Colour and surface temperature

6.5 Colour and surface temperature

If one spends a little time observing the night sky, it soon becomes apparent that stars can have different colours: Betelgeuse in Orion and Aldebaran in Taurus have an orange tint, Capella, in Auriga, is somewhat yellow in colour and Rigel, also in Orion, is a slightly bluish white. Brocchi’s Cluster, commonly called ‘The Coathanger’, is not a true cluster, but a chance arrangement of stars called an ‘asterism’. Figure 6.3 shows that star colours can range from red through to white and blue.

The colours we perceive are a function of the surface temperature of the star. As the surface temperature increases from ∼3000 up to ∼20 000 K, the colour of the star moves from red, through orange, yellow and white to blue. Stars act as approximate black body radiators and, as was described in Chapter 2, both the peak wavelength and total power output of a black body are related to their temperature, so giving us two ways of estimating a star’s surface

temperature. Let us calculate the surface temperature of Proxima Centauri. From the calcu- lation above of the luminosity of Proxima Centauri relative to our Sun, and given the Sun’s luminosity of ∼4 ⫻ 10 26 W, it has a luminosity given by:

L ⫽ ∼4 ⫻ 10 26 /19 000 W 22 ⫽ 2.1 ⫻ 10 W

Figure 6.3 Impression of Brocchi’s Cluster. The eff ect of a ‘soft-focus’ fi lter has been used to spread the stellar images so that star colours are more easily seen.

The Properties of Stars

In 2002, the Very Large Telescope used a special technique (to be described later) to measure an angular diameter of 1.02 ± 0.08 milliarcsec for Proxima Centauri. Given its distance of ∼1.3 pc the actual diameter of Proxima Centauri can be cal-

culated to be about 1/7th that of the Sun. Using the Stephan–Boltzmann Law: L ⫽ σAT 4

⫽ 5.671 ⫻ 10 ⫺8 ⫻ 4 ⫻ π ⫻ (1 ⫻ 10 8 ) 2 ⫻T 4

So, T ⫽ {2.1 ⫻ 10 22 /[5.671 ⫻ 10 ⫺8 ⫻ 4 ⫻ π ⫻ (1 ⫻ 10 8 ) 2 ]} ⫺4

⫽ 1300 K One can use a somewhat simplifi ed approach relating the temperature to that of

the Sun: Proxima Centauri is ∼1/7th the diameter of the Sun, thus its surface area, A, is ∼1/49 times smaller. Its luminosity, derived above, is ∼19 000 times

less. From the Stephan–Boltzmann Law L is proportional to AT 4 , so the ratio of its surface temperature, T PC , to that of the Sun is given by:

T PC /T Sun ⫽ [(1/19 000)/(1/49)] ⫺4

So,

T PC ⫽ 0.22 ⫻ 5800 K

⫽ 1300 K

We can use the same method for Rigel.

Rigel is 62 times the diameter of the Sun, thus its surface area is ∼3800 times greater. Its luminosity, derived above, is ∼45 000 times greater. From the Stephan– Boltzmann Law, L is proportional to AT 4 , so the ratio of its surface temperature, T Rigel , to that of the Sun is given by:

T Rigel /T Sun ⫽ (45 000/3800) ⫺4

So,

T Rigel ⫽ 1.85 ⫻ 5800 K

⫽ 10 700 K

This basic idea can make some calculations very easy. Here is an example:

A star has twice the diameter of the Sun and twice its surface temperature. What is its luminosity compared with that of the Sun? The star’s surface area will be 2 2 or 4 times that of the Sun. Each square metre of its surface will radiate 2 4 or 16 times more energy. Thus, in total, the star’s luminosity will be 64 times that of the Sun.

Introduction to Astronomy and Cosmology