Determination of the Solar Galactocentric distance from masers kinemics
aa r X i v : . [ a s t r o - ph . GA ] N ov Baltic Astronomy, vol. 99, 999–999, 2014
DETERMINATION OF THE SOLAR GALACTOCENTRICDISTANCE FROM MASERS KINEMATICS
A.T. Bajkova and V.V. Bobylev , Central (Pulkovo) Astronomical Observatory of RAS, 65/1 Pulkovskoye Ch.,St. Petersburg, Russia; [email protected] Sobolev Astronomical Institute, St. Petersburg State University, Bibliotechnayapl.2, St. Petersburg, Russia; [email protected]
Received: 2014 December 99; accepted: 2014 December 99
Abstract.
We have determined the Galactic rotation parameters and the solarGalactocentric distance R by simultaneously solving Bottlinger’s kinematicequations using data on masers with known line-of-sight velocities and highlyaccurate trigonometric parallaxes and proper motions measured by VLBI. Oursample includes 93 masers spanning the range of Galactocentric distances R from 3 to 15 kpc. The solutions found are Ω = 29 . ± . − kpc − , Ω ′ = − . ± .
11 km s − kpc − , Ω ′′ = 0 . ± .
029 km s − kpc − , and R = 8 . ± .
12 kpc. In this case, the linear rotation velocity at the solardistance R is V = 238 ± − . Key words:
Masers – Galaxy: kinematics and dynamics – galaxies: individ-ual: Solar distance.1. INTRODUCTIONBoth kinematic and geometric characteristics are important for studying theGalaxy, with the solar Galactocentric distance R being the most important amongthem. Various data are used to determine the Galactic rotation parameters. Theseinclude the line of sight velocities of neutral and ionized hydrogen clouds with theirdistances estimated by the tangential point method (Clemens 1985; McClure-Griffiths & Dickey 2007; Levine et al. 2008), Cepheids with the distance scalebased on the period–luminosity relation, open star clusters and OB associationswith photometric distances (Mishurov & Zenina 1999; Rastorguev et al. 1999;Zabolotskikh et al. 2002; Bobylev et al. 2008; Mel’nik and Dambis 2009), andmasers with their trigonometric parallaxes measured by VLBI (Reid et al. 2009a;McMillan & Binney 2010; Bobylev & Bajkova 2010; Bajkova & Bobylev 2012).The solar Galactocentric distance R is often assumed to be known in a kine-matic analysis of data, because not all of the kinematic data allow R to be reliablyestimated. In turn, different (including direct) methods of analysis give differentvalues of R .Reid (1993) published a review of the R measurements made by then by var-ious methods. He divided all measurements into primary, secondary, and indirectones and obtained the “best value” as a weighted mean of the published measure-ments over a period of 20 years: R = 8 . ± . etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance
Masers – Galaxy: kinematics and dynamics – galaxies: individ-ual: Solar distance.1. INTRODUCTIONBoth kinematic and geometric characteristics are important for studying theGalaxy, with the solar Galactocentric distance R being the most important amongthem. Various data are used to determine the Galactic rotation parameters. Theseinclude the line of sight velocities of neutral and ionized hydrogen clouds with theirdistances estimated by the tangential point method (Clemens 1985; McClure-Griffiths & Dickey 2007; Levine et al. 2008), Cepheids with the distance scalebased on the period–luminosity relation, open star clusters and OB associationswith photometric distances (Mishurov & Zenina 1999; Rastorguev et al. 1999;Zabolotskikh et al. 2002; Bobylev et al. 2008; Mel’nik and Dambis 2009), andmasers with their trigonometric parallaxes measured by VLBI (Reid et al. 2009a;McMillan & Binney 2010; Bobylev & Bajkova 2010; Bajkova & Bobylev 2012).The solar Galactocentric distance R is often assumed to be known in a kine-matic analysis of data, because not all of the kinematic data allow R to be reliablyestimated. In turn, different (including direct) methods of analysis give differentvalues of R .Reid (1993) published a review of the R measurements made by then by var-ious methods. He divided all measurements into primary, secondary, and indirectones and obtained the “best value” as a weighted mean of the published measure-ments over a period of 20 years: R = 8 . ± . etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance R deter-mination method, the method of finding the reference distances, and the type ofreference objects are taken into account. Taking into account