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Gravitational radiation from n-body systems
ANNALS
OF PHYSICS
103, 233-250 (1977)
Gravitational
Radiation from n-Body Systems T. T. CHIA*
Astronomy Group, Physics Department, Queen’s Unirrersity, Kingston, Ontario, Canada K7L 3N6 Received January 3, 1976
An expression for the quadrupole moment of any two-body system with structure is derived from a “parallel axes” theorem. Within the weak-held limit of the theory of general relativity, expressions for the gravitational radiation flux of energy and angular momentum from two particles or two spherically symmetric bodies in arbitrary plane motion arising from any type of forces are consequently obtained in terms of time derivatives of the relative coordinates of the system. An estimate of the gravitational flux from any plane motion follows. In particular, the flux from systems with Keplerian and straight-line motion are deduced as special cases. For the general problem of a two-body system with intrinsic quadrupole moment (due to deviation from spherical symmetry), it is found that in addition to the flux from the orbital and the spin motion there is another source of flux-the interaction flux. This is shown explicitly in two special cases-the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid, and that of two spinning rigid rods in plane circular motion with parallel spin and orbital angular momentum. The interaction flux is regarded as the result of interaction of the bodies with gravitational waves. An outline of the method for the calculation of gravitational radiation flux from an n-body system is given. For a three-body system-an astrophysically interesting situationthis is worked out in detail. It is seen that the presence of an unsuspected third body can, by virtue of the interaction power term, increase the generation of gravitational waves significantly.
1.
INTRODUCTION
In the weak-field limit of the theory of general relativity, expressionsfor the rates of energy and angular momentum carried by gravitational waves from a system of bodies are well known. The energy rate is given by [l]
PE = (G/4X5)
...
(D28Y,
(1)
while the angular momentum rate is [2]
* Present address: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2.
233 Copyright All rights
,D 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
234
T.
T.
CHIA
The quantity ~~~~ is the completely antisymmetric quadrupole moment tensor, is defined by D,, =
s
&)(3x,x,
-
unit pseudotensor
and D,, , the
6,,r2) dV.
In this paper, we shall use the above framework to study the rates of energy and angular momentum carried by gravitational radiation from systems consisting of n bodies, each of which may possessintrinsic quadrupole moment-a measure of deviation from spherical symmetry, originating from nonspherically symmetrical forces, e.g., those associated with tidal distortion, magnetic fields, rotation, and nonspherically symmetrical density of mass and radiation. Our treatment will be general enough to allow for any arbitrary relative motion arising from any type of forces within the n-body system. We shall first derive a “parallel axes theorem” for quadrupole moment from which the quadrupole moment of an arbitrary two-body system easily follows. For the specialcaseof two particles or spherically symmetrical bodies with arbitrary relative plane motion arising from any type of forces, expressions for the flux or energy and angular momentum carried by gravitational waves can be obtained in terms of time derivatives of the relative coordinates. From these general results, the well-known rates of gravitational radiation from Keplerian and straight-line motion can be easily deduced as special cases,while for general motion, an estimate of the rates can be obtained. By taking into account the intrinsic quadrupole moment of each body, we find that the values of the gravitational wave flux will be altered from that of the corresponding n-body point masssystem in two ways. Firstly, by considering the structure (deviation from spherical symmetry) of each body, we will be introducing more quadrupole moments which, if time dependent, will contribute to the flux. Secondly, asthe general expressionsfor the flux (Eqs. (1) and (2)) involve time derivatives of the quadrupole moment, the flux will depend on the motion of the system. As the equation of motion for an n-body system with structure clearly differs from that for the n-particle case, the value of the flux will be different. Further, we shall find that the inclusion of intrinsic quadrupole moments gives rise to another source of gravitational radiation, the interaction portion, in addition to the spin contribution. This is borne out explicitly by working out two special casesin detail-the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid and the system of two spinning rigid rods revolving about each other with the spin and orbital angular momentum being parallel. The interaction flux can be regarded as originating from the interaction of the bodies with gravitational waves. The gravitational radiation flux from an n-body system can be calculated conveniently from this scheme.This is illustrated by working out the flux from a threepoint mass system whose motion is restricted to a plane. We shall show that the existence of a third unsuspectedbody in the system can increase significantly the amount of gravitational radiation generated.
GRAVITATIONAL
2. QUADRUPOLE
MOMENT
235
WAVES FROM n-BODY SYSTEMS
OF A TWO-BODY
SYSTEM IN ARBITRARY
RELATIVE
MOTION
First we give a relation between the quadrupole moment tensor of any system of massM, evaluated at its center of mass,and that evaluated at a different origin about axes parallel to the first set. It is D:, = Dno + M(3hxb, - &%B),
(4)
where Da4is evaluated at the center of mass,D& at the new origin 0’, at a distance b away with the two setsof axes being parallel. This gives the parallel axes theorem for quadrupole moment, which is analogous to the parallel axes theorem for the moment of inertia. The result is easily obtained from the definition of the quadrupole moment tensor (Eq. (3)). We can usethis result to calculate the quadrupole moment of any two-body system in arbitrary relative motion arising from any type of forces. By choosing the origin of the inertial frame to coincide with the center of massof the two-body system, we get (5) where Dl, is the intrinsic quadrupole moment of body i evaluated about parallel axes through its center of mass. 0:;” is defined by
p
=
[MdfB/@fA
+
MB)],
(7)
cl = sin 8 cos I$,
(8)
5, = sin 0 sin c#,
(9)
c3 = c~s e.
(10)
The separation between the centers of massof the two bodies is r while 8 and 4 are the usual spherical coordinates. We note that in the absenceof any intrinsic quadrupole moment of the bodies, the only surviving term in Eq. (5) is D$” which may therefore be identified as the quadrupole moment of the orbital motion.
3. TWO-BODY
SYSTEMS WITH POINT OR SPHERICALLY SYMMETRIC MASSES IN ARBITRARY RELATIVE PLANE MOTION
For this special system, the quadrupole moment of the system is given by
In general, the motion of the two bodies need not be confined to a plane, as there may be noncentral forces, e.g., electromagnetic forces, acting between the bodies. The
236
T.
T.
CHIA
expressions for the energy and angular momentum flux obtained by substituting Eq. (11) into Eqs. (1) and (2) become complicated. However, when the arbitrary relative motion arising from any type of force is confined to a plane, the expressions become particularly simple. (SeeAppendix for details.) Choosing 6’ = n/2, we get PE = 4y12 + ys2 + y3%), pJ3
=
(+)bly.l
+
E = (32G/5c5) YI =
1-
YZ = i(6P1 +
Y4
= 1-
x/932
75
=
+
+
683'
3AA
-
+ I%),
+ i&h
t31'?12)(383~4
HA
(14)
6PlP3
12A -
=
(13)
yz?%),
p2r4w6,
tU% +
Y3
(12)
+
6P4),
(15)
(16) (17)
(18)
P*),
(19)
4P3h
p1 = GJj/oJ2,
(20)
/I, = w/w3,
(21)
= f/(rw),
(22)
I33
/3* = qrc2>,
(23)
p5 = ‘&J3),
(24)
CO= $2
(25)
where the three-axis is perpendicular to the plane of motion. The quantities & and yi are determined from the equation of motion. Once the motion is known, Eqs. (12) and (13) yield the flux. Even when the motion is not known, we can get an estimate of the flux. Since pi is small compared to unity, only yr and y4 are of the order of unity while the remaining yi are small. Thus PE - E and PJ - E/W. For the special casesof Keplerian motion and motion in a straight line, “we can use Eqs. (12) and (I 3) to determine the flux. 3(a) Circular
Keplerian
Orbit
For an orbit which is initially circular, the loss of energy by gravitational waves will change the size of the orbit. That is, r and w are not constants of motion. To a good approximation, we can neglect these variations. Consequently, when Kepler’s law is used, we get 1,
y1=y4=
Y2
=
Y3
=
y5
(26) = 0,
(27)
GRAVITATIONAL
WAVES
FROM
n-Bow
237
SYSTEMS
and p = 32G”MA’MB2(MA E 5cV pJ~ =
32G7/2M..,2M;z(M, + M#J’~ 5c5r7/2
to the same degree of approximation. Equations tional radiation flux from a circular orbit.
