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Astronomy Letters, Vol. 28, No. 7, 2002, pp. 454­464. Translated from Pis'ma v Astronomicheski Zhurnal, Vol. 28, No. 7, 2002, pp. 516­528. i Original Russian Text Copyright c 2002 by Zabolotskikh, Rastorguev, Dambis.

Kinematic Parameters of Young Subsystems and the Galactic Rotation Curve
M. V. Zabolotskikh1* , A. S. Rastorguev
1 2

1,2

, and A. K. Dambis

2

Moscow State University, Vorob'evy gory, Moscow, Russia Sternberg Astronomical Institute, Universitetskii pr. 13, Moscow, 119992 Russia
Received February 1, 2002

Abstract--We analyze the space velocities of blue supergiants, long-period Cepheids, and young open star clusters (OSCs), as well as the H I and H II radial-velocity fields by the maximum-likelihood method. The distance scales of the objects are matched both by comparing the first derivatives of the angular velocity determined separately from radial velocities and proper motions and by the statistical-parallax method. The former method yields a short distance scale (for R0 = 7.5 kpc, the assumed distances should be increased by 4%), whereas the latter method yields a long distance scale (for R0 = 8.5 kpc, the assumed distances should be increased by 16%). We cannot choose between these two methods. Similarly, the distance scale of blue supergiants should be shortened by 9% and lengthened by 3%, respectively. The H II distance scale is matched with the distance scale of Cepheids and OSCs by comparing the derivatives determined for H II from radial velocities and for Cepheids and OSCs from space velocities. As a result, the distances to H II regions should be increased by 5% in the short distance scale. We constructed the Galactic rotation curve in the Galactocentric distance range 2­14 kpc from the radial velocities of all objects with allowance for the difference between the residual-velocity distributions. The axial ratio of the Cepheid+OSC velocity ellipsoid is well described by the Lindblad relation, while u for gas. The following rotation-curve parameters were obtained: 0 = (27.5 ± 1.4) km s-1 kpc-1 and A = (17.1 ± 0.5) km s-1 kpc-1 for the short distance scale (R0 = 7.5 kpc); and 0 = (26.6 ± 1.4) km s-1 kpc-1 and A = (15.4 ± 0.5) km s-1 kpc-1 for the long distance scale (R0 = 8.5 kpc). We propose a new method for determining the angular velocity 0 from stellar radial velocities alone by using the Lindblad relation. Good agreement between the inferred 0 and our calculations based on space velocities suggests that the Lindblad relation holds throughout the entire sample volume. Our analysis of the heliocentric velocities for samples of young objects reveals noticeable streaming motions (with a velocity lag of 7 km s-1 relative to the LSR), whereas a direct computation of the perturbation amplitudes in terms of the linear density-wave theory yields a small amplitude for the tangential perturbations. c 2002 MAIK "Nauka/Interperiodica". Key words: Galactic kinematics, rotation curve, distance scale

INTRODUCTION The study of the kinematics of Galactic subsystems remains one of the most important fields of Galactic astronomy. The parameters of the Galactic rotation curve were determined repeatedly from HI and HII data (Clemens 1985; Fich et al. 1989; Merrifield 1992; Brandt and Blitz 1993; Nikiforov and Petrovskaya 1994; Honma and Sofue 1997; Nikiforov 1999) and stellar radial velocities (Karimova and Pavlovskaya 1973; Pont et al. 1994; Dambis et al. 1995; Glushkova et al. 1998). High-precision proper motions and trigonometric parallaxes that became available with the release of the HIPPARCOS catalog (The HIPARCOS and TYCHO catalogs, ESA SP-1200, 1997) stimulated further works aimed at
*

E-mail: ÷

ñïñü ïð ÿ½ºâ

ºÿâûºæû

refining the angular velocity 0 and the form of the rotation curve in the local solar neighborhood (Feast et al. 1998; Rastorguev et al. 1999; Dambis et al. 2001). It should be pointed out that the reliability of the resulting rotation curves depends first and foremost on the correctness of the adopted distance scale of objects under study. Objects with known distances--classical Cepheids, open star clusters (OSC), and OB-associations--allow the rotation curve to be determined only out to heliocentric distances of 4­5 kpc, whereas H I and H II kinematic data allow constructing the rotation curve over a considerably wider interval of Galactocentric distances. The main problem is that the distances of giant molecular clouds (GMC) and, consequently, those of H II-regions, are determined from their single hot exciting stars whose distance scale is prone not

1063-7737/02/2807-0454$22.00 c 2002 MAIK "Nauka/Interperiodica"


