A P O S S I B L E S E R V O M E C H A N I S M FOR .MATTER

D I S T R I B U T I O N Y I E L D I N G FLAT ROTATION CURVES IN

S P I R A L GALAXIES

A M I T A B H A G H O S H , S U D H E N D U R A I , a n d A J A Y G U P T A

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India

(Received 9 July, 1987)

Abstract. A dynamic model of gravitational interaction was proposed earlier which results in a velocity- dependent drag. This drag can quantitatively explain the cosmological red shift in a stationary universe, the secular acceleration of Phobos, the secular retardation of the Earth's rotation without any problem of the controversial close approach of the Moon and the extra red shift at the solar limb. In the present paper it has been shown that such a velocity dependent inertial induction can act as a servomechanism for distributing matter in rotating spiral galaxies in such a way that fiat rotation curves are obtained. A truncated thin disc model has been assumed, but the results are strongly in favour of the proposition.

1. Introduction

It has been now firmly established that almost all spiral galaxies have flat rotation characteristics. The stars in such galaxies have almost circular orbits with speeds almost constant not only for the visible disks but also beyond. The conspicuous absence of the Keplerian fall off has been explained by the theory that a large proportion of galaxy's matter is in dark form. At the same time a number of researchers have pointed out that a servomechanism to distribute the matter in the way required to yield fiat rotation curve must be present. Otherwise such a peculiar situation would not have been observed in almost all spiral galaxies. Till now such a servomechanism has not been reported. This paper presents a possible servomechanism which can explain the phenomenon.

2. Dynamic Model of Gravitation and Velocity-Dependent Inertial Induction

A dynamic model of gravitational interaction has been proposed (Ghosh, 1984; and subsequently modified in 1986a) based on the concept of inertial induction and extended Mach's principle. Application of this model to local interaction has successfully explained the observed large secular acceleration of Phobos (cf. Ghosh, 1986a). It also leads to a modification of the purely tidal friction theory of the secular retardation of the Earth's spin and the difficulty of the Moon's close approach is completely resolved (op. cir.). The model also explains the unexplained extra red shift at the limb of the solar disk (Ghosh, 1986b). When the dynamic inertial induction of an object with the rest of the matter in the Universe is considered it not only results in the exact equivalence of inertial and gravitational masses but also quantitatively yields the cosmological red shift without bringing in the concept of cosmological expansion (cf. Ghosh, 1984).

According to the proposed model the force arising out of inertial induction between

Astrophysics and Space Science 141 (1988) 1-7. �9 1988 by Kluwer Academic Publishers.

2 h. GHOSH ET AL.

two masses M and rn can be expressed as

GMm GMm GMm F - - f t , . - - v 2 f i , . f ( O ) - - aft,.f((p), (1)

?,2 C2F2 C2~,

where G is the coefficient of gravitation*, c is the velocity of light, r ( = a~r), v (= avv),

and a (-- ~,a) are the position, velocity, and acceleration o fm with respect to M, and F is the force on m. f(O) and f(~p) represent the inclination effects where cos 0 = ~r~ and cos 4~ = ~rfia, fir, fly, and ~,, being the unit vectors. As in the previous papers in this paper the following functions will be assumed:

f(O) = cos0 Icos01, f((p) = cos~p Icos~pl �9 (2)

The variation of G in case of local interaction problems can be ignored and the local value (= 6.67 x 10 -11 m 3 kg -1 s -a ) can be used.

3. Equilibrium Mass Distribution in Spiral Galaxies for Velocity-Dependent Inertial Induction

In this paper the spiral galaxies will be assumed to be axisymmetric thin disks truncated at a suitable radius within which most of the galactic matter is contained. This obviously makes the situation idealistic; but the main objective of the paper is only to indicate the feasibility of inertial induction to provide the necessary servomechanism.

As the first and the second terms of the fight-hand side of Equation (1) falls of as r 2 the main contribution will come from the interaction with the local matter present whereas the contribution of the third term represents the contribution of all the matters in the Universe. In the conventional Newtonian approach the contribution of the first term (the usual gravitational pull towards the centre) and that from the third term (the centrifugal force) are made to neutralize so that an equilibrium is achieved. However, the solution is not unique. When the velocity drag (the contribution of the second term) is considered another relationship between the mass distribution and the velocity profile is obtained so that the total tangential force on a mass particle is also zero. If the resultant effect is a pull the body will be pushed out till the pull is zero. Similarly if it is a drag, the body will spiral inwards till it reaches the equilibrium; and a stable

equilibrium is achieved. When both the radial and transverse equilibrium conditions are considered unique solutions for both the mass distribution and velocity profile are obtained.

Figures l(a) and l(b) show the interaction configuration of a point mass S with an elemental mass inside and outside the orbit of S, respectively. The resultant of the velocity-dependent interaction with all the mass inside the orbit of S is represented by a pull P and that for the mass outside the orbit by a drag D as indicated. When the radius

* In this model G is not a constant but depends on the distance between the interacting objects (Ghosh, 1984, 1986a). However, this variation becomes appreciable only when the distance involved is extremely large [G = G O exp( - 4 x 10-27 r), r in meters].

ROTATION CURVES IN SPIRAL GALAXIES 3

v(x) P(x)

(o)

Fig. 1.

