Gravitational Radiation from a particle in bound orbit around black hole Presentation for 11 th Amaldi Conference on Gravitational Waves By ASHOK TIWARI June 25, 2015 Tribhuvan University, Kathmandu, Nepal 1

Outline Statement of the Problem Introduction Theory Methodology Results and Conclusions 2

Introduction Bound orbit : For bound orbit e<1, E 2 <1 Gravitational waves: I.Perturbations of flat space-time II.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 3

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 n th harmonics 4

Methodology Inertia tensor method: (figure in rt. side shows coordinate system used for whole work ) Where, m 1 is mass of black hole and m 2 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, m 1 = 1 and m 2 = (for sake of simplicity) 5

Multipole expansion method : In quadrupole approximation the dominant type of radiation is magnetic quadrupole m 2M and, Where, ρ is mass density in the source Total power radiated in the n th harmonic s is, Where, 6

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 7

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) 8

we use relativistic correction in Newtonian results We calculate angular velocity with relativistic correction in following differential eq n : (S. Chandrasekhar, The Mathematical theory of Black Holes) Where, k 2 = 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= and get numerical values of this quantities. 9

Results: 1. Inertia Tensor Method Power 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 10

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: 11

2. Multipole expansion method Plot for different values of eccentricity (e) and g(n,e): which is called the relative power radiated into n th harmonics 12

Conclusion 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 = × 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 = × , 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.

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