the main types oferrors and correlations associated with the classes of measurements, he obtainedthe “best value” R = 7 . ± . R = 8 . ± . R determination between 1918 and 2013. Theyconcluded that the results obtained after 2000 give a mean value of R close to8.0 kpc.We have some experience of determining R by simultaneously solving Bot-tlinger’s kinematic equations with the Galactic rotation parameters. To this end,we used data on open star clusters (Bobylev et al. 2007) distributed within about 4kpc of the Sun. Clearly, using masers belonging to regions of active star formationand distributed in a much wider region of the Galaxy for this purpose is of greatinterest. However, the first such analysis for a sample of 18 masers performedby McMillan & Binney (2010) showed the probable value of R to be within afairly wide range, 6.7–8.9 kpc. At present, the number of masers with measuredtrigonometric parallaxes has increased (Reid et al. 2014), which must lead to asignificant narrowing of this range.The goal of this paper is to determine the Galactic rotation parameters andthe distance R using data on masers with measured trigonometric parallaxes.2. METHODHere, we use a rectangular Galactic coordinate system with the axes directedaway from the observer toward the Galactic center ( l =0 ◦ , b =0 ◦ , the X axis), inthe direction of Galactic rotation (( l =90 ◦ , b =0 ◦ , the Y axis), and toward the northGalactic pole ( b = 90 ◦ , the Z axis).The method of determining the kinematic parameters consists in minimizing aquadratic functional F :min F = P Nj =1 [ w jr ( V jr − ˆ V jr )] + P Nj =1 [ w jl ( V jl − ˆ V jl )] + P Nj =1 [ w jb ( V jb − ˆ V jb )] (1)provided that the following constraints derived from Bottlinger’s formulas with anexpansion of the angular velocity of Galactic rotation Ω into a series to terms ofthe second order of smallness with respect to r/R are fulfilled: V r = − u ⊙ cos b cos l − v ⊙ cos b sin l − w ⊙ sin b + R ( R − R ) sin l cos b Ω ′ + 0 . R ( R − R ) sin l cos b Ω ′′ , (2) V l = u ⊙ sin l − v ⊙ cos l + ( R − R )( R cos l − r cos b )Ω ′ +( R − R ) ( R cos l − r cos b )0 . ′′ − r Ω cos b, (3) V b = u ⊙ cos l sin b + v ⊙ sin l sin b − w ⊙ cos b − R ( R − R ) sin l sin b Ω ′ − . R ( R − R ) sin l sin b Ω ′′ , (4)where N is the number of objects used; j is the current object number; V r and V l ,V b are the model values of the three-dimensional velocity field: the line-of-sight02 A.T. Bajkova, V.V. Bobylev velocity and the proper motion velocity components in the l and b directions,respectively; V l = 4 . rµ l cos b , V b = 4 . rµ b are the measured components ofthe velocity field (data), with ˆ V jr , ˆ V jl and ˆ V jb , where the coefficient 4.74 is thequotient of the number of kilometers in an astronomical unit and the number ofseconds in a tropical year; w jr , w jl , w jb are the weight factors; r is the heliocentricdistance of the star calculated via the measured parallax π, r = 1 /π ; the star’sproper motion components µ l cos b and µ b are in mas yr − (milliarcseconds peryear), the line-of-sight velocity V r is in km s − ; u ⊙ , v ⊙ , w ⊙ are the stellar groupvelocity components relative to the Sun taken with the opposite sign (the velocity u is directed toward the Galactic center, v is in the direction of Galactic rotation, w is directed to the north Galactic pole); R is the Galactocentric distance of theSun; R is the distance from the star to the Galactic rotation axis, R = r cos b − R r cos b cos l + R . (5)Ω is the angular velocity of rotation at the distance R ; the parameters Ω ′ andΩ ′′ are the first and second derivatives of the angular velocity with respect to R, respectively.The weight factors in functional (1) are assigned according to the followingexpressions (for simplification, we omitted the index i ): w r = S / q S + σ V r , w l = βS / q S + σ V l , w b = γS / q S + σ V b , (6)where S denotes the dispersion averaged over all observations, which has themeaning of a “cosmic” dispersion taken to be 8 km s − ; β = σ V r /σ V l and γ = σ V r /σ V b are the scale factors, where σ V r , σ V l and σ V b denote the velocity disper-sions along the line of sight, the Galactic longitude, and the Galactic latitude,respectively. The system of weights (6) is close to that from Mishurov & Zenina(1999). We take β = γ = 1 according by Bobylev & Bajkova (2014).The errors of the velocities V l and V b are calculated from the formula σ ( V l ,V b ) = 4 . r vuut µ l,b σ r r ! + σ µ l,b . (7)The problem of optimizing functional (1), given Eqs. (2)–(4), is solved numeri-cally for the seven unknown parameters u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , Ω ′′ , and R froma necessary condition for the existence of an extremum. A sufficient conditionfor the existence of an extremum in a particular domain is the positive definite-ness of the Hessian matrix composed of the elements { a i,j } = d F/dx i dx j , where x i ( i = 1 , ...,
