CP-2156 Life In The Universe

 

[177] Detecting Planets in Binary Systems with Speckle Interferometry

SIMON P. WORDEN

 

Speckle interferometry is particularly suited for work with close-spaced binaries, where conventional astrometry has difficulties. The technique does not require special telescopes to study the 188 known close binary pairs nearer the Sun than 65 light years.

 

Several methods have been suggested for the detection of low-mass, nonluminous companions of nearby stars. Among the most promising are indirect methods, that is, methods for detecting the effects of an invisible object on its visible companion. Spectroscopic methods can be used to observe radial-velocity fluctuations in the visible star caused by orbital reflex motion relative to the invisible companion. Astrometric detection involves observing the positional perturbation caused by the orbit of the unseen object. Astrometry therefore requires establishing the position of the center of light in a stellar image and referencing this center to a fixed frame.

Astrometry is limited by our ability to establish the center of a star image and by uncertainties in the reference frame. Star images, perturbed by atmospheric seeing, are roughly gaussian in shape. Our ability to find the center of an image obviously depends on the stability of the gaussian profile. Long exposures are used to collect enough photons to define the center of each star image and to average out seeing for image motion. The reference frame of these measurements can in principle be established by the centers of three or more other star images. Since the proper motions of these stars can influence the coordinate frame, the reference stars are chosen to be distant background stars. Earth's atmosphere can also distort the reference frame as a function of atmospheric conditions and the colors of the reference stars. These uncertainties can be reduced by using a relatively large number of reference stars in conjunction with sophisticated mathematical error analysis schemes. Despite such difficulties, conventional astrometry [178] now yields positional accuracies of ±0.05 arcsec for a single exposure; this may be improved to better than ±0.001 arcsec (Gatewood, 1976) for the results of a year's observation.

Astrometric accuracy could be improved by getting smaller star images. Clearly, the smaller the image, the easier it is to find an image center. If images were improved to the diffraction limit, the increase in accuracy would be substantial: finding the center of a diffraction-limited spot of 0.05 arcsec from a 2-m telescope can be done with much higher precision than finding the center of a 1-arcsec seeing disk with the same number of photons. We can also increase accuracy by removing the atmospheric effects on the reference frame. Both improvements are possible using some form of interferometry.

Stellar interferometry, first demonstrated by Michelson (1920), makes it possible to approach the full theoretical (diffraction-limited) resolving power of large optical systems. Instruments with baselines up to hundreds of meters long have been demonstrated (Hanbury Brown, 1968). Smaller-scale instruments have been perfected to adapt existing telescopes into diffraction-limited systems using techniques known as amplitude interferometry (Currie, 1968a,b; Currie et al., 1974) and speckle interferometry (Gezari et al., 1972). These techniques have been extensively reviewed by Dainty (1975), Worden (1977), and Labeyrie (1978). Long-baseline interferometers have been proposed which would convert the image scale to the full diffraction limit and remove atmospheric effects on the reference frame. Such systems are ideal for detecting small-scale astrometric perturbations due to objects of planetary mass. However, these instruments are costly and have yet to be completely demonstrated in the field. On the other hand, amplitude and speckle interferometry have already demonstrated the ability to extend conventional telescopes to the diffraction limit.

Binary stars are suitable candidates for planetary searches. McAlister (1978) has used speckle interferometry with relatively simple defectors end calibration methods in a systematic program to determine binary separations to accuracies of a few thousandths of an arcsecond. If a planet orbits one component of a binary system, then the orbit of that component about the other star will "wobble." For the nearby stars and Jupiter-sized planets, this may be a modulation of up to 0.01 arcsec. Harrington (1977) has shown that most binary systems have potentially stable planetary orbits. Interferometric searches for planetary perturbations of binary star orbits are therefore important. Indeed, interferometry may be the only way to search small-separation binaries. Binary star interferometry is therefore important to assure a complete search for low-mass stellar companions through all nearby stars and all types of stellar systems.

