THE SECOND REALIZATION OF THE INTERNATIONAL CELESTIAL REFERENCE FRAME BY VERY LONG BASELINE INTERFEROMETRY

The geodetic/astrometric VLBI data set utilized for the construction of ICRF2 was obtained using a rich variety of stations and networks. The VLBI arrays used consisted of fixed and mobile antennas in configurations ranging from as few as two antennas (a single baseline) to as many as 20 antennas. Antenna sizes ranged from 3 up to 100 m. Unfortunately, the geographical distribution of the fixed VLBI antennas is very uneven. Out of approximately 53 antennas used over the past 30 years, only about ten were located in the southern hemisphere. Currently, there are about 34 fixed antennas that regularly or occasionally participate in geodetic/astrometric sessions, but only seven of those are in the southern hemisphere. This uneven geographical distribution directly affects the quality and quantity of data available in the southern hemisphere. The number of observations drops off rapidly for sources south of around −30° decl. In recent years, the IVS through its dedicated southern hemisphere observing program, has made great efforts to observe new sources in the southern hemisphere and to increase the number of observations of existing sources, with only moderate success due to limited resources.

The VLBA is an astronomical array of ten 25 m antennas. The VLBA antennas are some of the most sensitive and phase stable systems available. Details of their geodetic/astrometric use are given by Petrov et al. (2009). Use of the VLBA antenna at Pie Town, NM began in 1988 followed by the Los Alamos, NM antenna in 1991. Use of all ten VLBA antennas, and correlation on the VLBA correlator began in 1994. In a 2004 study, Gordon (2004) found that the regular VLBA (non-VCS) observations accounted for some 30% of the available geodetic/astrometric VLBI data and its usage improved the repeatability of the antenna positions at non-VLBA sites by typically 10%–40% and reduced the average source position formal uncertainties by ∼62% in R.A. and ∼54% in decl. for sources north of −30° decl. Currently, VLBA data comprises ∼28% of all the data used in this paper.

The VCS were a series of six multi-session S/X-band astrometric campaigns designed to image and find precise positions of as many new compact radio sources as possible for use by the radio astronomical community as VLBA phase reference calibrators. The first of these campaigns, VCS-1, was observed during 1994–1997. The resulting ten 24 hr sessions are described and analyzed by Beasley et al. (2002). An eleventh VCS-1 session, initially considered a failure, was later recovered and re-analyzed successfully. Five follow up VCS campaigns were made between 2002 and 2007 by Fomalont et al. (2003), Petrov et al. (2005, 2006, 2008) and Kovalev et al. (2007). These observing campaigns added another thirteen 24 hr sessions for a total of 24 VCS sessions. The observing mode was modified from that of regular geodetic/astrometric sessions. The VCS sessions concentrated on making short duration observations of many new sources as opposed to repeated observations of a smaller set of known sources. The sessions were not optimized for full sky coverage nor for atmospheric calibration, although the later sessions were better calibrated than the first sessions. The VCS sessions add nearly 2200 additional sources, with most of those observed in only one VCS session.

The Working Group had access to several VLBI software analysis systems including: CALC/Solve (Ma et al. 1986); OCCAM (Titov et al. 2004); SteelBreeze developed at the Main Astronomical Observatory of the National Academy of Sciences of Ukraine (MAO) and Quasar developed at the Institute of Applied Astronomy of the Russian Academy of Sciences (IAA). See Fey et al. (2009) for additional information on these VLBI analysis packages.

For ICRF2, two overarching questions needed to be answered: (1) would the source positions come from a single solution generated with a single software analysis package or would they be the result of a combination catalog (i.e., the positions would be the averaged values from those generated with different analysis packages appropriately weighted by the formal uncertainties); and (2) if a single solution, which software analysis package would best suit the requirements for ICRF2.