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455

only to random but also to systematic errors. In this paper we matched for the first time the distance scale of GMC to the most accurately determined (in the random and systematical sense) distances, i.e., those of long-period classical Cepheids and OSC, and inferred the kinematic parameters using an algorithm that allows for initial-data errors, for the ellipsoidal distribution of residual velocities, and for the errors of systemic radial velocities that result from the propagation of distance errors (Rastorguev 2001). Our second task was to compute the parameters of the spiral pattern of the Galaxy. Selection effects, i.e., the incompleteness of the sample due to interstellar extinction, makes it very difficult to localize spiral waves by analyzing only the space distribution of young objects. The use of kinematic data appears to be a more promising approach, because it is insensitive to selection effects (Mishurov et al. 1979). Thus Mishurov et al. (1997) determined, by analyzing the radial velocities of classical Cepheids exclusively, the principal parameters of the spiral pattern including the velocity-field perturbation amplitudes and concluded that the Sun is located near the corotation circle. The perturbations due to the spiral density wave are comparable in magnitude to the velocity dispersion of young subsystems. Therefore, only after the release of the HIPPARCOSand TYCHO-2 catalogs making high-precision proper motions available did it become possible to analyze the space velocity field of young objects. Torra et al. (2000) used the radial velocities of OB-stars and Cepheids and HIPPARCOS proper motions of these objects to infer a pattern speed of P (31 ± 4) km s-1 kpc-1 .Lepine et al. (2001) also concluded that the Sun is near the corotation circle by assuming a superposition of a two- and four-armed pattern. At the same time, Rastorguev et al. (2001) concluded that the Sun is inside the corotation circle by analyzing long-period Cepheids and young OSC, and Mel'nik et al. (2001) came to the same conclusion based on their study of the pattern of systematical noncircular motions of OB-associations. Here we explore this issue further. OBSERVATIONAL DATA We used young OSC and long-period Cepheids as a reference sample for matching the distance scales of various objects. Our reference sample included 89 young OSC with log T < 7.6 and heliocentric distances determined by Dambis (1999) by fitting Kholopov's (1980) ZAMS with an allowance for evolutionary deviations based on Geneva-group isochrones (Maeder and Meynet 1991). The radial velocities of cluster members were determined by Glushkova based on published data and can be found in the paper by Rastorguev et al. (1999). The proper
ASTRONOMY LETTERS Vol. 28 No. 7 2002

motions of clusters were computed from those of their member stars found in the HIPPARCOS catalog (Baumgardt et al. 2000). Our reference sample included 113 classical Cepheids with periods P > 9d (or ages log T < 7.6 as implied by the period-age relation of Efremov (1989)) and heliocentric distances computed using the fundamental-mode period-luminosity relation of Berdnikov et al. (1996): MK
I

= -5.46m - 3.52m log P

in accordance with the procedure described therein. An earlier statistical-parallax analysis (Rastorguev et al. 1999) showed that the sample of Cepheids with shorter periods is not homogeneous in terms of pulsation mode and may be contaminated by firstovertone pulsators. We used published Cepheid radial velocities and HIPPARCOS proper motions. Young OSC and long-period Cepheids make up a kinematically homogeneous sample consisting of 176 and 142 objects with radial velocities and proper motions, respectively, including 124 objects with space velocities. We performed a separate analysis of a bluesupergiant sample consisting of 102 stars with heliocentric distances tied to the OSC distance scale (Dambis 1990). The kinematic data for these stars were compiled by A.K. Dambis with the proper motions adopted from the HIPPARCOS catalog, and radial velocities, from the catalogs of BarbierBrossat and Figon (2000) and Wilson­Evans­ Batten (WEB) (Duflot et al. 1995). Brandt and Blitz (1993) published the distances and radial velocities for a total of 206 H II-regions. We selected 203 of these objects with spectroscopic or photometric distances inferred from their exciting stars. The radial velocities of H II-regions were determined from the CO (2.2.-mm) radio lines of their associated molecular clouds. We did not include three H II-regions in the final list because of their large residual velocities relative to the provisional rotationcurve solution. The catalog mentioned above also gives standard errors of individual distance and radial velocities. We adopted 150 tangent-point radial velocities of H I clouds from Fich et al. (1989). Note that published H I and H II radial velocities are traditionally corrected for the solar motion relative to the standard apex assumed to coincide with the local standard of rest (LSR), and we therefore first converted them into heliocentric radial velocities.


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METHOD OF ANALYSIS We used the techniques of maximum-likelihood and statistical parallax (including its simplified version) to compute the kinematic parameters and refine the distance scales involved. See Murray (1986) for a description of the principal ideas of the statisticalparallax method used in this paper. The tangential velocity of a star is computed from its proper motion and distance and therefore depends on the adopted distance scale, whereas radial velocities are distance independent. The essence of the method is to reconcile the fields of radial and tangential velocities in terms of some model of the field of systematic motions and ellipsoidal distribution of residual velocities. A number of authors applied this method with success. Hawley et al. (1986); Popovski and Gould (1998), Gould and Popowski (1998), Fernley et al. (1998), Popowski (1998), Fernley et al. (1998), Tsujimoto et al. (1998), and Dambis and Rastorguev (2001) used it to refine the distance scale of RR Lyrae type variables. In our previous paper (Rastorguev et al. 1999) we applied this method for the first time, albeit in a somewhat simplified form ignoring the scatter of absolute magnitudes, to analyze the space velocity field of young objects of the Galactic disk, which are characterized by small residual velocity dispersions. In this paper we also apply a simplified version of the statistical-parallax technique (as used, e.g., by Feast et al. 1998), based on reconciling the kinematic parameters inferred separately from radial velocities and proper motions. Thus, it is well known that Oort's constant A inferred from proper motions is much less sensitive to the adopted distance scale than is the value of the same constant inferred from radial velocities. This allows not only the kinematic parameters to be determined but also the distance scale of objects under study to be refined. Consider now a model of the field of space velocities that includes both differential rotation and effects due to a spiral density wave. The residual velocity of a star can be written in the form of the following column vector: V = Vobs - Vsys = Vobs - Vsun - Vrot - Vspir , where Vobs is the observed space velocity; Vsys , the total velocity of systematic motions including: Vsun , the mean heliocentric velocity of the sample studied; Vrot , the contribution of Galactic differential rotation; and Vspir , the perturbation due to the spiral density wave. To allow for spiral-pattern effects, we used a very simple kinematic model based on linear densitywave theory by Lin et al. (1969) with the perturbation of potential in the form of a running wave: S = A cos ,

