First steps to double star measurement using visible light interferometry.

Introduction

In the 1920s Michelson (1) and also Anderson (17) made pioneering interferometric measurements of double stars and also, with a longer baseline, of large single stars in order to estimate their diameters. Radio interferometry (and image synthesis) became a standard tool over 40 years ago, but it is only recently that visible light stellar interferometry has, with Michelson type interferometers, reached the same status, with very sophisticated equipment added to large telescopes. (2) In fact interferometry is now a widespread and powerful tool in professional astronomy, and has opened up orders of magnitude higher precision in some areas, but that is thanks to equipment which is way beyond amateur means. Is it nevertheless possible for amateur astronomers to achieve improved resolution, to a level approaching diffraction limited imaging (approx 0.3 arcsecs for our 40cm telescope), by similar methods? * This could not be used to estimate stellar diameters, but it may refine (amateur) measurements of the separation and position angle of double stars.

There are in fact two problems--minimising the obscuring effects of 'seeing', and analysing the interference. The standard approach to the first problem, for both Earth-bound professionals and amateurs, is to combine sequences of short exposures. Michelson, without the aid of fast photography, did not have this option so he (or to be exact his observing assistant, as he makes clear in his paper (1)) looked for secondary mirror separations where the interference disappears. (5) With electronic cameras, fast recording is now possible, and this also opens an approach to the second problem, by the process of fitting calculated patterns to recorded patterns.

Interferometry for the amateur (10,11) involves placing a mask over the telescope with a pattern (which can be cut from opaque cardboard) of slits of 1 to 2cm width. It seems counter-intuitive to attempt to improve resolution by throwing away light. The reason this can work is that the mask acts as a spatial filter. The interference pattern produced by the mask has a shape whose detail depends on the structure (if any) of the source.

The first stages are therefore:

1. Choose what structure to look for in a given object--let us say a double star with a separation of one arcsec.

2. Calculate the interference pattern that this structure would form with the given mask in ideal conditions (i.e. the 'theoretical' pattern.)

3. Take a sequence of short (~0.03 sec) exposures. A short exposure is of course essential to freeze the 'seeing' in a given frame, but then one must find a way of aligning the frames, which suffer random displacements due to the 'seeing' between frames, and that is the hard part. In fact what is normally attempted, as we describe, is to extract the spatial information in a displacement-independent way, rather than do alignment as such.

The first part of this paper describes the observations that we have made. The second section discusses possible analytical methods, analogous to the process of 'stacking' as used by Registax (3)--but as explained later the method commonly used very successfully for planetary images is not appropriate here (we show the results of our test of MaximDL vs Registax later). Our aim in publishing this paper is to pass on our experience with the analysis to any who wish to improve it and take it further.

Procedure

If the (double) stars of interest are so faint that they are difficult to see or record even with full aperture, there is no point in masking down for interferometry. How faint is faint? There is a quantitative answer to this, given by the number of photons detected by a pixel (see Appendix 1). The fractional error is the reciprocal of the square root of the number of photons. The number of photons must be high enough for this uncertainty to be acceptable--despite the short exposure (0.03 sec) which is essential to freeze seeing. Longer exposures cannot succeed. The number of photons can be estimated from the luminosity of the star. But why not just use the number given by the camera? It is important to realise that the number which a digital camera returns for each pixel is NOT a photon count but rather the charge accumulated on the pixel, amplified and digitised to fit into a (normally) 16-bit integer (let us call this the camera count). It is not the square root of that integer which gives the uncertainty, it is the square root of the photon count. For the Starlight Xpress SXVf-H9 camera we used, the detected photons were about one-third the camera count. (4) Other cameras will have different multiplying constants. The numbers we obtained (see Figures 10 to 14) were compatible with the stellar luminosity estimates.

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Knowing the number of photons, we can use the fact that Poisson statistics gives the simple rule that the error (spread) is the square root of the number of photons. Here we have both genuine signal counts and background counts, and the total random error is then the square root of the sum of the squares of the error on the background and of the error on the genuine photon count. The background is usually bigger than the signal.

