CONVERSION OF ASTRONOMICAL COORDINATES TO J2000.0,
METHODS FOR IPAC USE
As early as 1970, the requirement for improvements to the ephemerides
being used for the astronomical almanac publications and a need for a new
fundamental catalog of star positions to replace the FK4 became evident.
Better values were now known for many astronomical constants. The
International Astronomical Union (IAU) resolved that starting in 1984 the
new system shall be used (Astron.Alm. for 1984). The Astronomical Almanac
for 1984, pages S5 - S33 has a detailed discussion of the resolutions.
Pages S34 - S36 discuss a preliminary method of converting existing
reference data and was to be considered only an approximation until the
FK5 was published.
The FK5 has now been published and is available. Two of the major
changes being addressed in the FK5 is the elimination of the error in the
FK4 equinox and the introduction of the IAU (1976) System of Astronomical
constants. However, neither the Astronomical Almanac, nor the FK5, go
into all the exact details of implementation. Both the Almanac and the
FK5 agree that any method of reduction from existing catalog data is an
approximation. Fricke (1988), in the FK5, says
"The problem of transforming positions and proper motions from the FK4
system to the system of the FK5, and the transition to J2000.0, has
been discussed in the recent years in various papers by different
authors. At present there exists, however, no commonly adopted
procedure which can be applied to the various types observations
given in existing catalogues."
The methods used in the new IPAC J2000 conversion routines are similar
to those outlined in the FK5. To convert positions in observational
catalogues, where no proper motion values are given, to J2000, the
following steps are done. Let this series of steps be known as "Method
A".
(1) Compute and apply the systematic correction FK5-FK4 in right
ascension and declination (excluding the portion of the
correction that is a function of an object's magnitude for
magnitude 1 through 7 stars) at equinox B1950 with epoch of the
observation date. If the object's position is not at equinox
B1950, the position is precessed from its catalog equinox to
B1950 with NWCPRC before doing the correction. See the section
"FK5-FK4 Systematic Corrections, Definition and Handling"
below.
(2) Precess (using the precessional values adopted in that catalog)
each FK5-FK4 corrected object from B1950 to the mean equinox at
the object's mean epoch (i.e. date of observation, which for
IRAS objects is 1983.5). "The resulting position is practically
independent of any precessional quantities" (Fricke 1988). To
accomplish this step, the new IPAC routine NWCPRC subroutine
(Newcomb-type precession routine that uses the same precession
constants as IPAC's old JPRECJ routine) is used. Besselian
dates, based on tropical years, are used in this step and in
step (4).
(3) Remove the terms of elliptic aberration (so-called E-terms) from
each object's mean position.
See the section "E-terms, Definition, History and Handling" below.
(4) Apply the equinox correction, computed for the object's mean
epoch, to the object's right ascension.
eqxcor = 0.035s + 0.085s (T - 1950)/100 where T = mean epoch
year (Fricke 1988,p.6)
(5) Precess the corrected position to the new standard equinox J2000.0
using the precessional quantities adopted in the IAU (1976) System
of Astronomical Constants.
To accomplish this step, the new IPAC routine FK5PRC is used.
FK5PRC is based on the method described in the FK5 and ascribed
to Lieske (1979) (Fricke 1988, pages 10-11). Julian dates, based
on Julian years, are used in this step.
The procedure for handling objects with known proper motions is not so
clearly outlined in the FK5. Many new additional observations of the
stars in the FK4 went into the updating of fundamental catalog positions
for the FK5. However, from the text, it seems clear that the corrections
were applied at B1950 (not at 2000.0 as outlined in Astron.Alm. for
1984). To convert the positions and proper motions of objects (which in
fact have known proper motion values) to J2000, the following steps are
done. Let this series of steps be known as "Method B".
(1) Compute and apply the systematic correction FK5-FK4 in right
ascension, declination, proper motion in right ascension and
proper motion in declination (excluding the portion of the
correction that is a function of an object's magnitude for
magnitude 1 through 7 stars) at equinox and epoch B1950. If the
object's position is not at equinox and epoch B1950, the
position and associated proper motions must be precessed to
B1950 using the entry NWCPRP in subroutine NWCPRC before doing
the correction. See systematic correction note below.
(2) Remove the terms of elliptic aberration (so-called E-terms) from
each object's position.