where A < 0 is the amplitude of perturbations and = m(- +cot i ln(R/R0 )) + 0 , the phase angle of the object in the wave (it increases toward the Galactic center). Here m is the number of arms; , the position angle of the object (measured in the direction of rotation); i, the pitch angle of spiral arms (i < 0 for trailing spirals); 0 , the phase angle of the Sun; and R and R0 , the Galactocentric distances of the Sun and the object, respectively. The radial VR (which in the arm is directed toward the Galactic center) and azimuthal V (directed along differential rotation at the outer edge of the arm) components of velocity perturbation can be written in the following form: VR = fR cos , V = f sin , where fR and f are the amplitudes of velocity perturbations (Rohlfs 1977): kA F (1) (x), fR = 1 - 2 kA 1 F (2) (x). 2 1 - 2 We now use standard designations: f = - k= m cot i , R k
u 2

= 2

1-

A ,

x=

,

=

m(P - ) .

Here, k is the radial wavenumber; , the epicyclic (1) (2) frequency; A, Oort's constant; F (x) and F (x), the reduction factors; x, the Toomre instability parameter; u , the dispersion of radial velocities; , the relative frequency with which the object rotating in a circular orbit meets a passing spiral wave; , the angular velocity of differential Galactic rotation; and P , the angular velocity of the rigid rotation of the spiral pattern (i.e., the pattern speed). Residual space velocities are usually assumed to have a three- dimensional normal distribution: f (V) = (2 )
-3/2

|L

â exp -0.5VT â L

-1/2 obs | -1 obs â V

,

where Lobs is the matrix of covariances. The covariance matrix in our previous paper (Rastorguev et al. 1999) included only the ellipsoidal velocity distribution and the errors of radial velocities and proper motions, which is quite a justiable approach in the case of small errors in the adopted distances. The latter are related to the dispersion of absolute magnitudes as follows: 2 M = 4.71 (r /r )2 .
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KINEMATIC PARAMETERS OF YOUNG SUBSYSTEMS

457

L

obs

=L

loc,e

22 err +0.21M rt (P â L1 2 - 0.21M prt (M â L2 â P ) 2 +0.21M p2 (M â L3 â M ),

+L

âP )

V, km s

Here, angular braces mean averaging over the distribution of distance errors. The covariance matrix can be easily shown (Rastorguev 2001) to have the following form:

230
­1

200

Constant velocity dispersion Hr = 2 kpc Hr = 4 kpc Hr = 6 kpc

5

where matrices L1 ,L2 ,L3 ,M ,and P are equal to L1 =
T dVsys dVsys â , drt drt S,0

8 R, kpc

11

L2 = 2Vsys â

T dVsys , drt

L3 = GS â L

T â GT + Vsys â Vsys , S

Fig. 1. Galactic rotation curves inferred with different scale lengths Hr and assuming constant velocity dispersion.

0 0 0 M = 0 1 0 , 001

1 0 0 P = 0 p 0 , 00p

respectively, and the formulas for matrices GS , Lloc,e , Lerr ,and LS,0 can be found in the paper by Rastorguev et al. (1999). Here p is the distance-scale factor defined as: p = r0 /rt , where r0 and rt are the adopted (usually photometric) and true distance, respectively. We inferred the unknown parameters including the scale factor p using the maximum-likelihood method, i.e., by minimizing the following function with summation taken over all objects of the sample under study:
N

LF = -
i=1

ln f (V).

