Parameters of the Link between the Optical and Radio Frames from Gaia DR2 Data and VLBI Measurements
aa r X i v : . [ a s t r o - ph . GA ] D ec Astronomy Letters, 2019, Vol. 45, No 1, pp. 10–19.
Parameters of the Link between the Optical and Radio Framesfrom Gaia DR2 Data and VLBI Measurements
V.V. Bobylev Pulkovo Astronomical Observatory, Russian Academy of Sciences,Pulkovskoe sh. 65, St. Petersburg, 196140 Russia
Abstract —Based on published data, we have assembled a sample of 88 radio stars forwhich there are both trigonometric parallax and proper motion measurements in the GaiaDR2 catalogue and VLBI measurements. A new estimate of the systematic offset between theoptical and radio frames has been obtained by analyzing the GaiaDR2–VLBI trigonometricparallax differences: ∆ π = − . ± .
046 mas (with a dispersion of 0.156 mas). This meansthat the Gaia DR2 parallaxes should be increased by this correction. The parallax scalefactor is shown to be always very close to unity within ∼ b = 1 . ± . . Our analysis of the proper motion differences for the radio stars based on the model of solid-body mutual rotation has revealed no rotation components differing significantly from zero:( ω x , ω y , ω z ) = ( − . , . , − . ± (0 . , . , .
16) mas yr − . INTRODUCTION
Highly accurate stellar parallaxes are required to solve many stellar-astronomy problems.The trigonometric parallaxes are among the most reliable ones. However, it is necessary tocheck and eliminate the possible systematic offsets before using even the most reliable data.The first data release of the Gaia space experiment was published in September 2016(Prusti et al. 2016; Brown et al. 2016). The second data release of this experiment, GaiaDR2, appeared in April 2018 (Brown et al. 2018). This catalogue contains the trigonometricparallaxes and proper motions of ∼ G < m ) the parallax errors lie within the range0.04–0.02 milliarcseconds (mas), while for faint stars ( G = 20 m ) they are about 0.7 mas.Lindegren et al. (2018) pointed out the presence of a possible systematic offset ∆ π = − .
029 mas in the Gaia DR2 parallaxes with respect to the inertial reference frame. Atpresent, there are several reliable distance scales a comparison with which allows, in theopinion of their authors, the systematics of the Gaia trigonometric parallaxes to be checked.Stassun and Torres (2016) found quite a significant mean offset ∆ π = − . ± .
05 mas ofthe Gaia DR1 trigonometric parallaxes with respect to the parallaxes of a calibration sampleof eclipsing binaries. This result was soon confirmed by other authors based on an analysisof classical Cepheids close to the Sun (Casertano et al. 2017), the ground-based parallaxesof the nearest M dwarfs (Jao et al. 2016), and asteroseismology (Huber et al. 2017). e-mail: [email protected] π = − . ± .
033 mas. This value is also confirmed by other authors, in particular, whenanalyzing Cepheids (Riess et al. 2018) and asteroseismology (Zinn et al. 2018).Therefore, the distances to radio stars determined by very long baseline interferometry(VLBI) are of interest. Here we have in mind the absolute parallaxes that are absolutizedduring the observations using quasars. At present, the VLBI observations aimed at de-termining highly accurate trigonometric parallaxes and proper motions of radio sources, inparticular, galactic masers, are being performed by several research teams.The accuracy of astrometric VLBI measurements depends on many factors. For example,estimates of the contributions from the position of a calibration source, the Earth’s orienta-tion, the antenna position, and the tropospheric delay for radio sources located at differentdeclinations can be found in Pradel et al. (2006). Furthermore, the mean VLBI parallaxerror depends on the observation frequency: the higher the frequency, the smaller this error.As a result, the mean VLBI parallax error in observations at 22.2 GHz is ∼ DATA
In this paper we collected the VLBI observations of stellar trigonometric parallaxes andproper motions performed and published by various research teams. For example, theseinclude the Japanese VERA (VLBI Exploration of Radio Astrometry) project devoted tothe observations of H O masers at 22.2 GHz (Hirota et al. 2008) and a number of SiOmasers at 43 GHz (Kim et al. 2008). Methanol (CH OH) and H O masers are observed inthe USA on the VLBA (Reid et al. 2009). Similar observations are also performed withinthe framework of the European VLBI network (Rygl et al. 2010). The VLBI observationsof radio stars in continuum at 8.4 GHz are also carried out with the same goals (Torres etal. 2012).Table 1 gives the proper motion and trigonometric parallax differences for 88 stars. Thestars have a different evolutionary status. Some of them are very young stars with maseremission (H O and CH OH masers). Asymptotic giant branch stars observed as OH, H O,and SiO masers constitute the other part of the sample. Quite a few sources were observedin continuum. This is true for such objects as pulsars, Wolf–Rayet stars, systems with blackholes, and a number of T Tauri stars. 2able 1:
Gaia DR2–VLBI stellar proper motion and parallax differencesStar Type or ∆ µ α cos δ, σ ∆ µ α cos δ , ∆ µ δ , σ ∆ µ δ , ∆ π, σ ∆ π , Refspectrum mas yr − mas yr − mas yr − mas yr − mas masSY Scl Mira .
