The Astronomical Journal, 132:2478Y2488, 2006 December # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
CHARACTERIZING BINARY STARS BELOW THE DIFFRACTION LIMIT WITH CCD-BASED SPECKLE IMAGING Elliott P. Horch 1, 2 Department of Physics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747-2300;
[email protected]
Otto G. Franz 2 Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, AZ 86001;
[email protected]
and William F. van Altena1 Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101;
[email protected] Received 2006 July 22; accepted 2006 August 30
ABSTRACT An analysis of 15 speckle observations taken at the Lowell-Tololo Telescope at Cerro Tololo Inter-American Observatory and 6 speckle observations taken at the WIYN Telescope at Kitt Peak National Observatory indicates that it is possible to characterize the separations, position angles, and magnitude differences of binary stars down to at least one quarter of the diffraction limit with CCD-based speckle imaging. This is made possible by the fact that CCDbased speckle imaging permits the retrieval of reliable photometric information from speckle data, and therefore the elongation of the speckles due to a blended companion may be reliably measured. When observations in two colors are obtained, atmospheric dispersion, which also affects the speckle shape, can be distinguished from binarity in a large number of cases. A regimen for observing sub-diffraction-limited speckle binaries is proposed that could lead to efficient surveys of small-separation binary stars. Key words: astrometry — binaries: visual — techniques: photometric
We have also used the University of Toronto Southern Observatory and Lowell-Tololo 61 cm telescopes in the Southern Hemisphere. The main difference between this program and other large speckle programs is the use of large-format CCDs to capture a number of speckle images on the detector before readout. Because of the well-known properties of CCDs versus the microchannelplate-based devices typically used in speckle imaging, this presented an opportunity to reexamine the photometry issue. We have concluded that differential photometry of binary star systems is possible with CCD-based speckle imaging, both at WIYN (Horch et al. 2004a) and at the Cerro Tololo Inter-American Observatory (CTIO) ( Horch et al. 2001, 2006a). The precision obtained at WIYN is typically 0.13 mag per 2 minute observation, while at the Lowell-Tololo Telescope the figure is approximately 0.18 mag, on average. This leads directly to the use of speckle data in astrophysical studies of the component stars of binary systems, as noted already in Meyer (2002). In this work we suggest a further possibility of CCD-based speckle imaging for binary observations: the potential for characterizing binaries when the separation of the system is below the diffraction limit. This idea is not unlike studying a pair whose separation is below the seeing limit with normal astronomical imaging (so that the image is blended ) and deducing a separation and position angle based on the elongation of the blended image through, e.g., point-spread function fitting. One expects that the precision of the technique will be limited as the separation becomes smaller down to some point at which the method would not be possible, but an obvious prerequisite is reliable photometry. It is also true that, when applying such a method to speckle imaging, a special problem will exist at the separations of interest in the form of atmospheric dispersion and that this also could limit the utility of the method. Nonetheless, we have analyzed CCD-based
1. INTRODUCTION Speckle imaging has been used for many years to obtain diffraction-limited image information in the presence of atmospheric turbulence. It has been particularly successful in the area of binary star research, where some 50,000 position angle and separation measures have been published over the last 35 years, according to data in the Fourth Catalog of Interferometric Measures of Binary Stars (Fourth Interferometric Catalog; Hartkopf et al. 2001b).3 The technique gives astrometric precision that varies with the size of the telescope aperture, but, for example, at a 4 m class telescope, the best speckle observers routinely achieve precision on the order of 3 mas in separation measures. Speckle observations have not been as amenable to photometric analysis, even in the case of a simple object such as a binary star. Hartkopf et al. (1996b) concluded that for the CHARA speckle observations of that era, magnitude differences should be assigned an uncertainty of 0.5 mag due to a number of factors, such as the seeing of the observation, separation of the system, magnification of the image, and magnitude difference. In 1997 our collaboration began taking speckle observations at the WIYN 3.5 m Telescope4 at Kitt Peak National Observatory. 1 Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under cooperative agreement with the National Science Foundation. 2 Visiting Astronomer, Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by AURA, Inc., under cooperative agreement with the National Science Foundation. 3 See also http://ad.usno.navy.mil/wds/int4.html. 4 The WIYN Observatory is a joint facility of the University of Wisconsin— Madison, Indiana University, Yale University, and the National Optical Astronomy Observatory.