The Working Group examined the available software analysis systems and used them to study, characterize and analyze the available VLBI data. Preliminary analysis of the data included the generation and study of source position time series to identify stable and unstable sources, the generation and inter-comparison of preliminary source position catalogs generated independently using the different analysis systems and analysis options, and the creation and study of a combination catalog based on the preliminary catalogs. Detailed results of these studies can be found in Fey et al. (2009). In the end, the Working Group decided to use a single catalog rather than a combination catalog for several reasons. The position catalogs from the separate software analysis packages going into the combination catalog differed slightly due to small differences in the underlying geophysical modeling, in editing criteria of the raw data, and/or in the amount of data used and its overlap. Additionally, a combination catalog with existing software loses certain information such as the full covariance matrix and the links to EOP products and the ITRF. See Sokolova & Malkin (2007) for a more thorough discussion on comparison and combination of radio source position catalogs.

The Working Group decided that the solution used to estimate positions for ICRF2 would be made at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) using the CALC/Solve package, primarily as a matter of convenience and the fact that the GSFC system had access to the most complete set of VLBI data. Similar results could have been obtained with the other analysis systems but with greater effort. Although the final ICRF2 catalog is based on a single solution done at the GSFC, the generation of ICRF2 could not have been realized as accurately and with as much understanding of the limiting errors and noise levels without access to the OCCAM, SteelBreeze and Quasar VLBI software analysis packages. Detailed comparison of the analysis software, described in more detail in Fey et al. (2009) gives confidence in the correctness of the mechanical implementation of the VLBI modeling.

Variable intrinsic source structure can contribute to apparent source proper motion, e.g., due to changes of the position on the sky of the brightness peak of a core-jet source. In this section we attempt to identify the worst such sources so that their effect on the reference frame can be minimized.

A series of global solutions was made. In each solution the positions of all sources except for a small test set were estimated as time-invariant "global" parameters. The positions of the test sources were treated as "arc" parameters with a position estimated for each day the source was observed. Each source was treated as a test source in some solution. The complete set of source positions as functions of time was then analyzed to determine which sources had statistically significant variations in their positions.

Statistics of these position time series in R.A. and decl. were calculated and examined, such as weighted root mean square (wrms) variations about the mean, χ² per-degree of freedom and smoothed 2 year linear slopes. A total of 39 sources showing the largest position variations were selected in this manner for special handling. These sources show significant position variation in either R.A. and/or decl. and were observed historically in many sessions. Some of these are strong sources that have not been observed very often in recent years because of known adverse source structure effects on geodetic solutions (such as 3C84, 3C273B, 3C279, 3C345, and 3C454.3). A few are sources that are not very well observed overall, but still show convincing systematic position variations. Estimating the positions of these sources as global parameters would introduce significant systematic position errors and yield grossly underestimated position uncertainties that could possibly distort the overall reference frame. Therefore the positions of these sources were treated as arc parameters in the final global solution.

The 39 special handling sources are: 0014+813, 0106+013, 0202+149, 0208−512, 0212+735, 0235+164, 0238−084 (NGC 1052), 0316+413 (3C84), 0430+052 (3C120), 0438−436, 0451−282, 0528+134, 0607−157, 0637−752, 0711+356, 0738+313, 0919−260, 0923+392 (4C39.25), 0953+254 (OK 290), 1021−006, 1044+719, 1226+023 (3C273B), 1253−055 (3C279), 1308+326, 1404+286 (OQ 208), 1448+762, 1458+718 (3C309.1), 1611+343, 1610−771, 1641+399 (3C345), 1739+522, 2121+053, 2128−123, 2134+004 (2134+00), 2145+067, 2201+315, 2234+282, 2243−123, and 2251+158 (3C454.3). Seven of these sources (marked in italics) were original ICRF1 defining sources.

Note that it should not be assumed that there are only 39 unstable sources among the available sources. The vast majority of the sources have not been observed with the frequency necessary to detect the type of small systematic position variations seen in some of these sources. Many other sources showed smaller position variations that could, with more observation, show larger variability that would exclude them in the future. In such cases, it would seem that more data is not necessarily a good thing. We have attempted only to identify the worst such sources with the available data to minimize their effect on the reference frame.

Finally, there are a few sources excluded from the final global solution for various other reasons. Included in this category are three known gravitational lens sources and six known radio stars. The gravitational lens sources present analysis problems in assigning a single position and the radio stars exhibit real proper motion and are thus unsuitable for the purposes of a reference catalog. Also excluded from the final global solution were 795 sources with insufficient data for position determination. These sources were observed unsuccessfully for unknown reasons but are included in the VLBI data set for historical reasons and can be deselected when the analysis is done. Most of these sources were in the VCS sessions and are assumed to be either too weak or too spatially extended for reliable detection.

