New Astronomy ELSEVIER New Astronomy 3 (1998) 419-425

Determination of some galactic constants using planetary nebulae

A. Alia” ‘2, M.A. Sharafb’3 “Institut fiir Astronomic der Leopold-Franzens-Universittit Innsbruck, Technikerstrasse 25, A-6020 Innsbruck. Austria

hCairo University, Fact&y of Science, Astronomy Department, Giza, Egypt

Received 11 June 1998; accepted 14 July 1998

Communicated by Peter S. Conti

Abstract

In this paper, Planetary Nebulae (PNe) are considered as a stellar system by which we succeed in determining the following Galactic constants: the Galactic pole coordinates (LYE, S,), which are used for coordinate transformation in Galactic astronomy, the velocity and apex of the solar motion as an important kinematical parameter for stellar groups, Oorts’s first c&stant (A) and the rotation curve for the theory of Galactic rotation. An analytical method is introduced to determine the Galactic pole. This method depends on the constrained optimization technique. A general computational algorithm for the procedure was established in order to find exact solutions for the equations involved. The methods used for the solar apex and speed, and also the determinations of Oort’s first constant depend on the least squares technique. The velocity ellipsoid components are determined as a by-product from the solar elements. The resulting rotation curve derived by our sample is found to be similar to that obtained from planetary nebulae type I. We have derived the following constants: Galactic pole coordinates (Ye = 12h49m, S, = 27”22’, the solar apex an = 290”, S, = + 30”, solar speed S, = 18.6 kms-‘, and Oort’s first constant A = 15.2kms-‘kpc-‘, which are in good agreement with the standard and averaged values. 0 1998 Elsevier Science B.V. All rights reserved.

PACS: 98.38.L~ Keywords: Planetary nebulae: general; Galaxy: kinematics and dynamics; Galaxy: fundamental parameters

1. Introduction

The distribution of the PNe according to Galactic latitudes shows a remarkable concentration to the

Galactic plane (see Pottasch, 1992). Consequently this tempted us to consider PNe as a candidate for the determination of the Galactic equator. The north Galactic pole has been determined from Cepheid

‘Permanent address: Cairo University, Faculty of Science, As-

tronomy Department, Giza, Egypt.

*E-mail: AlaaOast7.uibk.ac.at

‘E-mail: MAS@frcu.eun.eg

variables (Ashbrook & Duncombe, 1952). Sharaf et

al. (1987) have develped an optimization technique to determine the plane containing the maxmium condensation of a group of objects. The optimization technique has been applied by Sharaf et al. (1997) to

determine the Galactic pole using Galactic Cepheids. Bukhari (1988) also used the same technique to determine the pole for a cluster of galaxies.

The solar motion has been determined by numer- ous authors with respect to a great range of stellar and interstellar constituents of the Galaxy, and there is extensive literature on the subject (e.g. Latypov, 1979; Jaschek & Valbousqet, 1991, 1992, 1993;

1384-1076/98 - see frontmatter 0 1998 Elsevier Science B.V. All rights reserved.

PII: S1384-1076(98)00022-O

420 A. Ali, M.A. Sharaf I New Astronomy 3 (1998) 419-425

Glushkova et al., 1998). The values vary, however, considerably from one author to another, and it is unclear whether this is due to the different methods

used or to the variety of samples. Determination of the position of the solar apex and the speed of the

solar motion using planetary nebulae was considered by Cudworth (1974), and Mihalas & Binney (1981).

The first author used the radial velocities for two classes of planetary nebulae (according to the classi- fication of Greig, 197 1, 1972). There was no signifi- cant differences between the solar velocities for the

two classes, in spite of large differences in the velocity ellipsoid components. Mihalas & Binney

(1981) mentioned the values of the apex coordinates. velocity of the solar motion, and the velocity ellip-

soed components. Their results differed much from those of Cudworth (1974).

2. Data samples and selection criteria

Our samples were collected from the Strasbourge- ES0 catalogue of planetary nebulae (Acker et al.,

1992). In determining the coordinates of the north Galactic pole the objects were selected to be near to

the Galactic plane, because the main criterion for determining the Galactic pole depends on choosing the objects that exhibit a strong concentration near the Galactic plane. We have selected PNe distributed in the range - 1.6 5 b 5 1.6, the sample consists of 98 PNe.

