TRANSFER OF SOLAR ANGULAR MOMENTUM

BY INERTIAL INDUCTION

AMITABHA GHOSH

Dept. of Mechanical Engineering, Indian Institute of Technology, Kanpur, India

(Received 14 January, 1988)

Abstract. Transfer of angular momentum from the Sun to the planetary system has been found to be inevitable in all evolutionary models for the origin of the solar system. In ‘cold’ theories it has been proposed to be achieved through friction whereas electromagnetic forces are considered to be the agent for this transfer in ‘hot’ theories. In the present paper it has been shown that the required order of magnitude of angular momentum can be transferred by another mechanism based on the principle of inertial induc- tion. In the previous theories most of the transfer had been assumed to have taken place during the pre-Main-Sequence period whereas in this proposed theory most of the transfer takes place during the Main-Sequence period of the Sun. The paper does not intend to go into the details of planet formation and the evolutionary process but confines itself only to the problem of angular momentum transfer.

1. Introduction

Nebular hypothesis for the origin of the solar system is philosophically most attrac- tive to the majority of the scientists but one of the major problems in this theory is to satisfactorily explain the distribution df angular momentum. Though the Sun possesses about 99.9% of the total mass of the solar system it possesses only about l/2% of the total angular momentum. Primarily because of this difficulty the theory remained almost abandoned till some mechanisms for the transfer of the Sun’s angular momentum were proposed. In modern times, the original theory of Kant and Laplace was revised by von Weizslcker, Kuiper and others in which turbulence and friction play important roles. On the other hand ‘hot’ theories have been developed by Alfvkn, Hoyle and others in which the ionization of the original cloud and the electromagnetic forces have been assumed to be responsible for the transfer of solar angular momentum.

In this paper a new mechanism for the transfer of the solar angular momentum is proposed. This transfer is based on the ‘velocity dependent inertial induction’ be- tween the spinning Sun and the planetary system* which is capable of generating a torque produced by the rotating Sun. It is not intended to discuss the evolution of the solar system; the aim is mainly to show that inertial induction is capable of transfer- ring the right order of magnitude of angular momentum from the Sun to the rest of the system in the available time. A simple and idealized model will be considered to keep the analysis simple without sacrificing the accuracy of the results to any signifi- cant extent.

*Estimate shows that the effect of the rest of the Galaxy (and the Universe) is much smaller than that of the local interaction

Earth, Moon, and Planets 42 (1988) 69-15. 0 1988 by Kluwer Academic Publishers.

70 AMITABHA GHOSH

2. Velocity-Dependent Inertial Induction

A dynamic model of gravitational interaction between bodies in relative motion has been earlier proposed (cf. Ghosh, 1984, 1986a) based on the concept of inertial induction. Application of this model to local interactions has successfully explained the observed large secular acceleration of Phobos (Ghosh, 1986a). It also leads to a modified tidal-friction theory for explaining the secular retardation of the Earth’s spin without facing any difficulty viz. the Moon’s close approach (Ghosh, 1986a).

The local interaction also provides an explanation for the extra red shift of the solar spectrum at the limbs (Ghosh, 1986b). This model has also been found to provide a servomechanism for distributing matter in spiral galaxies leading to flat rotation curves (Ghosh et al., 1988). When the dynamic model of inertial interaction of an object with the rest of the universe is considered it not only results in the exact equivalence of inertial and gravitational masses but also yields the cosmological red shift quantitatively without bringing in the concept of cosmological expansion (Ghosh, 1984).

According to this model the force arising out of dynamic inertial interaction be- tween two masses M and m can be expressed as

GMm * GMm F=- GMm

7 ur - c2r2 ~%f(Q - c2r a&f(4),

where G is the gravitational constant*, c is the velocity of light, r (=li,r), v (=&u) and a ( = &,a) are the position, velocity and acceleration of mass m with respect to M and F is the force on m. f(0) and f($) re p resent the inclination effects where cos ,9 = 6,. 2i, and cos C$ = z’i,. 1;,; ti,, &, and 6, are the unit vectors. As in the previous papers in this paper also the following functions will be assumed:

f(0) = cos 8 lcos 0 1 and f(4) = cos C#J (cos 4 I. (2)

3. Transfer of Solar Angular Momentum

In this simplified and somewhat idealistic model it will not be attempted to investigate the formation of the planetary disk and planets. Since the objective is to investigate the order of magnitude of the angular momentum which can be transferred we will consider a certain fraction of the original nebula to be dislodged from the equatorial region of the spinning protosun once the stage of rotational instability is reached. Subsequently the protosun reaches the main sequence in about 2 x 10’ yr and the angular momentum is transferred to the dislodged matter a part of which later forms the planets. From the analysis it is seen that an overwhelmingly large proportion of the total angular momentum is transferred during the main sequence period -i.e., about 4.7 x lo9 yr.

*Strictly speaking, G is not a constant in this model; but unless r approaches intergalactic distances its variation is not perceptible, and will be treated as a constant in this work.