3(b)
+ MB)
Elliptical
>
(29)
(28) and (29) give the usual gravita-
and Hyperbolic
Orbit
For two particles or spherically symmetric masses subjected only to mutual gravitational forces, the equation of motion is not strictly Keplerian. This is clear since the system, by emitting gravitational waves, behaves as if it is acted on by a gravitational radiation reaction force which is noncentral. Again to a good approximation, we could assume Kepler’s law to be valid. The equation of the orbit is therefore given by r=p/(l where for a hyperbola, p = a(e2 get
Substituting
+ecos+)=px, 1) and for a parabola, p = a(1
(30) e”). Thus, we
y1 = x,
(31)
y2 = (x2e sin 1$/4),
(32)
y3 = -(3112/12) x2 e sin 4,
(33)
y4 = x2[1 - (e”/2) + Z$ecos + + e2 cos2 $1,
(34)
y5 = xe sin +.
(35)
into Eqs. (12) and (13) we obtain p
E
= =G*MA~ME~“(M‘.I 5c5p5
+ MB)
. (1 + e Cm 4)” [(I + e cos 4)” + (e2/12) sin2 $1, p
J3
(36)
= 32G712M~2MB2(MA + MB)l12 5c5p712 . (1 + e cos 4)” [l - (e2/4) + $e cos + + $e2 cos2 ~$1,
the usual results [2, 31.
(37)
238
T. T.
3(c)
CHIA
Straight-line
Motion
The angle 4 is a constant for motion in a straight line. By absorbing w from E into the y’s, we immediately get PE = [8G/(l 5c5)] p2(rr’
+ 3ti)2
(38)
and P, = 0.
(39)
Neglecting radiative reaction, we have i: = -[G(M,
+ MB)/r2]
(40)
and pE = +$ where KO is an integrating
4.
A
TWO-BODY
( M~;F2
)( ~G(MA + MB) + KOj, r
(41)
constant.
SYSTEM
WITH
INTRINSIC
QUADRUPOLE
MOMENTS
When the bodies deviate from spherical symmetry, the gravitational radiation flux will differ from those of the two-point mass system (assuming that in both systems the other relevant physical parameters are identical). There is contribution from the rate of change of the intrinsic quadrupole moment of each body. Even when this is zero, e.g., in the case of a symmetrical top and a point mass whose motion is restricted to the plane of symmetry of the top, the flux will be altered because of a modified equation of motion. Further, as the expressions for the flux (Eqs. (1) and (2)) involve quadratic products of the time derivatives of the quadrupole moment tensor which consists of a sum of terms (Eq. (5)), we see that when the intrinsic quadrupole moment is time dependent, there will be contribution to the flux from the cross terms (products of the orbital and intrinsic quadrupole moment). This additional flux is identified as the interaction flux. In general, the motion of the two-body system may be complicated. Apart from the existence of other forces, the gravitational forces due to the intrinsic quadrupole moments will produce mutual torques resulting in complicated motion. To bring out the essence of the problem, we shall only consider two analytically tractable special cases. 5.
SYSTEM
OF A POINT
MASS OF
MOVING
A JACOBI
IN THE
SYMMETRIC
PLANE
ELLIPSOID
Consider a Jacobi ellipsoid of uniform density p and mass M, rotating uniformly with angular velocity Q about the smallest principal axis as. We assume that only
GRAVITATIONAL
WAVES
FROM
n-BODY
239
SYSTEMS
gravitational forces are present and restrict the motion of the particle, of mass MB , to the plane of symmetry of the Jacobi ellipsoid perpendicular to the a3 axis. The quadrupole moment of a homogeneous ellipsoid evaluated in the body axes is given by [4]. 0;; = (MA/5)(3a:
- a,,“),
(42)
where in this section i = 1, 2, 3, and is not summed over. The off-diagonal terms vanish because of reflection symmetry. Using Eqs. (5) to (lo), we find that the quadrupole moment of the whole system is given by D,, = 0;: cos’ I& + Dl,” sin* Bt + ~“(3 co? 4 -
I),
(43)
D,, = D$ sin* li’t + 0;: cos2 Qt + pr2(3 sin2 4 -
I),
(44)
D,, = 0;;’ -
pr2,
(45)
D,? = D,, = &(D;; - 0;;)
sin 2L?t + (3pr2/2) sin 24,
(46)
where we have chosen 8 = 7712and assumed that the body axes and the fixed inertial axes are parallel at t = 0. Substituting into Eqs. (1) and (2), we find that the gravitational radiation energy flux can be written as PE = PEA + PEAsB+ P,‘,
(47)
where PEA, Pi,’ and PEr denote the spin, orbital, and interaction portions of the flux, respectively. These are given by PEA = (32G/125c5) ,%fA”($” - ~7~“)~@,
A.B PE = 4Y12 +
Y22
+ Y32),
(48)
(49)
PE* = (64G/25c5) phfAr2(a12 - az2) w3Q3 * [yl cos 2(w - Q) f + y2 sin 2(w - Sz) t],
(50)
with E, p, w, and y defined by Eqs. (14) to (25) and the particle at a distance r from the center of the ellipsoid. Similarly, the angular momentum flux is given by (51) 595/103/x-16
240
T.
T.
CHIA
with Pi = PEA/Q,
(52)
AB PJ3 = (4WNYlY4 + Y*Y& P:, = (32G/251)
$fAr”(al”
(53) -
azz) w2J?
. KYP + y&3 cm 2(w - Q> f + (y2w + ~$9 sin 2(w
Q) tl. (54)
That is, it consists of the sum of three terms. In the above expressions, the variables r and w are not independent. They are related by the equation of motion. This can be determined in principle from the known expression for the gravitational acceleration at a point xi’ (measured in the body axes) external to a Jacobi ellipsoid. If the gravitational acceleration is denoted by gi, then [5] g, = -2rrGpxi’Bi, (55) with Bi = a1a2a3 sor [(a;* + 24)(a;2 + 24);: + u)]l/* (ai + 24) ’
(56)
a;” = ai + A,
(57)
T L42/(ai2+ 41 = 1,
(58)
where as before the dummy summation convention is not used for the Latin subscripts. The equation of motion is clearly non-Keplerian. We therefore see that with the inclusion of intrinsic quadrupole moment each of the energy and angular momentum fluxes consists of three terms: the spin, the orbital, and the interaction. We note that the interaction terms do not vanish when averaged over a long time interval, as the variables w, 52, and yi are functions of time. The ratio of the interaction to the orbital term is given by
. [
y1 cos 2(w - Q) t + y2 sin 2(w - Q) t Y12 + Y22 + Y3p
1
(59)
for the energy flux and by
cos 2(w . (Yl+ Y*GGJ)) [
- 9) t + (y2 + y,(Q/o~)) sin 2(w - Q) (YlY4 f Y2YJ
t
I (60)
GRAVITATIONAL
WAVES
FROM
II-BODY
SYSTEMS
241
for the angular momentum flux. Both ratios can be much bigger than unity under favorable conditions of large L?/w and large MA/MB , conditions realizable in the case of a planetary object moving in an orbit about a neutron star.