See the section "E-terms, Definition, History and Handling" below.
(3) Apply the equinox correction, computed for the object's equinox,
to the right ascension of the object.
See (4) in previous section.
(4) Proper motions in a catalog depend largely on the precessional
values which were adopted in that catalog (Fricke 1988,p.6).
Convert the proper motion units from units per tropical century
to units per Julian century; adjust the proper motions for change
in precession constant and apply the time-dependent portion of
the equinox correction to the proper motion in right ascension.
a. To convert to units per Julian century (Astron. Alm. for 1984,
p.S35):
Multiply the proper motions by 1.00002136
b. To adjust for change in precession constant (Astron. Alm. for
1984, p.S35):
where:
u = proper motion in seconds of time per Julian century in
right ascension
u' = proper motion in seconds of arc per Julian century in
declination
a = right ascension
d = declination
s = seconds of time
" = arcseconds
corr. u = u - 0.06912s - 0.0291s (sin(a)) (tan(d))
corr. u' = u' - 0.436" (cos(a))
Note: 0.06912s per Julian century is the difference (between
the IAU 1976 value and the Newcomb value) in the rate of
general precession in right ascension. 0.0291s (= 0.436")
per Julian century is the difference in the rate of general
precession in declination. As given in the Supplement (which
put forth a preliminary method) the constants were intended to
be applied using a and d at J2000. Therefore, these
differences in precession rates were computed at J2000.
However, the FK5 indicates any such corrections are to be
applied at B1950 and the IPAC method does so.
The differences were not recomputed for B1950 because, after
studying the Supplement and some of the explanatory text in
the Aoki paper, this author felt the difference between the
constants at B1950 and J2000 is within the error of the various
published "Newcomb" precession constants (e.g. Newcomb's
original, Andoyer's, etc.). The Astr. Alm. for 1983 only gives
Newcomb's constants to 0.001s per tropical century for rate in
right ascension and 0.01" per tropical century for rate in
declination.
c. Add the time dependent portion of equinox correction, 0.085
sec/century, to the proper motion in right ascension (Fricke
1988, p.6).
(5) Precess the corrected position and proper motions to the new
standard equinox J2000.0 using the precessional quantities
adopted in the IAU (1976) System of Astronomical Constants.
See section (5) in previous section. The FK5PRC routine has an
entry, FK5PRP, that incorporates the application of the proper
motions to epoch J2000 and precesses both the position and
proper motions to equinox J2000 using a matrix method.
It should be noted that converting an object's position from B1950 to
J2000 with proper motions equal to zero (Method B) and converting that
same object's position with "unknown" proper motions (Method A) will not
yield the same result. Non-zero proper motions are produced at J2000 when
using the method that deals with proper motions. This is consistent with
the application to the input B1950 zero proper motions of the
time-dependent portion of the equinox correction to proper motion in right
ascension, and the FK5-FK4 corrections and the change in precession
constants as applied to proper motions in right ascension and
declination. If an object's proper motions are "intrinsically" zero (e.g.
for a radio source), the method that ignores proper motions (Method A)
should be used.
The differences between Method A and Method B are illustrated below.
pma is proper motion in right ascension; pmd is proper motion in
declination; eqx is input equinox; tobs is observation epoch.
Method A vs. Method B
_______________________________________________________________________
| |
| delta ra = |raout ("B" w/ pma=pmd=0) - raout("A" w/ eqx=tobs=1950)| |
| delta dec= |decout("B" w/ pma=pmd=0) - decout("A" w/ eqx=tobs=1950)| |
| |
| |dec|<85 deg. |dec|=85 to 89 deg. |
| max delta ra = 3.8795" 25.4242" |
| max delta dec= .3757" .3398" |
| max (delta ra)*cos(dec)= .56494" for |dec| <= 89 deg. |
| max (delta dec) = .37573" for |dec| <= 89 deg. |
| |
| In addition, Method B computed the following proper motions at J2000 |
| for declination <= 89 degrees: |
| |
| |max(pmaout)| = 1.4869 sec/Julian century |
| |max(pmdout)| = 0.7515 "/Julian century |
|_______________________________________________________________________|
The test case: Input equinox B1950, epoch B1950; output equinox J2000.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
FK5-FK4 Systematic Corrections, Definition and Handling:
The FK5-FK4 systematic corrections represent regional errors in the
FK4 (Fricke 1988) and as such are independent of the changes in reduction
from B1950 to J2000 brought about by the IAU(1976) System of Astronomical
Constants. That is, even if the conversion to J2000 had not been
required, there would have been an FK5 Catalog with its associated FK5-FK4
systematic corrections.
