Introduction

With large ground-based telescopes, the resolution of an image is not set by the diffraction limit of the telescope (which is to say it is not set by the size of the primary aperture of the telescope), but instead it is determined by the properties of the atmosphere through which the astronomical objects are viewed. For resolution, the problems of the atmosphere are twofold. First it contains inhomogeneities in the refractive index above the telescope and second these inhomogeneities change rapidly with time.

For example, with a 4-m telescope, the two effects combine to give a point spread function for a long exposure image that is some 50 times broader than the diffraction limit of the telescope at visible wavelengths.

When "speckle interferometry" began, the term simply referred to a single technique of retrieving diffraction-limited image information in the presence of the dynamic, inhomogeneous atmosphere. The principle of this technique is to take many very short exposure images of a target. Within each exposure, or "frame," the atmosphere, while still inhomogeneous, is effectively frozen. By studying the autocorrelation function of frames, it is possible to extract diffraction-limited information about the object. It is not possible in "classical" speckle interferometry, as it is now sometimes called, to reconstruct a diffraction-limited image. Nonetheless, it represents a fundamental improvement in the resolution obtainable from large ground-based telescopes.

The "speckle imaging" refers to several methods of extending the technique of classical speckle interferometry in order to obtain the necessary extra information to produce a true reconstructed image. Generally speaking, these methods can be used on the same data taken in the same way as with classical speckle interferometry. Therefore it is convenient to say that both classical speckle interferometry and speckle imaging lie within the collection of techniques to which we may broadly apply the term "speckle interferometry."

Mignard et al. (1) have discussed that, in addition to measuring the trigonometric parallaxes of approximately 118,000 stars, the Hipparcos Satellite has discovered some 6000 new or newly suspected double star systems. A fraction of these systems are no doubt true (i.e., gravitationally bound) binaries that could provide valuable information concerning stellar masses and the main sequence mass-luminosity relation, if their orbits were known in sufficient detail. A distinct advantage of this set of stars in determining masses and luminosities is that their parallaxes and in many cases the magnitude differences have already been measured by the Hipparcos Satellite. However, with the exception of the relative astrometry appearing in the Hipparcos Catalog (European Space Agency, 1997(1)), no substantial orbital data exist for any of these objects at present.

Many of these newly discovered systems have separations and magnitude differences that are easily observable by way of speckle interferometry. Dr. Horch has therefore initiated a new program in 1997 of speckle observations of this set of stars at the WIYN Observatory, firstly to identify which systems exhibit orbital motion and secondly to determine the orbits of the true binaries en route to deriving their masses. The combination of relative astrometry derived from speckle observations and parallax can only provide total masses of the systems under study, but concurrent spectroscopic observations can be started in cases where orbital motion is detected. In this way, individual masses can eventually be obtained. The emphasis of this project is to discover new binary stars from the Hipparcos discovery objects.

Background

Binary Stars

A binary star consists of two stars that are gravitationally bound to each other. A double star consists of two stars that appear to be very close to each other in the sky and may be gravitationally bound and may be separated by large distances. Binary stars are a subset of double stars. The first double star was observed in 1650 by Mizar, the star in the crook in the handle of the Big Dipper. Sir William Herschel showed that some double star systems are gravitationally bound in his 1803 paper.

Binary stars are detected by observing newly discovered double stars over a long period of time to see if the secondary star in the system is moving around the primary. The secondary star in a binary system travels in an elliptical orbit around the primary. In a system that is not gravitationally bound the secondary's path is a straight line. Orbital motion is proven when the data points for the secondary star do not lie on a straight line. If two stars are in fact gravitationally bound then the secondary star position data will show that the secondary star travels in an arc around the primary star. Newton's stated that bodies move in straight lines with constant velocities unless acted upon by an outside force. Newton's Universal Law of Gravitation states that two bodies will act a force on each other equal to the masses of the bodies and the distances between them. When two stars are near enough together there gravitational attraction will causes the stars to accelerate towards each other, with the less massive star moving much faster than the primary star. When the secondary star moves in a straight line over time it means that stars are very far apart and are not gravitationally bound to each other.