5(a)
Circular
Motion
In the special caseof circular motion, the equation of motion can be determined easily and expressions for the interaction flux simplified. For circular orbits, we require that the force acting on the particle be radial. This is possible only if the particle lies on either of the extensions of the two remaining principal axes at all times. This implies that the orbital and spin angular velocity be equal. With these assumptionsand neglecting the ,8’s, as these are small, Eqs. (47) to (50) are replaced by PE = PEA + P$” PEA = +&
-
(62)
apz)2 Q6,
MA2MB2 (MA + M$
5C5
PE’ = g
(61)
MAz(al”
= ~32G
p$”
+ PE1,
pMAr2(a12
rW
-
’
a22) Q6,
(63)
(64)
where the particle is located on the extension of the diameter a, . The angular momentum flux is given by Eq. (51), each component being proportional to the corresponding component of the energy flux:
(65)
These results are derived with the assumption that the particle is located on the extension of the diameter a, . Corresponding to whether a, is greater or smaller than a, , two casesresult. When these are equal, both the spin and interaction terms vanish identically. 5(a)(i) Particle
along Extension
of Largest
Diameter
Here, as a, is bigger than a,, Eqs. (64) and (65) show that the interaction terms are positive. The equation of motion is easily obtained from Eqs. (55) to (58). It is Q2 = G(MA
+ MB) A, ,
(66)
242
T. T. CHIA
where (1, = 3 bwl 3 41) - w4 > 41)j 6%” - u22)(a12- a,2)1~z’
-W, $1 and W,
sin28, = (al2 - u~~)/(u~~- as2),
(68)
sin2 I& = (a,” - a3”)/r2.
(69)
4) are the usual incomplete elliptic integrals E(B, 4) = Jo’ (1 - sin2 0 sin2+‘)lj2 d$‘,
(70)
F(f?, 4) = job (1 - sin2 tI sin2 +‘)-1/2 d+‘.
(71)
As the equation of motion is now different from Kepler’s law, the orbital component of the flux will be different from that of the corresponding two-point masscase. S(a)(ii) Particle along Extension of Second-LargestDiameter The interaction terms are negative when a2 is greater than a,. The equation of motion now becomes ~-2’= G(M, + M&A,,
(72)
where A, =
3(a,2 - u3y2 (a2” - a12)(ff12- as2) . E(B, , 4,) - qe, , $2) (30~2 8, [
(a,2 -ra32)“2
cos &
sin2e,],
sin2 8, = (ffz2 - ~~~)/(a,~- a32), sin2 b2 = (uz2 - a32)l(r2+ a22-
(73) (74)
a12),
(75)
with the samedefinitions for E(O, 4) and F(0, 4). This equation again is different from Kepler’s law.
6. A SYSTEM OF Two GRAVITATIONAL
BOUND SPINNING
RODS
Consider two spinning rigid rods subjected only to mutual gravitational forces, moving in a circular orbit in a plane such that at all times, the center of massof the system and the two axes of the rods are collinear. This implies that the orbital angular velocity Q of one rod about the other and the spin angular velocity of each rod are identical. Let the length and massof rod A, B be denoted by 2LA,B and MR.B and the
GRAVITATIONAL
WAVES
FROM
n-BODY
243
SYSTEMS
separation between the centers of mass of each rod be Y. Using Eqs. (5) and (6), we get Da!3= Z(3lxs
- &3>,
(76)
where I= ZA.B=
IA,, + IA + IB ,
(77)
WAMBIWA + MidI r2,
(78)
IA = gif,L,2,
(7%
I, = +MBLB2.
W
Substituting Eq. (76) into Eqs. (1) and (2) we get, when the angular momentum parallel to the three-axis,
is
PE = (32G/5c5) 12Q6,
(81)
P, = (32G/5c5) Z2si5,
G32)
where we have to the same degree of accuracy neglected & . They can be expressed as (83) p:,
(84)
9
where PEA = (32G/5c5) IA2i?,
(85)
PEB = (32G/5c5) IB2@,
(86)
PisB = (32G/5c5) z;,,@,
(87)
PE’ =
+ IA,BIA
(64G/5C5)(I~I~
and P 25
P,“, pE--=-=-PEA
DI PJ”a” PEA.B= FE+ = J-i-1.
P,“, pBB
-
(8%
That is, the angular momentum and energy flux each consistsof the sum of spin and orbital, as well as interaction terms, all of which are positive. The equation of motion becomes c2
=
G(MA
+ 4LALBr
MB)
In
r2
-
(LB
-
LA)~
r2
-
(LB
+
LA)2
1 '
(90)
where the cross sectionsof the rods are assumedto be small compared to the lengths.
244
7. T. CHIA
We may regard our model as an extreme idealization of two tidally distorted synchronized stars. In reality, the distortion will be very small. We see that when the two stars are of equal mass M and equal radius R, pE’/pEA =
4(IA,B/zA)
+ 2
(91)
and
The interaction flux is therefore very small compared to the orbital flux. At most it can be equal to two-thirds of the orbital flux for the extreme case when the two stars are almost touching each other.
7. GRAVITATIONAL RADIATION FROM ~-BODY SYSTEMS The gravitational radiation flux from n-body systems in the weak-field limit can be calculated conveniently within our scheme. We use Eq. (5) to evaluate the quadrupole moments required for the flux determinations (Eqs. (1) and (2)). Though Eq. (5) refers to two bodies, it can be applied to n-body systems since each body in the twobody system can be regarded as comprising multiple bodies. Thus we first divide an n-body system into two subsystems, consisting of n, and (n - n,) bodies, respectively. Writing n2 for (n - n,), Eq. (5) yields DrB = Da”,’ + D?“B’+ D;;*n2,
(93)
where Dzi is the quadrupoIe moment of the n,-body system referred to its center of mass, and Dziqn2 , the orbital portion of the n, and n2 system, is given by Of;‘“”
= ~L1Z1,nzr~l,n2[3i,n1,nz5an1*n, - SE,] c y,;.y
(94)
The quantity r,l,,2 is the separation between the two centers of mass of the n, and n2 systems, while pnl,nz , the “reduced mass” of the two systems, can be written as (95) The quantities
GRAVITATIONAL
WAVES FROM n-BODY SYSTEMS
245
where 0’~~‘~~ and 4 n**~2are the spherical coordinates referred to the center of massof the combined system. Next we regard each of the 11~and n, systemsas comprising two bodies, and apply Eq. (5) again. The process can be repeated until we obtain n one-body systems. For example, a seven-body system labeled A, B, C, D, E, F, G yields, successively,
= i
D&, + Y;”
+ YsD + Y,EBF$ Y$*cD + YzsG + Y$‘CDgEFG. (99)
i=l
When the number of bodies is large, this method is tedious. It is useful particularly for small n.
8. SYSTEM OF THREE-POINT
OR SPHERICALLY MASSES IN PLANE MOTION
SYMMETRICAL
Consider a system of three bodies, A, B, C, each without intrinsic quadrupole moment, moving in a plane-a good model of a three-body stellar system. Grouping A and B, we obtain
where
PAB,C = [(MA + MB) MAM,
+ MB + &)I,
(102)
with the other variables defined in the preceding section. For convenience, we shall drop the indices AB, C and replace A, B by a prime. Thus, e.g., rAB,c = r and rA,E = r’. Choosing 6’ = 71.12, and using the resultsand notation of Section 3 and the Appendix, we get pE = p;s.c + Pi?” -I- PB’,
(103)
246
T. T.
CHIA
where (MA+
MB)MC
2
32G PEA.B =-[
MAMB MA
f’;
=
!$
+
1”
r’4u’6
(y’2
+
+
Y22
yL2
+
+
(104) ’
Y3%
(105)
y;“>,
MB
MA”BMC
[
r4W6(Y12
I
MA+Mi?+MC
]
r2r’2w3w’3
MA+MB+MC
. ((~1~1’i- YZYZ’ A- 3~~‘) -I- (y1’y2 - yIy2’) sin 2($ - d’)},
(106)
and P, = PfaBsC + PJ”;” i- p:, ,
(107)
with (MA+
1
MB)Mc
'
r4w5hy4
MA+MB+Mc-
32G
A.B p,
p;,
x
MAMB
=
-Q-
[
=
$
[
=
hY4’ +
b2Y4
r’4w’5h’y4’
72%7
w 7175’)
+
+ w
(y1’74 -
(109)
y2’y5’),
r2r’2w2w’2x,
]
MAMBMC MA+MB+Mc
-
(108)
YzY5),
2
I
MA+MB
+
+
+ b2’74
yZ’y5) -
(110) d
yl’y5)
cos d
2(+ sin
2(+
4’) -
6’).