The corrections were determined at equinox and epoch B1950. The
tables of corrections in the FK5 are claimed to hold for equinox and epoch
B1950 (Fricke 1988). The tabular values for the corrections are given in
units of 0.001 sec for right ascension, 0.01 arcsec for declination, 0.001
sec/century for proper motion in right ascension and 0.01 arcsec per
century for proper motion in declination. Formulas are given to adjust
equinox and epoch B1950 FK5-FK4 corrections to equivalent corrections at
another ("user- specified") epoch (but still equinox B1950 since no
precession is involved).
The FK5, in its outline of transforming positions to J2000, placed the
step for application of the correction in IPAC Method A's step 3 instead
of where this author put it (in step 1). In essence, the outline
indicated applying the correction to the position of the object at
Besselian equinox equal to the object's observation epoch date (TOBS)
instead of at equinox B1950 unless TOBS was B1950. However, it is not
clear if the order of the events presented in the FK5 is to be taken
literally. The order in which the correction and conversion operations
are done does affect the final position. Precession, itself, is very
sensitive to small changes in input in some areas of the sky (i.e. a small
change in input makes for a larger, or unexpected, change on output).
Since the corrections, with the material available in the FK5, can be
computed only for equinox B1950 (but with any epoch), it seems to this
author reasonable to apply the FK5-FK4 corrections only to the position of
an object at equinox B1950 and epoch TOBS. Using an IRAS object as an
example, applying the FK5-FK4 correction at equinox B1950, epoch B1983.5
and precessing the corrected position to equinox B1983.5 (the observation
epoch) will yield a different result at B1983.5 than precessing the
uncorrected position to equinox B1983.5 and then applying the FK5-FK4
correction. The difference can be as great as +/- 0.065" in ra(cos(dec))
and +/- 0.041" in dec by the time the position has been totally trans-
formed to J2000. Since coordinates of objects with known proper motions
can be precessed to equinox and epoch B1950, this problem of order does
not seem to apply to Method B.
Note: the systematic FK5-FK4 correction (with the exception of the
portion of the correction that is a function of magnitude for magnitude 1
through 7 stars) is applied by default. The omission of the portion
dealing with magnitude was done to simplify the argument list (magnitude
not needed), since the magnitudes of objects most commonly investigated at
IPAC are not bright enough to fall into the applicable magnitude range.
The Effect of FK5-FK4 Systematic Correction on Conversion to J2000
_______________________________________________________________________
| |
| For conversion of equinox B1950, epoch B1983.5 positions to equinox |
| J2000 ( using Method A) where: |
| |
| delta ra = |ra( w/ FK5-FK4 corr) - ra( w/o FK5-FK4 corr)| |
| delta dec= |dec(w/ FK5-FK4 corr) ) - dec(w/o FK5-FK4 corr)| |
| |
| |
| |dec|<85 deg. |dec|=85 to 89 deg. |
| max delta ra = 2.7983" 17.0422" |
| max delta dec= .2866" .2592" |
| |
| max (delta ra)*cos(dec)= .52452" for |dec| <= 89 deg. |
| max (delta dec) = .28657" for |dec| <= 89 deg. |
|_______________________________________________________________________|
The test case: Input equinox B1950, epoch B1983.5; output equinox J2000.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
E-terms, Definition, History and Handling:
E-terms (elliptic aberration) are part of the aberrational
displacements of a star in ecliptic longitude and latitude resulting from
the motion of the Earth in its elliptic orbit (Meeus 1991,p.129). The
mean positions in all star catalogs published before 1984 contain the
terms of elliptic aberration (Fricke 1988,p.9). Even though the variables
(longitude of perihelion of the Earth's orbit, eccentricity of the Earth's
orbit, ecliptic longitude and latitude) involved in the formulas for
E-terms are slowly varying quantities, the changes in the E-terms are so
small that the E-terms may be considered constant for a given star for a
period of several hundred years (Smart 1977,p.186). Apparently for this
reason, it had become standard practice to leave these terms in the mean
positions of stars (Meeus 1991,p129). Smart (1977) even stated that it
was more convenient to regard the quantity (ecliptic longitude + E-term in
longitude) as the true longitude of the star; and similarly for the
latitude.