541 .328 − .
155 .314 − .
075 .229 (1)S Per RSG .
480 .458 − .
380 .451 − .
191 .123 (2)HII 174 RS CVn .
020 .122 − .
105 .172 − .
111 .057 (3)HII 625 BY Dra .
409 .134 − .
121 .275 − .
008 .070 (3)HII 1136 RS CVn − .
800 .098 .
346 .246 − .
161 .057 (3)HII 2147 RS CVn − .
579 .112 .
879 .171 − .
119 .062 (3)V773 Tau T Tau − .
321 .929 − .
935 .391 .
113 .164 (4)HIP 20097 T Tau − .
020 .129 − .
115 .064 − .
084 .059 (5)HDE 283572 T Tau .
158 .150 .
106 .134 − .
107 .065 (5)T Tau N T Tau − .
994 .128 − .
037 .112 .
109 .066 (6)V1201 Tau T Tau − .
370 .115 − .
309 .096 − .
197 .083 (5)V807 Tau T Tau .
986 1.237 8 .
544 1.009 .
935 .667 (5)V1110 Tau RS CVn .
438 .112 − .
021 .096 − .
281 .154 (5)HIP 26233 B2/3V − .
421 .404 1 .
311 .409 − .
196 .187 (8)LSI +61 303 BH − .
146 .041 .
185 .067 — — (7)DG Tau T Tau − .
644 .876 − .
197 .932 — — (9)HD 118216 F2+K2 − .
013 .202 − .
031 .164 — — (10)WR 112 WR .
625 1.164 1 .
596 1.436 — — (11)WR 125 WR − .
964 .503 .
606 .604 — — (11)WR 140 WR .
377 .206 − .
847 .115 — — (11)WR 146 WR 2 .
284 .696 − .
375 2.242 — — (11)WR 147 WR − .
097 .803 − .
100 1.198 — — (11)PSR J0437–47 Pulsar 1 .
185 1.198 .
654 1.672 1 .
929 .679 (12)V999 Tau T Tau − .
040 .677 − .
306 .433 1 .
166 .438 (5)HD 282630 T Tau .
410 .191 .
078 .152 − .
798 .148 (5)T Lep Mira − .
887 .555 1 .
108 .739 − .
101 .193 (13)V1699 Ori YSO − .
209 .428 − .
144 .399 .
062 .267 (8)GMR G YSO − .
048 .139 .
745 .188 − .
242 .067 (8)GMR F YSO − .
250 .128 .
231 .164 − .
048 .074 (8)Parenago 1469 YSO .
026 .107 − .
047 .120 .
013 .049 (8)Parenago 1540 PMS .
162 .128 .
212 .109 − .
096 .063 (8)Parenago 1724 YSO .
199 .209 .
153 .170 − .
078 .057 (8)Parenago 1778 YSO .
332 .499 .
284 .728 − .