TRANSFER OF SOLAR ANGULAR MOMENTUM 71

Figure 1 shows a spinning sphere (representing the Sun) and another body of mass m in the equatorial plane. Using (1) and (2) ( an noting that only the velocity d dependent terms can produce a tangential force) the force on m due to the elemental mass (neglecting the effect of angle 6 as will be explained later)

Gmo2 dP%-

pr3 cos2 I#J c2 s2 + r2 - 2sr cos e

de dr dx,

Fig. 1.

12 AMITABHA GHOSH

where p is the density, o is the angular velocity of the Sun* and the other quantities are as shown in the figure. The tangential component of the force

dF w Gmo2 pr3 cos2 c++ sin I+G c2 s2+r2-2rscose

de dr dx.

Therefore the total moment about the Sun’s centre produced by the spinning sun is given by

where R is the outer radius of the Sun (or the proto-Sun in the pre-Main-Sequence period). Now since during most of the period s will be much larger than R, the following simplifying assumptions can be made:

6 E 0, cos2 4 z sin2(8 + $) z sin2 8, sin $ w (r/s) sin e and

s2 + r2 - 2sr cos e z s2.

With these simplifications, Equation (3) becomes

4Gmo2 R&i=2

z = 3c2s2 (J ss pr4 dr dx.

0 (4)

The density in the proto-Sun and the Sun varies quite significantly with the distance from the centre. However, the natures of this variation are also different. During the pre-Main-Sequence period the moment of inertia of the Sun can be expressed as 0.1 M,R& as suggested by Hoyle (1960) and 0.05 M,R& in the Main Sequence period where M, and R, are the Sun’s mass and radius, respectively. For the sake of simplicity the density variation in the Sun can be approximately expressed as follows using the available data (cf. Priest, 1982) that

P = pC exp( -8al&J, (5) where pc is the density at the centre and a ( =dm is the distance from the centre. Since the duration of the pre-Main-Sequence period is relatively short and does not contribute significantly to the transfer of angular momentum we will use the density function given by (5). Thus, from (4) and (5) we get

z z 6 x 1O-3 G”;;;2R3.

*The orbital angular velocity of the planetory matter is much smaller than o (except at the beginning) and so can be neglected. Since the torque is produced due to the difference between the angular velocity of the Sun’s rotation and the orbital angular velocity of the planetary mass the limiting situation will arise when both are equal. Therefore, the minimum distance of the inner edge of the planetary disk (or the nearest body in the case of a multiple-body planet system) will be that for which the orbital angular speed is equal to the angular velocity of solar rotation. This is about 0.3 AU or the radius of the orbit of the planet Mercury. This explains why Mercury should be the innermost planet.

TRANSFER OF SOLAR ANGULAR MOMENTUM 13

If L be the total angular momentum of the original cloud and fill, ( =m) be the fraction dislodged from the original body the starting angular momentum of the Sun

I so = L/( 1 + 1Of ).

If Z, be the angular momentum of the Sun at time t, then

I, cz 0.05M,R2m.

Again the distance of m can be expressed as

(L - L)* ’ z GMam2=

CL, + I,, - L>” GMom2 ’

where Z,, is the original angular momentum of the dislodged matter and is equal to lOfL/( 1 + 1Of ). Now z is equal to the rate of increase of angular momentum of m or the rate of decrease of Sun’s angular momentum. Hence, replacing o and s in (6) using the above relations, we have

dl,zp24G3Momi 1: dt ’ c= R(L - 1,)“’ (7)

R reduces to R, in about 2 x 10’ yr and such variation in R is incorporated by using

R z R,[ 1 + 1 exp( -at)],

where (1 + 1)R, is the protosun’s radius at the onset of instability and c( z 1.35 x 1014 (when t is expressed in seconds). Using the expression of R in (7) we finally get *

@+Eln[{l+iexp(-at)}/(l+i,)]z

= (1Of + 1)4(Z&/Z,) - 4( 1Of + l)‘Zl, ln(Z,,/Z,)

- 6( 1Of + l)“Z& Z, + 2( 1Of + l)Z,,Z; - 1,313

+ {6( lof + I)* + l/3 - (lof + 1)" - 2( 1of+ l>>z;,, (8)

where

p = 2.4 G3MomS =24 G3MV

c2Ro ’ c*R, ’

It can be easily seen that when t $2 x 10’ yr the contribution of the second term on the LHS of (8) will be negligible compared to that of the first term. Similarly when Z,,/Zs 9 1 the major contribution comes from the first term on the RHS of (8). So for the order of magnitude calculation we can use

pt - (1Of + 1)41:,/Z, = La/z,,

when t $2 x 10’ years and l,,/Z, 9 1. (9)

*It should be remembered that, as suggested by Hoyle, a large proportion of the planetary nebula evaporates, so that rn cannot be treated as constant. However, to keep the analysis tractable it has been assumed to be constant and equal to some average value.

