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Subsections

3 Discussion

3.1 Procedures

Traditional corrections were applied for the inclination of the micrometer, irregularities of the pivots and the micrometer screw, clamp and circle differences, circle diameter corrections, variation of latitude, and refraction, as well as collimation, level, azimuth, and nadir.

Measurements of the instrumental flexure determined from the horizontal collimators were made for each transit circle but these exhibited very large variations. Since a more consistent determination of the flexure can be determined from the star observations (Holdenried & Rafferty [1997]), the flexure determined from the horizontal collimators was not applied. Since the formation of the most recent absolute catalog from the Six-inch, the W1 $_{{\rm J}00}$, showed that a more consistent determination of the flexure can be extracted from star observations (Holdenried & Rafferty [1997]), this method was tried with the W2 $_{{\rm J}00}$ observations. Although excellent results were obtained from the Six-inch observations, the results from the Seven-inch were not satisfactory, showing a systematic error that was a function of zenith distance. The method involved using FK5 stars to solve for corrections to the flexure and refraction, and using circumpolar stars to solve for a correction to the latitude. This last step was necessary for an absolute program such as the W1 $_{{\rm J}00}$; however, in the case of the Seven-inch data, it was felt that the below pole observations of the circumpolar stars were the source of the systematic errors. Therefore, another method was employed, fitting a cubic spline to reduce the residuals of all Hipparcos stars but excluding the below pole observations (which are not necessary for the differential reductions). This new appoarch gave nearly identical results for the Six-inch as the other method and it removed the systematic differences seen in the Seven-inch observations. For consistency the cubic spline method was used on both the Six-inch and Seven-inch W2 $_{{\rm J}00}$ observations. Observations were grouped into "tours". Usually two tours were taken per night, dividing the night in half between two observers. Each tour contained determinations of the collimation, level, nadir, azimuth, and flexure taken at two to three hour intervals for the nighttime tours. For the Seven-inch transit circle, azimuth determinations were made hourly due to apparent motions of the piers. For each tour, observations were made of selected groups of stars to determine corrections to the sidereal time, azimuth, and refraction. In addition, a subset of stars, following a concept developed by Küstner and henced referred to as Küstner stars, distributed over the entire sky was observed during each tour to check for nightly variations of the instrument or atmosphere over large angles. The IRS were grouped in zones of 15$^\circ$ of declination and were observed with FK5 reference stars to allow differential reductions for each tour. As was explained previously, although the catalog was planned to be absolute, the IRS were observed in such a way as to allow differential reductions. Because the differential reductions could be carried out in almost real-time, they provided an opportunity to closely monitor the quality of the observations. Differential observations also are an effective method of reducing the random and systematic errors in the data.

The requirement imposed by the even distribution in time and zenith distance of the clock, azimuth, refraction, and Küstner stars as well as the need to choose IRS and their reference stars while maintaining a balance of all observations over the Clamps and Circles necessitated the development of an automatic method of selecting the stars to be observed for each tour. The logical criteria for this Star Selector software were constructed by T. Corbin, while the software and system development was done by F.S. Gauss.

3.2 Tour adjustments


Differential adjustments were applied to each tour from a least squares fit to a set of Hipparcos reference stars. Numerous models (9 in right ascension and 21 in declination) were tested for each tour, and the one providing the best fit was used. The models incorporated coefficients that depended on zenith distance, the tangent and sine of the zenith distance, and arguments of time. Tables 3 and 4 present the median, minimum, and maximum estimated standard deviations (average mean error of a single observation) of the selected models for each transit circle.


 