The determination of the Galactic rotation curve is

normally based on the position and kinematics of young population I objects, such as H II regions or H II/CO complexes. However, other classes of objects belonging to later populations may also give some contribution to this problem, especially when they

are bright, numerous and with reasonably accurate positions and velocities. As discussed by Schneider

& Terzian (1983) and Maciel (1987), (1989), planetary nebulae fulfil these conditions, provided that we have some means of selecting the younger

planetary nebulae populations. Maciel & Dutra ( 1992) studied a sample containing 150 planetary

nebulae classified according to the types of Peimbert (1978). They represented a sequence of decreasing mass (and increasing age) from type I (thin disc PNe) to IV (halo objects). Some kinematical consequences regarding the connection between planetary nebulae and H II regions are explored, leading to the determination of the Galactic rotation curve and the

Oort’s constants. Amaral et al. (1996) derived the Galactic rotation curve, using young planetary nebulae, oxygen-rich and carbon-rich asymptotic giant branch (AGB) stars as kinematic tracers.

For a determination of the coordinates of the solar

apex and the speed of the solar motion, we neglect the objects for which the observed and calculated

radial velocities (residuals) differ much. We found that the uncertainty of the solar elements depends much on this criterion. The sample in this case

consists of 72 PNe. The selection criterion has been used for the

determination of the first Oort’s constant and the rotation curve is the same as that considered for the determination of the solar apex, 69 PNe have been

chosen for this purpose. More than 30% of our sample is of type I and type II according to the

classification of Peimbert (1978), and the rest is unknown according to the same classification.

3. Methods of calculations

3.1. Determination of the coordinates of the galactic pole

Given the distribution of a class of objects over the celestial sphere, and determining the position of a great circle best fitting this distribution. The problem

being stated, we have to consider closely the follow- ing two questions: (1) Which group of objects shall we choose ? and (2) How shall a best-jitting circle be de$ned and determined?

The aim of the present paper is to explore the As an answer to the first question, we obviously advangates of using planetary nebulae as a stellar expect the best results by choosing objects which system to determine some of Galactic constants in exhibit the strongest concentration towards the Milky particular: the Galactic pole coordinates (cyp, S,), the Way. It is however, clear that, in a more general solar motion elements and speed, Oorts’s first con- investigation, other objects too must be considered. stant (A) and the rotation curve. Frequently, the determination based on the study of

A. Ali, M.A. Sharaf I New Astronomy 3 (1998) 419-425 421

the apparent distribution of the young Population I objects, and particularly the youngest among them: (a) Galagtic clusters, (b) Wolf-Rayet stars, (c) O-type stars, (d) classical Cepheids, can also be used.

The next question is how to define and determine the great circle that best fits the distribution of the

objects chosen over the sphere? It should be re- membered that the position of any great circle is

given either by: (a) the coordinates of its pole, or (b) if the circle is oriented, by the right ascension of its

ascending node and its inclination to the celestial equator. Our task will therefore be to deduce, from

the apparent distribution of our objects, one of the two pairs of quantities. In the present study, we shall consider method (a), that is the determination of the

Galactic equator through the determination of the Galactic pole.

The determination of the equatorial coordinates a,_, and SP of the Galactic pole is a typical constrained

minimization problem with the following elements.

The functionf, to be minimized, is the sum of the

N(say) squared distances of the objects from the plane of maximum concentration towards the

Milky Way, that is

N N

f,(I,m,n) =c Df = ~(ZX, + m yi +n~,)~, (1) , ,=I

Xi = r, cos S, cos a; , (2)

y, = ri cos S, sin cu, , i= 1,2,...,N (3)

7 = ri sin S, , ct (4)

where ri is the distance of the ith object and 1, m, and II are the direction cosines of the perpen- dicular to the plane drawn from the origin.

The constraint of the problem is

f,(l,m,n) = 1’ + rn2 + n2 - 1 = 0. (5)

Now, the solution of the constrained minimization problem is most easily found by Lagrange’s method of indeterminate multipliers and the objective func- tion is

F(E,m,n; A) =f, (l,m,n) - Af, (IJV). (6)

The necessary conditions for an extremum are

$0; $+); $0, (7)

which reduces to the matrix eigen value problem

AX=AX, (8)

where the elements of the symmetric matrix A are

given by

a22 =iJ’f ; ‘23 =iyi zi ; (9)

N

a33 =;z;

and the real (since A is symmetric) eigen values A,, A, and A, are the solution of the characterstic

equation

h3+k,A2+k2A+k3=0, (10)

where the coefficients k,, k,, and k, are given by

k, = -(a,, + a22 + a,,),

k2=u,, a22 + aI 1 a33 + a22 a33

- (4, + 43 + a;,,,

k, = ai, a33 + 43 a22 + 43 alI - UllU22 a33

-2a,2 a13 a23. (11)

The real solutions of Eq. (10) are given using the

following relations

A, =2p”3cos(+/3)-k,/3,

A,= -pL’3{cos(@/3)+&sin(&3)}-k,/3,

A,= -p”3{~os(~/3)-~sin(&3)}-k,/3,

where

p=G, 4=tan~‘(JEEIu),

u = p2 - v2, y=;k,-+k;,

v=+[k,k,-3k,]-&k;.

