CHINESE ASTRONOMY AND ASTROPHYSICS

PERGAMON Chinese Astronomy and Astrophysics 26 (2002) 267-275

A Study on the Sagittarius-Carina Arm and the Halo Mass of Our Galaxy+ *

ZHANG Ming’y2 HAN Jin-lin’ PENG Qiu-he213t4 1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012

2Department of Astronomy, Nanjing University, Nanjing 210093

3 Chinese Academy of Science-Peking University Joint Beijing Astrophysical Center,

Beijing 100871

40pen Lab of Co smic Rays and Space Astronomy, Chinese Academy of Science, Beijing

100039

Abstract On the basis of the new Galactic rotation curve and constants &

and Vo, and using HI1 regions, giant molecular clouds and other tracer objects,

we have revised the pattern of the Sagittarius-Carina arm of our Galaxy. We

have derived its pitch angle to be about 12”, and explored the dependence of the

pitch angle on the Galactic constants. The result indicates that with decreasing

Galactic constants, the pitch angle increases, and so does the number of spiral

arms, according to the symmetric logarithmic arms model. In addition, the halo

mass of our Galaxy will decrease with the flattening and downward turn of the

rotation curve.

Key words: Galaxy-spiral structure-halo

1. INTRODUCTION

Since Georgelin et al.1’1 presented the 4-spiral arm structure of our Galaxy based on HI1

regions, the controversy on the number of spiral arms has never stopped. On the assumption

that the spiral arm has the simple logarithmic spiral form, discriminating the number of arms

will rely on the determination of the pitch angle of the arms 121. Because our solar system is

situated inside the Galaxy, we can not observe the structure of our Galaxy as directly as we

t Supported by National Natural Science Foundation Received 2000-11-09; revised version 2001-05-08

l A translation of Acto Astrun. Sin. Vol. 43, No. 1, pp. 24-32, 2002

0275-1062/02/$-see front matter 0 2002 Elsevier Science B. V. All rights reserved. PII: SO275-1062(02)00066-8

268 ZHANG Ming et al. / Chinese Astronomy and Astrophysics 26 (2002) 267-275

observe extragalactic systems, but we can identify the spiral arms of the Galaxy by observing their tracer objects. In the optical band, the tracer objects are O-B stars, optical HI1 regions, cepheid variables etc. Because of light extinction in the Galactic disk, optical observations become ineffective for gaining an insight into the global structure of the Galaxy, so people have turned their attention from the optical to the radio. The 21 cm HI line has played an important role in the exploration of the Galactic structure, but the continuous nature of the distribution of neutral hydrogen makes it difficult to distinguish clearly features caused by flow, mass concentration, or temperature variation from the integrated spectrum profile131. The discrete HII regions do not have such a weakness, so they have gradually become the main tracer objects of Galactic structure. But when identifying Galactic spiral arms by the radio observations of HI1 regions or giant molecular clouds (GMCs), the determination of the distance is still an important and thorny problem. Excepting for a few HI1 regions and GMCs whose distances can be measured directly by optical measurements or associated objects, the distances can only be determined by kinematics. This in turn leads to two problems. Firstly, kinematic distance is model-dependent. Secondly, inside the revolution circle of the Sun, it is only for some of the objects that the distance is unambiguous, that is when the the line of sight is tangent to the object’s revolution circle, and the apparent velocity is the end velocity. In this case R = R,\sinII; but for most objects we can not, without some other criterion, select correctly one of the two possible (near and far) kinematic distances, i.e., there exists an ambiguity. With the help of other criteria, we are gradually getting acquainted with Galactic structure. The calculation of Galactic spiral arms requires Galactic constants and a rotation curve; the Galactic constants will influence the rotation curve, the calculation of kinematic distance, and hence the calculation of the spiral arms, in particular, their pitch angles. To examine the influence of the Galactic constants on the spiral arm structure, we have used new Galactic constants and rotation curves in a re-determination of the Sagittarius-Carina arm and a fitting of its pitch angle. In addition, using the 3-dimensional model of the Galactic structure given by Peng et al.141, the halo mass of the Galaxy has been fitted separately using the different Galactic rotation curves.