The following equations to compute E-terms, from Astron. Alm. for
1984, are used for the IPAC J2000 routines for objects with |declination|
< 89.999 degrees.
Where:
a = right ascension
d = declination
dela = delta(a) component of E-term to be added to a to remove
E-term.
deld = delta(d) component of E=term to be added to d to remove
E-term.
h = hours
s = seconds of time
" = seconds of arc
dela = 0.0227s sin(a+11.25h) sec(d)
deld = 0.341" cos(a+11.25h) sin(d) + 0.029" cos(d)
The above equations were also presented in Standish (1979) and the
Explanatory Supplement (1961), p.144. The latter presented them with the
following remark: "The form of the equations of condition and their
solution are not discussed here." However, they claimed the small errors
in this procedure are usually negligible.
The following equations, to compute E-terms, are used for the IPAC
J2000 routines for objects with |declination| >= 89.999 degrees
(Meeus 1991).
Where:
tobs = year of observation or B1950, as appropriate.
a = right ascension at tobs
d = declination at tobs
l = ecliptic longitude at tobs
b = ecliptic latitude at tobs
dl = delta(l) component of E-term to be subtracted from l to
remove the E-term.
db = delta(b) component of E-term to be subtracted from b to
remove the E-term.
t = time measured in Julian centuries from epoch J2000.0
h = hours
s = seconds of time
" = seconds of arc
k = constant of aberration (20.49552" used; Meeus 1991,p 139)
e = eccentricity of the Earth's orbit:
0.016708617 - 0.00004237 t - 0.0000001236 t**2
(Meeus 1991, p. 139)
p = longitude of the perihelion of the Earth's orbit (in degrees):
102.93735 + 0.71953 t + 0.00046 t**2
(Meeus 1991,p.139)
l and b are computed from a and d using a modified version of the
IPAC equatorial-to-ecliptic coordinate conversion routine. The
special version uses a computed obliquity (for tobs) of the ecliptic
so the ecliptic coordinates may be for equinox tobs. The obliquity
parameter = 84381.448" - 46.8150" t - 0.00059" t**2 + 0.001813" t**3
(Astron. Alm. for 1984, p.S26).
Then (Meeus 1991, p.129; also Green 1985, p.192; Smart 1977, p.186):
dl = e k cos(p - l) / cos(b)
db = e k sin(p - l) sin(b)
dl and db are applied to l and b. The ecliptic coordinates are
then converted back to equatorial using a special version of the
IPAC ecliptic-to-equatorial coordinate conversion routine that
allows for equinox other than just 1950.0. The actual implementa-
tion of this E-term removal involved some iterations in the program.
The way the correction is presented, it is the computation to be
made for an object without the E-terms; therefore one needs to
approximate the lambda and beta of the object without the E-terms
in order to get the best answer for the E-terms.
The Effect of E-terms on Position
_______________________________________________________________________
| |
| For conversion of B1950 positions to J2000 where: |
| |
| delta ra = |ra(1) - ra(2)| |
| delta dec= |dec(1) - dec(2)| |
| |
| and (1) indicates processing with E-term removal; (2) indicates |
| no E-term removal. Therefore the deltas are the absolute values |
| of the E-terms. |
| |
| |dec|<85 deg. |dec|=85 to 89 deg. |
| max delta ra = 3.2981" 22.354" |
| max delta dec= .3422" .3422" |
| |
| max (delta ra)*cos(dec)= .39013" for |dec| <= 89 deg. |
| max (delta dec) = .34223" for |dec| <= 89 deg. |
|_______________________________________________________________________|
The test case: Input equinox = B1950, epoch B1950; output equinox = J2000.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
Reproducibility of Inputs with IPAC Methods
Transforming J2000 positions from Besselian equinox positions is an
approximation. Such transformations are required only for old catalog
data. New data most likely would be reduced using the FK5 as a reference.
Ideally, the authors of a catalog are best able to transform their
positions to J2000, since they are aware of how the original data were
reduced.
The order in which corrections are applied and the values used for the
corrections and adjustments themselves affect results of the computation.