The speckle interferometric method to search binary star systems for planets was presented in some detail by McAlister (1977). In this paper, I [179] will discuss only briefly the physics behind atmospheric degradation of images and the speckle interferometric technique to remove this degradation. I will restrict myself to speckle systems, although much of what follows is also true of amplitude interferometers. A more detailed discussion of both Speckle and amplitude interferometric searches for planets has been given by Currie et al. (1980).

 

ATMOSPHERIC IMAGE DEGRADATION

 

Small-scale temperature inhomogeneities in Earth's atmosphere produce changes in the atmosphere's index of refraction. These changes cause phase delays along any incoming plane wave, such as light from a stellar point source (fig. 1). Without phase errors, optical systems would produce the image in figure 1 (a), which is said to be "diffraction-limited"; here the image of a point source is the classical Airy disk for a circular telescope aperture. The size of this image is inversely proportional to the telescope diameter. With phase errors, telescope resolution is degraded to that of an optical system only as large as the scale over which some phase coherence exists (i.e., over which the phase is the same). Since the atmosphere breaks an incoming plane wave into fragments of about 10 cm, all telescopes produce images with resolution no better than a 10-cm telescope, namely, 1 arcsec (fig.1 (b)).

Labeyrie (1970) proposed a method to recover information down to the diffraction limit of a large telescope. He pointed out that short-exposure photographs (~0.01 sec) "freeze" the turbulence in the atmosphere. Although the phase coherence size in this frozen system is still only 10 cm, there will be some 10-cm patches scattered over the entire aperture which are at the same phase. These portions act in concert as a form of multiple aperture interferometer that provides some information down to the diffraction limit of the entire telescope aperture. As shown in figure 1 (c), the image of a point source seen through a multiple-aperture interferometer is a series of nearly diffraction-limited images modulated by a 1-arcsec seeing disk. The process is known as speckle interferometry since the short-exposure photographs (fig. 2) look like laser speckle photographs.

 

SPECKLE INTERFEROMETRIC INSTRUMENTS AND DATA REDUCTION

 

A diagram of the Kitt Peak photographic speckle interferometer is shown in figure 3. There are six similar systems in use at the present time. The Kitt Peak camera was designed by Lynds (Lynds et al., 1976;

 


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180]

Figure 1. Schematic diagram of image formation through a turbulent atmosphere.

Figure 1. Schematic diagram of image formation through a turbulent atmosphere. (a) Image formation outside the atmosphere: diffraction-limited image. (b) Image formation through Earth's atmosphere; image is badly degraded. (c) Image formation using speckle techniques; envelope results from atmosphere seeing while the high-resolution features result from the resolution of the full aperture.


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181]

Figure 2. Speckle photographs for three stars from the Kitt Peak 4-m telescope

Figure 2. Speckle photographs for three stars from the Kitt Peak 4-m telescope; note the different characters of the three objects. (a) Resolved supergiant star Greek letter alphaOrionis (Betelgeuse), (b) Point source star Greek letter gammaOrionis (Bellatrix), (c) Close double star; separation, 0.05 arcsec, Greek letter alphaAuriga (Capella).

 

Breckinridge et al., 1979). As shown in the figure, light from the telescope passes through a shutter and focuses at the telescope image plane. The shutter is necessary to ensure exposures shorter than the atmospheric change time, typically 20 msec. The telescope image is relayed and magnified by a microscope objective. The magnification is set to provide a pixel resolution oversampling the telescope diffraction spot size by at least a factor of 4. For the Kitt Peak 4-m telescope, this provides a final image scale of 0.2 arcsec/mm. Atmospheric dispersion blurs speckle image patterns in the sense that the "red" portion of an image focuses at a slightly different place than the "blue" portion. Since this may be significant for even 200-Å bandpass photographs, a set of rotating atmospheric compensating prisms are included to counteract the dispersion. Since there are about 20 orders of optical interference across even a narrow band (Delta  Greek letter deltais equivalent to200 Å), an interference filter is used to preserve coherence across the entire speckle photograph. If this were not included, the "speckles" near the edge of the photographs would be elongated. A three-stage image tube intensifies the image enough to allow photographic data recording. A transfer lens relays the intensified image to a data recording system, in this case a 35-mm film camera.