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SUB-DIFFRACTION-LIMITED SPECKLE BINARIES speckle data of several systems known to be below the diffraction limit at the time of observation to demonstrate that sub-diffractionlimited separations, position angles, and magnitude differences can be estimated from CCD-based speckle imaging. 2. OBSERVATIONS AND ANALYSIS Speckle observations taken at two telescopes have been analyzed in cases in which a preexisting visual /speckle orbit of the system predicts a separation below the diffraction limit for the wavelength of observation. The first of these is the LowellTololo 61 cm Telescope, where speckle observations have been taken on two occasions with a fast-readout CCD camera. The work presented here only involves the second of these two runs, which occurred in 2001 November. The major group of astrometric and photometric results for systems above the diffraction limit from these observations is presented in Horch et al. (2006a). The second telescope is the WIYN 3.5 m Telescope, where observations analyzed here were all taken with the RYTSI speckle camera between 2002 and 2004. The RYTSI instrument is described in detail in Meyer et al. (2006), but, briefly, it is an optical package that can be used with any large-format CCD camera to obtain speckle observations. A grid of speckle images is collected sequentially within the area of the chip by moving the star image from location to location in a step-and-expose pattern. A dualaxis galvanometer scanning mirror system is used to execute the image motions. We have collected as many as 900 speckle images in one CCD frame using the Kitt Peak Mini-Mosaic Imager (Horch et al. 2004b), but we typically use a front-illuminated Kodak CCD that allows for 256 images per full-frame read of the chip. A speckle observation then usually consists of four consecutive full-frame images, meaning a total of 1024 individual speckle patterns of a target. The speckle observations have been analyzed using the same data reduction software used with binary stars above the diffraction limit. Our collaboration has used a Fourier-based approach where the power spectrum of a double point source exhibits a fringe pattern, the orientation and spacing of which is related to the separation and position angle of the binary star. The fringe depth is also determined and can be converted into a magnitude difference estimate of the two stars in the system. In the Fourier plane, that is, the plane of spatial frequencies of the image, the diffraction limit is represented by a circle of radius , where ¼
D : 1:22k ; 206265
ð1Þ
In the above, D is the telescope diameter (in meters), k is the wavelength of observation (in meters), and the units of are cycles per arcsecond. Frequencies below represent image features above the diffraction limit, and frequencies above represent image features below the diffraction limit. In the case of a system that has separation above the diffraction limit, a binary star will exhibit multiple fringes in the power spectrum before the diffraction limit is reached and the system is said to be ‘‘resolved.’’ On the other hand, a binary system that has a separation below the diffraction limit will not exhibit multiple fringes. Instead, the diffraction limit in the Fourier plane is reached before the first-order fringes peak, and the system is said to be ‘‘unresolved.’’ Examples of images and power spectra for model diffraction-limited data are shown in Figure 1. The Fourier data reduction scheme consists of five basic steps: (1) formation of the average spatial frequency power spectrum of the binary from the speckle images of the target; (2) formation
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of the same for a point source observed near in time and sky location; (3) division of the binary power spectrum by the pointsource power spectrum (which amounts to a deconvolution on the image plane); (4) use of a weighted least-squares power spectrum fitting algorithm to determine the position angle, separation, and magnitude difference of the binary; and (5) determination of the quadrant of the secondary by computing a reconstructed image via the method of bispectral analysis. This last step is necessary because the power spectrum analysis leaves a 180 ambiguity in the position angle of a binary star. (See Lohmann et al. [1983] for more information on the bispectrum.) In our previous work we have studied the precision of the fitted separation as a function of the separation itself and found no significant correlation for separations ranging from the diffraction limit to above 100 (see, e.g., Horch et al. 2002). This has led us to explore the capabilities of the same algorithm below the diffraction limit. It is important to emphasize that in using the fitting algorithm on a power spectrum obtained from speckle data, an assumption is made that a binary model is a good representation of the data set. This is usually obvious in the case of binaries with separations above the diffraction limit from the presence of multiple fringes in the power spectrum. In attempting to use the algorithm on a system with a separation below the diffraction limit, the critical reassurance that one is on solid ground in applying the binary model to make the fit (i.e., the presence of multiple fringes) is lost. In this case, we are no longer determining the separation of resolved speckles in the image but estimating the separation assuming that speckles are in fact elongated due to the presence of a sub-diffraction-limited component in the system. This is not the only explanation for the elongation of speckles, as is discussed more fully in x 4. For the moment, we simply make the analogy that what we wish to attempt is similar to estimating a sub-seeinglimited separation in the case of a blended or elongated stellar image, a process that (with limited precision) has been successfully used in order to identify potentially interesting targets for diffraction-limited imaging methods such as speckle and adaptive optics. Table 1 gives details for Lowell-Tololo observations for which results are presented in x 3, and Table 2 contains the same information for WIYN observations. The columns contain (1) the object name; (2) the Washington Double Star ( WDS) Catalog number, if the object is a binary (this also gives the right ascension and declination of the object); (3) the observation date; (4) the time in hours UT; (5) the local sidereal time during the observation; (6) the zenith angle during the observation; (7) the azimuth angle of the observation, with north through east defining the positive sense of the angle; (8) the filter center wavelength in nanometers; and (9) the full width at half-maximum (FWHM) of the filter passband in nanometers. In each case, the row below a binary contains the observational data for the object used as the pointsource calibrator in the analysis. Each observation took approximately 2 minutes to complete, so the times in columns (4) and (5) refer to the midpoint of the observation to the nearest minute. 3. RESULTS Table 3 shows the results of the 15 binary observations taken at the Lowell-Tololo Telescope and reduced as described above. The columns give (1) the Bright Star Catalogue (HR) number or, if none, the Aitken Double Star (ADS) number or, if none, the Durchmusterung (DM ) number; (2) the discoverer designation; (3) the Henry Draper Catalogue (HD) number; (4) the Hipparcos catalog (HIP) number (Perryman et al. 1997); (5) the WDS number; (6) the observation date in Besselian year; (7) the observed
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Fig. 1.— Progression of binary images and power spectra shown as shaded surfaces for noiseless diffraction-limited imaging. (a) Image of a binary star with separation equal to 2.8 times the diffraction limit. (b) Spatial frequency power spectrum of (a). (c) Image of a binary star with separation equal to 1.4 times the diffraction limit. (d ) Spatial frequency power spectrum of (c). (e) Image of a binary star with separation equal to 0.7 times the diffraction limit. ( f ) Spatial frequency power spectrum of (e). As the separation is reduced to the point that it is below the diffraction limit, the star images become blended, resulting in only the central fringe being visible in the frequency plane.
