Fermi Gamma-ray Space Telescope

Appendix B. Notes on Numerical Computation

1. Arrival Time Conversion

Precision in Arrival Time Conversions

Time precision needed in an analysis depends on various factors, including: pulse frequency (the faster it spins, the more precise arrival times must be); your intention (e.g., whether you are interested in pulse frequency only, or in absolute pulse arrival times as well); and your science goal. At the same time, various factors affect the time precision that can be achieved after arrival time conversions. Given below are some numbers to keep in mind when you consider the time precision needed in a pulsar analysis.

  • Uncertainty in spacecraft position and its relevance:

    30 km difference in spacecraft position may result in 0.1 ms difference (at maximum) in a geocentric time. The Fermi (formerly GLAST) spacecraft will travel 30 km in approximately 4 s.

  • Uncertainty in pulsar position and its relevance:

    0.1 arc sec difference in pulsar position may result in 0.24 ms difference (at maximum) in a barycentric time. A difference of 0.1 arc sec appears in the fifth decimal place of RA/Dec. That means, if RA is more than 100 degrees, the difference appears in the 8th digit of RA, which is beyond the last reliable digit of a 4-byte floating point variable in many platforms.

2. Pulsar Ephemeris Computation

Precision in calculation of pulse frequency and pulse phase

In Fermi (formerly GLAST), time duration of an observation can be very long; you might have to accumulate one year of data, just to detect pulsations. Long duration of an observation, however, makes it a little tricky to handle in your temporal analysis. How to handle it depends on the details and purpose of your analysis.

Below are some examples of what you should keep in mind:

  • Watch out for frequency derivatives.

    To perform periodicity tests on a long observation, you probably need to supply frequency derivatives, or at least an initial guess of what they might be. Incorrect values of frequency derivatives (f1 and f2) may result in periodic signals smeared out by going through the long duration of the observation.

    To see the smearing effect, simply calculate a pulse phase of the final moment of your observation as described in the previous section. If the terms that include f1 or f2 are in the order of unity, their contributions are significant and you cannot ignore them. So, you have to guess it correctly, or search for it.

  • Check the last digits of your ephemeris parameters.

    Both types of calculations, either pulse frequency or pulse phase, involve an elapsed time (factor (t - t0) for pulse frequency calculations and factor (ti - t0) for pulse phase calculations), which could be very large in Fermi (formerly GLAST) analyses.

    As a consequence, which pulsar parameter dominates uncertainty in resultant pulse frequencies and pulse phases really depends: sometimes uncertainty in f0 (frequency at ephemeris epoch) does; at other times f1 (1st frequency derivative) does; and at yet other times f2 (2nd frequency derivative) does.

    Before starting your analysis, it is recommended that you check out any uncertainties of pulsar parameters you are trying to use, and (try to) determine which pulsar parameters (f0, f1, or f2) determines the last reliable digit in your calculation of pulse frequencies or pulse phases for your observations.


Last updated by: Chuck Patterson 10/25/2005