THE PPMXL CATALOG OF POSITIONS AND PROPER MOTIONS ON THE ICRS. COMBINING USNO-B1.0 AND THE TWO MICRON ALL SKY SURVEY (2MASS)

To link a proper motion system of a sky survey (catalog) to the ICRS, two distinct approaches are possible. Either the "proper motions" of extragalactic objects are forced to vanish (this is the method adopted by Munn et al. 2004 or Gould & Kollmeier 2004 in their reduction of SDSS proper motions) or the optical representation of the ICRS, the Hipparcos catalog, is extended to fainter magnitudes. We chose the second alternative, as we did not try to identify point-source-like extragalactic objects in USNO-B1.0. Also, at low galactic latitudes this method can hardly work.

The Hipparcos catalog itself is, of course, unsuited for the link because of its low spatial density and its bright stars. The next obvious choice, Tycho-2, is unsuitable as well for reasons laid out in Section 3. This leaves the fainter part of the PPMX for the construction of the link between the USNO-B1.0 proper motions and the ICRS.

Before we come to this comparison, let us note that the proper motion system can be checked, to a certain degree, independent of a comparison with a reference catalog. The proper motions in an inertial reference system must, except for their peculiar motions, on average only reflect the physical motions of the stars in our Galaxy, i.e., the reflex of solar motion and the rotation of the Galaxy. Systematics parallel to the axes of right ascension or declination must not appear, nor should features be seen representing the plate lay-out of a photographic survey. In step 5, the stars in the preliminary catalog with 2MASS K_s-magnitudes between 12 and 13 have been chosen to represent the preliminary system (PS1). Doing so, we get a fair overlap with the faint stars in PPMX. The magnitudes in the visual range from USNO-B1.0 are not suited, because the magnitude system is very inhomogeneous from plate to plate.

Figures 1 and 2 show the proper motions of the PS1 in the right ascension, declination plane in bins of 025 × 025 averaged via a 3 × 3 bin moving average filter. Surprisingly, the proper motion system is remarkably smooth north of −20° declination. Only minor plate-dependent effects can be seen. This is a hint that the authors of USNO-B1.0 achieved very satisfactory results in their plate reductions in this portion of the sky. However, south of −20° declination, plate-dependent distortions of the proper motion system are obvious and need to be corrected for.

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In 2009 August, UCAC3 was released. UCAC3 contains some 100 million stars and therefore goes deeper than PPMX, and, in principle, could be used for the correction of PS1. To characterize the UCAC3 system (U3S) of proper motions, we chose stars with $14<m_{r_U}<15$ from UCAC3. This is well away from both the bright Tycho-2 stars and the magnitude limit of UCAC3.

Figures 3 and 4 show the proper motions of the U3S in bins of 025 × 025 averaged via a 3 × 3 bin moving average filter. White pixels show areas where UCAC3 (14 <r_U< 15) contains only stars without proper motions. These areas presumably largely coincide with those without first epochs in the UCAC3 project. We interpreted proper motion zero and negative errors on it (found for about 9 million stars) as signifying null values in the corresponding columns of UCAC3.

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While the proper motion system is well determined south of −20° declination, it clearly shows unphysical effects in both coordinates north of −20°, where pattern-dependent proper motions occur with amplitudes exceeding ±12 mas yr⁻¹. According to N. Zacharias (2009, private communication) the source of the systematic effects comes from the Schmidt plates used for the first-epoch positions, whereas in the south (δ < −20°) the plates from the SPM could be used. These systematic errors in UCAC3 north of δ = −20° are so large that UCAC3 cannot be considered to be on the ICRS system, and the reader is advised to take care when using it for kinematic investigations of our galaxy.

To summarize the situation, it happens that we have proper motion systems north and south of δ = −20° which do not show conspicuous plate-dependent structure. This enabled us to construct an intermediate system as a combination of both, i.e., the proper motions of the PS1 are left unchanged north of δ = −20°, and differences U3S − PS1 are applied south of δ = −20° on a grid with 0.25 × 0.25 deg² bins with a 3 × 3 bin moving average filter. The resulting system is called PS2. This completes step 5.