2. DETERMINATION OF SAGITTARIUS-CARINA ARM

2.1 The Basic Formulas On the assumption that Galactic matter revolves in circular orbits, we have

KSR = sV(R) - Vo ,

sin2 cosb R (1)

in which VLSR is the apparent velocity of the object. relative to the local standard of rest (LSR); R and V(R) are respectively the galactocentric distance and the circular r* tation velocity of the object; Re and V, are the galactocentric distance of the Sun and the circular rotation velocity at the position of the Sun; and Z&J are the galactic longitude and latitude of the object. For objects near the Galactic plane, lb1 < 2”, cosb N 1, so, as soon as the constants RQ,VO are given, by using the known rotation curve and the measured apparent velocity, the galactocentric distance R can be obtained from formula (1). Then from the geometry the kinematic distance between the tracer object and the Sun can be derived. If

ZHANG Ming et al. / Chinese Astronomy and Astrcphysics 26 (2002) 267-275 269

our Galaxy has a simple logarithmic spiral structure, then in the polar coordinate system

centered at Galactic center, we will have

13-f&= (tanp)-‘In k , 0

in which, p is the pitch angle of the logarithmic spiral arm, r, 8 are the polar radius and polar

angle, and TO, 00 their initial values. In the semi-log coordinates (In T, 19), the logarithmic

spiral is a straight line with slope tanp and intercept lnrs - 80 tan p. Thus, the pitch angle

of the spiral arm can be derived by linear fitting.

0 5 10

Wpc)

15 20

4(WW Fig. 1 Four rotation curves used to calculate the spiral structure of the Galaxy

2.2 Rotation Curves

Before 1985, the rotation curve used in the study of the Galaxy was mainly the Schmidt

model151, and the Galactic constants were taken as Rc, = 10 kpc, Vi = 250 km/s. The present

IAU recommendation (1985) is (8.5kpq 220km/s}. In 1993, Brand et a1.161, based on the

IAU recommended values, fitted the following analytic form to the rotation curve:

V R aa - = a1 vo ( > Ro + a3 , (3)

in which ai = 1.00767, a2 = 0.0394, and as = 0.00712. But now more studies have demon-

strated that the Galactic constants &, and VO should be smaller17-lo]. When the Galactic

constants {&, VO} take the values (8 kpc, 200 km/s}, the Galactic rotation curve becomes

flatter and approximately linear; and when they take the values (7.1 kpc, 184km/s}, the

curve even slopes downwardlgl. In this paper, the influence of the downward turn of the

270 ZHANG Ming et al. / Chinese Astronomy and Aetrophysics 26 (2002) 267-275

rotation curve on the pitch angle of the spiral arm is studied by using the linear approxi- mation 6 = 1.05 - 0.05(%) to a curve of this form given by Ollmg et a1.1~1 The 4 rotation curves mentioned so far are shown in Fig.1. From top to bottom, they are A: Schmidt161 model plus a flat outer part {&, Vs}={lOkpc, 25Okm/s}; B: The fitting solution of Brand et al. {&, Vo}={8.5 kpc, 22Okm/s}; C: Flat rotation curvel*~‘*l {&, Vs}={8.Okpc, 2OOkm/s); D: Approximation to a downward rotation curve used for simulation in this paper {&, Vi}= (7.1 kpc, 184km/s}.

2.3 Difficulties and Strategies As mentioned before, in the course of acquiring a knowledge of Galactic structure, the

biggest trouble is the uncertainty of the kinematic distance in the Galactic disk. To overcome this difficulty, we have consulted and compared the data of &swell et alP1, Downes et al.112i, Grabelsky et al.113i, and Dame et aLli to see to what extent the Galactic constants and rotation curve influence the Galactic structure. We use the following procedure to remove as much as possible the ambiguity in the kinematic distance:

log(Sd’ 10.1 Jy k$)

Fig. 2 The luminosity function of H II regions, with data taken from &swell et all”1

(1) If the tracer object has an optical counterpart with known distance, then we adopt the distance of the latter.

(2) For a source observed before, if its kinematic distance has been selected according to absorption or to the existence of an optical counterpart, then this selection will be noted in the removal of the ambiguity.