We performed CALC/Solve test solutions to determine the sensitivity of the source position estimation to different model choices, solution strategy and choice of included data.

The model choices investigated were: (1) application of atmospheric pressure loading, (2) the VMF1 (Böhm et al. 2006) versus the NMF (Niell Mapping Function) (Niell 1996) troposphere mapping functions, (3) application of antenna thermal deformation (Nothnagel 2008), and (4) estimation of troposphere gradients with relatively tight constraints using an a priori gradient model versus estimation of total gradients with very loose constraints. Results of these comparisons are listed in Table 1 as weighted mean and wrms differences in source position estimations between the various solutions. Also listed are the overall rotation angles between the positions from the various pairs of solutions.

Table 1.  Summary of Data and Model Comparisons

Data/Model Comparison	${\rm{\Delta }}$ αcosδ		${\rm{\Delta }}\delta$		Rotation Angles
	mean	wrms	mean	wrms	X	Y	Z
	(μas)	(μas)	(μas)	(μas)	(μas)	(μas)	(μas)
Pressure Loading: On versus Off	0	2	0	3	2	1	0
VMF1 versus NMF	−1	3	−3	5	−1	2	−1
Thermal Deformation: On versus Off	0	0	0	1	0	0	0
Gradients: a priori versus no a priori	0	7	6	12	8	5	3
TRF versus Baseline	−1	10	0	12	2	2	2
Start Time: 1990 versus All	1	8	1	11	0	2	1
Start Time: 1993 versus All	0	14	0	18	−1	5	4

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To assess the effect of solution type on source position estimates, we compared results from a baseline type solution with those from a TRF type solution. As discussed in Section 5, the ICRF2gsfc solution was chosen to be of the TRF type in which station positions and velocities are solved globally from the entire data set resulting in a single position and velocity estimate for each antenna at a specified epoch as opposed to a baseline type solution in which site positions are treated as arc parameters and separate positions are obtained for each session in which an antenna participated. This comparison essentially quantifies the effect of daily station motion on global source position estimates. These results are also listed in Table 1.

Finally, test solutions were made including data starting from 1990 or 1993 compared with the entire data set spanning 1979–2009. The chronologically earlier VLBI data is known to be considerably noisier than later data. This is due to many improvements over the past 30 years, such as increased individual channel bandwidths, increased spanned bandwidths, improved electronics, new and more sensitive antennas, larger networks, improved scheduling methods, and other factors. These results are also listed in Table 1.

The results presented in Table 1 show that uncertainties due to our choice of modeling options, the solution type and/or choice of included data are all well within the estimates of the noise floor and axis stability discussed in Sections 7.3 and 11.3, respectively.

Analysis noise refers to the cumulative effects of data editing and/or modeling errors in the resulting source position estimates from a global VLBI solution. One way of estimating the magnitude of this error is to compare solutions performed by different VLBI analysis centers using different software and/or modeling, where each center starts with the same observational data.

The wrms difference between source position estimates, s_i, from two global solutions after removing biases is

$$\begin{eqnarray*}&&{\sigma }^{2}=\langle {({s}_{1}-{s}_{2})}^{2}\rangle ={\sigma }_{1}^{2}+{\sigma }_{2}^{2}\end{eqnarray*}$$

where ${\sigma }_{i}^{2}$ are the estimated variances of the source positions and the position estimates from the two solutions are assumed to be uncorrelated. If we assume the two solutions have the same noise then we can get an estimate of the noise of each solution, ${\sigma }_{i}\sim \sigma /\sqrt{2}$.

Table 2 lists the wrms differences (scaled by a factor of 1/$\sqrt{2}$) between the ICRF2gsfc solution and several solutions from other analysis centers, namely USNO, IAA and MAO. VCS sources from these solutions were not included in the comparisons. The listed scale factors are the $\sqrt{{\chi }^{2}}$ of the differences scaled by the formal uncertainties of the solutions. These differences give a measure of the analysis noise inherent in the solutions. The GSFC/USNO differences are generally the smallest since both solutions used the CALC/Solve analysis software. The IAA and MAO differences tend to be larger presumably because these solutions used different analysis software, i.e., QUASAR for IAA and SteelBreeze for MAO.

