OB Stars and Cepheids From the Gaia TGAS Catalogue: Test of their Distances and Proper Motions

We consider young distant stars from the Gaia TGAS catalog. These are 250 classical Cepheids and 244 OB stars located at distances up to 4 kpc from the Sun. These stars are used to determine the Galactic rotation parameters using both trigonometric parallaxes and proper motions of the TGAS stars. In this case the considered stars have relative parallax errors less than 200%. Following the well-known statistical approach, we assume that the kinematic parameters found from the line-of-sight velocities $V_r$ are less dependent on errors of distances than the found from the velocity components $V_l$. From values of the first derivative of the Galactic rotation angular velocity $\Omega{'}_0$, found from the analysis of velocities $V_r$ and $V_l$ separately, the scale factor of distances is determined. We found that from the sample of Cepheids the scale of distances of the TGAS should be reduced by 3%, and from the sample of OB stars, on the contrary, the scale should be increased by 9%.


Introduction
The Gaia TGAS (the Tycho-Gaia astrometric solution, Prusti et al. 2016;Lindegren et al. 2016) catalogue was produced from a combination of data in the first year of Gaia observations with Hipparcos/Tycho (1997) stellar positions.The mean parallax errors are ∼ 0.3 mas.This means that a solar neighborhood with a radius of ∼300 pc can be covered with distance errors of about 10%.Therefore, when studying the structure and kinematics of the Galaxy at great heliocentric distances (3 kpc or more), the approach using the photometric or other distance scales is currently topical.For most TGAS stars the mean proper motion error is ∼ 1 mas yr −1 , but for quite a few Hipparcos stars this error is an order of magnitude smaller, ∼ 0.06 mas yr −1 (Brown et al. 2016).Therefore, analyzing their space velocities using highly accurate proper motions is of great interest.
Comparison of distances to Cepheids and RR Lyra variables from the TGAS catalog with the distances obtained by other methods, showed very good agreement up to distances 2 kpc (Casertano et al. 2017, Benedict et al. 2017;Clementini et al. 2017;Bobylev & Bajkova, 2017).But there are some reports about a zero-point offset of parallaxes of the TGAS stars.In particular, Stassun & Torres (2016) discovered the offset equal to −0.25 ± 0.05 mas with respect to 158 calibration eclipsing binary stars.From the analysis of the nearest stars to the Sun (<25 pc) Jao et al. (2017) found that, on average, the parallaxes of the Gaia TGAS stars are less by 0.24 ± 0.02 mas than trigonometric parallaxes of these stars measured by ground-based telescopes.
Since the properties of the distance scale of the TGAS catalog are not yet studied completely, the analysis of it with the use of independent approaches is actual.The purpose of this paper is to determine the Galactic rotation parameters using high-precision proper motions of the TGAS stars, a joint and separate solution of the main kinematic equations, both in terms of proper motions and line-of-sight velocities in order to investigate the distance scale of the TGAS catalog.

Method
From observations we know three components of the star velocity: the line-of-sight velocity V r and the two projections of the tangential velocity V l = 4.74rµ l cos b and V b = 4.74rµ b , directed along the Galactic longitude l and latitude b respectively and expressed in km s −1 .Here, the coefficient 4.74 is the ratio of the number of kilometers in astronomical unit to the number of seconds in a tropical year, and r is a heliocentric distance of the star in kpc.The components of a proper motion of µ l cos b and µ b are expressed in the mas yr −1 .
To determine the parameters of the Galactic rotation curve, we use the equations derived from Bottlinger's formulas in which the angular velocity Ω was expanded in a series to terms of the second order of smallness in r/R 0 : (1) where R is the distance from the star to the Galactic rotation axis, Ω 0 is the angular velocity of Galactic rotation at the solar distance R 0 , the parameters Ω ′ 0 and Ω ′′ are the corresponding derivatives of the angular velocity, and The Oort constants A and B can be found from the following expressions: written so that the following relations are hold: A − B = Ω 0 and A + B = −(Ω 0 + Ω ′ 0 R 0 ).In this paper, it is customary the value of R 0 = 8.0 ± 0.2 kpc, which Vallée (2017) in its recent survey found as the most probable.
There exists also the third Bottlinger's equation with the V b velocity on the left-hand side.The results of the analysis of Cepheids and OB stars on the basis of a system of three conditional equations with V r , V l , V b left-hand sides are described in the papers of Bobylev (2017) and Bobylev & Bajkova (2017).However, for distant young stars with the latitudes close to zero (b ≈ 0 • , ⇒ sin b ≈ 0) using the equation with V b left-hand side is ineffective.Therefore, in this paper we either solve a system of two equations ( 1)-( 2), or separately each of them.It is known (Zabolotskikh et al. 2002), that in such an approach the velocity W ⊙ can not be reliably determined only from equation ( 1), so we fix its value as The values Ω ′ 0 , obtained with separate solutions, are of great interest for controlling the distance scale.This method is based on the fact that the errors of the lineof-sight velocities are independent of the distance errors, but the errors of the tangential components of the proper motion depend on the distance errors.This is the basis for statistical methods for analyzing the distance scale.For

Data
In the present work we use the following two samples of OB stars: 1) the sample of spectroscopic binary OB stars (Bobylev & Bajkova 2015), 2) the sample of OB stars with distances determined by the spectral lines of the interstellar Ca II (Bobylev & Bajkova 2011), and 3)the sample of classical Cepheids described in Mel'nik et al. (2015).The latter consists of 290 classical Cepheids with distances, line-of-sight velocities and proper motions from the Gaia TGAS catalog.
The distributions on the Galactic plane XY of stars of three considered samples are given in figure 1.As it can be seen from this figure, all the stars are distant on no more than 5 kpc from the Sun.OB stars are very young, therefore they well trace a spiral pattern.It is also seen that among the older Cepheids, there are a sufficient number of stars well tracing the Carina-Sagittarius arm (R ≈ 7 kpc).

