Sizes of the Stars


Introduction

In 1920, A. A. Michelson and F. G. Pease measured the angular diameter of Betelgeuse, α Orionis, with the 100-inch reflector at Mount Wilson. From its distance, they directly inferred its diameter, confirming that it was a huge star. This was the first direct measurement of stellar diameter; all other methods had been indirect and subject to uncertainty. Attempts to enlarge the phase interferometer to make the method applicable to a larger number of stars undertaken by Pease in later years were unsuccessful. In 1956 Hanbury Brown and Twiss applied a method they had devised for radio astronomy to visual astronomy and measured the angular diameter of Sirius, α Canis Majoris. This was the intensity interferometer, which removed most of the limitations of the phase interferometer, allowing measurements on a much larger sample of stars. A large interferometer was built at Narrabri, NSW, Australia, which finally provided a number of accurate stellar diameters by direct measurement after 1968.

These important and interesting developments are largely ignored in astronomy textbooks. The Michelson experiment is usually only briefly acknowledged, and the Hanbury Brown inteferometer is not mentioned at all. One reason for this is the difficulty of explaining the method, especially without mathematics. Optics texts usually give a reasonable explanation, because the measurements are valuable examples of the wave nature of light, but the astronomy is slighted. For these reasons, I will attempt to give a thorough explanation of the methods of measuring stellar diameters by interferometry, together with the important dusky corners that they illuminate. First, we must review how stellar distances are found, which is a fundamental task of astronomy.

Stellar Distances

When we look at the stars through the years, we are impressed by their fixity, at least on human scales of time. Since Ptolemy looked at the stars, they have retained their places, except perhaps for slight differences in the case of a few stars such as Arcturus, which has moved about 1° since then. The lack of movement through the year, as the earth orbits the sun, and in secular time, can be considered evidence either of the fixity of the earth (if the stars are considered scattered in space) or that they are all at the same distance, or that they are extremely distant. Most old cosmologies put the stars on a spherical surface, the firmament, with heaven beyond, so their fixity was no problem. Other old cosmologies, which had stars scattered in space, interpreted the fixity as due to the fixity of the earth, though a few philosophers thought the stars greatly distant, a most unpalatable thought for most thinkers. There was no way to select between the alternatives.

As soon as the revolution of the earth about the sun was accepted, and the firmament banished, evidence for the apparent displacement of the nearer stars relative to the more distant, called parallax, was sought. Parallax is illustrated in the diagram at the right, together with some other definitions. A and B are the positions of the earth at times six months apart, so the base line AB is two astronomical units, 2a = 2 AU. The parallax is traditionally measured in arc-seconds, often written p" to emphasize the fact. The number 206,265 is the number of seconds in a radian. The distance from the sun O to the star S is d, measured in parsecs or light years. The light-year was originally for public consumption, to emphasize the inconceivably great distances involved, but remains a vivid and common unit.

Parallax, however, was not observed--a most unsatisfactory condition. William Herschel looked for it in vain. Not until 1838 was parallax finally found. Bessel thought that a star with large proper motion might be relatively close, so he chose 61 Cygni, a fifth-magnitude star in Cygnus, which appeared on a dense background of Milky Way stars that could form a good reference. By visual observation, using a special measuring instrument, he found a parallax of 0.296", which corresponded to a distance of 3.4 psc or 11 l.y. In the same year, Struve found 0.124" for Vega, α Lyrae, and Henderson 0.743" for α Centauri. These were huge distances, but were only to the apparently nearest stars, which surprised everyone. Henderson actually chose one of the nearest stars of all. Only dim, red Proxima Centauri has a larger parallax, at 0.786". It is about 3 l.y. distant.

The introduction of photography made it much easier to measure parallaxes. It was only necessary to compare plates taken 6 months apart, and special intstruments were developed to facilitate the task. When the two plates were presented alternately to the eye, nearby stars would jump back and forth, while the distant ones remained unmoved. In this way a large list of trigonometric parallaxes were determined, but they covered only the stars close to us, up to perhaps a parallax of 0.1", or a distance of 33 l.y.. These are merely our close neighbors in the vastness of space.

A star appears dimmer the farther it is away, according to m = M + 5 + 5 log p", where m is the visual magnitude of the star (smaller numbers mean brighter). Putting in p" = 0.1, we find m = M, called the absolute magnitude, the apparent magnitude if the star were viewed at a standard distance of 10 parsec. If by some magic we could infer the absolute magnitude M of any star we observed to have an apparent magnitude m, then the parallax could be determined by this formula based on the inverse-square law. There are many uncertainties here, the major ones the effect of interstellar absorption and, above all, the determination of M.

The spectroscope shows that the spectra of stars can be classified into a regular series based principally on surface temperature: the types O, B, A, F, G, K, M and C, each quite recognizable. The Sun has a type G spectrum, and a surface temperature of 5800-6000K, for example. By classifying the spectra of the sample of stars whose distances we know by trigonometrical parallax, it was found by Hertzsprung and Russell that they roughly followed a single path on a plot of absolute magnitude M against spectral type, called the main sequence. Many stars did not follow this path, but were usually easy to recognize as different. If a star was presumed to be on the main sequence by all available evidence, then a look at its spectrum established its spectral class, and the Hertzsprung-Russell diagram gave an estimate of its absolute magnitude. Then its parallax, and so its distance, could be estimated. This is called a spectroscopic parallax, which extended our knowledge of stellar distances to really large distances, albeit approximately.