Alternatively this whole process of calculating random (Poisson) errors can be sidestepped by estimating the errors from the statistics of repeated observations under the same conditions. This may sound easier, but here it means taking many more exposures (typically perhaps 10 sets of 20 rather than one set of 20), and in astronomy conditions are always changing due to seeing, thin cloud etc. which masks the inherent random spread.

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Given this level of uncertainties the statistics may sound meagre, but there are several factors in our favour. First we can (and must) add several--at least ten, and perhaps 100--frames together, a procedure that was not available to Michelson. Secondly we are going to be interested in the peaks in the distribution, with (by definition) more than the average number of counts. The third reason is more subtle but also very important. As mentioned above, if we were using Michelson's method of varying the slit/mirror separation, the 'question' we would be asking of each image is whether there is any structure visible. Using fixed slits the question is significantly different--it is 'how much of a known structure is present'. We know the expected structure, both by computation and by preparatory measurements on single stars. We have spent quite a lot of time making sure that we understand this structure. Estimating the fraction of a known structure is a statistically better-defined question than asking if any structure at all is present. By therefore being able, at least in principle, to use quite low photon counts, we sneak in what might even be regarded as a fourth advantage--the counting error only goes as the square root of the count, not the count itself. So in summary, we expect that looking for a well-defined interference structure of several peaks can be more revealing than looking at a straight image, provided that the stars are (in our case) brighter than apparent magnitude about 5. It is the ultimate purpose of this work to see if this is borne out in practice.

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The design of the slits is necessarily a compromise. Varying the width, number and separation of the slits affects the number, sharpness and intensity of the peaks. We have settled on four slits (arranged as shown in Figure 5) as a guess at the best experimental compromise.

Theoretical patterns

A single star viewed through two slits gives the theoretical intensity pattern shown in Figures 1 and 2. (See Appendix II and reference 6 for the mathematics used to calculate the patterns.) Note the peak intensity (y-axis) of approximately 0.003.

Using four slits (of the same width, see Figure 5) lets in twice the amount of light, and gives sharper main peaks than two slits with the same area (Figures 3 and 4). (This trend continues right up to the extreme case of a diffraction grating, which has a huge number of very narrow 'slits' and gives very sharp peaks.) Note the peak intensity of 0.01 and greater structure inside the three peak envelope (cf. Figure 2, which retains the same vertical scale for comparison).

The fine detail is unobservable (i.e. finer than the diffraction limit), and so needs to be smoothed. A frequency cutoff is easily imposed on the Fourier spatial transform (see Figure 1). The left-hand plots (Figures 1 and 3) are for the whole visible spectrum (400-700nm), however a wavelength filter mounted on the camera should make the interference plot cleaner (see Figures 2 and 4), but will let in less light so will need a longer total exposure. A broadband spectrum gives, in effect, a blurred superposition of patterns of the same shape but displaced and at different scales. Our camera was monochrome, so we did not see the colours.

We conducted theoretical tests with and without filters, and with different types of filter. We chose an OIII filter with bandwidth 490-510nm as being the best compromise without seriously reducing the intensity, and this was used throughout. Comparing Figure 1 with Figure 2, and Figure 3 with Figure 4, clearly shows the benefit of using a filter. **

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Calculated patterns on double stars from four slits and OIII filter

The observations and calculations were done with four slits placed as shown in Figure 5. The exact layout was chosen by trial and error. In principle one could calculate an optimum (i.e. minimised statistical error) design for a given stellar magnitude. The slits do not have to be of equal length, so they can be extended to the periphery of the telescope, but the computed pattern must then take the different weightings into account.

A single star produces a pattern of interference, due to the different path lengths from different slits to a given point in the image. We generally see just the central three peaks (see Figures 1 to 4). To be exact, every region in the star produces its own pattern, but if the star is effectively a point source they are all superimposed. A very few stars are large/ close enough for this not to be true--as Michelson demonstrated with long base lines.

With a double star, consider first the case where the line joining the two stars is perpendicular to the length of the slits. Each star produces such a pattern, but the two are displaced from each other and we see their (intensity) sum--typically six main peaks as in Figure 6.