The absence of plate-dependent distortions alone does not place a proper motion system onto the ICRS; the link to the Hipparcos system is mandatory. In step 6, we chose PPMX as representative for Hipparcos, and we added systematic differences PPMX-PS2 to PS2 on a grid with 1 × 1 deg² bins with a 3 × 3 bin moving average filter. This lay-out was chosen to have enough PPMX stars for the link. Their number varies between 1000 and 5000, with a few areas containing only 500 stars per bin. This link completes the proper motion system of PPMXL. The resulting system PS3 is shown in Figures 5 and 6. The link to PPMX introduced a negative systematic in Figure 5 compared to Figure 1 at +40° declination. This depression is already inherent in Tycho-2 as shown by Röser et al. (2008).

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So far, we only discussed the proper motion system of PPMXL for stars having 2MASS observations. To ensure that objects in PPMXL beyond the 2MASS limits are on the same proper motion system, we made a least-squares adjustment giving weight zero to 2MASS observations (step 7). Then, we compared this proper motion system with PS3 and applied the resulting corrections on a grid with 0.25 × 0.25 deg² bins with a 3 × 3 bin moving average filter. In other words, for the stars in PS3 we derived positions and proper motions with and without including 2MASS observations and, so, corrected the stars that happen to have no 2MASS observations.

It is well known that magnitude- and color-dependent systematic errors occur in astrometric observations, be they photographic, taken with CCDs or even with photoelectric meridian circles. In the case of USNO-B1.0 or PPMXL, they are hard to be detected, as there is no independent reference on ICRS at fainter magnitudes, e.g., at "blue" or "red" magnitudes between 19 and 20. Also, these stars have kinematics different from the bright stars in solar reflex and galactic motion. So, to a certain extent it is difficult at present to distinguish between systematic errors and different kinematics. There is only one thing one can check: the plate structure of the underlying surveys must not be seen. As an example, we show in Figure 7 the proper motion system of PPMXL in the I band of USNO-B1.0 in magnitudes 18 < I < 19. We chose the I band here as representative, because the I-magnitudes are more homogeneous over the full sky than are the "blue" or "red" magnitudes.

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Qualitatively, the faint stars show a similar pattern as the bright ones (12 <K_s< 13) in Figure 6. However, plate-dependent distortions still are not negligible. Empty areas display regions where the stars in PPMXL have no I-band observations in the range 18 < I < 19. We attribute this to inhomogeneities in the USNO-B1.0 I-band photometry, which leads to an apparent underdensities or missing stars compared to other fields.

In conclusion, the proper motions system at the faintest magnitudes is more uncertain than at bright ones. A remedy could come from a sophisticated new reduction of all the survey plates where care is taken to study and avoid all magnitude- and color-dependent effects. With Gaia at the horizon, this effort probably is not warranted.

Unlike the proper motions, the positions of PPMXL at J2000.0 can only be referred to the ICRS by comparison with a catalog which is on ICRS. The obvious choice would be UCAC3. However, UCAC3 does not publish the original observations made in the years 1998–2004 with the CCD camera on the USNO astrograph from CTIO and Flagstaff. In the published catalog, the positions at 2000.0 are given, which result from applying proper motions to the original positions. In the following, we show that the positional system of UCAC3 at 2000.0 is already corrupted by the systematically distorted proper motions even over the few years between observation and 2000.0. Indeed, this can best be seen in Figure 8 for the declination system. In Section 5.1, we showed the UCAC3 proper motions in declination (see Figure 4). If we average the proper motions over right ascension and let them pass through a high-pass filter (variations on scales larger than 10° are suppressed), we get on the northern sky 0° <δ< 90° a wave-like behavior with an amplitude of more than 5 mas yr⁻¹, a period of 5° and phase 0 at the equator. A similar behavior is seen on the southern hemisphere but with a much smaller amplitude of only 1 mas yr⁻¹. This is illustrated in the upper left panel of Figure 8. The upper right panel shows the same plot (proper motions in declination) for PPMXL. On the northern hemisphere, the difference is striking, whereas the coincidence south of δ < −20° is very good, as it should, because the system of PPMXL is strongly linked to UCAC3 in this area. This becomes much clearer in the plot in the lower left, where we show the proper motion differences PPMXL − UCAC3 for 30 million stars in common.

