As was mentioned above, the alignment of the instrumental frame with a dynamical frame was accomplished through an adjustment to the equinox point and the equatorial plane. The equinox correction assumed that the poles of the two frames were coincident, and thus a simple rotation was all that was necessary to align the zero point of the right ascensions. As a result of the way in which the right ascensions were determined, the zero point of the instrumental system, before any rotation was applied, was very close to that of the catalog which supplied the data for the apparent places of a special set of stars known as clock stars. Usually this catalog was one of the series of Fundamentalkatalogs (FK) from the Astronomisches Rechen-Institut in Heidelberg.
Traditionally the equator correction was an adjustment to the
declination system at the mean declination of the planetary
observations and was used to determine the
horizontal instrumental flexure (F)
and to develop corrections,(
) and (
),
to, respectively, the assumed latitude
and
refraction constant. These
three
quantities (, , and F)
can be related through an expression of the form:
where:
Because this method utilizes lower culmination observations it requires
stars that are circumpolar.
Since the mean epochs of the upper and lower culmination
observations purposely were kept nearly the same, the computed
positions canceled out and the solution was, therefore, considered
absolute. However, a solution including both the flexure (F) and
the correction to the latitude () was not
feasible because, over the small range of zenith distances
involved, F and
are strongly correlated. To alleviate this problem,
the value of the flexure empirically determined using two
horizontal collimating telescopes was substituted into the
equation.
Typically the resulting flexure
showed a very large scatter (for the W1
program, the
flexures had a standard deviation of 
).
Though the flexure measured employing the collimators
was poorly determined, it was necessary to use it because of the impossibility,
when
just the circumpolar stars were included in the solution, of solving
simultaneously for the flexure and
the corrections to
the constant of refraction and the assumed
latitude. Various theories have been proposed to explain
why the flexure determined from the collimators is so poorly
determined; probably the most reasonable postulates that
temperature gradients in the air in the tube of the instrument
introduce refraction effects that
distort the
measures. A study based on Washington, Six-inch data,
by Høg & Miller (1986), showed that flexures
measured using the collimators varied with the rate of change of
temperature leading to the conclusion that internal tube refraction
was responsible for this behavior.
A follow-up study, by
Høg & Fabricius (1988) involving the Carlsberg transit circle,
reinforced the conclusions about internal refraction
and suggested the
use of fans to mix the air. The construction of the Six-inch (specifically
the exposed wires in the micrometer), however, precluded the adoption
of that suggestion.
During the W1 program the application of flexure as
a function of the rate of change of temperature was tested; but
very little, if any, improvement could be detected (see Table 1 (click here), line 2).
| Empirical Values | Calculated Values | ||||
| no. | solutions | F( ) | F() |
 R() |
() |
| 1 | Circumpolar |  |  | +1.637 .161 | |
| 2 | Circumpolar | F(t) | +0.039 .017 | -0.228 .031 | |
| 3 | Circumpolar | +0.662.252 | +0.055 .017 | -0.292 .031 | |
| 4 | All-sky | +0.687 .083 | -0.024 .022 | -0.186 .007 | |
| 5a | Combined All-Sky (all-sky) | +0.687 .158 | -0.024 .085 | ||
| 5b | Combined All-Sky (circumpolar) | -0.172 .009 | |||
In any case, by substituting the mean flexure value measured
using the collimators for F in Eq. (1), and can be
determined. Then by using the following:
all the declinations were adjusted by the amount 
.
However, when this procedure was applied to the planets, it
frequently resulted in a sizeable systematic offset in the
's,
traditionally called the equator correction.
In the case of the W550 catalog (Hughes 1982) this offset
was 
.
Since the instrumental pole was regarded as absolutely fixed,
an additional step
was taken in which a solution was made for further corrections to
and , as well as a correction to F, by defining a
model that
smoothly distributed the offset between the pole and the equator
so that positions near the pole were unaffected while
the full offset was applied to those near the equator.
This would seem to have the effect of producing a system whose pole was
that of the instrument but with an equator that was not necessarily
connected
to that pole and with intermediate declinations that were only partially
linked to either the pole or the equator.
This method was employed
for the W550 (Hughes 1982), WL50 (Hughes 1992) and other Washington
absolute catalogs, and, because of the location of the stars involved,
will be referred to
as the circumpolar solution.
The circumpolar solution is very weakly defined because of the
sparseness of the set of stars contributing 's
and their
limited range of zenith distances. Since the objective is to
produce an instrumental system that is free of bias, such behavior
is a source of concern. For example, if a breakdown in the
solution produces an error in , this error will manifest itself
as a bias in the system that has a signature of the tangent of the zenith distance.