Speckle Interferometry

Consider an idealized astrophysical point source of monochromatic radiation. If this radiation were to enter the telescope in the absence of the atmosphere, it would be a plane wave and would therefore have uniform magnitude and phase across the telescope aperture. For the purposes of this discussion, it will be sufficient to use the model of a scalar electromagnetic field, so that the "aperture function" may be defined as the scalar (complex-valued) function representing the electromagnetic field entering the primary telescope aperture. An elementary theorem from optics states that the point spread function (henceforth called the PSF) of the telescope is the modulus square of the Fourier transform of the aperture function:

.........................................................................(1)

where S(x) is the PSF, A(z) is the aperture function, ^ denotes the Fourier transform, and the Fourier conjugate variable to z is x. The width of the PSF defines the resolution obtainable at the image plane. Since in astronomy the image corresponds to the irradiance distribution projected on the sky (where the positions of objects are measured in angular terms), it is common to refer to the width of the PSF in terms of arcseconds.

The atmosphere changes the character of the aperture function. Due to motion and density fluctuations in the air above the telescope aperture, inhomogeneities in the refractive index develop. These inhomogeneities have the effect of breaking the aperture into cells that have a typical length scale of about 10 cm and that are translated by a prevailing wind (in the simplest model) across the telescope aperture on a typical time scale of 10 ms. Within a cell, the phase of the aperture function remains roughly uniform but it takes on different values from cell to cell. Because of the relationship between the aperture function and the PSF, the PSF is likewise changed; in fact, the time-evolution of the aperture function implies that the PSF is also time-dependent. If the atmosphere is frozen at a particular instant in time, the resulting PSF may be described as the sum of interference patterns between pairs of sub-apertures defined by the cells arising from the inhomogeneities in the refractive index. Each such pair would produce a fringe pattern whose spacing is determined by the distance between the two sub-apertures (as in Michelson interferometry), but taken together, the various fringe patterns form a complicated interference pattern known as a speckle pattern. The speckles (or regions of constructive interference) fill a broad region typically 1 to 2 arcseconds in diameter in the image plane, but have a characteristic width similar to that of the diffraction-limited PSF of the telescope.

Over time, some of the phase cells pass out of the telescope aperture and new ones enter, and so the position and irradiance of the speckles changes on the image plane. On an exposure considerably longer than the time scale of atmospheric fluctuations, the speckle nature of the PSF washes out to leave only the shape of the broad envelope, called the "seeing disk" of the star. For this reason, the resolution possible with conventional ground-based astronomy is much poorer than the diffraction-limited resolution of large telescopes. While the speckle patterns contain structure on the scale of the diffraction limit, these structures are lost in any exposure of an object longer than a few tens of milliseconds.

The data for this project is being collected as a part of the ongoing speckle imaging program at the WIYN Telescope, in collaboration with Yale University. Bispectral analysis is used to analyze the data collected from WIYN. Speckle interferometry allows astronomers to achieve a diffraction-limited image, which is the theoretical image a telescope can take if there was no atmospheric interference. Speckle interferometry was discovered as a new technique in 1970 by Labeyrie (2) to try to achieve better images using conventional telescopes. He pointed out that there is a way to obtain the diffraction-limited spatial frequency power spectrum of an object from a collection of individual speckle patterns. For sources of sufficient simplicity such as binary stars, the power spectrum is a powerful tool for deducing high-resolution features. Because binary stars are extremely important in astronomy for several reasons, Labeyrie’s method held the potential for a kind of revolution.

The key was to be able to record the speckle patterns. The fluctuations of the atmosphere occur on a time scale of typically about 10 ms, so a speckle observation would consist of a sequence of many short exposures of this typical exposure length. Each frame is then a fixed quasi-time-independent speckle pattern. In this case, each frame preserves structure on scales much smaller that the seeing disk, as evidenced by the speckles themselves and to the extent that the speckles can be resolved by the imaging detector on this time scale, each frame contains high angular resolution about the source.

By way of showing how Labeyrie’s technique works, suppose that the object of interest is a binary star. Then the speckle patterns will exhibit speckle pairs at the vector separation of the two companions. Of course, these speckle pairs will occur at various places on the image plane, but they may be collected together by considering the autocorrelation function of the frame, which, up to a normalization factor, is equivalent to the histogram of the distance vectors between photon pairs. The autocorrelation function of a frame is defined by

..........................................................(2)

where I(x) is the frame image (or equivalently, the frame irradiance distribution). The Fourier transform of the image autocorrelation is the image power spectrum (defined as the modulus square of the Fourier transform of the frame image). The frame image, on the other hand, is the convolution of the instantaneous PSF S(x) with the actual irradiance distribution of the object O(x):

..............................................(3)

In the Fourier domain, a convolution becomes an ordinary product so that

.......................................................(4)

where ^ denotes the Fourier transform, and the spatial frequency variable conjugate is x to u. Taking the modulus square of this expression and averaging over many frames, the average image power spectrum is obtained:

........................................................(5)

where denotes the averaging over many frames. The function is known as the speckle transfer function. If a speckle observation is performed on the binary star, can be obtained by Fourier transforming the average autocorrelation of many frames of data. The same process on a point source (that is, an unresolved star) yields an estimate of . As equation (5) shows, the true binary power spectrum can then be obtained by division, where such a division is valid in the region where is nonzero. In fact the average PSF power spectrum is non-zero out to the diffraction limit of the telescope, and of course, the method works not only for binary stars but also for general objects. In the case of a binary system, reconstructing the object power spectrum is in principle sufficient to retrieve the separation, position angle (up to a 180° ambiguity), and the relative magnitude of the two companions.

The method is straightforward, but there are observational limitations. First, speckle patterns are wavelength dependent, so it is necessary to use a narrow band pass filter to maintain a high degree of coherence to the patterns on the image plane. Using a narrow filter significantly reduces the total irradiance of the speckle pattern, and limits the range of magnitudes for which the method is useful. In fact, the larger the telescope diameter, the narrower the band pass one must use, so that the limiting magnitude is only a weak function of collection area of the primary mirror of the telescope. Second, even with narrow band pass filters, the color dispersion of the atmosphere (which is a function of zenith angle) is large enough to elongate speckles on the image plane, and this effect must be removed. This is generally done with the use of two zero-deviation Risley prisms which can be rotated independently. The dispersions of the two prisms add vectorially, so it is possible to rotate the prisms to positions such that the resultant exactly counteracts the dispersion vector of the atmosphere. Third, because the atmosphere is changing rapidly, the imaging device must be able to read frames out quickly and have the capacity to simultaneously oversample the speckles and image the entire seeing disk. Fourth, the deconvolution is only valid when the object of interest is contained within the region of the sky known as the "isoplanatic patch." If the components of a binary star are widely separated on the sky for example, the path of the light from the primary through the atmosphere will be different enough from the secondary so that at the telescope they will produce different instantaneous aperture functions, and therefore different speckle patterns. In this case frames would not exhibit speckle doublets as discussed above. The isoplanatic patch describes the area of sky over which speckle patterns remain very much the same and is usually only a few arcseconds in diameter.

The speckle transfer function T(u) is defined by

......................................................................(6)

and relates the image power spectrum to the true object power spectrum. The theoretical models of T(u) generally break the function into two components, a seeing-limited (low frequency) portion approximated by

...................................................(7)

and a diffraction-limited (high-frequency) portion approximated by

..............................................................(8)

where , is the "Fried parameter" describing the length scale of coherence cells in the atmosphere, D is the telescope diameter, is the wavelength of observation, and is the (diffraction-limited) telescope transfer function. Though most of the power is in the low-frequency seeing-limited portion of the function, a high-frequency shoulder can also be seen out to the diffraction limit of the telescope. The low-frequency portion of the transfer function has a width which is determined by the Fried parameter. If is large, the seeing peak in frequency space is large, and the seeing disk on the image plane is therefore small, but if it is small, then the low-frequency portion of the transfer function is small and the seeing disk is large. The quantity , which is a measure of the width of the low-frequency transfer function, is sometimes called the "seeing cutoff" in frequency space. The high-frequency portion of the transfer function is an attenuated version of the transfer function of the telescope, which means that the speckle transfer function manifestly extends to the diffraction-limit, roughly given by the ratio in the frequency plane. The attenuation factor involves the quantity ; because of the relationship between the Fried parameter and the seeing, the value of the transfer function on the speckle shoulder is therefore a sensitive function of the seeing.

An important consideration of the method of speckle interferometry is the expected signal-to-noise ratio (SNR) of the image power spectrum. In the high frequency wing of the power spectrum, the SNR is given by

........................................................(9)

for the general object, where is the number of photons per speckle, and M is the number of frames averaged. When the number of photons per speckle is small compared to , the SNR increases linearly with , meaning essentially that the frames are limited by photon statistics, and when the number of photons per speckle is large compared to , the SNR becomes nearly independent of , and the frames are limited by the atmospheric process.

This interplay between photon statistics and the atmosphere also determines the frame integration time. The frame time should be chosen so that the motion of the speckles within the frame is small, but if too short a frame time is chosen, frames will be photon-starved and this leads to a loss of SNR. On the other hand, if too long a frame time is chosen, speckle motion will lead to a loss of contrast in the speckles, and again the SNR is reduced. The competition between these two effects results in an optimal frame time which depends on atmospheric (or equivalently, on seeing) conditions. Shorter frame times are required when the seeing is poor.

