Finding (or not) New Gamma-ray Pulsars with GLAST

TL;DR

This paper discusses the potential for detecting new gamma-ray pulsars with the GLAST telescope, emphasizing the challenges and limitations of blind searches in gamma-ray data.

Contribution

It provides an analysis of the expected number of pulsars detectable by GLAST and reviews the methods and constraints of blind pulsar searches in gamma-ray observations.

Findings

Estimated detection of a few to a few hundred pulsars by GLAST

Blind search limitations due to observational mode and computational constraints

Scaling of search statistics with observational parameters

Abstract

Young energetic pulsars will likely be the largest class of Galactic sources observed by GLAST, with many hundreds detected. Many will be unknown as radio pulsars, making pulsation detection dependent on radio and/or x-ray observations or on blind periodicity searches of the gamma-rays. Estimates for the number of pulsars GLAST will detect in blind searches have ranged from tens to many hundreds. I argue that the number will be near the low end of this range, partly due to observations being made in a scanning as opposed to a pointing mode. This paper briefly reviews how blind pulsar searches will be conducted using GLAST, what limits these searches, and how the computations and statistics scale with various parameters.

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Full text

1 Introduction

Pulsar emission mechanism(s) from radio to gamma-rays are poorly understood after 40 yrs of work. Yet despite this fact, pulsars are high-precision tools that probe a variety of topics in both fundamental physics and astrophysics. EGRET detected pulsed emission from at least 6, and probably 7$-$9, young pulsars at energies $\;\buildrel>\over{\sim}\;$100 MeV (Thompson, 2004). In addition, several tens of the (primarily Galactic) unidentified EGRET sources are likely pulsars (e.g. McLaughlin and Cordes, 2000, 2003; Kramer et al., 2003). With its much larger effective area and improved angular resolution, GLAST will almost certainly detect hundreds of pulsars. The majority of those pulsars will remain unidentified though (and useless as physics tools), unless pulsations are detected through 1) “folding” of the events using timing ephemerides for known radio pulsars; 2) searches of associated x-ray sources to find radio-quiet Geminga-like pulsars (Halpern and Holt, 1992), or very faint radio pulsars (Halpern et al., 2001; Roberts et al., 2002); or 3) “blind” searches of the gamma-ray events (Chandler et al., 2001). This paper discusses the latter, and perhaps most difficult, option.

The known gamma-ray pulsars have similar characteristics (for a review, see Thompson, 2004). They are mostly non-variable sources with fairly flat energy spectra in the gamma-ray regime, but with spectral cutoffs around or just above 1 GeV. The flat energy spectra imply that most of the photons detected by the Large Area Telescope (LAT) will be in the $\sim$100-300 MeV range, where the point spread function has a width $\;\buildrel>\over{\sim}\;$1∘. The gamma-ray pulse shapes are complex with two relatively sharp pulses which provide higher harmonic content in searches and make their detection easier.

Most of the GLAST mission will be spent in sky survey mode, where the whole sky is scanned every $\sim$3 hours111For details, see the GLAST mission website: http://glast.gsfc.nasa.gov. On average, a point in the sky will be within the LAT field of view $\sim 1/6$ of the time. Pointed observations would increase this fraction to $\sim 1/2$ (with Earth occultations preventing higher on-source efficiency). This loss in efficiency due to scanning (more specifically, the resulting decrease in the number of source events $N_{s}$ during a particular viewing period $T_{\rm view}$) will make coherent pulsation searches considerably more difficult and less sensitive.

2 Blind Searches for Gamma-ray Pulsars

Blind searches of low count rate event data are typically conducted using either Fourier techniques (i.e. binning events into a time series, computing one or more FFTs, and analyzing the resulting amplitude and possibly phase spectra; (e.g. van der Klis, 1989; Chandler et al., 2001)) or via brute-force epoch folding (i.e. assembling a pulse profile by determining the pulse phase of each event from a trial ephemeris and then computing a probability of non-uniformity for the profile; (e.g. de Jager et al., 1989; Gregory and Loredo, 1992)). Optimal sensitivity to pulsations comes from searches that treat all of the data in a “coherent” fashion, meaning that the pulsar’s rotational phase is accurately tracked over the full observation or analysis duration $T_{\rm view}$, from first event to last.