(b)

of S 's orbit is x, mass of S is m, the surface density of matter in the disk is tr(r) and the velocity profile is v(r) the pull*

P(x) 2Gmffo(r)r{v(r)cos~-v(x)sinq41v(r)cos'e-v(x)sin4~l c 2 r 2 + x 2 - 2rxcosO

0 0

x sinq~ d0 d r . (3)

If we take the truncation radius as R and use the nondimensional quantities r = r/R and rl = x/R, Equation (3) becomes

2Gin f f a ( r162 - v(t/)sinqSI x P(~) = c~2-- ~2 + t/2 _ 2~t/cos 0

0 0

x sin ~ d0 d~.

Integrating over 0 we get

e(n) -

3c 2 ~3(t/2 _ ~2) 0

a(~) {~v(~) - ev(r/)} I~v(~) - ev(~)l d{ . (4)

* Apparently the form of the second term in (2) and that in the integrand may look different but effectively they represent the same thing. The algebraic and trigonometric details are omitted to conserve space.

4 A. GHOSH ET AL.

In a similar fashion the expression for the drag becomes

1

8Gm f 1 D(rl) = 3c 2 ~(~2- rl 2) a(r {iv(t/) - r/v(~)} I~v(t/)- r/v(r d~. (5)

The equilibrium is obtained when P(r/) = D(r/) and the centrepetal acceleration is equal to the resultant acceleration due to gravity. The solution to these equations in analytical form is formidable. Therefore, an indirect approach has been adopted in this paper. We assume the velocity profiles as observed in case of spiral galaxies and show that there is a tendency of P(r/) and D(r/) to be identical. However, as the analysis is only an approximate one exact matching of P0/) and D(r/) is not expected.

4. Results and Discussion

To start with we present the idealistic situation where orbital velocity is constant. In such cases (cf. Fridman and Polyachenko, 1984)

v(r)= v o ~l 2R / '

M ~ r ( r ) = - -

2rcrR •/ r 2 M sin - i x / l - 4 2

sin- 1 1 - R 2 = 27zR 2

Substituting in (4), we find that

P(r/) - - -

D(t/) - - -

and, similarly,

rl

0

_ _ sin_ 1 ~ _ ~ de I _ 2GMmv~3rcc2R 2 ~(rl)

1 20 mvo [f 3 7zc2 R 2

- q ~z 1 2GMmv~ -D(q). ~ + r/) s in - l V / I - d~ - 3rcc2R z

Figure 2 shows the variation of P(~/) and D(t/). It is seen that the match is reasonably good in the mid-disk region. The velocity near the edge and the centre cannot remain constant (and does not remain constant in actual cases). Moreover, the model of a thin disk also does not fit near the centre. Therefore, the mismatch near the centre and the

edge is understandable. Another case has also been investigated in which the velocity increases with the radius

and then remains more or less constant. This is more like the characteristics actually observed. The velocity and the corresponding density functions are given (cf. Fridman

ROTATION CURVES IN SPIRAL GALAXIES 5

4 .0 --

.3.0

E"

I~ 2.0 E"

I~L

1.0

O'IV

v �9 constt.

0,2 0.4 0.6 0.8

~D(rl) v = constt.

I 0 1.0 0 .2 0 .4 0.6 0 ,8

rl "

Fig. 2.

,.'o

and Polyachenko, 1984) by

v2( r - - - 2 n + 1

4n

a({) - M ~ be,.(1 - ~2)e- 1/2 2 n R 2 e = 1

2 n + 1 4 ( k - 1 ) ( n - k + 1) bl . - . b e . =

' 2 n - 1 ' " ( 2 k - 1 ) ( 2 n - 2k + 1) b k - l ' "

The value of the parameter n determines the shape of the velocity curve. When n tends

to infinity the case approaches the ideal situation of constant velocity. Figure 3 indicates

6 A. GHOSH ET AL.

4 . 0 - -

3.0

2.0 r

I n

1.0

0

O',V

~ 5(n_)

0.2 0 . 4 0.6 rl

Fig. 3.

o" n = 1 0 2 4

0,2 0.4 0.6 0.8 1.0 q

] I 0 . 8 LO

the variation of P(q) and D(r/) for n = 1024 and it is seen that there is a sufficient closeness in their natures. It must be pointed out here that P(r/) and D(r/) curves are very sensitively dependent on v and or. In fact a number of other types of velocity curves have been tested but the natures of P(r/) and D(~/) become widely different unless v is constant. Some of these are the velocity models suggested by Toomre (1963) and Freeman (1970). Whatever mismatch we are observing, it is primarily because of assuming the Galaxy to be a thin disk and to some extent due to abrupt truncation at a particular radius. The authors intend to investigate more realistic models of spiral galaxies in the future.

5. Concluding Remarks

In spite of the very approximate nature of the analysis the results indicate that the velocity-dependent inertial induction arising out of the proposed model of dynamic

ROTATION CURVES IN SPIRAL GALAXIES 7

gravitational interaction can act as a servomechanism to distribute the matter in an unique manner. This particular matter distribution yields the observed flat rotation curves.

References

Freeman, K. C.: 1970, Astrophys. J. 160, 811. Fridman, A. M. and Polyachenko, V. L: 1984, Physics of Gravitating Systems L Equilibrium and Stability,

Springer-Verlag, New York, pp. 327-334. Ghosh, A.: 1984, Pram~n.a -J. Phys. 23 (No. 5), L671. Ghosh, A.: 1986a, PramKn.a-J. Phys. 26 (No. 1), 1. Ghosh, A.: 1986b, PramK~a -J. Phys. 27 (No. 6), 725. Toomre, A.: 1963, Astrophys. J. 138, 385.