Pulsars: Observation & Timing. Stephen Eikenberry 28 Jan 2014

Transcription

1 Pulsars: Observation & Timing Stephen Eikenberry 28 Jan 2014

2 Timing Pulsars Figure: D. Nice Pulsars send out pulsed signals We receive them Goal is to measure a Time of Arrival (TOA); this provides great accuracy

3 Timing Pulsars Figure: D. Nice Incoming data is a time series Often, S/N <1 per pulse Get high S/N pulse profile by folding the data on the pulse period Huh?

4 " Brightness as a function of time: b(t) Time series " Samples: b 1, b 2, b 3,..., b N [b i ; 0< i <N] " Time interval dt = time spacing between samples " Total duration T = N*dt " We'll deal with evenlyldtime series (dt is sampled constant)

5 Periodicities " Repeating pattern of brightness variations (e.g. binary orbit modulation; pulsar rotational modulation, etc.) " "Fold" the time series at the period add up all time bins (b i ) that are separated by one period " Turns a time series into a "lightcurve", or "pulse profile"

6 Pulsar Time Series and Pulse Profile

7 Finding periods " They're the little black dots at the ends of sentences, stupid. " Not quite " One solution: Periodigrams Take a range of "trial" periods Fold at each period and calculate variance of lightcurve around its mean No signal flat line (biophysics) Signal at different period flat line (signal smeared by period "error") Signal at trial period large variance

8 Why periodograms suck: " N evenly-spaced samples have N degrees of freedom N trial periods " Fold N samples N different ways N 2 calculations " If N ~10 9 ; N 2 ~ calcs " Fast tfourier Transforms (FFTs) " Do similar calculation w/ N log N scaling " N ~10 9 reqs "only" ~10 10 calcs " factor of 100 million savings (!)

9 Everything You Ever Wanted to Know About FFTs (And Then Some...) S Eikenberry S. Eikenberry 28 Jan 2013

10 FFTs " Just an algorithm to efficiently calculate the Fourier Transform of a function n " What's the Fourier Transform? " Complex (amplitude, phase) frequency response to function: " A k = j n j e -i2 jk/n k th frequency f = k/t " k " T=N*dt

11 Real FFTs " If n are real numbers (like number of photons), then FFT has special properties " FFT calculated for 0 k N " A k = A N-k * (complex conjugate) " So, can safely ignore second half of FFT (same info in reverse order) k = N/2 is the "Nyquist frequency" "

12 What does that mean? " A k = j n j e -i2 jk/n " Complex numbers above are unit vectors rotating in the complex plane " Rotation step is -2 k/n radians

13 Constant data " A k = j n j e -i2 jk/n A k j n j e " If are constant, the FFT makes k polygons with N/k sides " Each polygon returns to near origin " Last one returns exactly to origin " Thus FFT of constant data is all zeroes except for k=0 ("DC" frequency)

14 " FFTs are linear, so signal with noise is x=s+noise " FFT(x) = FFT(s)+ FFT(noise) " FFT of noise = fixed step directions with random length " Result = random walk Noisy data

15 Fourier Power " A k are complex numbers " P 0 =abs(a) 0 " P k = abs(a k ) 2 /P 0 is Fourier "power" " Proportional to vector length from origin squared " If noise is Poissonian (e.g. X-ray events) then prob(p) dp = e -P dp " Thus, a "3 " detection requires P 6.5

16 " Assume a signal with frequency f r = r/t y r FFT Response " n j = a cos(2 f r j t+ 0 ) = a cos(2 jr/n + 0 ) " If r is an integer, some k=r " A k = N a/2 e i 0 [derive] " k r gives A k 0

17 " If r not integer... FFT Response (cont) " Two terms, one near 0 and another large For k closest to r: A =a " k e -i (k-r) *sinc(k-r) [derive] " sinc(x) = sin( x)/ x

18 FFT Response (more) " Scalloped response " Power reduction Power reduction significant, but not HUGE

19 Fixing the response " Power is never "lost" just moved " Can recover power using Fourier interpolation or "finebinning" Just uses FFT " response to calculate response at smaller spacing

20 Using FFTs " Read in data to array x " Calculate FFT: y=fft(x,1) " NB: x should be 2^N elements to work simply [e.g. x=fltarr(2l^20)] " Calculate power: p= abs(y)^2 " Normalize it: p=p/abs(y(0)) " frequency: f= findgen(n/2)/(n*dt) ( ) " Plot it: plot, f, p(0:n/2-1) " Why the subscript?

21 Figure: D. Nice

22 Pulse Dispersion Figure: D. Nice Space is not empty (We are NOT alone!) Full of plasma free electrons light travel time is <c AND is frequency-dependent Dispersion Measure (DM) = n e (l) dl (units pc cm -3 ) Pulse delay is t (sec) = DM/[ f 2 ](fi in MHz) MH)

23 Pulse Dispersion (II) Time Figure: M. Bailes

24 Pulse Dispersion (III) Dispersion delays can exceed the pulse period for short periods and/or high dispersions This would effectively smear the signal into a constant t no pulsations (!!) Even for lower smearing, this would decrease S/N How do we fix it? (Especially for detecting new pulsars) Figure: Parkes

25 Pulse Dispersion (IV) Dispersion is painful Dispersion is our friend! Dispersion distances (combine measured dispersion i with model of free electrons in the Galaxy) Taylor & Cordes model NB: Dispersion distances are a tricky business! (why?) Figure: Parkes

26 Timing Residuals Figure: D. Nice Residuals to TOA fit: (t) = 0 + t + ½ t 2 + 1/3 t 3 + Note: over 7 years, this can be > 1 billion pulses Each measured to <1 s accuracy (precision = /N <10-13!!!)

27 Timing Residuals (II) Figure: D. Nice Figure shows error in P Phase (TOA) drifts linearly with time

28 Timing Residuals (III) Figure: D. Nice Figure shows error in Pdot g Phase (TOA) drifts quadratically with time

29 Timing Residuals (IV) Figure: D. Nice Earth moves telescope moves (!!) Introduces ~500s amplitude oscillation in TOA; amplitude and phase depend on direction to pulsar (RA, Dec)

30 Earth is a lousy place for an observatory It rotates AND orbits Changing gposition = changing time Need VERY accurate measurement of Earth s position, including GR effects of Sun/planet potential wells Barycentering

31 Timing Residuals (V) Figure: D. Nice Correct the TOAs to the Solar System Barycenter Called barycentering ; very sensitive to ACTUAL SS motions, incl. actual Earth rotation

32 Timing Residuals (VI) Earth s rotation is NOT steady (at all!) Must feed corrections in to the TOAs; often post-correct the data based on actual Earth rotation observations

33 Timing Astrometry Figure: D. Nice Barycenter corrections depend on RA, Dec (measured) Errors in position (incl. parallax!) change timing Good timing requires (or provides!) astrometry, incl. parallax/distance and proper motion

34 Some pulsars show sudden discontinuities in timing glitches / ~ 10-6 Crab glitch at right Followed by exponential recovery to almost pre- glitch state Often a residual long-term change in d /dt Timing: Glitches

35 Timing: Glitches Glitches mostly occur in young pulsars (Crab, Vela, poss , etc.) Note that these are also the ones with measurements of braking index n Usually, timescale to measure n is longer than time interval between glitches this is a problem!

36 Timing: Glitches Vela glitch history Vela braking index is n=1.4 (???)

37 Timing: Glitches Superfluid vortex pinning theory for glitches

38 Residuals to timing Timing: Timing noise Often show a random walk

39 Residuals to timing Often show a random walk Origin? (Vortex lattice oscillations??) Timing: Timing noise