Note: The following description is adapted from a paper by Hartkopf et al. (2001 AJ 122, 3472.)

The *Sixth Catalog of Orbits of Visual Binary Stars* continues the series of compilations of
visual binary star orbits published by William Finsen, Charles Worley, and Wulff Heintz from the
1930s to the 1980s. As of 27 July 2017 the new catalog included 2,739 orbits of 2,656 systems. All
orbits have been graded as in earlier catalogs, although the grading scheme was modified as of the
Fifth Catalog to be more objective.
Ephemerides are given for all orbits, as are plots including all associated data in the Washington
Double Star (WDS) database. A subset of orbits potentially useful for scale calibration is also
presented.

The 3+ decades since publication of the *Fourth Catalog* have seen revolutionary
changes in the field of visual double star work, primarily through the advent and
maturation of interferometry. Speckle interferometry, especially on large telescopes,
can produce astrometric results of very high accuracy (down to milliarcsecond level), even
for systems much closer and of shorter period than those available to micrometry and other
visual techniques. Although the speckle technique has been known since 1970, it did not
produce data in significant quantity until about 1975; at the time of publication of the
*Fourth Catalog* only a handful of orbits had been calculated in which speckle played
much of a role. Now, however, speckle interferometry is a mature field, and nearly all
orbits published since the 1980's have included speckle results, some exclusively.
Long-baseline interferometry [e.g., Mark III (cf., Pan et al. 1990) and the Navy Prototype
Optical Interferometer (NPOI; cf., Hummel et al. 1998)] is now perhaps in a similar degree
of maturation as was speckle in 1983; an increasing number of binaries once exclusively
the "property" of spectroscopists are now the targets of multi-aperture telescope arrays.
Indeed, the distinction between the *spectroscopic* and *visual* regimes will
largely disappear in the coming decades, as the magnitude sensitivity of these new
interferometers improves. Catalogs such as this will have to evolve as well; as
spectroscopic + visual "combined solutions" go from being rare to commonplace, the
argument for publishing only a subset of a binary's elements will grow increasingly
artificial. For the present, however, information on combined solutions is relegated to a
notes file.

Figure 1: Two examples each of grade 1 (left) and grade 5 (right) orbits. Factors used in determining these grades are discussed below. In these and all other orbit figures in this catalog, green plus signs indicate visual (micrometric) observations, violet asterisks photographic measures, and blue symbols various interferometric techniques (open circles, filled circles, and filled squares for eyepiece interferometry, speckle or other single-aperture techniques, and multi-aperture techniques, respectively). Finally, a red "H" or "T" indicates a measure from Hipparcos or Tycho. The dot-dash line indicates the line of nodes. Scales are in arcseconds, and the curved arrow at lower right indicates the direction of orbital motion.

- telescope aperture
- binary separation
- magnitude and magnitude difference: Since we are mainly interested in relative weights to be assigned for observations of a given binary, these factors are presumed constant and we have ignored them.
- "number of nights": Some observers publish individual measures, while others average
2 or more into means. A simple sqrt(
*N*) term handles this. - expertise of the observer: This is the most difficult factor to evaluate. Accuracy should improve with experience, but may decrease as, for example, a visual observer's eyesight deteriorates with age (some observers produced measurements for 40, 50, and even 70 years). We have ignored this age factor for the present, however.
- other factors, such as systematic errors in a given piece of equipment, quality of the scale calibration, seeing conditions at a given site, etc.: These are ignored as separate factors, but obviously are part of the "observer expertise" factor.

All grade 1 and 2 orbits from the *Fourth Catalog* were examined, together with all
more recently published orbits and numerous long-period orbits (such as GRB 34 in Figure
1). This last group was included in order to evaluate observers of wider systems. Many
of these systems show small orbit residuals over the covered orbit arc, although the
lack of phase coverage earns them a poor grade. From ~750 orbits and over 100,000
observations initially examined, some 450 orbits and ~66,000 observations were chosen for
evaluation of observer weights.

Since the number of "degrees of freedom" is large, we simplified the problem in two ways:

- Since binary resolution is a function of telescope aperture, we remove this
complication by scaling separation to the Rayleigh limit of the telescope used
(
*rho_lim*~*L*/*D*, where*L*is the wavelength and*D*is the telescope diameter. Assuming*L*= 550nm,*rho_lim*= 0.136/*D*, for*rho_lim*in arcseconds and*D*in meters). - Different observing techniques - micrometry, photography (including conventional CCD observations), and interferometry (plus adaptive optics, satellite observations, and other high-resolution techniques) - were evaluated separately. All data for a given technique were first studied, then relative weights for observers using that technique determined.

Figure 2: O-C errors in (left) relative separation and (right) position angle, versus separation (scaled to each telescope's Rayleigh limit).

