A D I A B A T I C R A D I A L P U L S A T I O N S O F A C O M P O S I T E

S T E L L A R M O D E L

MANMOHAN SINGH and SUDHA RAN1 AGARWAL

Department of Mathematics, University of Roorkee, Roorkee, India

(Received 9 March, 1983)

Abstract. Small adiabatic radial oscillations of composite models have been investigated. The effect of central condensation PcfP on the period of pulsation have also 'been examined. It has been shown that the second moment of mass concentration characterize the periods of pulsation more effectively than central condensation.

1. Introduction

All the stellar models considered these days are composite, in the sense that these

consists of a core and an envelope under different physical conditions. Jeans (1919) was the first to consider such a model known as generalized Roche's model. It consists of

a compressible homogeneous core of finite extent and density surrounded by an envelope of infinitesimal mass. The radial oscillations of this model were first investigated by Sen (1943). Later on, Kopal (1950) reconsidered this model after

eliminating the discrepancies pointed out by Cowling (1947). Many other authors, such

as Prasad (1949, 1953), Gurm (1960), Stothers and Frogal (1967), and Singh (1968), have investigated the radial oscillations of such models.

The present model was chosen to examine the effect of central condensation &/~,

where Pc is the density at the centre and ~ is the mean density of star, on the periods of pulsation. In view of this radial oscillation of four models with interfaces at x = 0.3,

0.5, 0.7, and 0.9 have been investigated. It has been found that second moment of mass

~ x 2 d M ( r ) / M characterize the periods more effectively concentration Jo �9 q, where q = than central condensation pc/p (Singh, 1968).

2. The Equilibrium of the Model

Let us take a composite model consisting of a spherical core of radius bR and of density pc(1 - rZ/R 2) at the distance r from the centre surrounded by a concentric spherical envelope of external radius R. The density in the envelope varies as 1/r 2. Thus for the core we have

p = pc(1 - r2 /R2) , (1)

and for the envelope

p = p c b 2 ( 1 - b 2 ) R 2 / r ~ . ( 2 )

Astrophysics and Space Science 94 (1983) 351-359. 0004-640X/83/0942-0351 $01.35. �9 1983 by D. Reidel Publishing Co., Dordrecht and Boston

352 M. SINGH AND S. R. AGARWAL

The constant of proportionality in (2) has been adjusted in such a manner that the density is continuous at the inferface.

The other relations for the equilibrium configuration of the star are: For the core (0 < r < bR)

4 rcpc G g - r(5 - 3r2 /R 2) (3)

15

and

( r4)} 2, op R2 f K _ r 2 5 - - - +

P - 15 ( R ~ R 2 ~ '

where

K = 5b 2 - 4b 4 + b 6 + 15P i

2nGp~R 2 (5 )

For the envelope (bR < r < R)

4 g = r2

and

P = 2rcGp~b4(1 - be)2 R 5 2L + - - - (2L + 3) , 3r 3 R

(7)

where

b(5 - 3b 2) L = - b . ( s )

15(1 - b 2)

in which p, g, P stand for the equilibrium values of density, gravity, and pressure,

respectively, at a distance r from the centre. The constants of integration have been so adjusted that the pressure at the interface (r = bR) is P,. and zero at the surface.

F rom the continuity o f the pressure across the interface we have

e,. = 27zGp~b(1 - ba) 2 Ra[(2L + 3b) - (2L + 3)b3] , 3

K = 5b 2 - 4b 4 + b 6 + 5b(1 - b2) 2 [(2L + 3b) - (2L + 3)b 3 ] .

(9)

(10)

Thus the model is completely determined by the parameters b, R, and m, or by b, R, and/7, the mean density of the star.

