Challenging a Newtonian prediction

Challenging a Newtonian prediction through Gaia wide binaries

X. Hernandez, R. A. M. Cortés, C. Allen and R. Scarpa
Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70–264 C.P. 04510 México D.F. México.
Instituto de Astrofsica de Canarias, C/O Via Lactea, s/n E38205—La Laguna (Tenerife), Spain.
Universidad de La Laguna, Dpto. Astrof´ısica, E-38206 La Laguna (Tenerife), Spain.
Released 19 October 2018
Abstract

Under Newtonian dynamics, the relative motion of the components of a binary star should follow Kepler’s laws and show a scaling with separation, . Once orientation effects and a distribution of ellipticities are accounted for, dynamical evolution can be modelled to include the effects of Galactic tides and stellar mass perturbers, over the lifetime of the solar neighbourhood. This furnishes a prediction for the relative velocity between the components of a binary and their projected separation. Taking a carefully selected small sample of 83 solar neighbourhood wide binaries from the work of Shaya & Olling (2011) for the Hipparcos catalogue, we identify these same stars in the recent Gaia DR2, to test the prediction mentioned using the latest and most accurate astrometry available. The results are consistent with the Newtonian prediction for projected separations below 7000 AU, but inconsistent with it at larger separations, where accelerations are expected to be lower than the critical value of MONDian gravity. This result challenges Newtonian gravity at low accelerations and shows clearly the appearance of gravitational anomalies of the type usually attributed to dark matter at galactic scales, now at much smaller stellar scales.

keywords:
gravitation — stars: kinematics and dynamics — (stars:) binaries: general
pagerange: Challenging a Newtonian prediction through Gaia wide binariesLABEL:lastpagepubyear: 2015

1 Introduction

In galactic dynamics, the range of systems over which gravitational anomalies appear in the low acceleration regime extend across vast astronomical scales. Ultra faint dwarf galaxies with scale radii of order a few tens of parsecs show stellar velocity dispersions implying Newtonian mass to light ratios in the hundreds or even thousands (e.g. Koposov et al. 2015). The classical dwarfs have sizes of order a kpc and mass to light ratios derived from observed stellar kinematics inconsistent with those of naked stellar populations under standard gravity by well over an order of magnitude (e.g. Salucci et al. 2012). This reflects what is observed in spiral galaxies at tens of kpc, where rotation curves (e.g. Lelli et al. 2017) again yield dynamics not corresponding to Newtonian dynamics given the empirically determined matter content. The trend has been extended to include elliptical galaxies observed out to their external low acceleration regions recently by Durazo et al. (2018), and even for the case of Galactic globular clusters where velocity dispersion profiles suggest a change away from Newtonian dynamics for low accelerations e.g. (Scarpa et al. 2003, Scarpa et al. 2011).

Empirically, the above gravitational anomalies can be described by MONDian dynamics (Milgrom 1984, Sanders & McGaugh 2002), where below an acceleration threshold of kinematics stop falling along Newtonian expectations of to flatten out at the Tull-Fisher values of for centrifugal equilibrium velocities or corresponding velocity dispersions for pressure supported systems, where is the total baryonic mass of the system in question. The standard interpretation of this being the presence of dominant halos of a yet undetected hypothetical dark matter component surrounding the astrophysical systems being observed.

Wide binary pairs in the solar neighbourhood offer an opportunity to probe dynamics at low accelerations on the smallest astrophysical scales. In principle these can yield crucial restrictions on the structure of gravity at low accelerations and lengths where the presence of dark matter is not expected. For a solar mass binary, at separations of above pc, AU, accelerations will fall below under Newtonian expectations. A first attempt in this direction was made by two of us in Hernandez, Jiménez & Allen (2012) where we used the Shaya & Olling (2011) -henceforth SO11- carefully selected sample of Hipparcos wide binaries. This catalogue includes a full Bayesian model and use of local 5 dimensional phase space density to identify wide binary candidates and rigorously assign a probability that each candidate forms a physical system, rather than being the result of chance associations.

Retaining a sample of binaries from SO11 were contamination was limited to less than , in Hernandez, Jiménez & Allen (2012) results indicated relative velocities for the binary pairs studied above the Newtonian expectations for accelerations below . This, even after accounting for projection effects, ellipticity distributions and disruption and evolution of ionised binaries due to the Galactic tidal field and encounters with field stars and stellar remnants, as modelled by Jiang & Tremaine (2010), albeit the large errors in proper motions present in the Hipparcos catalogue.

One of us, in Scarpa et al. (2017), explored the problem using a small sample of 60 candidate wide binaries with projected separations between 0.004 and 1.0 pc. That study found that a number of wide binaries are capable of surviving the galactic tides and stellar encounters of the solar neighbourhood dynamical environment, with a small sub-sample of the widest pairs showing kinematics more consistent with MONDian dynamics than Newtonian ones. More recently, theoretical studies by Pittordis & Sutherland (2018) and Banik & Zhao (2018) have confirmed that Gaia data, in terms of expected number of detected wide binaries and confidence intervals for the relevant proper motions and parallaxes, are sufficient to detect MONDian deviations from Newtonian dynamics in the low acceleration regime probed by these systems, should they be present.

The obvious next step is to reproduce the careful and detailed procedure presented in SO11, but using this time the Gaia DR2 catalogue. This painstaking approach will ultimately furnish a definitive answer regarding the presence of gravitational anomalies at stellar scales in the low acceleration regime, but is currently hampered by our incomplete understanding of the problems still present in the data of the very novel Gaia DR2. For example, only about two thirds of the Hipparcos2 sources have Gaia DR2 counterparts (Marrese et al. 2018). Also, Gaia treats all binaries closer than about 1” (depending on the magnitude difference) as single sources, which may give anomalous parallaxes, and the parallax solutions may be more sensitive to duplicity in certain areas, etc. (see Gaia Collaboration (2018) A1 for some of the known issues.)