7) denote the sought-for parameters, everywhere in this domain. Wecalculated the Hessian matrix in a wide domain of parameters or, more specifically, ±
50% of the nominal values of the parameters.Our analysis of the Hessian matrix for both cases of weighting showed itspositive definiteness, suggesting the existence of a global minimum in this domainand, as a consequence, the uniqueness of the solution. As an example, Fig. 1 showsthe two-dimensional residuals, or the square root of the functional F, with one ofthe measurements being specified by the parameter R and one of the parameters u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , and Ω ′′ , acting as the second measurement, provided that the etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance
50% of the nominal values of the parameters.Our analysis of the Hessian matrix for both cases of weighting showed itspositive definiteness, suggesting the existence of a global minimum in this domainand, as a consequence, the uniqueness of the solution. As an example, Fig. 1 showsthe two-dimensional residuals, or the square root of the functional F, with one ofthe measurements being specified by the parameter R and one of the parameters u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , and Ω ′′ , acting as the second measurement, provided that the etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance Fig. 1.
Graphical representation of the two-dimensional residuals δ = √ F correspond-ing to solution (9); one of the measurements is specified by the parameter R ; one of theparameters u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , and Ω ′′ acts as the second measurement; the remainingparameters are fixed at the level of the solutions obtained. remaining parameters from the series are fixed at the level of the solution obtained.The presented pictures clearly demonstrate a global minimum in a wide domain ofparameters. In the case of unit weight factors, the Hessian matrix is also positivelydefined far beyond this domain. However, as will be shown below, the adoptedweighting allowed the accuracy of the solutions obtained to be increased.We estimated the errors of the sought-for parameters through Monte Carlosimulations. The errors were estimated by performing 100 cycles of computations.For this number of cycles, the mean values of the solutions virtually coincidewith the solutions obtained purely from the initial data, i.e., without adding anymeasurement errors.04 A.T. Bajkova, V.V. Bobylev
3. DATABased on published data, we gathered information about the coordinates, line-of-sight velocities, proper motions, and trigonometric parallaxes of Galactic masersmeasured by VLBI with an error, on average, less than 10%. These masers areassociated with very young objects, protostars of mostly high masses located inregions of active star formation.One of the projects to measure the trigonometric parallaxes and proper motionsis the Japanese VERA (VLBI Exploration of Radio Astrometry) project devotedto the observations of H O masers at 22.2 GHz (Hirota et al. 2007) and a numberof SiO masers (which are very few among young objects) at 43 GHz (Kim et al.2008).Methanol (CH OH, 6.7 and 12.2 GHz) and H O masers are observed in theUSA on VLBA (Reid et al. 2009a). Similar observations are also being carried outwithin the framework of the European VLBI network (Rygl et al. 2010), in whichthree Russian antennas are involved: Svetloe, Zelenchukskaya, and Badary. Thesetwo programs enter into the BeSSeL project (Bar and Spiral Structure LegacySurvey, Brunthaler et al. 2011).Initial data on 103 masers was taken from Reid et al. (2014).4. RESULTSUsing the three-dimensional maser velocity field for sample of 103 masers for equa-tions with seven unknowns and weights (6) we obtained the following solution:( u ⊙ , v ⊙ , w ⊙ ) = (5 . , . , . ± (0 . , . , .