position angle ( ) of the pair, with north through east defining the positive sense of ; (8) the observed separation () in arcseconds; (9) the observed magnitude difference in the filter used; (10) the center wavelength of the filter used; and (11) the FWHM of the filter used. For these observations, the Bessel R and V filters were used, with the 644 nm filter corresponding to R and 541 nm corresponding to V. In Table 4 we show the results of the six observations taken at the WIYN Telescope. The column headings are identical to those in Table 3, although interference filters are generally used in our speckle observations at WIYN (with one exception, a Bessel V measure in one case for MCA 47). The separations in these two tables range from just below the diffraction limit to approximately 0.25 times the diffraction limit. All of the objects selected have an orbit determination that appears in the Sixth Catalog of Visual Orbits of Binary Stars
(Hartkopf et al. 2001a).5 In Tables 5 and 6 we compare our observations with the orbit prediction for the epochs of observation based on the orbital parameters. Table 5 shows these results for the Lowell-Tololo observations, while Table 6 shows the same for the WIYN observations. If the quadrant of the secondary was noted as inconsistent with previous measures in Tables 3 and 4, then the position angle was flipped by 180 prior to this comparison. Specifically, the columns in Tables 5 and 6 give (1) the discoverer designation (the same as col. [2] of Tables 3 and 4); (2) the orbit reference appearing in the Sixth Orbit Catalog; (3) the observation date in Besselian year (the same as col. [6] in Tables 3 and 4); (4) the ephemeris prediction for the position angle in degrees; (5) the ephemeris prediction for the separation in arcseconds; (6) the observed minus ephemeris difference in position 5
See also http://ad.usno.navy.mil/wds/orb6.html.
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TABLE 1 Journal of Observations, Lowell-Tololo Telescope
Object (1)
WDS (, J2000.0) (2)
KUI 7 ................................ HR 366.............................. KUI 7 ................................ HR 497.............................. FIN 333............................. HR 852.............................. FIN 333............................. HR 852.............................. FIN 333............................. HR 714.............................. FIN 333............................. HR 714.............................. FIN 333............................. HR 852.............................. FIN 333............................. HR 852.............................. B52.................................... HR 1214............................ B52.................................... HR 1214............................ B52.................................... HR 1214............................ B52.................................... HR 1214............................ BU 1032AB ...................... HR 1746............................ BU 1032AB ...................... HR 1657............................ BU 1032AB ...................... HR 1787............................
013760924 ... 013760924 ... 024346643 ... 024346643 ... 024346643 ... 024346643 ... 024346643 ... 024346643 ... 033393105 ... 033393105 ... 033393105 ... 033393105 ... 053870236 ... 053870236 ... 053870236 ...
Date (3) 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001 2001
Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov Nov
12 12 19 19 9 9 9 9 10 10 10 10 20 20 20 20 18 18 18 18 21 21 21 21 11 11 19 19 21 21
Time ( UT) (4)
Local Sidereal Time (5)
Zenith Angle (deg) (6)
Azimuth (deg) (7)
k (nm) (8)
k (nm) (9)
03:55 03:26 03:09 03:43 05:00 05:07 04:57 05:10 04:17 04:11 04:20 04:08 03:29 03:45 03:32 03:42 04:15 04:20 04:10 04:22 03:57 04:13 04:00 04:10 08:26 08:07 07:10 06:47 07:18 07:02
02:37 02:08 02:19 02:53 03:30 03:37 03:27 03:40 02:51 02:45 02:54 02:42 02:43 02:59 02:46 02:55 03:21 03:26 03:16 03:28 03:15 03:31 03:18 03:28 07:05 06:46 06:20 05:57 06:36 06:20
24.3 25.6 22.8 15.1 37.2 33.5 37.1 33.7 36.6 30.3 36.6 30.3 36.5 32.7 36.4 32.7 2.9 7.4 4.0 7.1 4.2 6.6 3.6 7.1 34.3 38.8 29.2 28.1 30.7 32.1
324.1 327.8 333.7 257.3 187.6 189.9 187.1 190.5 181.3 184.9 181.7 184.2 179.9 182.1 180.4 181.3 109.0 129.9 104.5 131.8 103.8 135.2 105.9 131.8 319.4 323.8 338.5 332.6 331.1 333.2