(3) Referring to the luminosity function of HI1 regions displayed in Fig.2, a far or near solution is selected so as to make the resulting luminosity to have a greater probability. In the x-axis of Fig.2, S stands for the radiation flux of the HI1 region, d, its distance from the observer, so Sdl is its luminosity. It is apparent that most HII regions are concentrated in

ZHANG Ming et al. / Chinese Astronomy and Astrophysics 26 (2002) 267-275 271

the neighborhood of log(Sd2/0.1 Jy kpc2)=3.5.

(4) We take into consideration the continuity of apparent velocity and distance along

the arm.

2.4 Determination of Spiral Arm Using HI1 Regions and CO GMCs When we select a spiral arm for fitting, the brightness of HI1 regions and the mass of

CO GMCs are our most important targets. As Georgelinsl’l pointed out, only the bright,

extended sources are tracers of spiral arms. But, from a jumble of gas clouds it is very

difficult to identify which ones belong to a given arm. In the semi-log coordinates of Fig.3,

we marked out inclined (inclination angle ll”), 1.2 kpc-wide strips and counted the number

of tracer objects within each strip. We then selected the strip with the maximal count (i.e.,

when the counts in its neighboring strips are less than one-half of its value) as the possible

location of an arm. In this figure, squares represent CO GMCs, circles represent HI1 regions,

and the size of these symbols is proportional to the stellar magnitude. The upper diagram is

for Galactic constants {&, Vo}={ 10 kpc, 250 km/s}l”-‘41, and the lower diagram is newly

calculated using the IAU recommended values {&,, V0}={8.5 kpc, 220 km/s} and the new

rotation curve161. The area between the two parallel lines 1 kpc apart is the most likely

location of the Sagittarius-Carina arm. In Fig.3, we can see this arm very clearly. Therefore,

using the data of HI1 regions given by Caswell et al.ll’l, Downes et a1.[121, and the data of

CO GMCs given by Grabelsky et a1.1131, Dame et al.1141, positions are calculated for points

in this area with different Galactic constants and rotation curves and a weighted fitting is

made. The selection of the area of the spiral arm is shown in Fig.3 and Fig.4. The pitch

angle is fitted separately for the HI1 regions and the CO GMCs, and the results are listed

in Table 1. The results show that the spiral arm structures traced by the HI1 regions and

the CO GMCs are quite consistent with each other. Hence, below we will use both the HI1

regions and CO GMCs as the arm tracers when studying the dependence of the spiral arm

structure on the Galactic constants.

Table 1 The consistency between the pitch angles of the Sagittarius-Carina arm determined by

HI1 regions and CO GMCs

tRo,Vol { 10, 250) (8.5. 2201

H II regions go.9 f 1.3 lOO.7 f 1.5

CO GMCs

9O.9 f 0.6 llO.2 f 0.6

rotation curv Schmidt model plus outer flat

Brand & Blitz

2.5 Influence of Galactic Constants on the Pitch Angle

Using HI1 regions and CO GMCs, we have defined and fitted the Sagittarius-Carina

arm separately with 4 different rotation curves. The results are summarized in Table 2, so

showing how the pitch angle varies as the shape of the rotation curve is changed from a

slight upward turn, through flat, to a slight downward turn. From Table 2, it is apparent

that as the Galactic constants become smaller and the rotation curve becomes flatter or

even turning downward, the pitch angle is gradually increased.

272 ZHANG Ming et al. / Chinese Astrvnomy and Astrophysics 26 (2002) 267-275

MWW 0 WJY b& 0 IO’ 0 OICW

0 106 0 low

0 IO0

0 10* - 10

50 c

100 150 200 250 300 35 e(*)

0

0

~=OGJ 0

0

0 l!J@* 0 @CJy b’) 0 10’ 0 Klooo

0 106 0 loo0

t-4. “8 0 100

0 105 - 10

k

0 50 100 150 200 250 300 39

8(‘)

Fig. 3 The Sagittarius-Carina arm plotted in semi-log coordinates

ZHANG Ming et al. / Chinese Astronomy and Astrophysics 26 (2002) 267-275 273

Distance from the Sun: X (kpc)

Fig. 4 Sagittarius-Carina arm traced by HI1 regions and CO GM&. See keys in Fig.3. The distances are determined by the rotation curve fitted by Brand & Blitz. The dotted lines are possible areas of Sagittarius-Carina arm, and the full line is the fitted logarithmatic spiral.