A first order sampling of the issue can be more directly probed by taking advantage of the correspondence between the Gaia and the Hipparcos catalogues. One can take the sample selection from the accurate Bayesian analysis of Shaya & Olling (2011), and the actual astrometric data from the Gaia DR2. In this letter we present results of such an approach, yielding a small sample of 83 wide binaries from the original SO11 catalogue having a probability of being chance associations, Gaia DR2 data consistent with the original Hipparcos reported quantities, and consistent parallaxes for both components in the Gaia DR2. Although it is only a reduced sample, the superior quality of the Gaia satellite allows to infer relative velocities for the binaries in question to a much higher degree of accuracy than what was available to Hernandez et al. (2012). Interestingly, the results are consistent and show a departure from Newtonian predictions as projected separations grow beyond the critical 0.034 pc. Indeed, our new results are consistent with what was reported in Hernandez et al. (2012), the mean values of the inferred relative velocities are essentially unchanged, with the error bars showing a dramatic reduction. This effectively rules out the Newtonian prediction of Jiang & Tremaine (2010) and provides solid evidence for a gravitational anomaly in the low acceleration regime, this time at stellar scales.

Figure 1: Comparison of SO11 Hipparcos and Gaia DR2 proper motion data in right ascension for each of the two components of the binaries studied. The agreement with the identity line in the majority of cases shows the stars in question have been successfully identified from the first catalogue to the Gaia DR2 sample.

2 Sample selection

As outlined in the introduction, the sample selection is based on the wide binary catalogue of SO11, which was constructed using a very detailed Bayesian procedure. This identifies and quantifies the probability of each binary pair being an actual physical system, rather than the result of projection effects or chance associations, including also the Tycho-2 and the Tycho double star catalogues (Hg et al. 2000 and Fabricius et al. 2002). To that end a 5 dimensional space of spatial positions and proper motions was cross correlated with a galactic phase-space density library including local groups, to identify binary candidates as significant local over-densities in phase space. Corrections due to spherical projections effects were also considered, to yield a catalogue of 840 wide binaries with projected separations of between 0.003 and 10 pc and crucially, a well determined probability of chance association, . Taking only those pairs where this probability satisfies , reduces the original SO11 sample to 359 wide binaries. This catalogue is also narrowly restricted in spectral type for both primaries and companions of each binary, yielding stars in a narrow range of masses centred on . This last is important as it allows a clean comparison to the fixed mass binary simulations of Jiang & Tremaine (2010), see below.

The SO11 search criteria ensure the absence of near neighbours, and results in binary candidates with separations which are always many times smaller than the typical interstellar separations at the location of the binaries in question. Extensive testing with synthetic samples in SO11 guarantees the catalogue includes very few multiple systems with undetected extra companions and is highly complete in the 6 to 100 pc distance range from the sun we treat.

Figure 2: Comparison of SO11 Hipparcos and Gaia DR2 proper motion data in declination for each of the two components of the binaries studied. The agreement with the identity line in the majority of cases shows the stars in question have been successfully identified from the first catalogue to the Gaia DR2 sample.

We now take advantage of the Hipparcos to Gaia DR2 correspondence availability to search for the updated astrometry of the 359 wide binary SO11 sample in the Gaia DR2. The search returns only 151 pairs where each component of the SO11 binaries appears in the Gaia DR2 with two proper motion parameters and measured parallax. This is not surprising, since only about two thirds of the Hipparcos2 sources have Gaia DR2 counterparts (Marrese et al. 2018). It is not yet clear why there are so many sources missing, according to the above authors a combination of effects may be present. As each binary is excluded if either component is absent from the DR2, the fraction we obtain is typical. Next, the SO11 catalogue returns a few systems where more than one secondary is identified as the companion to a given primary, and cases where a single secondary is identified as the companion to more than one primary. We remove all these cases of multiple identifications, bringing the sample down to 134 binaries.

Figure 3: Comparison of the Gaia DR2 reported parallax for the two distinct components of each of the binaries studied. The disagreement of only a handful of cases with the identity line allows to exclude such discordant pairs as part of the expected misidentified binaries in the original SO11 catalogue, or as misidentified stars in going from the Hipparcos catalogue to the Gaia DR2.

A first test of the reliability with which the Hipparcos binaries have been identified in the Gaia DR2 comes from comparing the proper motion measurements reported in SO11 with the corresponding measurements reported in the Gaia DR2. This is shown in figures 1 and 2, were the ranges shown in the axes were chosen so as to display clearly most of our sample, a handful of very discordant systems do not appear in the plots, as they are very far from the identity line shown. It is clear that in most cases, the proper motion values reported by both satellites are in agreement with each other, to within their respective confidence intervals, being the Hipparcos error ranges much larger than the Gaia DR2 ones. Still, we introduce a cut to remove from consideration any binary candidate were for any component the Hipparcos and Gaia DR2 data are more than from each other. Our final results are not sensitive to this threshold, provided the few discordant misidentified binaries are removed. This cut leaves us with only 120 stellar pairs. Next, in figure 3 we check that the Gaia DR2 reported parallax measurements for each of the primaries and companions are not discordant, and remove from the sample any binary where the distances to each component are not within 13% of each other. This exclusion criterion identifies only 18 candidates, to reduce the sample to 102 pairs. Notice that the test shown in figures 1 and 2 effectively gives us a 10 year baseline which, on top of the robustness in the SO11 catalogue towards multiple systems, where no radial velocities are involved, removes any remaining binary where a third component might be altering proper motions with timescales shorter than 10 years (see Banki & Zhao 2018).

Finally, we take advantage of the Gaia radial velocity measurements (when available) and remove any binaries where the radial velocity difference, , between both components is larger than . The resulting cut is not very sensitive to this velocity threshold, as the removed binary pairs typically have much larger and discordant values with an average value for those removed of . Our final sample comprises 83 binary pairs.

Thus, we have prefered very strict cuts to our final sample which leave us with modest numbers, but guarantee the exclusion of misidentified stars in going from Hipparcos data to Gaia DR2 and chance alignment contamination in the original SO11 catalogue, all of which become conspicuous in the comparisons presented in this section. The accompanying electronic table summarises the Gaia DR2 properties used and catalogue numbers from both Hipparcos and Gaia for the primaries and companions of all the binaries used, together with the results of the exclusion criteria described.

Figure 4: One dimensional relative velocities for the final binary sample, showing both results from using only right ascension data, and from using only declination values, with corresponding error bars.

3 Gaia wide binary projected kinematic results

In Hernandez et al. (2012) we calculated the projected separation in the plane of the sky using only the parallax to the primary of each binary, but given the higher quality Gaia DR2 data, we now compute the projected separation between the components of each binary using explicitly the observed Gaia positions and parallaxes to each component of the binary. The average parallax to both components is used to gauge the distance to each pair. Using reported Gaia proper motions, the relative velocity difference in one dimension is calculated for each binary twice, once considering only right ascension proper motions, and once considering only declination proper motions.