32) km s − , Ω = 29 . ± .
45 km s − kpc − , Ω ′ = − . ± .
11 km s − kpc − , Ω ′′ = 0 . ± .
029 km s − kpc − ,R = 8 . ± .
12 kpc ,σ = 10 .
59 km s − ,N ⋆ = 93 . (8)Note, that in this case, ten sources (G000.67-00.03, G010.47+00.02, G010.62-00.38, G023.70-00.19, G025.70+00.04, G027.36-00.16, G009.62+00.19, G012.02-00.03, G078.12+03.63, G168.06+00.82) were rejected according to the 3 σ crite-rion. From this solution the linear rotation velocity at the solar distance R is V = 238 ± − and the Oort constants A = 0 . R Ω ′ and B = Ω + 0 . R Ω ′ are A = − . ± .
45 km s − kpc − and B = 12 . ± .
63 km s − kpc − .As a clear illustration of the uniqueness of the solution obtained (i.e., theexistence of a global minimum of the functional F in a wide range of soughtfor parameters), Fig. 1 presents the two-dimensional dependence of the residuals δ = √ F (see (1)) on R and one of the parameters u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , and Ω ′′ ,provided that the remaining parameters are fixed at the level of solutions (8).Figure 2 presents the Galactic rotation curve constructed with parameters (8)using the value of R = 8 .
03 kpc found; when calculating the boundaries of theconfidence region, we took into account the uncertainty in estimating R of 0.12kpc. etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance
03 kpc found; when calculating the boundaries of theconfidence region, we took into account the uncertainty in estimating R of 0.12kpc. etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance Table 1.
Kinematic parameters found by using the three-dimensional velocity field.
Parameters 4 < R <
12 kpc e π /π < u ⊙ , km s − . ± .
75 5 . ± .
72 6 . ± .
77 7 . ± . v ⊙ , km s − . ± .
65 15 . ± .
73 14 . ± .
81 13 . ± . w ⊙ , km s − . ± .
35 8 . ± .
38 8 . ± .
42 9 . ± . , km s − kpc − . ± .
45 29 . ± .
43 29 . ± .
48 29 . ± . ′ , km s − kpc − − . ± . − . ± . − . ± . − . ± . ′′ , km s − kpc − . ± .
02 0 . ± .
05 0 . ± .
03 0 . ± . R , kpc 8 . ± .
41 7 . ± .
13 8 . ± .
13 8 . ± . σ , km s − N ⋆
101 88 78 80
We also obtained a few solutions satisfying various limitations on data (seeTable 1). The results shown might be of some interest. In the second column ofthe table we have the solution, obtained with the use of nearly all masers ( N =101), only two masers (G000.67-00.03, G010.47+00.02) with the most unreliablevelocities were rejected. In the third and fourth columns there are the solutionsobtained with the limitation on the star galactocentric distance R and with alimited precision of parallaxes, e π /π . The first solution has a significant error unitof weight σ , in other cases, this value is significantly less. Therefore we considerthe first solution as the most unreliable. In the last column we give the resultsobtained when neglecting 23 masers that were flagged as outliers by Reid et al.(2014). As it is seen other three solutions do not have principal differences. Inour opinion, the solution (8) obtained from maximum amount of initial masers( N = 103) is of most interest.5. DISCUSSIONThe parameters of the Galactic rotation curve we found (8) are in good agreementwith the results of analyzing such young Galactic disk objects as OB associations,Ω = 31 ± − kpc − (Mel’nik & Dambis 2009), blue supergiants, Ω =29 . ± . − kpc − and Ω ′ = − . ± .
