A Study of the Gravitational Wave Form from Pulsars John Drozd, Dr. Valluri, Dr. Mckeon Two Colliding Black Holes
What is a Gravitational Wave? Ripples in Space Time Sources: Pulsars and Black Holes
Accreting neutron star in a low-mass x-ray binary system
Albert Einstein – 1916 – predicted the existence of gravitational waves: Amount of gravitational radiation emitted from a binary star system which had a long period was so small that Einstein concluded that the radiation had a negligible practical effect. Felix Pirani, professor at King’s College and a former gold medal physics UWO student (1948): completely unlikely that gravitational waves will be the subject of direct observation (1962)
Gravitational Waves the evidence Delay in the Time of Periastron Of Binary Pulsar Gravitational Waves the evidence Emission of gravitational waves Neutron Binary System – Hulse & Taylor PSR 1913 + 16 -- Timing of pulsars 17 / sec · · ~ 8 hr Neutron Binary System separated by 106 miles m1 = 1.4m; m2 = 1.36m; e = 0.617 Prediction from general relativity spiral in by 3 mm/orbit rate of change orbital period
Direct Detection astrophysical sources Gravitational Wave Astrophysical Source Terrestrial detectors LIGO, TAMA, Virgo,AIGO Detectors in space LISA
Livingston Observatory Obtain LIGO data, Data Mining using Neural Networks Adopt Principal Component Analysis and a Differential Geometric formulation for defining a metric to examine parameters of the pulsars in an all sky search Livingston Observatory Hanford Observatory
Analytic Formulation for The Study of The Gravitational Wave Form from A Pulsar Fourier Transform of GW signal Account for rotation and orbital motion of earth Doppler shifted signal for single sky search Plane wave expansion of rotational and orbital parts Zak-Gelfand Transform Fast Chirp Transform (to be published) Spin-Down of Pulsar (to be published) Jupiter Perturbations (to be studied)
Angles
Doppler Shifted Received Signal
Plane Wave Expansion for the Rotational Part
Plane Wave Expansion for the Orbital Part
Fourier Transform of the Signal
Defining allows the integral from 0 to 2R to be split as a succession of sums from 2(j1) to 2j with j from 1 to R
The integrand with the summation becomes where Z[] is the Zak-Gelfand Transform of
Analytical Formulation (without Spindown)
Log10{ | Snlm | 2 } vs k and l for = / 2, = / 4, = / 4, Borb = 0
Re ( Snlm ) vs and for l = 2, k = 270, = / 4, Borb = 2.1
Re ( Snlm ) vs l and Borb for k = 45, = / 2, = / 4, = / 4
Implementation Prototype Maple programs C / GNU Scientific Library Asymptotic analysis of Bessel, Hypergeometric and Gamma Functions MPI Parallel implementation on SHARCNET by Adam Vajda Further work required on Spherical Harmonics for simultaneously large l and m Pulsar Spindown formulation to be implemented
Fast Chirp Transform For Analyzing signals of varying frequency such as gravity wave burst oscillations from rotating neutron stars or pulsars For I superimposed chirp signals embedded in noise
Fast Chirp Transform and Jacobi Theta Function Prince & Jenet: Jacobi Theta Function: Fast Chirp Transform and Jacobi Theta Function
Fast Chirp Transform and Fresnel Integrals
Intrinsic frequency modes for pulsar rotation: Pulsar Spindown Formulation Intrinsic frequency modes for pulsar rotation:
Gravitational wave frequency measured at detection:
The integrals now involve additional trigonometric functions besides
Using a Binomial Expansion:
Define the Spindown Moment Integrals:
The kth derivative gives the corresponding Spindown moment intregral. Igeneric has been analytically evaluated. Its derivatives with respect to fo can be done by Maple.
Conclusions Looking for specific sources of continuous GW – has been tried As yet, no `Search` of a large area of the sky attempted. Would increase the chances of discovering GW source or invisible pulsars. Unimagined sources can exist and can lead to an exciting GW astronomy! Will open a new window to the universe. It is a truly interdisciplinary problem of Mathematics, Physics, Astrophysics, Computer Science!