Equation (9) can be rewritten for the case of a point source as follows:

...........................................................(10)

Here, n is the number of photons observed per frame and again the spatial frequency variable is . Equation (8) shows that the value of is dependent on the ratio of the Fried parameter to the diameter of the telescope, and so the SNR also depends very much on seeing conditions. However, it is also clear from equation (10) that the maximum SNR per frame in the high frequency of the power spectrum is 1. For a typical frame integration time of 10 ms, this means that in 6000 frames, or 1 minute of observing, the SNR in the high frequency portion of the power spectrum will be . A similar type of calculation shows that a good figure to hold in mind for the limiting magnitude of the method is something on the order of 18 in the visible. In general, the SNR per frame for good speckle observations can be a substantial fraction of the maximum value, especially for bright objects.

Bispectral Analysis

In principle, if one had a diffraction-limited estimate of the (generally complex valued) function , it would be simple to Fourier invert the function to obtain a reconstructed image. In order to do that, however, one needs to have estimates of both the magnitude and the phase of . Classical speckle interferometry as discussed above falls short of providing true image reconstructions because the recovered function is the the object's power spectrum, i.e. , and manifestly has no phase information. This is known as the "phase problem" in speckle interferometry. Bispectral Analysis is a technique to solve the 180 degree ambiguity problem.

The standard way to calculate the image bispectrum is to start from the triple correlation of a frame image, which is defined by

....................................(11)

The Fourier transform of the triple correlation is called the image bispectrum and can be written in the form

.......................................(12)

The right side of equation (12) may be rewritten in terms of the instantaneous PSF and the true object irradiance distribution, just as with the cross-spectrum. Averaging over many frames, this becomes

.................(13)

The quantity is known as the bispectral transfer function. It can be shown that the bispectral transfer function has zero phase, so that the phase of Equation (13) is simply

....................................(14)

Consider the region of the bispectrum such that and with some small fixed increment in frequency space. The bispectrum is a four-dimensional function for a two-dimensional image, so the requirement fixed defines a subplane of the bispectrum close to the plane . Let the phase of at u be denoted by , and Equation (14) may be rewritten as

........................................(15)

If it were not for the term , this would again be a difference equation for the function as with the cross-spectrum, and therefore, an estimate of this function could be built up over the entire frequency plane using two nonparallel vectors (or equivalently, using at least two subplanes from the bispectrum). The added term complicates this, however. If one chooses arbitrarily, the effect on the recovered function is to add a linear term to it. A linear phase term added to a function in frequency space merely translates the position of the inverse transform in image space, but image features are not affected. It is therefore sufficient to choose arbitrarily for the purposes of image reconstruction. A standard initial condition to integrate the difference equation is to set .

Bispectral analysis produces, from the speckle data, an estimate of the phase derivative of and this function must be integrated before it is combined with a modulus estimate which is obtained from the standard autocorrelation analysis of classical speckle interferometry. In a practical sense, the two methods can be compared in the following way. The cross spectrum is derivable from a two-point correlation function of the image irradiance distribution (just as the power spectrum is derivable from the autocorrelation function). The bispectrum on the other hand is the Fourier transform of a three-point correlation function, namely the triple correlation. This means the bispectrum takes longer to compute, because each photon is correlated with combinations of two other photons, not just one. There are ways to reduce the amount of computation in reconstructing images, but nonetheless, the bispectrum is more cumbersome to deal with than the cross spectrum. However, the bispectrum has the advantage of high SNR, and it is relatively insensitive to telescope aberrations.

Wirntizer (3) first studied the SNR of the bispectrum, and Nakajima (4) gave the first general expression for the SNR at arbitrary light levels. At low light levels, the SNR is proportional to , where n is the number of photons per frame. This can be compared to the power spectrum, where in the low light level regime, the SNR is proportional to n. In the low light regime, only the so-called "near axis" subplanes (subplanes with small) have high SNR. For simple images, these subplanes are usually sufficient to derive good reconstructed images in the way discussed above. For more complicated images, other regions of the bispectrum may play a more important role in determining the phase map. Nakajima found that the limiting magnitude for bispectral analysis was between 13th and 15th magnitude, depending on telescope aperture size and seeing conditions.

Phase Reconstruction and Image Reconstruction

In the discussion above of the cross spectrum and the bispectrum, we have seen that these functions give an estimate of the phase derivative of the function , but from this point there are two steps before arriving at a reconstructed image: 1) phase reconstruction and 2) Fourier inversion of . Neither step is trivial in speckle image reconstruction.