14 AMITABHA GHOSH

The condition for rotational instability of the proto-Sun is given by R3a2 = GM,, or, R20 = ,/w. Using this condition and noting that at the end of the process of dislodgement (i.e., disk formation) the angular momentum of the proto-Sun is given by ls, = L/( 1 + 1Of) the radius of the proto-Sun (or, the inner radius of disk) can be expressed as

R = L&/GM& zz L2/{( lOj-+ l)‘GM&}. (10)

Now (6) can be used to find out the rate at which the angular momentum of an orbiting mass Am increases by replacing m by Am. However, o continuously changes depending on the total interaction with the whole planetary (or disk) system. How- ever, if we take the average value of a2 and treat it as a constant along with R3 = R& (as during the most of the period R = R,), Equation (6) can be easily solved to yield s as a function of the time. Since lAp = Amsi R (Q is the orbital angular velocity and s, is the orbital radius of Am) and Q2 = GM/s; (for Keplerian orbit)

I,, = Am,/GMos,.

Using this in (6) we find that

5/2 3 x 10-2JGM, SA %

c2 o;,R&t+A. (11)

From (10) the initial inner radius of the disk can be found out which will give the value of A in (11). After 4.7 x lo9 yr the current value of the inner radius can be found out from (11).

4. Results and Discussion

No definite value of I is available, but as per Hoyle’s estimate it is about 4 x 1O44 kg m2 SC’. He further assumed that 10% of the original nebula formed the disk most of which again evaporated during the long Main-Sequence period. It implies that though 99% of the disk evaporated only 90% of the disk’s angular momentum was taken away leaving about 3 x 104’ kg m2 s-i for the existing planet- ary system. Similar propositions were earlier made by von WeizsLcker (1943, 1947).

If we assume L - 1O44 kg m2 SC’ and m - 0.023 M, (i.e., about 2.3% of the orig- inal cloud) substituting t = 4.7 x lo9 yr the present angular momentum of the Sun

1, - L4/pt - 1.4 x 104’ kg m2 ssi,

which is close to the present value. Most of the original angular momentum is transferred to the planetary disk which evaporates (as suggested by Hoyle) leaving behind 4% of the disk mass and 30% of the momentum to constitute the present planetary system.

Taking L - 4 x 1O44 kg m2 s-l as suggested by Hoyle it can be shown that the present value of the solar angular momentum (1.5 x 104’ kg m2 s-i) is obtained after 4.7 billion years iffis taken as 0.07. Using these values of L andfthe radius at which

TRANSFER OF SOLAR ANGULAR MOMENTUM 75

dislodgement of matter from the Sun ends (or the initial value of the disk’s inner radius) is found from (10) to be 10.38 x lo9 m. The current radius of the innermost particle Am can be found out using (11) as about 2.1 x 10” m which is close to the orbital radius of Mercury.

It must be mentioned that the values are only very approximate ones and these only indicate the feasibility of the mechanism suggested. There is one major difference with the previous theories though other basic ideas are not much different. In the previous theories by von Weizsacker and Hoyle most of the transfer of angular momentum took place in the short pre-Main-Sequence period, whereas with the proposed mechanism this happens during the long Main-Sequence period.

In reality more than one mechanisms may be responsible for the transfer in which inertial induction may play a major role in the Main-Sequence period.

5. Conclusions

Though the analysis is a very approximate one it clearly indicates that inertial induc- tion can transfer the required order of magnitude of angular momentum in the available time (4.7 x lo9 yr). The angular momentum transfer is quite sensitively dependent on the mass of the dislodged matter. The present estimate uses the dis- lodged matter as a single body and more elaborate analysis is required for extending the analysis to a disk model.

Acknowledgement

The author wishes to thank sincerely Mr Arunabha Ghosh, Physics Department, for his assistance in this work.

References

Ghosh, A.: 1984, ‘Velocity-Dependent Inertial Induction -An Extension of Mach’s Principle’, Prumuna (Jr. of Physics) 23, L611.

Ghosh, A.: 1986a, ‘Velocity-Dependent Inertia1 Induction and Secular Retardation of the Earth’s Rota- tion’, Pramana (Jr. of Physics) 26, 1.

Ghosh, A.: 1986b, ‘Velocity-Dependent Inertial Induction-Possible Explanation for Supergravity Shift at Solar Limb’, Pramana (Jr. of Physics) 21, 725.

Ghosh, A., Rai, S., and Gupta, A: 1988. ‘A Possible Servo-mechanism for Matter Distribution Yielding Flat Rotation Curves in Spiral Galaxies’, Astroph. Space Sci. (in press).

Hoyle, F : 1960, ‘On the Origin of Solar Nebula’, Quart., J. Royal Astronomical Society 1, 28. Priest, E. R.: 1982, ‘Solar Magnetohydrodynamics’, D. Reidel Publ. Co., Dordrecht, Holland. Von Weizsacker, C. F.: 1943, ‘tiber die Entstehung des Planetensystems’, Zeit f: Astrophysik 22, 319. Von Weizsacker, C. F.: 1947, ‘Zur Kosmogonie’, Zeitf Astrophysik, 24, 181.