644 644 644 644 644 644 541 541 644 644 541 541 644 644 541 541 644 644 541 541 644 644 541 541 644 644 644 644 644 644
128 128 128 128 128 128 88 88 128 128 88 88 128 128 88 88 128 128 88 88 128 128 88 88 128 128 128 128 128 128
angle in degrees; (7) the observed minus ephemeris difference in separation in arcseconds; and (8) the ephemeris separation divided by the diffraction limit at the center wavelength of the observation, which gives the fraction of the diffraction limit represented by the predicted separation. In Figures 2 and 3 we show how these results compare with other measures appearing in the Fourth Interferometric Catalog. For the Lowell-Tololo observations, shown in Figure 2, many of the other observations shown were taken from larger telescopes
with diffraction limits much smaller than that of the LowellTololo Telescope. Even without an orbit the other observations give an indication that the sub-diffraction-limited results obtained here are sensible, but the orbital ephemerides also show that Lowell-Tololo observations do follow the orbit predictions. In the case of WIYN observations ( Fig. 3), there are two cases in which the observations presented are near the minimum separation expected for the systems (BU 1163 and MCA 47) and two other systems of small separation throughout their orbital motion
TABLE 2 Journal of Observations, WIYN Telescope
Object (1)
WDS ( , J2000.0) (2)
Date (3)
Time (UT) (4)
Local Sidereal Time (5)
Zenith Angle (deg) (6)
Azimuth (deg) (7)
k (nm) (8)
k (nm) (9)
BU 1163............................ HR 366.............................. MCA 13 ............................ HR 1180............................ MCA 13 ............................ HR 1180............................ A1634................................ HR 5800............................ MCA 47 ............................ HR 6388............................ MCA 47 ............................ HR 6388............................
012430655 ... 04044+2406 ... 04044+2406 ... 15318+4054 ... 17217+3958 ... 17217+3958 ...
2004 Dec 21 2004 Dec 21 2004 Dec 19 2004 Dec 19 2004 Dec 19 2004 Dec 19 2004 Feb 10 2004 Feb 10 2002 Apr 28 2002 Apr 28 2002 Apr 28 2002 Apr 28
03:20 03:18 06:37 06:27 06:39 06:30 13:36 13.41 10:27 10:24 10:30 10:11
01:54 01:52 05:03 04:53 05:05 04:56 15:29 15:35 17:25 17:22 17:28 17:09
39.5 40.9 15.1 16.1 15.5 16.7 9.0 7.1 6.9 9.2 7.1 8.8
191.6 194.3 242.4 244.9 243.3 246.2 3.4 0.4 352.7 345.0 344.9 0.7
754 754 698 698 698 698 698 698 539 539 550 550
44 44 39 39 39 39 39 39 88 88 40 40
TABLE 3 Binary Star Speckle Measures, 2001 November, Lowell-Tololo Telescope
HR, ADS, DM, etc. (1)
Discoverer Designation (2)
HD (3)
HIP (4)
WDS ( , J2000.0) (5)
Date (6)
(deg) (7)
(arcsec) (8)
m (9)
k (nm) (10)
k (nm) (11)
HR 466............................. HR 466............................. CP 67 181..................... CP 67 181..................... CP 67 181..................... CP 67 181..................... CP 67 181..................... CP 67 181..................... HR 1093........................... HR 1093........................... HR 1093........................... HR 1093........................... ADS 4241 ........................ ADS 4241 ........................ ADS 4241 ........................
KUI 7 KUI 7 FIN 333 FIN 333 FIN 333 FIN 333 FIN 333 FIN 333 B52 B52 B52 B52 BU 1032AB BU 1032AB BU 1032AB
10009 10009 17326 17326 17326 17326 17326 17326 22262 22262 22262 22262 37468 37468 37468
7580 7580 12717 12717 12717 12717 12717 12717 16628 16628 16628 16628 26549 26549 26549
013760924 013760924 024346643 024346643 024346643 024346643 024346643 024346643 033393105 033393105 033393105 033393105 053870236 053870236 053870236
2001.8649 2001.8839 2001.8568 2001.8568 2001.8594 2001.8594 2001.8867 2001.8867 2001.8813 2001.8813 2001.8895 2001.8895 2001.8626 2001.8844 2001.8899
340.4 149.6 211.8 209.0 211.1 207.4 213.6 214.9 331.1 326.4 322.5 322.0 285.2 282.0 291.7
0.259 0.237 0.216 0.214 0.221 0.222 0.220 0.219 0.177 0.173 0.177 0.182 0.261 0.248 0.246
0.85 0.49 0.36 0.40 0.05 0.17 0.05 0.16 0.44 0.44 0.41 0.27 1.06 1.17 1.34
644 644 644 541 644 541 644 541 644 541 644 541 644 644 644
128a,b 128a 128 88 128a 88a 128a 88a 128 88 128 88 128b 128b 128b
a b
Quadrant is ambiguous. Quadrant is inconsistent with previous measures in the Fourth Interferometric Catalog.
