Chin.Astron.Astrophys. 5 (1982) 115-120 ActAstronSinica 23 /1982)10-16 -

Pergamon Press. Printed in Great Britain 02~5-1062~82~020115-06~07.501~

MODEL OF SPIRAL GALAXIES AND DISTRIBUTION OF MATTER IN THE MILKY WAY SYSTEM

TONG Yi, ZHENG Xue-tang Department of Beijing University.

Received 1981 July 22.

ABSTRACT Assuming an axisymmetric spiral galaxy to consist of 3 components, s plane-parallel distribution with an exponentially decreasing density in the z-direction, a polytropic distribution about the centre and a highly concentrated nuclear ball, we give analytic expressions for the mass density in such a system. Specific results for Milky Way are given. Our results agree with the latest observations.

1. INTRODUCTION

Earlier studies of the mass distribution in spiral galaxies were confined to the galactic disk, e.g. Toomre /I/ regarded the galaxy as an infinitely thin disk; later, PENG @i-he et al. /2/ extended the investigations to disks of finite thickness and Miyamento f3/ obtained a three-dimensional model for the distribution of matter in the Milky Way system. However, they have taken into account neither the galactic nucleus nor the galactic halo. In order to make a more comprehensive investigation, we shall suppose on the basis of observed facts, that the galactic mass distribution consists of the following three components.

1. Galactic Halo. This is supposed to follow a polytropic distribution symmetric about the centre, i.e., the density is a function of the radius R only. We shall adopt a basic solution of Emden's equation given by Plummer /4/ in 1911, in which the gravitational potential @a(R) and the mass density o,(R) have the forms

where EI~ is the total mass of the haloR- dG<r,z beingthe cylindrical coordinates), bl is a non-zero constant length. The potential satisfies the Poisson equation.

v'+,(R)- 4rGdR). (2)

This model corresponds to Schuster's gas sphere of polytropic index 5.

2. Plane-Parallel Distribution. The density in each layer is assumed to decrease exponently with the height s, fS/,

drr s> - dr, o)e-a”t. (3)

where o is the reciprocal half-thickness. The density is related to the projected surface density up(r) through

p*(r) z) - +(++". (4)

The gravitational potentail for this distribution, Jlp(rt zf, must satisfy the Poisson' equation

V%(r, a> - 4*cp,, (5)

3. Nuclear Ball. This is again supposed to follow a polytropic distribution,

116

P.(R) 36% - e (R’ : y)a”’ I

TONG and ZHENG

(6)

satisfying the Poisson's equation

V'&(R) - hGp,(R). (7)

The boundary of the galaxy is defined by that surface of equal density on which the

combined density would be just observable. Lastly, we suppose the system to be in a stationary state, so that

YY?,Z) a&$ a>* (8)

,

where v(r, z) is the rotational velocity about the centre and $(r, z) is the total p&aa4&a

potential.

2. INTEGRAL EXPRESSION FOR THE SURFACE DENSITY

In the general case under the above assumptions, the Poisson's equation is

V'&r,s)- 4rGp, (9)

where

cc,-+,+(I"+&, (10)

P-P, + P, + PI. (11)

Ihe forms for Jls, Jln, ps, pn are given at (1) and (6) and we now seek the variation of the layered distribution with r. From (4) and (5) we have

V'& - 4xGpp(r,a)- 2,Gsu,,(r)e+='. (12)

According to the method given in 161, we solve the Poisson's equation by taking the potential of the infinitely thin disk as Green's function, that is, we start with the form

V’[-J&7r)e-~‘=‘l = qM3r)6(rr~, (13)

Where u(z) is Dirac's function. We develop the up(r) in (12) as a Fourier-Bessel integral

using Bessel functions of order zero

Z'(P)- \:(,~~;;~ J1(BlM, (14)

where

We then obtain

(15)

(16)

where

H%l?,E)=- (17)

(18)

The gravitational force in the radial direction on unit mass is -m, and from (16) we have Br

SJlp(r,r)_Ga m'S(B>J,(Br)@7(s, 89 a)@. (19)

Br 0

On the galactic plane, we have

(20)

Model of Spiral Galaxies 117

First, we determine the mass distrubution from the rotational curve in the galactic plane. From (l), (6), (S), (lo), we have,

(21)

On the plane of symmetry (z=o), this becomes

a+,(r,o> sz(r) GMI~ GMor

& r (rl i tir/3 - (9 + b:)"' (22)

We develop the above as a Fourier-Bessel integral using Bessel functions of order 1,

ti- G"lr - ($J& - \:~I@)I,(B+B. r (r' + b:)"

(23) 0

where

Equating the corresponding coefficients under the integral sign in (20) and (23), we found

(24)

(25)

where

I"@) - 1: s'(r)&+&, (26)

(27)

(28)

Inserting (25) in (14), we obtaib the projected surface density of the layered distribution,

u,(r)-a&)+&) -u>(r) -dp(r) -u;(r)-&!(r). (29)

where

oT(r) is the surface density of Toomre's infinitely thin disk Ill, o&) is the correction term due to finite thickness given in /2/, and ok(r), ok(r), and o?(r), u'!T<r) are the further corrections obtained here for the galactic nucleus and halo.