116 .312 (8)Parenago 1955 YSO − .
249 .694 − .
038 1.053 − .
594 .215 (8)Parenago 2148 YSO 2 .
276 .347 .
909 .530 .
606 .429 (8)V621 Ori YSO .
475 .463 − .
335 .293 .
269 .115 (8)HIP 26220 HAe/Be − .
274 .187 2 .
653 .184 − .
253 .145 (8)HIP 26314 B3III .
392 .142 .
366 .160 .
170 .076 (8)RW Lep Mira 1 .
139 .634 − .
792 .724 .
735 .209 (14)HD 294300 T Tau 7 .
695 .682 − .
858 1.376 − .
514 .373 (8) able 1. Contd.Star Type or ∆ µ α cos δ, σ ∆ µ α cos δ , ∆ µ δ , σ ∆ µ δ , ∆ π, σ ∆ π , Refspectrum mas yr − mas yr − mas yr − mas yr − mas masTYC 5346-538-1 B8.1 .
147 .171 .
188 .290 .
045 .091 (8)HD 290862 B3/5 − .
607 .291 − .
482 .836 − .
020 .549 (8)U Lyn Mira − .
257 .607 − .
297 .602 − .
690 .232 (15)R UMa Mira 1 .
436 .551 .
691 .517 .
075 .208 (16)RT Vir M8III — — — — − .
367 .320 (17)FV Boo Mira — — — — − .
397 .191 (18)S Crt M6III − .
869 .327 .
460 .268 .
316 .195 (19)R Cas Mira 1 .
400 2.384 1 .
660 1.786 − .
328 1.965 (20)RX Boo M7.5/8 − .
572 1.178 1 .
809 2.432 .
519 .583 (21)S CrB Mira − .
671 .526 1 .
172 .467 − .
038 .366 (22)U Her Mira − .
261 .360 − .
911 .392 − .
991 .628 (22)WLY 2–11 T Tau 2 .
725 .361 − .
388 .301 .
231 .181 (23)YLW 24 T Tau − .
083 .213 .
268 .148 − .
143 .166 (23)DoAr21 T Tau − .
554 .269 .
155 .176 .
061 .243 (23)rho Oph S1 T Tau − .
120 .254 3 .
163 .167 .
917 .145 (23)VSSG11 T Tau .
739 1.118 14 .
217 .776 − .
523 .521 (23)DROXO 71 PMS − .
799 .640 1 .
376 .525 − .
812 .312 (23)SFAM 87 T Tau 1 .
143 .142 − .
653 .111 .
345 .115 (23)GWAYL 5 T Tau − .
188 .568 .
532 .428 − .
669 .342 (23)DoAr51 T Tau − .
396 1.071 1 .
572 .726 .
265 .387 (23)VX Sgr RSG 2 .
091 .883 3 .
691 .875 .
147 .232 (24)[GFM2007] 11 YSO − .
654 .139 .
862 .170 − .
072 .109 (25)[GFM2007] 65 YSO 3 .
397 1.873 2 .
574 2.086 − .
780 .852 (25)W 40 IRS 5 B1 .
360 .404 − .
487 .360 − .
249 .221 (25)W 40 IRS 1c YSO − .
102 .892 2 .
848 .752 .
840 .476 (25)[KGF2010] 133 YSO − .
177 .472 − .
881 .520 − .
379 .245 (25)PN K 3–35 PN .
545 .157 2 .
459 .194 .
123 .131 (26)RR Aql Mira 3 .
713 .883 1 .
077 .614 1 .
566 .499 (22)Cyg X–1 BH − .
102 .077 .
229 .132 − .
117 .046 (27)IRAS 20126+4104 YSO − .
853 .790 − .
558 .861 .
275 .369 (28)IRAS 20143+3634 YSO − .
123 .193 1 .
447 .454 − .
047 .080 (29)V404 Cyg BH − .
729 .176 − .
205 .176 .
021 .103 (30)HIP 101341 O6.5+ − .
443 .985 3 .
075 1.282 .
028 .227 (31)NML Cyg RSG 1 .
