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

2. Synopsis “The Matrix”: polarisation and pulsar timing
invariant interval & matrix template matching
“The One”: PSR J
use bright source to calibrate itself
“The Reload”: PSR J
use bright source to calibrate weak sources
“The Future”

3. The Matrix
Statistical definition of polarisation in terms of the second moments of the electric field, variance on diagonal, covariance off diagonal
Rho = coherency matrix E = column vector
Outer product is implied, Hermitian transpose, ensemble average …
Rho is Hermitian -> 4 degrees of freedom

4. Stokes Parameters
S_k = four Stokes parameters (Einstein notation is used)
Sigma_0 = 2x2 identity matrix; Sigma_1-3 are the Pauli matrices,
Tr is the matrix trace operator.
Coherency matrix is a linear combination of hermitian basis matrices; Stokes parameters are projections onto those basis matrices

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
Linear transformations of electric field
Congruence transformation of coherency matrix
Lorentz transformation of Stokes parameters

7. Rotations and Boosts Rotations: Boosts:
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

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  P() = aS()e-i2 + ()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: The ugly:
closest and brightest MSP known
narrow pulse peak
The ugly:
OPM sweep near pulse peak
TOA strongly distorted by calibration errors

12. PSR J profile

13. PSR J spectrum
Fluctuation power spectra (power in Fourier transform of pulse profile)
Black = total intensity, Red = polarized intensity
At high frequencies (power on shortest time scales), polarized flux provides more power

14. Parallactic Variation of Arrival Time
Polarisation also impacts on high-precision timing
6.5 hours of time-of-arrival
~9 microsecond peak-to-peak
Due to parallactic rotation of receiver

15. Even After Calibration
Ideal feed assumption
2 us peak-to-peak

16. Enter the Invariant
Form the invariant profile, Sinv(), where Sinv2 = S02 - |S|2
Dramatically reduced systematic error in PSR J data (Britton 2000; van Straten et al 2001)

17. Caveats Sinv  0 as |S|/S0  1 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 Pulsar MTM invariant J0437-4715 0.8318(1)
1.4314(3) J
0.669(4)
1.80(2) J
0.877(2)
1.543(5) B
0.9314(9)
1.430(2) J
0.959(4)
1.464(8) B
0.862(4)
1.451(9)
Relative uncertainty with respect to timing total intensity profile
Does not consider systematic error

20. Limitations Also limited by S/N: Calibration before integration …
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
How is J determined?
Variation of Stokes parameters as a function of parallactic angle
(pulsars provide many constraints, one page for each pulse phase bin)

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 J profile

33. PSR J spectrum

34. Arrival Time Comparison
Intensity TM
Matrix TM
Postfit r.m.s.
1.453 s
0.942 s
Reduced 2
2.03
1.01
two CPSR2 bands at 20cm
234 TOAS (Nfree=217)
r.m.s reduced by 39% (predicted 33%)

35. Conclusions
Long-term stability of PSR J 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
* MTM-only data from 2006 have underestimated errors
* Variations in MSP polarization would appear as phase-dependent residuals
* Well-fit model with fewer parameters than data increases instantaneous precision