Fermi Gamma-ray Space Telescope

Limited-Frequency Search

In some cases you will be analyzing a LAT source coincident with a known pulsar using an ephemeris known approximately at other wavelengths. Thus, the frequency (or period) search can be restricted to ephemerides around a trial ephemeris.

A number of statistics have been created to test whether a time series is periodic with a specified frequency or period:

  • Chi-squared test (Leahy et al. 1983, ApJ, 266, 160; Vaughan et al. 1994, ApJ, 435, 362)
  • Zn2 test (Buccheri et al. 1983, A&A, 128, 245)
  • Rayleigh test (equivalent to Zn2 test for n = 1)
  • H test (De Jager et al. 1989, A&A, 221, 180)
  • Bayesian approach (Gregory and Loredo 1992, ApJ, 398, 146; Gregory and Loredo 1996, ApJ, 473, 1059)

Currently the first four tests have been implemented in the Fermitools. These tests calculate the phase for each count, and then evaluate the distribution of phases for non-uniformity; if the signal is not periodic at the tested frequency, then the distribution should be uniform (same number of counts in each phase interval). A phase can be calculated even when the frequency (or period) varies, either because of intrinsic spin variability or the pulsar is in a binary. As a result, each of the above tests can be applied to a specific set of values for the frequency and frequency derivatives (or period and period derivatives). Of course, if you are searching for the set of values most consistent with the data, the more parameters considered, the larger the space that must be searched.

These tests determine whether the signal can be described by a specific ephemeris that may involve more than one parameter. As implemented in the gtpsearch tool, the user-specified test is carried out on a grid of frequencies (or periods) centered on a trial value with user-specified constant values of the first and second derivatives of the frequency (or period). For example, a user may apply the chi-squared test to counts from a suspected pulsar at 100 frequencies around 30 Hz with a frequency derivative of 0, then at the same 100 frequencies with a frequency derivative of 10-9 s-2, etc.

Running gtpsearch

The Fermi tool for a limited frequency search is gtpsearch; examples of running the tool can be found in the Period Search analysis thread. This tool can be run by the user in either frequency or period space. The discussion here assumes that the search is in frequency space; running the tool in period space is similar. Internally the tool translates a period search into a frequency search.

The limited-frequency search is applied to the arrival times of the counts contained in an event file; these counts should have been extracted from the region around the source suspected of being a pulsar. The gtpsearch tool does the barycentric time correction at run time.

The limited-frequency search will be centered on a trial frequency. This frequency can be input manually, or may be extracted from a pulsar ephemerides database. The Fermi pulsar tools use a specific FITS pulsar ephemerides database file format (D4).

The default is to input the central value of the frequency grid that will be tested, and the spacing and number of gridpoints. Frequencies are given in units of Hz, and periods in units of seconds. The gridpoint spacing is in units of the Fourier frequency, the inverse of the dataset's total time range. If you include 'cancelpdot=yes' on the command line when calling gtpsearch, you will also input the first and second derivatives of the frequency. But remember, the periodicity test is carried out on a grid of frequencies at these constant values of the frequency derivatives.

An ephemeris includes a time origin, which determines when the phase is 0 and when the instantaneous frequency is equal to the constant part of frequency (i.e., the frequency is a Taylor series around the time origin). When searching for the frequency the absolute phase is unimportant, and therefore the time origin is not necessary for the phase. Similarly, if the frequency derivatives are zero then the time origin is not necessary for specifying the frequency. However, if there are frequency derivatives, then the time origin must be specified. Regardless of whether it is necessary, you will always input the time origin; you may enter a specific time, or choose the beginning (TSTART), end (TSTOP), or midpoint (MIDDLE) of the data's time range.

You will input parameters for your choice of periodicity test:

  • Chi2—The chi-squared test bins the phases of the counts and compares the number of counts in each phase bin to the average (the total divided by the number of bins). Therefore the user has to specify the number of phase bins to be used in the analysis.
  • Zn2—This test determines the strength of the harmonics across the range of phases. Thus, the n=1 component looks for a simple sinusoidal variation, n=2 tests for two evenly spaced peaks, etc. The n=1 test is also called the Rayleigh test.
  • H—This test is the Zn2-test with n automatically optimized for the data.

The output consists of text with the result of the periodicity search and an optional plot. The output includes the significance of the maximum value of the periodicity statistic, which is the probability of obtaining this maximum value if no periodicity is present (i.e., the null hypothesis); the probability includes the number of trials, the frequency range tested in units of the Fourier frequency. This tool does not create an output file.

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