[Page 1] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
Lecture 5. Supermassive binary black holes and gravitational waves
We see from observations that time to time galaxies collide. When it happens the supermassive black holes at their centers interact and merge (see Fig.5.1.) This image of NGC4038 and NGC4039 shows two spiral galaxies, known as the �antennae� galaxies, that are in the process of colliding with each other. Images like this are evidence that galaxies do collide, thus providing an opportunity for the supermassive black holes at their centers to merge. Merging supermassive black holes are expected to be among the strongest sources that LISA (Laser Interferometric Space Antenna) will be able to detect (Fig.5.2.)
This lecture is an example of some sort of a toy research project which has the following scientific objective: What conditions should be satisfied by parameters of a binary black hole in order some future detector of gravitational waves could detect gravitational radiation from the binary. During this lecture we will make several simplifying assumptions which make this project doable during such short period as the duration of this lecture, trying, nevertheless, to present the general spirit of research in this field.
[Page 2] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
Let us assume that we have a black hole binary consisting of two black holes of mass M and m correspondingly, such that
m = αM, where α ≤ 1. According to the quadrupole formula(see lecture 6 of Relativistic Astrophysics), the amplitude of gravitational wave emitted by such a system is
hαβ = −
2G d2 Dαβ (t˜) , 3c4 R dt2
where R is the distance to the binary, t˜ is retarded time
R t˜ = t − c and
Dαβ =
Z
(3xα xβ − r2 δαβ )dM
is the quadrupole tensor. We assume for simplicity that the black holes in the binary move along circular orbits. Let us choose a frame of reference such that orbital plane coincides with (x1 , x2 )- plane, see Fig.5.3.. Assuming that the separation distance between the two black holes, r, is much larger then their gravitational radii, we can describe the motion of the two black holes with the help of the Newtonian theory (not primed values correspond to the black hole of mass M and primed values correspond to the black hole of mass m):
x1 = a cos ω0 t, x01 = a0 cos ω0 t, x2 = a sin ω0 t, x02 = a0 sin ω0 t, x3 = x03 = 0, where ω0 = 2π/T and T is the orbital period. Choosing the frame of coordinate with its origin in the center of mass of the binary, we have
M a = ma0 = αM a0 , and a + a0 = r, hence
a = αa0 and
a0 =
r αr and a = . 1+α 1+α
Then we can calculate ω0 from:
ω02 a0 =
GM . r2
[Page 3] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
Introducing dimensionless separation
x=
r , rg
we have r ω0 =
GM = a0 r2
r
s r 2GM 1+α GM (1 + α) 1 + α c −3/2 c2 ( 2 ) = c = x , r3 rg3 x3 2 rg
where rg is the gravitational radius of the black hole of mass M . Now we can calculate all nonzero components hik :
h11 = −
d2 2G d2 D11 2 ˜ − 1 ) = Aω02 sin 2ω0 t˜ = h0 sin 2ω0 t˜, = −A (cos ω t 0 3c4 R dt2 dt2 3
2G d2 D22 d2 1 h22 = − 4 = −A 2 (sin2 ω0 t˜ − ) = −Aω02 sin 2ω0 t˜ = −h0 sin 2ω0 t˜, 2 3c R dt dt 3 2 2 d 2G d D12 = −A 2 (cos ω0 t˜sin ω0 t˜) = Aω02 cos 2ω0 t˜ = h0 cos 2ω0 t˜, h12 = − 4 2 3c R dt dt where
A=
x2 rg3 α 2G(M a2 + ma02 ) 2Gr2 M (α2 + α) 2Gr2 M α r 2 rg α = = = = c4 R c4 R(1 + α)2 c4 R(1 + α) c2 R(1 + α) c2 R(1 + α)
and
h0 =
Aω02
x2 rg3 α 1 + α c2 rg α = 2 = . 2 3 c R(1 + α) 2 rg x 2xR
we can see that the frequency of the gravitational waves from the binary is equal to
ω = 2ω0 and ν =
ω ω0 = = ν0 x−3/2 , 2π π
where s
ν0 =
1+α c , 2 πrg
hence, the amplitude of the gravitational waves
h0 =
rg α αrg = 2xR 2R
ν ν0
2/3
.
[Page 4] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
Let us assume for simplicity that some future detector of gravitational waves, for example LISA, will be able to detect gravitational waves with h0 > h∗ in the range of frequencies ν∗ < ν < ν∗∗ (seeFig.5.4.)[Blue lines correspond to different h0 , red area corresponds to the detection of gravitational waves.] Now we can put and answer the whole variety of questions:
EXAMPLE 1 From what distance will it be possible to detect gravitational radiation from the binary system, containing the black holes of mass M and αM if the orbital period, T , is given.