(I
1 l)
The equation of motion of the system must be solved to yield the time derivatives of the relative coordinates. However, an approximation of the gravitational radiation flux can be obtained for the cases when r > r’. Since & and pi’ are both much less than one, we equate y1 , yl’, y4, and y4’ to unity and neglect the remaining y’s. Further, we put
G(M,
With this approximation,
G(MA + MB) N w’2r’3,
(112)
+ MB + MC) N w2r3.
(113)
Eqs. (103) to (106) yield the energy flux P,‘/P$”
PE1/P$y p~‘CIP$B
E
2K1
(114)
,
II 2/q
(115)
N
(116)
K12,
with KI = [(MA i- MB)(MA i- MB + MC)]“”
(Mc/MaMB)(r’/r)5/2.
(117)
GRAVITATIONAL
WAVES
FROM
n-BODY
Similarly, the angular momentum flux can be written
(from Eqs. (107) to (111)) as
P;,/P$B = K2(1+ K3)c0s 2(4 - $0 P:JPfy p~‘“/Pf;”
247
SYSTEMS
(118)
= ((1 + K3YK2K3) cos 2(4 - 4')Y
(119)
=
VW
K22K3,
where
K3
=
[(cMA
+
MB
+
&)/(MA
+
MB>)(r’/r)31”2.
(122)
The interaction components of the angular momentum flux have an angular dependence which may average to zero if the time variations of w, w’, r, and r’ are small. However, the interaction portion of the energy flux is always finite. The possible ranges of or are (a)
or > 2. This condition
implies that
which is not an interesting case. On the contrary, (b)
the next three are.
For2>K1>1,
PE1> P$= > PisB.
(d)
For small
(124)
K1 ,
Cases (b) and (c) are of astrophysical importance. Consider a three-body system in which body A and B are close to each other compared with their separation from C. Observationally, the existence of A may not be suspected because of its underluminosity compared to the other two bodies or its sepration from B being too small to be resolved. The expected flux is then given by PiB,=, since AB is viewed as a composite system. However, the actual flux should be given by Eq. (103). For definiteness, take r/r’ to be equal to four, the mass of C and the composite
248
T. T. CHIA
system AB in units of solar massto be six and two; respectively. If the massof A is taken to be @f, , we have PE’ ‘v (13/8) P$C, P$B ‘v (169/256) PppC,
(127) (128)
while if MA = +M, , we obtain (129) and (130) Thus the existence of a third unsuspected body in a binary system can contribute a significant increase of gravitational radiation compared to that generated by the expected observed composite system AB, C.
9. DISCUSSION We have seenthat when the intrinsic quadrupole moments of the two bodies of any system are taken into account, the rate of gravitational radiation differs from that of the corresponding two-point or spherically symmetric mass system. Firstly, as the equation of motion is no longer Keplerian (compare Eqs. (66) and (72) for the particleJacobi ellipsoid system and Eq. (90) for the rod-rod system with the usual Kepler’s equation) the value of the orbital portion of the flux will be different. Secondly, the intrinsic quadrupole moments, if time dependent, will emit an additional gravitational radiation termed the spin portion of the flux. Finally, the products of the orbital and intrinsic quadruple moment will result in another component of gravitational radiation-the interaction portion. For the particle-Jacobi ellipsoid system, the interaction flux can be positive or negative depending on whether the point massis located along the extension of the longest or second longest diameter, whereasfor the double-rod system, it is always positive. It is clear from Eqs. (1) and (2) that mathematically, the interaction comes from the cross products of the time derivates of the quadrupole moments. However, intuitively, without recourse to these equations, its presence might not be readily obvious. For example, in the caseof the double-rod system, the energy of the system can be written as
i.e., the sum of the rotational energy of each rod and orbital energy of the system. Since the rate of loss of energy by gravitational radiation from an isolated spinning
GRAVITATIONAL
WAVES
FROM
N-BODY
249
SYSTEMS
rod, A, and from an isolated orbital system are given by PEA and P,“*” one tends to think PE = Py
+ PEA -+- PEB
(132)
for the whole system instead of Eq. (83), in which an additional interaction term appears. The apparent paradox can be resolved as follows. In the derivation of Eqs. (1) and (2). the origin of the coordinate system coincides with the center of mass of the whole system. By intuitively treating the rods independently and then jointly, we are making use of three different coordinate systems with origins at the centers of mass of A and of B as well as of the system A&--an unacceptable situation. Just as the spin and orbital portions of the flux are said to come from the spins of the bodies and the orbital motion, respectively, we would like to know from where the interaction term comes. The physical origin of this interaction term is apparent by resorting to the energy momentum conservation law [6], T& = 0.
(133)
Integration over a volume V,, enclosing only the mth body of the system and use of the Gauss divergence theorem yields (apt)
J (-(-g)‘/” = (-c/2)
TOO)dP J’ (-g)‘iz
gPr,,z,oT(D’dC’ f c f, (-g)“2
TOwnadS.
(134)
The expression on the left-hand side denotes the rate of change of the energy of matter (excluding gravitational field energy) within the volume, V,,, . Since the first integral on the right-hand side of this equation involves a finite gl,,o which represents gravitational waves, the energy within the volume V,, , is modified by incident gravitational waves coming from other bodies. That is, due to the interaction of gravitational waves with the bodies there is transfer of energy among the components [6]. This implies that the flux of gravitational waves radiated to infinity will differ by the amount of gravitational wave energy flux that is transferred among the bodies. We can, therefore, attribute the interaction term to the process of interaction of the bodies with gravitational waves. A negative interaction power means that energy from the gravitational waves is absorbed by the bodies, while a positive interaction power implies that the bodies lose more energy as a result of interacting with gravitational waves. We can make similar observations about the interaction portion of the angular momentum flux. Finally, our scheme can be used for the calculation of gravitational radiation flux from n-body systems. In particular, for a three-body system the presence of an unsuspected body in a system hitherto assumed as binary can under certain conditions, increase significantly the gravitational radiation flux. Consequently, the upper limit of gravitational radiation flux from astrophysical sources needs to be revised to allow for multiple stellar systems.
250
T.
APPENDIX: TWO-POINT
T. CHIA
TIME DERIVATIVES OF QUADRUPOLE MOMENTS OF SYSTEMS COMPRISING OR SPHERICALLY SYMMETRIC MASSES IN ARBITRARY RELATIVE PLANE
MOTION
From Eqs. (6) to (IO), the nonvanishing time derivatives of the quadrupole moments are %, = 6cLr2w2[i - (yd3)
(Al)
- y,C - yJ1,
... D,, = 12vr24(y3/31/2)-
y2C
+ yJ1,
642)
ky = 6cLr2w2[i - (yd3) + y,C + @I, ... D,, = l&r2~3Ky3/31’2) + y,C - ylSl, O;,, = 4pr2w2[y4 -
(A3) (A4)
11,
(A5)
... D,, = -S(3)1/2 pr2u3y,,
W)
B,, = figz = 6pr2c02[y,C - yJ],
(A7)
...
D,, = --12pr2W3[y2S + yICl,
648)
where S = sin 24,
(A91
c = cos 24,
(AW
with the notation of Section 3.
REFERENCES 1. L. D. LANDAU AND E. M. LIFSCHITZ, “The Classical Theory Mass., 1962. 2. P. C. PETERS, Phys. Rev. B136 (1964), 12241232. 3. R. 0. HANSEN, Phys. Rev. D 5 (1972), 1021-1023. 4. W. Y. CHAU, Astrophys. J. 147 (1967), 664-671. 5. S. CHANDRASEKHAR, “Ellipsoidal Figures of Equilibrium,” 1969. 6. F. I. COOPERSTOCK, Phys. Rev. 163 (1967), 1368-1373.
of Fields,”
Yale Univ.
