Basic Structure of the High-energy Observations. Preparation of the Data for the Analysis. Methods to Examine the Periodic Signal

Transcription

1 Timing Analysis of the Fermi Observations Basic Structure of the High-energy Observations Preparation of the Data for the Analysis Methods to Examine the Periodic Signal The related commands to perform timing analysis with Fermi Science Tools

2

3

4

5 Caveats about analyzing LAT data: (Event Selection) The latest release of data is P(ass)7 observation!!

6 Region selection: Only the events from close to the source direction (typically in the radius of 1 degree or less) are required because we want to reduce the contamination from the background. =4 Time selection: We need to clarify for Good Time Interval (GTI). 1. It is necessary to use DATA_QUAL==1 as a filter. 2. We The need default to consider is to select the time science not configuration in Southern Atlantic of the instruments Anomaly (LAT_CONFIG==1) (SAA), i.e. IN_SAA!=T and eliminate pointed observations from the dataset (ROCK_ANGLE<52 ) counts 1821 counts

7 Please use gtbin to give a quick review for your data. 3C 279 CTA 1

8 For the targets which is not bright enough or close to the galactic plane, we need to generate the TS map to obtain the precise position of the source. Ex: PSR B (12 MSPs in M28; Bagdanov et al., 2011) 1FGL J

9 Before doing the temporal analysis, we need to perform the barycentric time correction to convert the photon arrival time. TT: terrestrial time; TDB: Barycentric Dynamical Time ( A Time Tutorial ) The location of the pulsar is needed to calculate photon travel time between the spacecraft and the geocenter. EX: Error: Caught St13runtime_error at the top level: Cannot get Fermi spacecraft position for Fermi MET (TT): the time is not covered by spacecraft file. In order to avoid this problem, it is recommended to make a spacecraft file that covers a time range wide enough to contain everything in an event file.

10 Fourier Transform, also known as the task of power spectrum in the HEAsoft package, is the widely used method to perform the blind search for periodicity. (HEAsoft) (Fermi Science Tools)

11 A suitably scaled plot of the complex modulus of a discrete Fourier transform (DFT) is commonly known as power spectrum. For a given signal, the power spectrum gives a plot of the portion of a signal s power falling within given frequency bins. The complex numbers X k represent the amplitude and the phase of different sinusoidal components of the input signal x n. DFTs are extremely useful and reveal periodic components of the input data by relative strengths. According to the unitary normalization, the power (P) can be presented as:

12 Epoch-folding is the most direct approach which simply bin the phase and use the Pearson s χ 2 test to determine if the resulting plot is consistent with a random flat probability distribution. If not, the trial period to bin the data can be believed as the true signal. (efsearch/heasoft) (Fermi Science Tools)

13 Window effect: Periodic signal is generated from the combination of a real detection and the effects of the window function/periodic noise.

14 If we assume N j is the number of events in the jth bin, the background should be Poisson distributed with a mean that is the same for each bin. The test statistic for uniformity of phase is: n is the number of bins and The statistic S is a measure for the goodness of fit of the measured data N j by a flat distribution. Anomalously large value of S indicates a bad fit and hence a possible signal. When N j is large, the Poisson distribution for N j becomes Gaussian. In this case S is approximately distributed as for a flat prob. density. Leahy (1983) also gave the uncertainty of the period based on this method: The binned phase distribution gives an estimate of the folded light curve.

15 The Rayleigh test is a basically conceptual starting point to introduce -test and H-test. It is simple to apply and computationally fast. The related content to apply the method in an astronomical observation can be referred to Gibson et al. (1982). If the phase prob. for the test is flat, the displacements of phasors are a uniform two-dimensional random walk. Phases: Prob. density of the phase:

16 For the no-signal (uniform phase) case this is essentially a twodimensional diffusion problem with fixed step length. The Cartesian components of the sum can be represented as : If a particular set of measurements gives a point that is far away from the origin, the displacement appears to be anomalously large. This might indicate a true signal.