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In Figure 6 the two stars are separated by 10 arcsec and there are different path lengths to a given point in the image from the two stars (i.e. rather than from two slits), but since the light from the two stars is incoherent, this does not produce any further interference peaks. (To be exact, they will theoretically produce instantaneous patterns, but these shimmer and merge into a background on a timescale of picoseconds). If on the other hand the line joining the stars is parallel to the length of the slits, the patterns from the two stars superimpose exactly and the (projected) pattern is the same shape as for a single star (see Figure 4). There is no extra structure arising from the two (incoherent) stars--so no further calculation is needed to understand the pattern.

Thus, if a series of observations is made with changing angles of the slits (i.e. rotating them on the front of the telescope, about the axis of the telescope), a single star will give the same pattern at every angle, whereas a double star will give a changing pattern, as the angle passes from the parallel to perpendicular configurations described above. This is a method of detecting double stars, and can work even if they are unresolved. It should also provide an estimate of the position angle. In addition, if the distribution of light in the pattern is modelled as two displaced distributions, it provides an estimate of the angular separation of the two stars (as projected onto the line perpendicular to the slits). These patterns are seen by projecting the pixel counts onto a line perpendicular to the slits (the image is always elongated in this direction so this is unambiguous). As noted earlier there is no interference pattern in the projection onto the line parallel to the slits. The angular position of the camera has no effect on the pattern seen, but in practical terms it is easier to make the projection if the elongation of the image is parallel to (the long side of) the frame. It also saves rotating the image mathematically, which can (for angles not equal to multiples of 90[degrees]) lead to loss of resolution.

For double stars with closer and closer separation the peaks merge, as shown in Figures 7 and 8.

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Observations

We used the Meade LX200 40cm f/10 Schmidt-Cassegrain telescope in the George Abell observatory at Milton Keynes, jointly run by the Open University Astronomy Club and the Department of Physics and Astronomy. The choice of CCD camera was important as we needed to use exposures of around 0.03s, and a pixel size that would give the resolution we needed. We started by using the SBIG ST1001 before we realised that it had a mechanical shutter and would not allow exposures shorter than 0.1s. (Our version of MaximDL (14) did not make us aware of the fact that the camera was not responding to our chosen 0.03s exposure until we did some other tests.) These limitations were overcome by using the Starlight Xpress HX516 and later the SXVf-H9, both with electronic shutters.

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As the interference pattern only takes a tiny part of the frame, flat-fielding was unnecessary; however in the final results, dark frame subtraction was found to give better results. General practice, and also some observations of our Own (12) showed that we needed an exposure of 0.03s or less in order to freeze the 'seeing'.

The CCD images were taken through the 4-slit mask using an OIII filter, with exposure time of 0.03s. The horizontal axis is shown in pixel widths, with one pixel = 6.5[micro]m= 0.5 arcsec, so a CCD camera with smaller pixels than 6.5[micro]m would not be an advantage. We have taken a total of hundreds of such images.

Single star--Arcturus

Figures 9 and 10 show the total value for each column of pixels within the white box. Comparing this with the theoretical pattern (Figure 4) shows the 3 main peaks and the 2 outer small peaks. In addition the central peak shows some structure. (As the data points on the graph are discrete, we have not changed the scale to arcsec.) The rough estimate from stellar power output would be about 10,000 camera counts--a similar order of magnitude to the actual values shown here.

Double stars--[zeta] UMa and [zeta] Her

The image and plot in Figures 11 and 12 (Zeta [[zeta]] UMa) show less well defined peaks, although comparing this with the theoretical pattern (Figures 6 and 7) one sees that there are two main peaks corresponding well with the known separation of 14 arcsec.

Figures 13 and 14 ([zeta] Her) are much more muddled than one would expect from a close separation double star, although the peaks each side of the central maximum compare with the theoretical pattern in Figure 8. The crucial point is to be able to identify and quantify the amount of muddle. Notice that in this double the secondary is about 10 times fainter than the primary, so that typical photon counts per pixel might be about 500 from the primary and 50 from the secondary. Remembering the 1/square root (counts) rule, they should still be distinguishable, but bigger luminosity differences become rapidly more difficult. We have many hundreds of such images, for a dozen different stars taken on many different occasions, to the stage where there was no point in continuing the observations until we had a better understanding of the requirements of the analysis.