ADIABATIC RADIAL PULSATIONS OF A COMPOSITE STELLAR MODEL 353

3. The Rad ia l Osc i l la t ions

The differential equation for small radial oscillations of a star (Rosseland, 1949) is

d2r/+ + r/= 0 (11) dr e r dr k 7P

where ~/is the amplitude of the relative displacement ~r/r; # = pgr/P, g, p, P being

respectively the equilibrium values of gravity, density, and pressure at a distance r from the centre, ~ = 3 - (4/7), 7 being the effective ratio of specific heats; and n = 2~rv where v is the frequency of oscillation. The boundary conditions require the solution to be

non-singular. T h e particular frequency in which a star oscillates is now determined from the

condition that the motion should be continuous across the interface. The continuity of

the motion requires that br/r and bP/P should both have the same values on the two sides of the interface at any instant. This will be so, provided ~/and dq/dr are both

continuous across the interface. In practice, it is found more convenient to ensure the continuity of q ' / q (Prasad, 1953) across the interface, since the quantity is independent

of one of the constants of integration. Substituting for g and P from (3) and (4) in Equation (11), we have the following

equation of pulsation for the core

x[K - x2(5 - 4x 2 + x4)] d2~q + [4{K - x2(5 - 4x 2 + X4)} - dx 2

where

f l c l

- 2x2(1 - x 2) (5 - x2)] ~.1 + dx

+ [x(1 - xZ)J,, - 2c~x(1 - x 2) (5 - 3xZ)] r / : 0 , (12)

J , = 15n2/2rcTGPc and x = r / R . (13)

A series solution of Equation (12) is

= ~ a2jx2j , (14) o

where the a 's are given by the relation

K(n + 1)(n + 4)an+l + [ J , - 1 0 e - 5(n - 1)(n + 4)]an_l +

+ [ - J n + 1 6 e + 4 ( n - 3 ) ( n + 4 ) ] a n - 3 -

- [6c~ + (n - 5)(n + 4)]an_5 = 0. (15)

The points of singularity for the Equation (12) are given by

x = 0 and x 6 - 4x 4 + 5x 2 - K = 0. (16)

354 M . S I N G H A N D S, R . A G A R W A L

For the particular cases considered it is found that the moduli of the roots of Equation (16) are all greater than b. So the series (14) for r/converges throughout the core and at the interface whatever be the value of jn.

Again substituting for g and P from Equations (6) and (7) in the Equation (11), we have the following equation of pulsation for the envelope

x2[(3x(1 - X 2) + 2L(1 - x3)] d2- r /+ [4x {3x(1 - X 2) + dx 2

dr/ + 2 L ( 1 - x 3 ) } - 6 x ( L + x)]~-xx + [ J ' x 3 - 6 ~ ( L + x ) ] r / = O , (17)

where

J ' = 3n2 /2nyGpcb2(1 - b 2) and x = r / R .

Equation (17) has singularities at

1 1 x = - ( K ~ + ~ ) , 0 , ( K I - ~ ) and 1, where

(18)

2L KI -- _ / -

~/4 3 + 2 L

For b = 0.3, 0.5, 0.7, 0.9 the values K 1 - � 8 9 are, respectively, -0 .023 56, 0.218 12, 0.265 31, 0.0591 so that solutions which are convergent obtain in the range b < x < 1. To do this, we substitute x = 1 - t in (17), obtaining

(1 - t) 2 [(3 + 2L)t 3 - 3t2(3 + 2L) + 6t (L + 1)] d 2 ~ - (1 - t) • dt 2

x [4{t3(3 + 2L) 3t2(3 + 2L) + 6 t (L + 1)} {6(L + 1) 6t}] dr/ . . . . +

dt

+ [ - t 3 j ' + 3t2J ' - 3 t ( J ' - 2c 0 - 6~(L + 1) + J ' ] r /= 0 . (19)

The series solution of the Equation (19), finite at t = 0, is

r/ = ~ Cn tn , 0

where o's are given by the relations

[6ne(L + 1)]c, + [ - 3(n - 1) t,,n- 2) (3 + 2L) -

- 6 ( n - 1)(2n+ 1 ) (L+ 1 )+J ' -

- 6 ~ ( L + 1 ) - 6 ( n - 1)]c._l + [ ( n - 2 ) ( 3 + 2 L ) ( 7 n - 9 ) +

+ 6(n - 2) (n + 1) (L + 1) - 3 ( J ' - 2~) + 6(n - 2)]Cn-Z +

+ [3J ' - (n - 3)(5n - 4 ) (2L + 3) ]c ,_3 +

+ [(n - 4 ) (n - 1 ) ( 3 + 2L) - J ' ] c n - 4 = 0 . (20)

ADIABATIC RADIAL PULSATIONS OF A COMPOSITE STELLAR MODEL 355

No singularity of Equation (19) lies within a circle of radius (1 - b). Therefore, the series (20) converges at every point of the envelope and the interface, whatever be the frequency of oscillations.