 
Table 7: Mean declination positional errors for each transit circle as well as the mean errors and epochs for the final positions
Declination errors
  Six-inch Seven-inch Total
Declination $\sigma$ $\bar{\sigma}$ n $\sigma$ $\bar{\sigma}$ n mean $\sigma$ $\bar{\sigma}$ n
Range mas mas stars mas mas stars epoch mas mas stars
+90 to +85 251 72 97       1990.76 251 72 97
+85 to +80 226 73 265       1990.72 226 73 265
+80 to +75 216 74 435       1990.68 216 74 435
+75 to +70 212 75 577       1990.71 212 75 577
+70 to +65 215 76 736       1990.73 215 76 736
+65 to +60 213 78 874       1990.76 213 78 874
+60 to +55 206 74 1018       1990.74 206 74 1018
+55 to +50 209 76 1156       1990.83 209 76 1156
+50 to +45 201 74 1277       1990.80 201 74 1277
+45 to +40 194 71 1416       1990.80 194 71 1416
+40 to +35 200 73 1533       1990.83 200 73 1533
+35 to +30 201 75 1579 142 189 16 1990.76 201 75 1579
+30 to +25 201 75 1765 210 114 191 1990.80 202 75 1765
+25 to +20 207 76 1748 227 91 236 1990.82 209 76 1748
+20 to +15 206 77 1782 219 72 228 1990.80 208 77 1782
+15 to +10 209 81 1811 218 65 210 1990.82 211 80 1811
+10 to $\;\:$+5 201 79 1827 216 58 244 1990.85 203 78 1827
$\;\:$+5 to $\;\;\;\:\:$0 194 80 1820 217 93 1792 1991.45 215 64 1822
$\;\;\;\:\:$0 to $\;\:-$5 186 80 1727 218 87 1796 1991.58 211 63 1802
$\;\:-$5 to -10 183 59 282 227 86 1825 1992.15 225 84 1822
-10 to -15 179 57 207 220 80 1829 1992.16 218 80 1831
-15 to -20 173 62 204 223 80 1842 1992.16 221 79 1841
-20 to -25 173 70 200 221 78 1701 1992.14 219 77 1701
-25 to -30 165 86 186 221 76 1596 1992.14 220 76 1596
-30 to -35 124 148 67 217 77 1787 1992.18 217 77 1787
-35 to -40       220 77 1832 1992.17 220 77 1832
-40 to -45       225 78 1657 1992.25 225 78 1657
-45 to -50       222 78 1644 1992.22 222 78 1644
-50 to -55       222 77 1329 1992.16 222 77 1329
-55 to -60       227 79 1184 1992.27 227 79 1184
-60 to -65       228 79 1034 1992.22 228 79 1034
-65 to -70       235 81 799 1992.25 235 81 799
-70 to -75       247 83 648 1992.21 247 83 648
-75 to -80       266 87 494 1992.26 266 87 494
-80 to -85       275 84 310 1992.30 275 84 310
-85 to -90       298 79 111 1992.23 298 79 111


3.3 Combined observations

The locations of the two transit circles allowed nearly 70$^\circ$ overlap in the declinations accessible to each telescope. For those stars in this overlap region, the observations were combined in a weighted mean. The weights (given in Table 5) were based on the mean standard deviation of a single observation as a function of zenith distance and were an attempt to account for the degradation suffered by observations made through large air masses.

3.4 Mean errors of the observation and positions

The weighted standard deviation of the mean is given with the position of each star. For the stars observed with both transit circles, the mean positions and their standard deviation of the mean as determined by each instrument are given as well as the weighted mean and weighted standard deviation of the mean of the combined data.

Tables 6 and 7 group the average standard deviations of a single observation, the average standard deviation of the mean, and the number of stars into five degree zones of declination. The average standard deviation of a single observation was close to 200 mas in right ascension and 215 mas in declination. The average standard deviation of the mean position for a star varied by the number of observations. Since the majority of stars in each zone were IRS, which averaged six (two on Circle One and four on Circle Two) observations each, the average standard deviation of the mean was close to 70 mas in right ascension and 77 mas in declination. In the declination zone -5$^\circ$ to +5$^\circ$ the Six-inch and Seven-inch observed the same IRS stars thus doubling the number of observations each received, and this manifests itself in a sharp drop in the average standard deviation of the mean.

3.5 Epochs


The average epoch of a position in right ascension is 1991.53 and in declination 1991.52. However, because the Seven-inch started observing about a year after the Six-inch, there is a pronounced dependence on declination of the epochs of individual stars. Tables 6 and 7 show the epochs averaged over the same 5$^\circ$ zones in declination that were used in the groupings for the errors.

3.6 Double stars

A few double stars observed by both transit circles showed significant differences. For example, in some cases the image dissector on the Seven-inch transit circle could not split doubles that the observers on the Six-inch transit circle were able to resolve. In those situations, where it was clear that each telescope observed a particular double differently, the observations by one instrument or the other were dropped. Double stars outside the overlap zone for the two telescopes, of course, can not be compared in this way and may have undetected errors in their positions.

