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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 Presentation
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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
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”
The Matrix
Stokes Parameters
Stokes 4-vector
• Total intensity (time-like)
S0 = I
• Polarization vector (space-like)
S = (S1, S2, S3)
Transformations
Jones matrix, J
Mueller matrix, M
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
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
Template Matching
P() = aS(+ ) + b
P() = aS()e-i2 + ()b
FFT
Frequency Domain
• High frequency power yields greatest constraint on slope,
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
PSR J0437-4715 profile
PSR J0437-4715 spectrum
Parallactic Variation of Arrival Time
Even After Calibration
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)
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
Matrix Template Matching (MTM)
• Replace scalars with matrices
• 6 new degrees of freedom
• 4 times the number of constraints
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)
Limitations
• Also limited by S/N:– depolarization due to time/frequency
variations of instrumental response
• Calibration before integration …
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
Parallactic variation of
Stokes parameters
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
Receiver solution 2003
Receiver solution 2008
Calibrator solution 2003
Calibrator solution 2008
Ellipticity time variation 54
Ellipticity time variation 74
Calibrator time variation 54
Calibrator time variation 74
PSR J1022+1001 profile
PSR J1022+1001 spectrum
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
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
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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