Yuki David Takahashi

Mentor: Dr. Shri R. Kulkarni

ABSTRACT

1. PROLOGUE

The purpose of this project is to find new millisecond pulsars and possibly even submillisecond pulsars. Pulsars are rapidly rotating neutron stars that appear to be pulsating. A pulsar is observed to be pulsating because it emits beams of radiation in directions other than their spinning axis so that the beam sweeps across the observer periodically as the neutron star rotates. The period of the pulse therefore equals the rotation period of the star. Out of almost 800 pulsars currently known, most rotate once every 0.25 to 2 seconds. About 30 of the known pulsars have periods under 10 milliseconds and make up a separate population called "millisecond pulsars". The fastest one known is PSR B1937+21, which was discovered by my mentor Kulkarni’s group in 1982. Its period is measured to be 1.557806468819794(2) millisecond, implying that the neutron star, whose mass is comparable to that of our Sun, is rotating more than 600 times per second. What allows us to determine the period to such high accuracy is the stability of millisecond pulsars’ periods. Observers can use these pulsars across the sky as very stable clocks for a variety of astrophysical studies including tests of general theory of relativity. Some suggest that long-period gravitational waves can be detected using these clocks.

What would be even more remarkable is discovery of submillisecond pulsars — pulsars with periods under one millisecond. Existence of neutron stars that spin so rapidly would allow physicists to narrow down the equation of state of the densest matter known.

There will be an implication of a whole new dimension if this search discovers a submillisecond pulsar. The existence of neutron stars that spin more than a thousand times per second would constrain some theories about the state of neutron star matter. More specifically, it would narrow down the possible equation of state that describes the behavior of nuclear matter at very high densities of order 10¹⁴ g/cm³ . If a submillisecond pulsar had a typical radius of 10 km, its equatorial speed would be over 20 % of the speed of light. Such a star would be very unstable if not impossible. Therefore, submillisecond pulsars must be smaller so that its surface speed is small enough for its gravitational force to hold itself together. Then, the neutron star material must be soft enough to become that small. Therefore the smaller minimum rotation period of neutron stars imply "softer" equation of state. Thus, this pulsar search is trying to help some physicists prove that something with a density of 10 million tons per cm³ is "soft". However, if this and all other searches with sufficiently short sampling intervals cannot find pulsars with periods under 1.6 ms, then the equation of state of neutron stars may be very stiff.

Before attacking the problem of searching for millisecond pulsars, searchers should understand the basics of pulsar astronomy. Since pulsar astronomy is a relatively new field, there are only two comprehensive books on pulsars–Pulsars by Manchester & Taylor and Pulsar Astronomy by Lyne & Graham-Smith. For more specific introduction to pulsar searches, a recent review by Lorimer¹ is highly recommended.

Now the main problem of this project is deciding how to search for new millisecond pulsars using an existing search program and finally finding them using given data. The data come from the Arecibo 305-meter antenna — the largest radio telescope on Earth — and cover a little over 3% of the sky. More specifics about the data and justification for the choice of survey parameters are presented in section 2.1. Section 2.2 describes step by step how the search method using existing search routines was developed. Then section 2.3 describes the actual search and examination of candidates. The actual search has not begun, but section 3 presents the result of testing the search program on data containing known pulsars. In section 4, the implication of existence of submillisecond pulsars is discussed. As summarized in section 5, this survey has a high potential for discovering new millisecond pulsars and even a reasonable chance of discovering submillisecond pulsars.

2. DATA AND SEARCH

Finding pulsars involves three main steps, described in the following three subsections. Each stage involves many decision-makings.

2.1. Data

2.1.1. Survey Specifications

Since the ultimate goal is to find pulsars with the shortest periods, the most important parameter in this search is the sampling interval. The sampling interval is the time interval at which the telescope takes power measurements. Of course sampling at shorter interval is more desirable, but the finite recording speed of the data storage device limits the possible sampling rate. This survey used a sampling interval of 80 µs. This is short enough to possibly detect pulsars with periods as short as 160 µs (the period cannot be lower than twice the sampling interval because the signal must be identifiable as changing instead of continuous).

The next decision to make is at which frequency to look for pulsars so that the greatest number of millisecond pulsars is detectable. Pulsar signals are generally stronger at lower frequency. However, electromagnetic waves at lower frequency are scattered more, resulting in greater smearing of pulse signals (see equation 4). Taking these two factors into account, this survey made observations at frequencies around 430 MHz. For this observing frequency, the best telescope to use is the Arecibo 305-meter antenna because it has the largest collecting area of all radio telescopes.

Bandwidth and frequency resolution are also important factors in observation, especially at radio wavelengths. Wider bandwidth increases the signal-to-noise ratio because pulsar signals generally span the whole frequency range while noise is usually in a narrow range of frequency. The observation for this search was in frequency range between 426 and 434 MHz so the bandwidth was 8 MHz, which is about the widest for Arecibo Telescope at this frequency. To reduce smearing due to dispersion (see equation 2), the 8-MHz bandwidth was divided into 128 channels, each 60-kHz wide.