3.7 Planetary observations

The purpose of the daytime observations of the Sun, Mercury, Venus, Mars, and bright stars was to create an absolute catalog tied to the dynamical reference frame. Because these observations were not necessary for the link to the ICRF and the quality of these observations makes it difficult to adjust them to the nighttime system, the daytime observations were not reduced.


 

 
Table 8: Additional phase corrections applied to Mars, Jupiter, and Saturn
 
Additional phase corrections
Mars - Six-inch - $\pm 0.15 + {\rm phase\:corr}_{\alpha} \times 0.282$
  -   - $\pm 0.10 + {\rm phase\:corr}_{\delta} \times 1.290$
  - Seven-inch - $\pm 0.88$
  -   - ${\rm phase\:corr}_{\delta} \times 3.450$
Jupiter - Six-inch - ${\rm phase\:corr}_{\alpha} \times 1.901$
  -   - ${\rm phase\:corr}_{\delta} \times 6.426$
  - Seven-inch - $\pm 0.25 + {\rm phase\:corr}_{\alpha} \times 4.250$
  -   - $\pm 0.15 + {\rm phase\:corr}_{\delta} \times 4.445$
Saturn - Six-inch - ${\rm phase\:corr}_{\alpha} \times 7.000$
  - (limb obs) - none
  - Six-inch - none
  - (ring obs) - none
  - Seven-inch - none
  -   - none


The same corrections that were developed for the observations of the stars also were applied to the nighttime planetary observations. It is necessary to apply additional corrections to the observations of most of the planets due to their orbital motions, appearances, and distances. These additional corrections must be calculated using data from an ephemeris. For the major planets, ephemeris data from JPL's DE405 ([1998]) were used, and for the minor planets, USNO (Hilton [1999]) provided the ephemerides.

Corrections for orbital motion were applied to bring the mean, measured position into coincidence with the meridian.

$\displaystyle {\rm OMCorr}_{\alpha}$     (1)
  = $\displaystyle ({\rm TimeObs}_{\alpha} - {\rm ObsRA})\times\frac{{\rm SSMTD}}{{\rm SSMTD}\times {\rm OM}_{\alpha}}$  
       

and


$\displaystyle {\rm OMCorr}_{\delta}$     (2)
  = $\displaystyle \frac{{\rm ClpSw}\times {\rm MicEq}_{\alpha}\times {\rm MPoB}\times {\rm OM}_{\delta}}{{\rm SSMTD}
\times {\rm cos(ObsDec)}}$  
       

where:


$\displaystyle {\rm OMcorr}_{\alpha}$ = $\displaystyle \mbox{right ascension orbital motion correction;}$  
$\displaystyle {\rm TimeObs}_{\alpha}$ = $\displaystyle \mbox{time of observation;}$  
$\displaystyle {\rm ObsRA}$ = $\displaystyle \mbox{observed right ascension;}$  
$\displaystyle {\rm SSMTD}$ = $\displaystyle \mbox{sidereal seconds per Mean Time Day;}$  
$\displaystyle {\rm OM}_{\alpha}$ = $\displaystyle \mbox{motion in RA per Mean Time Day;}$  
$\displaystyle {\rm OMcorr}_{\delta}$ = $\displaystyle \mbox{declination orbital motion correction;}$  
$\displaystyle {\rm ClpSw}$ = $\displaystyle \mbox{clamp switch;}$  
    $\displaystyle \mbox{(+1 for East Clamp and $-$ 1 for West);}$  
$\displaystyle {\rm MicEq}_{\alpha}$ = $\displaystyle \mbox{right ascension micrometer screw equivalent;}$  
$\displaystyle {\rm MPoB}$ = $\displaystyle \mbox{mean place of bisection (mean measured}$  
    $\displaystyle \mbox{position minus the collimation);}$  
$\displaystyle {\rm ObsDec}$ = $\displaystyle \mbox{observed declination;}$  
$\displaystyle {\rm OM}_{\delta}$ = $\displaystyle \mbox{motion in Dec per Mean Time Day.}$  
       

Declinations were corrected for horizontal parallax using the following:


$\displaystyle {\rm ParCorr}$     (3)
  = $\displaystyle {\rm EarthRV\times HorPar\times sin(gLat - ObsDec)}$  
       

where:


$\displaystyle {\rm ParCorr}$ = $\displaystyle \mbox{parallax correction;}$  
$\displaystyle {\rm EarthRV}$ = $\displaystyle \mbox{Earth's radius vector;}$  
$\displaystyle {\rm HorPar}$ = $\displaystyle \mbox{horizontal parallax;}$  
$\displaystyle {\rm gLat}$ = $\displaystyle \mbox{geocentric latitude;}$  
$\displaystyle {\rm ObsDec}$ = $\displaystyle \mbox{observed declination.}$  
       

Corrections for the visual appearance of each solar system object were based on their appearance in the transit circle and the method of measurement used.

The Seven-inch, observing with the image dissector, used digital centering algorithms developed by Stone ([1990]). Changes to these algorithms have caused the observations of Mars, Jupiter, and Saturn made between 1987 and 1992 to be dropped. The algorithm also had difficulty with Saturn as the rings tilted edge on during the last year of the program and these observations were also dropped. For Uranus, Neptune, and the minor planets the center of light was observed.

The Six-inch, observing visually, dealt with the planetary objects as follows:

Mars - The four limbs were observed for all the nighttime observations, except for three when the center of light was taken. Corrections for phase were applied using:

$\displaystyle {\rm phase\:corr}_{\alpha}$     (4)
  = $\displaystyle \frac{240}{(1-{\lambda})({\rm cos}\:{\delta})}\times q$  
    $\displaystyle \times\left(\frac{1}{2}{\rm sin}^{2}Q-\frac{1}{16}(1-{\rm cos}\:i){\rm sin}^{2}2Q\right)$  
       

and


$\displaystyle {\rm phase\:corr}_{\delta}$     (5)
  = $\displaystyle q\times\left(\frac{1}{2}{\rm cos}^{2}Q-\frac{1}{16}(1-{\rm cos}\:i){\rm sin}^{2}2Q\right)$  
       

where:


$\displaystyle {\rm phase\:corr}_{\alpha}$ = $\displaystyle \mbox{right ascension correction for phase;}$  
$\displaystyle {\rm phase\:corr}_{\delta}$ = $\displaystyle \mbox{declination correction for phase;}$  
$\displaystyle \lambda$ = $\displaystyle \mbox{planet's orbital motion;}$  
$\displaystyle \delta$ = $\displaystyle \mbox{declination;}$  
q = $\displaystyle \mbox{defect of illumination;}$  
Q = $\displaystyle \mbox{position\,angle\,of\,the\,defect\,of\,illumination;}$  
i = $\displaystyle \mbox{angle at planet between Earth and Sun.}$  
       

Minor Planets - No visual appearance corrections were applied as all presented point source images. Jupiter - The four limbs were observed. Corrections for phase were applied using the same equations as were given for Mars. Saturn - The four limbs of Saturn were observed about 65% of the time, otherwise the edges of the rings were taken. Even though in previous catalogs no phase corrections were applied to the observations of Saturn, plots of the (O-C)s from the limb observations showed a systematic offset symmetrical around oppositions indicating the need for such an adjustment. The (O-C)s from the ring observations showed no such systematic offsets. Corrections for phase were applied to the limb observations using the same equations used for Mars and Jupiter. Uranus and Neptune - Center of light was observed and no corrections for phase were applied. Plots of the (O-C)s as functions of the phase corrections determined from equations above show systematic offsets symmetrical around opposition even after phase corrections given above were applied. The equations used for the Six-inch data, as well as the algorithms developed for the Seven-inch data, are based on the geometric changes in the appearances of these planets. The failures to account for all the phase effects are likely the result of limb darkening or other illumination effects. Empirically correcting for these residual effects is the cause for some concern in that the effects may be in the ephemeris rather than in the observations themselves. However, the Six-inch results for Saturn were able to clarify the situation when the observations of Saturn's limbs showed the systematic offsets while the observations of the rings did not (no such phase corrections could be determined for the Seven-inch Saturn observations because the algorithm used was fitted to both the limbs and rings). The empirically determined, additional phase corrections for Mars, Jupiter, and Saturn are shown in Table 8.


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