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Fourier transformation

The rest of this paper considers the work in progress.

The Fourier Transform is an important image processing tool which is used to decompose an image into its sine (real) and cosine (imaginary) components. The output of the transformation represents the image in the Fourier or spatial frequency domain, while the input image is the spatial domain equivalent. In the Fourier domain image, each point represents a component with given amplitude and phase.

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The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression. (13) A common use of Fourier transforms is to find the frequency components of a signal buried in a noisy time domain signal.

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Professional instruments may be used to produce two dimensional images through speckle imaging (visible light) or image synthesis (radio). We only need to analyse the one-dimensional interference pattern. We therefore need a one-dimensional Fourier transform, to produce a (spatial) power spectrum for each 30ms image up to a spatial frequency of about 1 per pixel--higher spatial frequencies can have little significance. Since this spectrum is independent of translational shifts, the power spectra can then be added, sidestepping 'seeing'. This is the basic idea of some of the various algorithms available in Registax. (3) [Notice however that we would not expect the common way of using Registax for planetary observations, relying on a sharp reference feature, to work here--there is no such sharp feature available. However to check the point we tried taking one of the clearest sets of six images we have (saved in uncompressed FITS format) and manually aligned and processed them through Registax; it gave a degraded image (Figure 15). We also tried the straightforward manual alignment process with MaximDL and this did give a sharper composite image (Figure 16). Of course these are both very extensive programs with many options and we have only tried the most obvious ones.]

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At this point there is a choice. We could compare this composite spatial frequency spectrum with the spatial frequency spectrum of a calculated interference pattern, up to the same frequency cutoff, for a given hypothesis (e.g. a double star with 1 arcsec separation at a given position angle), and repeat for different hypotheses, comparing them via some goodness-of-fit parameter (15) such as [chi]-squared. Alternatively we could apply an inverse Fourier transform to the composite frequency spectrum, to yield a (hopefully) cleaned up spatial image, and compare this directly to the calculated spatial pattern (for a given hypothesis).

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We can either use proprietary Fourier routines (but these output with unspecified scales which we have found hard to unravel); or we can write our own Fourier routine with a proprietary integration routine, but the output also seems baffling. However, to indicate how Fourier analysis can help, Figures 17-22 are pairs of plots of derived spatial frequency where the first is the theoretical pattern, and the second is the plot obtained from the real image. The scale on the vertical axis of each plot should be ignored and the patterns compared. There are some similarities, but not enough to reach a conclusion. The horizontal axis is spatial frequency on an arbitrary scale, with its power density on the vertical scale.

When this method is working we plan to attempt to determine the position angle by comparing the plots for different orientations of the slits. When the slit pattern is parallel to the line between the stars the pattern should be the same as for a single star, and it might be useful to observe a single star as a check.

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Understanding the axes

In order to understand the scale of the horizontal axis (and the meaning of the results) and apply a spatial frequency cut-off we need to feed calculated (simulated) data into a Fourier integral of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where k and x are conjugate variables (e.g. time and frequency, or wavelength and spatial frequency) and f and F are the Fourier and Inverse Fourier transforms.

The authors were unable to evaluate an integral between infinite limits using the commercial program MatLab. (16) The specific problem is how to embed the second integral function into the first. It needs an analytic form for the first integral, not just a set of discrete values. One way forward would be to write our own routine to do the 'integration within the integration', and associated fitted polynomial. The problem of the infinite limits can be tackled by trying increasingly large limits until there are only small differences in the results (and before there is any divergence).