32 km s − kpc − (Zabolotskikh etal. 2002), or distant OB3 stars ( R = 8 kpc), Ω = 31 . ± . − kpc − ,Ω ′ = − . ± .
16 km s − kpc − and Ω ′′ = 1 . ± .
35 km s − kpc − (Bobylev &Bajkova 2013). The solution (8) is in good agreement with V = 254 ±
16 km s − at R = 8 . V = 244 ±
13 km s − for R = 8 . V having been foundto be close to 240 km s − and R ≈ . R with an error of 10–15%. Note several important isolated measurements. Based on POPULATION-IICepheids and RR Lyr stars belonging to the bulge and using improved calibrationsderived from Hipparcos data and 2MASS photometry, Feast et al. (2008) obtainedan estimate of R = 7 . ± .
21 kpc. Having analyzed the orbits of stars movingaround a supermassive black hole at the Galactic center (the method of dynamicalparallaxes), Gillessen et al. (2009) obtained an estimate of R = 8 . ± .
35 kpc.According to VLBI measurements, the radio source Sqr A* has a proper motionrelative to extragalactic sources of 6 . ± .
026 mas yr − (Reid & Brunthaler06 A.T. Bajkova, V.V. Bobylev
Fig. 2.
Galactic rotation curve constructed with parameters (8) (thick line); the thinlines mark the 1 σ confidence region; the vertical straight line marks the Sun’s position. R = 8 . ± .
29 kpc and V =238 ± − . Two H O maser sources (Sgr B2), are in the immediate vicinity ofthe Galactic center, where the radio source Sqr A* is located. Based on their directtrigonometric VLBI measurements, Reid et al. (2009b) obtained an estimate of R = 7 . +0 . − . kpc.Using 73 masers Bobylev & Bajkova (2014) obtained an kinematic estimateof R = 8 . ± . V = 241 ± − . Based on 80 maser sources,Reid et al. (2014) obtained an kinematic estimate of R = 8 . ± .
16 kpc and V = 240 ± − . Thus, our kinematic estimate of R = 8 . ± .
12 kpc is ingood agreement with the known estimates and surpasses some of them in accuracy.6. CONCLUSIONSBased on published data, we produced a sample of masers with known line-of-sightvelocities and highly accurate trigonometric parallaxes and proper motions mea-sured by VLBI. This allowed the maser velocity field needed to solve Bottlinger’skinematic equations to be formed. Bottlinger’s kinematic equations we consid-ered relate the Galactic rotation parameters (Ω and its derivatives), the solarGalactocentric distance ( R ) , the object group velocity components relative to theSun ( u ⊙ , v ⊙ , w ⊙ ). The method of minimizing the quadratic functional that is thesum of the weighted squares of the residuals of measurements and model velocitieswas used to find the unknown parameters. Solutions were found in the cases ofboth three-dimensional and two-dimensional velocity fields for various numbers ofsought-for parameters when various weighting methods were applied. We estab-lished that the solution obtained from the three-dimensional maser velocity fieldfor seven sought-for parameters ( u ⊙ , v ⊙ , w ⊙ , Ω , Ω ′ , Ω ′′ , and R ) corresponding tothe global minimum of the functional in a wide range of their variations is mostreliable. This solution is (8). The linear Galactic rotation velocity at the solardistance R is V = 238 ± − . The solar Galactocentric distance R isthe most important and debatable parameter. Our value R = 8 . ± .
12 kpcis in good agreement with the most recent estimates and even surpasses them inaccuracy. etermination of the Solar Galactocentric Distanceetermination of the Solar Galactocentric Distance