A Study of the Gravitational Wave Form from Pulsars

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• A Study of the Gravitational Wave Form from
  Pulsars
• John Drozd, Dr. Valluri, Dr. Mckeon

Two Colliding Black Holes
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What is a Gravitational Wave?

• Ripples in Space Time
• Sources Pulsars and Black Holes

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Accreting neutron star in a low-mass x-ray binary
system
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• 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 Kings College and a
  former gold medal physics UWO student (1948)
  completely unlikely that gravitational waves will
  be the subject of direct observation (1962)

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Gravitational Waves the evidence
Delay in the Time of Periastron Of Binary Pulsar
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

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Direct Detectionastrophysical sources
Gravitational Wave Astrophysical Source
Terrestrial detectors LIGO, TAMA, Virgo,AIGO
Detectors in space LISA
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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

Hanford Observatory
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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)

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Angles
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Doppler Shifted Received Signal
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Plane Wave Expansion for the Rotational Part
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Plane Wave Expansion for the Orbital Part
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Fourier Transform of the Signal
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• 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

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• The integrand with the summation becomes
• where Z is the Zak-Gelfand Transform of

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• Analytical Formulation (without Spindown)

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Log10 Snlm 2 vs k and l for ? ? / 2, ?
? / 4, ? ? / 4, Borb 0
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Re ( Snlm ) vs ? and ? for l 2, k 270, ? ?
/ 4, Borb 2.1
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Re ( Snlm ) vs l and Borb for k 45, ? ? / 2,
? ? / 4, ? ? / 4
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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

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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

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Fast Chirp Transform and Jacobi Theta Function

• Prince Jenet
• Jacobi Theta Function

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Fast Chirp Transform and Fresnel Integrals
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Pulsar Spindown Formulation

• Intrinsic frequency modes for pulsar rotation

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• Gravitational wave frequency measured at
  detection

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• The integrals now involve additional
  trigonometric functions besides

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• Using a Binomial Expansion

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• Define the Spindown Moment Integrals

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• 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.

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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!