Addison-Wesley,
Press, New
Reading,
Haven/London,
OF PHYSICS
103, 233-250 (1977)
Gravitational
Radiation from n-Body Systems T. T. CHIA*
Astronomy Group, Physics Department, Queen’s Unirrersity, Kingston, Ontario, Canada K7L 3N6 Received January 3, 1976
An expression for the quadrupole moment of any two-body system with structure is derived from a “parallel axes” theorem. Within the weak-held limit of the theory of general relativity, expressions for the gravitational radiation flux of energy and angular momentum from two particles or two spherically symmetric bodies in arbitrary plane motion arising from any type of forces are consequently obtained in terms of time derivatives of the relative coordinates of the system. An estimate of the gravitational flux from any plane motion follows. In particular, the flux from systems with Keplerian and straight-line motion are deduced as special cases. For the general problem of a two-body system with intrinsic quadrupole moment (due to deviation from spherical symmetry), it is found that in addition to the flux from the orbital and the spin motion there is another source of flux-the interaction flux. This is shown explicitly in two special cases-the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid, and that of two spinning rigid rods in plane circular motion with parallel spin and orbital angular momentum. The interaction flux is regarded as the result of interaction of the bodies with gravitational waves. An outline of the method for the calculation of gravitational radiation flux from an n-body system is given. For a three-body system-an astrophysically interesting situationthis is worked out in detail. It is seen that the presence of an unsuspected third body can, by virtue of the interaction power term, increase the generation of gravitational waves significantly.
1.
INTRODUCTION
In the weak-field limit of the theory of general relativity, expressionsfor the rates of energy and angular momentum carried by gravitational waves from a system of bodies are well known. The energy rate is given by [l]
PE = (G/4X5)
...
(D28Y,
(1)
while the angular momentum rate is [2]
* Present address: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2.
233 Copyright All rights
,D 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
234
T.
T.
CHIA
The quantity ~~~~ is the completely antisymmetric quadrupole moment tensor, is defined by D,, =
s
&)(3x,x,
-
unit pseudotensor
and D,, , the
6,,r2) dV.
In this paper, we shall use the above framework to study the rates of energy and angular momentum carried by gravitational radiation from systems consisting of n bodies, each of which may possessintrinsic quadrupole moment-a measure of deviation from spherical symmetry, originating from nonspherically symmetrical forces, e.g., those associated with tidal distortion, magnetic fields, rotation, and nonspherically symmetrical density of mass and radiation. Our treatment will be general enough to allow for any arbitrary relative motion arising from any type of forces within the n-body system. We shall first derive a “parallel axes theorem” for quadrupole moment from which the quadrupole moment of an arbitrary two-body system easily follows. For the specialcaseof two particles or spherically symmetrical bodies with arbitrary relative plane motion arising from any type of forces, expressions for the flux or energy and angular momentum carried by gravitational waves can be obtained in terms of time derivatives of the relative coordinates. From these general results, the well-known rates of gravitational radiation from Keplerian and straight-line motion can be easily deduced as special cases,while for general motion, an estimate of the rates can be obtained. By taking into account the intrinsic quadrupole moment of each body, we find that the values of the gravitational wave flux will be altered from that of the corresponding n-body point masssystem in two ways. Firstly, by considering the structure (deviation from spherical symmetry) of each body, we will be introducing more quadrupole moments which, if time dependent, will contribute to the flux. Secondly, asthe general expressionsfor the flux (Eqs. (1) and (2)) involve time derivatives of the quadrupole moment, the flux will depend on the motion of the system. As the equation of motion for an n-body system with structure clearly differs from that for the n-particle case, the value of the flux will be different. Further, we shall find that the inclusion of intrinsic quadrupole moments gives rise to another source of gravitational radiation, the interaction portion, in addition to the spin contribution. This is borne out explicitly by working out two special casesin detail-the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid and the system of two spinning rigid rods revolving about each other with the spin and orbital angular momentum being parallel. The interaction flux can be regarded as originating from the interaction of the bodies with gravitational waves. The gravitational radiation flux from an n-body system can be calculated conveniently from this scheme.This is illustrated by working out the flux from a threepoint mass system whose motion is restricted to a plane. We shall show that the existence of a third unsuspectedbody in the system can increase significantly the amount of gravitational radiation generated.
GRAVITATIONAL
2. QUADRUPOLE
MOMENT
235
WAVES FROM n-BODY SYSTEMS
OF A TWO-BODY
SYSTEM IN ARBITRARY
RELATIVE
MOTION
First we give a relation between the quadrupole moment tensor of any system of massM, evaluated at its center of mass,and that evaluated at a different origin about axes parallel to the first set. It is D:, = Dno + M(3hxb, - &%B),
(4)
where Da4is evaluated at the center of mass,D& at the new origin 0’, at a distance b away with the two setsof axes being parallel. This gives the parallel axes theorem for quadrupole moment, which is analogous to the parallel axes theorem for the moment of inertia. The result is easily obtained from the definition of the quadrupole moment tensor (Eq. (3)). We can usethis result to calculate the quadrupole moment of any two-body system in arbitrary relative motion arising from any type of forces. By choosing the origin of the inertial frame to coincide with the center of massof the two-body system, we get (5) where Dl, is the intrinsic quadrupole moment of body i evaluated about parallel axes through its center of mass. 0:;” is defined by
p
=
[MdfB/@fA
+
MB)],
(7)
cl = sin 8 cos I$,
(8)
5, = sin 0 sin c#,
(9)
c3 = c~s e.
(10)
The separation between the centers of massof the two bodies is r while 8 and 4 are the usual spherical coordinates. We note that in the absenceof any intrinsic quadrupole moment of the bodies, the only surviving term in Eq. (5) is D$” which may therefore be identified as the quadrupole moment of the orbital motion.
3. TWO-BODY
SYSTEMS WITH POINT OR SPHERICALLY SYMMETRIC MASSES IN ARBITRARY RELATIVE PLANE MOTION
For this special system, the quadrupole moment of the system is given by
In general, the motion of the two bodies need not be confined to a plane, as there may be noncentral forces, e.g., electromagnetic forces, acting between the bodies. The
236
T.
T.
CHIA
expressions for the energy and angular momentum flux obtained by substituting Eq. (11) into Eqs. (1) and (2) become complicated. However, when the arbitrary relative motion arising from any type of force is confined to a plane, the expressions become particularly simple. (SeeAppendix for details.) Choosing 6’ = n/2, we get PE = 4y12 + ys2 + y3%), pJ3
=
(+)bly.l
+
E = (32G/5c5) YI =
1-
YZ = i(6P1 +
Y4
= 1-
x/932
75
=
+
+
683'
3AA
-
+ I%),
+ i&h
t31'?12)(383~4
HA
(14)
6PlP3
12A -
=
(13)
yz?%),
p2r4w6,
tU% +
Y3
(12)
+
6P4),
(15)
(16) (17)
(18)
P*),
(19)
4P3h
p1 = GJj/oJ2,
(20)
/I, = w/w3,
(21)
= f/(rw),
(22)
I33
/3* = qrc2>,
(23)
p5 = ‘&J3),
(24)
CO= $2
(25)
where the three-axis is perpendicular to the plane of motion. The quantities & and yi are determined from the equation of motion. Once the motion is known, Eqs. (12) and (13) yield the flux. Even when the motion is not known, we can get an estimate of the flux. Since pi is small compared to unity, only yr and y4 are of the order of unity while the remaining yi are small. Thus PE - E and PJ - E/W. For the special casesof Keplerian motion and motion in a straight line, “we can use Eqs. (12) and (I 3) to determine the flux. 3(a) Circular
Keplerian
Orbit
For an orbit which is initially circular, the loss of energy by gravitational waves will change the size of the orbit. That is, r and w are not constants of motion. To a good approximation, we can neglect these variations. Consequently, when Kepler’s law is used, we get 1,
y1=y4=
Y2
=
Y3
=
y5
(26) = 0,
(27)
GRAVITATIONAL
WAVES
FROM
n-Bow
237
SYSTEMS
and p = 32G”MA’MB2(MA E 5cV pJ~ =
32G7/2M..,2M;z(M, + M#J’~ 5c5r7/2
to the same degree of approximation. Equations tional radiation flux from a circular orbit.