Finally, for resolution of power measurement, 4 bits were used to digitize the amplitude for each sample. This produced 0.8 megabytes of data each second. To collect data at this rate, the Penn-State Pulsar Machine was used.

Table 1 presents the summary of survey parameters:

2.1.2. Survey Area

Another decision to make is what direction in the sky, or in our galaxy, to do the survey. Because millisecond pulsars have weak magnetic fields and therefore low radio luminosity, most would be detectable only within 1 kilo-parsec or so. And, their life expectancies are long enough for them to move away from galactic plane⁸. Therefore, readily detectable millisecond pulsars are distributed nearly isotropically. So observation at any direction in the sky would give similar results for our purpose.

All the data were taken between November 1996 and May 1998 during Arecibo‘s major upgrade when drift surveys were still possible. The telescope remained pointed at the same azimuth and zenith angle so that the right ascension of the telescope’s beam changed as the Earth rotated. With Arecibo telescope’s beam width being 10 arc-minutes, the transit time at declination d was about 42/cos)d) seconds. This search covers a total of about 50,000 beams. Figure 1 shows the 45 strips in the sky between declinations of 6 and 28 degrees north where data were taken. The area covered totaled 1300 degrees², or 3 % of the entire sky. Figure 2 shows the directions in our galaxy the data come from. The amount of data totaled 1.8 terabytes.

2.2. Search Method

2.2.1. Range of Pulsar Periods and Dispersion Measures Covered

To efficiently develop the search method, we should be aware of what range of pulsar periods the survey is sensitive to for a given dispersion measure (DM). Dispersion measure is the integrated column density of free electrons in the line of sight up to the distance of the source. It has a unit of pc/cm³. A pulsar becomes undetectable when its observed pulse width smears out to be longer than its period. So, the shortest period the survey can detect at a given DM corresponds to the observed pulse width at that DM. The observed width of a pulse is calculated by¹:

, (1)

where W_i is the intrinsic pulse width, t is the sampling interval, �t_DM is the smearing due to dispersion, and �t_scatter is pulse broadening due to interstellar scattering. To see the minimum detectable period in relation to the DM of the source, the observed pulse width is plotted as a function of DM in figure 3. In calculating the pulse widths for this plot, W_i is ignored because this differs from pulsar to pulsar and it is very short for fast pulsars. The other three factors in the above equation are discussed below.

First, the sampling interval limits the lowest period the survey can detect. The factor of two accounts for the fact that the pulsar period must be at least twice the sampling interval before it can be detected as a pulsating signal. This constraint is significant only up to DMs of about 13 pc/cm³.

. (2)

In this survey using frequency of 430 MHz and channel width of 60 kHz, smearing across each channel is 6.3 microseconds per DM.

Finally, for DMs above about 200 pc/cm³, scattering of signals by interstellar particles prevents detection of many millisecond pulsars. Scattering broadens the pulse because scattered signals arrive the Earth at different times. For frequency of 1 GHz, this broadening of pulse is calculated as a function of DM by:

. (3)

2.2.2. Development of Steps for Processing Data

Now the most time-consuming stage was deciding the most efficient and productive way to process the data to achieve the goal of finding new millisecond and submillisecond pulsars. Routines from the Caltech Pulsar Package (PSRPACK) written by William Deich in 1993 were used for basic pulsar search processes. The data will be processed in chunks of beam corresponding to integration time of 42 seconds. Consecutive chunks are overlapped by 50% to improve the signal-to-noise ratios of pulsars that may be at the edges of chunks.

The first step in analyzing the raw data is identifying noise local to the antenna. Because such noise will not have dispersion, time series before de-dispersion will be used in this step. For sporadic noise, outlying regions are recorded to be masked later. Outlying regions will be defined as time interval of about 1.5 millisecond around each points that are more than 9 root-mean-squares away from the mean. For periodic noise, periodic signals that are present across 9 beams will be recorded as noise frequencies.

The second step is de-dispersing the raw data at many trial dispersion measures to take into account the dispersion of signals across different frequency channels. To make a list of trial dispersion measures, the equation for observed pulse width was used. The trial values were separated from each other so that at least half of the original pulse would stay within the observed pulse width for one of the trial dispersion measures. Since scattering broadens pulse signals significantly at DMs higher than about 200, the highest trial DM was 238 pc/cm3. The raw data will be de-dispersed at 299 different values of dispersion measures using a PSRPACK routine.