As an additional estimate of realistic source position noise levels, we did a decimation test in which all included sessions were ordered chronologically and divided into two sets selected by even or odd session. Results of this test are also listed in Table 2 but discussed in Section 7.3.

If all errors were Gaussian then the uncertainty of position estimates should fall off as 1/$\sqrt{N}$ where N is the number of observations. To determine a realistic level of source position errors, we ran a decimation test in which all VLBI experiment sessions were ordered temporally and divided into two independent data sets selected by even or odd session. This was done for each well-defined session type, where the session type refers to a series of experiments with the same core network of observing stations. This should help ensure that the two full sets of sessions were equivalent in terms of networks and sources observed. The remaining group of sessions not in an obvious category were similarly divided. Global solutions were performed with each set of sessions. Source position estimates from the two solutions are independent and the solution position differences yield estimates of the noise of each solution as well as how much the formal uncertainties should be scaled. The differences in source position estimates from the two decimation solutions were scaled by their formal errors and then the standard deviation of the scaled differences was computed. The resulting scaling factors (standard deviations) of 1.6 and 1.5 for R.A. and decl., respectively, are listed in Table 2 and can be compared to the scale factors arising from differencing solutions from different analysis centers, also given in Table 2.

To obtain a better estimate of the systematic noise floor, the sources were ordered by the average number of experiment sessions in which a source was observed in the two decimation solution. The differences in position were analyzed for a running window of 50 sources in this ordered sequence of sources. We computed the wrms difference of positions from the two solutions for each 50 source subset of all the sources common to both decimation solutions. Figure 1 shows the dependence of the wrms difference (scaled by 1/$\sqrt{2}$) as a function of the minimum number of sessions in each subset. This is compared to the median formal uncertainty in the subset. The wrms differences are larger and fall off less steeply than the median formal errors, which fall off as 1/$\sqrt{N}$. The observed minimum error of 25 μas for decl. and 15 μas for R.A. is reached for sources that have been observed in more than 200 sessions. If one applies an overall scaling factor of 1.5 based on all source position differences, one still needs to add additional noise to account for residual scaling errors that are as large as 1.5 for sources observed in less than 75 sessions. At the expense of overly conservative uncertainties for the most well observed sources, we chose a noise floor of 40 μas, which is the noise level in Figure 1 for both R.A. and decl. for sources observed in at least 100 sessions.

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Consequently, the formal errors of the ICRF2gsfc catalog were inflated following the formula:

$$\begin{eqnarray}&&{\sigma }_{\alpha \mathrm{cos}\delta }^{2}={(1.5\;{\sigma }_{\alpha \mathrm{cos}\delta ,0})}^{2}+{(0.04\ \mathrm{mas})}^{2}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\sigma }_{\delta }^{2}={(1.5\;{\sigma }_{\delta ,0})}^{2}+{(0.04\ \mathrm{mas})}^{2}\end{eqnarray} \tag{ 2 }$$

where the subscript zero indicates the formal errors. If we inflate the formal errors in this way, the average residual scaling factors are 0.95 for decl. and 0.88 for R.A.

The International GNSS Service (IGS), through observation and analysis of the Global Positioning System (GPS) satellites, provides a suitable external data set with the same or better quality for external validation of VLBI EOP products, but only for polar motion cf., Böckmann et al. (2010b). The other components of EOP, UT1-UTC and offsets in nutation angles, are the almost exclusive domain of VLBI and as such, for these components, no suitable external, i.e., non-VLBI, comparison is available.

We compared polar motion X and Y between the IGS EOP series and ICRF2gsfc together with those from five other VLBI global solutions contributed from the various VLBI analysis groups including BKG, USNO, MAO, IAA and the Observatoire de Paris (OPA). After subtracting an overall bias and rate between the IGS and VLBI series, the six VLBI solutions considered exhibit wrms agreement with the IGS series at the roughly 100 μas level in both the X and Y components. Noticeable systematic variations evident in the Y component from all the VLBI series have been identified with changes in the VLBI network arrays (Artz et al. 2008), but in general, the scatter between the different VLBI solution results and the systematic network effects are at the same level indicating that the wrms values are representative of the overall agreement between the IGS and VLBI polar motion values.