4. Solutions of the Equations

With the aid of the condition that the motion (i.e., (dtl/dr)/tl) should be continuous across interface, numerical solutions have been carried out for few cases with different values of b. The solutions are obtained by trial and error. The values of t/' /t/at the interface are obtained for the core and envelope for two different frequencies, and the correct frequency of oscillation inferred from them by interpolation. The values of t / / t / recalculated for the frequency and a better approximation to the frequency obtained by interpolation again. Repetition of this process gives ultimately a correct value.

T A B L E I

Values of Tx/GIS/6~ for composite model (c~ = 0.6)

• Model with "--.~terfaces

M o d e b = 0.3 b = 0,5 b = 0.7 b = 0.9

Ze ro th 0 .5349282 0.5823994 0.6089448 0.5734573 First 0.2162690 0.2182200 0 .2142157 0.2154571

Second 0.1395331 0.1358995 0.1343216 0.1371860

2.2

2.0

1.8

1.o

1.4

1.2

l,O

0-8

o.6

0.4

0.2

o

-o.2

-0.4

-0.8

-o-8

-I.o

-u2

-l.4

-I-8

Fig. 1.

i

o lh

/

o.~

\ " \ ~.d

\ \

I

'\ / / / -

\ . / .

\ / / \ /

o!< i i

• ~~ii.._.i./i I

I I

i s t ~ , / / /

/ /

Variation of the amplitudes for ease b = 0.5.

356 M. SINGH AND S. R. AGARWAL

In all the cases the series solution provided values of t/correct to seven figures from x -- 0.02 to x = 0.10 near the centre and from x = 0.95 to x = 1.00 near the surface. For the intermediate values, the integrations were carried out numerically.

The characteristic periods T ~ (Prasad, 1949), where T is period of oscilla- tions, for the first three modes &oscillation, for the models with interfaces b = 0.3, 0.5,

0.7, 0.9 are given in Table I. The amplitudes are given in the Table IV. The variation of amplitudes for different modes for the case b = 0.5 have been shown in Figure 1.

5. Conclusions

In Table II, the periods T x/G}/6rc for the fundamental mode of the present models alongwith Prasad (1949) model, Stothers and Frogal's (1967) models and homogeneous model, have been arranged in the order of increasing Pc~}. The present models with interfaces b = 0.3, 0.5 however do not fit in this table. They support the conclusion

T A B L E II

M o d e l p~/ff T . / ~ / 6 re

H o m o g e n e o u s ~ 1.000 000 0 0 . 7 0 7 1 0 0 0

b = 0.7 b 2 . 1 4 1 0 1 5 9 0 . 6 0 8 9 4 4 8

b = 0.9 b 2 . 3 7 5 9 9 6 7 0 . 5 7 3 4 5 7 3

p = pc(1 - x 2 ) r 2 . 5 0 0 0 0 0 0 0.549 3000

b = 0.5 b 2.580 645 2 0.582 3 9 9 4

p = pc(1 -- X) d 4 . 0 0 0 0 0 0 0 0 . 5 2 1 8 3 7 7

b = 0.3 b 5 . 0 6 2 4 7 0 8 0 . 5 3 4 9 2 8 2

S te rne (1937). b P resen t .

~ P r a s a d (1949).

d S t o t h e r s a n d Froge l (1967).

~ P r a s a d (1949).

d S to the r s a n d Froge l (1967).

T A B L E III

M o d e l M2= ~x2dq r,,/~/6~

p = pc(1 - x) d 0 . 4 0 0 0 0 0 0 0.521 8377

b = 0.3 b 0.410 3283 0 . 5 3 4 9 2 8 2

p = pc(1 - x2) ~ 0 . 4 2 8 5 7 1 4 0.549 3000

b = 0.9 b 0.453 854 6 0,573 4 5 7 3

b = 0.58 0.463 1336 0 . 5 8 2 3 9 9 4

b = 0.7 b 0.491 8590 0 . 6 0 8 9 4 4 8

H o m o g e n e o u s a 0 . 6 0 0 0 0 0 0 0 . 7 0 7 1 0 0 0

a S te rne (1937). b Presen t .