Also, because of the aforementioned correlation between the F and
, the circumpolar solution requires an initial value for the
flexure. Although, in principle, the flexure can be measured using the
collimators, it is
evident from long experience that this
flexure cannot be applied in a simple manner to other pointings of
the instrument at the level of accuracy required. This is
underscored by the fact that in past Washington transit circle
catalogs the final adopted flexure differed significantly from the
empirical. In the case of the W550, the measured flexure was
while the adopted value derived from the circumpolar solution was
.
For the reasons enumerated above, we felt that a new approach
would be of benefit in the discussion of the future catalogs. It
was judged that F and could be determined more accurately from
star observations that were not restricted to just those that
were circumpolar. Using the equation (for upper culmination observations)
and (for lower culmination observations)
for the stars observed at all zenith distances, all three unknowns
(
and F) can be solved for by least squares.
Because zenith
distances are not restricted as they are in the circumpolar
solution, the strong correlation between and F is broken.
However, the explicit introduction of C, the star's computed place,
will threaten the absolute quality of a catalog. It can be said though,
in general, that the systematic errors in the C's are not correlated
with the zenith distance or, more precisely, with the sine and tangent
of the zenith distance. Furthermore, by grouping the 's
into
zones arranged symmetrically about the zenith, we can take advantage
of a useful feature of odd, trigonometric functions (such as sine and
tangent).
A least squares solution for the coefficients of such functions
have the property of being
completely independent of the intercept as long as the data points are
distributed absolutely symmetrically about the origin.
Thus a solution for
and F, from data properly arranged,
need not be dependent on .
This fact can be seen in Table 1 (click here), lines 4 and 5a; whether is
included in the solution does not affect the values of and F.
However, systematic
errors in the C's will impact directly on
in a cumulative fashion
unless steps are taken to mitigate them. To accomplish this we have solved
for separately using only circumpolar observations.
By restricting
the observations in this way, the systematic errors in the C's should
practically (to the extent that the mean epochs of the upper and lower
culmination
observations are the same)
cancel out. We shall refer to these methods, collectively,
(i.e. both solving for all three terms simultaneously and solving for F
and separately from ) as the
all-sky solution.
To summarize, the values of F and from
the all-sky solution
can be considered independent of the C's, but this is not the case for
. The value of
necessarily will link the instrumental system to the system of the C's.
As a regression problem, the
values for F and can be obtained, without including ,
from the all-sky method using the model:
where 
for upper culmination observations
and for lower culmination 
.
These values may then be applied
to the observations and a solution for may be determined
from the circumpolar stars using:
By combining the all-sky and circumpolar solutions, the absolute nature
of the instrumental system is retained. We shall refer to this method
as the combined all-sky solution.
In the discussion of the W1, only the FK5, Part I (Fricke 1988)
stars were used in the combined
all-sky solution. Comparisons were made between the circumpolar
(Table 1 (click here), lines 2-3),
the all-sky (Table 1 (click here), line 4), and the combined
all-sky solutions (Table 1 (click here), lines 5a and 5b).
For the W1, the value of the flexure produced by the all-sky
solution is
nearly identical to the value measured with respect to the collimators.
This is probably coincidence for we have not seen this happen
in other Washington catalogs.
The major difference
between the two methods is in the value for . Since the
correction to the constant of refraction
is a function of 
, even small differences in will have a
large effect at the extreme zenith distances. To the north
the effect is not significant because lower culmination
observations, which have the largest zenith distances, are not
usually reported in the final catalog; but to the south the effect
can be much more of a problem. To ascertain which method results
in the "best" , comparisons were made with positions of FK5 stars
observed by the Carlsberg transit circle between 1986 and 1990
(Carlsberg 1989, 1991 and 1992) located at La Palma
in the Canary Islands. This
transit circle is at a latitude 
south of that of the
instrument that observed the W1, which allows the Carlsberg
instrument to observe at moderate zenith distances the same stars
that are very near the southern horizon of the Washington instrument.
Thus the Carlsberg observations, for these stars, are considerably
less impacted by refraction. When comparing the positions for the
southern-most stars, those adjusted by the values from the all-sky
solution agreed more closely with the Carlsberg results than did
the positions adjusted by values from the circumpolar solution. A
constant was applied to the Carlsberg observations so that the mean
's were the same for the two instruments thus
eliminating the effect of . It is interesting that
the 's differ
significantly between the all-sky and the circumpolar solutions even though
in each case they seem well determined. This is probably
a result of the differing values for .