When values puted posed

refining the distance scale by reconciling the of Oort's constant A, we set p = 1. We comthe parameter errors using the method proby Hawley et al. (1986). RESULTS AND DISCUSSION

Kinematics of the Sample of Long-Period Cepheids and OSC
Our main task was twofold: to refine the distance scale of objects considered and to construct the rotation curve of the corresponding subsystem. We first applied the maximum-likelihood method to our sample of Cepheids and OSC with heliocentric distances r < 4 kpc and ignored spiral-pattern effects in the velocity field. Because the eventual correlation between the solar Galactocentric distance R0 and
ASTRONOMY LETTERS Vol. 28 No. 7 2002

the distance- scale factor prevents simultaneous determination of these parameters, and because of the uncertainty in the determination of R0 , we performed our computations twice with the two most commonly adopted values: R0 = 7.5 and 8.5 kpc. We determined the angular velocity of Galactic rotation 0 from space velocities of Cepheids and OSC and then used it to construct the Galactic rotation curve based on the radial velocities of all objects considered. How distance errors affect the results. To elucidate the effect of the distance errors on the results obtained, we repeated our computations with three different standard errors of absolute-magnitude cal. . . ibration: 0m1, 0m15, and 0m2 for Cepheids and OSC. We set R0 = 7.5 kpc and p = 1 in all three cases. The results are listed in Table 1. The columns of this table give the standard error of the absolute magnitude; heliocentric velocity components of the sample; velocity-ellipsoid axes; and rotation-curve parameters. The inferred kinematic parameters can be seen to be virtually independent of the adopted M , and therefore in the following computations we used a . compromise value of M = 0m15, which agrees with the scatter of the period-luminosity relation for the Cepheids members of open clusters (Berdnikov et al. 1996). The effect of the variation of velocity dispersion with galactocentric distance. The study of the kinematics and space distribution of objects in the disks of other galaxies showed that the disk surface brightness and velocity dispersion decrease exponentially with galactocentric distance, and the squared velocity dispersion is proportional to the surface density (van den Kruit and Freeman 1986; Bottema 1993). The corresponding scale length for our Galaxy can be estimated only indirectly and is most likely confined between 2 and 6 kpc depending on the age of the subsystem studied (Lewis and Freeman 1989; Kent et al. 1991; Malhotra 1995; Dehnen and Binney 1998; Freudenreich 1998; Drimmel and Spergel 2001). Let us assume that radial velocity


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Table 1. Kinematic parameters of the Cepheid + OSC sample inferred adopting different standard errors of absolute magnitudes M 0m10 . 0 0
m15 . m20 .

u

0

0 ­12.39 ­12.33 ­12.25

w0 km s
-1

u 12.90 12.82 12.72

8.02 7.93 7.81

w km s 7.21 7.19 7.15

0 -1

, kpc

-1

km s

-1

, kpc

-2

km s

-1

, kpc

-3

­6.34 ­6.31 ­6.27

­6.95 ­6.96 ­6.98

27.71 27.71 27.71

­4.66 ­4.66 ­4.65

1.18 1.17 1.15

Table 2. Kinematic parameters of the Cepheid + OSC sample inferred with different scale lengths of the assumed exponential radial decrease of velocity dispersions Hr , kpc 2 4 6 u0 , ­6.77 ­6.35 ­6.30 0 , ­12.45 ­12.34 ­12.32 w0 , km s ­6.94 ­6.95 ­6.95
-1

u , 14.40 13.25 13.04

, 7.88 7.93 7.94

w , km s 7.15 7.17 7.18

0 -1

, kpc

-1

km s

-1

, kpc

-2

km s

-1

, kpc

-3

28.76 28.22 28.05

­4.84 ­4.74 ­4.71

1.23 1.18 1.17

Table 3. Kinematic parameters and the distance-scale factor for the Cepheid + OSC sample inferred via statistical parallaxes R0 ,kpc 7.5 8.5 p u0 , 0 , w0 , km s 0.86 ­7.24 ­11.51 ­8.06 0.84 ­7.21 ­12.33 ­8.24 ±1.76 ±1.61 u ,
-1

, 8.03 8.15

w , km s 8.55 8.76

0 -1

, kpc

-1

km s

-1

, kpc

-2

km s

-1

, kpc 0.94 0.73

-3

13.70 13.65

26.93 26.61 ±1.35

­4.27 ­3.66 ±0.24

Standard errors ±0.05 ±2.10

±1.62 ±1.18 ±1.68

±0.19

dispersion varies exponentially with Galactocentric radius:
0 u = u exp -

R - R0 2Hr

,

Table 4. The first derivative of angular velocity inferred separately from radial velocities Vr and proper motions µ of the Cepheid + OSC sample and the resulting distancescale factor Method R0 ,kpc 1 2 1 2 7.5 7.5 8.5 8.5 (Vr ), km s-1 kpc ­4.67 ­4.68 ­4.04 ­4.06 ±0.26
-2

(µ), km s-1 kpc ­4.53 ­4.42 ­3.97 ­3.88 ±0.34

0 where u is the radial velocity dispersion in the solar neighborhood and Hr , the disk scale length parameter. As is evident from our Table 1 (see also results of Rastorguev et al. (1999) and Dehnen and Binney (1998)), in the neighborhood of the Sun the components of the velocity dispersion tensors of both the classical Cepheids + OSC and local MS-star sample obey the following Lindblad relation to a good accuracy: = u . 2