~ P r a s a d (1949). d S to the r s a n d F roge l (1967).

ADIABATIC RADIAL PULSATIONS OF A COMPOSITE STELLAR MODEL 357

TABLE IV

r/'s for the composite models

Model (I) b = 0.3 (II) b = 0.5

x ~ d e Zeroth First Second Zeroth First Second

1 2 3 4 5 6 7

0 . 0 2 0.6619053 -1.3680044 1.7832117 0.8390437 -1.4312274 1.9902490 0 . 0 4 0.6622311 -1.3668607 1.7777539 0.8391852 -1.4294982 1.9828176 0 . 0 6 0.6627645 - 1.3649408 1 . 7 6 8 6 1 6 9 0.8394215 - 1.4266096 1.9704349 0 .08 0.6635507 -1.3622244 1 . 7 5 5 7 3 8 1 0.8397533 -1.4225513 1.9531064 0 . 1 0 0.6645581 - 1.3586813 1.739028 1 0.840 1813 - 1.4173093 1.930839 1 0 . 1 2 0.6658096 -1.3542763 1.7183814 0.8407068 -1.4108675 1.9036417 0 . 1 4 0.6673196 -1.3489556 1.6936388 0.8413311 -1.4032019 1.8715278 0 . 1 6 0.6691063 -1.3426528 1 . 6 6 4 6 0 6 8 0.8420562 -1.3942845 1.8345139 0.18 0.671 1923 -1.3352857 1.6310453 0.8428840 -1.3840827 1.7926210 0 . 2 0 0.6736059 -1.3267518 1.5926629 0.8438171 -1.3725587 1.7458744 0 . 2 2 0.6763827 - 1.3169228 1.5490986 0.8448582 - 1.3596688 1.6943060 0 . 2 4 0.6795678 -1.3056382 1.4999068 0.8460106 -1.3453630 1.6379547 0 . 2 6 0.6832189 -1.2926934 1.4445315 0,8472782 -1.3295851 1.5768672 0 .28 0.6874111 - 1.2778258 1.3822707 0.8486649 -1.3122711 1.5111007 0 . 3 0 0.69224390 -1.26069050 1,31222290 0.8501759 -1.2934343 1.4407239 0 . 3 2 0.69781157 -1.24098240 1.23377270 0.8518165 -1.2727355 . 1.3658201 0 . 3 4 0.70405667 -1.21879040 1.14806230 0.8535929 -1.2503404 1.2864883 0 . 3 6 0.71087763 -1.19423670 1.05646900 0.8555124 -1.2260595 1.2028482 0 .38 0.71818350 -1.16736090 0.96011771 0.8575831 -1.1997751 1.1150421 0 . 4 0 0.72589532 -1.13815870 0.85999000 0.8598144 -1.1713546 1.0232405 0 . 4 2 0.73394559 -1.10660210 0.75698804 0.8622172 -1.1406464 0.9276473 0 . 4 4 0.74227703 -1.072651 16 0.65197434 0.8648040 -1.1074781 0.8285053 0.46 0.750 84122 - 1.03626090 0.545 796 73 0.8675894 - 1.071 6519 0.726105 5 0 . 4 8 0.75959719 -0.99738481 0.43930458 0.8705909 -1.0329397 0.6207955 0 . 5 0 0.76851033 -0.95597655 0.33335951 0.87382912 -0.99107613 0.51299249 0 . 5 2 0.77755130 -0.91199153 0.22884267 0.87732157 -0.94584432 0.40336724 0 . 5 4 0.78669523 -0.86538716 0.12665965 0.88105847 -0.89733342 0.29327607 0 . 5 6 0.79592098 -0.81612314 0.02774398 0.88502067 -0,84568595 0.18418715 0 .58 0.80521053 -0.76416142 -0.06694047 0.88918865 -0,79100898 0.07752418 0 . 6 0 0.81454851 -0.70946621 -0.15639768 0.89354331 -0.73338339 -0.02531349 0 , 6 2 0.82392176 -0.65200377 -0.23959822 0.89806650 -0.67287050 -0.12293622 0 , 6 4 0.83331905 -0.59174226 -0.31547832 0.90274119 -0.60951681 -0.21395303 0 .66 0.84273070 -0.52865165 -0.38293917 0.90755165 -0.54335757 -0.29696331 0 . 6 8 0.85214844 -0.46270347 -0.44084632 0.91248339 -0.47441934 -0.37055055 0 .70 0.86156514 -0.39387071 -0.48802934 0.91752315 -0.40272198 -0.43327735 0 . 7 2 0.87097467 -0.32212767 -0.52328143 0.92265888 -0.32828012 -0.48368164 0 . 7 4 0.88037177 -0.24744984 -0.54535916 0.92787961 -0.25110427 -0.52027359 0 . 7 6 0,88975190 -0.16981378 -0.55298220 0.93317537 -0.17120173 -0.54153319 0 . 7 8 0.89911115 -0.08919703 -0.54483316 0.93853713 -0.08857721 -0.54590826 0 . 8 0 0.90844619 -0.00557790 -0.51955728 0.94395671 -0.00323344 -0.53181279 0 . 8 2 0.91775415 0.08106415 -0.47576225 0.94942670 0.08482853 -0.49762556 0 . 8 4 0.92703256 0.17074944 -0.41201786 0.95494039 0.17560891 -0.44168882 0 . 8 6 0.93627932 0.26349732 -0.32685567 0.96049171 0.26910895 -0.36230714 0 . 8 8 0.94549265 0.35932660 -0,21876836 0.96607514 0.36533075 -0.25774610 0 . 9 0 0.95467103 0.45825559 -0.08620873 0.97168571 0.46427707 -0.12623064 0 . 9 2 0.96381321 0.56030211 0.07241216 0.97731888 0.56595121 0.03405741 0 . 9 4 0.97291792 0.665483 31 0.258 675 05 0.98297030 0.67035621 0.224 925 82 0 . 9 6 0.98198440 0.77381586 0.47422963 0.98863643 0.77749639 0.44825333 0 .98 0.99101194 0.88531610 0.72076306 0.99431401 0.88737614 0.70595857 1 .00 1.00000000 1.00000000 1.00000000 1,00000000 1.00000000 1.00000000