The third step is taking the fast Fourier transform of the de-dispersed time series to find strong periodic signals. Before taking the fast Fourier transform, the time series will be flattened to eliminate any fluctuations with periods over 0.7 seconds. This process also makes the mean of the time series zero. Then, the outlying regions found in the first step will be masked by replacing them with zero. Now, the fast Fourier transform of the time series is computed and the resulting power spectrum will be harmonically folded to find periodic signals. In this stage, frequencies lower than 1 Hz will be ignored since this search is interested in fast pulsars. Frequencies above the Nyquist frequency — frequency corresponding to twice the sampling interval — are ignored as well. Also, harmonics of 60±0.1 Hz will be ignored to eliminate noises associated with the AC power supply. In addition, all the noise frequencies recorded in the first step will be ignored. Then, this process produces a list of 128 strongest periodic signals per trial dispersion measure in each beam.

The forth step is selecting pulsar candidates and producing visual representations of their signals for later manual confirmation. Periodic signals with "confidence level" above a certain threshold are selected as candidates, where confidence level depends on signal-to-noise ratio and appearance of harmonics in the list of periodic signals. The threshold used corresponds to signal-to-noise ratio of approximately 7 or so. To visualize each of the pulsar candidates, the appropriately de-dispersed time series will be folded at the candidate's period to produce a pulse profile. The raw data will also be folded at the candidate's period to produce a plot of folded filterbank data.

The entire process of data reduction was automated by making shell-scripts to run appropriate search routines in order. Josh Shapiro helped with writing the shell-scripts especially for memory management and also wrote a program to make the examination of pulsar candidates easier.

2.3. Search

When the program for automated search is developed, running it on available data is easy. All 1.8 terabytes of the data were copied to the High Performance Storage System. Data reduction will be done using the HP Exemplar at Caltech’s Center for Advanced Computing Research. This parallel machine is the world’s largest cache-coherent shared-memory computer, with 256 processors and 64 GB of memory. Each processor takes about 2 hours to de-disperse each beam at 299 values of dispersion measures, and about 30 minutes to find periodic signals from each beam. With the other processes combined, the program takes about 3 hours to process each beam using one processor.

3. RESULTS

The actual search has not been started yet; however, the program has detected all the known pulsars that it was expected to detect, including two millisecond pulsars. The sky coverage of this survey includes locations of 21 known pulsars, two of which are millisecond pulsars. To test the program, it was run on data that should contain these known pulsars. Out of 21, 17 pulsars were detected. Three of the four undetected pulsars were buried in high noise so that their signal-to-noise ratios were very low. Another one, J1742+27, was indistinguishable from the nearby pulsar J1742+2758. Tables 2 and 3 show the list of known millisecond and slow pulsars, respectively, that are in the area covered by this survey. The pulsars are listed in order of increasing period. Figures 4 and 5 show the folded filterbank plot and the pulse profiles of the two millisecond pulsars.

For this survey, search sensitivity (S_min) for fast pulsars is approximately 1 mJy. Search sensitivity is the minimum flux density required to detect a pulsar with a radio telescope, and is given by¹:

, (5)

In this equation, T_sys is the system temperature (50 K); G is the forward gain of the antenna (12 K/Jy); n is the number of polarization summed (2); �v is the observing bandwidth (8 MHz); t is the integration time (42 s); P is the period of the pulsar; and W is its observed pulse width. Using P = 1 ms and W = 0.2 ms for a typical fast pulsar, S_min turns out to be roughly 1 mJy. This is good as the sensitivity for millisecond pulsars, compared to past searches.

4. EPILOGUE

For the next several months, we may safely expect discoveries of more millisecond pulsars because the data reduction for this search will now begin. Optimists may also expect that a submillisecond pulsar will be discovered soon if they exist. The data used in this search comes from a survey with very promising parameters including a very high sampling rate and a fine frequency resolution. The search program used to process the data has shown its capability by detecting the known millisecond pulsars. It is expected to find new millisecond pulsars not only because it can, but also because the population of millisecond pulsars appears to be comparable to that of slow pulsars. Finally, our enthusiasm as searchers of pulsars will prevent any real pulsars from being overlooked in the process of examining the candidates.

The number of new millisecond pulsars found in this search will give us a better idea of the real population of millisecond pulsars. In addition, new millisecond pulsars will be useful as probes for astrophysical observations. If this search finds more pulsars in binary systems or with planets, they would be intrinsically interesting subjects to study further.

This search is about finding something that rotates hundreds or thousands of times every second while being as massive as the Sun. It is about finding something more precisely periodic than anything else we know other than the best atomic clocks. It is about finding something that will let us learn the nature of matter at its highest density. We like searching for extremes.

ACKNOWLDGMENTS

I would like to thank Professor Shri Kulkarni for guiding me as my mentor while keeping a busy schedule as a great astronomer. I also thank Dr. Stuart Anderson for helping me learn this summer, and being available to answer my questions. Thanks go to Richard Crown and all others who financially supported the SURF program to help students like me have a valuable experience during the summer.

REFERENCES