Figure 5: The solid curve shows the rms. value for the one dimensional relative velocity between the components of a present day Solar Neighbourhood binary as a function of projected separation, for the Newtonian prediction of Jiang & Tremaine (2010). The same quantity for the SO11 wide binary catalogue stars, this time using Gaia DR2 data, is shown by the points with error bars, circles for right ascension data and triangles for declination. The inconsistency of the observed data with the Newtonian prediction is obvious, for separations greater than the 7000 AU at which accelerations are expected to drop below . This threshold is indicated by the dashed vertical line.

Figure 4 gives the two measurements for each binary pair in the final sample, with corresponding error bars. A clear flat upper envelope is evident. In figure 5 we show the rms. value for the one dimensional velocity differences described above, plotted against projected separations in a binned logarithmic scale, circles and triangles for right ascension and declination data, respectively. The horizontal error bars give the bin sizes, while the vertical ones show the contribution of Gaia reported errors and error propagation, to which a Poisson contribution has been added, and which given the small numbers of binaries in each bin (21, 24, 17, 8 and 13, from left to right), actually dominates the error budget in most cases. The dashed vertical line appears at 7000 AU, the approximate scale where acceleration is expected to drop below .

Also shown in figure 5 are the Newtonian predictions for this same quantity from Jiang & Tremaine (2010), where large collections of 50,000 simulated binaries are modelled for a range of plausible distributions of ellipticities, and followed dynamically under Newtonian expectations within the local Galactic tidal field. These are also subject to the effects of field star and field stellar remnant bombardment for a 10 Gyr period. Finally, the resulting bound and un-bound stellar pairs are projected along a fiducial line of sight to yield a robust prediction for the expected as a function of , solid curve. This results are easy to understand; a trend is apparent, down to the tidal radius of the problem which appears at 1.7 pc. Beyond this point, ionised binaries continue to move along practically common Galactic orbits, with relative velocities which show a mild enhancement which then levels off at close to 0.1 km/s.

It is clear that to the left of the dashed line, our results are consistent with the Newtonian expectations. However, in going to separations larger than 7000AU, the observed points stop following the expected trend and actually level off at a amplitude close to the values seen at the point, reproducing qualitatively the phenomenology seen in galactic rotation curves.

This result for the binary sample presented is extremely challenging to a Newtonian point of view, where the relative velocities are expected to be much lower than observed. Given the construction of the SO11 sample, binaries with small velocity differences would appear as stronger local over densities in phase space, and hence, selection criteria, if anything, are biased against binaries with large velocity differences, not small ones. Thus, from a Newtonian point of view, bound binaries with separations smaller that the tidal radius of 1.7 pc and larger than 7000 AU are unexpectedly missing. Also, a population of non-chance associated binaries appears at scales above 7000AU having relative velocities over an order of magnitude above bound expectations, and which should fly apart over timescales much shorter than the lifetime of the system…. what sustains and replenishes these populations? At separations below the 1.7 pc tidal radius of the problem, bound binaries should appear, under a Newtonian framework.

Furthermore, the results shown in figure 5 confirm what was presented in Hernandez et al. (2012), though the much coarser Hipparcos data of that first study yielded significantly larger error bars. That those first results might have been the result of missed biases or simply the error structure of the Hipparcos data now appears very unlikely, as we see two consistent results coming from data obtained by two completely independent satellites. Indeed, the error bars have significantly shrunk, with central results changing little. Note also that the two estimates we obtain, using only right ascension or only declination data, are consistent with each other.

Although the gravitational anomaly detected appears on crossing the threshold of MOND, in MOND as such, the results are equally unexpected as the external field effect of MOND (or AQUAL e.g. Sanders & McGaugh 2002) should dominate. Given that the orbital acceleration of the solar neighbourhood is higher than the internal acceleration of the binaries in question, in MOND as such, only a very modest enhancement of the effective value of G would be expected (e.g. Pittordis & Sutherland 2018). Thus, within a MONDian frame our results imply not the most well studied version, but rather a variant where the external field effect does not appear, or is much suppressed e.g. as in Milgrom (2011). In terms of covariant extensions to GR having a MONDian low velocity limit, it is hard to know to what extent an external field effect might be present in many of the recently developed options (e.g. the f(R) variants reviewed in Capozziello & De Laurentis 2011, the emergent gravity of Verlinde 2017 or the F(R,L) proposal of Barrientos & Mendoza 2018), so our results then serve as constraints in terms of requiring a minimal external field effect, at least for the sub-parsec scales in the solar neighbourhood explored.

4 Final remarks

We have presented a sample of 83 wide binaries which were very carefully selected against chance associations or projection effects through the cross correlation of the Hipparcos, Tycho-2 and the Tycho double star catalogues, amongst others, with the detailed 5 dimensional phase space structure of the solar neighbourhood by SO11. By taking advantage of the cross identification of the Hipparcos catalogue and the Gaia DR2 data, we updated the parallax and proper motion observations of SO11 to use exclusively Gaia DR2 astrometry.

These binaries are then compared to Newtonian predictions for the expected one dimensional rms. relative velocity between the components of each binary and their projected separations, including modeling orientation effects, a number of plausible distributions of ellipticities and crucially and the effects of Galactic tides and stellar and stellar remnant perturbers over a 10 Gyr period, by Jiang & Tremaine (2010).

For separations below 7000 AU, where accelerations are expected to be above the of MOND, we find the data to be consistent with the Newtonian predictions. For projected separations above 7000 AU however, the data are strongly inconsistent with Newtonian predictions. This strongly challenges the validity of Newtonian dynamics at the low acceleration regime, and shows the existence of gravitational anomalies of the type generally attributed to the presence of a hypothetical and dominant dark matter component, this time down to the relatively tiny sub-parsec stellar scales.

acknowledgements

XH and RAMC acknowledge the support of DGAPA-UNAM PAPIIT IN-104517 and CONACyT.