282 1.260 3 .
727 1.310 .
906 .570 (32)UX Cyg Mira 3 .
381 .810 .
254 1.621 − .
364 .178 (33)SS Cyg Df Nova − .
047 .133 .
209 .117 − .
076 .130 (34)IRAS 22480+6002 RSG − .
075 .354 − .
250 .212 .
079 .082 (35)IM Peg RS CVn .
111 .164 .
419 .159 − .
320 .114 (36)R Aqr M6.5e − .
800 .632 − .
239 .593 − .
578 .847 (37)PZ Cas RSG .
590 .232 .
192 .320 .
064 .085 (38) able 1. end.Star Type or ∆ µ α cos δ, σ ∆ µ α cos δ , ∆ µ δ , σ ∆ µ δ , ∆ π, σ ∆ π , Refspectrum mas yr − mas yr − mas yr − mas yr − mas masUX Ari RS CVn 5 .
089 .525 2 .
111 .411 .
443 .452 (39)HR 1099 RS CVn − .
304 .355 − .
082 .332 − .
127 .478 (39)HIP 79607 RS CVn − .
275 .104 − .
265 .154 .
205 .119 (39)HD 199178 G5III − .
277 .415 .
498 .435 .
312 .332 (39)AR Lac RS CVn − .
110 .137 .
160 .195 − .
537 .371 (39)AM Her polar .
063 .223 − .
784 .183 .
105 .082 (40)W Hya Mira − .
533 2.418 − .
408 3.234 − .
089 2.497 (20)VY CMa RSG 3 .
726 1.865 − .
074 1.847 − .
772 .827 (41)Mira—Mira Ceti variable; RSG—red supergiant; RS CVn—Canes Venatici variable; BY Dra—BYDraconis variable; T Tau— T Tauri variable; PMS—pre-main-sequence star; HAe/Be—HerbigAe/Be star; YSO—young stellar object; PN—planetary nebula; Df Nova—dwarf nova; BH—oneof the binary components is a black hole; WR—Wolf–Rayet star.(1) Nyu et al. (2011); (2) Asaki et al. (2010); (3) Melis et al. (2014); (4) Torres et al. (2012);(5) Galli et al. (2018); (6) Loinard et al. (2007); (7) Dhawan et al. (2006); (8) Kounkel et al.(2017); (9) Rivera et al. (2015); (10) Abbuhl et al. (2015); (11) Dzib and Rodriguez (2009); (12)Deller et al. (2008); (13) Nakagawa et al. (2014); (14) Kamezaki et al. (2014); (15) Kamezaki etal. (2016a); (16) Nakagawa et al. (2016); (17) Zhang et al. (2017); (18) Kamezaki et al. (2016b);(19) Nakagawa et al. (2008); (20) Vlemmings et al. (2003); (21) Kamezaki et al. (2012); (22)Vlemmings, and van Langevelde (2007); (23) Ortiz-Leon et al. (2017a); (24) Xu et al. (2018); (25)Ortiz-Leon et al. (2017b); (26) Tafoya et al. (2011); (27) Reid et al. (2011); (28) Xu et al. (2013);(29) Burns et al. (2014); (30) Miller-Jones et al. (2009); (31) Dzib et al. (2013); (32) Zhang et al.(2012a); (33) Kurayama et al. (2005); (34) Miller-Jones et al. (2013); (35) Imai et al. (2012); (36)Ratner et al. (2012); (37) Min et al. (2014); (38) Kusuno et al. (2013); (39) Lestrade et al. (1999);(40) Gawro´nski et al. (2018); (41) Zhang et al. (2012b).
