1. A Study of the Gravitational Wave Form from Pulsars
John Drozd, Dr. Valluri, Dr. Mckeon
Two Colliding Black Holes

2. What is a Gravitational Wave?
Ripples in Space Time
Sources: Pulsars and Black Holes

3. Accreting neutron star in a low-mass x-ray binary system

4. 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)

5. 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 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

6. Direct Detection astrophysical sources
Gravitational Wave Astrophysical Source
Terrestrial detectors
LIGO, TAMA, Virgo,AIGO
Detectors in space
LISA

7. 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

8. 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)

9. Angles

13. Doppler Shifted Received Signal

15. Plane Wave Expansion for the Rotational Part

16. Plane Wave Expansion for the Orbital Part

17. Fourier Transform of the Signal

20. Defining
allows the integral from 0 to 2R to be split as a succession of sums from 2(j1) to 2j with j from 1 to R

21. The integrand with the summation becomes
where Z[] is the Zak-Gelfand Transform of

26. Analytical Formulation (without Spindown)

27. Log10{ | Snlm | 2 } vs k and l for  =  / 2,  =  / 4,  =  / 4, Borb = 0

28. Re ( Snlm ) vs  and  for l = 2, k = 270,  =  / 4, Borb = 2.1

29. Re ( Snlm ) vs l and Borb for k = 45,  =  / 2,  =  / 4,  =  / 4

30. 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

31. 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

32. Fast Chirp Transform and Jacobi Theta Function
Prince & Jenet:
Jacobi Theta Function:
Fast Chirp Transform and Jacobi Theta Function

34. Fast Chirp Transform and Fresnel Integrals

35. Intrinsic frequency modes for pulsar rotation:
Pulsar Spindown Formulation
Intrinsic frequency modes for pulsar rotation:

36. Gravitational wave frequency measured at detection:

37. The integrals now involve additional trigonometric functions besides

40. Using a Binomial Expansion:

41. Define the Spindown Moment Integrals:

42. 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.

43. 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!