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Matrix Template Matching Reloaded Third ATNF Gravitational Wave Workshop 2009 Willem van Straten

Matrix Template Matching Reloaded Third ATNF Gravitational Wave Workshop 2009 Willem van Straten. Synopsis. “The Matrix”: polarisation and pulsar timing invariant interval & matrix template matching “The One”: PSR J0437-4715 use bright source to calibrate itself “The Reload”: PSR J1022+1001

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Matrix Template Matching Reloaded Third ATNF Gravitational Wave Workshop 2009 Willem van Straten

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  1. Matrix Template Matching ReloadedThird ATNF Gravitational Wave Workshop 2009Willem van Straten

  2. Synopsis • “The Matrix”: polarisation and pulsar timing • invariant interval & matrix template matching • “The One”: PSR J0437-4715 • use bright source to calibrate itself • “The Reload”: PSR J1022+1001 • use bright source to calibrate weak sources • “The Future”

  3. The Matrix

  4. Stokes Parameters

  5. Stokes 4-vector • Total intensity (time-like) S0 = I • Polarization vector (space-like) S = (S1, S2, S3)

  6. Transformations Jones matrix, J Mueller matrix, M

  7. Rotations and Boosts • Rotations: • rotate Sabout an axis by an angle • leave S0 unchanged • Boosts: • boost along an axis by a Lorentz factor • modify both S0 and S

  8. Boosts: Instrumental • Non-orthonormality of receptors: • differential gain (along S1 axis) • cross-coupling (along S2 & S3 axes) • Mixing of S0 and S depends on S• • Produce pulse phase dependent distortions of total intensity profile

  9. Template Matching P() = aS(+ ) + b  FFT P() = aS()e-i2 + ()b

  10. Frequency Domain • High frequency power yields greatest constraint on slope,     

  11. The One: PSR J0437-4715 • The good: • closest and brightest MSP known • narrow pulse peak • The ugly: • OPM sweep near pulse peak • TOA strongly distorted by calibration errors

  12. PSR J0437-4715 profile

  13. PSR J0437-4715 spectrum

  14. Parallactic Variation of Arrival Time

  15. Even After Calibration

  16. Enter the Invariant • Form the invariant profile, Sinv(), whereSinv2 = S02 - |S|2 • Dramatically reduced systematic error in PSR J0437-4715 data(Britton 2000; van Straten et al 2001)

  17. Caveats • Limited by S/N: • depolarization due to time/frequency variations of instrumental response • baseline estimation error/bias • no longer normally distributed error • Limited by degree of polarization: • Sinv 0 as|S|/S0  1

  18. Matrix Template Matching (MTM) • Replace scalars with matrices • 6 new degrees of freedom • 4 times the number of constraints

  19. Relative TOA Errors

  20. Limitations • Also limited by S/N: • depolarization due to time/frequency variations of instrumental response • Calibration before integration …

  21. Calibration Options (2007) • Measurement equation modeling(MEM; van Straten 2004) • unknown source and calibrators observed over a range of parallactic angles • Matrix template matching(MTM; van Straten 2006) • known source matched to a single (short) observation; no calibrator input

  22. Parallactic variation of Stokes parameters

  23. MTM Reloaded • Merge MEM and MTM methods • use template to also constrain calibrator • model time-variations and multiple observations • Derive 20 MEM solutions (~8hr each) • Integrate into standard (~150hr) • Derive 150 MEM/MTM solutions (~3.5hr) • both receiver and calibrator

  24. Receiver solution 2003

  25. Receiver solution 2008

  26. Calibrator solution 2003

  27. Calibrator solution 2008

  28. Ellipticity time variation 54

  29. Ellipticity time variation 74

  30. Calibrator time variation 54

  31. Calibrator time variation 74

  32. PSR J1022+1001 profile

  33. PSR J1022+1001 spectrum

  34. Arrival Time Comparison • 234 TOAS (Nfree=217) • r.m.s reduced by 39% (predicted 33%)

  35. Conclusions • Long-term stability of PSR J0437-4715 can be exploited to calibrate instrumental polarization • Matrix Template Matching can do better than doubling integration length

  36. The Future • Turn MEM/MTM solutions back on PSR J0437-4715 • Quantify long-term stability of MSP polarization • Fit time- & frequency-varying receiver & calibrator model to MEM/MTM solutions

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