J ' 5.05724120 30.93968700 74.32725200 3.65577720 26.03947600 67.14028000

358 M. SINGH AND S, R. AGARWAL

Table I V (continued)

Model (III) b = 0.7

x ~ d e

(IV) b = 0.9

Zeroth First Second Zeroth First Second

1 2 3 4 5 6 7

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0,76 0,78 0,80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1 . 0 0

j '

0.8908467 - 1.5146464 2.4584577 0.8909581 -1.5125511 2.4485321 0.8911441 -1.5090538 2.4320090 0.8914048 -1.5041470 2.4089178 0.8917405 -1.4978198 2.3793008 0.8921517 -1.4900597 2.3432058 0.892 6389 - 1.480 849 0 2.300704 6 0.893 2029 - 1.4701671 2.251 8809 0.8938442 -1.4579898 2.1968333 0.8945639 - 1.4442898 2.1356770 0.8953627 - 1.4290367 2.0685456 0.896 2420 - 1.412195 7 1.995 591 8 0.8972026 -1.3937289 1.9169908 0.8982462 -1.3735939 1.8329406 0.8993739 -1.3517447 1.7436663 0.9005873 - 1.3311619 1.6494214 0.901 8882 - 1.302 698 8 1.5504916 0.9032782 -1.2753886 1.4471977 0.9047592 - 1.246 1369 1.3398996 0.9063332 -1.2148758 1.2290004 0.9080022 -1.1815323 1.1149498 0.9097685 - 1.1460290 0.9982512 0.9116343 -1.1082834 0.8794642 0.9136021 - 1.0682089 0.7592130 0.9156741 - 1.0257141 0.6381900 0.9178528 -0.9807045 0.5171628 0.9201407 -0.9330813 0.3969813 0.9225400 -0.8827447 0.2785820 0.9250528 -0.8295934 0.1629956 0.9276811 -0.7735273 0.0513500 0.9304262 -0.7144510 - 0.055 125 6 0.9332891 -0.6522760 -0.1551022 0.9362695 -0.5869271 -0.2471516 0.9393666 -0.5183487 -0.3297587 0.94257741 -0.446 513 82 -0.40133845 0.94589682 -0.37145783 -0.46026188 0.94931750 -0.293 21792 -0.50490540 0.95283200 -0.21184230 -0.53362431 0.95643304 -0.127 37048 -0.54475090 0.96011359 -0.039 83488 -0.53659226 0.96386696 0.05073789 - 0.50742749 0.96768682 0.14432600 - 0.455 50541 0.97156722 0.24091158 - 0.379 04260 0.97550257 0.34048005 -0.27622145 0.97948764 0.44301959 -0.145 18800 0.98351756 0.54852068 0.01595102 0.98758771 0.65697381 0.20907518 0.991694 00 0.768373 99 0.436129 98 0.995 83262 0.882717 14 0.6~9098 24 1.000 00000 1.000 000 00 1.000 00000

