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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
New precession formula
T. Fukushima
National Astronomical Observatory, Tokyo, Japan
We modified J. G. Williams' formulation of the precession and the nutation
by using the 3­1­3­1 rotation [1] so as to express them in an arbitrary inertial
frame of reference. It gives the precession­nutation matrix as the product of
four rotation matrices as NP = R 1 (\Gammaffl)R 3 (\Gamma/)R 1 (')R 3 (fl), and the pre­
cession one similarly as P = R 1 (\Gamma¯ffl)R 3 (\Gamma ¯
/)R 1 (')R 3 (fl). Here ' and fl are
the angles to specify the location of the ecliptic pole of date in the given
inertial frame, / and ¯
/ are the true and mean ecliptic angles of precession,
respectively, and ffl and ¯ffl are the true and mean obliquities of the ecliptic,
respectively. As a result, the pole coordinates of the true and mean equators
are explicitly given in terms of the newly introduced precession angles. Al­
though the expression of nutation matrix is unchanged, we recommend the
usage of the above form of NP instead of preparing P and N separately
because of faster evaluation. The formulation is robust in the sense it avoids
a singularity caused by finite pole offsets near the epoch. Facing the singu­
larity is inevitable in the current IAU formulation. By using a recent theory
of the forced nutation of the non­rigid Earth, SF2001 [2], we converted the
true pole offsets referred to the ICRF, observed by VLBI for 1979--2000, and
compiled by USNO, to the offests in the above three angles of precession,
¯
/, ', and fl, while we fixed ¯ffl as the combination of the linear part pro­
vided in SF2001 and the quadratic and higher terms derived by Williams
(1994). From the converted offsets, we determined the best­fit polynomial
expressions of the three precession angles in the ICRF by a weighted least
square method where we kept the quadratic and cubic terms as the same as
in Williams (1994). These constitute a new set of fundamental expressions
of the precessional quantities. The combination of the new precession for­
mula and the periodic part of SF2001 serves a good approximation of the
precession­nutation matrix in the ICRF.
1. Introduction
It is well known that the combination of the IAU formula of precession [3] and
the IAU theory of nutation [4] is incorrect at the level of 10 mas when referred
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to the International Celestial Reference Frame (ICRF). Apart from the periodic
feature of the difference, which shows the error of the IAU nutation theory, we
notice the existence of secular trends, especially that in longitude. Such secular
departures exist mainly due to the error of the adopted polynomial coefficients of
the IAU precession formula and partly due to the difference between the ICRF
and the reference frame referred to the mean equator and equinox of the epoch,
J2000.0.
There have been many studies on the revision of nutation theories. Refer the
reports of IAU/IUGG Joint WG on this issue [5] and the attached comprehensive
lists of the articles related. Most of these works treated the revision of precession
in the form of linear correction terms added to their nutation theories. Then their
results naturally lead to a possible revision of precession constant as given in the
reports of IAU WG on Astronomical Standards [6]. See also Table 8 and Figure
5 of our previous work [2].
From a practical point of view, however, it is unsatisfactory to revise the
precession constant only. Rather demanded is the replacement of the IAU pre­
cession formula as a whole. There was such an example [7]. Unfortunately their
results gave no good agreement with the observation. On the other hand, there
is a movement to bypass the concept of ecliptic and equinox in expressing the
motion of the equatorial pole. At the last General Assembly of the IAU, this was
adopted as the basic policy to establish the new IAU formulation to connect the
ICRF and the International Terrestrial Reference Frame by way of an intermedi­
ate reference frame using the concept of the non­rotating origin. A formulation
on this line was already given [8].
Anyway, what we need is a formulation predicting the secular motion of the
equatorial pole in the ICRF such that its combination with a suitable theory
computing the periodic feature provides satisfactory agreement with the obser­
vations. In this article, we report such an example created from an analysis of
the VLBI observation mentioned.
2. Weakness of IAU Formulation
Now a considerable amount of the observed pole offsets are availlable. Then it
would be thought easy to revise the IAU precession formula to fit to the obervation
by updating the numerical values of its polynomial expressions. Unfortunately it
is not so straightforward. When we plotted the corrections in the true equatorial
precession angles, (ffii; ffi `; ffi z), which are directly converted from the observed
corrections in nutation, large departures appeared in ffii near the epoch. This is
clear from the differential relation ffi i ú \Gammaffi ffl= sin ` A , which shows the possibility
of divergence due to a small divisor, sin ` A . Of course, the periodic features in ffi `
and in the sum ffi i + ffiz reflect the periodic motion of the observed correction in
nutation.