Any good astronomy text will show how larger and larger distances were estimated by other methods, such as Cepheid variables (the period gave a clue to the absolute magnitude, and these were very bright stars that could be seen a long ways) and the red shift, which extended distance knowledge far into the realm of the galaxies. However, trigonometric and spectroscopic parallaxes are sufficient for our present purposes. Accurate determination of distance is essential to accurate determination of stellar diameter.

It is worth remembering that no star has a parallax as large as 1", and that spectroscopic parallaxes are uncertain and subject to revision. For example, the parallax of Betelgeuse, α Orionis, was taken as 0.018" in 1920, but now a figure of 0.0055" is accepted. Of course, the contemplation of the vast distances in space is an essential part of the appreciation of astronomy, so very different from the views popularized by fiction, in which the Enterprise flits about space like a Portuguese trader in the Indian Ocean. Betelgeuse is 411 l.y. away. To get there about now, at the speed of light, you would have had to have departed when Shakespeare was born.

Interference

The other important preparation for our task is the understanding of inteference, interferometry and coherence. This is a big job, which requires reference to Optics texts for a thorough attack. Here we can only present the fundamentals in an abbreviated form. This should be sufficient for our purpose, however.

Observation teaches that if one candle gives a certain amount of illumination, then two candles give twice as much. The energy comes out on the light rays and just piles up where it is received, like so much snow. This reasonable and common-sense view is, of course, totally wrong, like the concept that matter is continuous, like cheese. It is useful in practice, but does not lead to understanding, only off into the weeds of speculation.

The truth is that light has an amplitude that moves on propagating wavefronts from its source. The amplitudes from different sources add at any point, and the energy received is the average value of the square of the resultant amplitude. Any effects caused by the adding of amplitudes are traditionally called interference, but amplitudes do not "interfere" with each other in the usual sense of the word, but seem blissfully independent of each other. We find that the wavelength of the amplitudes is quite small, only 500 nm for green visible light. Combined with the large velocity of light, 3 x 108 m/s, it turns out that the frequency is 0.606 x 1015 Hz, a really large value. This makes it impossible to observe the amplitude directly. All we can measure are time averages of functions of the amplitude. If A is the amplitude, then the intensity I = <AA*> is one such function. The angle brackets imply a time average over some suitable interval, much longer than the period of the oscillation of the amplitude. There are units to be considered, which introduce numerical factors, but we shall usually neglect them.

An amplitude can be represented by A(t) = Ae2πiνt aee2πiνt, where the complex amplitude A has been given in polar form, with an amplitude a and phase φ. Unfortunately, we have to use the same word for what we have called a generalized amplitude and the modulus of a complex number. It would have been better to call a the modulus, but this is not usually done. The two meanings for "amplitude" are not easy to confuse, fortunately. This form of the amplitude is not typical of most light sources, and is a kind of idealization. However, it is approached closely by laser light, so it is easy for us to experience. When we do use laser illumination, the light-as-snow illusion is shattered, and there are fringes and spots everywhere. These are, of course, results of interference, and show that our amplitude picture is correct. Complex values are the easiest way to reflect the phase properties of an amplitude (in the general sense), and we need not be dismayed by their appearance.

Let's suppose we have two amplitudes, A = aei0 and B = be2πix/λ, where λ is the wavelength λ = c/ν, and x is a linear distance. When x = 0, the two amplitudes will be in phase, and the net amplitude will be A + B = a + b. The intensity I = (a + b)2, while the intensities in the unmixed beams are a2 and b2. The intensity in the mixed beam is the sum of the intensities in each beam alone, plus the amount 2ab, the interference term. If the two beams have equal amplitudes, then when the two beams fall together, the total intensity is four times the intensity of one beam, or (one candle) + (one candle) = (four candles), or 1 + 1 = 4. We never see this with candles, but we do with lasers, so the strange mathematics is quite valid. Energy is conserved, of course, so this extra intensity must come from somewhere else, where the intensity is less

If x = λ/2, then A = a and B = be = -b. Now when we superimpose the two beams, the resultant amplitude is a - b, and the intensity is I = (a - b)2. The intensity is the sum of the separate intensities plus the interference term -2ab. If a = b, the intensity I = 0. Here, we have (one candle) + (one candle) = (zero candles), or 1 + 1 = 0. It is clear where the intensity came from for x = 0. As x increases steadily, the intensity forms bright fringes for x = 0, λ, 2λ, etc. and dark fringes for x = λ/2, 3λ/2, etc. If the amplitudes of the two beams are equal, the dark fringes are black, and the bright fringes are 4 times the average value. This gives the maximum contrast or visibility to the fringes. If b is less than a, the maxima are not as bright and the minima are not as dark. If b = 0, then the fringes disappear, and their visibility is zero. Michelson defined the visibility of fringes as V = (Imax - Imin)/(Imax + Imin), which ranges from 0 to 1.

Fringes had been observed since the early 17th century, when objects were illuminated by light coming from pinholes. Newton's rings are only one of the interference phenomena discovered by him. It was the explanation of the fringes that was lacking. No explanation was satisfactory until Thomas Young's experiments around 1801. Young did not discover fringes, but explained them in the current manner, which was no small accomplishment. He observed the interference of two beams, of the type just described, and measured the wavelength of light in terms of the fringe spacing. His experiment, in an abstract form, is a standard introduction to interference. It is not an easy experiment, especially when performed with a candle. Young actually used a wire, not two slits, which would have given an impossibly low illumination. When he held it before a pinhole through which a candle shone, fringes were seen in the shadow of the wire by direct observation, and their spacing could be compared with the diameter of the wire.