-2

p 0.97 0.94 0.98 0.96 ±0.09

Standard errors

It is possible, assuming that this relation is obeyed at every point of the disk for the current values of angular rotation velocity and epicyclic frequency, to determine how the inferred kinematic parameters depend on the adopted disk scale length. In this analysis we can neglect the effect of the variation of the vertical velocity dispersion w with Galactocentric distance, because, first, it is insignificant compared to the errors of tangential velocities Vb , and, second,
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459

the allowance for the dependence of vertical velocity dispersion on Galactocentric distance has virtually no effect on the results obtained. Table 2 presents the kinematic parameters computed with R0 = 7.5 kpc, and p = 1. The tabulated velocity-ellipsoid axes refer to the solar neighborhood. Note that 0 is sensitive to the adopted scale length parameter. Figure 1 shows how the inferred rotation curve changes with the adopted scale length. Although the exact scale for our sample is unknown, an analysis of the results of Kent et al. (1991) leads us to conclude that young subsystems are characterized by a relatively shallower decrease of radial dispersion with Galactocentric distance. According to our results, the rotation curve for Hr = 2 kpc is 9 km s-1 higher than if computed for constant velocity dispersion. Dehnen and Binney (1998) inferred a scale length of 2­ 2.5 kpc for old main-sequence stars; Drimmel and Spergel (2001) found a scale length of 0.28R0 by analyzing COBE/DIRBE data (note that both old and young stars contribute to infrared radiation). Since no accurate data are available about the relation between surface brightness and velocity dispersion, hereafter we assume that velocity dispersion remains constant along Galactocentric radius. Refining the distance scale. Table 3 lists the kinematic parameters inferred treating the distancescale factor as an unknown parameter. The initial distances to Cepheids and OSC are on the short distance scale. As is evident from the table, the distancescale factor depends only slightly on the adopted R0 . Judging by these results, the adopted distance scale should be increased by 14­16%. Besides the rigorous method of statistical parallaxes, we also used its simplified version, which involves comparing the values of the first derivative of angular velocity inferred separately from radial velocities with proper motions. We determined the kinematic and rotation-curve parameters of the sample under study from independent maximum-likelihood solutions based on radial velocities and proper motions. It can be easily seen that radial velocities of stars of flat subsystems allow neither w0 nor w to be accurately constrained. We therefore inferred in two ways: (1) by computing the heliocentric space velocity components u0 ,0 ,w0 and vertical velocity dispersion w of the sample under study from space velocities, and then fixing these values in separate radial velocity and proper-motion solutions; and (2) by substituting u0 and 0 inferred from radial velocities into the proper-motion solution and substituting w0 and w inferred from proper motions into the radial velocity solution. Table 4 lists the resulting and distance-scale factors p = (µ)/ (Vr ).
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The resulting mean distance-scale factor for the Cepheid+OSC subsystem is equal to p = 0.96 for R0 = 7.5 kpc (with = -4.50 km s-1 kpc-2 ) and kpc (with = p = 0.97 for R0 = 8.5 -3.95 km s-1 kpc-2 ). Again, we note a weak dependence of the distance-scale factor on the adopted solar Galactocentric distance (see Table 5). Noteworthy are (see Tables 3 and 5) systematic differences between the distance-scale factors given by the statistical-parallax technique ( 0.86) and by its simplified modification ( 0.96). We analyzed the problem for possible biases using numerical simulations. To this end, we used the real coordinates and initial distances to the objects of our sample and simulated their "true" space velocities based on the earlier determined values of kinematic and rotation-curve parameters. We then added normally distributed errors to the "true" distances and space velocities and redetermined the kinematic parameters and distance-scale factor using both the rigorous statistical-parallax technique (space velocities) and its simplified modification. We set velocity errors based on the typical errors of observational data and ellipsoidal distribution of residual velocities. One hundred numerical simulations yielded a mean distance-scale factor of p = 1.00 ± 0.05 and p = 1.01 ± 0.07 by making inferences from space velocities or by comparing the first derivatives of angular velocity, respectively. The possible distance-scale factors were confined to the (0.85­1.15) interval, with, on the average, correlated deviations of the two values from unity. We cannot unambiguously choose between the two approaches to the distancescale refinement. Since it is logical to associate the short and long distance scales with R0 = 7.5 kpc and R0 = 8.5 kpc, respectively, hereafter we inferred the kinematic parameters assuming that the Cepheid+OSC distance-scale factors of p = 0.96 and 0.84 correspond to R0 = 7.5 and 8.5 kpc, respectively. In support of this conclusion, we determined R0 from space velocities with fixed p. Our analysis yielded R0 = (7.4 ± 1.0) and (8.3 ± 1.0) kpc for p = 0.96 and 0.84, respectively. The large errors of the resulting R0 are due to the small size of the data sample used. Determination of 0 from radial velocities using Lundblad's relation. Note that the fact that the velocity dispersions of Cepheids and OSC obey the Lindblad relation allows the angular velocity of rotation 0 at the solar Galactocentric distance to be estimated independently from radial velocities exclusively. When computing the kinematic parameters, the idea is to set as unknown only the radial velocity 0 dispersion u at the solar Galactocentric distance and to determine the ratio of velocity ellipsoid axes from the Lindblad relation while setting the angular velocity and its derivative equal to their local values for each


