Time-variable Dispersion Research Articles

The NANOGrav 15 yr Data Set: Detector Characterization and Noise Budget

Pulsar timing arrays (PTAs) are galactic-scale gravitational wave (GW) detectors. Each individual arm, composed of a millisecond pulsar, a radio telescope, and a kiloparsecs-long path, differs in its properties but, in aggregate, can be used to extract low-frequency GW signals. We present a noise and sensitivity analysis to accompany the NANOGrav 15 yr data release and associated papers, along with an in-depth introduction to PTA noise models. As a first step in our analysis, we characterize each individual pulsar data set with three types of white-noise parameters and two red-noise parameters. These parameters, along with the timing model and, particularly, a piecewise-constant model for the time-variable dispersion measure, determine the sensitivity curve over the low-frequency GW band we are searching. We tabulate information for all of the pulsars in this data release and present some representative sensitivity curves. We then combine the individual pulsar sensitivities using a signal-to-noise ratio statistic to calculate the global sensitivity of the PTA to a stochastic background of GWs, obtaining a minimum noise characteristic strain of 7 × 10−15 at 5 nHz. A power-law-integrated analysis shows rough agreement with the amplitudes recovered in NANOGrav’s 15 yr GW background analysis. While our phenomenological noise model does not model all known physical effects explicitly, it provides an accurate characterization of the noise in the data while preserving sensitivity to multiple classes of GW signals.

Dispersion measure variability for 36 millisecond pulsars at 150 MHz with LOFAR

Context. Radio pulses from pulsars are affected by plasma dispersion, which results in a frequency-dependent propagation delay. Variations in the magnitude of this effect lead to an additional source of red noise in pulsar timing experiments, including pulsar timing arrays (PTAs) that aim to detect nanohertz gravitational waves. Aims. We aim to quantify the time-variable dispersion with much improved precision and characterise the spectrum of these variations. Methods. We use the pulsar timing technique to obtain highly precise dispersion measure (DM) time series. Our dataset consists of observations of 36 millisecond pulsars, which were observed for up to 7.1 yr with the LOw Frequency ARray (LOFAR) telescope at a centre frequency of ~150 MHz. Seventeen of these sources were observed with a weekly cadence, while the rest were observed at monthly cadence. Results. We achieve a median DM precision of the order of 10−5 cm−3 pc for a significant fraction of our sources. We detect significant variations of the DM in all pulsars with a median DM uncertainty of less than 2 × 10−4 cm−3 pc. The noise contribution to pulsar timing experiments at higher frequencies is calculated to be at a level of 0.1–10 μs at 1.4 GHz over a timespan of a few years, which is in many cases larger than the typical timing precision of 1 μs or better that PTAs aim for. We found no evidence for a dependence of DM on radio frequency for any of the sources in our sample. Conclusions. The DM time series we obtained using LOFAR could in principle be used to correct higher-frequency data for the variations of the dispersive delay. However, there is currently the practical restriction that pulsars tend to provide either highly precise times of arrival (ToAs) at 1.4 GHz or a high DM precision at low frequencies, but not both, due to spectral properties. Combining the higher-frequency ToAs with those from LOFAR to measure the infinite-frequency ToA and DM would improve the result.

