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Kinematics of the Milky Way from the Gaia EDR3 Red Giants and Sub-Giants

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Kinematics of the Milky Way from the Gaia EDR3 Red Giants and Sub-Giants MNRAS 000, 1–11 (2021) Preprint July 30, 2021 Compiled using MNRAS LATEX style file v3.0 Kinematics of the Milky Way from the Gaia EDR3 red giants and sub-giants. P. N. Fedorov,1★ V. S. Akhmetov,1 A. B. Velichko,1 A. M. Dmytrenko1 and S. I. Denischenko1 1Institute of astronomy of V. N. Karazin Kharkiv national university, Svobody sq. 4, 61022, Kharkiv, Ukraine Accepted XXX. Received YYY; in original form ZZZ ABSTRACT We present the results of the kinematic investigations carried out with the use of spatial velocities of red giants and sub-giants containing in the 080 EDR3 catalogue. The twelve kinematic parameters of the Ogorodnikov–Milne model have been derived for stellar systems with radii 0.5 and 1.0 kpc, located along the direction the Galactic center – the Sun – the Galactic anticenter within the range of Galactocentric distances ' 0–8–16 kpc. By combining someof thelocal parameterstheinformationrelatedto theGalaxy as a wholehasbeen received in the distance range 4–12 kpc, in particular the Galactic rotational curve, its slope, velocity gradients. We show that when using this approach, there is an alternative possibility to infer the behaviour of the Galactic rotational curve and its slope without using the Galactocentric distance '⊙. The kinematic parameters derived within the Solar vicinityof 1 kpc radiusare in good agreement with those given in literature. Key words: stars: kinematics and dynamics–Galaxy: kinematics and dynamics–solar neighborhood–methods: data analysis–proper motions 1 INTRODUCTION This approach is effective, as demonstrated in a number of pa- pers, for instance Vityazev & Shuksto (2004); Vityazev & Tsvetkov Investigation of stellar kinematics is an important instrument for (2005); Makarov & Murphy (2007); Mignard & Klioner (2012); the Galactic study. It is based on the analysis of stellar proper mo- Velichko et al. (2020). tions and radial velocities within various physical and mathemati- cal models. The Ogorodnikov–Milne (O–M, Ogorodnikov (1932); Milne (1935); Ogorodnikov (1965)) model is most commonly used for the stellar kinematics analysis. The model provides the ability to investigate the velocity field in a deformable stellar system. The 2 THE O–M MODEL EQUATIONS AND FORMING THE analysis of the stellar velocity field is usually confined to solving the LOCAL COORDINATE SYSTEMS basic kinematic equations to compute the O–M model parameters To write mathematically the basic kinematic equations of various in the local coordinate system, which moves together with the Sun arXiv:2107.13979v1 [astro-ph.GA] 29 Jul 2021 models, the Cartesian coordinate system G,H,I is usually applied. about the Galactic centre. The origin of the system coincides with the Solar system barycenter. In the last time, some new models are often used to analyse The G axis points to the Galactic center, H axis coincides with the the stellar kinematics. Since stellar proper motions and radial ve- direction of the Galactic rotation, while I axis is perpendicular to locities are the components of the velocity vector field, it would the Galactic plane and complements the right-handed Cartesian be reasonable to use the methods of decomposition of the cor- coordinate system (see Fig. 1). The given system is called the local responding stellar velocity field on a set of the vector spherical rectangular Galactic coordinate system. harmonics (VSH). The approach allows to detect all the system- In this paper, we use the O–M model which describes the atic constituents present in the stellar velocity field under study, as systematic differential stellar velocity field within the vicinity of well as make the mathematically complete kinematic model. Com- a chosen point. According to the Helmholtz theorem, the stellar parison of the decomposition coefficients with model parameters motion may be represented by the sum of velocities, namely the shows whether the models are complete as well as allows to reveal translational motion of the centroid of the corresponding stellar all significant systematics which is not included into the models. system, as well as its rigid-body rotation and deformation velocities. By the term "centroid" we mean a point that coincides with that moving at a velocity equal to the mean velocity of stars belonging ★ E-mail: [email protected] (PNF) to the given stellar system. This approach has been used, for instance, © 2021 The Authors 2 Fedorov et al. Components of the antisymmetric matrix − 1 mE8 mE: " = − = l8: = −l:8 (4) 2 mG: mG8 0 constitute antisymmetric tensor which is referred to as the tensor of local rotation velocities. It