You can reproduce the experiment with an LED, a piece of #22 wire (diameter 0.6439 mm), and a hand lens of about 100 mm focal length, as shown in the diagram at the left. I used a yellow high-intensity LED in a clear envelope, viewed from the side where the source is practically a pinhole. The shield only reduces glare. I did not actually count the fringes (a micrometer eyepiece would make this easy) because I was holding the wire by hand, but the fine fringes were quite visible. The center fringe was a bright one.

This experiment uses some little-known characteristics of diffraction. If you look at a wire held at some distance from a pinhole, with your eye in the shadow of the wire, two short bright lines will be seen at the top and bottom edges of the wire. These act a line sources of light, producing two beams that intefere to make fringes in the shadow. There are also fringes outside the shadow, with different phase relations (the pattern is not continuous at the shadow edges), but are difficult to see in the glare. This gives a much larger intensity than two slits would in the same places, and made the experiment possible for Young. Of course, one could use two clear lines scratched carefully on a blackened photographic emulsion, as is done in schools, but the effect is not as good.

The geometry of a two-beam interference experiment is shown at the right. The source S is behind a pinhole that makes the illumination beyond the pinhole spatially coherent. That is, it all comes from the same direction and meets the two apertures with equal amplitudes. There may also be filtering that makes the light monochromatic, or temporally coherent, so that it approximates the model that we have been using. The distance D must be sufficient if the light at the apertures is to be coherent, something we will have much to say about below. However, the fringe spacing does not depend on D in any way. The screen is at a distance f from the apertures, presumed much larger than the separation a of the apertures. A lens of focal length f placed at its focal length from the screen makes the geometry exact in a short distance, and prevents too much spreading of the illumination. The fringe spacing is fλ/a, a useful relation to remember.

For the suggested Young's experiment, λ = 600 nm (roughly), f = 100 mm, and a = 0.6439 mm, giving a spacing of 0.093 mm, which seems roughly in agreement with observation. The wire will be approximately 7 fringes wide as seen through the lens. This is a way of measuring the wavelength if you know the wire diameter, or the wire diameter if you know the wavelength. Try #30 wire and observe that the fringes are not only wider, but fewer fit into the shadow.

An interference problem of interest to us is what happens with a circular aperture of diameter 2a. We must now superimpose the amplitudes from each area element of the aperture, and this calls for a double integral, using polar coordinates. Such problems are called diffraction, but there is no difference in principle with interference. The amplitude from each element of area will be the same, but its phase will differ depending on the distance from the element to the screen. This integral is done in all texts on physical optics, and the result is what is most important to us. This result is I(r) = I(0)[2J1(z)/z]2, where z = 2π[r/(2λf/d)], where d is the diameter of the aperture. We see the same factor λf/a as for the two apertures, where now a = d/2. The function J1(x) is the Bessel function of order 1, which behaves like x/2 - (x/2)3/2 + (x/2)5/12 - ... for small x. This is the famous Airy pattern, first derived by G. B. Airy, Astronomer Royal, on the basis of Fresnel's new wave theory. The intensity is strongly concentrated in the central maximum. 91% of the intensity is in the central maximum and the first bright ring surrounding it. If the intensity distribution across the disc is not uniform, the diffraction pattern will change slightly, but the general characteristics will be the same.

The Bessel function involved is zero when z = 3.83, so the radius of the first dark ring a = [(2)(3.83)/(2π)](λf/d) = 1.22λf/d. The angle (in radians) subtended by this radius at the aperture is θ = 1.22λ/d. The image of a star in a perfect telescope is an Airy disc. The d = 100" (2.54 m) Mount Wilson reflector has a Cassegrain focal length f = 40.84 m. If the effective wavelength is 575 nm, then θ = 2.76 x 10-7 rad = 0.057". Since neither the telescope nor the seeing can be perfect, this is a limit that can only be approached more or less closely. As we shall see, it is of the order of the angle subtended by the diameters of the largest stars, so actually seeing the disc of a star in a telescope is a vain hope. There are now some larger telescopes, including the 200" Hale reflector and the 5 m reflector in Russia, but this does not change the situation very significantly. Photographs have been taken with the 4 m telescope at Kitt Peak that with special processing have seemed to show some details of Betelgeuse. Direct observation of stellar discs seems just beyond practicality, unfortunately.

Stellar Diameters from Luminosity

Stellar diameters are such an important parameter in theories that some way to estimate them before they could be directly measured was sought. The usual method was to estimate the total radiated energy from the absolute magnitude. Magnitudes relative to total radiated energy are called bolometric magnitudes, and can often be obtained by adding a (negative) correction to the visual magnitude. The observed magnitudes may be affected by interstellar extinction as well. The total radiation was set equal to the known rate of radiation from a black body at the surface temperature of the star, which could be inferred from its spectrum. The effective temperature is defined in terms of Stefan's Law with unit emissivity, which bypasses the problem of emissivity without solving it. Since the emission of energy is proportional to the fourth power of the effective absolute temperature T, and the area is proportional to the square of the diameter of the star, we have the diameter D proportional to the square root of the luminosity L (an exponential function of the bolometric magnitude) and the square of the effective temperature T. If D', L' and T' = 5800K are the same quantities for the Sun, then D/D' = √(L/L')(5800/T)2. The diameter of the Sun is D' = 1.392 x 106 km, and its luminosity L' = 3.90 x 1033 erg/s.

Let's consider Betelgeuse, α Orionis. This M2-spectrum red star is said to have a luminosity 13,500 times that of the sun (it is variable, but this is a typical value at maximum). Its surface temperature is about 3000K. This gives D/D' = 434, or a diameter of 6.04 x 108 km, or 376 x 106 miles. A star as bright and as cool as Betelgeuse has to be large.