3.02419050 24.43777600 62.15446100

0.867 9484 0.8680592 0.8682440 0.8685032 0.8688370 0.869 2459 0.8697307 0.870 2922 0.8709311 0.8716484 0.8724454 0.873 323 2 0.8742834 0.875 327 6 0.876457 1 0.877 674 5 0.8789814 0.8803804 0.881873 7 0.883 4643 0.885 155 1 0.8869492 0.8888502 0.890 8620 0.8929885 0.8952344 0.897 6045 0.9001040 0.9027386 0.905 5147 0.908439 0 0.911 5187 0.9147617 0.9181768 0.9217729 0.925 5599 0.929 548 2 0.933 7483 0.9381709 0.942 825 9 0.947 7214 0.9528614 0.9582424 0.963 847 2 0.969 634 77 0.975 54409 0.98154600 0.987 62919 0.99378332 1 . 0 0 0 0 0 0 O0

4.98958590

- 1.003 319 9 - 1,002 0623 - 0.999 9624 -0.9970148 -0.9932112 - 0.9885426 -0.9829957 - 0.976 5550 - 0.969 2025 -0.9609174 - 0.9516764 -0.9414529 -0.9302172 -0.9179362 - 0.904 5733 -0.8900879 -0.8744352 -0.857566 1 -0.8394265 -0.8199573 - 0.799 0933 -0.7767632 -0.7528891 - 0.727 3853 -0,7001583 - 0.671 1053 -0.6401137 -0.6070602 -0.571 8092 -0.5342124 - 0.494106 8 -0.4513135 - 0,405 6369 -0.3568628 - 0.304 757 8 - 0.249 0687 -0.1895228 -0.125 831 1 - 0.057 693 2

0.0151903 0,0930974 0.1762445 0.264720 0 0.3583680 0.456600 31 0.558496 18 0.66375618 0.772408 66 0.88448171 1 . 0 0 0 000 O0

35.34633400

1.8322607 1.825 6469 1.8146304 1,7992207 1.779 4321 1.7552805 1,726 7905 1.693990 9 1.6569169 1.6156103 1.570120 3 1.5205049 1.4668325 1.409 1824 L347646 5 1.2823322 1.2133636 1.1408840 1.0650589 0,986079 6 0,904166 1 0.819 5721 0,7325892 0.643553 4 0.552850 3 0,460923 5 0,368283 1 0.275 5034 0.1832899 0.0923816 0.003 6757

-0,081 8116 -0.1629167 - 0,238306 6 - 0,306 4598 - 0.365 643 9 -0.413 8948 -0.449001 1 - 0.468499 7 - 0.469 6906 - 0.449 6944 - 0.405 580 2 - 0.334 620 9 -0.2347615 -0.10542090

0.05258871 0.240079 83 0.45907917 0.71172394 1 . 0 0 0 000 00

87.18583300

ADIABATIC RADIAL PULSATIONS OF A COMPOSITE STELLAR MODEL 359

(Ledoux and Walraven, 1958; Singh, 1968) that Pc/P is not sufficient to characterize the effect of density distribution on the periods.

We then calculated the second moment of mass concentration M 2 = So ~ x 2 dq, where q = M(r) /M, of all the model given in Table II. In Table III, we observe that the periods of all the models monotonically increase with the increase of second moment of mass concentration M 2. So we confirm that second moment of mass concentration characterize the periods more effectively than Pc~P, the central condensation.

The amplitudes increases more rapidly in the envelope than the core as shown in Figure 1.

Acknowledgement

One of the authors (S.R.A.) is grateful to U.G.C. for research fellowship.

References

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