3(b)
+ MB)
Elliptical
>
(29)
(28) and (29) give the usual gravita-
and Hyperbolic
Orbit
For two particles or spherically symmetric masses subjected only to mutual gravitational forces, the equation of motion is not strictly Keplerian. This is clear since the system, by emitting gravitational waves, behaves as if it is acted on by a gravitational radiation reaction force which is noncentral. Again to a good approximation, we could assume Kepler’s law to be valid. The equation of the orbit is therefore given by r=p/(l where for a hyperbola, p = a(e2 get
Substituting
+ecos+)=px, 1) and for a parabola, p = a(1
(30) e”). Thus, we
y1 = x,
(31)
y2 = (x2e sin 1$/4),
(32)
y3 = -(3112/12) x2 e sin 4,
(33)
y4 = x2[1 - (e”/2) + Z$ecos + + e2 cos2 $1,
(34)
y5 = xe sin +.
(35)
into Eqs. (12) and (13) we obtain p
E
= =G*MA~ME~“(M‘.I 5c5p5
+ MB)
. (1 + e Cm 4)” [(I + e cos 4)” + (e2/12) sin2 $1, p
J3
(36)
= 32G712M~2MB2(MA + MB)l12 5c5p712 . (1 + e cos 4)” [l - (e2/4) + $e cos + + $e2 cos2 ~$1,
the usual results [2, 31.
(37)
238
T. T.
3(c)
CHIA
Straight-line
Motion
The angle 4 is a constant for motion in a straight line. By absorbing w from E into the y’s, we immediately get PE = [8G/(l 5c5)] p2(rr’
+ 3ti)2
(38)
and P, = 0.
(39)
Neglecting radiative reaction, we have i: = -[G(M,
+ MB)/r2]
(40)
and pE = +$ where KO is an integrating
4.
A
TWO-BODY
( M~;F2
)( ~G(MA + MB) + KOj, r
(41)
constant.
SYSTEM
WITH
INTRINSIC
QUADRUPOLE
MOMENTS
When the bodies deviate from spherical symmetry, the gravitational radiation flux will differ from those of the two-point mass system (assuming that in both systems the other relevant physical parameters are identical). There is contribution from the rate of change of the intrinsic quadrupole moment of each body. Even when this is zero, e.g., in the case of a symmetrical top and a point mass whose motion is restricted to the plane of symmetry of the top, the flux will be altered because of a modified equation of motion. Further, as the expressions for the flux (Eqs. (1) and (2)) involve quadratic products of the time derivatives of the quadrupole moment tensor which consists of a sum of terms (Eq. (5)), we see that when the intrinsic quadrupole moment is time dependent, there will be contribution to the flux from the cross terms (products of the orbital and intrinsic quadrupole moment). This additional flux is identified as the interaction flux. In general, the motion of the two-body system may be complicated. Apart from the existence of other forces, the gravitational forces due to the intrinsic quadrupole moments will produce mutual torques resulting in complicated motion. To bring out the essence of the problem, we shall only consider two analytically tractable special cases. 5.
SYSTEM
OF A POINT
MASS OF
MOVING
A JACOBI
IN THE
SYMMETRIC
PLANE
ELLIPSOID
Consider a Jacobi ellipsoid of uniform density p and mass M, rotating uniformly with angular velocity Q about the smallest principal axis as. We assume that only
GRAVITATIONAL
WAVES
FROM
n-BODY
239
SYSTEMS
gravitational forces are present and restrict the motion of the particle, of mass MB , to the plane of symmetry of the Jacobi ellipsoid perpendicular to the a3 axis. The quadrupole moment of a homogeneous ellipsoid evaluated in the body axes is given by [4]. 0;; = (MA/5)(3a:
- a,,“),
(42)
where in this section i = 1, 2, 3, and is not summed over. The off-diagonal terms vanish because of reflection symmetry. Using Eqs. (5) to (lo), we find that the quadrupole moment of the whole system is given by D,, = 0;: cos’ I& + Dl,” sin* Bt + ~“(3 co? 4 -
I),
(43)
D,, = D$ sin* li’t + 0;: cos2 Qt + pr2(3 sin2 4 -
I),
(44)
D,, = 0;;’ -
pr2,
(45)
D,? = D,, = &(D;; - 0;;)
sin 2L?t + (3pr2/2) sin 24,
(46)
where we have chosen 8 = 7712and assumed that the body axes and the fixed inertial axes are parallel at t = 0. Substituting into Eqs. (1) and (2), we find that the gravitational radiation energy flux can be written as PE = PEA + PEAsB+ P,‘,
(47)
where PEA, Pi,’ and PEr denote the spin, orbital, and interaction portions of the flux, respectively. These are given by PEA = (32G/125c5) ,%fA”($” - ~7~“)~@,
A.B PE = 4Y12 +
Y22
+ Y32),
(48)
(49)
PE* = (64G/25c5) phfAr2(a12 - az2) w3Q3 * [yl cos 2(w - Q) f + y2 sin 2(w - Sz) t],
(50)
with E, p, w, and y defined by Eqs. (14) to (25) and the particle at a distance r from the center of the ellipsoid. Similarly, the angular momentum flux is given by (51) 595/103/x-16
240
T.
T.
CHIA
with Pi = PEA/Q,
(52)
AB PJ3 = (4WNYlY4 + Y*Y& P:, = (32G/251)
$fAr”(al”
(53) -
azz) w2J?
. KYP + y&3 cm 2(w - Q> f + (y2w + ~$9 sin 2(w
Q) tl. (54)
That is, it consists of the sum of three terms. In the above expressions, the variables r and w are not independent. They are related by the equation of motion. This can be determined in principle from the known expression for the gravitational acceleration at a point xi’ (measured in the body axes) external to a Jacobi ellipsoid. If the gravitational acceleration is denoted by gi, then [5] g, = -2rrGpxi’Bi, (55) with Bi = a1a2a3 sor [(a;* + 24)(a;2 + 24);: + u)]l/* (ai + 24) ’
(56)
a;” = ai + A,
(57)
T L42/(ai2+ 41 = 1,
(58)
where as before the dummy summation convention is not used for the Latin subscripts. The equation of motion is clearly non-Keplerian. We therefore see that with the inclusion of intrinsic quadrupole moment each of the energy and angular momentum fluxes consists of three terms: the spin, the orbital, and the interaction. We note that the interaction terms do not vanish when averaged over a long time interval, as the variables w, 52, and yi are functions of time. The ratio of the interaction to the orbital term is given by
. [
y1 cos 2(w - Q) t + y2 sin 2(w - Q) t Y12 + Y22 + Y3p
1
(59)
for the energy flux and by
cos 2(w . (Yl+ Y*GGJ)) [
- 9) t + (y2 + y,(Q/o~)) sin 2(w - Q) (YlY4 f Y2YJ
t
I (60)
GRAVITATIONAL
WAVES
FROM
II-BODY
SYSTEMS
241
for the angular momentum flux. Both ratios can be much bigger than unity under favorable conditions of large L?/w and large MA/MB , conditions realizable in the case of a planetary object moving in an orbit about a neutron star.