The speckle photographs in figure 2 were taken with the Kitt Peak system. The different character of these photographs is readily apparent. This is understandable from the analogy to a multiple-aperture interferometer. Each speckle should be a diffraction-limited image of the object. Indeed, the binary-star (Greek letter alphaAuriga) speckles are double, the point-source speckles are roughly diffraction spots, and the resolved-star (or Greek letter alphaOrionis) speckles are somewhat larger. This aspect led Lynds et al. (1976) to a direct speckle-image reconstruction scheme whereby individual speckles were identified and....

 


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182]

Figure 3. Diagram of the Kitt Peak photographic speckle interferometry camera.

Figure 3. Diagram of the Kitt Peak photographic speckle interferometry camera.

 

[183] .... co-added to produce a nearly diffraction-limited image for the special case of stars like Greek letter alpha Orionis.

A number of methods can be used to reduce speckle interferometry data. Labeyrie's original method is widely used-particularly for measurements of binary stars. Individual speckle photographs are Fourier-transformed either optically or digitally, and the Fourier modulus is computed. If the speckle image is represented in one dimension as i(x) and its transform as I(s), this process is mathematically represented by

 

mathematical equation(1)

 

The modulus or power spectrum, |I(s)|2, of this transform contains the diffraction-limited information in an easily extractable form. Examples of mean power spectra for several binary systems are shown in figure 4. The power spectra for such systems show banding that represents the separation of the elements: the farther apart the bands, the closer the elements. The orientation of these bands represents the position angle of the binary system. Superimposed on this signal is a background attributable to the residual effects of seeing. For stars brighter than about +7 visual magnitudes, about 50 individual speckle snapshots are transformed to produce a mean power spectrum. A least-square fit to the spacing and orientation angle of the bands in this power spectrum yields the binary separation and position angle-all accomplished from less than 1 sec actual exposure time at the telescope!

The residual effects of seeing must be removed to achieve maximum accuracy. Even though the bands (fringes) are readily visible in raw speckle power spectra, their spacing is affected by the residual seeing effects. Labeyrie's method uses observation of point-source stars to determine and remove these seeing effects. If pi(x) are point-source speckle photographs with a mean power spectrum <|P(s)|2>, and <|I(s)|2> is the mean power spectrum of the object speckle photographs ii(x), the diffraction-limited power spectrum of the object is

 

mathematical equation(2)

 

Point-source data are usually derived from speckle observations of pointsource stars situated near on the sky to the program objects. Since these point-source objects are not generally observed within the same isoplanatic angle or at the same time, their power spectrum can only represent the residual seeing effects in a statistical sense. Worden et al. (1977) have developed a method to calibrate for residual seeing effects using the same set of speckle photographs as were used to study the program objects.

 


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Figure 4. Mean speckle power spectra for two binary stars

Figure 4. Mean speckle power spectra for two binary stars. Larger separated fringes are for i Serpentis (separation, 0.7 arcsec), the smaller for Greek letter betaCephei (separation, 0.25 arcsec).

 

Current photographic speckle cameras are generally limited to objects brighter than +7. The photographic recording systems are therefore being replaced with high-quantum-efficiency, digital recording systems that record individual photon events. The University of Arizona's speckle camera uses a CID (charge-injected device) television system to record photon arrivals. This system simply replaces the photographic emulsion, and it can record data for objects faint enough so that only a few photons arrive in a 20-msec exposure. Figure 5 shows data from this system for Saturn's moon Rhea, which is a 10th magnitude object. For faint objects like this, only the fey' hundred photon locations are recorded rather than the entire frame. This allows such systems to run at the maximum speckle data rate of one speckle.....