An important feature of the combined all-sky solution is that it does
not utilize observations of solar system objects, thus leaving them
free to be used to align the axes of the instrumental frame with the dynamical
reference system. We will posit that
the instrumental coordinate frame is a rigid system of orthogonal
axes, and that the alignment with the dynamical frame should be
accomplished by simple orthogonal transformations. If a rectangular
coordinate system is employed
the transformation can be accomplished by three
rotations, one about each of the the axes. We will choose the axes in the
usually manner so that the z axis is parallel to the celestial pole, the
x axis points to the Vernal Equinox, and rotations are defined as positive
in the right-handed sense. A rotation about the x axis is described by the
angle i, about the y axis by the angle j and about the z axis by the
angle k.
To make the problem amenable to solution by linear least squares
it is customary to utilize the small angle
approximations of; 
, 
, and 
(where a and b are
small angles). In this way
the rotation matrix reduces to
The equations of condition can be formulated
as
where:
and the primed variables are the rectangular coordinates after
being rotated. This procedure has become so common in problems in
astrometry involving coordinate transformations that it has been
called the standard method (Vityazev 1994), a term that we
shall also use.
We tested the ability of the standard method to determine
the rotational angles in the transformation between two frames and also
to apply the rotations.
This was done by
running numerical experiments with data from a synthetic, observed
catalog in which random errors could be injected that were typical
of a transit circle along with known rotations between the catalog axes
and the
dynamical frame. Two types of data sets were treated; planetary
data, which were restricted to "observations" located near the
ecliptic, and star data, which included "observations" at all
declinations. The standard method proved to be accurate (in
the sense of mean residual) to better than 
over the range of expected rotation angles
(0 - 10).
The angles resulting from the solution by the standard method using 2062
planetary observations (Sun, Mercury, Venus and Mars)
from the W1 and the ephemeris from DE200 are:
For the purposes of comparison, the same set of data was treated by
what may be described as the "traditional" method for aligning an
instrumental transit circle frame with the dynamical. That is, a
least squares solution for an equator and equinox correction was
performed employing the same programs that in the past were used to
solve for corrections to the orbital
elements. Forcing the corrections to the orbital elements to be
zero resulted in an equinox correction of
. As expected this value is very
close to k, a rotation about the z-axis, since the equinox correction
represents a rotation about the celestial pole, which is parallel to
the z-axis. The angles i and j, representing an
inclination of the celestial pole,
have no analogs in the traditional solution.
After the rotations determined by the standard method
have been applied to the planet observations, a significant offset in
declination (i.e. the mean of the planetary
's)
frequently remains.
It could be argued that, to align the transit circle's
instrumental system to the dynamical system,
this declination offset must be accounted for in some fashion.
However to do so, two problems must be solved.
First, as others have demonstrated convincingly (Vityazev 1994), non-rotational differences between coordinate systems can be cross-correlated with the rotational ones making them very difficult to separate. The declination offset is clearly a non-rotational term and as such may confound the determination of rotation angles. Vityazev has shown that non-rotational terms should be removed before a solution for rotational terms can be attempted. Unfortunately, in the problem we are addressing here, there is no a priori source of information that would allow us to remove the declination offset from the data. However, it may be possible to carefully choose the data in such a way as to break the correlation between the offset and the rotation angles. That this might be so, is intuitively evident in the case of "perfect" data. Perfect data would be distributed evenly in right ascension and declination. With this data the determination of rotation angles is completely uncorrelated with a declination offset.
Our study has shown that if one chooses to solve for the rotation angles
and offset simultaneously by least squares using the equations of condition
where 
= declination offset
that cross correlations of greater than 
(from the correlation matrix)
indicate unstable solutions and poorly distributed data.
It is the distribution of the observations in right ascension that is most
important.
We have found if cross-correlations are greater than
, then that data set cannot be used to
solve for rotation angles unless
the declination offset can be estimated and removed
beforehand.
The second problem involving the declination
offset is demonstrated in Tables 2 (click here)-4 (click here).
These tables summarize solutions, by the standard method, for rotations
(between the instrumental frame and DE200), and
after the rotations have been applied, a determination of for
data sets comprised of various combinations of solar system observations
(the Sun and major planets) from the three
absolute catalogs of transit circles observations, W1J00,
Capetown observations between 1951 and 1959 (Stoy 1968), and
Greenwich observations between 1942 and 1954 (Tucker 1983).