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Table 5. Kinematic parameters of the Cepheid + OSC sample inferred from space velocities using the mean distancescale factors (determined by comparing the first derivatives of angular velocity) R0 ,kpc 7.5 8.5 u
0

0

w0 km s

u
-1

7.92 8.04

w km s 7.55 7.44

0 -1

, kpc

-1

km s

-1

, kpc

-2

km s

-1

, kpc 1.09 0.90

-3

-6.55 -12.11 -7.25 -6.38 -12.98 -7.18

13.04 12.80

27.47 27.37 ±1.39

-4.54 -3.99 ±0.24

Standard errors ±1.77 ± 1.71 ±1.24 ±1.49 ±1.10 ±1.22 Table 6. Kinematic parameters of the blue-supergiant sample R0 ,kpc 7.5 8.5 u
0

±0.19

0

w0 km s

u
-1

8.96 9.41

w km s 5.13 5.71

0 -1

, kpc

-1

km s

-1

, kpc

-2

km s

-1

, kpc 0.89 0.60

-3

-6.04 -10.92 -7.12 -6.18 -11.33 -7.86

11.49 11.63

29.60 29.14 ±1.62

-4.76 -4.00 ±0.32

Standard errors ±1.93 ±1.73 ±1.08 ±1.42 ±1.14 ±1.03 Table 7. Kinematic and rotation-curve parameters inferred from H II data R0 ,kpc 7.5 8.5 Standard errors u
0

±0.53

0 km s
-1

u 6.70 6.56 ±2.07

km s 6.91 7.03 ±1.14

-1

, kpc

-2

km s

-1

, kpc 1.26 1.01

-3

-8.11 -7.92 ±1.71

-14.88 -15.73 ±1.19

-4.77 -4.08 ±0.27

±0.22

object. The resulting likelihood function therefore depends explicitly on the unknown angular velocity 0 . We applied this method with fixed w0 = -7 km s-1 and w = 7 km s-1 to the radial velocities of Cepheids and OSC with p = 0.96 and R0 = 7.5 kpc to obtain 0 = (26.5 ± 8.7) km s-1 kpc-1 . The large error of the inferred angular velocity is fully explained by the errors of the inferred velocity dispersions, which are equal to 1.2­1.7 km s-1 (see Table 5). The surprisingly good agreement between the angular velocity values inferred from space and radial velocities indicates that the Lindblad relation is obeyed accurately enough throughout the entire space region studied.

Kinematics of the Blue-Supergiant Sample
We applied the maximum-likelihood technique to a sample of 102 blue supergiants. The dispersion of the inferred absolute magnitudes for these stars is higher than for Cepheids and OSC and is equal to M 0m38 (Dambis 1990). We estimated . the distance-scale factor using the two methods

