1

Gravitational Radiation from a particle in

bound orbit around black hole

Presentation for 11th Amaldi Conference on Gravitational Waves

ByASHOK TIWARI

June 25, 2015

Tribhuvan University, Kathmandu, Nepal

2

Outline • Statement of the Problem• Introduction• Theory• Methodology• Results and Conclusions

3

Introduction• Bound orbit: For bound orbit e<1, E2 <1• Gravitational waves:

I. Perturbations of flat space-timeII. Highly prized carriers of information from distant regions of the

universe.

• Black Hole: is a relativistic analog of Newtonian point particle.• Geodesics: are the analogues of st. lines in curved space

geometries• Sources of Gravitational waves: Supernovae and gravitational

collapse, Binary collapse, Chirping binary system, Pulsar and neutron stars, random background and other unexpected sources

• Detection: Resonant bar detectors, Beam detectors (LISA, LIGO, VIRGO, GEO600, KAGRA, EGO) and other detectors

4

Objectives

• To study the power of gravitational radiation emitted from a particle in bound orbit in Schwarzschild geometry

• To study the variation of power with the eccentricity of the bound orbit (e < 1)

• To study the spectrum of the emitted gravitational radiation according to multipole expansion, g(n,e); relative power radiated in nth harmonics

5

MethodologyInertia tensor method:(figure in rt. side shows coordinate system used for whole work )

Where, m1 is mass of black hole and m2 is mass of test particle

We assume binary system (BH - test particle) as shown in figure

The power radiated by the binary system over one period at the elliptical motion is,

And average radiated power is,

Where, f(e) is called enhancement factor and f(e)

We put, m1 = 1 and m2 = 0.001 (for sake of simplicity)

6

Multipole expansion method:In quadrupole approximation the dominant type of radiation is magnetic quadrupole m2M and,

Where, ρ is mass density in the source

Total power radiated in the nth harmonic s is,

Where,

7

Plots of time-like geodesics in Schwarzschild space-time

For: e=1/2, M=3/14,l=3

For: e=1/2, M=3/14, l=11

2

4

6

30

210

60

240

90

270

120

300

150

330

180 0

5

10

15

20

25

30

210

60

240

90

270

120

300

150

330

180 0

Time like geodesics for latus rectum l = 11

8

For: e=1/2, M=3/14, l=7

For e=1/2, M=3/14, l=1.5 For e=1/2, M=2/14, l=1(unstable orbit)

5

10

15

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

9

we use relativistic correction in Newtonian resultsWe calculate angular velocity with relativistic correction in following differential eqn :

(S. Chandrasekhar, The Mathematical theory of Black Holes)

Where, k2= 4 µ e/ (1- 6 µ e + 2 µ e)

We get,

Calculation of latus rectum, angular mom. and time period by:

We put, G=c=1 and a=11.458 and get numerical values of this quantities.

10

Results: 1. Inertia Tensor MethodPower calculation for different values of eccentricity, which is

shown in table: Unit of power is relative, because we put G=c=1, for sake of simplicity

11

Our result: Local minimum is seen when e=0.7, we believe that this is due to relativistic correction.

P.C peters and J. Mathews result:

12

2. Multipole expansion method

Plot for different values of eccentricity (e) and g(n,e): which is called the relative power radiated into nth harmonics

13

Conclusion1. Inertia Tensor Method

• With increasing eccentricity (e), power also going to be increasing as shown in table

• When e = 0.35, the calculated value of average power radiated P = 3.2076 × 10 -16 begin to decrease. we made the range small between e = 0.6 and e = 0.8, because the relative total power radiated is decrease to local minimum in this region. For e = 0.70, P = 1.5808 × 10 -16 , again when we increase e, then power going to rise steeply.

• The nature of the plot is similar to the results of P.C. Peter's and J. Mathew's results except that the minimum doesn't appear in Newtonian result. We believe this local minimum is due to relativistic correction.

2. Multipole Expansion Method

• We apply quadrupole approximation, the dominant type of the radiation is magnetic quadrupole (m 2m).

• We calculate total radiated power in terms of the g(n,e); the relative power radiated into the n th harmonics. With the increasing of harmonics (n) and eccentricity (e), we get large and smooth curves.

• Fourier components of large (n) must be present to give such a peaking of the radiation at one part of the path.

14

Thank you