Each line in the tables gives
the angles of rotation (i,j,k) as well as produced by treating
the 
's of different
combinations of solar system objects. Also, the columns
labeled "traditional"
give the results obtained by code that solved for
equator and equinox corrections along with corrections to the orbital
elements, but, here for comparison purposes only, the orbital elements
were forced to zero.
As the tables show,
the angle k is equivalent to the traditional equinox correction
and the is equivalent to the traditional equator
correction. The tables are divided into two parts, the upper part
contains data sets that were judged to have acceptable distributions;
acceptable in the sense, as explained previously, of being suitable for
solutions for rotations that are not correlated with . The
results shown in the lower part of the table and marked by asterisks
were judged, by this criteria, not to be acceptable.
Note that agrees most closely with the traditional equator
correction for the acceptable data.
It is perhaps more important to note, however,
that even among the acceptable data sets
varies significantly. All three catalogs suffer from this
behavior.
| W1J00 (1977-1982) | ||||||||||
| Rotation Angles | Declination Offset | |||||||||
| traditional | standard | traditional | new | |||||||
| method | no. | |||||||||
| solar system objects | E() | k() |
j() | i() | Eq Corr() |
() | obs | |||
| S | -.130.023 | -.134.033 | +.059.042 | -.033.044 | +.093.024 | +.106.022 | 861 | |||
| M | -.469 .044 | -.438 .063 | +.131 .080 | +.156 .083 | -.222 .044 | -.150 .040 | 328 | |||
| v | -.256 .034 | -.256 .050 | -.020 .065 | +.223 .063 | -.265 .035 | -.220 .033 | 621 | |||
| M, v | -.329 .022 | -.317 .039 | +.037 .051 | +.198 .050 | -.250 .023 | -.195 .026 | 949 | |||
| S, M, v | -.235 .010 | -.230 .026 | +.047 .033 | +.094 .034 | -.090 .011 | -.056 .018 | 1810 | |||
| S, M, v, m | -.225 .008 | -.223 .023 | +.044 .029 | +.097 .031 | -.078 .009 | -.051 .016 | 2062 | |||
| S, M, v, m, j | -.227 .007 | -.221 .021 | +.055 .026 | +.086 .028 | -.064 .007 | -.052 .014 | 2389 | |||
| S, m | -.220 .030 | +.076 .038 | +.021 .039 | +.031 .020 | 1189 | |||||
| S, m, j | -.155 .022 | +.072 .027 | -.010 .031 | +.072 .015 | 1440 | |||||
| S, m, j, s, u, n | -.210 .016 | +.077 .020 | +.010 .021 | +.045 .011 | 2254 | |||||
| S, M, v, m, j, s, u, n | -.241 .016 | +.068 .020 | +.070 .021 | -.021 .011 | 3203 | |||||
| m* | -.156 .026 | -.177 .040 | +.007 .048 | +.146 .075 | +.010 .027 | +.003 .026 | 252 | |||
| j* | -.233 .021 | -.210 .031 | +.125 .038 | -.002 .044 | +.026 .022 | +.000 .019 | 327 | |||
| m, j* | -.190 .025 | +.082 .029 | +.038 .038 | +.001 .016 | 579 | |||||
| s, u, n* | -.303 .021 | +.091 .028 | +.029 .027 | +.007 .014 | 814 | |||||
| m, j, s, u, n* | -.255 .016 | +.091 .020 | +.039 .021 | +.004 .010 | 1393 | |||||
| S = Sun | j = Jupiter | |||||||||
| M = Mercury | s = Saturn | |||||||||
| v = Venus | u = Uranus | |||||||||
| m = Mars | n = Neptune. | |||||||||
| Greenwich (1942-1954) | ||||||||||
| Rotation Angles | Declination Offset | |||||||||
| traditional | standard | traditional | new | |||||||
| method | no. | |||||||||
| solar system objects | E() | k() |
j() | i() | Eq Corr() |
() | obs | |||
| S | -.335.031 | -.311.043 | +.132.053 | +.021.059 | +.007.030 | +.002.025 | 1429 | |||