TABLE 4 Binary Star Speckle Measures, WIYN Telescope
HR, ADS, DM, etc. (1)
Discoverer Designation (2)
HD (3)
HIP (4)
WDS (, J2000.0) (5)
Date (6)
(deg) (7)
(arcsec) (8)
m (9)
k (nm) (10)
k (nm) (11)
ADS 1123 ........................ ADS 2965 ........................ ADS 2965 ........................ ADS 9688 ........................ HR 6469........................... HR 6469...........................
BU 1163 MCA 13 MCA 13 A1634 MCA 47 MCA 47
8556 25555 25555 138629 157482 157482
6564 19009 19009 76041 84849 84849
012430655 04044+2406 04044+2406 15318+4054 17217+3958 17217+3958
2004.9723 2004.9762 2004.9762 2004.1111 2002.3228 2002.3229
345.9 331.1 324.1 46.3 66.7 248.4
0.013 0.033 0.036 0.045 0.039 0.035
1.53 1.62 2.03 3.50 0.27 0.37
754 698 698 698 539 550
44 39a,b 39a,b 39a 88a,b 40
a b
Quadrant is ambiguous. Quadrant is inconsistent with previous measures in the Fourth Interferometric Catalog.
TABLE 5 Orbits and Residuals, Lowell-Tololo Telescope
Discoverer Designation (1)
Orbit Reference (2)
Date (3)
Eph. (deg) (4)
Eph. (arcsec) (5)
(deg) (6)
(arcsec) (7)
Fraction of Diff. Limit (8)
KUI 7 ................................... KUI 7 ................................... FIN 333................................ FIN 333................................ FIN 333................................ FIN 333................................ FIN 333................................ FIN 333................................ B52....................................... B52....................................... B52....................................... B52....................................... BU 1032AB ......................... BU 1032AB ......................... BU 1032AB .........................
Tokovinin 1993 Tokovinin 1993 So¨derhjelm 1999 So¨derhjelm 1999 So¨derhjelm 1999 So¨derhjelm 1999 So¨derhjelm 1999 So¨derhjelm 1999 Heintz 1996 Heintz 1996 Heintz 1996 Heintz 1996 Hartkopf et al. 1996a Hartkopf et al. 1996a Hartkopf et al. 1996a
2001.8649 2001.8649 2001.8568 2001.8568 2001.8594 2001.8594 2001.8867 2001.8567 2001.8813 2001.8813 2001.8895 2001.8895 2001.8626 2001.8844 2001.8899
147.5 147.4 217.9 217.9 217.9 217.9 217.9 217.9 316.5 316.5 316.5 316.5 107.0 106.9 106.9
0.241 0.241 0.208 0.208 0.208 0.208 0.210 0.210 0.208 0.208 0.209 0.209 0.247 0.247 0.247
+12.9 +2.1 6.1 8.9 6.8 10.5 4.3 3.0 +14.6 +9.9 +6.0 +5.5 1.8 4.9 +4.8
+0.018 0.004 +0.008 +0.006 +0.013 +0.014 +0.010 +0.009 0.031 0.035 0.032 0.027 +0.014 +0.001 0.001
0.91 0.91 0.78 0.93 0.78 0.93 0.79 0.94 0.78 0.93 0.79 0.94 0.93 0.93 0.93
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TABLE 6 Orbits and Residuals, WIYN Telescope
Discoverer Designation (1)
Orbit Reference (2)
Date (3)
Eph. (deg) (4)
Eph. (arcsec) (5)
(deg) (6)
(arcsec) (7)
Fraction of Diff. Limit (8)
BU 1163................................. MCA 13 ................................. MCA 13 ................................. A1634..................................... MCA 47 ................................. MCA 47 .................................
So¨derhjelm 1999 Mason et al. 1997 Mason et al. 1997 Hartkopf et al. 1989 Scarfe et al. 1994 Scarfe et al. 1994
2004.9723 2004.9762 2004.9762 2004.1111 2002.3228 2002.3229
18.0 149.2 149.2 46.3 247.4 247.4
0.014 0.042 0.042 0.039 0.037 0.037
32.1 +1.9 5.1 6.1 0.7 +1.0
0.001 0.009 0.006 +0.006 +0.002 0.002
0.26 0.84 0.84 0.78 0.95 0.93
Fig. 2.— Comparison of the astrometry obtained in the analysis presented here from Lowell-Tololo observations ( filled circles) with previous observations appearing in the Fourth Interferometric Catalog (open circles) and the orbit determination referred to in Table 5. In all plots, line segments are drawn from the ephemeris position for the epoch of observation to the observed secondary location. The dashed circle represents the FWHM for the image of a perfect point source located at the origin, and the dotted circle shows the locus of separations at the diffraction limit; that is, two diffraction-limited point sources would just satisfy the Rayleigh criterion for being marginally resolved if the secondary were to lie on the dotted circle. In cases in which observations in two filters appear in Table 3, two dashed and two dotted circles are drawn, with the inner contour in each case corresponding to the bluer wavelength. In cases in which our quadrant determination is noted as inconsistent with previous measures in Table 3, we have flipped the quadrant prior to plotting. In all plots, north is down and east is to the right. (a) KUI 7, (b) FIN 333, (c) B52, (d) BU 1032AB.
