A&A 431, 721-727 (2005)
H. G. Walter - R. Hering
Astronomisches Rechen-Institut, Mönchhofstr. 12-14, 69120 Heidelberg, Germany
Received 21 June 2004 / Accepted 13 October 2004
The comparison of Hipparcos and FK5 proper motions points to an inconsistency with the correction, , of the luni-solar precession derived from VLBI and LLR observations with unprecedented accuracy. An attempt is made to explain this inconsistency of approximately -1.3 mas/yr by rotational offsets of the Hipparcos and FK5 proper motion systems. In terms of right ascension and declination, these offsets give rise to proper motion offsets in the range of 1 mas/yr on average which is not exceptional given the FK5 error budget. In the case of Hipparcos it is proven that the proper motions are not affected by rotational offsets larger than those indicated by the errors of the proper motion link to the ICRF. This result is obtained by analysing the Hipparcos proper motions in view of the existence of additional systematic motions other than those caused by galactic rotation and the parallactic motion of stars due to the solar motion with respect to the LSR. It is concluded that the Hipparcos proper motions are nearly free of unmodelled rotations, confirming that the Hipparcos frame is inertial at the accuracy level of the proper motion link to the ICRF. The gap between the estimated precessional corrections is bridged primarily by minor changes in the FK5 proper motions of the order of their errors, and only to a small extent by the elimination of a bias in the Hipparcos proper motions.
Key words: astrometry - reference systems
Table 1: Angular rates of rotation and internal errors obtained from the comparison of FK5 and Hipparcos proper motions for different sets of stars (d stands for stellar distance).First let us recall the determination of the final Hipparcos positions and proper motions from those related to the original Hipparcos working frame, , established prior to the linking to the ICRF. Once the orientation and spin matrices, M0and , had been determined (Kovalevsky et al. 1997), the stellar positions and proper motions of the Hipparcos Catalogue (HCAT) were derived by
Some insight into the exactness of the correction of the luni-solar precession adopted in 1976 (IAU 1977) may be expected from the comparison of the FK5 proper motions (Fricke et al. 1988) and the Hipparcos proper motions (ESA 1997) since the latter ones refer to a basically non-rotating coordinate frame. The problem, therefore, consists of finding out whether a residual rotation of the FK5 proper motions exists relative to the Hipparcos proper motions.
The basic principles for establishing the orientation and spin differences
between reference frames are well known and have been discussed in several
papers (e.g., Froeschlé & Kovalevsky 1982; Lindegren & Kovalevsky 1995). Applying these
principles to the FK5 and Hipparcos catalogues we get for small angles of
rotation that, by analogy to Eqs. (1) and (2), the FK5 and
Hipparcos positions and proper motions are related through the matrices R and
by the following formulae:
The well known equations in right ascension ()
and declination ()
can be deduced from Eq. (5)
If, in Table 1, we take account only of the sets with more than 1100
stars the average values of
and 0.78 mas/yr, respectively, thus yielding
the VLBI-determined precession correction
This quantity is in close agreement with the current optimum value
of the precession correction of
(McCarthy & Capitaine 2002). The most recent comparisons of high precision precession models
(Capitaine et al. 2004) indicate marginal improvements of the value adopted above.
They are, however, of no significant influence on the results inferred below.
value as a yardstick (Sôma 2000) we infer from Eqs. (9) to (11) the corresponding nominal values for the angular rates of
rotation. They are
One way of reconciling the solutions of Eqs. (7) and (8) given
in Table 1 with the nominal values above consists of tentatively adding
offsets to the angular rotations so that the nominal values are obtained. These
offsets, for example, are in the first case of Table 1 as follows:
For the assessment of the spin matrix we propose a method based exclusively on Hipparcos data and on the assumption that the Hipparcos proper motions referring to the quasi-inertial frame, ICRF, are composed of the peculiar motion of the stars and the following systematic contributions:
On the basis of the Oort-Lindblad model of galactic rotation the formulae of
our approach read as follows (Green 1985, p. 346; Fricke 1977):
Table 2: Angular rates of rotation estimated for 4 different sets of HCRF proper motions referred to the ICRF. X, Y, Z denote the components of solar motion in equatorial coordinates, "P'' is Oort's constant related to differential galactic rotation, "d'' stands for stellar distance, "elim'' for elimination of residuals and "no'' refers to the number of HCRF stars.