There is no agreement as to a general method of transformation at
present. For many purposes, whether a method is theoretically right or
not is not as important as having positions in standardized form in order
to facilitate communication in the astronomical community. That is, when
one group of astronomers speaks of an object at a particular J2000
position, other groups need to also recognize the same object at that same
"address" in order to speak a common language. What if later there is a
generally accepted routine that produces results different than the
methods presented here do? In essence, how can we recompute positions to
agree with a new method. First, the original input (e.g. B1950 positions)
is needed. The IPAC method provides routines to undo both Method A and
Method B to recompute the "original" Besselian equinox positions.
However, Method A is sensitive to the observation epoch value and that
value must be remembered in order to recompute the original input
accurately. For Method B, only the J2000 ra, dec, and proper motions are
required to reproduce B1950 ra, dec, and proper motions. Results of
Method A and Method B "undo"s are presented below.
Method A Reproducibility Test
_________________________________________________________________________
| |
| The results of two unknown proper motion (Method A) comparisons (each |
| with a different observation epoch) where: |
| |
| delta ra = |
| |ra(B1950 input) - ra(B1950 recomputed from Method A J2000 output)| |
| |
| delta dec= |
| |dec(B1950 input) - dec(B1950 recomputed from Method A J2000 output)| |
| |
| ------epoch 1950--------- -----epoch 1983.5------ |
| |dec| |dec| |dec| |dec| |
| <85 deg. 85 to 89 deg. <85 deg. 85 to 89 deg. |
| |
| max delta ra = 4.093e-10" 11.255e-10" .00015" .00069" |
| max delta dec= 1.023e-10" 1.023e-10" .00001" .00001" |
| |
| max (delta ra)*cos(dec)= 4.093e-10" .00005" for |dec| <= 89 deg. |
| max (delta dec) = 1.023e-10" .00001" for |dec| <= 89 deg. |
|_________________________________________________________________________|
The test cases: Input equinox = B1950.; output equinox = J2000.
FK5-FK4 correction applied for this test.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
Note: The test case with epoch 1983.5 does not yield differences as
small as in the case with epoch 1950.0. This is because, in the
former case, the Newcomb precession routine must be used to precess
between the equinoxes of B1950 and B1983.5. The only precession
routine required in the latter case is the one using the IAU 1976
constants which are carried to sufficient places to make input very
accurately reproducible. The Newcomb constants were probably not
known to as many places.
Method B Reproducibility Test
_________________________________________________________________________
| |
| The results of the known proper motion (Method B) comparison where: |
| |
| dra = delta ra = |
| |ra(B1950 input) - ra(B1950 recomputed from Method B J2000 output)| |
| |
| ddec = delta dec= |
| |dec(B1950 input) - dec(B1950 recomputed from Method B J2000 output)| |
| |
| dpma = delta proper motion in ra = |
| |pma(B1950 input) - pma(B1950 recomputed from Method B J2000 output)| |
| |
| dpmd = delta proper motion in dec = |
| |pmd(B1950 input) - pmd(B1950 recomputed from Method B J2000 output)| |
| |
| |dec| |dec| |dec| |
| <85 deg. 85 to 89 deg. < 89 deg. |
| |
| max dra = 1.115e-08" 1.228e-09" max dpma = 4.409e-07" per century|
| max ddec= 1.094e-09" 1.023e-10" max dpmd = 4.333e-09" per century|
| max(dpma)*cos(dec) = 4.376e-08" per century|
| max (dra)*cos(dec)= 4.093e-10" |dec| <= 89 deg. |
| max (ddec) = 1.023e-10" |dec| <= 89 deg. |
|_________________________________________________________________________|
The test case: Input equinox = B1950.; output equinox = J2000.
FK5-FK4 correction applied for this test.
Coordinate computation grid as previous (no proper motion) test.
Input proper motion in ra = 1.0 sec/tropical century; input proper
motion in dec = 1.0"/tropical century.
Notes on time:
The letter J, in notations such as J2000.0, in star catalogs indicates
that the unit of time is the Julian year. The Julian year is 365.25
(exactly) days. The letter B, indicates the Besselian (= tropical) year.