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Fig. 3.— Comparison of the astrometry obtained in the analysis presented here from WIYN observations ( filled circles) with previous observations appearing in the Fourth Interferometric Catalog (open circles) and the orbit determination referred to in Table 6. In all plots, line segments are drawn from the ephemeris position for the epoch of observation to the observed secondary location. The dashed circle represents the FWHM for the image of a perfect point source located at the origin, and the dotted circle shows the locus of separations at the diffraction limit; that is, two diffraction-limited point sources would just satisfy the Rayleigh criterion for being marginally resolved if the secondary were to lie on the dotted circle. In cases in which our quadrant determination is noted as inconsistent with previous measures in Table 4, we have flipped the quadrant prior to plotting. North is down and east is to the right. (a) BU 1163, (b) MCA 13, (c) A1634, (d ) MCA 47.
( MCA 13 and A1634). In all cases, the sub-diffraction-limited analysis here is consistent with the orbit prediction and /or previous observations at similar orbital phase. The smallest-separation object with a measure in Table 4 is BU 1163; the ephemeris separation is only one quarter of the diffraction limit. Although this is the only object presented here with a separation below 0.78 of the diffraction limit, the successful reduction of another WIYN observation in which one companion in a quadruple system had a sub-diffraction-limited separation was discussed in Horch et al. (2006b). In that paper a visual orbit for HD 157948 was determined based on seven Hubble Space Telescope Fine Guidance Sensor observations and the radial velocity information in Goldberg et al. (2002). Based on the analysis presented there, the visual orbital parameters of HD 157948 are
not known as precisely as those of the objects selected for the current work, but the ephemeris prediction gives a separation at the time of the WIYN observation of 0.58 times the diffraction limit, with residuals from the orbit prediction of 8N9 in position angle and 5.0 mas in separation. We illustrate the data reduction of BU 1163 in Figures 4Y 6. Figure 4 shows shaded-surface plots of the power spectra of both BU 1163 and the point-source calibrator object HR 366. North is to the left and east is into the page in these figures. The ephemeris position angle is 18N0, and the value obtained from our fitting procedure is 345N9. In either case, this indicates that only a modest angle is made with the axes in the plot and therefore that central fringe should run close to the v-axis. There is slightly less signal in the binary power spectrum compared to that of the point
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Fig. 4.— Shaded-surface plots of the power spectrum of (a) BU 1163 and (b) HR 366. Both plots show a prominent peak centered on zero spatial frequency; this is due to the seeing envelope of the speckle patterns. The speckle shoulder extends out to the diffraction limit, at approximately 18 cycles arcsec1. North is in the u-direction, and east is in the +v-direction.
source in the left and right regions of the diagram, as expected if the binary power spectrum has a wide fringe pattern superposed on the power spectrum of a point source. In Figure 5, cuts along the u- and v-axes are explicitly compared, and it is seen that there is a loss of power in the binary power spectrum at high spatial frequencies relative to that of the point source in the u-axis cut, whereas this is much less evident in the v-axis cut. This is again consistent with a broad fringe positioned nearly parallel to the v-axis. In Figure 6 we show a comparison of cuts running parallel to the u-axis through the data and fit arrays for two values of v, five and eight cycles per arcsecond. (In this context, the ‘‘data’’ array is the result of dividing the power spectrum of BU 1163 by that of HR 366.) Also plotted in each case are the (data minus fit) residuals. It is interesting to note that the peak of the fringe fit occurs slightly farther from u ¼ 0 cycles arcsec1 in the case of the cut at v ¼ 8 cycles arcsec1. This is consistent with a fringe slightly tilted from the v-axis, leading to a position angle of a few degrees less than 360 , as determined by the fitting routine. 4. DISCUSSION 4.1. Comments on Measurement Precision Although the number of measures in Tables 3 and 4 is small, there are enough repeat measures that we can form some preliminary estimates of the standard deviation of the separation and position angle measures. Of course, these will be biased toward
low values due to the small samples, but we use an unbiased estimator of the standard deviation to account for this. In most cases, using the well-known unbiased estimator of the variance, s 2 , given by s2 ¼ 2
N ; N 1
ð2Þ
would be sufficient for deriving the standard deviation. [ Here is the variance, N is the number of independent measurements, and the factor N /ð N 1Þ is known as Bessel’s correction.] However, for very small samples, the standard deviation calculated by taking the square root of the above still underestimates the average standard deviation. Although the error is small, we wish to account for this, and so a more conservative approach for our data set is to use the unbiased estimator of the standard deviation itself, corr ¼ =c; where the constant c is given by rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½ð N =2Þ 1! : c¼ N 1 ½ð N 1Þ=2 1!
ð3Þ
ð4Þ
Further information, including the definition of the half-integer factorials, can be found in the NIST/SEMATECH e-Handbook
Fig. 5.— Cuts along the (a) u- and (b) v-axes for the power spectra in Fig. 4. The solid line shows the power spectrum of BU 1163, and the dashed line shows that of the point-source calibration object HR 366.