The phase is obtained from the bispectrum or cross spectrum by integration, but it is possible to integrate the phase derivative map in several different ways, the simplest being via iteration, proposed by Lohmann et al. in 1983. From the initial condition of zero phase at the origin (pixel 0) and the difference equation, the phase at pixel 1 is calculated, then from the value at pixel 1 and the difference equation, the value at pixel 2 is calculated, and so on. However, this method has the disadvantage that errors increase as the spatial frequency increases, making the phase estimate least reliable in the high frequency region of frequency space. Another technique which was used by Meng et al. (5) is a relaxation method, where zero phase or some initial guess at the phase map is assumed at the start and the phase at a particular pixel in the frequency plane is assigned by using the difference equation and the starting values of the phase at nearby pixels. This produces a new phase map which replaces the initial guess and the process is repeated. After many iterations, the phases converge to the "correct" values.

Methods

Measurement of the Pixel Scale

For the CCD data, the fundamental scale calibration was obtained by attaching a slit mask to the tertiary mirror baffle support structure of the telescope, which is located approximately 84 cm above the tertiary mirror. The mask is therefore placed in the converging beam between the secondary mirror and the image plane at the focus. When the telescope is pointed at a single star, a diffraction pattern is produced on the detector such that the spacing of fringes uniquely determines the scale in arc seconds per pixel, once the slit spacing, the distance from the Nasmyth focus to the mask and the f/number of the converging beam are known. The WIYN has an altitude-azimuth mount, with image ports at the Cassegrain and Nasmyth foci.

The position angle offset was determined by taking a series of short exposure 1 s images of stars, where the telescope was offset in right ascension and declination by a known amount (10", 15", or 20") between exposures. Computing the centroid position of each image and comparing these coordinates can then yield the detector orientation. This method was found to be faster and more practical than the star trail method when using the integrating detector. The pixel scale detection is more fully described in Horch et al. 1997 (1).

Figure 1: Speckle Images from the WIYN telescope.

Actual data from the WIYN telescope. These are 5 samples of the .fit files in which the data is stored. Each observation from the star contains hundreds of frames like each column above. These are the raw speckle frames that are captured by the telescope.

Figure 2: The Image of Hip 689

The image of Hip 689 below is diffraction-limited. The central peak is central star of system. The secondary star is the peak to the right of the central star. The third peak to the left of the central star is an artifact of the image processing technique. The artifact is caused because only two subplanes of the bispectrum are used to calculate the spatial image. Only the location of the secondary is critical, so calculating the entire bispectrum is unnecessary and wastes a large amount of computing time. This method of only calculating the first two subplanes is the most efficient method of finding the position of the secondary star.

Derivation of Relative Astrometry from the Raw Data

The Images were stored on CDs as .fit files, which are in the form of long strings of images. An example of these speckle images is shown above. The images are then transformed into a single image in the Fourier domain using C programs written by Dr. Horch. If fringes are seen in the Fourier domain images, then a second star is located very close to the target star. The spacing of the fringes demonstrate how close the two stars are together in degrees. The images are then transformed into a spatial image which shows a bright peak in the center of the image, which is the primary star, and possibly two dimmer peaks on opposite sides of the central peak. The two peaks, in they are present, represent a secondary star. The position of the secondary star is recorded to the nearest pixel by identifying the peak location in the image. The position angle (i.e. the angle between the vector connecting the primary and secondary and the vector pointing to celestial north) can then be calculated. The separation of the two stars (in arcseconds) is obtained by noting the number of pixels between the primary and secondary in the image and applying the plate scale determined as discussed in the previous section.

If the secondary star can be detected, then past measurements of the system are examined to determine if the stars are gravitationally bound. The position data for systems in which I detected a secondary star and for which past position data exists are plotted. Some of the position data from several of the systems showed the position of the secondary tracing out an arc around the primary star. This can only be caused by a gravitational attraction between the two stars. This indicates that the two stars from a binary system. If the secondary star traced a straight line across the primary star, then the stars are not gravitationally bound are merely behind each other in the sky from our view of the stars.

Results

Of the 15 stars that were studied in this project, four of the double star systems are confirmed to be gravitationally bound. The orbital data for these stars is shown in figures 3-6 below. Four of the 15 stars that were observed had no detectable secondary. This lack of detection could be due to the very large magnitude difference between the primary and secondary stars, or could be the result of noise, poor visibility, or some other cause that we did not examine yet. The other stars that were observed could not be proven to be binary stars.

