LISALISA - iucaa

1.1 Theory of gravitational radiation
where the integral is over the entire volume of the source, then the standard trace-free quadrupole
tensor is
Q jk = I jk − 1 3 Iδ jk , (1.6)
where I is the trace of the moment tensor. (The tensor Q is sometimes called I– in textbooks.
Note that I jk is not the moment of inertia tensor, despite the notation.) The radiation amplitude
is, for a nonrelativistic source at a distance r,
h = 2G
c 4
¨Q
r , (1.7)
where we have left off indices because we have not been quantitative about the antenna and
radiation patterns. The total luminosity in gravitational waves is given by
〈
L GW = G ∑
( d 3 ) 2
〉
Q jk
c 5 dt 3 , (1.8)
jk
where the angle brackets 〈...〉 denote an average over one cycle of the motion of the source. In
this formula we have put in all the correct factors and indices.
There are simple order-of-magnitude approximations for these formulas, which are both easy to
use and instructive to look at. For example, one can write
∫
∫
Ï jk = d2
dt 2 ϱx j x k d 3 x ∼ ϱv j v k d 3 x.
Now, the quantity v j v k will, by the virial theorem, be less than or of the order of the internal
gravitational potential Φ int . Combining this with Equation 1.7 gives
h ≤ G ∫
Φ int
c 4 ϱd 3 x = Φ ext Φ int
r c 2 c 2 , (1.9)
where Φ ext is the external gravitational potential of the source at the observer’s position, GM/r.
This simple expression provides an upper bound. It is attained for binary systems where all the
mass is participating in asymmetrical motions. The exact formula was first derived by Peters
and Mathews [10]. For a circular orbit the radiation is a sinusoid whose maximum amplitude
can be expressed in terms of the frequency of the emitted waves and the masses of the stars by
( ) f 2/3 ( ) r −1 ( ) M 5/3
h 0 = 1.5×10 −21 10 −3 , (1.10)
Hz 1 kpc M ⊙
where f is the gravitational wave frequency (twice the binary orbital frequency), r is the distance
from source to detector, and M is the so-called “chirp mass”, defined in terms of the two stellar
masses M 1 and M 2 by
M = (M 1M 2 ) 3/5
. (1.11)
(M 1 + M 2 ) 1/5
Equation 1.10 can be derived, to within factors of order unity, by eliminating the orbital radius
from Equation 1.9 in favour of the orbital frequency and the masses using Kepler’s orbit equation.
For equal-mass binaries, for example, one uses
ω orbit =
( GMT
d 3 ) 1/2
, (1.12)
Corrected version 1.04 15 13-9-2000 11:47