5n our previous paper (Bobylev 2010) we used 23 radio stars from this list to study thetie-in of the Hipparcos catalogue (1997) to the inertial reference frame. In this paper the listwas expanded significantly both through an increase in the number of VLBI observationsand owing to a great density of the Gaia DR2 catalogue.The first column in the table gives the names of the radio stars using which they are easilyfound in the SIMBAD electronic search system. The second column lists the types of thestars or their spectral types. The next columns provide the proper motion and trigonometricparallax differences. The dispersions of the differences are given for each type of differences.For example, for the proper motions the formula to calculate the dispersions of the differencesis as follows: σ ∆ µ = q σ µ Gaia + σ µ V LBI , (1)for the dispersions of the parallax differences the expression is similar in form after anappropriate substitution.There are no parallax differences for eight stars in Table 1. This is either due to negativeparallaxes in the Gaia DR2 catalogue or the absence (for example, for Wolf–Rayet stars)of VLBI measurements. At the same time, we used these eight stars to analyze the propermotion differences. For two stars, RT Vir and FV Boo, there are data only on their VLBIparallax measurements.It can be seen from Table 1 that several stars have differences that differ significantlyfrom the expected zero. For example, these include the stars R Aqr with ∆ µ α cos δ = − . ± .
632 mas yr − , VSSG11 with ∆ µ δ = 14 . ± .
776 mas yr − , or VY CMa with∆ π = − . ± .
827 mas. Note that the presence of a long tail in the distribution of radiosource position differences was established by Petrov and Kovalev (2017) when analyzing alarge sample of quasars from the Gaia catalogue with VLBI measurements.When the optical and radio images of stars are compared, the size and pattern of theradio-emitting region can play an important role. The supergiant S Per can serve as anexample of a “good”, symmetric radio image. As can be seen from Fig. 5 in Asaki et al.(2010), more than 40maser spots are distributed quite uniformly in a region with a radiusof about 50 mas, while, according to Fig. 10 in the cited paper, the residual velocity vectorsexcellently pinpoint the position of the image center. As can be seen from our table, alldifferences for the star S Per are close to zero.On the other hand, the radio emission can be associated with the jets or vast diskstructures surrounding the radio star. In that case, the probability of the appearance of asignificant offset when comparing the optical and radio images of a star is great.Finally, the optical image of a radio star can also be asymmetric. The well-known starVY CMa can serve as such an example. This is a red supergiant; the star has a record size.It is actually a presupernova and is surrounded by a nebula with a highly asymmetric shape.All of this necessitates using constraints on the differences being investigated when solvingour problems. Such constraints were selected through several iterations to eliminate thelargest discrepancies. 6igure 1: Gaia–VLBI stellar proper motion differences.
RESULTS
Comparison of the Proper Motions
We use the following coupling equations to determine the three angular velocities of mutualrotation of the two frames around the equatorial coordinate axes ω x , ω y , ω z :∆ µ α cos δ = − ω x cos α sin δ − ω y sin α sin δ + ω z cos δ, ∆ µ δ = ω x sin α − ω y cos α, (2)where the Gaia–VLBI differences are on the left-hand sides of the equations. We use thestellar proper motion differences whose absolute values do not exceed 6 mas yr − . There area total of 81 such differences; their distribution is given in Fig. 1.As can be seen from the table, the data are unequally accurate. Therefore, we solvethe system of conditional equations (2) both with unit weights ( p = 1) and with weightsinversely proportional to the measurement errors p = 1 / q σ µ Gaia + σ µ V LBI , (3)where the dispersions σ ∆ listed in the corresponding columns of the table are in the denom-inator (see Eq. (1)).Having solved the system of 162 conditional equations (2) by the least-squares methodwith unit weights, we obtained the rotation components ω x = − . ± .
20 mas yr − ,ω y = − . ± .
29 mas yr − ,ω z = − . ± .
21 mas yr − . (4)At the same time, with weights (3) we obtained the rotation components ω x = − . ± .
15 mas yr − ,ω y = +0 . ± .
22 mas yr − ,ω z = − . ± .
16 mas yr − , (5)7igure 2: The histogram of Gaia–VLBI parallax differences constructed from all differences:a Gaussian with an expectation value of − .
30 mas and a dispersion of 0.40 mas (a) and aGaussian with the constraint on the difference σ ∆ π < .
25 mas (here it has an expectationvalue of − .