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The lower right panel of Figure 8 shows the differences in positions at epoch 2000.0 between PPMXL and UCAC3. Here, we find again a 5° wave with an amplitude increasing from the equator to the north pole (from 15 mas to 30 mas). We note that the difference in proper motions (lower left panel) and the difference in declination (lower right panel) between PPMXL and UCAC3 are in anti-correlation; the explanation is given below.

At epoch 2000.0, the following equations hold (individually as well as averaged over right ascension). Here, the subscript U stands for UCAC3 and P for PPMXL:

$$\begin{equation} \delta _{U,2000} = \delta _{U,{\rm orig}} + \mu _{\delta,U}\times (2000.0 - T_{U,{\rm orig}}) \end{equation} \tag{ 3 }$$

$$\begin{equation} \delta _{P,2000} = \delta _{P,{\rm orig}} + \mu _{\delta,P}\times (2000.0 - T_{P,{\rm orig}}). \end{equation} \tag{ 4 }$$

Subtracting both equations

$$\begin{eqnarray} \delta _{P,2000} - \delta _{U,2000} &=& \delta _{P,{\rm orig}}- \delta _{U,{\rm orig}}+ \mu _{\delta,P}\nonumber\\ &&\times (2000.0 - T_{P,{\rm orig}})- \mu _{\delta,U}\nonumber\\ &&\times (2000.0 - T_{U,{\rm orig}}). \end{eqnarray} \tag{ 5 }$$

Note that in this magnitude range, δ_P,orig essentially coincides with the 2MASS declination, because 2MASS has the highest weight and is almost at epoch 2000.0. The difference δ_P,orig − δ_U,orig hence gives the difference between an original 2MASS position and an original UCAC3 position where no proper motions (i.e., old epochs) are involved. Both are reduced with Tycho-2, are almost coeval, and the mean difference should be zero, at least not showing 5° waves. Also, having filtered out the large scale effects, μ_δ,P is negligibly small (smaller than 1 mas yr⁻¹, Figure 8 (upper right panel)), so we get

$$\begin{equation} \delta _{P,2000} - \delta _{U,2000} \approx -\mu _{\delta,U}\times (2000.0 - T_{U,{\rm orig}}). \end{equation} \tag{ 6 }$$

For δ>0°, we find that the position difference at 2000.0 is in phase with the UCAC3 proper motions as long as (2000.0 − T_U,orig) < 0 and in anti-phase otherwise. For the northern sky, (2000.0 − T_U,orig) is negative because UCAC3 was observed from 2002 to 2004 from equator to pole, exactly what we observe in Figure 8. The increase in amplitude from equator to pole even reveals that the observations at the pole came later than at the equator. Although position differences of 15–30 mas are small, modern UCAC3 observations, accurate at epoch to 15–70 mas (Zacharias et al. 2004), are considerably deteriorated systematically in only a few years. In consequence, we could not use UCAC3 as a reference catalog on ICRS for the position at epoch 2000.0.

The system PS2 of PPMXL consists of the proper motion system and the position system at J2000.0. As in the final step of the construction of the proper motion system (step 7), we chose PPMX as representative of the positional system at epoch 2000.0, so we added systematic differences PPMX-PS2 at epoch 2000.0 to PS2 on a grid with 1 × 1 deg² bins with a 3 × 3 bin moving average filter to achieve the final positional system of PPMXL at 2000.0. In the case of stars having no 2MASS observations, we proceeded analogously to the proper motion system. This completes step 8.

PPMXL contains 910,468,710 entries, including stars, galaxies, and bogus entries. Of these, 412,410,368 are in 2MASS, i.e., 2MASS is used to determine proper motions and the J, H, K_s magnitudes are given in the catalog. In total, 6,268,118 stars are taken from PPMX, so PPMXL aims to be complete from the brightest stars down to about 20th magnitude in V.

The covariance matrix obtained in the least-squares adjustment in Section 4 gives (per coordinate and per star) the mean epoch, the mean error of position at mean epoch, and the mean error of proper motions. All these quantities are published in the catalog.

Mean errors of the positions at the reference epoch 2000.0 can be computed star by star. On average, the mean errors of position 2000.0 are between 80 and 120 mas if 2MASS astrometry is available, and range from 150 mas to 300 mas else.