References

  • [1] Banki I. & Zhao H., 2018, MNRAS, 480, 2660
  • [2] Barrientos, E., & Mendoza, S. 2018, Phys. Rev. D, 98, 084033
  • [3] Capozziello, S., & De Laurentis, M. 2011, PhR, 509, 167
  • [4] Durazo R., Hernandez X., Cervantes Sodi B., & Sanchez, S. F., 2018, ApJ, 863, 107
  • [5] Fabricius C., Høg E., Makarov V.V., Mason B.D., Wycoff G.L. & Urban S.E., 2002, A&A, 384, 180
  • [6] Gaia Collaboration 2018, A&A, 616
  • [7] Hernandez X., Jimenez M. A. & Allen C., 2012, Eur. Phys. J. C, 72, 1884
  • [8] Høg E., et al., 2000, A&A 357, 367
  • [9] Jiang Y.-F. & Tremaine S., 2010, MNRAS, 401, 977
  • [10] Koposov S. E. et al., 2015b, ApJ, 811, 62
  • [11] Lelli F., McGaugh S. S. & Schombert J. M. 2017, MNRAS, 468, L68
  • [12] Marrese P. M., Marinoni S., Fabrizio M. & Altavilla G., 2018, arXiv:180809151
  • [13] Milgrom M., 1984, ApJ, 287, 571
  • [14] Milgrom, M. 2011, AcPPB, 42, 2175
  • [15] Pittordis C. & Sutherland W., 2018, MNRAS, 480, 1778
  • [16] Salucci P., Wilkinson M. I., Walker M. G., Gilmore G. F., Grebel E. K., Koch A., Frigerio M. C. & Wyse R. F. G., 2012, MNRAS, 420, 2034
  • [17] Sanders R. H. & McGaugh S. S., 2002, ARA&A, 40, 263
  • [18] Scarpa R., Marconi G. & Gilmozzi R., 2003, A&A, 405, L15
  • [19] Scarpa R., Marconi G., Carraro G., Falomo R. & Villanova S., 2011, A&A, 525 A148
  • [20] Shaya E. J. & Olling R. P., 2011, ApJS, 192, 2
  • [21] Verlinde, E. 2017, SciPostPhys, 2, 016
Index HIP2 GDR2 Exclusion
SO11 (mas/yr) (km/s) (pc) Test
16 15371 4722135642226356736 1331.1510.355 648.5230.431 12.01 0.32 12.0460.027
15330 4722111590409480064 1337.5910.142 649.9300.154 12.21 0.17 12.0390.011 -1.7440.002
17 17414 43335880716390784 157.9450.086 -316.3260.052 34.18 0.15 17.2090.012
17405 43335537119008896 156.2150.089 -310.2910.064 34.22 1.08 17.2390.014 -2.0510.001
21 19859 3285218186904332288 -109.7000.095 -107.3680.079 -7.21 0.15 22.0870.026
19855 3285218255623808640 -101.7630.080 -111.9820.055 -7.93 0.16 22.1040.022 -2.1630.001
22 23693 4763906879239461632 -32.1400.276 117.4170.310 -1.15 0.22 11.6250.020
23708 4763897739549071744 -32.7840.049 119.6330.054 -0.88 0.16 11.6980.003 -1.7400.001
25 25278 3400292798990117888 250.7650.316 -7.3320.209 37.67 0.24 14.5850.037
25220 3394298532176344960 251.0000.092 -5.7780.068 37.94 0.13 14.5650.011 -1.3010.001
28 26690 3395863205942142976 2.5510.100 -36.0580.080 167.9532.110
26844 3347826784173590656 79.1150.087 -43.0710.068 22.4230.025 -0.1720.010 c
29 26779 263916708025623680 2.7840.075 -523.6020.072 1.07 0.18 12.2800.007
26801 263916742385357056 3.9150.078 -515.9380.078 1.92 0.27 12.2780.008 -2.2340.001
65 62229 6060965699625586176 -201.0330.052 -131.5480.046 14.76 0.16 19.8080.018
69570 6092573252981419776 -102.2180.095 -121.8970.120 -23.92 0.13 37.8420.087 0.9100.002 c,d
73 70529 1254695603704323712 792.5480.092 -1116.6010.111 8.78 0.21 16.3460.014
70536 1254694882149817728 793.4870.085 -1119.0100.095 16.3390.014 -2.4450.004
80 79755 1642641410934267008 -498.0180.050 84.1100.053 -19.86 0.19 10.7680.004
79762 1642642957122493824 -483.1680.066 89.2660.072 10.7650.005 -2.4730.001
81 80337 6018047019138644480 74.1460.306 3.6660.226 12.89 0.13 12.9080.023
80300 6018034958869558912 77.1350.147 0.3340.110 12.9140.012 -1.6660.001
85 83591 4364527594192166400 -916.5620.155 -1138.8040.104 34.14 0.15 10.4660.007
83599 4364480521350598144 -917.2760.098 -1131.9470.065 34.44 0.44 10.4560.006 -2.0290.001
87 84405 4109030160308317312 -466.5410.646 -1142.0630.451 5.9600.008
84478 4109034455276324608 -479.8500.101 -1124.5450.068 -0.04 0.22 5.9500.003 -1.6750.001 b
112 493 2797111130991722240 -150.9360.121 -147.9850.089 -45.70 0.16 37.4290.102
495 2773086595766697856 -147.6140.094 -145.7980.079 -45.00 0.74 37.6830.085 -0.9820.002
130 7699 4911275281704066048 91.1460.041 -31.6520.041 9.48 0.31 46.8550.061
6485 4909846500703006976 92.7900.045 -36.0820.038 8.62 0.21 45.3150.053 0.2840.001
132 9487 2517584007848935808 32.6920.940 -2.8950.821 50.5061.707 b
9519 2517585927699042944 36.3680.126 -10.8110.098 4.58 0.16 48.4040.137 -1.0140.016
140 11477 4967177781457918976 18.5260.121 5.1630.193 47.2580.315
11448 4967153630858709120 18.3830.030 4.4810.052 12.26 0.17 47.7080.075 -1.0460.004 b
155 15304 10584899657116672 166.3600.071 -6.2370.062 31.35 0.20 47.2880.092
15310 10608573516849536 168.7490.073 -6.3140.063 31.90 0.15 47.2690.088 -1.4470.002
157 15527 5060104351007433472 349.0570.036 -65.2990.041 39.89 0.13 36.0240.041
15526 5060105892897388288 348.8470.065 -66.6630.076 40.31 0.14 36.0090.052 -1.3550.001
173 22534 4777112872882315264 -80.7690.929 85.6580.968 6.44 3.13 38.9700.624
22562 4777119126354782592 -82.4130.043 83.6840.052 10.70 0.49 38.0060.032 -1.2680.008 d