Figure 3: O-C separation errors versus separation, for (left) visual, (center) interferometric, and (right) photographic observing techniques. The relative accuracies of the three techniques are apparent.

As a second step, we wish to determine the relative qualities of each observer who uses a given technique. We do this by removing the overall error versus separation fit derived above, then determining rms residuals for each observer. Observers having too few measures for individual weighting are averaged together. Relative weights for each observer are then defined as the inverse square of their rms residual (with the weighted mean weight for each technique scaled to 1). We find:

- Visual observers received a wide range in relative weight, from about 0.1 to 4.5. It
must be noted that this is not really a measure of an observer's competence; a person
who only looks at bright, wide, low zenith-distance, small d
*m*pairs will tend to receive a better grade than does someone who pushes his instrument to its limits in magnitude, d*m*, etc. These more difficult observations are usually the more important, however. - Photographic observers were of fairly uniform quality. Observers having significant numbers of measures ranged in relative weight from about 0.5 to 1.4, while the observers making smaller contributions received a weight of 0.3. Since photographic techniques are presumably somewhat more objective than visual measurement, this finding seems reasonable.
- Eyepiece interferometry tends to get rather low marks (0.01 to 0.3) compared to other interferometric techniques. Speckle and other single-aperture techniques garnered weights of 0.02 to 1.4, with the larger speckle efforts generally receiving the higher weights. The Mark III received a comparable weight of about 0.9. NPOI received a very high weight (13.3), but this is rather misleading, as the number of measures is small and the separation regime of this instrument is such that this is largely an indication of internal consistency.

- A
- C
- E (mostly E2)
- Eu
- H (mostly Hf,Ht,Hw)
- Hh
- J
- K
- M (dates 1750-1829)
- M (dates 1830-1849)
- M (dates 1850-present)
- P (mostly Pa)
- Po
- S/I (tel 0.00 - 1.49m)
- S/I (tel 1.50 - 2.49m)
- S/I (tel 2.50 - 99.99m)
- T
- misc (other techniques)

Micrometry data were subdivided into various date ranges (since the earliest micrometry measures are known to be of significantly lower quality than later measures). After some experimentation, the date ranges given in the above list were found to be reasonable subdivisions for the technique. In a similar manner, speckle interferometry data were subdivided by telescope aperture, and the three aperture ranges given above were found to be appropriate. Subcategories for space-based (code H) and wide-field imaging (technique E) were checked, as well.

Results are as one might expect. Assuming a mean weight for CCD measures of 1.0, post-1850 micrometry has a comparative weight of about 0.1, with the earliest micrometry measures only 0.01. Large-aperture speckle measures have a relative weight of just over 10, and long-baseline interferometry yields a relative weight on this scale of about 140.

As mentioned above, measures obtained at separations near an instrument's Rayleigh limit will typically be of much lower quality than those obtained at larger separations. In order to determine relative weight for a given technique as a function of separation, residuals in each technique category were converted to weights and sorted by separation (in "rayleighs", as described above). Running mean weights were determined, and quadratic or cubic fits made to each running mean. These were finally scaled to a maximum weight of one. As expected, these weights begin at a value of zero at zero separation, and rise to a value of 1.0 at some 10s of "rayleighs" in separation.

As described earlier, the overall weight of a given observation, then, is determined by the product of four terms, now slightly modified to:

One additional modification should be noted. Speckle obserations made at separations beyond the size of the isoplanatic patch will be of degraded quality; W_quality is therefore set to 0.5 for any speckle measure made at a separation greater than 3 arcseconds.

**1 = Definitive**.- Well-distributed coverage exceeding one revolution; no revisions expected except for minor adjustments.
**2 = Good**.- Most of a revolution, well observed, with sufficient curvature to give considerable confidence in the derived elements. No major changes in the elements likely.
**3 = Reliable**.- At least half of the orbit defined, but the lesser coverage (in number or distribution) or data consistency leaves the possibility of larger errors than in Grade 2.
**4 = Preliminary**.- Individual elements entitled to little weight,and may be subject to substantial
revisions. The quantity 3 log(
*a*) - 2 log(*P*) should not be grossly erroneous. This class contains: orbits with less than half the ellipse defined; orbits with weak or inconsistent data; orbits showing deteriorating representations of recent data... **5 = Indeterminate**.- The elements may not even be approximately correct. The observed arc is usually too short, with little curvature, and frequently there are large residuals associated with the computations.