First detection of frequency-dependent, time-variable dispersion measures

Context. High-precision pulsar-timing experiments are affected by temporal variations of the dispersion measure (DM), which are related to spatial variations in the interstellar electron content and the varying line of sight to the source. Correcting for DM variations relies on the cold-plasma dispersion law which states that the dispersive delay varies with the squared inverse of the observing frequency. This may, however, give incorrect measurements if the probed electron content (and therefore the DM) varies with observing frequency, as is predicted theoretically due to the different refraction angles at different frequencies. Aims. We study small-scale density variations in the ionised interstellar medium. These structures may lead to frequency-dependent DMs in pulsar signals. Such an effect could inhibit the use of lower-frequency pulsar observations as tools to correct time-variable interstellar dispersion in higher-frequency pulsar-timing data. Methods. We used high-cadence, low-frequency observations with three stations from the German LOng-Wavelength (GLOW) consortium, which are part of the LOw-Frequency ARray (LOFAR). Specifically, 3.5 yr of weekly observations of PSR J2219+4754 are presented. Results. We present the first detection of frequency-dependent DMs towards any interstellar object and a precise multi-year time-series of the time- and frequency-dependence of the measured DMs. The observed DM variability is significant and may be caused by extreme scattering events. Potential causes for frequency-dependent DMs are quantified and evaluated. Conclusions. We conclude that frequency dependence of DMs has been reliably detected and is indeed caused by small-scale (up to tens of AUs) but steep density variations in the interstellar electron content. We find that long-term trends in DM variability equally affect DMs measured at both ends of our frequency band and hence the negative impact on long-term high-precision timing projects is expected to be limited.

The NANOGrav 11-year Data Set: High-precision Timing of 45 Millisecond Pulsars

Abstract We present high-precision timing data over time spans of up to 11 years for 45 millisecond pulsars observed as part of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) project, aimed at detecting and characterizing low-frequency gravitational waves. The pulsars were observed with the Arecibo Observatory and/or the Green Bank Telescope at frequencies ranging from 327 MHz to 2.3 GHz. Most pulsars were observed with approximately monthly cadence, and six high-timing-precision pulsars were observed weekly. All were observed at widely separated frequencies at each observing epoch in order to fit for time-variable dispersion delays. We describe our methods for data processing, time-of-arrival (TOA) calculation, and the implementation of a new, automated method for removing outlier TOAs. We fit a timing model for each pulsar that includes spin, astrometric, and (for binary pulsars) orbital parameters; time-variable dispersion delays; and parameters that quantify pulse-profile evolution with frequency. The timing solutions provide three new parallax measurements, two new Shapiro delay measurements, and two new measurements of significant orbital-period variations. We fit models that characterize sources of noise for each pulsar. We find that 11 pulsars show significant red noise, with generally smaller spectral indices than typically measured for non-recycled pulsars, possibly suggesting a different origin. A companion paper uses these data to constrain the strength of the gravitational-wave background.

THE NANOGRAV NINE-YEAR DATA SET: OBSERVATIONS, ARRIVAL TIME MEASUREMENTS, AND ANALYSIS OF 37 MILLISECOND PULSARS

We present high-precision timing observations spanning up to nine years for 37 millisecond pulsars monitored with the Green Bank and Arecibo radio telescopes as part of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) project. We describe the observational and instrumental setups used to collect the data, and methodology applied for calculating pulse times of arrival; these include novel methods for measuring instrumental offsets and characterizing low signal-to-noise ratio timing results. The time of arrival data are fit to a physical timing model for each source, including terms that characterize time-variable dispersion measure and frequency-dependent pulse shape evolution. In conjunction with the timing model fit, we have performed a Bayesian analysis of a parameterized timing noise model for each source, and detect evidence for excess low-frequency, or "red," timing noise in 10 of the pulsars. For 5 of these cases this is likely due to interstellar medium propagation effects rather than intrisic spin variations. Subsequent papers in this series will present further analysis of this data set aimed at detecting or limiting the presence of nanohertz-frequency gravitational wave signals.

GRAVITATIONAL WAVES FROM INDIVIDUAL SUPERMASSIVE BLACK HOLE BINARIES IN CIRCULAR ORBITS: LIMITS FROM THE NORTH AMERICAN NANOHERTZ OBSERVATORY FOR GRAVITATIONAL WAVES