Coherence

Actual light disturbances are not as simple as the sinusoidal variations with constant amplitude and phase that we have discussed above. Laser light may approximate such disturbances, but not the light from candles or stars. This light is the resultant of a multitude of amplitudes from individual atomic emissions, which take place independently. Two signals of different frequencies get out of step quickly, the more quickly the more they are different in frequency. Random phase changes between two signals of the same frequency cause interference fringes to move. The light from thermal sources--candles and stars--is in the nature of a noise signal, with a wide frequency spectrum and constantly fluctuating phase. It is no surprise that we do not observe interference fringes in the usual conditions. What is surprising is that we can devise arrangements in which fringes appear. To do this, we must arrange that the phase relations between signals coming from the same atomic sources are constant. One way of doing this was described above, where we used a pinhole to define the source, and a filter to reduce the bandwidth. When we illuminated the two apertures with this light, stable fringes were then produced.

The light from two different thermal sources cannot be made to produce fringes. Two different lasers can produce fringes, but the experiment is rather difficult even for such ideal sources. Fringes can be made in white light, but only a few colored fringes are seen near the point where the phase difference is zero.

When the light from two points can be made to form fringes, the signals are said to be coherent. When no fringes are seen, the signals are called incoherent. In the two-aperture experiment, if we make the pinhole larger and larger, the fringes lose contrast or visibility, and eventually disappear. The light falling on the apertures becomes less and less coherent as this takes place. This simple observation shows the basis for determining stellar diameters by interferometry. We only have to find the limits of the region where the light from the star is coherent, using interference, and this is directly related to the apparent angular extent of the source. We take apertures farther and farther apart, and find out where the fringes disappear.

To analyze this quantitatively, we introduce a quantity called the degree of coherence, γ12 = γ(P1,P2,τ) = γ(r12,τ). P1 and P2 are the two points considered, r12 is the distance between them, and τ is the time difference in arrival at the screen (observing position). The degree of coherence is a complex number, though we usually consider its modulus, |γ 12|. The modulus of the degree of coherence varies between 0 (incoherent) to 1 (completely coherent).

To find out how γ is defined in terms of the light disturbances, we return to the two-aperture experiment. If A1 and A2 are the complex time-dependent signals from the two apertures, then the signal at the observation point Q is K1A1 + K2A2, where the K's are propagators that describe the changes in amplitude and phase as we go from an aperture to the screen. They are of the form K = ie2πi(t - t1)/r, the form typically used in diffraction integrals. We will not use these expressions explicitly, so do not worry about them. The curious nonintuitive factor "i" makes the phases come out properly. To find the intensity at Q, we multiply the signal by its complex conjugate and take the time average. The intensity of beam 1 alone is I1 = <A1A*1>, with a similar expression for I2. We find I = |K1|2I1 + |K2|2I2 + 2 Re[K1K2*<A1A2*>], where Re stands for "real part." If z is a complex number, Re(z) = (z + z*)/2.

We now define Γ12 = <A1(t + τ)A2*(t)> and call it the mutual coherence of the light signal at the two points. In statistical language, it is the cross-correlation of the two signals. The complex degree of coherence is simply the normalized value of this quantity, γ12 = Γ12/√(I1I2). Using Schwartz's Inequality, we find that 0 ≤ |γ| ≤ 1.

Now, using the intensities at Q (including the K's) we have the interference formula I(Q) = I1 + I2 + 2√(I1I2)Re[γ12(τ)], where τ is the time difference (s2 - s1)/c between the paths P1Q and P2Q. The visibility of the fringes, V = |γ|, so if we measure V, we then know |γ|.

To better understand what this means, let the two signals at Q be ae2πiνt and be2πiν(t + τ). Then, Γ12 = abe-2πiτ and γ12 = e-2πiντ, so Re(γ) = cos(2πντ). This gives I(Q) = I1 + I2 + 2√(I1I2)cos(2πΔs/λ), where Δs is the path difference. This is just the formula we found earlier for the two-aperture problem. We see that the degree of coherence is unity here.

We also see that the phase of γ has a rapidly-varying part for monochromatic light, 2πντ. Ordinary narrow-band or quasimonochromatic light is very much like narrow-band noise. The frequency varies randomly over a small range centered on the average value, while the amplitude varies up and down irregularly. For such signals, it is useful to separate the rapidly varying part (at the average frequency) from the more slowly varying part. Hence, we write γ = |γ(&tau)|ei[α(&tau) - δ], where δ is the part we have just looked at that involves the path difference, and α(τ) is the rest. The dependence on τ reflects what is often called the temporal coherence, while the dependence on the location of the source points reflects spatial coherence. In either case, we remember that coherence is the ability to produce stable interference fringes.

A signal with a frequency bandwidth Δν shows incoherence after a time interval of the order of Δτ. Visible light has an effective bandwidth of roughly 500 nm to 600 nm (not the extremes of visual sensitivity, of course), so that Δν = 1 x 1014 Hz, and Δτ = 1 x 10-14 s. The light has barely time to wiggle once before coherence is destroyed. It is no wonder that fringes are not seen in white light except in special cases, and even then only one or two. This is a result of temporal coherence alone. By restricting the frequency bandwidth, the coherence time Δτ can be increased to more comfortable values.

Van Cittert-Zernike Theorem

This theorem tells us the dependence of the coherence on distance for an extended source, such as a pinhole or a star. We will apply it only to a circular source of uniform brightness, but it can also be used for much more general sources.