describes the rigid-body rotation of the stellar system under examination about the axis which passes through the stellar centroid with the instant angular velocity l, and l1,l2,l3 are projections of l on the Galactic axes G,H,I respectively. The matrix + 1 mE8 mE: " = + = D8: (5) 2 mG: mG8 0 is symmetric, and its components E8: = E:8 form the symmetric second-rank tensor which is referred to as the tensor of local de- formation velocities. It defines the velocity of deformation motion in the stellar system under study. Usually, it is these components of the tensor that correspond to the forces inducing deformations in the dynamical investigations (Tarapov (2002); Sedov (1970)). Figure 1. The local rectangular Galactic G,H,I and cylindrical Galactocen- + + The diagonal components of the symmetric matrix "11, "22 tric ', \,I coordinate systems. ( is an arbitrary star with the heliocentric + and "33 characterize velocities of relative contractions/expansions distance A. of the stellar system along G,H,I axes, while the components + = + + = + + = + "12 "21, "23 "32, "13 "31 do velocities of the angular in the papers by Clube (1972); du Mont (1977); Miyamoto & Soma deformations in the (G, H), (H, I) and (G, I) planes, respectively. Here (1993); Miyamoto & Zhu (1998); Bobylev & Khovrichev (2011). the velocity of angular deformation means the velocity of changing In general, representation of the velocity field of a deformable angles between segments in the planes. stellar system is based on the general form of expanding the con- To set a relation between the velocity components V(r) and the tinuous vector function V(r) which is defined in the vicinity of any observational data, namely proper motions `; and `1 on the Galac- point r0 (G0, H0, I0). In the Cartesian Galactic coordinate system tic longitude and latitude, respectively, as well as radial velocity +A G,H,I the general form of the stellar velocity field expansion V(r) of a star, we project the vector V(r) on the unit vectors of the spher- is as follows: ical Galactic coordinate system e;, e1, eA and introduce the factor : = 4.74057 to transform the dimension of stellar proper motions from mas yr−1 to km s−1 kpc−1. Then, the conditional equations for 2 m+ 1 m + proper motions and radial velocities in the local Galactic coordinate V(r) = V(r0+3r) = V(r0)+ 3G: + 3G: 3G<+... mG: 0 2! mG: mG< 0 system have the following form: (1) = where the indexes :,< 1,2,3, and the partial derivatives are :`; cos 1 = -⊙/A sin ; − .⊙/A cos ; − l1 sin 1 cos ; − l2 sin 1 sin ;+ evaluated at the point r0 (G0, H0, I0). + + + l3 cos 1 + " cos 1 cos 2; − " sin 1 sin ;+ When analysing the solar vicinity, we restrict the expansion 1 12 13 + "+ sin 1 cos ; − 0.5 "+ cos 1 sin 2;+ to the first-order terms and derive the components E8 of the linear 23 11 velocity field V at the point located at the heliocentric distance from + + 0.5 "22cos 1 sin 2; (6) the Sun r = r0 + 3r as follows: :`1 = -⊙/A cos ; sin 1 + .⊙/A sin ; sin 1 − /⊙/A cos 1 + l1 sin ;− 1 mE8 mE: − l cos ; − 0.5 "+ sin 21 sin 2; + "+ cos 21 cos ;+ E8 (r0 + 3r) = E8 (r0) + − 3G: + (2) 2 12 13 2 mG: mG8 0 + + 2 + "23 cos 21 sin ; − 0.5 "11 sin 21 cos ;− 1 mE8 mE: = − + + + 3G: E8 (r0) + " 3G: + " 3G: − 0.5 "+ sin 21 sin2 ; + 0.5 "+ sin 21 (7) 2 mG: mG8 0 22 33 where 8, : = 1, 2, 3. +A /A = −-⊙/A cos ;cos 1 − .⊙/A sin ; cos 1 − /⊙/A sin 1+ In this equation, E8 (r0) is usually interpreted as the mean + + + 2 velocity of the stellar system relative to the Sun, or on the contrary, + "13 sin 21 cos ; + "23 sin 21 sin ; + "12 cos 1 sin 2;+ as the Solar velocity relative to the chosen centroid with the opposite + 2 2 + 2 2 + 2 + "11 cos 1 cos ; + "22 cos 1 sin ; + "33 sin 1 (8) signs, where the components are E8 (r0) = −-⊙eG − .⊙eH − /⊙eI . Thus, finally one can write: The equations given above define the differential stellar veloc- ity field and contain 12 unknown parameters. The local Galactic coordinate system can be introduced at any E (A + 3A) = −- e − . e − / e + "−3G + "+3G (3) 8 0 ⊙ G ⊙ H ⊙ I : : arbitrary point of the Galactic plane provided that one know coordi- − + The matrix elements " and " are the partial derivatives of nates G,H,I, spacial velocities +G,+H,+I of the point (for instance, a the projections (E1, E2, E3) of the velocity vector on the axes of the star) and for all stars located within the corresponding vicinity. Tak- rectangular Galactic coordinate system, and they are usually called ing the Galactocentric distance of the Sun to be equal to '⊙ = 8.0 kinematic parameters. kpc (Valleé (2017)), one can make the transition from the local MNRAS 000, 1–11 (2021) Kinematics of the Milky Way from the Gaia EDR3 data 3 Galactic coordinate system with the origin at the barycenter of the -2 Solar system to one with the origin at the chosen point.
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