 


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Figure 5. Speckle data showing individual photons for Saturn's moon Rhea taken with the University of Arizona's CID speckle camera.

Figure 5. Speckle data showing individual photons for Saturn's moon Rhea taken with the University of Arizona's CID speckle camera.

 

....frame every 20 msec. This form of data is ideal for fast computer reduction. Data reduction is simple enough that a direct computer interface can compute the results in real time at the telescope. The limiting requirement for the method is that at least two photons arrive in a 20-msec exposure. This translates to about a +16 stellar magnitude limit. Although angular diameters are more difficult to derive than binary separations, we have used this system to derive angular diameters for 13th magnitude stars accurate to ±5% with less than 5 min total observing time.

 

ACCURACY

 

McAlister has initiated a substantial speckle program to derive binary star parameters at Kitt Peak. The first results (McAlister, 1978) give a basis for estimating the precision possible with speckle interferometry.

Internal errors in speckle interferometry include the basic uncertainty in the data itself and the error due to uncertain calibration of the image scale. McAlister has computed errors based on 46 pairs of observations for 5 binary systems, with each pair separated in time by one day to one month. An observation is defined as the result from a single 50-frame set of speckle photographs. For these data (with binary separations of 0.2 to 3.25 arcsec), McAlister concludes that the error due to basic uncertainty in the data is ±0.3% in separation and ±0.2° in position angle for each 50-frame data set. If the calibration errors are included, the angular separation error is reduced to ±0.6%.

Calibrations of image scale and position angle are made by placing a double slit with known slit separation over the telescope aperture. Since the telescope is then effectively a two-slit interferometer, the fringe spacing and position angle in the power spectra of data taken through this slit provide [186] accurate calibrations of angular separations and position angles. Calibrations are generally done only a few times each night. If a set of built-in double slits were used to calibrate each star after every observation, calibration errors could be reduced to much less than the inherent error in the data. For binaries with separations less than 1 arcsec, accuracies of ±0.002 arcsec are already obtainable, and accuracies of ±0.1 arcsec are obtainable for stars of 5 arcsec separation, in single observations.

McAlister has computed possible external errors in his results by comparing binary orbits derived from speckle interferometry with high-quality published orbits. He concludes that speckle orbits match the published orbits to within the accuracies of these orbits; this result precludes large systematic errors in speckle measurements of binary stars.

The above analysis for speckle interferometry is based on photographic data-recording systems. Advanced photoelectric data acquisition systems have several advantages. Since the new systems run at essentially television rates (60 frames/sec), a single 50-frame sample takes less than 1 sec to obtain! McAlister observes about 150 stars per night, spending several minutes on each star. We might expect that 50 observations of 5-min duration would be possible in an observing session with a dedicated telescope. If we assume that errors are reduced as the square root of observing time, then the over 104 50-frame data sets obtained per year would refine the accuracies by a factor of 102 over the McAlister values. This corresponds to 2 x 10-5 arcsec/yr on binaries with separations smaller than 1 arcsec, and 10-4 arcsec/yr on a binary with 5-arcsec separation. The higher quantum efficiency and linearity of the digital system indicate that these numbers should apply to stars brighter than about +9, compared to McAlister's limit of +7. The accuracies on binaries near the faint limit at +14 would probably be a factor of 10 worse for the same observing time.

Another limitation is the isoplanatic angle-the maximum binary separation at which speckle interferometry will work. Conventional wisdom, not based on any real observations, places the isoplanatic angle at about 3 arcsec, meaning that interferometry of binary stars with separations much larger than this would be impossible. Recent measurements by Hubbard et al. (1979) of binary stars with larger separations indicate that this angle is closer to 6 arcsec and may be as large as 10 arcsec. It may therefore be possible to use as a reference star an unrelated background star rather than the other binary component. This may extend interferometric position determinations to wider binaries and some single stars.