The theorem states that: "The complex degree of coherence between P1 and P2 in a plane illuminated by an extended quasi-monochromatic source is equal to the normalized complex amplitude in the diffraction pattern centered on P2 that would be obtained by replacing the source by an aperture of the same size and illuminating it by a spherical wave converging on P2, the amplitude distribution proportional to the intensity distribution across the source." We have already discussed diffraction from a circular aperture, which is precisely the diffraction pattern we require in the case of a uniformly bright disc. The coherence is 1 when P1 and P2 coincide, and decreases according to J1(z)/z as P1 moves outwards, becoming zero where the diffraction pattern has its first dark ring.

The distance for γ = 0 is, therefore, given by a = 1.22λ/θ, where θ is the whole angle subtended by the source at P2, measured in radians. If we take the effective wavelength as 575 nm, and measure the angle in seconds, we find a = 0.145/θ" m. This holds for pinholes and stars. The smaller the angle subtended by the source, the larger the radius of coherence a. A pinhole 0.1 mm in diameter subtends an angle of 69" at a screen 300 mm distant, so a = 2.1 mm. The Sun or Moon subtend an angle of about 0.5° or 1800" at the surface of the Earth, so a = 0.08 mm. Venus subtends an angle between 9" and 60" (its distance from the Earth varies greatly), so a = 2.3 mm to 14.4 mm. Venus is often not a disc, especially when closest to the Earth, but this gives an idea of its radius of coherence. Jupiter subtends an angle about the same as the maximum for Venus, so the radius of coherence of its light is 2 mm or so. On the other hand, Betelgeuse subtends an angle of 0.047" when it is at its largest, so a = 3.1 m or about 10 ft. The discs of most stars subtend much smaller angles, so for all stars a > 3 m, and often hundreds of metres.

We now have all the theory we need to understand stellar interferometry. It is clear that we are looking for the radius at which the illumination has zero coherence, and this radius gives us the angular diameter. The linear diameter is obtained by multiplying by the distance. If we knew the diameter to start with, inverting this method would give us the distance.

Scintillation

One subject we can take up before describing stellar interferometers is the familiar phenomenon of twinkling, or scintillation of celestial bodies. This is caused by slight variations in the density of the atmosphere due to turbulence, wind shear and other causes, and is closely related to the "heat shimmer" seen on summer days over hot surfaces. The stars seem to jitter in position, become brighter or dimmer, and show flashes of color. The amount of scintillation varies greatly depending on elevation and weather, and sometimes is almost absent. Scintillation is the cause of good or poor telescopic "seeing." When the seeing is bad, stellar images wobble and jump, and the resolving power is reduced.

It is often observed that the planets do not scintillate to the same degree as the stars, but the effect is variable. Venus scintillates and shows flashes of color when a thin crescent and closest to us, but mostly the planets show a serene and calm face even when nearby stars are twinkling. The reason for this difference is often ascribed to the fact that the planets show an apparent disc, while the stars do not. However, even the edges of the planet images do not wiggle and jump, so this is probably not the reason for the difference.

It is much more reasonable that since a planet's light is incoherent over any but a very small distance, interference effects do not occur. The image may still move slightly depending on its refraction by changes in density. With a star, however, the area of coherence may include whole turbulence cells, and the randomly deflected light may exhibit interference, causing the variations in brightness and the colors. So scintillation does depend on the apparent size of the body, but in a more esoteric way. Exactly the same effects occur for terrestrial light sources, though they must be quite distant to create large areas of coherence.

The Michelson Stellar Interferometer

A. A. Michelson was led to the stellar interferometer through his experiences with using his original interferometer, now named after him, with light consisting of a narrow line or line, such as sodium light with its D line doublet. The fringes go through cycles of visibility as the path length is varied, and from the variations the structure of the line can be unraveled. Doing this with sodium light was a popular laboratory exercise in optics.

If the end of the telescope tube is closed with a mask with two apertures, fringes are produced at the focus, showing that the light is coherent, if the two apertures are close enough together. With the Sun, or Jupiter, fringes would not appear at all because of the small coherence radius. With stars, however, the decrease in fringe visibility would be evident, and from the separation of the apertures for zero visibility the angular diameter could be found.

No telescope was large enough in aperture to give γ = 0 for even the stars with the largest angular diameters, even giant Betelgeuse, so light had to be collected from a greater separation by means of mirrors mounted on a transverse beam. A 20 ft beam was selected for the initial experiments, which should be sufficient for measuring Betelgeuse. The telescope selected was the 100-inch reflector at Mount Wilson, not because of its large aperture, but because of its mechanical stability. A 20-ft steel beam of two 10-inch channels is certainly not light, although its weight was reduced as much as possible by removing superfluous metal. The two outer mirrors directed the light to two central mirrors 45 in. apart, which then sent the light toward the paraboloidal mirror.

The 100-inch (2.54 m) telescope could be used at a prime (Newtonian) focus at a focal length of 45 ft (13.72 m), the beam diverted to the side by a plane mirror near the top, or at a Cassegrain focus at a focal length of 134 ft. (40.84 m) after reflection from a hyperboloidal mirror also near the top, and diversion to the side near the bottom of the telescope. The latter was chosen to give greater magnification, 1600X with a 1" efl eyepiece. The average wavelength for Betelgeuse was taken as 575 nm. With the 45 in. separation of the apertures and 40.84 m focal length, the fringe spacing is 0.02 mm, as you can easily check from these figures. The fringes were observed visually, and were easily seen, even when the image was unsteady but could still be followed by the eye.