According to Newcomb the tropical year is 365.2421988 days. JDE is the
Julian Day number that has been in astronomical use. JDE is a common,
less confusing reference point to use than fractional parts of a Julian
or Besselian year. When doing conversions between the Besselian and
Julian systems, it is easiest to think in Julian Day numbers. Two useful
reference points are given by Meeus (1991),p.125:
B1950.0 = JDE 2433282.4235 = 1950 January 0.9235
J2000.0 = JDE 2451545.00 exactly = 2000 January 1 at 12h TD
Therefore:
In the Julian system of time, B1950 is
(2433282.4235 - 2451545.00)/365.25 = -50.00021 Julian years from
J2000 (or equivalent to J1949.99979 in time scale only; more than
time is involved in converting B1950.0 coordinates to J2000.0 based
coordinates).
Comparison of IPAC J2000.0 Algorithm with Another Algorithm
The Astron. Alm. for 1992, on page B42, presents "a matrix method for
calculating mean places of a star at J2000.0 on the FK5 system from the
mean place at B1950.0 on the FK4 system, ignoring the systematic
corrections FK5-FK4....". This method, using a pre-computed 6 x 6 matrix,
is presented with no explanation other than to reference the Aoki (1983)
and Standish (1982) papers. The method seems to be the Aoki one. The
matrix values in the Astron. Alm. (which are used in the comparison
program) vary very slightly from the Aoki values in a few elements. The
method in the almanac appears to be useful only when proper motions are
known since corrections seem to be split up in such a way that B1950 input
proper motions of zero yield non-zero J2000 output proper motions and the
position itself seems to be affected in an unexpected way. The Aoki paper
describes a way of manipulating the matrix so the objects without proper
motions may be handled properly. B1950.0 is the only allowed input
equinox to these Aoki methods, but the observation epoch may be used in
the method for unknown proper motions.
Exact agreement between the IPAC and Aoki methods is not expected
since there are some philosophical differences (see Aoki 1983 for
discussion of his method) between the two methods. However, the results
should end up in the same universe, and they do. The IPAC method is
slightly more flexible in that E-term removal is an option and that
individual elements involved in the conversion may be better understood
since they are not pre-combined in a pre-computed 6 x 6 matrix presented
in a publication.
The Aoki method, itself, does not include the FK5-FK4 systematic
corrections. The paper indicates that such corrections should be added
to the positions when the corrections become available (the Aoki paper
pre-dates the FK5 publication).
IPAC vs. Aoki (Method A)
_________________________________________________________________________
| |
| The results of two unknown proper motion comparisons using Method A |
| (each with a different observation epoch) where: |
| |
| delta ra = |ra(IPAC:J2000) - ra(Aoki:J2000)| |
| delta dec= |dec(IPAC:J2000) - dec(Aoki:J2000)| |
| |
| ------epoch 1950--------- -----epoch 1983.5------ |
| |dec| |dec| |dec| |dec| |
| <85 deg. 85 to 89 deg. <85 deg. 85 to 89 deg. |
| |
|max delta ra = .02248" .17668" .02410" .18860" |
|max delta dec= .00236" .00229" .00254" .00245" |
| |
|max (delta ra)*cos(dec)= .00308" .00329" for |dec| <= 89 deg.|
|max (delta dec) = .00236" .00254" for |dec| <= 89 deg.|
| |
|_________________________________________________________________________|
IPAC vs. Aoki (Method B)
_________________________________________________________________________
| |
| The results of known proper motion comparisons using Method B |
| (input proper motion in ra = 1 sec (i.e. 15 arcsec); in dec = 1 sec |
| of arc per tropical century) where: |
| |
| dra = | ra(IPAC:J2000,epoch 2000) - ra(Aoki:J2000,epoch 2000)||
| ddec = |dec(IPAC:J2000,epoch 2000) - dec(Aoki:J2000,epoch 2000)||
| |
| dpma = delta proper motion in ra = |pma(IPAC:J2000) - pma(Aoki:J2000)||
| |
| dpmd = delta proper motion in dec = |pmd(IPAC:J2000) - pmd(Aoki:J2000)||
| |
| |dec| |dec| |dec| |dec| |
| <85 deg. 85 to 89 deg. < 85 deg. 85 to 89 deg. |
| |
|max dra = 0.05411" 0.42986" max dpma = 0.1485" 1.1865" per century|
|max ddec= 0.00546" 0.00548" max dpmd = 0.0151" 0.0151" per century|
| max(dpma)*cos(dec) = 0.0156" 0.0207" per century|
| |
|max (dra)*cos(dec)= 0.00752" |dec| <= 89 deg. |
|max (ddec) = 0.00548" |dec| <= 89 deg. |
|_________________________________________________________________________|
The test cases: Input equinox = B1950.; output equinox = J2000.