Summary and future work

In the past, the astronomical use of optical interferometry has necessarily involved the subjective appraisal of fringe clarity. With digital imaging, fringes can be objectively recorded, which makes the technique far more useful and attractive. The really exciting thing to do with interferometry is to measure stellar diameters, but that requires large scale, very high precision engineering which puts it firmly in the realm of the professional. For the amateur, interferometry offers a challenge of managing to achieve objective analysis as well as objective recording, with the satisfaction of slaying the monster of 'seeing' with short exposures and achieving a modest improvement in resolution, albeit only with simple structures such as double stars. The techniques are new, for both amateurs and professionals, in the sense that they have only become practicable with digital imaging. The techniques involve mathematical analysis; that is what makes them objective, there are no short cuts. It is important to realise that four slits cannot be treated as a sort of very simple diffraction grating, as the techniques are completely different.

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The next and most important step for us is to take the analysis to its conclusion, which, as explained in the text, we have not yet been able to do. When that is achieved, a practical method of handling and combining a large number of images--hundreds rather than tens--will be needed. On the experimental side, we see the next step as making properly machined metal slits. Ours were very crude, made in cardboard; the only reason one can get away with this at all is because the angles involved are very small. So there is no need to chase quarter wave optics (which would be impracticable anyway), but ours were so crude that they may have caused un-reproducible conditions and so obscured the results. Among factors that we varied, but not systematically enough to optimise, were the spacing of the slits, exposure time (which was nearly always 0.03sec), and magnification by Barlow lenses. It might be useful to try a prism to produce dispersion at right angles to the interference pattern.

Appendix I: the practical minimum magnitude limit

The direct calculation is easy and of some interest. A star of absolute magnitude 5 (like the Sun) emits about 4x10 (26) watts and at 10 parsecs this provides a flux of visible light photons of roughly [10.sup.7][m.sup.-2][s.sup.-1] at the telescope. An exposure of 0.03s therefore gives about 3x[10.sup.5] photons [m.sup.-2]. Our slits have a total area of about 0.0160[m.sup.2], so that gives about 5000 photons per frame.

A star of apparent magnitude 1 would therefore give about 270,000 photons per frame. This should be regarded as an estimate of upper limits, since there will be losses along the path. These are going to be spread into the interference pattern, which in our setup was typically 30 to 50 pixels long and a few pixels wide. So this calculation, rough as it is, indicates that with at best tens or hundreds of photons per pixel, random count errors will be considerable.

Appendix II: diffraction in the Fraunhofer plane

1. Single-slit

To describe diffraction and hence interference, it is normal to revert to Huygens' principle of 1678 which describes how every point on a wavefront can be thought of as contributing isotropic radiation to a secondary forward travelling spherical wavelet. The formal treatment of this is the Kirchhoff integral. Remarkably, all the unwanted waves cancel out. In the case of a single slit, any part of the slit can be thought of as such a point. The coherent superposition of these wavelets produced at the slit results in the diffraction pattern seen at the focal length of the lens (the Fraunhofer plane (6)). This diffraction pattern can be explained by the Fresnel-Kirchhoff formula.

2. Multi-slit

Mathematically, the problem is an extension of the single and double slit examples in two dimensions. (8) The interference patterns shown in Figures 1-8 were generated by summing the complex integral obtained from a single slit across the chosen number of slits, in our case 4 (see diagram).

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In the formula below U is the field vector (i.e. amplitude and phase carried as a complex number) at a (diffracted) angle [theta] for a beam of wave number k and is given by: (6,7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which when summed (coherently) for multiple slits becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where:

[theta] is the angle of deflection to a point on the pattern, as measured over the path length from the centre of the mask to the focal plane;

y is the distance across the slit pattern from outer edge of first slit ('the point of origin of the forward travelling wavelet');

b is the slit width (in our case 0.015m);

h is the distance between slit centres (in our case 0.05m);

k is 2[pi]/wavelength of the incoming light;

L is the length of the slit.

(See also refs. 5, 8, 9, 10). The intensity is the real part of U2.

Acknowledgments

We wish to thank Dr T. Smith for general discussions on interference phenomena and Dr Jonti Horner for very useful comments on a draft of the paper; both are of the Physics and Astronomy Department of the Open University. The authors are responsible for any remaining errors. John Yewen, a member of OUAC, was the first to raise awareness of stellar interference in the Club, with some intriguing demonstrations.