The path length from each of the outer mirrors to the center mirrors had to be carefully adjusted for equality, because of the limited temporal coherence of the light (as mentioned above; these are white-light fringes). This was done by glass wedges in one beam. A direct-vision prism to observe the fringes in a restricted bandwidth made them easier to locate. The interferometer was very difficult to align, but satisfactory fringes were seen. When the outer mirrors were moved to cause the fringes in the image of Betelgeuse to disappear, it still had to be verified that other stars gave fringes under the same conditions, in case the absence of fringes was due to some other cause. One star chosen for this test was Sirius, which gave prominent fringes.

For Betelgeuse, a separation of 121 in (3.073 m) caused disappearance of the fringes, so the angular diameter was 0.047". At the time, the parallax of Betelgeuse was thought to be 0.018", but it now seems to be closer to 0.0055", for a distance of 182 psc or 593 l.y.. From these numbers we find the diameter of Betelgeuse to be 1.28 x 109 km, or 797 x 106 miles. This is about twice as large as the diameter estimated from luminosity, which implies that the star is cooler than expected, or its emissivity is lower for some reason. It is often stated that Betelgeuse would fit within the orbit of Mars. This was from the older figures; it actually would extend halfway to Jupiter, well within the asteroid belt. The angular diameter of Betelgeuse varies from 0.047" at maximum down to about 0.034" at minimum, since the star pulsates irregularly.

The verification of the large size of Betelgeuse was one of the principal results of the 20-ft interferometer. This makes its average density quite small, but of course it becomes more concentrated toward the center where the thermonuclear reactions are taking place. The star long ago exhausted the hydrogen in its core, and began burning hydrogen in an expanding shell as it swelled and cooled to a red supergiant. Now the pulsations show that it is beginning to light its helium fire at the center, which is fighting with the hydrogen reactions further up. The helium will be consumed in a relatively short time, and the star will shrink and cool to a white dwarf. That seems to be the history according to the stellar theorists, at least.

The diameters of seven stars in all were measured by the 20-ft interferometer, down to an angular diameter of 0.020", where some extrapolation had to be made. All these stars were red supergiants with spectra from K1 to M6, including the remarkable ο Ceti, Mira, that pulsates more deeply and regularly than Betelgeuse. The angular diameter of Mira at maximum was .047", the same as Betelgeuse's, but it is five times closer, so its linear diameter is about 160 x 106 miles, so Venus could revolve within it.

In hopes of measuring smaller diameters, perhaps even those of main-sequence stars, a larger interferometer was designed and built by Pease, with a 50-foot beam and mounted on the 200-inch Hale telescope at Palomar. This instrument was very difficult to operate, but the measurements on Betelgeuse and Arcturus agreed with those from the earlier instrument, while those on Antares differed significantly. Little was added by the new instrument, but it showed that a limit had been reached, largely because of the difficulty of keeping the two paths equal and the bad effects of scintillation. Modern techniques might overcome these limitations to some degree, but no great improvement is to be expected.

The Correlation Interferometer

The correlation or intensity stellar interferometer was invented in about 1954 by two remarkable investigators, R. Hanbury Brown and R. Q. Twiss. A large interferometer was completed in 1965 at Narrabri, Australia, and by the end of the decade had measured the angular diameters of more than 20 stars, including main sequence stars, down to magnitude +2.0. This interferometer was equivalent to a 617-foot Michelson stellar interferometer, was much easier to use, and gave repeatable, accurate results.

Hanbury Brown was a radio astronomer at the University of Manchester's Jodrell Bank observatory, and Twiss was at the U.K. Services Electronics Research Laboratory at Baldock. They united the resources necessary to conceive and execute the project between them. They seem to have been not very much appreciated in their native country, but prospered in Australia, which offered them the opportunity to develop their ideas.

The intensity interferometer was introduced as a new type of interferometer for radio astronomy, but it was soon realized that it could be applied to the problem of stellar angular diameters as a successor to the Michelson interferometer of thirty years before. It works on the same fundamental principle of determining the coherence of starlight as a function of the distance between two points, but the means of finding the coherence is totally different, and relies on some esoteric properties of quasi-monochromatic light. The diameter of Sirius, the first main-sequence star whose diameter was measured, was determined in preliminary tests at Jodrell Bank in 1956, under difficult observing conditions. This was not an accurate result, but it was a milestone. To explain the interferometer, the best way to start is to look at its construction.

A diagram of the Narrabri interferometer is shown at the right. The two mirrors direct the starlight to the photomultipliers PM (RCA Type 8575, and others). Each mirror is a mosaic of 252 small hexagonal mirrors, 38 cm over flats, with a three-point support and an electrical heater to eliminate condensation. They are aluminized and coated with SiO. The focal lengths are selected from the range naturally produced by the manufacturing processes to make the large mirrors approximate paraboloids. Great accuracy is not necessary, since a good image is not required, only that the starlight be directed onto the photocathodes. The starlight is filtered through a narrow-band interference filter. The most-used filter is 443 nm ± 5 nm. The photocathode is 42 mm diameter, and the stellar image is about 25 x 25 mm. This is all of the optical part of the interferometer; all the rest is electronics.

The mirrors are mounted on two carriages that run on a circular railway of 188 m diameter and 5.5 m gauge. At the southern end of the circle is the garage where the mirrors spend the day and maintenance can be carried out. A central cabin is connected to the carriages by wires from a tower. This cabin contains the controls and the electronics. Note that the separation of the mirrors can be varied from 10 m up to 188 m. The mirrors rotate on three axes to follow the star. One of the small mirrors in each large mirror is devoted to the star-guiding system, that consists of a photocell and a chopper. This keeps the mirrors locked on the star under study, without moving the mirror carriages. The light-gathering power of the 6.5 m diameter mirrors is much greater than that of the small mirrors in the Michelson stellar interferometer, allowing the Narrabri interferometer to operate down to magnitude +2.0. The available baseline distances permit measurements of angular diameters from 0.011" to 0.0006".