In all four cases of Table 2 the components X, Y, Z of the solar motion relative to stars in the solar neighberhood are of the same order of magnitude. For Case 3 we obtain km s-1 for the total velocity relative to the local standard of rest (LSR), compatible with values derived independently (cf. Cox 2000, p. 493). Expressed in galactic coordinates, Case 3, Table 2 results in km s-1 in fair agreement with Feast & Whitelock (1997) and Mignard (2000). With regard to the angular rates of rotation, Cases 2 to 4 give evidence of robustness and stability of the results, which applies even more to Cases 3 and 4 yielding errors of unit weight tolerable in the present circumstances. In view of Eqs. (20) to (22), Case 3, for instance, yields for Oort's constant Q, in turn mas/yr, mas/yr and mas/yr. Only the z component falls distinctly outside the range of the widely accepted value of Q by Kerr & Lynden-Bell (1986), which is mas/yr and will be called the "standard value'' in the following text. Possible reasons for this deviation in z are discussed in Sect. 3.3. For Oort's constant P we get without exception acceptable values between 2.56 and 2.82 mas/yr ; the widely accepted value is mas/yr (Kerr & Lynden-Bell 1986). Recent determinations of the Oort constants based on more modern data (e.g., Feast & Whitelock 1997) including those of the Hipparcos Catalogue are in close agreement with the above standard values.
When we adopt in Eqs. (20)-(22)
the standard value of Q as a benchmark, we conclude from
Table 2 that in these circumstances
the Hipparcos proper motions cause surplus angular rotations given by:
For further analysis we suggest the hypothesis that the Hipparcos proper motions, at least partly, are responsible for the surplus angular rates of rotation and that the source of these rotational rates are associated with the spin matrix, , of the Hipparcos link. In other words, we suppose that the surplus angular rates, which in the ideal case are expected to be zero, are indicators of a bias of the spin matrix.
Table 3: Surplus angular rates of rotation produced by the subset of Hipparcos proper motions of Case 3 based on the standard value of the Oort constant, Q=-2.53 mas/yr, and its variations for comparison (errors are only given for the standard value of Q as those of the comparative cases are almost identical).
The surplus angular rotations depend on the Oort constant Q, the value of which, however, is not known with great precision. We assume the true value of Q within the error range of its standard value and we believe it to be on the safe side if we extend the estimate of over the whole error range of Q, as illustrated in Table 3 by the variety of surplus rotations as a function of Q. Whatever way we choose Q in Table 3, the modification of the Hipparcos proper motions by the corresponding surplus rotations and the application of Eqs. (18) and (19) to these modified proper motions results in iterated surplus angular rates that tend to approach zero. This is outlined by the following procedure:
In a first step it is shown that the surplus angular rates approach zero if the
Hipparcos proper motions are subjected to rotations by
By analogy to Eqs. (7) and (8) the corrected proper
motions are related to the original ones through
Note that the iterated surplus angular rates fall almost entirely inside the error range of the Hipparcos proper motion link.
It remains to trace back, in a second step, the correction of the
Hipparcos proper motions given in Eqs. (26) and (27) to
a bias of the spin matrix. Since
requires the correction
the corrected proper motions, to a first
approximation, follow from Eq. (2) and are given by
Starting from Eq. (31) and by analogy to Eq. (5) as well as
Eqs. (7) and (8) one arrives at
Eqs. (26) and (27) provided that
are chosen as elements of
conclude that the surplus angular rates are caused
by a bias of the spin matrix of the Hipparcos link. The
components of the correction matrix,
are of the order of the
surplus angular rates of rotation.
For example, for Case 3, Table 2 with Q=-2.53 mas/yr
the correction matrix becomes
When the corrections above are applied to the Hipparcos proper motions, the estimates by means of Eqs. (18) and (19) yield Qx=-2.47, Qy=-2.27, Qz=-2.06 mas/yr in good agreement with the standard value of the Oort constant Qand its error bars. Good agreement is also obtained for Qx, Qy and Qzwith the variations of the standard values of Q when the cases Q=-2.0 mas/yr and Q=-3.0 mas/yr are treated.