The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) project currently observes 43 pulsars using the Green Bank and Arecibo radio telescopes. In this work we use a subset of 17 pulsars timed for a span of roughly five years (2005--2010). We analyze these data using standard pulsar timing models, with the addition of time-variable dispersion measure and frequency-variable pulse shape terms. Within the timing data, we perform a search for continuous gravitational waves from individual supermassive black hole binaries in circular orbits using robust frequentist and Bayesian techniques. We find that there is no evidence for the presence of a detectable continuous gravitational wave; however, we can use these data to place the most constraining upper limits to date on the strength of such gravitational waves. Using the full 17 pulsar dataset we place a 95% upper limit on the sky-averaged strain amplitude of $h_0\lesssim 3.8\times 10^{-14}$ at a frequency of 10 nHz. Furthermore, we place 95% \emph{all sky} lower limits on the luminosity distance to such gravitational wave sources finding that the $d_L \gtrsim 425$ Mpc for sources at a frequency of 10 nHz and chirp mass $10^{10}{\rm M}_{\odot}$. We find that for gravitational wave sources near our best timed pulsars in the sky, the sensitivity of the pulsar timing array is increased by a factor of $\sim$4 over the sky-averaged sensitivity. Finally we place limits on the coalescence rate of the most massive supermassive black hole binaries.

Phase errors in diffraction-limited imaging: contrast limits for sparse aperture masking

Bispectrum phase, closure phase and their generalisation to kernel-phase are all independent of pupil-plane phase errors to first-order. This property, when used with Sparse Aperture Masking (SAM) behind adaptive optics, has been used recently in high-contrast observations at or inside the formal diffraction limit of large telescopes. Finding the limitations to these techniques requires an understanding of spatial and temporal third-order phase effects, as well as effects such as time-variable dispersion when coupled with the non-zero bandwidths in real observations. In this paper, formulae describing many of these errors are developed, so that a comparison can be made to fundamental noise processes of photon- and background-noise. I show that the current generation of aperture-masking observations of young solar-type stars, taken carefully in excellent observing conditions, are consistent with being limited by temporal phase noise and photon noise. This has relevance for plans to combine pupil-remapping with spatial filtering. Finally, I describe calibration strategies for kernel-phase, including the optimised calibrator weighting as used for LkCa 15, and the restricted kernel-phase POISE technique that avoids explicit dependence on calibrators.

LIMITS ON THE STOCHASTIC GRAVITATIONAL WAVE BACKGROUND FROM THE NORTH AMERICAN NANOHERTZ OBSERVATORY FOR GRAVITATIONAL WAVES

We present an analysis of high-precision pulsar timing data taken as part of the North American Nanohertz Observatory for Gravitational waves (NANOGrav) project. We have observed 17 pulsars for a span of roughly five years using the Green Bank and Arecibo radio telescopes. We analyze these data using standard pulsar timing models, with the addition of time-variable dispersion measure and frequency-variable pulse shape terms. Sub-microsecond timing residuals are obtained in nearly all cases, and the best root-mean-square timing residuals in this set are ~30-50 ns. We present methods for analyzing post-fit timing residuals for the presence of a gravitational wave signal with a specified spectral shape. These optimally take into account the timing fluctuation power removed by the model fit, and can be applied to either data from a single pulsar, or to a set of pulsars to detect a correlated signal. We apply these methods to our dataset to set an upper limit on the strength of the nHz-frequency stochastic supermassive black hole gravitational wave background of h_c (1 yr^-1) < 7x10^-15 (95%). This result is dominated by the timing of the two best pulsars in the set, PSRs J1713+0747 and J1909-3744.

Oxygen-Sag in a Tidal River

In the present work, freshwater flow in a tidal river is incorporated in a dimensionless solution of the derived equations for BOD and DOD and a comparison is made between the profiles obtained with both a constant and a time variable dispersion coefficient, E. The time variation of E is expressed as the ratio of the absolute value of the oscillating velocity to a reference velocity. Flow data from the Potomac River, Washington are used in the computer programs and illustrative BOD and DOD profiles are presented. Increases in the river discharge decrease and translate downstream the points of maximum BOD and DOD. Time variation of the dispersion coefficient smooths both BOD and DOD profiles, as opposed to a constant value of E, and reduces their peak values. Decisions on the maximum DOD associated with a specific effluent outfall which are based on a solution with a constant E appear, therefore, to be conservative.