ANSWER 1 In order to be in the range of delectability
ν∗ < ν < ν∗∗ , hence,
ν∗ <
2 2 < ν∗∗ , f rac2ν∗∗ < T < . T ν∗
In this case from
h0 > h∗ follows that the distance, R, should satisfy to the inequality
R < Rmax
αrg = 2h∗
ν ν0
2/3
.
[Page 5] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
EXAMPLE 2 How small should be separation between the black holes in the binary in order to detect gravitational waves from the binary if the distance and masses are given.
ANSWER 2: In order to be in the range of delectability
ν∗ < ν < ν∗∗ , hence,
ν∗ < ν0 x
−3/2
< ν∗∗ ,
ν0 ν∗∗
2/3
ν0 <x< ν∗∗
2/3
and
rg
ν0 ν∗∗
2/3
< r < rg
ν0 ν∗∗
2/3
.
In this case from
h0 > h∗ follows that the separation, r, should satisfy also to the inequality
r < rmax
αrg2 = . 2h∗ R
Combining with the previous inequalities, we obtain that the separation r should satisfy to the following inequality:
rg
ν0 ν∗∗
2/3
(
< r < min rg
ν0 ν∗∗
2/3
αrg2 , . 2h∗ R )
[Page 6] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
EXAMPLE 3 a) Assume that the number of binary black holes at the distance R in the range (R, R+ dR), emitting gravitational waves with the frequency ν in the range (ν, ν + dν), with masses of companions M and m in the ranges (M, M +dM ) (m, m+dm) correspondingly, is 4πn(R, ν, M, m)R2 dRdνdM dm such that the total number of the binaries, N , is
N = 4π
∞
Z
dν
∞
Z
0
dM
M
Z
0
dm
∞
Z
0
R2 dR n(R, ν, M, m).
0
How many sources of gravitational radiation could be detected if we know the distribution of the binaries over masses, frequencies and distances. b) Consider the following toy model: (
B n(R, ν, M, m) = 2 ν
1 if m = M = M∗ 0 if m 6= M 6= M∗
)(
0 if ν < νmin ν∗ 1 if ν ≥ νmin
)
.
ANSWER 3 a) The number of binaries emitting detectable gravitational waves, Ndet , is
Ndet = 4π
Z
ν∗∗
dν
Z
∞
dM
M
dm
0
0
ν∗
Z
Z
αrg 2h∗
ν ν0
2/3
R2 dR n(R, ν, M, m),
0
where α=
m . M
b) In this case
Ndet = 4πB
Z
ν∗∗
ν∗
rg dν Z 2h∗ ν2 0
ν ν0
2/3
R2 dR =
4πB 3
rg 2h∗
3 Z
ν∗∗
ν∗
dν ν2
ν ν0
2
=
4πB 3ν02
rg 2h∗
3
(ν∗∗ −ν∗ ).
[Page 7] A G Polnarev. Extragalactic Astrophysics (ASTM-052), 2008, Lecture 5. Supermassive binary black holes and gravitational waves
At the moment of binary black holes formation the separation between them is of order the typical size of merging galaxies. To have detectable gravitational waves this separation should be much much smaller. Hence the following problem arises: what are mechanisms of spiraling in of the black holes in the binaries. The first mechanism which is efficient enough at least for large separation is dynamical friction. An intuition for the effect can be obtained by thinking of a massive object moving through a cloud of smaller lighter bodies. The effect of gravity causes the light bodies to accelerate and gain momentum and kinetic energy. By conservation of energy and momentum, we may conclude that the heavier body will be slowed by an amount to compensate. Since there is a loss of momentum and kinetic energy for the body under consideration, the effect is called dynamical friction. Another equivalent way of thinking about this process is that the light bodies are attracted by gravity toward the larger body moving through the cloud, and therefore the density at that location increases and is referred to as a gravitational wake. In the meantime, object under consideration has moved forward. Therefore, the gravitational attraction of the wake pulls it backward and slows it down. The greater the density of the surrounding media, the stronger the force from dynamical friction. Similarly, the force is proportional to the square of the mass of the object. One of these terms is from the gravitational force between the object and the wake. The second term is because the more massive the object, the more matter will be pulled into the wake. The force is also proportional to the inverse square of the velocity.
When the separation between black hole in the binaries is smaller than some critical distance bur large than the separation required for detectability of gravitational waves, other mechanisms should be considered. For example, gravitational interaction of the smaller black hole with the accretion disk around the larger black hole. Not going into details, it is possible to say that this mechanism is very similar to the mechanism responsible for migration of planets in protoplanetary disks (see ASTM003, Angular Momentum and Accretion Processes in Astrophysics).
Finally, when the separation is small enough, however still larger than what is required for the detectability of gravitational waves, the main mechanism of spiraling in is due to losses of the orbital energy of the binary due to the emission of gravitational waves themselves.