described above. The maximum-likelihood method applied to space velocities of stars yielded p = 0.97 ± 0.08, whereas a comparison of the first derivatives of angular velocity determined separately from radial velocities and proper motions yielded p = 1.09 ± 0.16 (error estimated approximately). Both results agree fairly well with the correction factor to the bluesupergiant distance scale (p = 1.03 ± 0.04) inferred from practically the same sample by comparing photometric and HIPPARCOS trigonometric parallaxes (Dambis et al. 2001). As in the case of the Cepheid and OSC sample, the distance-scale correction factors given by the two methods differ systematically by 0.1. Table 6 lists the final kinematic parameters for the blue-sueprgiant sample. Blue supergiants yielded somewhat higher angular velocity 0 compared to what we inferred from the Cepheid and OSC sample, but the difference is within the quoted errors. The systematic difference between the two angular velocity values is partly due to the specifics of the space distribution of objects involved. The most reliable estimates of angular velocity are those inferred from objects lying in the vicinity of the
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KINEMATIC PARAMETERS OF YOUNG SUBSYSTEMS Table 8. Galactic rotation curve V (R) for the short and long distance scales R,kpc 2. 2. 2. 2. 2. 2. 2. 3. 3. 3. 3. 3. 3. 3. 4. 4. 4. 4. 4. 4. 4. 4. 5. 5. 5. 5. 5. 5. 5. 6. 6. 6. 6. 6. 6. 6. 7. 7. 7. 7. 7. 7. 7. 8. 0 1 3 4 6 7 8 0 1 3 4 5 7 8 0 1 2 4 5 7 8 9 1 2 4 5 6 8 9 1 2 3 5 6 8 9 0 2 3 5 6 7 9 0 V (R), km s-1 R0 = 7.5 kpc 198. 199. 201. 202. 203. 204. 204. 205. 205. 206. 206. 207. 207. 208. 208. 209. 210. 210. 211. 211. 212. 212. 212. 213. 213. 213. 213. 213. 213. 213. 212. 212. 211. 211. 210. 209. 208. 208. 207. 206. 205. 204. 203. 202. 1 9 3 4 3 0 6 2 7 2 7 2 8 3 9 5 1 6 2 7 2 6 9 2 4 5 5 4 3 0 7 2 7 1 5 7 9 1 2 3 4 4 5 5 V (R), km s-1 R0 = 8.5 kpc 198. 201. 203. 205. 207. 209. 210. 211. 212. 213. 213. 214. 215. 216. 216. 217. 218. 219. 219. 220. 221. 222. 223. 223. 224. 225. 226. 226. 227. 227. 228. 228. 229. 229. 229. 229. 229. 229. 229. 229. 229. 228. 228. 228. 2 3 8 9 6 0 3 4 3 2 9 7 4 2 9 6 4 1 9 7 5 3 1 8 6 3 0 6 2 7 2 6 0 2 4 5 6 6 5 3 0 8 4 0 R,kpc 8. 8. 8. 8. 8. 8. 9. 9. 9. 9. 9. 9. 9. 10. 10. 10. 10. 10. 10. 10. 11. 11. 11. 11. 11. 11. 11. 11. 12. 12. 12. 12. 12. 12. 12. 13. 13. 13. 13. 13. 13. 13. 14. 2 3 4 6 7 9 0 1 3 4 6 7 8 0 1 3 4 5 7 8 0 1 2 4 5 7 8 9 1 2 4 5 6 8 9 1 2 3 5 6 8 9 0 V (R), km s-1 R0 = 7.5 kpc 201. 200. 199. 199. 198. 197. 196. 196. 195. 195. 195. 194. 194. 194. 194. 194. 194. 195. 195. 195. 196. 196. 197. 198. 199. 200. 201. 202. 203. 205. 206. 208. 209. 211. 213. 215. 218. 220. 223. 226. 230. 233. 237. 6 7 8 0 3 6 9 3 8 4 1 8 6 5 5 6 8 0 4 8 3 9 6 4 2 2 2 4 6 0 4 0 8 6 7 9 3 9 7 7 0 6 4 V (R), km s-1 R0 = 8.5 kpc 227. 227. 226. 225. 225. 224. 224. 223. 222. 222. 221. 220. 220. 219. 219. 219. 218. 218. 218. 218. 218. 218. 218. 218. 218. 219. 219. 220. 221. 221. 222. 223. 224. 226. 227. 228. 229. 231. 233. 234. 236. 238. 239. 5 0 4 8 2 6 0 3 7 1 5 9 4 9 4 0 7 5 3 2 1 2 3 5 9 3 8 4 1 9 8 7 8 0 2 5 9 4 0 6 3 0 8

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462
­1

ZABOLOTSKIKH et al.

V, km s

200 150 4

R0 = 7.5 kpc R0 = 8.5 kpc

250

V, km s­1

8 R, kpc

12
150
Rotation curve HI H II Cepheids + OSC Supergiants

Fig. 2. Galactic rotation curve V (R) for the short (R0 = 7.5 kpc) and long (R0 = 8.5 kpc) distance scales.

"tangent circle" (it is the circle in the Galactic plane with the interval connecting the Sun and the Galactic center as its diameter), because, in the corresponding conditional proper-motion equations, the coefficients at the angular-velocity derivatives are close to zero (Glushkova et al. 1999). The proper motions of 23 blue supergiants lying in the vicinity of the "tangent circle" yielded 0 = (27.91 ± 2.79) km s-1 kpc-1 ; i.e., the angular velocities inferred from two samples agree well with each other. This result justifies the subsequent use of blue supergiants for constructing the combined rotation curve over a wide interval of Galactocentric distances.

50

4

8 R, kpc

12

Fig. 3. Galactic rotation curve V (R)for the short distance scale (R0 = 7.5 kpc) with young-object data points superimposed.

Kinematics of Ionized Hydrogen
The only way to match the distance scales of H II and stars is to compare the first derivatives of angular velocity inferred from line-of-sight and space velocities for gas and stars, respectively. Given that the scatter of velocities along the z -coordinate has virtually no effect on the radial velocities of the thin-disk objects, we fixed w0 = -7 km/s and w = . Table 7 lists the kinematic and rotationcurve parameters inferred from H II data for r < 4 kpc. In this interval of Galactocentric distances the first derivatives of angular velocity for gas (inferred from radial velocities) and stars (inferred from space velocities) are estimated at either ­4.77 and ­4.54 km s-1 kpc-2 , respectively (if R0 = 7.5 kpc) implying the H II distance-scale correction factor p = 0.95, or ­4.08 and ­3.66 km s-1 kpc-2 , respectively (if R0 = 8.5 kpc) implying the H II distance-scale correction factor p = 0.90. Note that, as expected, the velocity ellipsoid axes inferred for gas do not obey the Lindblad relation, but u .