175 24046 3422042582096699520 197.8600.093 -89.5790.056 15.64 0.16 40.4700.074
24035 3422047495539178496 198.1790.596 -88.5910.357 21.24 0.43 40.2950.457 -1.2100.006 a,b,d
187 33705 5607190344506642432 18.7890.043 35.5960.049 16.61 0.17 38.1230.043
33691 5607189485513198208 18.8930.061 36.1190.065 16.83 0.21 38.3560.061 -1.2220.001
190 34714 890422213103244544 -110.5100.075 -7.4950.066 3.05 0.24 45.1250.095
34700 890346243721923968 -110.8660.102 -6.7890.088 3.02 0.33 44.7480.143 -0.4090.002
195 37718 5493209501673364736 -114.4350.047 143.4590.042 8.69 0.37 30.0840.022
37727 5493209437253410432 -111.7830.060 142.6030.053 9.16 0.30 30.0800.027 -2.1200.002
201 42401 5746824674801810816 -63.8580.051 38.3740.041 20.53 0.25 31.5960.032
41662 5751951182125903872 -56.5560.064 34.1220.050 2.30 0.24 35.3280.055 0.3640.001 d
204 43970 610526719204475136 60.3370.292 19.7480.221 48.4010.415
44001 610549499710989440 60.4400.161 20.5090.106 48.8010.233 -0.6380.006
207 44858 692119656035933568 -53.2390.128 71.6590.096 30.02 0.22 48.8910.237
44864 692120029700390912 -51.8190.113 73.5240.074 30.31 0.14 49.1240.230 -1.9100.007
215 51312 749786562715192320 -111.1860.069 -62.0170.069 4.11 0.17 49.3440.126
52140 748360706587700352 -110.3380.106 -58.9390.087 -0.60 0.17 50.3820.138 0.4110.002 d
218 52787 3550081879381593728 -124.6900.115 -28.3410.084 23.85 0.25 33.6120.085
52776 3550084490721711872 -124.5390.112 -29.8370.070 23.24 0.52 33.6030.079 -1.4170.002
224 55765 3967618155853506304 -142.3470.204 -5.6450.179 18.88 0.67 47.1980.296
55262 3965063921622777856 -146.5890.080 -0.0060.069 8.33 0.21 47.4720.093 0.4540.004 d
229 58067 3975129194660883328 -450.5020.095 -16.5540.069 5.94 0.29 39.6220.082
58073 3975223065466473216 -450.6000.083 -15.4970.058 5.76 0.22 39.6010.071 -1.8510.002
230 58085 5236197322996128128 13.7920.037 -65.8160.036 -6.80 0.37 44.0780.047 b
58121 5236196498362394112 12.7240.048 -63.2790.045 -7.63 0.13 44.0160.063 -1.3280.001
245 64057 3945118265299248128 -261.6380.077 148.0030.059 -1.68 0.15 37.4420.053
64059 3945118643256370688 -262.4550.078 146.0420.054 -1.70 0.13 37.4330.051 -2.1500.003
253 67246 3721126409323324416 -510.4470.071 -110.2250.064 -30.42 0.20 31.4890.039
67291 3721114933170707328 -509.4400.083 -111.0220.061 -30.67 0.15 31.3320.045 -1.1280.001
264 73365 1586977844504488576 -33.6800.043 100.0460.044 -18.94 0.14 33.7450.027
73360 1586977737129182848 -35.3210.037 101.0740.039 -18.99 0.26 33.7680.024 -1.9860.002
270 74666 1278391075716738560 82.5490.554 -111.9090.560 37.3420.531 a
74674 1278392381386793856 82.7760.040 -110.0400.044 -11.83 0.18 36.9480.042 -1.7240.009
271 75104 1277115023753077248 -179.3390.036 139.0580.039 -26.82 0.17 45.8850.055
75011 1277185633015465856 -181.0210.033 139.5730.042 -26.32 0.19 45.4330.057 -0.4560.001
284 79958 5931674608438449792 5.6690.067 18.4120.051 -0.31 0.52 28.1130.029
80365 5831891213733627264 -3.2050.063 -5.5720.052 76502501 2.4170.283 c
298 85620 1440518669436791296 1.5330.049 -182.0610.053 -34.02 0.16 45.7380.051
85575 1440425863783337856 0.8640.053 -181.9670.060 -33.72 0.18 45.6630.060 -1.3680.001
309 92388 4505477838068264064 -199.3130.096 -223.9790.102 -54.54 0.21 37.7600.068
92638 4312046495498949888 -189.8300.070 -224.4770.079 -22.45 0.90 38.3020.065 0.1500.002 d
314 94150 6421542154150684160 155.0880.122 -42.9760.141 -9.61 0.12 36.9820.122
94154 6421555485729063424 155.7100.036 -41.9370.046 -9.28 0.13 36.6550.041 -1.0960.002
322 97384 4240508901699614976 -28.7130.067 -226.7540.046 47.1160.093
95829 4287506873404614784 -25.9320.072 -237.5550.070 44.0670.083 0.5600.002
323 97940 4240626686883261184 -1.5620.085 -270.2060.052 9.55 0.16 40.1510.097
97950 4240625896609242624 -0.1860.077 -269.8310.047 10.20 0.20 40.0470.087 -1.5020.002
325 99171 4236276194243977344 115.5200.095 -67.5930.060 -3.27 0.14 46.6780.137
99100 4235895935019949952 115.2760.104 -67.9150.050 -3.08 0.14 46.6330.144 -0.6210.003
326 99572 6879764552737781888 193.0680.111 -195.4970.064 27.30 0.16 28.2490.054
99550 6879662263797806976 192.6270.068 -193.9570.045 27.53 0.25 28.2480.033 -0.8540.001
331 100896 4228891667990334976 -64.7790.352 -67.5230.216 49.4640.479
100895 4228891221313732864 -61.6300.099 -69.0390.057 -14.70 0.17 46.0780.108 -1.8580.006
342 106353 6830027182179257472 -279.0740.103 -124.2410.086 32.57 0.16 28.7400.061
106350 6830027143525634432 -284.6650.091 -123.3110.075 33.62 0.28 28.7380.053 -2.1070.004
344 107299 6458951765971500672 -116.5630.062 -52.5750.054 36.95 0.20 44.4620.089
107300 6458952345790198144 -115.2980.049 -52.8650.045 37.16 0.16 44.3730.066 -1.4840.002