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In order to reduce the magnitude of the periodic features and hopefully large
departures in ffii as well, we transformed the corrections from the true sense to
the mean sense by replacing the IAU nutation theory with the periodic part 1
of a recent nutation theory [2]. The resulting corrections are the corrections in
the mean equatorial precession angles, (ffii A ; ffi ` A ; ffiz A ). This time ffi ` A and the
sum ffii A + ffi z A are sufficiently small and are appropriate to be expressed as the
corrections in polynomial of time. However, unchanged is the ill determination of
ffi i A near the epoch although its magnitude is significantly smaller than the case
of the true precession angles.
Thus the appearance of large departures in ffi i near the epoch hardly depends
on the incorrectness of the nutation theory adopted in the conversion. This sit­
uation is the same as we face in the determination of Keplerian elements for a
low inclination orbit. In that case, the longitude of
node\Omega and the argument of
pericenter ! are ill­defined while the inclination I and the longitude of pericenter
$
j\Omega + ! are well­defined. This phenomenon appears whether periodic pertur­
bations are excluded or not. Therefore we should find another and robust way to
express the precession matrix.
3. J. G. Williams' Formulation
Let us consider the cause of the ill determination we faced in the previous
section. Careful examination of the procedure of the conversion reveals that the ill
determination is caused by the fact that the ICRF does not satisfy the assumption
of IAU formulation, namely the exact coincidence of the precession matrix and
the unit matrix at the epoch, P 0 = I. To overcome this fragile property of the
IAU formulation, we should adopt a new formulation which works well in an
arbitrary inertial frame of reference.
Once J. G. Williams derived a new formulation by skipping the intermediate
process of routing the ecliptic pole at the epoch, C 0 , and directly moving from
the mean equatorial pole at the epoch, ¯
P 0 , to the ecliptic pole of date, C, first [1].
In order to deal with non­zero pole offset at the epoch, we modify his formulation
by replacing the starting point from ¯
P 0 , to the z­axis of the given inertial frame,
Z. As a result the precession is described as Z ! C ! ¯
P while the precession­
nutation is done similarly as Z ! C ! P. In other words, we (1) first specify the
ecliptic pole of date, C, in the given inertial reference frame and shift from the
inertial frame to an ecliptic reference frame of date, (2) then specify the mean or
true equatorial pole of date, ¯
P or P, in the ecliptic reference frame and shift from
it to the mean or true equatorial reference frame of date. Since two angles are
necessary to specify the direction of a pole in the given reference frame, we need
1 The contribution of FCN was omitted.
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four rotational operations in total. Thus we express the precession matrix as
PNEW j R 1 (\Gamma¯ffl) R 3
\Gamma
\Gamma ¯
/ \Delta
R 1 (')R 3 (fl):
The new polar diagram leads to the following definition of these new angles as
fl j 180 ffi \Gamma
6 YZC =
6 CZ “
Y; ' j ZC; ¯
/ j
6 ¯
PCZ; ¯ffl j C ¯
P;
where Y and “
Y j \GammaY denote the positive and negative y­axes of the given inertial
frame of reference. Here we adopted different notations from Williams' one since
the starting point is different. Also we assigned a different notaion ¯ffl to the angle
ffl A so as to discriminate their realizations in polynomial forms. Since ¯ffl is the
mean obliquity of the ecliptic of date, we can use the same form in expressing the
precession­nutation matrix as
(NP) NEW j R 1 (\Gammaffl)R 3 (\Gamma/)R 1 (')R 3 (fl)
where / j ¯
/ + \Delta/; ffl j ¯ ffl + \Deltaffl. As Williams stressed, this unified treatment is a
merit of the original Williams' formulation, which is inherited to the modified one.
Also note that the usual nutations are used in this formulation. This is another
merit. The recent formulation [8] to describe the equatorial pole coordinates, X
and Y , requires the preparation of a little different nutations, \Delta 1 / and \Delta 1 ffl, the
nutations referred to the eclptic of the epoch. They must be correctly converted
from the existing theories of nutation giving the usual nutations, \Delta/ and \Deltaffl,
the nutations referred to the ecliptic of date 2 . Further note that the number of
rotational operations needed to express the precession­nutation matrix reduces
from six of the IAU formulation to four of the new one. This is yet another merit
of the new formulation.