V (R) inferred from the entire sample of young objects (OSC + Cepheids + supergiants + H I + H II) for R0 = 7.5 and 8.5 kpc. These rotation curves are tabulated in Table 8. Here we expanded the difference of angular velocities into a seventh- order Taylor series in the vicinity of R0 , and computed the velocity ellipsoid axes separately for neutral and ionized hydrogen, blue supergiants, and for Cepheids + OSC (see Table 9) with the distances to all objects matched to each other. The resulting local centroid velocity and Oort's constant A are equal to V (R0 ) = (206 ± 10) km s-1 , A = (17.1 ± 0.5) km s-1 kpc-1 and V (R0 ) = (226 ± 12) km s-1 , A = (15.4 ± 0.6) km s-1 kpc-1 for the short and long distance scale, respectively. Figure 3 shows the rotation curve for the short distance scale (R0 = 7.5 kpc) with the data points for individual objects computed using the following formula: R V = R 0 + (Vr - Vsun,r ), R0 sin l cos b where Vsun,r is the radial projection of the heliocentric velocity of the sample considered. As is evident from the figure, the scatter of V about the rotation curve is due mainly to small sin l. Note that systematic differences between HI velocities may be manifestations of a barlike structure at the center of the Galaxy (Freudenreich 1998). The gas-stellar disk of the Galaxy is well known to show appreciable warp in the direction l 90 at Galactocentric distances > 10 kpc. Our rotation curve therefore applies only to the part of the Galactic disk where warp is insignificant.
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Constructing the Rotation Curve
The good agreement between the mean heliocentric velocity components of different young-object samples allows us to construct the rotation curve over a sufficiently wide interval of Galactocentric distances, 2­14 kpc, using radial velocities of both stars and gas. Figure 2 shows the rotation curves


KINEMATIC PARAMETERS OF YOUNG SUBSYSTEMS Table 9. Heliocentric velocity components and velocity-ellipsoid axes for the young-object sample Objects Cepheids + OSC Standard errors Supergiants Standard errors HII Standard errors HI Standard errors


463

u , 13.30 ±1.79 14.17 ±0.51 6.71 ±0.60 6.60 ­

, 7.59 ±0.41 10.00 ±2.12 7.19 ±0.93 6.05 ±0.34

w ,

u km s
-1

0

0 -12.98 » » » » » » ±0.78

w0

7.55 ­ 5.13 ­ 5.0 ­ 5.0 ­

-9.17 » » » » » » ±0.48

-7.25 » » » » » » ­

Parameters fixed at values inferred from the space-velocity solution.

Table 10. Parameters of the spiral pattern (R0 = 7.5 kpc) Objects Cepheids OSC Standard errors -stars Standard errors 2 m 2 4 f
R

f km s
-1



i deg -6.02 -12.18 ±0.72 -6.55 ±0.86



0

-6.66 -5.51 ±2.34 -6.64 ±2.51

-1.40 -0.16 ±1.56 0.42 ±2.31

-85.19 -88.05 ±14.50 -97.28 ±18.30

Allowing for Spiral-Arm Effects
Our computations showed that young-object samples lag behind the Sun on the average by 13 km s-1 . Dehnen and Binney (1998) used HIPPARCOS proper motions and parallaxes of nearby MS stars to find out that the Sun moves ahead of the LSR by 5.25 km s-1 . It follows from this that young subsystems lag behind the LSR by 8 km s-1 , whereas their velocity dispersion should imply a velocity lag of 1.5 km s-1 . This discrepancy may be due, among other things, to streaming motions induced by spiral arms. To allow for the spiral-arm effects in the velocity field, we performed our computations in terms of twoand four-armed models of the spiral pattern (see Table 10). The phase of the Sun with respect to the spiral wave, which is close to -/2, indicates that the Sun is situated at the outer edge of the arm; V 1.4 km s-1 (for the Cepheid + OSC sample). We therefore face a discrepancy between the magnitude and direction of the tangential disturbance as inferred from the centroid velocity lag behind the LSR (V
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-6.5 km s-1 ) and the value of the same quantity computed directly in terms of a model of spiral-pattern effects in the velocity field. The discrepancy is beyond the quoted errors. The velocity lag of the centroid of young objects relative to the LSR can be explained by, among other things, noncircular motions of the LSR discussed by a number of authors (Schuter 1982; Clemens 1985) based on their analyses of H I radial velocities. Thus Clemens (1985) inferred an LSR tangential velocity of 7 km s-1 from an analysis of H I motions in the local solar neighborhood. However, Dehnen and Binney (1998) showed that, despite their different ages and velocity dispersions, all mainsequence stars (except late-type -stars) closely follow a unified theoretical dependence of the sample tangential velocity on velocity dispersion. Streaming motions should be "washed out" by ever increasing velocity dispersion, and we therefore consider the above determinations of the solar velocity relative to the LSR to be quite correct, and thus the discrepancy in question is left unexplained. Interestingly, the heliocentric velocity of -type stars inferred by Dehnen and Binney agrees well with the velocities we inferred for the young-object samples.


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ACKNOWLEDGMENTS We are grateful to A.M. Mel'nik, E.V. Glushkova, and V.G. Surdin for their assistance and valuable discussions. The work was supported by the Russian Foundation for Basic Research, grant nos. 0102-06012, 00-02-17804, 99-02-17842, and 01-0216086; Astronomy State Research and Technology Program; and the Council for the Support of Leading Scientific Schools, grant no. 00-15-96627. REFERENCES
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Translated by A. Dambis
ASTRONOMY LETTERS Vol. 28 No. 7 2002