Further discussions require us to know how
into the estimates of the angular rates of rotation of the HCAT relative
to the FK5 as formally expressed by Eqs. (5)-(8). Taking account of corrected Hipparcos proper motions
we obtain from Eq. (5)
Table 4: Rotational offsets of Hipparcos and FK5 proper motions required to constrain the comparison of the two proper motion systems to the VLBI-determined correction of the luni-solar precession, mas/yr.Moreover, the correction of the z component in has repercussions on the correction of the motion of the equinox, . By comparing FK5 and Hipparcos proper motions based on Eq. (33) we infer, following the analogy of Eqs. (10) and (11),
Having set up the correction of the spin matrix one could speculate about the share of the FK5 proper motions that is required to explain the discrepancy of the corrections, , referred to in Sect. 2. In pursuing this goal the inclusion of the equinox motion is inevitable. We adopt for it the plausible correction mas/yr suggested by the preceding considerations. As a consequence, the z component of the nominal angular rates of rotation in Eqs. (12) to (14) reads now mas/yr. To estimate the share of the FK5 proper motions we employ Eq. (33) where is the matrix introduced in Eq. (6). Note that one part of is made up of ; another part will be ascribed to the FK5 proper motions as outlined below.
In the ideal case the elements of
consist of the nominal angular rates
For a better insight into the combined corrections we show, in Table 4, the aforementioned procedure by turning to the numerical values in Table 1, first case, and Table 3, case Q=-2.53 mas/yr. The first line of Table 4 refers to the angular rates of rotation resulting from comparisons of the catalogued FK5 and Hipparcos proper motions, see Eq. (5). Rotational offsets caused by a bias of the spin matrix of the Hipparcos link are taken from Table 3 and are given in the second line. The third line of Table 4 contains the hypothetical offsets of the FK5 proper motions that are needed to reach agreement with the precessional correction adopted here of mas/yr, see Eqs. (41) and (42). The sizes of these offsets are within the errors of the FK5 proper motions and the systematic differences between the FK5 and Hipparcos proper motions (Schwan 2001).
As a benchmark we adopted the correction of the luni-solar precession determined by VLBI and LLR techniques, i.e. mas/yr, a value which is also supported by the latest high precision precession models; furthermore, we used as a guideline the widely accepted value of the Oort constant Q, i.e. Q=-2.53 mas/yr. On this basis we have shown that the discrepancy between this correction of and the one derived from the comparison of FK5 and Hipparcos proper motions cannot be explained by a systematic rotation of the Hipparcos system of proper motions relative to the extragalactic reference frame (ICRF). In particular, it could be confirmed that the proper motion link of Hipparcos to the ICRF is responsible for this discrepancy only to a small extent. On the contrary, the results indicate that the Hipparcos proper motions referring to the Hipparcos Celestial Reference Frame (HCRF) are aligned to the ICRF within the order of the errors of 0.25 mas/yr per axis, perhaps with a minor exception in the direction of the z axis. Instead, we conclude that the comparison of the FK5 and Hipparcos proper motions is impaired by systematic errors of the FK5 proper motions amounting in extreme cases to about 1 mas/yr depending on the region of the sky.
As a by-product we obtained results on the solar motion relative to the LSR as well as on Oort's constants A and B that compare well with widely accepted values. Furthermore, the study supports evidence for the amendment of the recent correction of the equinox motion (Miyamoto & Sôma 1993) which apparently should be changed from -1.2 mas/yr to about -1.6 or even -2 mas/yr, which has the desirable consequence of diminishing the unusually large offset of the angular rotation about the z axis given in Eq. (17). This seems to be plausible for the z axis because it provides more compatibility with the undisputed FK5 error budget of proper motions and, which should not be overlooked, the systematic differences between FK5 and Hipparcos proper motions would easily absorb this smaller offset. On these grounds it could be proved that the corrections to the luni-solar precession obtained by VLBI and LLR on the one side and by comparing FK5 and Hipparcos proper motions on the other are not contradictory. They are explained primarily by errors of the FK5 proper motions caused by rotational effects within the error ranges of the FK5, and to a lesser extent by slightly biased Hipparcos proper motions that originate from a rotational imperfection about the z axis of the spin matrix of the Hipparcos link to the ICRF.
We thank the referee and the editor of this paper for their suggestions for improvement. Discussions with B. Fuchs and H. Lenhardt were particularly helpful.