Received 2007 June 18; accepted 2008 March 26

(Open University Astronomy Club, Milton Keynes)

* We are concerned here with phase interferometry. Intensity interferometry is a (smaller) second order effect but brings the possibility of post-detection correlation, vital for all very long baseline interferometry. Such correlation was widely dismissed as physically impossible until Robert Hanbury Brown showed in 1956 (both observationally and analytically) that it was both possible and useful. Speckle interferometry is 2-dimensional phase interferometry. All these techniques are easier in the infrared because seeing is much better.

** A cleverer method might be to have a prism a short distance from the camera, oriented to spread the interference pattern by colour at right angles to its length. Then each line in the interference pattern appears as a short slanting spectrum, which can be projected out (i.e. summed) along its length, so that there is no light loss, which would be a marked improvement. The angle of slant varies along the pattern but in a way which is easily calculable beforehand and identical for every observation, for a given prism, so it could be made automatic.

Notes and references

(1) Michelson A. & Pease F. G., Ap.J. 51, 257-262, and 53, 249-259 (1920)

(2) Quirrenbach A., Ann.Rvw.Astron.&Astrophys., 39, 353-401 (2002) provides an overview and bibliography.

(3) Registax by Cor Berrevoets: http://www.astronomie.be/registax/

(4) Our interpretation of data given in a private communication from Terry Platt (of Starlight Xpress) in 2007 May.

(5) Andreas Glindemann VLTI tutorial website--The figures are particularly good; that on p.6 is a very clear explanation of the Michelson original method. http://www.eso.org/projects/vlti/general/ tutorial_introduction_to_stellar_interf.pdf

(6) Brooker G., Modern Classical Optics, Oxford University Press, 2003, p.51 and pp.199-248

(7) Fowles G. R., Introduction to Modern Optics, Dover Publications, New York, 1975, and http://www.phy.davidson.edu/StuHome/sethvc/Diffraction/pages/ experiments.htm

(8) Grievenkamp J. E., Handbook of Optics, McGraw Hill, New York, 1995, ch. 2

(9) Hecht E. & Zajac A., Optics, 4th edn., Addison Wesley, 2002 contains more graphical treatments.

(10) Maurer A., Sky and Telescope, 1997 March, p.91. This is a Michelson type of stellar interferometer in that the slit separation is the variable, and the observer seeks the separation(s) at which the interference pattern disappears.

(11) Maurer A., Chapter 14 in Argyle R., Observing and measuring double stars, Springer, 2004. This gives details of construction for an elegant four slit setup, but the explanation seems to treat this as if it acts as a diffraction grating and as is obvious from our paper (e.g. Figure 2) this is not true with broadband light, though the structure becomes simpler with OIII light (Figure 4). It is therefore difficult to give any meaning to the words 'order' and 'spectra' used by Maurer. References 10 and 11 both refer to purely visual observations, so there are no recorded images, and there are also no comments on the effects of 'seeing' in reducing resolution.

(12) Chambers P. et al., J. Brit. Astron. Assoc., 115(3), (2005). Video recording at the Open University telescope of the 2004 transit of Venus provided an excellent opportunity to measure both translation and distortion at 20ms intervals. (But this was of course during the daytime, when seeing is worse than at night).

(13) Harburn G. et al., Atlas of Optical Transforms, G. Bell, London, 1975. This shows the effects of a Fourier transform for various two dimensional patterns, and also for various data cut-offs.

(14) MaximDL--Astronomy image capture and processing software: http://www.cyanogen.com/

(15) Bevington P. R. & Robinson D. K., Data reduction and error analysis for the physical sciences, McGraw Hill, 2nd edn. (1994), ch. 11

(16) MATLAB by MathWorks is one of several programs that handles arrays, integration, etc. in an easy-to-use programmable way. It was used to save time in manual coding.

(17) Anderson J. J., 'On the application of Michelson's interferometer method to the measurement of close double stars', ApJ. 51, 263-275 (1920)

Sheridan Williams & Alan Cooper

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