422 A. Ali. M.A. Sharaf I New Astronomy 3 (1998) 419-425

The eigen vectors Xj corresponding to A,, A1 and A, are

X,’ = {l,,m,,n,}; j = 1,2,3, (12)

which can be obtained from the solution of the

homogenous linear system (Eq. (8))

(A - Aj’)X, = 0; j = 1,2,3 (13)

as

1, = G\j’lRj ; m, = Gk”IR, ; nj = GY’IR, (14)

where Vj = 1,2,3;

(J) _ G, - (~22 - A,) (a33 - A,) - a;,> (15)

(1) _ G2 - -a,2 (~33 -A,)+~23 ~13. (16)

(11 _ G3 - a12 a23 - a13 (a22 - Ai ), (17)

R,’ = (G;“)’ + (Gy’)’ + (G:j’)‘. (18)

Let (L&I) be the set of global minimum values of F given by Eq. (6), then the equatorial coordinates 9 and S, of the Galactic pole are given by

(Ye = tan -‘(m/i), (19)

sP = sin-‘(n). (20)

3.2. Determination of the solar upex and the speed of the solar motion

Depending on the observational data, three differ- ent methods can be used to determine the position of the apex and the speed of the solar motion. These are the radial velocity data alone, the proper motion data alone or the space motions (both radial and trans- verse components). The first two methods can be used for objects whose distances are not known. In our calculations we have used the radial velocity method.

Since

v~=Xcos6cosa +YcosSsincr+ZsinS, (21)

where (Y, S, and V~ are the right ascension, declina- tion, and the radial velocity of the object respective- ly, then for a group of N objects, Eq. (21) can be considered as the condition equation for the least-

squares solution X, Y, Z. Having obtained X, Y, Z then the components of the Sun’s velocity with respect to the same group and referred to the same axes as X, Y, and Z are given as

x0= -x; Y,= -Y; zo= -z; (22)

Then, the position of the solar apex in equatorial

coordinates (tu,, 8,) and the speed of solar motion (So) can be determined from

X,=S,cos6,coscu,, Y,=S,cos6,sinff,,

Z, = S, sin S, .

That is,

ff* = tan-‘(YolXo)

S, = tan-‘(Z,/dXR),

S, = (X:, + Y:, + z;)“*. (23)

Finally, (f,, b,), the position of the apex in Galactic coordinates, is easily obtained from (cu,, 6,) by standard formulae of spherical trigonometry. As a

by-product of the solar elements, we can compute the velocity ellipsoid components as follows:

U, = - S, cos 4 cos bA ,

V, = S, cos lA sin bA ,

W0 = S, sin bA . (24)

3.3. Determination of Oort ‘s first constant (A) and the rotation curve

There are contraversies regarding the rotation curve of the Galaxy outside the solar circle, and it is not clear whether the curve decreases, remains constant or even increases (see, e.g., Blitz et al., 1980; Knapp, 1983; Clemens, 1985; Rohlfs et al., 1986).

Since the pioneering work of Schmidt (1965), the Galactic rotation curve has been used for studies of the Galactic structure. Most determinations of the rotation are based on observations of the gaseous components of the disc (H I, CO), and rely on the

A. Ali, M.A. Sharaf I New Astronomy 3 (1998) 419-425 423

assumption that the maximum velocity observed at a given longitude corresponds to the velocity of the subcentral point (Clemens, 1985; Rohlfs &

Kreitschmann, 1987). It is also important to obtain the rotation curve from traces of young massive stars, since they have velocties close to that of the

surrounding gas. Among the studies of the rotation

curve are those based on stellar tracers; open clusters (Hron, 1986), and classical Cepheids (Pont et al., 1994). OH/IR stars are particularly tempting candi- date tracers, since they can be recognized at large distances, based solely on radio and infrared ob-

servations.

5 $iJ ri sin 21, cos2 bi

A= i=’ hi (27)

s rf sin2 21, cos4 bi

Given the radial velocity (v,) of an object at a distance r from the Sun in a direction of the Galactic

coordinates 1, b, the distance from the centre project

on the plane is given by

R={R~+(rcosb)2-2R,rcosbcosl}“2, (28)

the rotation velocity is

Oort’s constants A and B can be determined from

local kinematics, and should be regarded as local quantities. The constant A can be detrmined from

either radial velocties or proper motion. The only direct method for calculating the rotation constant B is from proper motion,

8(R) ={t?,, + v,/(sin 1 cos b)}RIR, , (29)

where R, and 0, are the distance to the centre and circular velocity of the local standard of rest, respec- tively.