5(a)
Circular
Motion
In the special caseof circular motion, the equation of motion can be determined easily and expressions for the interaction flux simplified. For circular orbits, we require that the force acting on the particle be radial. This is possible only if the particle lies on either of the extensions of the two remaining principal axes at all times. This implies that the orbital and spin angular velocity be equal. With these assumptionsand neglecting the ,8’s, as these are small, Eqs. (47) to (50) are replaced by PE = PEA + P$” PEA = +&
-
(62)
apz)2 Q6,
MA2MB2 (MA + M$
5C5
PE’ = g
(61)
MAz(al”
= ~32G
p$”
+ PE1,
pMAr2(a12
rW
-
’
a22) Q6,
(63)
(64)
where the particle is located on the extension of the diameter a, . The angular momentum flux is given by Eq. (51), each component being proportional to the corresponding component of the energy flux:
(65)
These results are derived with the assumption that the particle is located on the extension of the diameter a, . Corresponding to whether a, is greater or smaller than a, , two casesresult. When these are equal, both the spin and interaction terms vanish identically. 5(a)(i) Particle
along Extension
of Largest
Diameter
Here, as a, is bigger than a,, Eqs. (64) and (65) show that the interaction terms are positive. The equation of motion is easily obtained from Eqs. (55) to (58). It is Q2 = G(MA
+ MB) A, ,
(66)
242
T. T. CHIA
where (1, = 3 bwl 3 41) - w4 > 41)j 6%” - u22)(a12- a,2)1~z’
-W, $1 and W,
sin28, = (al2 - u~~)/(u~~- as2),
(68)
sin2 I& = (a,” - a3”)/r2.
(69)
4) are the usual incomplete elliptic integrals E(B, 4) = Jo’ (1 - sin2 0 sin2+‘)lj2 d$‘,
(70)
F(f?, 4) = job (1 - sin2 tI sin2 +‘)-1/2 d+‘.
(71)
As the equation of motion is now different from Kepler’s law, the orbital component of the flux will be different from that of the corresponding two-point masscase. S(a)(ii) Particle along Extension of Second-LargestDiameter The interaction terms are negative when a2 is greater than a,. The equation of motion now becomes ~-2’= G(M, + M&A,,
(72)
where A, =
3(a,2 - u3y2 (a2” - a12)(ff12- as2) . E(B, , 4,) - qe, , $2) (30~2 8, [
(a,2 -ra32)“2
cos &
sin2e,],
sin2 8, = (ffz2 - ~~~)/(a,~- a32), sin2 b2 = (uz2 - a32)l(r2+ a22-
(73) (74)
a12),
(75)
with the samedefinitions for E(O, 4) and F(0, 4). This equation again is different from Kepler’s law.
6. A SYSTEM OF Two GRAVITATIONAL
BOUND SPINNING
RODS
Consider two spinning rigid rods subjected only to mutual gravitational forces, moving in a circular orbit in a plane such that at all times, the center of massof the system and the two axes of the rods are collinear. This implies that the orbital angular velocity Q of one rod about the other and the spin angular velocity of each rod are identical. Let the length and massof rod A, B be denoted by 2LA,B and MR.B and the
GRAVITATIONAL
WAVES
FROM
n-BODY
243
SYSTEMS
separation between the centers of mass of each rod be Y. Using Eqs. (5) and (6), we get Da!3= Z(3lxs
- &3>,
(76)
where I= ZA.B=
IA,, + IA + IB ,
(77)
WAMBIWA + MidI r2,
(78)
IA = gif,L,2,
(7%
I, = +MBLB2.
W
Substituting Eq. (76) into Eqs. (1) and (2) we get, when the angular momentum parallel to the three-axis,
is
PE = (32G/5c5) 12Q6,
(81)
P, = (32G/5c5) Z2si5,
G32)
where we have to the same degree of accuracy neglected & . They can be expressed as (83) p:,
(84)
9
where PEA = (32G/5c5) IA2i?,
(85)
PEB = (32G/5c5) IB2@,
(86)
PisB = (32G/5c5) z;,,@,
(87)
PE’ =
+ IA,BIA
(64G/5C5)(I~I~
and P 25
P,“, pE--=-=-PEA
DI PJ”a” PEA.B= FE+ = J-i-1.
P,“, pBB
-
(8%
That is, the angular momentum and energy flux each consistsof the sum of spin and orbital, as well as interaction terms, all of which are positive. The equation of motion becomes c2
=
G(MA
+ 4LALBr
MB)
In
r2
-
(LB
-
LA)~
r2
-
(LB
+
LA)2
1 '
(90)
where the cross sectionsof the rods are assumedto be small compared to the lengths.
244
7. T. CHIA
We may regard our model as an extreme idealization of two tidally distorted synchronized stars. In reality, the distortion will be very small. We see that when the two stars are of equal mass M and equal radius R, pE’/pEA =
4(IA,B/zA)
+ 2
(91)
and
The interaction flux is therefore very small compared to the orbital flux. At most it can be equal to two-thirds of the orbital flux for the extreme case when the two stars are almost touching each other.
7. GRAVITATIONAL RADIATION FROM ~-BODY SYSTEMS The gravitational radiation flux from n-body systems in the weak-field limit can be calculated conveniently within our scheme. We use Eq. (5) to evaluate the quadrupole moments required for the flux determinations (Eqs. (1) and (2)). Though Eq. (5) refers to two bodies, it can be applied to n-body systems since each body in the twobody system can be regarded as comprising multiple bodies. Thus we first divide an n-body system into two subsystems, consisting of n, and (n - n,) bodies, respectively. Writing n2 for (n - n,), Eq. (5) yields DrB = Da”,’ + D?“B’+ D;;*n2,
(93)
where Dzi is the quadrupoIe moment of the n,-body system referred to its center of mass, and Dziqn2 , the orbital portion of the n, and n2 system, is given by Of;‘“”
= ~L1Z1,nzr~l,n2[3i,n1,nz5an1*n, - SE,] c y,;.y
(94)
The quantity r,l,,2 is the separation between the two centers of mass of the n, and n2 systems, while pnl,nz , the “reduced mass” of the two systems, can be written as (95) The quantities
GRAVITATIONAL
WAVES FROM n-BODY SYSTEMS
245
where 0’~~‘~~ and 4 n**~2are the spherical coordinates referred to the center of massof the combined system. Next we regard each of the 11~and n, systemsas comprising two bodies, and apply Eq. (5) again. The process can be repeated until we obtain n one-body systems. For example, a seven-body system labeled A, B, C, D, E, F, G yields, successively,
= i
D&, + Y;”
+ YsD + Y,EBF$ Y$*cD + YzsG + Y$‘CDgEFG. (99)
i=l
When the number of bodies is large, this method is tedious. It is useful particularly for small n.
8. SYSTEM OF THREE-POINT
OR SPHERICALLY MASSES IN PLANE MOTION
SYMMETRICAL
Consider a system of three bodies, A, B, C, each without intrinsic quadrupole moment, moving in a plane-a good model of a three-body stellar system. Grouping A and B, we obtain
where
PAB,C = [(MA + MB) MAM,
+ MB + &)I,
(102)
with the other variables defined in the preceding section. For convenience, we shall drop the indices AB, C and replace A, B by a prime. Thus, e.g., rAB,c = r and rA,E = r’. Choosing 6’ = 71.12, and using the resultsand notation of Section 3 and the Appendix, we get pE = p;s.c + Pi?” -I- PB’,
(103)
246
T. T.
CHIA
where (MA+
MB)MC
2
32G PEA.B =-[
MAMB MA
f’;
=
!$
+
1”
r’4u’6
(y’2
+
+
Y22
yL2
+
+
(104) ’
Y3%
(105)
y;“>,
MB
MA”BMC
[
r4W6(Y12
I
MA+Mi?+MC
]
r2r’2w3w’3
MA+MB+MC
. ((~1~1’i- YZYZ’ A- 3~~‘) -I- (y1’y2 - yIy2’) sin 2($ - d’)},
(106)
and P, = PfaBsC + PJ”;” i- p:, ,
(107)
with (MA+
1
MB)Mc
'
r4w5hy4
MA+MB+Mc-
32G
A.B p,
p;,
x
MAMB
=
-Q-
[
=
$
[
=
hY4’ +
b2Y4
r’4w’5h’y4’
72%7
w 7175’)
+
+ w
(y1’74 -
(109)
y2’y5’),
r2r’2w2w’2x,
]
MAMBMC MA+MB+Mc
-
(108)
YzY5),
2
I
MA+MB
+
+
+ b2’74
yZ’y5) -
(110) d
yl’y5)
cos d
2(+ sin
2(+
4’) -
6’).
