How do They Measure the Diameter of a Star - part ii

In the previous article we saw in the experiment by T Young how two rays of light were passed through two narrow slits, very close: together. The result obtained, was:

or = 2xd/D where is the wave length;
                                                  x = the distance to the first dark fringe;
                                                  d = distance between the two slits;
                                                  D = distance from the centre to the first dark fringe.

If x, d and D are measured the wave length , can be calculated. For example, if D = 5 metres; and d = 1 millimetre i.e. 10^-3 metres, and x is found to be 1,5 millimetre ( or 1,5 x 10^-3 metres ), then:

                                                 = 6000 x 10^-10 metres
                                                 = 6000 angstrom units.

This is the wave length of red light,

If the wave length on which the star radiates most strongly known, the distance d between the slits can be calculated if x and D are measured. d can then be the distance between two extremities of the star's disc, namely its diameter in circular measure (arc seconds) if the above mentioned formula is altered to . If the distance of the star from the Earth is known, its diameter in kilometres can be calculated.

In 1920 the new Hooker telescope of 2½ metres aperture on Mount Wilson was used for this purpose. A 6-metre long beam was placed across the top of the telescope mounting. Two plane mirrors A and B were placed at 45° and 3 metres apart so as to reflect the light coming from the edges of the star towards another two plane mirrors C and D also at 45° as shown in the diagram. A shield was placed across the top of the telescope so as to cut off the light from the centre of the star and to allow only the light from the edges of the star to reach the primary mirror. The rays of light from the two extremities of the diameter of the star thus followed the routes shown in the diagram. By moving the two mirrors C and D symmetrically away from or towards each other. Interference fringes were obtained in the false disc. This happened when the mirrors C and D were 114 centimetres apart .

The two rays of light were thus equivalent to the rays through the narrow slits. The interference fringes showed that the array was in alignment. A microscope with a magnification of 1600 x was used to measure the distance apart of the fringes. The star Betelgeuse formed fringes that were 0,005 millimetres apart. i.e. 0,0005 cm apart.

However, the disc of a star is not identical to the two narrow slits. Physicists proved that the circular measure of a stellar disc is equal to 1,22d, the Rayleigh limit, where d is the distance between the mirrors A and B and is the wave length of light used.

In the case of Betelgeuse the interference fringes just vanished when the distance between the mirrors A and B was 307,34 centimetres. To ensure that the vanishing of the fringes was not due to loss of alignment, the telescope was directed on to another star - reappearance of the fringes proved that the alignment had been maintained..

Before the angular diameter of Betelgeuse could be measured, the wave length of the star's peak radiation of light had to be determined. When CD was 114 cm, the distance apart of the fringes was 0,0005 cm.

= x/f where f is the focal length FO of the telescope, i.e. = 114 x 0,0005 ÷ 1000 where f, the focal length is 1000 cm. This gave a wave length of 0,000057, or 5700 x 10^-8 cm and this is the wave length of the orange-coloured light of Betelgeuse.

The angular diameter of Betelgeuse could thus be worked out from:

The Rayleigh formula   = 0,000 000 2262 radians
                                                              = 0,000 000 2262 x 206 265
                                                              = 0,04667 arc seconds
                                                              = 0,047 correct to two significant figures.

The distance of Betelgeuse from the Earth is 200 parsecs and therefore its diameter is 0,047 x 200 = 9,4 astronomical units (AU).

This equals 9,4 x 149 597 870 = 1406 x 106 km or slightly more than 1000 times the diameter of the Sun!

Diameters of stars have also been determined by means of occultations of stars by the Moon. Just as a star vanishes behind the disc of the Moon the same tell-tale interference fringes are formed in the light from the star. This method was perfected by P Bartholdi of Haute Provence Observatory.

D S Evans, T G Barnes and C H Lacy argued that when they determine the bolometric magnitude of a star, they are actually measuring the total light output of the star. By making use of the (V - R) colour index of the star, they could calculate the diameter of the star. By this means they could determine the diameters of small stars as easily as those of giants.

For example, they found that the diameter of the nearby star Epsilon Eridani is 1 200 000 km or 0,86 times that of the Sun. This fits very well with measurements made by the mass-luminosity, law.

Jan Eben van Zyl