The real problem with gamma-ray pulsation searches comes from the fact that the sources have very low count rates of 10*-7*$-$10*-8* photons($>$100 MeV) cm*-2* s*-1*, which for the LAT corresponds to roughly 0.2$-$2 events per hour. Therefore, very long observations lasting from weeks to years are required to make significant detections. And unfortunately, young pulsars are notoriously badly behaved over such timescales.

Ideally, we would only need to search over the unknown spin frequency $f$ of a pulsar. However, young pulsars rapidly spin-down (i.e. have large frequency derivatives $\dot{f}$), they exhibit timing noise which manifests itself in searches as significant frequency 2nd derivatives $\ddot{f}$ (and for the noisiest pulsars, many higher order derivatives as well), and they occasionally, on month or year timescales, “glitch” and instantaneously change their observed $f$ and $\dot{f}$. Figure 1 shows the number of independent frequency bins of width $1/T_{\rm view}$ in Hz, corresponding to an accumulated phase error of one rotation, over which a pulsar signal will “drift” during a coherent search for a variety of realistic values of $f$ and $\ddot{f}$. Pulsar spin-down becomes important for energetic young pulsars in a day or two, and so all gamma-ray searches must account for an unknown $\dot{f}$. Timing noise can become important after a few weeks or months. Since gamma-ray pulse profiles contain sharp features, and therefore their pulsed signals contain many harmonics, search codes must significantly oversample the $\sim 1/T_{\rm view}$ or $\sim 1/T_{\rm view}^{2}$ spacings in $f$ and $\dot{f}$ to maintain optimal sensitivity.

A potentially bigger problem, as shown by Chandler et al. (2001), are positional errors or uncertainties $\epsilon$, which cause apparent changes in the observed spin frequency $\delta\!f\sim 10^{-6}\epsilon_{\rm mrad}f_{10}\sin\theta$ Hz, frequency derivative $\delta\!\dot{f}\sim 2\times 10^{-13}\epsilon_{\rm mrad}f_{10}\cos\theta$ Hz s*-1*, and frequency 2nd derivative $\delta\!\ddot{f}\sim 4\times 10^{-20}\epsilon_{\rm mrad}f_{10}\sin\theta$ Hz s*-2*, where $\epsilon_{\rm mrad}$ is the position error in milliradians, $f_{10}$ is the spin frequency in tens of Hz, and $\theta$ is the time varying angle between the Earth’s velocity vector and the line of sight to the pulsar. For candidate GLAST pulsar sources, $\epsilon$ will be milliradians (arcminutes) in size, making the blind detection of even stable Geminga-like pulsars much more difficult.

The intrinsic $\dot{f}$, timing noise, glitch frequency, and position error effects place upper limits on the useful $T_{\rm view}$ for coherent pulsation searches to durations as short as $\sim$10 days for the most energetic and active pulsars, or a few months for older, slower, and more stable pulsars. Because of the scanning nature of the GLAST mission, there will be fewer counts accumulated for a particular source over the specified $T_{\rm view}$ by a factor of $\sim$3 as compared to a pointed observation. This decrease in accumulated counts per unit time greatly impacts coherent search sensitivity and is the reason why scanning is not the optimal observing mode for pulsar studies.