| v | -.060 .060 | -.055 .086 | +.019 .105 | +.077 .118 | -.186 .059 | -.198 .055 | 406 | |||
| M, v | -.056 .054 | -.056 .080 | -.007 .098 | +.124 .109 | -.173 .052 | 453 | ||||
| m, j | -.897 .088 | -.051 .109 | -.460 .119 | -.142 .064 | 194 | |||||
| S, M, v | -.250 .038 | +.098 .047 | +.045 .052 | -.038 .023 | 1882 | |||||
| S, M, v, m | -.284 .022 | -.269 .037 | +.081 .046 | +.031 .051 | -.036 .022 | -.044 .022 | 1941 | |||
| S, M, v, m, j | -.328 .021 | -.313 .036 | +.083 .044 | -.007 .049 | -.042 .020 | -.041 .022 | 2076 | |||
| m, j, s, u, n | -.709 .054 | -.153 .065 | -.355 .078 | -.052 .039 | 447 | |||||
| S, m | -.334 .042 | +.107 .052 | +.003 .057 | -.008 .025 | 1488 | |||||
| S, m, j | -.384 .040 | +.108 .049 | -.042 .054 | -.009 .024 | 1623 | |||||
| S, m, j, s, u, n | -.411 .036 | +.059 .044 | -.063 .049 | -.031 .022 | 1876 | |||||
| S, M, v, m, j, s, u, n | -.350 .018 | -.343 .033 | +.046 .040 | -.027 .045 | -.055 .018 | -.054 .020 | 2329 | |||
| M* | -.021 .153 | -.081 .208 | -.236 .263 | +.513 .273 | +.196 .152 | -.018 .137 | 47 | |||
| m* | -.767 .089 | -.813 .126 | -.348 .145 | -.421 .192 | -.229 .089 | +.111 .098 | 59 | |||
| j* | -.962 .080 | -.923 .114 | +.117 .146 | -.472 .145 | -.129 .080 | -.091 .081 | 135 | |||
| s,u,n* | -.562 .065 | -.219 .077 | -.258 .101 | +.048 .048 | 253 | |||||
| S = Sun | j = Jupiter | |||||||||
| M = Mercury | s = Saturn | |||||||||
| v = Venus | u = Uranus | |||||||||
| m = Mars | n = Neptune. | |||||||||
| Capetown (1951-1959) | ||||||||||
| Rotation Angles | Declination Offset | |||||||||
| traditional | standard | traditional | new | |||||||
| method | no. | |||||||||
| solar system objects | E() | k() |
j() | i() | Eq Corr() |
() | obs | |||
| S | -.549.052 | -.548.049 | +.146.061 | -.305.064 | -.409.032 | -.350.033 | 944 | |||
| M | -.449 .108 | -.450 .109 | -.013 .138 | -.558 .140 | +.216 .072 | +.229 .078 | 376 | |||
| v | -.354 .117 | -.353 .103 | +.271 .132 | -.149 .131 | -.301 .071 | -.206 .070 | 658 | |||
| M, v | -.451 .044 | -.392 .077 | +.159 .098 | -.300 .098 | -.113 .045 | -.032 .053 | 1034 | |||
| m, j | -.548 .073 | -.019 .091 | -.119 .101 | -.064 .051 | 247 | |||||
| S, M, v | -.511 .015 | -.466 .046 | +.152 .059 | -.303 .060 | -.225 .015 | -.184 .032 | 1978 | |||
| S, M, v, m | -.514 .014 | -.472 .044 | +.139 .056 | -.281 .058 | -.249 .014 | -.184 .031 | 2073 | |||
| S, m | -.567 .046 | +.108 .058 | -.314 .062 | -.344 .032 | 1003 | |||||
| S, m, j | -.549 .041 | +.106 .052 | -.265 .055 | -.290 .029 | 1191 | |||||
| S, M, v, m, j | -.516 .013 | -.477 .042 | +.129 .053 | -.281 .055 | -.236 .013 | -.170 .029 | 2225 | |||
| m* | -.561 .076 | -.547 .116 | +.001 .148 | +.114 .158 | -.132 .075 | -.089 .073 | 95 | |||
| j* | -.539 .065 | -.535 .095 | +.004 .116 | -.278 .133 | -.051 .066 | +.069 .069 | 152 | |||
| S = Sun | m = Mars | |||||||||
| M = Mercury | j = Jupiter | |||||||||
| v = Venus. | ||||||||||
Such behavior might be explained by problems with DE200, but we think
this is unlikely since the three catalogs show variations in
that are not consistent between the instruments. We think it more
probable that
the observations themselves are the cause.
For transit circle observations, a non-zero may be
introduced through an erroneous correction to the assumed latitude.