4. Determination of Precession Angles
Now the pole offset at the epoch is sufficiently small as around 0:04 00 , we
approximate the new angles of precession to be almost the same as Williams'
original precession angles as
fl ú ¸ A ; ' ú ffl 0
A ; ¯
/ ú j A ; ¯ffl ú ffl A
whose numerical expressions are given in Table 5 of Williams (1994). Note that
Williams' ffl A is significantly different from that of IAU 1976 theory. In fact, we
previously estimated ¯ffl from the VLBI observations [2] as
¯ffl SF (t) = 84 381:442 8 \Gamma 46:838 8 t
2 Note that Eq.(15) of [8] describing the procedure of conversion is linear and ignores the
second and higher order effects. The formula of rigorous transformation becomes tedious.
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where the unit is arcsecond and t is the time since J2000.0 measured in Julian
centuries. Its linear part is quite close to that of Williams' ffl A . Adopting the
second and higher order terms from Williams' result and keeping the cut­off level
of the coefficients as 0.1mas at 1 century apart from the epoch, we fixed the
polynomial expression of ¯ffl as
¯ ffl(t) = 84 381:442 8 \Gamma 46:838 8 t \Gamma 0:000 2t 2 + 0:002 0t 3 :
By using this formula for ¯ffl, we transformed the observed corrections in nutation
to those of the new precession angles after removing the polynomial forms of
their counterparts in Williams' expressions. Thus converted corrections in terms
of the new precession angles distribute so smoothly as to be well approximated
by polynomials of time.
In order to find the best polynomial approximation of the obtained corrections
in the new precession angles, we executed the weighted linear least square method
for linear functions. Quadratic and higher order fittings failed to be meaningful.
Then the adopted solutions are linear as
¯
/ \Gamma j A = \Gamma(0:0431 \Sigma 0:0006) + (0:0174 \Sigma 0:0100) t;
' \Gamma ffl 0
A = (0:0389 \Sigma 0:0003) \Gamma (0:0044 \Sigma 0:0040) t;
fl \Gamma ¸ A = \Gamma(0:0000 \Sigma 0:0000 1 ) \Gamma (0:0052 \Sigma 0:0000 2 ) t:
Note that the constant term of fl \Gamma ¸ A , and therefore 3 that of fl, too, is practically
equal to zero. This means that the ecliptic pole of the epoch precisely lies on the
yz­plane of the ICRF. By adding the Williams' original expressions subtracted
prior to the determination, we obtained the final results as
¯
/(t) = \Gamma0:043 1 + 5 038:473 9 t + 1:558 4 t 2 \Gamma 0:000 2 t 3 ;
'(t) = 84 381:447 9 \Gamma 46:814 0 t + 0:051 1 t 2 + 0:000 5 t 3 ;
fl(t) = 10:552 5 t + 0:493 2 t 2 \Gamma 0:000 3 t 3 ;
where again we omit the terms not exceeding 0.1 mas in 1900--2100. We confirmed
that small and of no sign of apparent secular trends are the residuals in nutations
when the precession­nutation matrix is computed by the combination of the new
precession formula whose coefficients are determined as above and the periodic
part of the nutation theory of SF2001.
Note that the procedure to determine the polynomial forms of the new pre­
cession angles described here is applicable to any combination of the observation
and the nutation theory as long as the latter gives the corrections to the IAU
precession formula in polynomial form of the corrections in nutation. Thus the
results presented here will be easily updated.
3 Note that ¸A(0) = 0 by its definition.
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References
[1] Williams J. G. Astron. J., 1994, 108, 711.
[2] Shirai T., Fukushima T. Astron. J., 2001, 121, 3270.
[3] Lieske J. H., Lederle T., Fricke W., Morando B. Astron. Astrophys., 1977,
58, 1.
[4] Seidelmann P. K. Celes. Mech., 1982, 27, 79.
[5] Dehant V. et al. Cel. Mech. & Dyn. Astron., 1998, 72, 245.
[6] Fukushima T. in IAU Coll. 180, Towards Models and Constants for Sub­
Microarcsecond Astrometry, ed. K. J. Johnston et al. (Washington, D. C.,
USNO), 2000, 417.
[7] Simon J. L., Bretagnon P., Chapront J., Chapront­Touz'e M., Francou G.,
Laskar J. Astron. Astrophys., 1994, 282, 663.
[8] Capitaine N., Guinot B., McCarthy D. D. Astron. Astrophys., 2000, 355,
398.
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