Photographic speckle systems have been limited to binary stars in which the components are within 5 magnitudes of each other. Photoelectric systems may extend this limit to 7 or 8 magnitudes. However, the requirement of two photons in each exposure practically limits us to systems in which both stars are brighter than +16 magnitudes.

 

[187] SUITABLE TARGET BINARY SYSTEMS

 

This section will examine a set of possible program binary systems and discuss detection probabilities. As a data source I have used Gliese's Catalogue of Nearby Stars (1969), which includes all stars with known parallaxes equal to or greater than 0.045 arcsec, plus borderline cases.

There is some conjecture that the formation of binary stars inhibits planet formation. Since definitive models for planet formation are not available, however, there is absolutely no reason to dismiss binary systems a priori as possible planetary systems. Knowledge of the frequency of binaries and the mass distribution of companions would further illuminate the physical processes of star and planetary-system formation. The extensive satellite systems of Jupiter and Saturn point strongly to the hierarchical formation of such systems. There are, however, dynamical constraints on possible planetary orbits in binary systems. Harrington (1977) examined the dynamical stability of a planetary body in a binary system in terms of the restricted three-body problem. He concluded that stable planetary orbits are possible in two classes of binary systems: those in which the planetary orbit is large compared to the binary orbit and those in which it is small. In both cases, the planetary orbit must be a factor of 3 to 4 larger (or smaller) than the maximum (or minimum) binary separation. Since there are no detectable effects of a planet in the case where the binary separation is small compared to the planetary orbit, we restrict our discussion to the opposite case. If we use Jupiter's orbit at approximately 5 AU radius as a benchmark, we can examine which binaries may have a stable Jovian orbit. Table 1 lists the separations for a stable Jovian orbit as a function of parallax. Table 2 shows the effects of a Jovian planet (m = 10-3 symbol for solar mass in a Jovian orbit (5 AU) and a large terrestrial planet (m = 10-5 symbol for solar mass

= 3symbol for earth mass

) at 1 AU. These effects are shown as a function of the reflex motion on a Sun-type primary (1symbol for solar mass

) and a late type-M dwarf (0.15symbol for solar mass

). Based on our previous calculations of 10-4 arcsec.....

 

TABLE 1. ANGULAR SIZE OF BINARY ORBITS FOR A STABLE JOVIAN ORBIT (in arcsec).

Parallax

Size of Jovian orbit

Minimum binary orbit for stable Jovian orbit

.

0.2

1.000

4.000

0.1

0.500

2.00

0.075

0.375

1.500

0.050

0.250

1.00

 

[188] TABLE 2. EFFECTS OF PLANETS ON PRIMARY-STAR ORBITAL AMPLITUDE AS A FUNCTION OF DISTANCE (in arcsec)

 

Parallax

10-3 symbol for solar mass

                  effect on 1symbol for solar mass

10-3 symbol for solar mass

                  effect on 0.15symbol for solar mass

10-5 symbol for solar mass

                  effect on 1symbol for solar mass

10-5 symbol for solar mass

                  effect on 0.15symbol for solar mass

.

0.2

2 x 10-3

1.3 x 10-2

2 x 10-6

2.6 x 10-5

0.1

1 x 10-3

6.7 x 10-3

1 x 10-6

1.3 x 10-5

0.075

7.7 x 10-4

5 x 10-3

7.5 x 10-7

9.7 x 10-6

0.050

5 x 10-4

3.3 x 10-3

5 x 10-7

6.5 x 10-6

 

.....accuracy, we see that Jovian planets are detectable for all binary separations out to 20 parsecs. In the special case of nearby M dwarf stars, it may even be possible to detect large terrestrial planets.