The photocurrent, which is about 100 μA, is a measure of the total intensity, required for normalizing the correlation coefficient. This is measured and sent to the data-handling devices (connections not shown). The photocurrent is sent to a wide-band amplifier, then through a phase-reversing switch, and then through a wide-band filter that passes 10-110 MHz. This bandwidth excludes scintillation frequencies, eliminating their effects. The signals from the two photomultipliers then are multiplied in the correlator. The phase of one of the photocurrents is reversed at a 5 kHz rate, which makes the correlation signal change sign at the same rate, but leaving the noise unchanged. A tuned 5 kHz amplifier at the output of the multiplier selects just this signal, which is then synchronously rectified. This is a standard method of increasing the signal-to-noise ratio in situations such as these. The signal-to-noise ratio in the photocurrent is about 1 to 105. The other channel is reversed at a much slower rate, once every 10 seconds, and the correlation for each state is separately recorded. When these values are subtracted, the changes in gain and other effects are eliminated, and the result is the desired correlation.

The electrical bandwidth of 100 MHz implies that the signal paths from the photomultipliers to the correlator must be equal to about 1 ns to avoid loss of correlation due to temporal coherence. This seems like a very tight requirement at first view, but it is much easier to equalize electrical transmission lines that optical paths. The 1 ns corresponds to about 1 ft in length, which now does not seem as bad. In the case of the Michelson stellar interferometer, the paths must be equal to a wavelength or so, and this was the most important factor limiting its size.

Small lamps in the photomultiplier housings can be turned on when the shutters are closed. These lamps give uncorrelated light, so any correlation that is recorded when they are on is false. In another test, perfectly correlated noise is supplied to both channels from a wide-band noise generator for measuring the gain of the correlator. These and other tests are carried out during an observing session. The correlator is the most critical part of the interferometer, and most of the effort went in to making it as accurate and reliable as possible.

Skylight is allowed for by measuring the intensity and correlation with the mirrors pointing to the sky near the star. One contribution to the correlation was anticipated, that of the Cherenkov radiation from cosmic rays. This is a faint blue streak of light (that both mirrors would see simultaneously, and would thus correlate) that is produced when the cosmic ray is moving at greater than the speed of light (c/n) in the atmosphere. This proved to be unobservable. Meteors would have the same effect, but they are so rare that this is ruled out. Observations were not carried out when the Moon increased the skylight to an unacceptable level.

The theory of how the correlation in this case is related to the degree of coherence is similar to what we explained in connection with the Michelson instrument, but happens to be more involved, so only the idea will be sketched here. The filtered starlight is a quasi-monochromatic signal, in which the closely-space frequency components can be considered to beat against one another to create fluctuations in intensity, <AA*>. This is a general and familiar aspect of narrow-band noise. There are also accompanying fluctuations in phase, but these are not important here. The correlation measured in the intensity interferometer is proportional to <ΔI1ΔI2>, where ΔI = I = Iav is the fluctuation in I. If expressions for the quantities are inserted in terms of the amplitudes, it is found that the normalized correlation is proportional to |γ12|2, the square of the fringe visibility in the Michelson case. The phase information is gone, but the magnitude of the degree of coherence is still there, and that is enough for the measurement of diameters.

Advantages of the Brown and Twiss interferometer include: larger light-gathering capacity permitting use on dimmer stars; ease of adjusting the time delays of the channels to equality; electronic instead of visual observation; immunity to scintillation; much larger practical separations; and the elimination of the need for a large, sturdy telescope as a mount.

The Photon Question

The photoelectric effect has long been evidence for what has been called the "particle" nature of light. Einstein demonstrated that the probability of emission of a photoelectron was proportional to the average intensity of the light, what we have represented by <AA*>, that the kinetic energy of the emitted electron was E = hν - φ, where φ is the work function, and that the emission of photoelectrons occurred instantaneously, however feeble the illumination. It was seen as a kind of collision of a "photon" with an electron, ejecting the electron as the photon was absorbed. A photocathode is called a square law detector because of its dependence on the square of the amplitude. Actually, all this is perfectly well described in quantum mechanics, and there are no surprises. What is incorrect is thinking of photons as classical particles (even classical particles obeying quantum mechanics) instead of constructs reflecting the nature of quantum transitions. Those who thought of photons as marbles, and there were many, thought Brown and Twiss were full of rubbish, since whether the light was coherent or not, the random emission of electrons by "photon collisions" would erase all correlations. The photocurrents of two separate detectors would be uncorrelated whether they were illuminated coherently or incoherently. One would simply have the well-known statistics of photoelectrons. If Brown and Twiss were correct, then quantum mechanics "would be in need of thorough revision," or so they thought.

The experiment that Brown and Twiss performed to verify that correlations could be measured between the outputs of two photomultipliers is shown at the right. The source was a mercury arc, focused on a rectangular aperture, 0.13 x 0.15 mm. The 435.8 nm line was isolated by filters. The photocathodes were 2.65 m from the source, and masked by a 9.0 x 8.5 mm aperture. Since the illumination had reasonable temporal coherence, the two light paths were only made equal to about 1 cm. A horizontal slide allowed one photomultiplier to be moved so that the cathode apertures could be superimposed or separated as seen from the source, varying the degree of coherence from 1 to 0. The electrical bandwidth, determined by the amplifiers, was 3-27 MHz. The output of the multiplier was integrated for periods of about one hour. If repeated today, the experiment could not be done with a laser, because the source incoherence is essential to the effect. The experiment clearly showed that correlation was observed when the cathodes were superimposed, which disappeared when they were separated.