352 110084 6613555642140332672 163.4850.077 -203.6900.078 -77.34 0.20 47.6920.111
111520 6620882890706789248 198.1811.765 -163.7621.749 1.78 0.54 51.4672.529 0.7000.023 d
359 112724 2211820991079869312 -63.6630.625 -125.7720.633 37.1630.525
112324 2007431640735691776 -62.5650.062 -135.7670.059 -10.25 0.18 35.3530.046 0.7080.007
366 114131 6541802406664428672 -49.0370.621 -13.4540.651 36.0460.634
114112 6541802578463122688 -47.9210.088 -22.1510.100 15.37 0.23 40.2370.143 -1.5310.010
381 1266 2428524528072046464 42.6920.099 10.1210.050 7.21 0.26 98.4010.465
118 2428948114926672128 46.8990.117 1.4300.085 1.66 0.26 78.5790.471 0.7420.005 d
384 2292 2526899001640197248 -110.3510.145 -221.1850.071 9.54 0.18 54.8770.174
2350 2526925389919277056 -107.0190.094 -223.0390.062 9.77 0.27 55.0490.155 -0.6500.003
392 4702 2356290256259997696 128.4690.101 -54.5040.062 5.31 0.16 82.2250.309
4833 2356080043380354816 129.3400.085 -54.5290.061 5.62 0.19 81.4360.274 -0.1650.003
402 6668 315635261093206656 156.2960.086 -48.6020.076 18.27 0.20 63.0890.207
6675 315635192373731328 155.3190.086 -50.5080.075 63.3430.183 -1.7600.004
404 6772 4916890556306664192 294.8670.051 75.3270.049 49.99 0.18 56.7050.096
6804 4916887395210737024 294.0460.032 75.0680.033 50.23 0.29 56.7640.068 -0.9710.001
405 7189 4929377881661762944 -28.2520.047 -73.6860.044 3.37 0.37 81.1530.209
7086 4929369360446613248 -28.3690.040 -75.4050.044 81.0000.207 -0.4060.002 b
417 11137 76300510625993344 112.3960.074 184.3000.061 27.02 0.15 60.1200.132
11134 76300476266255488 111.0980.064 183.1740.050 27.30 0.24 60.1050.118 -1.9940.004
426 13223 31986240656172800 59.1500.093 -8.6510.090 -14.72 15.28 94.8650.546
13122 33216147491735808 58.0130.088 -6.6090.085 30.25 0.21 76.8070.280 0.1610.004 d
427 13271 4949158198924394496 41.7600.052 -9.9370.057 96.6940.329
13499 4949081572411575552 42.1020.043 -6.4090.047 18.12 0.91 96.4530.291 0.0350.003
441 15432 5047006006423053440 37.0170.040 -5.1050.059 105.6800.336
15152 5046487311812732544 38.9720.039 -5.3180.057 17.59 0.97 100.6490.266 0.3220.003
445 16410 117916235464382336 38.5610.106 -31.9660.079 104.5410.602
16742 69733883588579840 28.0330.088 -31.2480.069 195.9151.768 0.4780.007 c
448 16959 4728825002249947904 26.0940.042 43.3560.047 76.9490.139
16942 4728825036609672576 25.6980.199 44.6550.207 9.03 2.40 77.6150.735 -1.6680.008
450 17486 4666907551119833984 -10.2340.043 -97.1510.049 27.65 0.14 55.1180.069
17515 4666907516760096512 -9.2490.048 -97.3580.050 27.94 0.19 54.9880.074 -1.6930.002
462 20109 4780193841901310336 47.5380.086 72.6620.154 75.9210.271 a
20074 4780194185498710144 49.7050.061 73.6460.100 21.51 0.25 74.9620.192 -1.0050.003
465 21537 3230677565443833088 16.6530.102 11.6030.057 38.60 0.17 64.1320.214
21534 3230677870385455232 15.9530.100 12.1630.054 38.97 0.17 64.1190.204 -1.6150.003
467 21704 2976981131534077056 12.7190.049 22.0370.077 9.57 0.16 83.3750.300
21702 2976981337692506240 12.4110.042 22.8920.046 7.96 0.23 83.0510.224 -1.7880.005
478 25453 3238066592819780608 9.1670.097 -32.4650.080 101.8340.560
25483 3334161160308637312 9.3410.102 -33.1640.089 98.4590.715 -0.2600.006
481 25711 3447142233536837376 -0.5610.099 -46.0800.069 75.5650.284
25614 3447107495841475712 -2.3990.088 -53.5670.066 -21.34 0.30 132.9691.180 -0.0940.006 c
484 26309 2907397747897070336 25.3250.054 -3.0820.073 58.8970.154
26453 2907308172059100544 24.0800.034 -3.1000.041 26.05 0.45 59.1140.083 -0.4600.002
487 26680 3339560999352744960 -7.5410.119 -28.7270.092 29.53 0.54 82.9250.369
26646 3339565461821955712 -8.3120.085 -29.0080.069 30.69 0.68 82.5080.328 -0.7180.004
489 27069 3023084272561678976 14.6700.080 -23.0360.085 87.3120.462
26588 3017087261266087552 15.7290.086 -21.7790.080 25.40 0.74 89.5940.410 0.3330.004
492 27791 3454470100579668992 -8.8340.110 -71.5740.089 -37.59 5.29 89.4280.675 a,b
28872 3453690993510788480 -8.2600.080 -70.5930.066 25.30 0.50 109.9590.579 0.7980.005 d
499 31323 3382205557837836288 -8.3010.087 -35.5550.073 94.4440.524 a
31316 3382205592197580928 -7.3160.092 -34.3480.076 -26.28 0.30 94.8560.692 -1.6100.008
510 35341 948509515477490560 -12.7840.165 12.0650.155 95.7251.022
35799 974093501788080384 -11.5630.064 -5.9040.039 32.57 0.27 102939 1.6170.031 c
525 41081 5322206718812246656 -23.0390.140 19.5480.117 73.2490.382
40916 5320716335102626048 -22.0950.056 19.6440.059 72.7130.138 -0.4020.003
527 41880 3076534861386561536 -133.6830.073 83.5190.053 15.31 0.29 69.0990.201
41881 3076535243639085440 -135.6080.076 81.9680.056 14.85 0.35 69.3560.222 -1.8780.003
540 45811 5742852762061629056 -12.8960.462 -27.7070.536 25.14 0.16 68.5391.707
45802 5742848123496954112 -13.9820.178 -29.1100.155 31.99 1.32 66.9170.523 -1.1240.015 d