2.1 Pulsation Detection and Computational Considerations

The power $P$ in a periodic gamma-ray signal is $P\sim 1+\alpha N_{s}^{2}/N_{t}$, where $\alpha$ is a pulse-shape dependent factor (0.4$-$0.9 for the known gamma-ray pulsars)(e.g. Buccheri et al., 1987). The total number of events $N_{t}$ is the sum of the source (pulsed) events $N_{s}$ and background (unpulsed) events $N_{b}$. The probability that noise fluctuations produce a certain search power $P_{o}$ is exponentially related to the power: ${\rm Prob}(P\geq P_{o})\propto\exp(-P_{o})N_{\rm trials}$, where $N_{\rm trials}$ is the number of independent trials searched. $N_{\rm trials}$ is proportional to the range of $f$ and $\dot{f}$ searched, but more importantly, to $T_{\rm view}^{3}$.

In general then, to optimize search sensitivity, we want large $N_{s}$ to maximize $P$, yet a relatively short $T_{\rm view}$ (appropriate for the limits discussed in the last section) to reduce $N_{\rm trials}$ and the computational complexity. This rule exposes another problem for scanning-mode observations of pulsars: if $T_{\rm view}$ is increased by a factor of $\sim$3 to recover the events “lost” compared to a pointed observation (assuming the pulsations remain well-behaved over the longer $T_{\rm view}$), a detected signal will have the same power as from the shorter pointed observation, yet the probability that the signal is significant will decrease by a factor of $\sim 3^{3}=27$ due to the larger $N_{\rm trials}$ searched.

Computationally, the complexity of epoch-folding searches scales roughly as $T_{\rm view}^{4}$ (i.e. $N_{t}\times N_{\rm trials}$), while FFT-based techniques fare slightly better and scale as $T_{\rm view}^{3}\log T_{\rm view}$. Moderate sized computer clusters can currently handle epoch-folding searches of hundreds to thousands of events for viewing periods of up to several weeks (see Figure 1). For longer $T_{\rm view}$ or more limited computing resources, one can use evolutionary Brazier and Kanbach (1996) and/or heirarchical Ransom (2002) techniques to greatly reduce the computational burden of coherent searches without greatly impacting the overall sensitivity, if the average pulsation flux is time invariant. Coherent searches with GLAST with T_{\rm view}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1$$-2 months should be sensitive to pulsars with photon fluxes ($>$100 MeV) of a few times 10*-8* to a few times 10*-7* photons cm*-2* s*-1* for scanning-mode observations, or as low as $\sim$2$\times$10*-8* photons cm*-2* s*-1* for pointed observations.

Finally, incoherent searches are possible where the absolute phase of pulsations is not tracked over the full $T_{\rm view}$, but only over much shorter “windows” $T_{\rm win}$. Incoherent searches are less sensitive than coherent ones for the same $T_{\rm view}$, but they use many fewer computing resources and enable searches with much longer $T_{\rm view}$ van der Klis (1989); Atwood et al. (2006). The basic idea limits the duration of the coherently processed intervals $T_{\rm win}$ such that a signal with any reasonable $\dot{f}$ or $\ddot{f}$ stays within a single independent frequency bin (of width $1/T_{\rm win}$ Hz) or fraction thereof. These windowed analyses are then combined without phase information, but with corrections for $\dot{f}$ and/or $\ddot{f}$ effects occurring between the intervals, and analyzed. Initial simulations show that one might detect relatively stable pulsars with photon fluxes ($>$100 MeV) as low as 1$-$2$\times$10*-8* photons cm*-2* s*-1* with $T_{\rm view}\sim 1$ yr and $T_{\rm win}\sim 1$ day. Without phase information, however, the folding of all events over such long durations to accumulate high-quality pulse profiles may be difficult or even impossible.

In summary, instabilities of young pulsars, source positional uncertainties, and the default survey/scanning mode for the mission will make blind pulsation searches with GLAST much more difficult than early estimates had predicted. It is therefore likely that GLAST will blindly discover only tens of new pulsars rather than hundreds.

Acknowledgements: Thanks go to Paul Ray, Julie McEnery, Mallory Roberts, Maura McLaughlin, and Kent Wood for useful discussions.