35 mas and a dispersion of 0.18 mas) (b) are shown.where ω x decreased greatly compared to the solution (4); the errors in the parameters beingdetermined also decreased. Comparison of the Parallaxes
To compare the parallaxes, we use 75 stars selected in such a way that the relative parallaxerrors from the Gaia DR2 catalogue and the VLBI parallax errors do not exceed 50%.First, we found the mean ∆ π = − . ± .
073 (0 . qP ( x − x ) /n ( n − , is given, and the dispersion σ = P ( x − x ) /n (herethe square of the rms deviation) is given in parentheses. Then, we calculated the weightedmean with weights (3) ∆ π = − . ± .
046 (0 . , (6)where the error of the weighted mean is given and the corresponding dispersion is given inparentheses. We see that the errors and dispersions differ greatly. This effect is explained bythe fact that we used significantly inhomogeneous data. Very broad distribution wings canbe seen already from the distribution of stellar proper motion differences (Fig. 1), namely(a) a central clump that can be described by a Gaussian with a small dispersion and (b)broad wings that can be described by a Gaussian with a considerably larger dispersion.The effect is more pronounced in the distribution of stellar parallax differences. Thehistogram of differences for 75 stars is presented in Fig. 2a. This figure shows a Gaussianwith an expectation value of − .
30 mas and a dispersion of 0 .
40 mas that poorly describesthe distribution. Two Gaussians with significantly differing dispersions would be bettersuited for the description of this distribution. However, we did otherwise. To construct thehistogram in Fig. 2b, we used 49 stars that were selected under constraints on the error inthe differences (see (1) and the table): σ ∆ π < .
25 mas. The parameters of the Gaussianfound (an expectation value of − .
35 mas and a dispersion of 0.18 mas) are now in excellentagreement with the result (6). On this basis we conclude that the application of weights8igure 3: Parallaxes of the radio stars from the Gaia DR2 catalogue versus their parallaxesmeasured by VLBI; the solid and dotted lines correspond to a correlation with a coefficientof 1 and the solution (8), respectively.(3) gives a result consistent with the available data; this approach allows the entire set ofavailable data to be used.To determine the scale factor b, we set up a system of conditional linear equations π Gaia = a + b · π V LBI , (7)from the solution of which we can estimate two parameters, a and b. As above, we use 75stars with relative parallax errors less than 50%. Solving the system of conditional equations(7) by the least-squares method with weights (3) yields the following result: a = − . ± .
059 mas ,b = +1 . ± . . (8)In Fig. 3 the parallaxes of the radio stars from the Gaia DR2 catalogue are plotted againsttheir VLBI parallaxes. The scales are clearly seen to be virtually identical within about3 kpc of the Sun, and only at greater distances does the Gaia DR2 parallax scale becomelonger than the VLBI parallax one. DISCUSSION
Liu et al. (2017) studied the frame of the Gaia DR1 catalogue (Brown et al. 2016). In partic-ular, the TGAS (Tycho-Gaia Astrometric Solution) version was compared with the Tycho2catalogue (Høg et al. 2000) and the version of the Hipparcos catalogue (1997) improvedby van Leeuwen (2007) using the model of solid-body rotation (2). These authors foundthe rotation vector components ( ω x , ω y , ω z ) = (0 . , . , − . ± (0 . , . , . − from the Hipparcos-TGAS proper motion differences for ∼
87 000 stars and( ω x , ω y , ω z ) = (0 . , . , . ± (0 . , . , . − from the Tycho2-TGAS9roper motion differences for ∼ ∼
23 000 K–M giants from the TGAS catalogue and found nonzero componentspointing to a possible residual rotation in the Gaia DR1 frame or the presence of problemsin the kinematic model. The rotation components were found to be ω Y G = − . ± . − and ω ′ Y G = − . ± .