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Vol. 132
Fig. 6.— Cuts parallel to the u-axis after division of the binary power spectrum (BU 1163) by the point-source power spectrum (HR 366). (a) Data cut for v ¼ 5 cycles arcsec1. (b) Data cut for v ¼ 8 cycles arcsec1. In each plot, the solid line shows the data, the dashed line shows the fit, and the dotted line shows the residuals (data minus fit).
of Statistical Methods, x 6.3.2.6 Using these formulae, an unbiased standard deviation estimate may be at least formally obtained for all objects appearing in Tables 3 and 4, except BU 1163 and A1634, where only one measure is available. Table 7 shows the result of computing these corrected standard deviation estimates for a given filter. ( In the case of MCA 47, the filters were not the same but were very similar in central wavelength, and so these two observations were grouped together.) The averages of these values are ¯ ¼ 5N5 1N3 and ¯ ¼ 7:2 3:0 mas for the Lowell-Tololo data and ¯ ¼ 3N8 3N3 and ¯ ¼ 3:1 0:6 mas for the WIYN data. Both the separation and position angle values show reasonably good agreement with the nominal figures from Horch et al. (2006a) in the case of the CTIO data and Horch et al. (2002) in the case of the WIYN data. Turning now to the photometric results, we now use equations (2) and (3) above to form unbiased estimates of the standard deviation of the magnitude difference, m , from Tables 3 and 4, and these appear in the final column of Table 7. These m values compare favorably with results obtained above the diffraction limit, e.g., in Figure 4b of Horch et al. (2001) and Figure 7 of Horch et al. (2006a) for CTIO data and Figure 5b of Horch et al. (2004a) for WIYN data. Looking at previous observations of the Lowell-Tololo objects, the magnitude differences of KUI 7 and BU 1032 have also been measured from speckle data taken at WIYN (where the separations are well above the diffraction limit), and at 648 nm, KUI 7 has a magnitude difference of 1:10 0:06, while BU 1032 has a magnitude difference 6
See http://www.itl.nist.gov/div898/handbook/.
of 1:26 0:06 (see Horch et al. 2004a). The former is approximately 0.4 mag larger than the average of the two measures presented in Table 3 (although these lead to an estimated unbiased standard deviation of 0.32 mag in Table 7 ), and the latter is in good agreement with the work presented here. For the WIYN objects, the m of MCA 47 is somewhat lower than expected from Figure 5b in Horch et al. (2004a), while that of MCA 13 is higher than expected. However, a result as high as m ¼ 0:3 was seen for one comparable system in that work. From the Fourth Interferometric Catalog, MCA 13 has a magnitude difference estimate at 720 nm from lunar occultation, which is 1:74 0:45, in good agreement with the values here. From Horch et al. (2004a), MCA 47 has three magnitude difference measures when the system had a separation above the diffraction limit, two at 648 nm (i.e., redder than the observations presented here), which are 0.56 and 0.80, and one at 550 nm, which is 1.04. The latter would appear to be inconsistent with the two measures presented in Table 4, but the primary of this system is almost certainly a giant, given the apparent V magnitude (8.37) and parallax (1:73 0:77 mas) appearing in the HIP. It is entirely reasonable in this case to have a smaller magnitude difference at 550 nm than at 648 nm if the secondary is closer to the main sequence and therefore presumably bluer than the primary. Further observations are needed before more definitive statements can be made. Taking both the data from CTIO and those of WIYN together, there is no compelling evidence that the uncertainty of the speckle magnitude differences should be substantially higher below the diffraction limit than values obtained above the diffraction
TABLE 7 Internal Repeatability Estimates
Discoverer Designation
Telescope
Filter k0 (nm)
Number of Measures
(deg)
(arcsec)
m
KUI 7 ................................................. FIN 333.............................................. FIN 333.............................................. B52..................................................... B52..................................................... BU 1032............................................. MCA 13 ............................................. MCA 47 .............................................