Since the systematic effect, due to Galactic rota- tion on the radial velocities is given by (e.g. Smart, 1968)

4. Numerical applications

VI = A r sin 21 cos2b

=v,+U,cosbcosl+V,cosbsinl+W,sinb

l The method described in Section 3.1 was applied

to the data sample (Section 2), it gives for the equatorial coordinates of the Galactic pole, the

values

(25)

where U,, V,, W, are the components of the Sun’s

velocity with respect to the centroid at the Sun’s position. Hence, Eq. (25), represents the correction to be applied to the radial velocities relative to the Sun, in order to get radial velocities relative to the centroid coinciding with the position of the Sun.

Assuming that the values of the components U,, V, and WI are known, consequently, from the observed values of the radial velocities v, and their corrections

computed from Eq. (25) the left hand side of Eq. (25) is known for each of the selected objects. Hence, our equation of condition is

CY’, = 12h 49”) ~3~ = 27”22’

These values are in full agreement with 1950.0 equatorial coordinates. To the best of our knowl-

edge, no reference mentioned the determination of the Galactic pole using PNe, so our result seems to be the first trial to use PNe for this objective.

l Regarding the solar motion, we obtained the following values:

cu,=290”, S,= +30”, S,=18.6kms-‘,

U, = - 7.83kms-‘ , V, = 16.6kms-‘ ,

and Wo = 2.4kms~‘.

v,. = r A sin 21 cos’ b . (26)

It should be mentioned that the PNe were used for determining the solar motion (e.g. Cudworth, 1974; Mihalas & Binney, 1981).

From the known values of the Galactic coodinates l In 1964, the IAU adopted the values A = (I,, b;), radial velocities v, (” and the distances from 15kms-’ kpc-‘, B = - lokrns--’ kpcc’. Ac- the Sun r, of N(say) selected objects where, i = cording to recent work, the value of A is broadly 1,2,. . . ,N, we can solve Eq. (26) by the least- averaged in the range 12-17kms-’ kpc-‘. With squares method for A to get U, = - 9, V, = 12 and W, = 7 km s-’ (Delhaye,

424

Table 1

A. Ali, M.A. Sharaf 1 New Astronomy 3 (1998) 419-425

First Oort’s constant (A) from the radial velocities using different obiects

Reference

Crampton & Femie ( 1969)

Humphreys ( 1970)

Georgelin & Georgelin ( 1970)

Barkhatova & Gerasimenko ( 1976)

Clube & Dawe ( 1980)

Blitz et al. (1980)

Cudworth (1974)

Cudworth (1974)

Present work

Type of objects A (kms-‘kpc-‘)

Cepheids 12.5

Supergiants 14.0

H II regions 14.2

Open clusters 15.0

RR Lyraes 16.0

co 13.3

PNe (type B) 12.0

PNe (type C) 08.0

PNe 15.2

.

1965), solving Eq. (27) and using the data sample (Section 2) yields A = 15.2, a value which is in a good agreement with the above range.

Table 1 shows a comparison between our

calculation of the constant A using PNe and other estimates using different objects. Clearly, our result is better than that given by Cudworth

( 1974) for type B and C. Finally for the rotation curve (Fig. 1) we have

adopted the values R,, = 8.5 kpc and 4, = 220 km s-‘. The curve shows a similar behaviour to those of type I Maciel & Dutra (1992). It is

important to mention that more than 30% of our

sample is of type I and type II, and the rest is unkown according to the classification of Peim- bert (1978). Since type I (2.4-8 M,) and type II

(1.2-2.4Mo) have high and intermediate mass progenitor. This means that the used sample represents a young population I, therefore it was a convenient sample for determining the rotation

curve.

Summarizing the present work, efficient al- gorithims are established for determing the Galactic pole coordinates, solar apex coordinates, solar mo-

tion speed, Oort’s constant (A), and the rotation

400

300- .

. .

5?

g 200- .

1 -

. .

lOO-

4 5 8 7 0 9 10 11 12

R (WI

Fig. 1. Rotation curve based on our data sample.

A. Ali. M.A. Sharaf I New Astronomy 3 (1998) 419-425 425

curve. We have derived the following constants: the Galactic pole coordinates 9 = 12h49m, S, = 27”22’,

the solar apex LY* = 290”, 6 = + 30”, solar speed S, = 18.6km s-‘, and Oort’s first constant A = 15.2 km s-’ kpc-‘. Moreover, The equations in- volved are solved exactly without any approxima-

tions. The PNe data appears to be remarkably useful in our application of the algorithms, and the results

are in a good agreement with the standard and the averaged values.

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