Cepheids
In table 1 the parameters of Galactic rotation obtained from Cepheids with the TGAS proper motions and parallaxes are given.We used the Cepheids with relative errors of trigonometric parallaxes σ π /π < 200%.In the table the error of unit weight σ 0 , obtained by solving conditional equations ( 1)-( 2) using a well-known least squares method is given.This error is close to the dispersion of residual velocities of the analyzed sample stars averaged over all directions.The Oort constants A and B calculated using (4) are also given in the table.According to this table, we find the coefficient of the distance scale p = (−4.03)/(−4.16)= 0.97 ± 0.07.The error of p was calculated as follows where Ω ′ 0t is Ω ′ 0(l,b) .Such an estimate from Cepheids was obtained, apparently, for the first time.
It is possible to estimate the distance at which the accuracy of the tangential V t velocity is equal to the accuracy of the line-of-sight velocity V r .Using the upper bound of the accuracy estimate of the tangential component, from the equality σ Vt = 4.74r σ 2 µα cos δ + σ 2 µ δ we find the value of the critical distance, near which the tangential velocities are more accurate than the line-ofsight ones.Based on the fact that σ Vr = 3 km s −1 and σ µα cos δ = σ µ δ = 1 mas yr −1 for TGAS stars we obtain r = 0.45 kpc.Therefore, it is most correct to apply our method as follows: for distances greater than the critical value, it is necessary to consider the kinematic parameters obtained from the line-of-sight velocities of the stars to be more accurate, and at distances less than the critical one, it is necessary to consider the kinematic parameters obtained from the proper motions of the stars to be more accurate.
It is important to note that in our sample average distance r = 2 kpc.Only 9 Cepheids are located closer than 0.45 kpc from the Sun.Their exclusion from the sample does not affect an estimate of the coefficient p.
In paper by Mel'nik et al. (2015) were found the following parameters of Galactic rotation from 274 Cepheids with line-of-sight velocities taken from various sources and proper motions taken from the HIPPARCOS catalog (ESA 1997;van Leeuwen 2007) 5) are in good agreement with the latter values.Number of used Cepheids and equations are approximately the same in both cases, but thanks to the use of proper motions from the TGAS catalog, the kinematic parameters (5) are determined with smaller errors.

OB Stars
In table 2 the parameters of Galactic rotation found from OB stars with proper motions and parallaxes from the TGAS catalog are given.We used the stars with relative errors of trigonometric parallaxes σ π /π < 200%.According to this table, we can find the distance scale factor p = (−4.68)/(−4.29)= 1.09 ± 0.07, those the TGAS distances should be increased on 9%.In this sample average distance r = 1.9 kpc and only 20 stars are located closer than 0.45 kpc from the Sun.Their exclusion from the sample does not affect an estimate of the coefficient p.
Note that Bobylev & Bajkova (2017) found the value p = 1.04 from kinematic analysis of OB stars with proper motions and parallaxes from the TGAS catalog.

CONCLUSIONS
Galactic rotation parameters are determined using three samples of young stars with different distance scales.These samples were studied previously using data from various catalogs.The first sample contains massive spectral-double OB stars with photometric distances, the second one consists of OB stars, which distances are determined along the lines of interstellar Calcium, the third sample consists of classical Cepheids, the distances to which are determined using the period-luminosity relation.In this paper, we use those stars from these samples that are included in the Gaia TGAS catalog.
According to the samples of Cepheids and OB stars, kinematic parameters were determined using also trigonometric parallaxes.In this case we used the stars with relative errors of parallaxes less than 200%.From the comparison of values Ω ′ 0 , found as a result of solutions of equations for V r and V l separately, the distance scale coefficient p has been determined.Here, we start from the assumption that the kinematic parameters, found from line-of-sight velocities, are less dependent on distance errors than those found from proper motions.According to Cepheids, we found that the distance scale of the TGAS catalog should be reduced by 3% (p = 0.97 ± 0.07), and for OB stars, on the contrary, it should be increased by 9% (p = 1.09 ± 0.07).

Acknowledgment
We are grateful to the referee for the helpful remarks that contributed to an improvement of this paper.This study was supported by the "Transient and explosive processes in astrophysics" Program of the Presidium of Russian Academy of Sciences (P-7).

Table 1 .
Parameters of Galactic rotation, found from Cepheids with the TGAS proper motions and trigonometric parallaxes

Table 2 .
Parameters of Galactic rotation, found from OB stars with the TGAS proper motions and trigonometric parallaxes