19 mas yr − , which are interpreted as an additional rotationaround the Galactic Y axis.Note the paper by Fedorov et al. (2017), where it was found from a comparison ofthe stellar proper motions from the Gaia DR1 catalogue with a number of ground-basedcatalogues based on the model (2) that the component ω y changes dramatically from +0 . − . − with magnitude. In our case (5) this component is small, ω y = 0 . ± . − .It has been shown by Lindegren et al. (2018) that the optical reference frame defined byGaia DR2 is aligned with ICRS and is non-rotating with respect to the quasars to within0.15 mas yr − . Since a large number of stars were used, the random errors of rotationalparameters are small, less than 10%. The dependence of ω x , ω y , ω z on magnitude is clearlyseen from Fig. 4 of cited publication. For example, for G ≈ m , which is typical for thesample of stars considered in this paper, we will have ( ω x , ω y , ω z ) ≈ (0 . , − . , − .
15) masyr − . We see good agreement of these values with our estimates (5).As has already been noted in the Introduction, from a comparison with the Gaia DR2data for 89 detached eclipsing binaries Stassun and Torres (2018) found a correction ∆ π = − . ± .
033 mas. Here the dispersion of the Gaussian 0.033 mas should be comparedwith our value of 0.156 mas in the solution (6). These stars are interesting in that theywere selected from published data using very rigorous criteria imposed on the photometriccharacteristics. As a result, the relative errors in the stellar radii, effective temperatures, andbolometric luminosities (from which the distances are estimated) do not exceed 3%. Thespectral types of the stars in this sample lie in a wide range, from late O to M; most of thestars belong to the main sequence and there are also a few giants. According to Stassun andTorres (2016), the relative parallax errors for eclipsing binaries, on average, do not exceed5% and do not depend on the distance.Riess et al. (2018) estimated ∆ π = − . ± .
013 mas based on a sample of 50 long-period Cepheids by comparing their parallaxes with those from the Gaia DR2 catalogue.They used the photometric characteristics of these Cepheids measured onboard the HubbleSpace Telescope. Interestingly, relative to the highly accurate calibration scale of Riess etal. (2016), in which the relative Cepheid distance errors are 1–2%, these authors determinedthe scale factor b = 1 . ± .
033 that differs little from that found by us in the solution (8).One might expect that the stellar parallaxes from the Gaia DR1 and DR2 cataloguesdo not greatly differ systematically. For example, based on a kinematic analysis of starsfrom the Gaia TGAS catalogue, Bobylev and Bajkova (2018) concluded that the distancesto them calculated from their trigonometric parallaxes do not require using any additionalcorrection factor. This conclusion is also confirmed by our study with regard to the stellarparallaxes from the Gaia DR2 catalogue.Zinn et al. (2018) found ∆ π = − . ± .
002 mas by comparing the distances of ∼ π to be determined with a highaccuracy.Young stars from the Gould Belt, the distances to which have been measured by VLBI,constitute a significant fraction of our sample. Using data on 55 such stars (they are allpresented in our table as PMS, YSO, and T Tau), Kounkel et al. (2018) found the followingparameters based on relation (7): a = − . ± .
034 mas and b = +0 . ± . . Thevalue of these parameters are in excellent agreement with our estimates (8).
CONCLUSIONS
Based on published data, we produced a sample of 88 radio stars for which there are bothtrigonometric parallax measurements in the Gaia DR2 catalogue and VLBI measurements.A new estimate of the systematic offset between the optical and radio frames of theparallaxes, ∆ π = − . ± .
046 (0 . b, whose value differs from 1 by no more than 1%, is determined withconfidence. Such a situation is observed within 3 kpc of the Sun, and only at greater distancesis the Gaia DR2 parallax scale slightly extended compared to the VLBI parallax one.Based on the model of solid-body mutual rotation, we determined the rotation vectorcomponents in equatorial coordinates from the Gaia-VLBI proper motion differences forradio stars, ( ω x , ω y , ω z ) = ( − . , . , − . ± (0 . , . , .
16) mas yr − . ACKNOWLEDGMENTS
I am grateful to the referee for the useful remarks that contributed to an improvement of thepaper. This work was supported by Basic Research Program P–28 of the Presidium of theRussian Academy of Sciences, the subprogram “Cosmos: Studies of Fundamental Processesand their Interrelations”.
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