- weighted rms residual in separation (d
*R*) - weighted rms residual in relative separation (i.e., d
*R*/*R*) *theta*coverage: measures were sorted in order of*theta*, then the rms difference in angle [i.e.,*theta*(n) -*theta*(n-1)] was calculated- maximum "gap" in
*theta*: also from above*theta*sort - phase coverage: calculated from period (
*P*) and periastron epoch (*T*), then sorted and rms differences determined as done with*theta* - maximum "gap" in phase (Why analyze both
*theta*and phase coverage? While both position angle and phase are equivalent for a circular, face-on orbit, position angle coverage becomes increasingly meaningless for inclinations approaching 90 degrees, while uniform phase coverage may undersample periastron passage for a high-eccentricity orbit.) - number of revolutions from first to last observation
- total number of observations

Figure 4: Fitting WH4 grades to rms residuals and other factors, as described in the text.

As is apparent in both the orbit examples in Figure 1 and the Figure 4 results, no one factor alone is sufficient for determining the grade. For example, some poorer orbits show very small separation residuals (as evidenced by the turnover in the rms d

Simple polynomial fits were made between each set of means and their corresponding grades,
and the best fit to the *Fourth Catalog* grades was found by averaging results for
the number of observations, the number of revolutions, the maximum angle and phase gaps,
and the weighted rms separation residual. New grades were then calculated for each of the
901 orbits based on all these factors; Figure 5 illustrates the degree of correlation
between our new, "objective" grades and the *Fourth Catalog* originals. Some 98% of
the grades matched to within one grade level. A check of those systems where our grades
disagreed by more than 1 grade found that in nearly all cases the *Fourth Catalog*
grades appeared to be in error. It therefore appears that our quantitative method for
grading orbits gives a reasonably good match to Worley & Heintz' originals.

Figure 5: Comparison of grades determined by the method described here with those determined by Worley & Heintz. Circle sizes are scaled to the corresponding number of orbits with these grades; the numbers themselves are given inside all but the smallest circles, where

One last adjustment was made before grades were determined for all orbits. Thanks to the higher astrometric accuracy achievable by interferometric techniques, we now have the ability to determine orbital elements with higher accuracy than previously considered possible. Since an old "grade 1" orbit may no longer be considered

It is worth noting that combined astrometric/spectroscopic solutions are graded only on the number,
quality, and distribution of their differential astrometric measures. These solutions typically
have *P*, *T*, *e*, and *w* (or at least a subset of these elements) known to
higher accuracy than is reflected in only the visual data. The quality of a combined solution orbit
is therefore higher than the grade indicates, although the extent of the improvement depends on the
quality of the spectroscopic data, the evaluation of which is beyond the scope of this catalog. The
fact that an orbit is a combined solution is certainly taken into consideration when evaluating
which orbit of a given pair is considered "best".

A handful of orbits could not be graded, due to a lack of *rho* and *theta* measures in
the WDS. The first class of these are the few interferometric binaries observed by the Mark III or
the Palomar Testbed Interferometer (c.f., Boden et al. 1999) for whom only visibilities were
published. These orbits, given a grade of "8" in the catalog, are usually of quite high quality.
More common are astrometric orbits, which receive a grade of "9"; these orbits tend to give rather
poor fits to any later resolved measures.

A final note: we do not consider this grading method optimum; a visual inspection of competing orbits is still necessary if their grades are within a few tenths of each other. Other schemes, such as the "efficiency" technique of Eichhorn (c.f., Eichhorn & Cole 1985), may be superior and worth investigating. For the present, however, we think this method gives reasonably reliable results.

(Note: All observer weights were reevaluated in March 2004, and all orbits were regraded and new
figures generated on a regular basis to reflect measures recently added to the WDS. The reevaulation
weights led to minor changes for most orbits - typically 0.0 to 0.2 grades - but the inclusion of
new data has occasionally resulted in major revisions of grades for individual systems, as might be
expected.)

Names and orbital elements for a given system are tabulated on two lines, with a blank line separating each orbit for ease in reading. A text version of the catalog (one line per orbit) is also available, as described in the format description page. Columns in the main catalog are as follows:

Line 1:

- Epoch-2000 coordinates, usually given to 0.01s accuracy in right ascension and 0.1" in declination. Most coordinates were extracted from the Tycho-2 catalog.
- Washington Double Star (WDS) designation. Many of these coordinates (given to 0.1m and 1') were generated by precessing lower precision B1900 positions to J2000; therefore the WDS designations may vary slightly from the coordinates in column 1.
- Discoverer designation and components involved. If no components are listed, the orbit is of the AB pair.
- HD (Henry Draper catalog) number.
- Magnitude of the primary component. Codes ">" and "<" preceding the magnitude indicate the value is an upper or lower limit. Codes "v" or "k" following the value indicates a star of variable magnitude or a K-band magnitude.
- The period (
*P*), followed by code indicating units: "c" = centuries, "d" = days, "h" = hours, "m" = minutes, "y" = years. - The semi-major axis (
*a*), followed by code indicating units: "a" = arcseconds, "m" = milliarcseconds, "u" = microarcseconds. - The inclination (
*i*), in degrees. - The node (
*Omega*), in degrees. An identified ascending node is indicated by an asterisk following the value. - The time of periastron passage (
*T*), followed by code indicating units: "d" = Julian date, with first two digits truncated (i.e., JD-2400000), "m" = modified Julian date (JD-2400000.5), "y" = fractional Besselian year. - The eccentricity (
*e*). - The longitude of periastron (
*w*), in degrees, reckoned from the node as listed. - Equinox, if any, to which the node refers.
- The grade (to nearest integer), as previously discussed.
- A link "N" to any notes for this system.
- A link "P" to a figure illustrating the orbit and all data for this object currently tabulated in the WDS database. Symbols used in these figures are as in Figure 1.
- A link "E" to appropriate entries in the ephemeris file.
- A code for the reference (usually based on the name of the first author and the date of publication), with a link to a reference file.

- ADS (Aitken Double Star catalog) number.
- Hipparcos number.
- Magnitude of the secondary component. Codes are as described for the primary.
- Published formal errors in
*P*. Units are the same as for*P*. - Published formal errors in
*a*. Units are the same as for*i*. - Published formal errors in
*i*. - Published formal errors in
*Omega*. - Published formal errors in
*T*. Units are the same as for*T*. - Published formal errors in
*e*. - Published formal errors in
*w*. - The date of the last observation used in the orbit calculation, when given.

- The WDS designation, as above.
- The discoverer designation, as above.
- The orbit grade, as above.
- The reference code, as above.
- Predicted values of
*theta*and*rho*for the years 2005-2009.

This subset of the *Sixth Catalog* presently includes 81 orbits of 80 systems. As in
the main catalog, figures are included in order to allow the user to visually inspect each
orbit's quality prior to use. An expanded set of ephemerides has also been generated,
giving predicted separations and position angles with finer time resolution than in the
main catalog (although these ephemerides will obviously still be of little use for very
short-period systems).

Note that all "calibration candidate" orbits are **not** of the same quality. Before
adopting a set of elements it is recommended that users examine the elements, figures,
etc. carefully to check whether that orbit appears to be of proper scale and sufficient
quality for their purposes. Also, using measurements of double stars to calibrate the
measurement of other double stars is certainly circular (or, if you will, Keplerian). We
strongly advocate the use of other absolute calibration techniques such as a slit mask
(cf., McAlister et al. 1987, Hartkopf et al. 1997, Douglass et al. 1997) or at least
star trails (for east-west orientation) if at all possible. When double stars are
necessary for scale calibration, the set provided should be adequate; however, the
measures determined will only be as accurate as the calibration systems used. The use of
these systems for identification of higher-order motions or submotions is discouraged.

Figure 8 (above) shows a plot of log(*P*) versus *e* for the orbits in the
*Sixth Catalog*. The dramatic circularization seen in similar plots for
spectroscopic binaries is not seen in resolved binaries, due to their longer periods.

Finally, Figure 9 (above) is a plot of [Periastron_(outer binary) / Apastron_(inner binary)] versus |i_(outer binary) - i_(inner binary)| for the twenty hierarchical systems with visual orbits determined for both hierarchies. Harrington (1992) quantified a value of 3 for the ratio of periastron(outer binary) to apastron(inner binary) of the inner binary as the critical factor for long-term stability (assuming equal mass companions). Hierarchies to the left of the dotted line may demonstrate long-term instabilities, while those to the right of the dashed line have ratios that may allow intermediate hierarchies.

Systems which are *mildly* interacting may show a tendency towards co-planarity, so
the tendency of smaller inclination differences having smaller period ratios is not
surprising. Of the five systems lying between the dotted and dashed lines:

- 02291+6724 = CHR 6Aa + STF 262Aa-B : Both orbits are of grade 5 and the ratio here is suspect.
- 06003-3102 = HU 1399AB + HJ 3823AB-C : this one is the closest to the line and the incomplete orbital coverage of the wider system may be the culprit.
- 08592+4803 = HU 628BC + HJ 2477A-BC : the orbit for the wider system is highly
suspect. There seems to be a systematic drift in the
*O-C*values for this orbit and it may not be even a physical association. - 23019+4220 = BLA 12Aa + WRH 37Aa-B : The short-period orbit is based on only five data points (one of those with a large residual) and needs more data.
- 23393+4543 = CHR 149Aa + 3A 643Aa-B : Also suffering from a paucity of data, the short-period system here has four differential measures and needs more data, too.

We dedicate this catalog to the memory of Charles Edmund Worley, our colleague and friend.

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