Lowell-Tololo Lowell-Tololo Lowell-Tololo Lowell-Tololo Lowell-Tololo Lowell-Tololo WIYN WIYN
644 644 541 644 541 644 698 539/550
2 3 3 2 2 3 2 2
9.6 1.5 4.5 7.6 3.9 5.6 6.2 1.5
0.019 0.003 0.004 0.000 0.008 0.009 0.0027 0.0035
0.32 0.20 0.15 0.03 0.15 0.16 0.36 0.09
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limit. However, the statistics are small and we suggest that, in principle, the precision of the magnitude difference measure should degrade as one goes to sub-diffraction-limited separations. In terms of measurement precision, it is probably wise for the moment to assume that sub-diffraction-limited observations should be treated as ‘‘poor-quality’’ super-diffraction-limited observations that would have uncertainties near the upper edge of the data presented in Figure 5b of Horch et al. (2004a) in the case of, e.g., WIYN data. 4.2. Effects of Dispersion Because the analysis procedure described here associates a separation of two components to elongated speckles, it is worth considering what other effects might produce a similar signature. One obvious and undesirable effect is that of atmospheric dispersion. Dispersion, if uncorrected, would elongate speckles along a line leading to the zenith, and the magnitude of the elongation would be more severe at larger zenith distances. At midsized telescopes and larger, most speckle observations are corrected for dispersion with the use of two zero-deviation (Risley) prisms that can be independently rotated to yield a vector dispersion opposite of that produced by the atmosphere, thereby canceling its effect. Of course, in practical terms, there may still be some ‘‘residual’’ dispersion after compensation, but this is generally small. In the above analysis, one important defense against misinterpreting elongation due to dispersion as elongation due to a companion of small separation is the observation of a known point source near in sky position and in time to the binary observation. The dispersion present in the point-source data will be similar to that of the science target, and therefore when the power spectrum of the science object is divided by that of the point source, the effect will, in principle, be calibrated out. Another way to distinguish the effects of dispersion from binarity would be to require observations of both the science target and the point source in two colors. Dispersion is both colordependent and dependent on the width of the filter used, whereas the binary signature is not. If one uses two filters of the same width but with a substantial separation in center wavelength, then the dispersion will be smaller in the case of the redder filter, leading to a derived separation (if we assume the object to be binary) that is smaller than the separation derived in the bluer filter. Thus, observations producing different separations in the two filters could be rejected as inconsistent with the binary model. To illustrate this point, one can compare differences in position angle and separation measures for the contemporaneous observations in Table 3 taken in the R and V filters. There are five such measurement pairs, three for FIN 333 and two for B52. We have then taken point-source observations at different zenith angles (representing different amounts of dispersion in the data) and analyzed them exactly as described above using another point source as the calibration object. In some cases a point-source calibrator near in sky location was used, and in other cases a point-source calibrator near the zenith was used. In this way the ‘‘science’’ targets sometimes had dispersion differing from that of the calibrator and sometimes similar to it. At the LowellTololo Telescope, no dispersion correction was used, since the diameter of the telescope is small enough that dispersion effects only become substantial above zenith angles of approximately 40 . In Figure 7 we plot the difference in the separation value obtained between Vand R observations of the target as a function of the difference in position angle for the same two observations. We find in comparing the results of the point sources with those of the true binaries that the data points for the true binaries cluster near the origin (as expected, since binary separation does not
Fig. 7.— Two-color regimen for identifying binary stars below the diffraction limit in speckle observations. For the data sets described in the text, the absolute value of the difference in separation between the R and Vobservations for a target is plotted against the difference in position angle between the observations in the two filters. Open circles represent the collection of point sources at different zenith angles (resulting in dispersion-induced elongation of speckles), and filled circles represent a subset of the binaries from Table 3 as described in the text (where the elongation of speckles is caused by the presence of the companion).
depend on color), whereas the data for the dispersion-elongated point sources are not located near the origin. Although a more detailed study is warranted, perhaps using WIYN data, in which the effect of residual dispersion could be studied in more detail, these initial results indicate that observing in two colors may provide excellent discrimination between binarity and atmospheric dispersion effects in the sub-diffraction-limited regime. Most of the data points representing the case of dispersion in Figure 7 show reasonably good agreement in the position angle between the two filters; this is expected because the dispersion direction in both filters will be the same, namely, along the line containing the sky position of the target and the zenith. However, large differences in position angle can sometimes occur, in which case the object would easily be rejected from consideration as a binary. These results suggest that it might be possible to conduct a speckle survey of stars to identify sub-diffraction-limited separation binary systems. One would first require that CCD-based speckle imaging be used for the most precise information on the shape of speckles, and preferably one would have a speckle camera that would be capable of detecting two speckle patterns simultaneously in two colors. (Taking observations in two colors simultaneously is not a requirement for success but would serve to minimize any differences in residual dispersion between the two observations.) Then, by observing many targets, one could search for the signature of binarity: a consistent separation and position angle in both observations. Targets identified as having a high probability of being binary could then be passed to longbaseline optical interferometry teams to follow up and observe the systems with greater resolution and precision. Because of the large number of targets that speckle imaging permits one to observe per night, this method could efficiently produce a list of targets, each of which could eventually provide important information concerning the mass-luminosity relation and stellar structure. 5. CONCLUSIONS We have presented and discussed a total of 21 speckle measures of binary stars in which the separation obtained is below the diffraction limit. These measures are consistent with ephemeris
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predictions based on previous visual and speckle orbit determinations and indicate that CCD-based speckle imaging can be useful in characterizing binary stars to separations as small as 0.25 of the diffraction limit. Initial indications suggest that binarity can be distinguished from dispersion in most cases by observing in two colors and using point-source calibration objects, and that this technique may therefore be useful in providing important information on a large number of small-separation binary systems. Results from a survey of this kind could significantly im-
prove the efficiency of binary star work at long-baseline optical interferometers.
This work was funded by NSF grant AST 05-04010. It made use of the Washington Double Star Catalog, maintained at the US Naval Observatory, and the SIMBAD database, operated at CDS, Strasbourg, France.
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