541 45937 5310444491348428544 18.1120.055 120.9470.055 91.12 0.16 67.8990.139
45952 5310446621652246144 22.5110.108 114.5440.121 67.4930.238 -0.9770.002
542 46298 1131772432806417280 30.4880.071 26.0210.069 -10.71 0.18 53.8530.117
46299 1131771646828736000 34.5020.053 25.9670.066 -11.18 0.21 53.8770.096 -2.1170.005
546 47017 5249310751464090112 -34.6980.058 43.9700.051 44.93 14.31 87.4420.228
47351 5249282164160875136 -34.9050.051 41.3690.050 22.62 1.01 90.5480.222 -0.1870.002 d
549 47231 1050821477623210368 -11.8800.065 -32.0460.053 9.19 0.13 112.6550.488
46170 1038770314786277248 -17.6080.054 -20.5200.055 -19.63 0.28 103.2500.378 0.4780.003 d
550 47335 5247416705243609088 -36.6130.052 45.0890.053 85.5340.230
47017 5249310751464090112 -34.6980.058 43.9700.051 44.93 14.31 87.4420.228 0.4580.002
551 47399 813906786608830464 -6.1190.063 -6.5870.065 -3.36 0.48 74.0170.272
47851 801608096216800000 -1.9020.089 -3.8640.100 370.7519.574 0.7820.019 c
558 51608 5229614856076359168 -56.7980.047 -45.8610.046 0.37 0.20 70.9270.136
51611 5229612622693362048 -56.6980.043 -46.4890.037 0.40 0.54 70.9180.112 -1.7710.002
582 57950 5343610331876174336 -39.4100.058 -10.2270.052 13.36 0.66 102.1280.460
59716 6075548419252906368 -35.7220.050 -11.0410.042 14.22 1.23 114.8520.518 0.7760.004
588 58751 3575541998835891456 33.0410.105 -18.7630.077 -12.39 0.20 57.2620.183
58722 3575542995268304512 33.0670.840 -20.6640.747 -16.16 0.70 54.4261.115 -1.0470.010 d
589 58813 6147474037517738624 98.9700.059 18.1730.039 6.41 0.12 53.0890.131
58815 6147474071877478144 101.3610.055 15.7240.036 6.14 0.17 52.9520.126 -2.2240.010 b
592 59243 5788355123068307328 -43.3640.062 -7.6190.060 99.8080.327
58490 5224879500011370496 -41.0250.043 -6.1900.038 12.63 1.40 99.7460.231 0.6890.002
595 59865 6054216931644194304 -145.4070.046 6.3290.041 -9.93 0.16 68.9190.157
59913 6054219165008718848 -145.8290.047 7.7840.045 -9.15 0.19 69.0460.166 -0.9740.002
596 59898 5860789299912353024 -38.3310.083 -10.0730.077 105.9430.604
60183 5860587917472173056 -41.4340.100 -11.3220.084 97.3630.587 -0.1750.005
597 59960 6072902994276659200 -39.0140.057 -12.8720.046 103.3340.458
59716 6075548419252906368 -35.7220.050 -11.0410.042 14.22 1.23 114.8520.518 -0.0530.004
607 61135 3907637494456207360 -57.3440.128 3.7190.078 106.8250.700
61416 3931759748777206272 -57.2920.155 6.3640.073 17.46 0.19 107.1510.850 0.3390.006
610 61937 3927438771159865856 -106.5180.092 -0.1930.057 71.1600.247 b
62350 3927604694336986368 -112.5440.114 -0.4720.059 -17.49 0.14 65.4050.297 0.2670.003
620 65466 1446271417351852544 -10.8360.220 -7.5960.194 90.6500.841
65508 1448260262087566848 -10.3090.088 -8.6660.071 -0.35 0.66 86.7550.292 0.4530.006
624 65779 5863709946382545280 -127.9020.048 11.3680.056 -24.72 0.29 78.9090.228
65785 5863709946382541952 -127.3180.041 11.0560.050 -24.52 0.35 78.8380.207 -1.9890.008
626 65899 1442870284289795968 25.4000.076 -45.4360.034 -35.48 0.18 83.7560.263
65884 1441682158897145600 22.9570.075 -49.3660.032 58.12 0.30 87.0910.277 0.1790.003 d
630 66749 6293670601102816128 -36.5210.086 -11.3330.079 -8.19 0.46 90.5400.458
66717 6293764540627556224 -45.8440.116 -13.5940.112 -1.02 0.51 108.4161.021 -0.1030.006 d
633 67250 1496459690754708736 -134.2900.088 -22.4640.115 -11.20 0.13 92.4920.788
67041 1496708798857641600 -141.2620.026 -23.2930.038 -32.94 0.18 95.9460.296 -0.0220.005 d
635 67470 3728726646011331328 -147.3960.067 25.3520.059 56.2810.141 b
67673 3728039760481678592 -149.4960.077 25.6580.055 -2.80 0.48 58.2190.162 0.0810.002
636 67639 6094716308525121408 -114.0760.108 -27.2360.101 5.75 0.18 68.4840.284
67645 6094716239805635072 -113.7400.104 -26.7930.102 4.72 0.22 68.0850.278 -1.9960.013
642 68830 6291206045789315328 39.4250.094 -45.4240.076 15.18 12.84 88.3820.436
68833 6291205977069837696 37.2700.094 -44.5590.081 28.05 0.18 89.1250.420 -1.8710.008 d
655 71726 1282815063829295360 93.4100.064 -44.3630.086 -11.80 0.15 54.9830.136
71737 1282817022334383232 93.4700.073 -43.6780.095 -11.49 0.14 54.7240.156 -1.0910.002
677 76046 4414349489701257984 -79.8270.083 -30.3710.071 -9.52 0.26 76.0370.302
75923 4416093315142832128 -70.1500.072 -28.9830.060 -25.82 1.27 67.4630.203 0.0140.003 d
678 76133 4416110082695309184 -17.7650.214 -42.2170.205 -15.85 0.15 82.9440.887
77728 4403070149671286656 -15.4080.093 -43.9810.066 -3.50 0.26 88.6300.468 0.8680.007 d
691 78283 1202709349617637632 -59.4450.065 2.5200.058 -61.82 0.16 71.0790.306
78067 1204270110670555136 -56.6580.058 9.3150.061 -18.21 0.71 80.1840.321 0.4600.004 d
704 80886 1299204074916904320 -30.6830.041 31.9310.050 -20.02 0.23 74.9580.181
