matrix template matching reloaded third atnf gravitational wave workshop 2009 willem van straten
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Matrix Template Matching ReloadedThird ATNF Gravitational Wave Workshop 2009
Willem van Straten
Matrix Template Matching ReloadedThird ATNF Gravitational Wave Workshop 2009
Willem van Straten
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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”
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The Matrix
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Stokes Parameters
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Stokes 4-vector
• Total intensity (time-like)
S0 = I
• Polarization vector (space-like)
S = (S1, S2, S3)
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Transformations
Jones matrix, J
Mueller matrix, M
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Rotations and Boosts
• Rotations:
– rotate S about an axis by an angle
– leave S0 unchanged
• Boosts:– boost along an axis by a Lorentz factor
– modify both S0 and S
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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
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Template Matching
P() = aS(+ ) + b
P() = aS()e-i2 + ()b
FFT
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Frequency Domain
• High frequency power yields greatest constraint on slope,
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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
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PSR J0437-4715 profile
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PSR J0437-4715 spectrum
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Parallactic Variation of Arrival Time
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Even After Calibration
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Enter the Invariant
• Form the invariant profile, Sinv(), where
Sinv2 = S0
2 - |S|2
• Dramatically reduced systematic error in PSR J0437-4715 data
(Britton 2000; van Straten et al 2001)
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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
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Matrix Template Matching (MTM)
• Replace scalars with matrices
• 6 new degrees of freedom
• 4 times the number of constraints
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Relative TOA Errors
Pulsar MTM invariant
J0437-4715 0.8318(1) 1.4314(3)
J1022+1001 0.669(4) 1.80(2)
J1713+0747 0.877(2) 1.543(5)
B1855+09 0.9314(9) 1.430(2)
J1909-3744 0.959(4) 1.464(8)
B1937+21 0.862(4) 1.451(9)
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Limitations
• Also limited by S/N:– depolarization due to time/frequency
variations of instrumental response
• Calibration before integration …
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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
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Parallactic variation of
Stokes parameters
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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
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Receiver solution 2003
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Receiver solution 2008
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Calibrator solution 2003
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Calibrator solution 2008
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Ellipticity time variation 54
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Ellipticity time variation 74
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Calibrator time variation 54
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Calibrator time variation 74
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PSR J1022+1001 profile
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PSR J1022+1001 spectrum
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Arrival Time Comparison
• 234 TOAS (Nfree=217)
• r.m.s reduced by 39% (predicted 33%)
Intensity TM Matrix TM
Postfit r.m.s. 1.453 s 0.942 s
Reduced 2 2.03 1.01
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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
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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