An evaluation of the other three components of the standard Earth orientation representation, UT1-UTC and nutation in dX and dY, can only be carried out by inter-comparison of the results of the six VLBI solutions mentioned in the previous paragraph. This is a valid approach here since the six VLBI EOP time series were generated using the three different and independent software packages CALC/Solve (BKG, GSFC, OPA, USNO), SteelBreeze (MAO) and Quasar (IAA). The IERS EOP 05 C04²⁶ (hereafter IERS C04) series was used as a standard reference to subtract a common signal from the six VLBI series. Note that this comparison does not represent a qualitative analysis of the differences between the VLBI series and IERS C04 in an absolute sense since the IERS C04 series is mainly derived from VLBI results but does represent the level of relative agreement.

Examination of the differences between the six VLBI series and IERS C04 show that there are no systematic differences in nutation angles between the four CALC/Solve solutions and the SteelBreeze solution. However, a very obvious systematic effect with an irregular period is present in the IAA time series. This effect was traced to errors in the relativistic model used in the submitted IAA EOP series. Since the MAO and IAA time series do not show strong correlations but the MAO series follows the four Calc/Solve solutions, albeit with some excess noise, it can be concluded that the inter-comparisons provide a reliable relative indication of the quality of each VLBI input series. The wrms agreement between the MAO nutation angle series and the IERS C04 series agrees at the ≈100 μas level in both nutation angles whereas the CALC/Solve solutions range in agreement at the 60–95 μas level. Since these time series all agree with the reference series at a similar level, the absolute accuracy of the nutation estimates should not be worse than by about a factor of $\sqrt{2}$, thus indicating that the nutation angle accuracy is at the same level as that of polar motion.

A comparison of the six VLBI series for UT1-UTC shows that the reference series IERS C04 exhibits a long term drift after about epoch 2002.5. Nevertheless, the VLBI solutions all agree with each other at the few μs level. The wrms differences in UT1-UTC with respect to the reference series is at the level of about 9 μs which corresponds to 135 μas. This number is driven by the systematic effect in the differences and does not characterize the agreement of the six series as such. The agreement between the six VLBI series is rather at the 4–5 μs level corresponding to 100 μas which is the same level of agreement as the polar motion results and can, therefore, be considered as the upper limit also of this component of Earth rotation.

From the comparisons of the EOP results, it can be concluded that the six VLBI solutions agree with each other at the level of better than 100 μas excluding known systematic effects. The polar motion results of the VLBI series agree with the IGS GPS results at the ≈100 μas level and considering that the other EOP components do not exhibit any obvious systematic effects, it can be concluded that their accuracy is at the same level. Consequently, an upper bound of 110 μas or 3.3 mm at the Earthʼs surface can thus be inferred for the overall accuracy of each observing session contributing to the VLBI determination of ICRF2.

A second option for external validation of the ICRF2gsfc solution is to investigate the accuracy of the station positions and velocities in terms of the quality of the TRF that they represent. As discussed in Section 6, station positions and velocities are estimated in the same solution as the source positions.

We compared the TRF from the ICRF2gsfc solution with VTRF2008 (Böckmann et al. 2010a), the VLBI determined input to ITRF2008. Ideally, a comparison should be made to ITRF2008, but unfortunately ITRF2008 had not been released at the time this work was done. VTRF2008 is a combination product with input from several IVS Analysis Centers and should provide a very reliable reference due to the stabilizing effect of the combination. Six of the seven input solutions used the CALC/Solve analysis software and one solution was generated using the OCCAM analysis software. Although it would be better to have more solutions from different independent software packages, the agreement of all the input solutions in general and the agreement of the TRF between the independent software packages of CALC/Solve and OCCAM in particular, should preclude any serious deficiencies in the combined TRF.