In the Gliese catalog there are 248 star systems with binary separations between 0.2 and 15.0 arcsec. Of these 248 systems, 188 lie north of -30°. Table 3 lists the separation distributions for these 188 star systems; while table 4 lists their parallax distributions. Using Harrington's stability criterion,

 

TABLE 3. BINARY SEPARATIONS OF STARS NORTH OF -30° IN THE GLIESE CATALOG.

Separation, arcsec

Number of systems

.

0.2 - 0.5

23

0.5 - 2.0

64

2.0 - 6.0

64

6.0 - 10.0

19

10.0 - 15.0

18

 

TABLE 4. PARALLAXES OF NORTHERN BINARY STARS WITH SEPARATIONS LESS THAN 15 ARCSEC IN THE GLIESE CATALOG.

Parallax, arcsec

Number of systems

.

>0.200

9

0.200 - 0.100

24

0.100 - 0.075

21

0.075 - 0.050

76

<0.050

58

 

[189] .....we find that roughly half the 188 systems could have a stable Jovian orbit, while almost all could have a stable terrestrial orbit. The sample therefore permits large numbers of planetary orbits at distances similar to those in the Solar System.

For the 182 systems that list magnitudes for both components, we found the following magnitude differences between the two components (see table 5). Almost half the systems have nearly equal magnitudes, while 82% have less than the 5th magnitude difference needed for the present photographic system. These magnitude differences should be about 1 magnitude less if observations were made around 8000 Å since the secondary is almost invariably redder than the primary. The magnitudes for 375 of the component stars are shown in table 6. Of these, 50% are brighter than +9, the magnitude limit for the maximum accuracy. The spectral classification of 251 of these stars is given in table 7. As might be expected, the listings, which are generally for the primary component only, are weighted heavily toward late spectral types. The secondary components should be weighted even more heavily toward later spectral types. Almost all of these stars are main-sequence (luminosity class V), but 16 are subgiants (luminosity class IV) and 2 are giants (luminosity class III).

 

TABLE 5. MAGNITUDE DIFFERENCES FOR 182 NORTHERN BINARY SYSTEMS IN THE GLIESE CATALOG.

Magnitude difference

Number of systems

.

0 - 2

74

1 - 3

48

3 - 5

28

>5

32

 

TABLE 6. MAGNITUDE OF COMPONENT STARS IN NORTHERN GLIESE CATALOG BINARY STARS.

Apparent visual magnitude

Number of stars

.

0 - 5

35

5 - 7

79

7 - 9

68

9 - 11

90

11 - 13

74

>13

29

 

[190] TABLE 7. SPECTRAL TYPES OF NORTHERN GLIESE CATALOG BINARY STARS.

Spectral type

Number of stars

.

A

10

F

40

G

43

K

59

M

99

 

I conclude that there is a sizable sample of binary candidates for planetary search. Even if we apply the restrictive requirements that one component be brighter than +9, that the system have a stable Jovian orbit, that the magnitude difference be less than 5, and that the separation be less than 6 arcsec, we have nearly 40 candidate systems.

The results of systematic long-term searches of these systems would be extremely valuable. The binary orbits that would be a by-product of a planetary search would allow a very accurate calibration of lower main-sequence masses and solar neighborhood distance scales. The problem should also turn up large numbers of low-mass, but not planetary, stellar components. These data will be essential for calibrating binary star mass functions.

 

SUMMARY

 

A small-aperture speckle interferometer can obtain accuracies of better than 10-4 arcsec/yr on binary star orbits. There are 188 accessible binary systems within 20 parsecs of the Sun. About half should have stable orbits for a Jovian planet, which would be easily detectable within the accuracies possible.

I conclude that stellar speckle interferometry is a viable option for detecting planets, and it is an option that could be easily implemented. In fact, interferometry is ideal for precisely those systems in which conventional astrometry has some difficulty, namely, the close binary systems.