Table 2 The pitch angle of the Sagittarius-Carina arm fitted with different galactic constants

and rotation curves

{&,vol (10, 250) (8.5, 220) {8, 200) {7.1,184}

pitch angle 9O.7 f 0.6 11O.3 f 0.7 12O .3 f 0.7 12O.7 f 0.7

rotation curve Schmidt model plus outer flat

Brand & Blitz flat

declined

VallCe121 has collected the results of Galactic spiral arm studies from 1980 to 1995, and

given 12’ as an average estimate for the pitch angle of the arms. Based on a symmetric

logarithmic spiral model, he concluded that the pitch angle for an 8-arm structure is 25”.8,

for a g-arm structure is 19”.9, for a 4-arm structure is 13”.6, and for a a-arm structure is

6O.9. The size of the pitch angle reflects the degree of tightness of the arms, a larger pitch

angle goes with a larger number of arms. The pitch angle we derived is 12”.

274 ZHANG Ming et al. / Chinese Astnwzomy and Astrophysics 26 (2002) 267-275

3. HALO MASS OF OUR GALAXY

Most astronomers believe that in order to explain the rotation curve and to satisfy the virial theorem, the Galaxy needs a halo which contains most part of dark matter[‘51. Therefore, the Galactic rotation curve is especially important for estimating the mass of the Galactic halo. Based on the 5-component model (bulge, nuclear bulge, thin disk, thick disk, halo) given by Peng et al. 141, the mass of Galactic halo is fitted separately with the fitted rotation curve of Brand et al. and the flat rotation curve. The result showing the trend of its variation is given in Table 3, in which 7 is defined as the ratio between the halo mass inside the border of the optically visible disk (r, = 16 kpc) and the mass of the thin disk, and & (50 kpc) is the halo mass within the 50 kpc Galactic radiusf41. From Table 3, we find that the flat rotation curve demands a smaller halo mass.

Table 3 The halo mass on different Galactic constants and rotation curves

{&I, vo} (10.250)

rl Mh(5Okpc)(lO” MO) rotation curve 3.42 20.87 Schmidt model ~1~s outer flat 2.06 11.68 Brand & Blitz 1.78 10.54 flat 1.23 4.91 declined

4. DISCUSSION

In the study of Galactic structure, we have followed usual assumptions. For example, the formula (1) for calculating the rotation curve is based on the model of circular orbits, al- though non-circular movements certainly exist 13J61. When we estimate the number of spiral arms in our Galaxy, a simple, symmetric logarithmic spiral structure is assumed, but re cent studies have indicated that our Galaxy has a more complicated structure, such as the structure of 2+4 arms1171. This paper is not intended to solve these problems thoroughly, because to describe precisely the details of Galactic structure, a more complicated physi- cal model and more elaborate computations are required: our discussion is limited to the generally accepted structure. In a traditional way, this paper has studied the dependence of the Galactic structure on some determined Galactic constants. The result indicates that although a flattening or a downward turn of the rotation curve decreases the kinematic distances, so causing the Galaxy a tendency to tighten up, the simultaneous decrease in the Galactic constant l& causes the pitch angle of the arms a tendency to increase rather than to decrease. For the symmetric simple logarithmic spiral structure, the fitted pitch angle of the spiral arm is about 12”.

Based on the weighted average of the results obtained by different authors and tracer objects, Vallde has derived the value of the pitch angle of Galactic spiral arm, and it is also about 12’. The Galactic rotation curve used at that time was still based on the values of the Galactic constants of {&, Vo)={lOkpc,250km/s}. For the present Galactic constants and rotation curve, the pitch angle of Galactic spiral arm should be 14”, but es stated above, it turned out to be not so large. In our study on the spiral arm structure of the Galaxy, only the Sagittarius-Carina arm is considered. For future work we plan to collect more data to

ZHANG Ming et al. / Chinese Astronomy and Astrophysics 26 (2002) 267-275 275

determine the other spiral arms and to study the global structure of the Galaxy. And further

determinations of the Galactic rotation curve will help us with a more exact knowledge of

Galactic structure and of the characteristics of dark matter.

ACKNOWLEDGEMENT We thank Dr. CHEN Yang of Beijing Normal University

and Mr. SUN Xiao-hui of Peking University for their enthusiastic help in the preparation

of this paper.

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