The size of stars

6.10 The size of stars

6.10.1 Direct measurement

The angular sizes of relatively nearby stars can be measured directly. The dia- meter of our Sun, calculated earlier, comes from knowledge of its angular size and distance. There is only one star, Betelgeuse (a red supergiant in Orion), whose angular size can be directly observed with a normal telescope. In 1995, the Faint Object Camera of the Hubble Space Telescope (HST) was used to cap- ture the fi rst conventional telescope image of Betelgeuse and measured an angu- lar size of ∼0.05 arcsec (Figure 6.8). (Betelgeuse has a very diffuse outer envelope so it is rather hard to estimate the angular size.) The distance to Betelgeuse is not

precisely known but, if it is assumed to be 131 pc, then one can calculate the diameter:

d ⫽Dθ

Figure 6.8 Hubble Space Telescope image of Betelgeuse. Image: A. Dupree (CfA), NASA, ESA.

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With θ in radians and d (the diameter) and D (the distance) in kilometres we get: d ⫽ 131 ⫻ 3.1 ⫻ 10 13 ⫻ [0.05/(3600 ⫻ 57.3)] km

⫽ 4.1 ⫻ 10 15 ⫻ 2.4 ⫻ 10 −7 km

So the diameter is ∼1 billion km, about 700 times that of the Sun.

A method called optical interferometry uses two or more mirrors separated by distances of order tens of metres and combines their light to give the effect of one giant optical telescope so giving far higher resolution than single telescopes such as the HST which has a mirror of 2.4 m diameter. They have been used to measure the angular diameters of nearby stars, so that, given their distances as measured by the method of parallax, their diameters can be derived (Figure 6.9).

In 2002, the light from two 8.2-m telescopes of the Very Large Telescope (VLT) array in Chile was combined to form an interferometer with a baseline of 102.4 m. Its resolution was thus equivalent to an optical telescope ∼100 m in diameter. It measured an angular diameter of Proxima Centauri, the nearest star to the Earth,

Figure 6.9 An image of the double star Capella made with the COAST optical interferometer

The Properties of Stars

of 1.02 ⫹/⫺ 0.08 thousandth’s of an arcsecond – incidentally about the angular size of an astronaut on the surface of the Moon as seen from the Earth!

Proxima Centauri is a red dwarf star that lies at a distance of 1.3 pc so its angu- lar diameter is given by:

d ⫽ D θ (where θ is in radians) ⫽ 1.3 ⫻ 3.1 ⫻ 10 16 ⫻ 1.02/(1000 ⫻ 3600 ⫻ 57.3) m

⫽ 2 ⫻ 10 8 m

The Sun has a diameter of 1.4 ⫻ 10 9 m, so Proxima Centauri has a diameter ∼1/7th that of the Sun.

6.10.2 Using binary star systems to calculate stellar sizes

Over 80% of all stars are in binary or multiple star systems. If the plane of the orbit of a binary pair is close to the line of sight, each star will occult the other in turn and brightness of the system will be seen to drop. Such a system is called an eclipsing binary. If the orbital parameters of the system are known, the time during which the larger star is occulted by the smaller gives a measure of the diameter of the occulting star.

The most famous, and the fi rst to be discovered, eclipsing binary is the star sys- tem Beta Persei, Algol. It was called the ‘demon star’ as it appears to ‘wink’ – its brightness drops by 30% (from magnitude 2.1 down to 3.4) for a total of ∼10 h

precisely every 2.86739 days or ∼68.8 h (Figure 6.10).

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The two stars are separated by 0.062 AU or 9.275 million km. These facts allow us to estimate the diameter of the occulting star. The circumference of the orbit is 29.1 million km, (π ⫻ D), so if the eclipse lasts 10 h, which is 14.5% of the

68.8 h period, the diameter will be approximately (29.1 ⫻ 14.5)/100. This equals

4.2 million km. Our Sun has a diameter of 1.4 million km so the occulting star has a diameter ∼3 times that of our Sun.

6.10.3 Using the Stephan–Boltzman Law to estimate stellar sizes

Given a star’s position in the H–R diagram, one can use the Stephan–Boltzman Law to calculate its radius. The luminosity of a star compared with our Sun is pro- portional to the square of the star’s diameter relative to the Sun and to the fourth power of its temperature relative to the Sun’s temperature.

Rigel has an absolute magnitude of ⫺6.7, our Sun ⫹4.83, a difference of

11.53 magnitudes corresponding to a luminosity difference of:

Rigel has a surface temperature of 10 700 K, or 1.84 times that of the Sun. Each square metre of Rigel’s surface will thus radiate 1.84 4 or 11.5 times that of the Sun. Its surface area must thus be 41 000/11.5 or ∼3500 times that of the Sun, so its diameter will be 3500 1/2 , that is ∼59 times that of the Sun.

We can also use this method to derive a diameter of Betelgeuse. It is ∼60 000 times brighter than our Sun and its surface temperature is 3500 K which is 0.6 that of our Sun. Each square metre of Betelgeuse’s surface will thus radiate 0.6 4 or

0.13 times that of the Sun. Its surface area must thus be 60 000/0.13 or 460 000 times that of the Sun so its diameter will be 460 000 1/2 or ∼679 times that of the Sun. This agrees quite well with the direct measurement of ∼700 times that of our Sun derived earlier.

The star with the brightest apparent magnitude in the northern hemisphere is Sirius A, which is in a binary system with a white dwarf star Sirius B (Figure 6.11). Sirius A is ∼26 times brighter than our Sun and Sirius B is 416 times less bright than our Sun. Sirius A has a surface temperature of 90 K, whilst Sirius B has a surface temperature of 15 000 K. Each square metre of Sirius A’s surface will thus

radiate 1.72 4 or 8.7 times that of the Sun. Its surface area must thus be 26/8.7 or

3 times that of the Sun so its diameter will be 3 1/2 or ∼1.7 times that of the Sun. In contrast, each square metre of Sirius B’s surface will radiate (15 000/5800) 4 or

45 times that of the Sun. Its surface area must thus be (1/416)/45 or 5.3 ⫻ 10 ⫺5 times that of the Sun so its diameter will be (5.3 ⫻ 10 ⫺5 ) 1/2 or 0.007 times that of the Sun. White dwarf stars are very small and of comparable size with our Earth

The Properties of Stars

Figure 6.11 Hubble Space Telescope image of Sirius and its companion white dwarf star, Sirius B (lower left). Image: NASA, H.E. Bond, E. Nelan, M. Barstow, M. Burleigh, J.B. Holberg, STScI, U. Leicester, U. Ariz.

(Estimates of the surface temperature of Sirius B vary rather widely. Due to the 4th power dependence on temperature, this has a major effect on the calculated diameter.)

In general, the radii of stars on the main sequence range from ∼20 times that of the Sun at the upper left of the H–R diagram down to 0.1 that of the Sun at the lower right. Giant stars lie in the region of ∼10–100 times that of the Sun; an example is Aldebaran, in Taurus, which has a radius 45 times that of the Sun. Supergiant stars such as Betelgeuse often pulsate, its radius varying between around 700–1000 times that of the Sun with a period of ∼2100 days.

As we have seen, in complete contrast, the diameters of white dwarfs are com- parable with that of the Earth, not the Sun (Figure 6.12).