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