17 a and b are normally distributed if N is large and hence twice the Rayleigh power can be described as the sum of the squares of two independent standard normal variables if a uniform phase distribution is assumed. The probability that P exceeds a thread P 0 is: The power, P, is the test statistic to assess the significance of any putative signal. The Rayleigh test is free of the binning uncertainties associated with epoch-folding, and this method is sensitive only to the first harmonic of the light curve. In order to detect the sharp peak that would produce large amplitudes in many harmonics, a generalization of the Rayleigh test with greater sensitivity for light curves with narrow structures and multiple peaks are taken into consideration. (Buccheri et al., 1983) ν is the assumed frequency of the signal; m is the harmonics.

18 -statistic is twice of the sum of Rayleigh powers for the first m harmonics. m=1 is appropriate to detect a single-peak light curve, m=2 can be used for the detection of a broad light curve with two peaks and large m is appropriate for examining the light curves with more very narrow peaks. Since the distribution of the -statistic is the extension of the Rayleigh statistic, is distributed as χ 2 with 2m degrees of freedom w h i l e 2 P ( t w i c e t h e R a y l e i g h p o w e r ) is d i s t r i b u t e d approximately as (Fermi Science Tools)

19 H-test was developed by de Jager (1989, 1994). The H-test involves the Z m2 -statistic as basis with chosen smoothing parameter (m=1,,20). The H-statistic is defined as H M ax Z m 2 4m 4 1 m 20 And the Z m2 -statistic is given by : Z 2 m 2 [( cos2 k N i ) 2 ( sin2 k i ) 2 ] k 1 i 1 i 1 N is total number of detected photons, and φ is the arrival phase of the ith photon. Transform the Photon Arrival time to the Arrival Phase fractional part of( 0 (t t 0 ) 1 2 & 0 (t t 0 )2 1 6 & 0 (t t 0 ) 3...) ν is the trial frequency, t is the barycentric corrected arrival time of the photon, and t 0 is time-zero epoch The H-test offers some advantages over the other tests when searching for an unknown pulse shape in sparse data. It is independent of the binning of the data, and it is sensitive to a wide variety of pulse shapes. Advantages of H-test: m N N

20 GTI start ~ MJD

21 New probability distribution of H was provided by de Jager (2010). H-statistics follows a simple exponential with a parameter of 0.4 and yield more significant results comparing to the old distribution. E.g., for H=50, a probability of 4x10-8 is generally quoted in the article, whereas the true probability is 2x10-9 a factor 20 smaller. (de Jager, 2010)

22 The searching step is always chosen to be a small fraction (typical 1/5) of the corresponding independent Fourier width/spacing. The Fourier spacing is P 2 /T obs in the period domain where P is the trial period and T obs is the total duration of the investigation. (The precision in the frequency domain is 1/T obs ). (Fermi Science Tools)

23 Check the known pulsar timing database: The timing information and the parameter of the ephemerides for the pulsar can be found from the link. The sample of the ephemeris data file can be found: The above link also provides the way to create a ephemeris file in fits format for the new pulsar. Pulsation search using power spectrum: barycentric time correction

24 For example: With the timing parameters provided by Fermi Data Center, we can obtain the ephemeris of PSR J : The definitions of all the parameters can be obtained by the link:

25 Performing a period search using another schemes: (with the database) 100 independent trials The way to perform other schemes (e.g. chi-squared test or Z m2 test) just follow the same way.

26 At first, we generate the information to record the (pulsed) phase of the data. (without the database for the pulsar) barycentric time correction Then we generate the fits file of the pulse profile. We will have 20 bins in one cycle.

27

28

29 De Jager (1994) provided the way to estimate the upper limit of the pulsed fraction when no significant detection was found for the periodicity search of the source. Upper limit: x2 H 10 H exp log exp log x3 H 10 H x p N p x N p p N N N t t t b p: pulsed fraction of the real source; p t : pulsed fraction of the extracted region N: total photons; N b : background photons; N-N b : source photons With the loading of the duty cycle (δ) and largest H-value in the search, we can estimate the pulsed fraction (p) with the number of events N. If we want to search the spin-period of the pulsar in a binary s y s t e m, Fermi Science Tools (FST) also provide the related commands to correct the orbital effect in the time series (Blandford & Teukolsky,1976). In reality, the commands provided by FST are not practical since we can not consider the effect from unknown period/freq. derivative for the observation of a long time span (Atwood et al., 2006; Ziegler et al., 2008). In most cases, try to write your own program to deal with the timing 2 2