The statistics of the mean errors of the proper motions resembles the observational history rather than the poorer signal to noise at fainter magnitudes. In Figure 9, we present the distributions of the mean errors of the proper motions in four declination zones. The following can be drawn from this figure. Including the measurements from 2MASS yields a considerable improvement, both because of its good accuracy and its recent epoch. The latter effect is most pronounced on the southern sky. At δ < −30°, the proper motions without 2MASS are of poor quality, because the first epoch is much later than in the other zones, and hence the epoch difference is rather small. 2MASS improves the situation, but a contemporary (2010) survey such as the Sky Mapper Southern Sky survey (Keller et al. 2007) will give a considerable improvement.

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PPMXL is a catalog that is nominally on the ICRS system. It is linked to the Hipparcos catalog, the optical representation of the ICRS, via Tycho-2 and PPMX. A word about the inertiality of PPMXL is therefore appropriate. The uncertainty of a residual rotation of Hipparcos itself is 0.25 mas yr⁻¹ (Kovalevsky et al. 1997). This is a global quantity; on smaller scales the uncertainty is larger. On a typical field of the sky of 1 deg² we find about three faint Hipparcos stars with an rms error of the proper motion of, say, 1.7 mas yr⁻¹ each. Therefore, their "average motion" has a mean error of roughly 1 mas yr⁻¹, a figure representative of the uncertainty of the deviation of Hipparcos from a truly inertial system on a 1° scale. The actual value of such a deviation can be determined only with the results from Gaia or other space missions with limiting magnitude deeper than Hipparcos such as JMAPS and, perhaps, nano-JASMINE. Also, all-sky block-adjustment procedures applied to old and new surveys can help. The two intermediate steps from Hipparcos to PPMXL introduce additional systematic errors which cannot be estimated rigorously. It is therefore not unreasonable to state that the absolute proper motions given in PPMXL have an underlying systematic uncertainty of at least 1–2 mas yr⁻¹, which is small compared to the random error for the vast majority of PPMXL stars. The mean motion of an ensemble of stars, however, cannot be determined better than the 1–2 mas yr⁻¹ mentioned above. These arguments, of course, hold similarly to Tycho-2, PPMX, and the UCAC series of catalogs.

Stars with n_obs = 2 or with "P" flags. About 146 million stars (or 16%) have proper motions based on two observations only; and some 77 million (or 8.4%) carry the P flag defined in Section 4 denoting bad χ² sums of the residuals after the least-squares adjustment. Both cases concentrate toward the edges of plates and in dense regions of the southern sky.

Magnitude system. We made no attempt to recalibrate the USNO B1.0 magnitude system. There are discrepancies in the magnitude system from field to field and from early to late epoch. In principle, the magnitudes can be calibrated using the Guide Star Photometric Catalog (Bucciarelli et al. 2001).

Double entries. Cross-matching with 2MASS has been performed using a cone of 3 arcsec radius. Given the spatial resolution on the Schmidt plates underlying USNO-B1.0 only a single match between a 2MASS and a USNO-B1.0 entry should occur. However, on the northern hemisphere a single match was found in 93.6% of all cases, on the southern hemisphere in only 88.7%. There are also triple and multiple matches, but their number is smaller than 0.03% north and 0.16% south. The number of doubles and multiples strongly increases at plate boundaries and in dense regions on the southern sky (LMC, inner Galactic plane). No complete auto-cross-match has been made with PPMXL, but extrapolating the matches with 2MASS to the full 900 million objects, we estimate that about 90 million (10%) are false doubles or multiples from USNO-B1.0. Turning the cross-match around we also found double matches of 2MASS stars with a USNO-B1.0 star. This amounts to 1.5% of all cases. So, 2MASS has between 6 and 7 million doubles.

"High proper motion" stars. There is a huge number of stars with high proper motions, e.g., on the northern hemisphere about 24.5 million objects have proper motions larger than 150 mas yr⁻¹. The vast majority of them must be fakes; a practically flat proper motion distribution function between 130 and 430 mas yr⁻¹ is a hint to this. Also, the LSPM-NORTH Catalog (Lépine & Shara 2005) lists only some 61,000 stars on the northern hemisphere with proper motions larger than 150 mas yr⁻¹ and claims to be completed to V = 19. An attempt to solve the problem with these large proper motions (already inherent in USNO-B1.0) is far beyond this paper; we only note in passing that many cases occur among the above-mentioned doubles or multiples.