A similar experiment was performed by Brannen and Ferguson in which the coincidences of photoelectrons emitted from two cathodes were observed. No extra coincidences, or correlation, were observed when the cathodes were illuminated coherently, and this, it seemed, proved that the Brown and Twiss interferometer could not work (although, of course, it confounded them by working anyway). Some thought maybe light wasn't described well by quantum mechanics at all, and that the classical theory predicted what was observed. This is very nearly true, since the amplitudes of wave theory include a lot of quantum mechanical characteristics by their very nature. However, light is quite properly and correctly described by quantum mechanics when it is done properly, and not by naive intuition.

With the concurrence of E. M. Purcell, Brown and Twiss showed that the coincidence experiment was much too insensitive to show the effect as the experiment was designed, and instead would have required years of data to show any correlation by photon counting. They later demonstrated correlations using photon counting, resolving the problem. Of course, their method using electronic correlation, as in the stellar interferometer, was much more efficient and gave much better results than photon counting.

Information on this interesting controversy can be found in the References.

Results of the Measurements

The best test of the interferometer would be the measurement of a star of known diameter. However, there are no such stars. Therefore, the only tests are the consistency of repeated measurements. The interferometer measures the angular diameter directly, and the linear diameter depends on knowing the distance, which in many cases is uncertain. All astronomical data is subject to error, revision and misinterpretation, though the current quoted figures always look firm and reliable enough.

The problem with using a terrestrial source for a test is seen from the fact that a source of diameter a mm has an angular diameter of 0.2a" at a distance of 1 km. The maximum angular diameter that the Narrabri interferometer can measure is 0.011", so a source diameter of only 0.05 mm would be required at 1 km, or 5 mm at 100 km. It would be very difficult to push enough light to be seen through such a small aperture!

The angular diameter can be used directly to find the exitance (energy emitted by the stellar surface per unit area) without knowing the distance, and the exitance can be used to find the temperature. Therefore, the interferometer data has been used to refine the temperature scale of the stars, which previously was estimated only from the spectrum. The monochromatic flux F at the surface of a star is related to the monochromatic flux received outside the Earth's atmosphere by F = 4f/θ2, where θ is the angular diameter, as illustrated in the diagram. This does not include corrections for interstellar extinction. Then, &integ;F d&lambda = σTe4, where σ is Stefan's constant.

By 1967, measurements had been made on 15 stars from spectral type B0 to F5, including a number of main sequence stars, including Regulus (3.8), Sirius (1.76), Vega (3.03), Fomalhaut (1.56), Altair (1.65) and Procyon (2.17), for which reliable parallaxes were known. The number in parentheses is the diameter in solar diameters. Measurements could not be made on Betelgeuse, since the mirrors could not be brought closer than 10 m apart, and besides the 6.5 m mirrors would themselves resolve the star, reducing the correlation to a trifle.

References

E. Hecht and A. Zajac, Optics (Reading, MA: Addison-Wesley, 1974). Section 12.4 covers the application of coherence theory to stellar interferometry. This is also a good reference for the other optical matters discussed above.

M. Born and E. Wolf, Principles of Optics (London: Pergamon Press, 1959). Chapter X treats partial coherence. Section 10.4 is especially relevant to our subject. The Michelson stellar interferometer is covered in Section 7.3.6, pp 270-276, with a mention of the intensity interferometer, which was quite new when this book was written.

A. A. Michelson and F. G. Pease, "Measurement of the Diameter of α Orionis With The Interferometer," Astrophysical J. 53, 249-259 (1920).

R. Hanbury Brown and R. Q. Twiss, "A New Type of Interferometer for Use in Radio Astronomy," Philosophical Magazine (7)45, 663 (1954).

R. Hanbury Brown and R. Q. Twiss, "A Test of a New Type of Stellar Interferometer on Sirius," Nature 178, 1046-1048 (1956).

E. Brannen and H. I. S. Ferguson, "The Question of Correlation Between Photons in Coherent Light Rays," Nature 178, 481-482 (1956).

R. Hanbury Brown and R. Q. Twiss, "The Question of Correlation Between Photons in Coherent Light Rays," Nature 178, 1447-1448 (1956).

E. M. Purcell, (Same title as previous reference) Nature 178, 1449-1450 (1956).

R. Q Twiss and A. G. Little, and R. Hanbury Brown, "Correlation Between Photons, in Coherent Beams of Light, Detected by a Coincidence Counting Technique," Nature 180, 324-326 (1957).

R. Hanbury Brown, "The Stellar Interferometer at Narrabri Observatory," Sky and Telescope, 27(2) August 1964, 64-69.

R. Hanbury Brown, J. Davis and L. R. Allen, "The Stellar Interferometer at Narrabri Observatory, I and II," Monthly Notices of the Royal Astronomical Society, 137, 375-417 (1967).

R. Hanbury Brown, "Measurement of Stellar Diameters," Annual Reviews of Astronomy and Astrophysics, 1968, 13-38. With bibliography.

_________, "Star Sizes Measured," Sky and Telescope 38(3), March 1968, 1 and 155.


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Composed by J. B. Calvert
Created 25 September 2002
Last revised 19 November 2008