 

REFERENCES

 

- Breckinridge, J. B.; McAlister, H. A.; and Robinson, W. G.: Kitt Peak Speckle Camera. Applied Optics, vol. 18, April 1979, pp. 1034-1041.

[191] - Currie, D. G.: On a Detection Scheme for an Amplitude Interferometer. Woods Hole Summer Study on Synthetic Aperture Optics. NAS-NRC, Washington, D.C., 1968a.

- Currie, D. G.: On the Atmospheric Properties Affecting an Amplitude Interferometer. Woods Hole Summer Study on Synthetic Aperture Optics, NAS-NRC, Washington, D.C.,1968b.

- Currie, D. G.; Knapp, S. L.; and Liewer, K. M.: Four Stellar-Diameter Measurements by a New Technique - Amplitude Interferometry. Astrophys. J., vol.187, Jan.1974, pp.131-134.

- Currie, Douglas G.; McAlister, Harold A.; Schneeberger, Timothy, J.; and Worden, Simon P.: Using Small Aperture Interferometry to Detect Planets in Nearby Binary Star Systems. In: An Assessment of Ground-Based Techniques for Detecting Other Planetary Systems, Volume II: Position Papers. NASA CP-2124,1980.

- Dainty, J. C., ed.: Laser Speckle and Related Phenomena. Springer-Verlag, Berlin, 1975.

- Gatewood, G.: On the Astrometric Detection of Neighboring Planetary Systems. Icarus, vol. 27, Jan.1976, pp.1-12.

- Gezari, D. Y.; Labeyrie, A.; and Stachnik, R. V.: Speckle Interferometry: Diffraction-Limited Measurements of Nine Stars with the 200-inch Telescope. Astrophys. J., vol. 173, April 1972, pp. L1 -L5.

- Gliese, Wilhelm: Catalog of Nearby Stars, vol. 22. Veroff des Astron. Rechen-Institut, Heidelberg, 1969.

- Hanbury Brown, R.: Measurement of Stellar Diameters. Ann. Rev. Astron. Astrophys., vol. 6,1968, pp.13-38.

- Harrington, R. S.: Planetary Orbits in Binary Stars. Astrophys. J., vol. 82, Sept.1977, pp. 753-756.

- Hubbard, G.; Hege, K.; Reed, M. A.; Strittmatter, P. A.; Woolf, N. J.; and Worden, S. P.: Speckle Interferometry, I -The Steward Observatory Speckle Camera. Astronom. J ., vol. 84, Sept. 1979, pp. 1437-1442.

- Labeyrie, A.: Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysing Speckle Patterns in Star Images. Astron. Astrophys, vol. 6, May 1970, pp. 85-87.

[192] - Labeyrie, A.: Stellar Interferometry Methods. Ann. Rev. Astron. Astrophys., vol. 16,1978, pp. 77-102.

- Lynds, C. R.; Worden, S. P.; and Harvey, J. W.: Digital Image Reconstruction Applied to Alpha Orionis. Astrophys. J., vol. 207, 1976, pp.174-180.

- McAlister, H. A.: Speckle Interferometry as a Method for Detecting Nearby Extrasolar Planets. Icarus, vol. 30,1977, pp. 789-792.

- McAlister, H. A.: In: Modern Astrometry, F. V. Prochazka and R. H. Tucker, eds., Proceedings of IAU Colloquium 48, Vienna, Austria, Sept. 1978.

- Michelson, A. A.: On the Applications of Interference Methods to Astronomical Measurements. Astrophys. J., vol. 51,1920, p. 257.

- Worden, S. P.: Astronomical Image Reconstruction -Speckle Interferometry. In: Vistas in Astronomy, vol. 20, pt. 3, 1977, pp. 301-311 and 313-318.

- Worden, S. P.; Stein, M. K.; Schmidt, G. D.; and Angel, J. R. P.: The Angular Diameter of Vesta from Speckle Interferometry. Icarus, vol. 32, Dec. 1977, pp. 450-457.


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