Positive Results: Four True Binary Stars

The four stars systems that were confirmed in be binary in this project are Hipparcos catalog numbers 689, 11352, 17891, and 28019. The orbital position data for these four stars is given below.

Figure 3: Hip 689 is a binary star 79 parsecs from Earth and the orbital period is estimated at 25 years.

Figure 4: Hip 11352 is 43 parsecs from Earth and has a estimated orbital period of 12 years.

Figure 5: Hip 17891 is 71 parsecs from Earth and has an orbit estimated at 10 years.

Figure 6: If the system is gravitationlly bound, the secondary in the system Hip 28019 is a blue subdwarf, probably cooling down to bacome a white dwarf.

The three data points have indicate that the white dwarf is orbiting the primary star, but more data will need to be collected be sure. Anaylsis of the three data points indicate that the 11/19/99 data point is more that sigma from a straight line running through the first two data points, which means that the probability that the secondary star is traveling in a straight line is less that 2.5%.

Position data for the other star systems

The results of the observations of eight of the fifteen double stars are shown in table 1. The secondary star in the other studied double star systems could not be identified using the data collected in November.

Table 1: Positions of Secondary Stars

The secondary stars were detected and were shown to have movements around the primary star. For some of the double stars the secondary is close to the diffraction limit of the telescope and the exact position of the secondary star is difficult to calculate.

Discussion

The binary star H 689 is proven to be gravitationally bound by the fact that the position points trace an arc about the primary star. The first point labeled Hipparcos 1991 was collected by the Hipparcos satellite. These data were combined with data published by Dr. Horch in 1997 (1). That datum point is labeled 8/14/97. The last two data points were found using recent images in which I had to determine the position angle and separation distance using the 128 X 128 pixel images created by Dr. Horch for the January 1999 datum and by me for the November '99 datum. These four points are plotted in Figure 3 and show a portion of the orbit of the secondary star around the primary star. The orbit for this binary star was not calculated due to the fact that there is still a lot of uncertainly in the data points. Any orbital calculation made now would only have to be revised in a few years. The orbit for these stars will be published when the star has completed one full orbit in about 16 years.

The binary stars Hip 11352 and 17891 are also proven to be binary as shown in the arcs that the orbits trace out. The secondary stars in these systems both have shown movements around the primary stars as shown in figures 4 and 5. Both of the secondary stars in these systems have completed nearly a full orbit since the Hipparcos discovery observations. The orbits for these two systems appear to be less that 15 years. The data for these two star systems are collected from the Hipparcos satellite in 1991 and data from the WIYN telescope over the last four years. These data were used to plot the position of the secondary star versus the primary star and an arc of the orbit was found.

Hip 28019 an interesting system because the secondary star is believed to be a hot subdwarf. The secondary star in classified as this type of star because of its strong emmitance of UV as detected by Schmidt and Carruthers in 1993. The mass of hot subdwarfs are not known because of their short life span and their low luminance. A hot subdwarf is theorized to be the connection between a planetary nova and a white dwarf. This stage of stellar evolution lasts only a few million years, which is very short lived in the billion year life of stars.

The other double stars that were studied in this project have not been proven to be binary stars. The secondary stars in most of the 15 systems that were studied were detected using this technique. Other that the four stars shown above, the other systems could not be proven to be gravitationally bound. The secondary star position data were plotted versus the primary star data for the double stars systems. Although movement was detected for the secondary star, the data either plotted a straight line indicating that the system is actually two separate stars or the position data were not taken over a long enough baseline.

In some of the systems that were studied the secondary star could not be detected. This is due to the large magnitude differences between the primary and secondary stars. These systems were know to be double because they were found by the Hipparcos satellite in 1991. In the next few months, these stars will be studied using the full bispectrum to try to detect the faint secondary star in these systems.

Conclusions

The binary stars Hip 689, 11352 and 17891 are gravitationally bound as proved using speckle interferometry. This is the first time any of these stars have ever been analyzed and they have now been shown to be true binary stars. The orbits for two the the three systems shown to be binary have orbits of only 10 - 12 years.

Hip 28019 is an interesting object to astronomers because the secondary object is believed to be a hot subdwarf. If the hot subdwarf is proved to be in orbit around the primary, then the mass of the hot subdwarf can be calculated from its orbit. The detection of the orbital motion of these binary stars is an important step to understanding the creation of solar systems and for better understanding stellar evolution.