(I
1 l)
The equation of motion of the system must be solved to yield the time derivatives of the relative coordinates. However, an approximation of the gravitational radiation flux can be obtained for the cases when r > r’. Since & and pi’ are both much less than one, we equate y1 , yl’, y4, and y4’ to unity and neglect the remaining y’s. Further, we put
G(M,
With this approximation,
G(MA + MB) N w’2r’3,
(112)
+ MB + MC) N w2r3.
(113)
Eqs. (103) to (106) yield the energy flux P,‘/P$”
PE1/P$y p~‘CIP$B
E
2K1
(114)
,
II 2/q
(115)
N
(116)
K12,
with KI = [(MA i- MB)(MA i- MB + MC)]“”
(Mc/MaMB)(r’/r)5/2.
(117)
GRAVITATIONAL
WAVES
FROM
n-BODY
Similarly, the angular momentum flux can be written
(from Eqs. (107) to (111)) as
P;,/P$B = K2(1+ K3)c0s 2(4 - $0 P:JPfy p~‘“/Pf;”
247
SYSTEMS
(118)
= ((1 + K3YK2K3) cos 2(4 - 4')Y
(119)
=
VW
K22K3,
where
K3
=
[(cMA
+
MB
+
&)/(MA
+
MB>)(r’/r)31”2.
(122)
The interaction components of the angular momentum flux have an angular dependence which may average to zero if the time variations of w, w’, r, and r’ are small. However, the interaction portion of the energy flux is always finite. The possible ranges of or are (a)
or > 2. This condition
implies that
which is not an interesting case. On the contrary, (b)
the next three are.
For2>K1>1,
PE1> P$= > PisB.
(d)
For small
(124)
K1 ,
Cases (b) and (c) are of astrophysical importance. Consider a three-body system in which body A and B are close to each other compared with their separation from C. Observationally, the existence of A may not be suspected because of its underluminosity compared to the other two bodies or its sepration from B being too small to be resolved. The expected flux is then given by PiB,=, since AB is viewed as a composite system. However, the actual flux should be given by Eq. (103). For definiteness, take r/r’ to be equal to four, the mass of C and the composite
248
T. T. CHIA
system AB in units of solar massto be six and two; respectively. If the massof A is taken to be @f, , we have PE’ ‘v (13/8) P$C, P$B ‘v (169/256) PppC,
(127) (128)
while if MA = +M, , we obtain (129) and (130) Thus the existence of a third unsuspected body in a binary system can contribute a significant increase of gravitational radiation compared to that generated by the expected observed composite system AB, C.
9. DISCUSSION We have seenthat when the intrinsic quadrupole moments of the two bodies of any system are taken into account, the rate of gravitational radiation differs from that of the corresponding two-point or spherically symmetric mass system. Firstly, as the equation of motion is no longer Keplerian (compare Eqs. (66) and (72) for the particleJacobi ellipsoid system and Eq. (90) for the rod-rod system with the usual Kepler’s equation) the value of the orbital portion of the flux will be different. Secondly, the intrinsic quadrupole moments, if time dependent, will emit an additional gravitational radiation termed the spin portion of the flux. Finally, the products of the orbital and intrinsic quadruple moment will result in another component of gravitational radiation-the interaction portion. For the particle-Jacobi ellipsoid system, the interaction flux can be positive or negative depending on whether the point massis located along the extension of the longest or second longest diameter, whereasfor the double-rod system, it is always positive. It is clear from Eqs. (1) and (2) that mathematically, the interaction comes from the cross products of the time derivates of the quadrupole moments. However, intuitively, without recourse to these equations, its presence might not be readily obvious. For example, in the caseof the double-rod system, the energy of the system can be written as
i.e., the sum of the rotational energy of each rod and orbital energy of the system. Since the rate of loss of energy by gravitational radiation from an isolated spinning
GRAVITATIONAL
WAVES
FROM
N-BODY
249
SYSTEMS
rod, A, and from an isolated orbital system are given by PEA and P,“*” one tends to think PE = Py
+ PEA -+- PEB
(132)
for the whole system instead of Eq. (83), in which an additional interaction term appears. The apparent paradox can be resolved as follows. In the derivation of Eqs. (1) and (2). the origin of the coordinate system coincides with the center of mass of the whole system. By intuitively treating the rods independently and then jointly, we are making use of three different coordinate systems with origins at the centers of mass of A and of B as well as of the system A&--an unacceptable situation. Just as the spin and orbital portions of the flux are said to come from the spins of the bodies and the orbital motion, respectively, we would like to know from where the interaction term comes. The physical origin of this interaction term is apparent by resorting to the energy momentum conservation law [6], T& = 0.
(133)
Integration over a volume V,, enclosing only the mth body of the system and use of the Gauss divergence theorem yields (apt)
J (-(-g)‘/” = (-c/2)
TOO)dP J’ (-g)‘iz
gPr,,z,oT(D’dC’ f c f, (-g)“2
TOwnadS.
(134)
The expression on the left-hand side denotes the rate of change of the energy of matter (excluding gravitational field energy) within the volume, V,,, . Since the first integral on the right-hand side of this equation involves a finite gl,,o which represents gravitational waves, the energy within the volume V,, , is modified by incident gravitational waves coming from other bodies. That is, due to the interaction of gravitational waves with the bodies there is transfer of energy among the components [6]. This implies that the flux of gravitational waves radiated to infinity will differ by the amount of gravitational wave energy flux that is transferred among the bodies. We can, therefore, attribute the interaction term to the process of interaction of the bodies with gravitational waves. A negative interaction power means that energy from the gravitational waves is absorbed by the bodies, while a positive interaction power implies that the bodies lose more energy as a result of interacting with gravitational waves. We can make similar observations about the interaction portion of the angular momentum flux. Finally, our scheme can be used for the calculation of gravitational radiation flux from n-body systems. In particular, for a three-body system the presence of an unsuspected body in a system hitherto assumed as binary can under certain conditions, increase significantly the gravitational radiation flux. Consequently, the upper limit of gravitational radiation flux from astrophysical sources needs to be revised to allow for multiple stellar systems.
250
T.
APPENDIX: TWO-POINT
T. CHIA
TIME DERIVATIVES OF QUADRUPOLE MOMENTS OF SYSTEMS COMPRISING OR SPHERICALLY SYMMETRIC MASSES IN ARBITRARY RELATIVE PLANE
MOTION
From Eqs. (6) to (IO), the nonvanishing time derivatives of the quadrupole moments are %, = 6cLr2w2[i - (yd3)
(Al)
- y,C - yJ1,
... D,, = 12vr24(y3/31/2)-
y2C
+ yJ1,
642)
ky = 6cLr2w2[i - (yd3) + y,C + @I, ... D,, = l&r2~3Ky3/31’2) + y,C - ylSl, O;,, = 4pr2w2[y4 -
(A3) (A4)
11,
(A5)
... D,, = -S(3)1/2 pr2u3y,,
W)
B,, = figz = 6pr2c02[y,C - yJ],
(A7)
...
D,, = --12pr2W3[y2S + yICl,
648)
where S = sin 24,
(A91
c = cos 24,
(AW
with the notation of Section 3.
REFERENCES 1. L. D. LANDAU AND E. M. LIFSCHITZ, “The Classical Theory Mass., 1962. 2. P. C. PETERS, Phys. Rev. B136 (1964), 12241232. 3. R. 0. HANSEN, Phys. Rev. D 5 (1972), 1021-1023. 4. W. Y. CHAU, Astrophys. J. 147 (1967), 664-671. 5. S. CHANDRASEKHAR, “Ellipsoidal Figures of Equilibrium,” 1969. 6. F. I. COOPERSTOCK, Phys. Rev. 163 (1967), 1368-1373.
of Fields,”
Yale Univ.
Addison-Wesley,
Press, New
Reading,
Haven/London,