FK5-FK4 correction turned off for this test.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
Comparison of "IPAC" J2000.0 vs. IPAC B2000 Converted/Precessed from B1950
Since the question often arises as to how much difference exists
between coordinates converted to J2000 and those precessed using the old
(Newcomb) precession routine to B2000, the following statistics are
presented.
J2000 vs. B2000
_________________________________________________________________________
| |
| delta ra = |ra(IPAC:J2000) - ra(IPAC:B2000) |
| delta dec= |dec(IPAC:J2000) - dec(IPAC:B2000) |
| |
| ----------CASE 1--------- --------CASE 2--------- |
| |dec| |dec| |dec| |dec| |
| <85 deg. 85 to 89 deg. <85 deg. 85 to 89 deg. |
| |
| max delta ra = 6.6352" 36.1219" 6.9881" 42.1606" |
| max delta dec= .5827" .5828" .6798" .6146" |
| |
| max (delta ra)*cos(dec)= 1.4687" 1.5286" for |dec| <= 89 deg.|
| max (delta dec) = .5828" .6798" for |dec| <= 89 deg.|
| |
| CASE 1 is comparison of conversion of B1950 to J2000 using |
| Method A without FK5-FK4 systematic correction vs. |
| straight precession of B1950 to B2000 positions. |
| |
| CASE 2 is comparison of conversion of B1950 to J2000 using |
| Method A with FK5-FK4 systematic correction vs. straight |
| precession of B1950 to B2000 positions. |
|_________________________________________________________________________|
The test cases: Input equinox = B1950, epoch 1950; output equinox = 2000.
Coordinate computation at grid line intersections defined by:
RA 0 thru 360, by 1 degree increments.
Dec = -85 to +85 by 1 degree increments for first column.
|dec| = 85 to 89 by 1 degree increments for second column.
Note: FK5-FK4 systematic correction is not applicable to straight
precession.
Notes on Comparison of Positions in the FK5
The FK5 contains columns labelled B1950 and J2000. The positions in
the B1950 column were computed from the J2000.0 data per the text in the
FK5 (p. 10). These positions contain the same corrections as their
J2000.0 counterparts (i.e. equinox correction is included, Julian time is
used, E-terms are not present, etc.). Essentially, it appears that the
J2000.0 positions were precessed to 1949.99979 (i.e. JDE of B1950
expressed as a function of Julian years and JDE of J2000) using the new
(e.g. as described on pages 10-11 of the FK5) precession constants. The
B1950.0 positions here are not the same as input B1950 positions, which
would, of course, contain no equinox correction, but would contain
E-terms.
References
Aoki, S., Soma, M., Kinoshita, H., Inoue, K. 1983, Astron. Astrophys. 128,
263-267.
Astronomical Almanac for the year 1983, (Washington,D.C.,1982), p. L8.
Astronomical Almanac for the year 1984, (Washington,D.C.,1983), pp. S5-S33,
S34-36.
Astronomical Almanac for the year 1992, (Washington,D.C.,1990), pp. B42-B43.
Explanatory Supplement to the Astronomical Ephemeris and the American
Ephemeris and Nautical Almanac:1961,Her Majesty's Stationery Office,
London, p. 144.
Fricke, W., Schwan, H., Lederle, T. 1988, Veroff. Astron. Rechen-Institut
Heidelberg, No. 32 (Fifth Fundamental Catalogue (FK5) Part I), pp. 5-12,
85.
Green, R. 1985, Spherical Astronomy, Cambidge University Press, pp. 192-193.
Leiske, J. 1979, Astron. Astrophys. 73, 282-284.
Meeus, J. 1991, Astronomical Algorithms, Willmann-Bell Inc., pp. 125,
129-130, 139-140.
Smart, W. 1977, Textbook on Spherical Astronomy, Cambidge University Press,
pp. 184-187.
Standish, E. 1982, Astron. Astrophys. 115, 20.
Author:
J. Bennett
30Sep92