In order to facilitate comparison, a 14-parameter Helmert transformation between the ICRF2gsfc solution and the VTRF2008 were calculated. In the context of ICRF2, the rotations and their time evolution are of particular importance. The TRF from the ICRF2gsfc solution is rotated with respect to VTRF2008 but not by more than ∼40 μas. The rotation rates are at the level of a few μas year⁻¹ with formal errors at about the same level. The scale difference and its rate with respect to VTRF2008 is not significant.

The quality of the coordinates and velocities of individual VLBI antenna sites can best be discussed by looking at the residuals of the antenna positions at epoch and their velocities. Antenna positions at sites actively observing at the reference epoch (J2000.0) generally show differences with respect to VTRF2008 below about 5 mm in their horizontal topocentric positions with matching discrepancies in the station velocity components. Notable exceptions are SYOWA and OHIGGINS, two VLBI stations located in Antarctica, the TIGOCONC station located in Chile and the NYALES20 station located in Norway, with residuals in the horizontal components being slightly larger. However, the differences in the vertical components of these four sites fit very well to VTRF2008. Other stations with larger residuals are older radio telescopes which either no longer exist or are no longer in use and hence do not have current data.

On the basis of the Helmert parameters of the ICRF2gsfc solution with respect to VTRF2008 it can be stated that the orientation of the TRF axes from the ICRF2gsfc solution matches the orientation of the VTRF2008 within ∼40 μas and a few μas year⁻¹, respectively. The residuals of horizontal and vertical components of the station coordinates as well as of the velocities confirm the overall accuracy of the ICRF2gsfc solution at the level of about 3.5 mm.

Source ranking based on positional stability is derived using the position time series used to identify the 39 special handling sources in Section 3.2 and from the positions and uncertainties from the ICRF2gsfc catalog, after removing the 39 special handling sources. We start by keeping only the 593 sources whose positions were (1) estimated as global parameters in the ICRF2gsfc solution, (2) were observed in at least ten sessions, and (3) have an observational history of more than 2 years.

From the position time series, one can compute a positional stability index as

$$\begin{eqnarray}&&r=\sqrt{{\mathrm{wrms}}_{\alpha \mathrm{cos}\delta }^{2}{\chi }_{\nu ,\alpha }^{2}+{\mathrm{wrms}}_{\delta }^{2}{\chi }_{\nu ,\delta }^{2}}\end{eqnarray} \tag{ 4 }$$

where ${\mathrm{wrms}}_{\alpha \mathrm{cos}\delta }$ and ${\mathrm{wrms}}_{\delta }$ are the wrms position variations about the weighted mean position for R.A. and decl., respectively, and ${\chi }_{\nu ,\alpha }^{2}$ and ${\chi }_{\nu ,\delta }^{2}$ are the reduced ${\chi }^{2}$ of the fit to the mean positions.

From the ICRF2gsfc catalog, a combined formal error on the position estimate can be computed as

$$\begin{eqnarray}&&d=\sqrt{{\sigma }_{\alpha \mathrm{cos}\delta }^{2}+{\sigma }_{\delta }^{2}+{\sigma }_{\alpha \mathrm{cos}\delta }{\sigma }_{\delta }C(\alpha ,\delta )}\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{\alpha \mathrm{cos}\delta }$ and ${\sigma }_{\delta }$ are the formal uncertainties in R.A. and decl., respectively, and $C(\alpha ,\delta )$ is the correlation between estimates of α and δ.

If we naively define an overall positional stability index as $p=r+d$, we find that r is generally larger than d by about a factor of ten, further illustrating that the formal errors from the global solution are significantly underestimated. Thus a ranking based on p alone would be dominated by information primarily from the time series. Further, a ranking based on p alone rejects many of the southern hemisphere sources due to their poor observing histories. Consequently, we implement a method suggested by Fey et al. (2001) to select candidate defining sources which gives more equal weight to the wrms and formal uncertainties and which selects sources more uniformly distributed as a function of decl.

First, the 593 sources under consideration are divided into five approximately equal number bins based on decl. with boundaries at −31°, 0°, 18°, and 40°. Then, separately in each decl. band, sources are ranked on a normalized scale based on r alone. This normalized ranking is repeated but instead this time using the quantity d alone. Finally, the scaled r and d for each source are summed and re-normalized to a 0–100 scale. This final normalization constitutes the "quality" index P. The distribution of P is displayed in Figure 3.

