue light and has a negative $B - V$ , and a cool red star a large positive one. The colour index does not depend on the distance, since the distance dims both filters equally.

ResultTemperature from colour(4.13)

\begin{aligned} T \approx 4600 K \times ( & \frac{1}{0.92 (B - V) + 1.7} \\ + \frac{1}{0.92 (B - V) + 0.62}) \end{aligned}

$T$	the effective temperature of the star
$B - V$	its colour index, corrected for dust

This is a fit to real stars by C. Ballesteros (2012), good to a few hundred kelvin for ordinary stars from about 4000 to 10 000 K. A blackbody gives a slightly different relation, because the absorption lines in real spectra take more light from the blue than from the yellow-green. In a problem the link between colour and temperature is usually given as a table or a graph.

Quick checkThe Sun's temperature from its colour

The Sun has $B - V = 0.65$ . What temperature does (4.13) give?

4.7.2Reddening by dust

Dust between the stars dims blue light more than red, much as the air does (Section 4.4). A star seen through dust therefore looks redder than it is, and its measured $B - V$ is larger than its true, or intrinsic, value $(B - V)_{0}$ . The difference is the colour excess. Since the dimming and the reddening come from the same dust, the extinction in V is close to a fixed multiple of the colour excess.

ResultColour excess and extinction(4.14)

\begin{aligned} E (B - V) & = (B - V) - (B - V)_{0} \\ A_{V} & = R_{V} E (B - V) \end{aligned}

$E (B - V)$	the colour excess, the reddening caused by dust
$(B - V)_{0}$	the intrinsic colour index the star would have with no dust
$A_{V}$	the extinction in the V band, in magnitudes
$R_{V}$	the ratio of total to selective extinction, about 3.1 for the dust of our Galaxy

The intrinsic colour comes from the star's spectrum, which shows its temperature through its lines (Section 4.9). With

A_{V}

found from the colour, the distance modulus of Section 4.3 gives the distance, corrected for dust.

Example 4.13A star behind dust

A star of spectral type A0, a white star near 9600 K (Section 4.9), should have $(B - V)_{0} = 0.00$ , but it is measured with $B = 10.80$ and $V = 10.30$ . Taking $R_{V} = 3.1$ , find the extinction and the magnitude $V_{0}$ the star would have without dust.

Solution

The measured colour is $B - V = 0.50$ , so by (4.14), $E (B - V) = 0.50 - 0.00 = 0.50$ and

A_{V} = 3.1 \times 0.50 = 1.55 .

Without dust the star would be brighter by $A_{V}$ , so $V_{0} = 10.30 - 1.55 = 8.75$ .

AnswerThe dust dims the star by 1.55 magnitudes in V, and without it the star would have

V_{0} = 8.75

4.7.3Bolometric magnitude

A magnitude through a filter measures only the light in that band. The bolometric magnitude $m_{bol}$ measures the flux at all wavelengths together, and the absolute bolometric magnitude $M_{bol}$ measures the luminosity of (4.12). The difference between the bolometric and the visual magnitude is the bolometric correction.

ResultBolometric correction and luminosity(4.15)

\begin{aligned} M_{bol} & = M_{V} + B C \\ \frac{L}{L_{⊙}} & = 10^{- 0.4 (M_{bol} - 4.74)} \end{aligned}

$M_{bol}, M_{V}$	the absolute bolometric and visual magnitudes
$B C$	the bolometric correction, which depends on the temperature of the star
$4.74$	the absolute bolometric magnitude of the Sun

The same correction links apparent magnitudes,

m_{bol} = V + B C

. With this convention

B C

is negative or close to zero. It is near zero for stars of about 6000 to 7500 K, for which the V band catches the largest share of the total light, and large and negative for very hot stars, which give out most of their light in the ultraviolet, and for very cool ones, which give it out in the infrared. For the Sun,

B C = 4.74 - 4.83 = - 0.09

Example 4.14A hot starfinish it

A star of 25 000 K has absolute visual magnitude −3.0 and bolometric correction −2.4. What is its luminosity in units of the Sun's?

The start

By (4.15), $M_{bol} = - 3.0 - 2.4 = - 5.4$ .

Your stepNow find

L / L_{⊙}

solar luminosities

Common errorsSection 4.7

A larger $B - V$ means a redder, cooler star. The colour index says nothing about how bright a star is.
Dust makes a star look both fainter and redder. Its measured colour must be corrected for the colour excess before it is used to find the temperature.
The bolometric magnitude of the Sun, 4.74, is not its visual absolute magnitude, 4.83. Luminosities from bolometric magnitudes must be compared with the bolometric value.

ExercisesSection 4.74 questions

4.7.1
A star has $B = 7.45$ and $V = 7.15$ . What is its colour index $B - V$ ?
4.7.2
A star has colour excess $E (B - V) = 0.30$ . What is its extinction in V, with $R_{V} = 3.1$ ?
magnitudes
4.7.3
A star has absolute bolometric magnitude −0.26. How many times as luminous as the Sun is it?
times
4.7.4
Which star has the most negative $B - V$ ?

Olympiad problemsPast IOAA problems, restated2 problems

2013What kind of star is it?10 points

Adapted from IOAA 2013 (Volos), theory problem S7. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

A star has visual apparent magnitude 12.2, parallax 0.001″ and effective temperature 4000 K. Its bolometric correction is −0.6. (a) Find its luminosity in units of the Sun's. (b) Is it a red giant, a blue giant or a red dwarf?

I solved it without the solution

AnswerThe star is about 18 times as luminous as the Sun and about nine times its size, so it is a red giant.

2008A star seen in three colours

Adapted from IOAA 2008 (Bandung), theory problem L2. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

A star is measured in the U, B and V filters as $U = 8.15$ , $B = 8.50$ and $V = 8.14$ . Its spectrum shows that its intrinsic colour is $(U - B)_{0} = - 0.45$ . The star has radius $2.3 R_{⊙}$ , absolute bolometric magnitude −0.25 and bolometric correction −0.15. Take $R_{V} = 3.2$ , and use the relation $E (U - B) = 0.72 E (B - V)$ between the two colour excesses. Find:

(a) the magnitudes $U_{0}$ , $B_{0}$ and $V_{0}$ the star would have without dust,

(b) its effective temperature,

I solved it without the solution

AnswerWithout dust the star would have $U_{0} = 7.47$ , $B_{0} = 7.92$ and $V_{0} = 7.70$ . Its effective temperature is about 12 000 K and it is about 360 pc away.