3. RESULTS FOR SPECIFIC MODELS OF ROTATION.

First we take the same rotation model as Toomre did, namely,

118 TONG and ZHENG

t&r) - cxi + r~/e?P”, (36)

where CO and a are two constants to be determined from observations. Substituting this in (26) and looking up Tables of Bessel Transfer /7/, we find

Na,8) - $(l -e-q. (37)

From eon. (26) we see that rg(a,B) is a linear functional of the function v', and hence from (30), (31) that o&), UT(r) are also linear functional8 of v'. Hence the derivative of (36) with respect to a' must also lead to the derivatives of I:(B), o&r), and OR(~) with respect to aa. Differentiating (36) gives "Model 1" ,

d(r) ( 4 - 42).

The corresponding I:(8) is

Q((B)-m-& c+ 6n' 0: *

From (30), (31) we find the corresponding

(38)

surface densities

(39)

(40)

(41)

But "Modell" does not agree with the actual situations. We now differentiate once again with respect to the Parameter (-a') and obtain "Model2"

Here, CL g

is the angular velocity at the galactic centre. The expression (42) represents *rather etter the rotational velocitites at vari9us points on the plane of symmetry, when r<<a, it tends to,a solid-body tot&tion v(r)=fiG, when r>>a, it tends to the Kepler rotation vfr)=r-' *

Integrating (27) gives

Substituting this in (32), (33) and integrating, we have

u;(r) - M* 2*6: (1 + &w-*,

(l;(r) - 2s 1 + $ -w Zxab: ( > 6,

* (2 - r'/#).

Similarly, integrating (28) gives

I"(@) - GM+ -59,

and substituting in (34), (35) and then integrating give

(42)

(43)

(44)

(45)

(46)

(47)

(48)

Substituting (43), (44), (46), (47), (49), (50) in (29) goves the projected density of the layered distribution up(r), hence, from (4), its volumne density pp(r,z). Hence, and finally, we obtain the total density distribution of the complete galaxy

Model of Spiral Galaxies 119

(51)

4. DETERMINATION OF THE MASS DISTRIBUTION OF THE MILKY WAY SYSTEM

We have derived above the analytic expressions for the mass density distribution in a galaxy. To determine completely the mass distribution, the parameters in these expressions must be specified. We now proceed to do this for the Milky Way system, to which the most accurate observational data pertain

1. The Density in the Central Nucleus. Following technical advances, we now have available the measured rotational velocities at points within 100.~~ of the central nucleus /8/. From these we can now fix the values of & and bo. Since the degree of central concentration of the nucleus is several orders of magnitude greater than that of the halo, the gravitational potential at the centre can be regarded as due to the nucleus only, that is, near the centre, (10) can be rewritten as

and according to (8), at z=O, we have

v’(r) - GM.t’

(9 + bp ’

(52)

(53)

Using the velocity values given in 181, we found by least squares, ~~=l.l5xlO'~e, bo=0.61pc Inserting these values in the second expression of (6) then gives the distribution of the nuclear mass. Our density values differ little from those given in /a/, which were calculated from a different model.

2. Density outside the Nucleus. We use the known densities at various points near the Sun /9/ in a least-squares solution of (51) and find M1 =1.2~1O'~Ele, br=l.lkpc, u=2.lkpc-? After inserting these values back in (51, the mass distribution of the entire Milky Way system is completely specified.

3. A Model Calculation . We shall suppose that the boundary of the Milky Way system corresponds to where the total density from all three components has decreased to l/e times the solar neighbourhood value of 0.13Mep~-~. Our calculated result for this boumdary using (51) is shown in Fig.1. It is in overall agreement with observations.

z(b)

- 6.0

- 4.5

Fig.1. A calculated profile of the Milky Way system.

120 TONG and ZHENG

5. CONCLUSIONS

Stimulated by recent advances in observational techniques, we tried to improve on previous galactic models in order to fit better the observations. We think that it is inappropriate not to take into consideration the galactic nucleus and the galactic halo, especially the latter, as its mass may be comparable to the galactic disk, and its effect on the total mass distribution may be very large. The final result calculated in accordance with our proposed model appears satisfactory.

For other galaxies, it may be possible to determine their mass distribution from the shape of their disk boundary.

ACKNOWLEDGEMENT We wish to thank LI Jing for providing much valuable data and WU Sheng-gu for participating in the computing work.

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