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Tests to determine the minimum number of sources required to define stable frame axes discussed in Fey et al. (2009) suggest that the minimum number of sources should be around 200 with an optimal value about 380, after which the stability is degraded. Thus we chose a value of quality index P ≥ 40 to give a list of 423 candidate defining sources.

The final list of defining sources results from the cross-correlation between the ranked list of sources described in Section 10.1, based on positional stability, and the ranked list of sources based on SI described in Section 9. Overall, the two criteria (positional stability and source SI) show good correlation with positional stability increasing as the SI decreases.

The initial list of 423 candidate defining sources was further narrowed by setting the threshold for the continuous SI to 3.0 (i.e., all sources with SI values larger than or equal to this threshold value were removed from the list). Thus, the final set of defining sources was reduced to 295 sources. Unfortunately, about 25% of these 295 sources, with most of them located in the southern hemisphere, had no SI available. These sources were kept on the basis of their positional stability alone.

The distribution on the sky of the 295 ICRF2 defining sources is shown in Figure 4.

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Of the 295 sources selected as ICRF2 defining in Section 10, only 97 were also defining sources in ICRF1 (as realized by ICRF-Ext.2). Most of the overlap defining sources are in the northern hemisphere and hence are poorly distributed on the sky for reliable estimation of rotation angles between the two frames. To address this situation, an additional 41 of the 295 defining sources, mostly in the southern hemisphere, were selected resulting in 138 common objects for computation of rotation angles with the ICRF-Ext.2 catalog. The status of the additional 41 sources in ICRF-Ext.2 is: 24 candidate sources, 16 other, and 1 new.

The ICRF2gsfc catalog (with inflated formal errors as described in Section 7) was compared to ICRF-Ext.2 using a 4-parameter transformation in which the coordinate differences are modeled by three rotation angles A₁, A₂ and A₃, around the X, Y and Z axes of the celestial frame, respectively, and a parameter dz accounting for a global translation of the source coordinates in decl., e.g., see Feissel-Vernier et al. (2006):

$$\begin{eqnarray}&&{\rm{\Delta }}\alpha ={A}_{1}\mathrm{tan}\delta \mathrm{cos}\alpha +{A}_{2}\mathrm{tan}\delta \mathrm{sin}\alpha -{A}_{3}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\delta =-{A}_{1}\mathrm{sin}\alpha +{A}_{2}\mathrm{cos}\alpha +{dz}.\end{eqnarray} \tag{ 7 }$$

The additional two deformation parameters used in the transformation formula for the alignment of ICRF1 (Ma et al. 1998) were found negligible and are not estimated here. Values of estimated parameters are listed in Table 3. The quantities ${r}_{\alpha \mathrm{cos}\delta }$ and ${r}_{\delta }$ in Table 3 are the wrms residuals in $\alpha \mathrm{cos}\delta$ and δ, respectively.

Table 3.  Parameters to Transform ICRF2gsfc Onto ICRF-Ext.2

A1	A2	A3	dz	${r}_{\alpha \mathrm{cos}\delta }$	${r}_{\delta }$
23.3 ± 19.2	−33.5 ± 19.5	7.8 ± 18.4	11.2 ± 16.6	9.2	12.4

Note. Unit is μas.

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Improvements in the models and procedures applied in the ICRF2gsfc catalog solution resulted in a frame with fewer deformations than ICRF-Ext.2, but with a slight mis-orientation. In the procedure applied to rotate the ICRF2gsfc catalog positions into the ICRS, care was taken not to transfer the deformations of ICRF-Ext.2 to ICRF2. Consequently the radio source coordinates of the ICRF2gsfc catalog were rotated onto the ICRS using only the three rotation angles A₁, A₂, and A₃ listed in Table 3. The rotated ICRF2gsfc catalog constitutes the final ICRF2 catalog.

The stability of the ICRF2 axes were tested by estimating the relative orientation between ICRF2 and ICRF-Ext.2, using Equations (6) and (7), on the basis of various sets and subsets of sources. The results of a sample of these comparisons is listed in Table 4. The scatter of the rotation parameters obtained in the different comparisons indicate that the axes are stable to within about 10 μas.