81875 1299508639637961344 -30.1220.032 30.2840.045 -40.40 0.29 99.9690.249 0.6550.002 d
707 81641 4434301983614947072 0.3530.175 -10.4760.134 101.2091.084
81634 4434301841878055296 0.3640.124 -9.8590.095 99.8320.640 -1.4690.008
713 84054 4571879578631697024 -11.7820.054 30.6550.056 80.8480.272
84070 4571879475552172544 -11.2300.037 29.7700.042 79.6670.190 -1.1190.003
715 84515 4388409330344765056 -60.2490.072 19.5680.063 -14.44 0.22 82.6610.311
85911 4485937214320449280 -54.9480.083 21.7690.071 -22.79 0.29 73.4600.234 0.8120.003 d
726 85940 4487578544660527488 -16.0920.099 -59.0110.087 -28.07 0.18 90.9780.397 b
85944 4487581018562340992 -13.1340.087 -60.2750.084 90.1330.337 -0.9970.004
737 88782 6719536193567989120 -32.6170.086 -106.9320.081 -83.59 0.12 71.0530.322
87887 5954345541675673728 -43.0370.096 -102.5480.085 19.85 0.25 84.9840.363 0.4190.004 d
740 89373 6721441784663730816 -0.8730.104 -53.6330.091 -44.16 0.33 103.3060.605
89274 6724728980838829952 -5.7290.101 -51.2860.087 109.8260.798 0.2230.006
743 90026 6649398690418063232 77.7170.075 1.5860.076 96.0830.534
90028 6649398690418059392 79.5340.073 1.7480.068 -5.14 0.16 96.9340.538 -1.8360.013
749 93029 2133995118527034496 43.6380.057 -5.6170.055 1.76 0.46 84.2360.185
92467 2143913950359678464 45.0000.042 -1.7730.051 -12.52 0.40 86.5320.173 0.5160.002 d
756 96979 4301163357576782464 23.9890.081 53.9120.053 -40.83 0.17 60.9530.220
96976 4301163323217046912 23.9640.074 52.3460.050 -41.00 0.16 60.9500.186 -2.0950.005
759 97646 6447036152303430528 23.3000.463 -12.4330.407 85.9472.503
97581 6447030139349235200 24.0310.114 -15.8990.082 91.4020.617 -0.7570.015
761 98174 6447100091479968256 23.2480.230 -14.7830.246 90.1491.382
97646 6447036152303430528 23.3000.463 -12.4330.407 85.9472.503 0.1250.019
768 99689 1808691203160200576 -7.2801.319 5.6311.216 12512
100451 1808808571723073152 1.5380.067 0.4970.060 328.7374.829 0.9080.033 c
770 99729 4249652990144051840 -132.5280.101 -58.4450.064 -0.18 0.20 63.0780.208
99727 4249652783985617920 -129.7290.096 -59.3880.065 -0.21 0.19 63.0930.203 -1.8780.004
776 100941 6679307846231740032 94.6620.082 -122.8580.057 104.9080.685
100937 6679308018030433536 93.7960.078 -122.5110.055 -62.19 0.23 105.0160.629 -1.7430.009
777 101082 2298101352139398144 66.2430.082 221.3810.105 -14.38 0.17 64.3640.211
101166 2298101901895214720 66.4940.058 221.0430.070 -14.24 0.27 64.0490.124 -1.1760.002
780 101719 2195226749280378368 23.8340.051 -42.6210.048 3.21 0.18 57.4930.082
102727 2194321816851242624 19.4280.053 -40.4210.050 -9.36 0.18 65.3320.122 0.1890.001 d
782 102725 6376446646806738816 42.0260.043 -61.5300.069 85.9880.270
103917 6375793502540188800 44.1040.053 -59.4480.055 -10.04 0.17 87.8080.446 0.3090.004
784 103107 6481251098731588224 43.5990.064 -30.4380.051 -16.94 0.28 89.0530.300
103139 6481246906845850880 43.8750.051 -27.4280.044 -17.98 0.43 89.9930.292 -0.8210.003
793 104536 1839746393680655232 22.2390.137 -11.8880.138 116.7591.188
102981 1838905954482116480 20.7050.081 -11.9910.072 -11.55 0.18 108.3780.582 0.9330.007
802 109306 6611824083124944384 13.2610.078 24.4470.081 78.6540.344
110433 6599619950733282304 11.6700.060 25.9500.066 86.4380.281 0.6490.003
815 112537 2396293134977302272 24.5760.071 -10.5830.065 -3.25 0.37 78.7260.286
111596 6628071944405827712 31.3190.072 -4.8370.062 11.06 1.60 106.2880.444 0.6880.003 d
822 113381 6602965661537192704 -8.7830.108 5.4110.104 5.81 0.22 83.8210.430
113374 6602965622882422144 -7.8700.080 6.8080.073 84.4390.289 -1.6160.006
823 113532 6606118786008302720 -7.2280.296 5.1180.214 94.5591.278
113381 6602965661537192704 -8.7830.108 5.4110.104 5.81 0.22 83.8210.430 0.7670.008
839 117454 2337899270721594752 -13.1770.093 -42.9660.111 -3.30 0.83 85.7140.686
117720 2338428513772452480 -15.3550.095 -39.4810.069 44.28 0.67 113.4580.771 0.3430.006 d
Table 1: Hipparcos and Gaia identification numbers for the 133 binary stars used, together with Gaia DR2 data and results of the exclusion tests performed: (a) shows an individual star excluded (which results in the exclusion of the respective binary) for having an inconsistent proper motion in right ascension between Hipparcos and the DR2, (b) the corresponding result for declination, (c) shows a binary excluded for having discordant parallaxes to both components (distance differences larger than 13 %) in the DR2 data, and (d) gives binaries removed for having DR2 radial velocity differences , with the 83 retained binaries appearing blank in this column. Upper rows give primaries and lower ones secondaries.
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This is a comment super asjknd jkasnjk adsnkj
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