However, in the case of the W1,
the correction to the assumed latitude
is very well determined either from the all-sky solution
(solving for the flexure, correction to the constant of refraction,
and the correction to the assumed latitude, Table 1 (click here), lines 4 and 5b)
or the circumpolar
solution (solving for just the correction to the assumed latitude,
Table 1 (click here), lines 2 and 3). Another source of uncertainty is the effect of making
observations in two very different environments, daytime and nighttime.
The Sun, Mercury, and Venus can only be observed in the daytime
while the remainder of the solar system objects are
usually observed only at night.
Systematic differences between the daytime
and nighttime observations are known to exist and are
corrected for by an analysis of the observations of the same stars
observed during the day and the night.
In the case of the W1, this analysis produced
night-minus-day corrections as function of hour angle of the
Sun, declination, and temperature.
A further source of uncertainty is the method of observing the
solar system objects.
For Mercury, the "center of
light" is observed with corrections applied for the
phase. For Venus, the two illuminated limbs are observed and
corrections applied to reduce the observations to
the center of the planet. For the W1, the W550, and other
Washington catalogs,
additional
corrections to both Mercury and Venus due to phase were determined
empirically and also applied. Tables 2-4 show the rotation angles and
values determined from Mercury and Venus to have
the greatest differences from among the various solar system objects.
Unfortunately DE200 has been constructed in such a way as to exacerbate
these problems. That is,
the solar system objects given the highest
weight in defining the dynamical frame are daytime and/or extended objects,
these are the
Sun (i.e the reflex of the Earth's motion), and (because of radar ranging
and spacecraft fly-bys) Mercury, Venus, and Mars.
For the W1, it
is surprising how consistent the results are for the outer planets.
However for these planets there is evidence of systematic errors in DE200
which can be as large as 
, depending on date.
It might
be possible to utilize observations of outer planets
if an ephemeris was developed based on more
recent data than was included in DE200, and if that ephemeris
was adopted as the dynamical standard.
Currently, though, it is the general consensus, that the combination
of planets interior to Jupiter represent the
best realization of the coordinate frame of DE200.
The solar system objects
that would be least effected by any of the above described problems
are the minor
planets. Lamentably, although 5 minor planets are used in the
perturbation models, there are no ephemerides for them
included in DE200 or any of its current successors.
If we had more confidence in the planetary observations from the transit
circle, we would apply
to complete the alignment of the instrumental
system to the dynamical system. At this time we hesitate to
do this because of the problems discussed above and because
the value of the offset varies so much between the individual and
combined groups of solar system objects.
The Pole-to-pole program (Corbin 1988), for which the observations were
recently completed (1985-1996),
using transit circles in Washington and New Zealand might
resolve the problems with the daytime observations. The
Washington transit circle made visual observations whereas the
New Zealand transit circle used an image dissector, thus the
reductions of observations of extended objects such as Mercury and Venus
will differ considerably between the two instruments. The
New Zealand transit circle, also, was able to observe a great many
more daytime stars and should be able to improve
the determination of the night-minus-day
correction (Rafferty & Loader 1995). Both transit circles
observed Mars during the daytime and nighttime, which also should
improve the determination of the night-minus-day correction.
The observations of each instrument can be reduced using the
methods introduced to the W1 and comparisons made to see
if applying improves the agreement
between the two programs. Finally there are efforts
under way by others to include minor planets in the dynamical frame.
It has been pointed out that, in the near future, the IAU will adopt the VLBI positions of some 600 extragalactic radio sources as the definition of the new International Celestial Reference Frame (ICRF) (Morrison 1997). The optical counterpart of this frame will be realized by the Hipparcos Catalogue, which promises to be co-aligned with the ICRF to better than 0.6 mas at epoch J1991.25. Of course, the standard method of alignment can just as easily be used to rotate transit circle catalogs into this system (i.e. the ICRF as represented by the Hipparcos Catalog) as it was used to rotate them into a dynamical system. Indeed, such a scheme offers the possibility of determining the rotational offsets between the dynamical reference system the ICRF. However, because of the degradation of the Hipparcos Catalog over time caused by proper motions, the treatment of transit circle catalogs at epochs more that 30-40 years from the mean epoch of Hipparcos (1991.25) will require the use of some other system. At present the best system for covering the widest range in epochs is DE200. It is planned that the Washington Fundamental Catalog (Corbin 1995), which will include absolute transit circle catalogs from the early 1900's, will make use of the standard method to align all the catalogs to a common system, probably DE200 or a more current version of it. It should then be possible, using the same algorithm, to rotate everything into the system of Hipparcos.
).
Solution 1 simultaneously solves for F,