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MNRAS479,3393-3404 (2018)doi:10.1093/mnras/sty1705 The decomposition of temporal variations of pulsar dispersion measures

P. F. Wang

1,3‹

andJ.L.Han

1,2,3‹

1

National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, China

2

School of Astronomy and Space Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China

3 CAS Key Laboratory of FAST, NAOC, Chinese Academy of Sciences, Beijing 100101, China Accepted 2018 June 25. Received 2018 June 25; in original form 2017 August 11

ABSTRACT

The pulsar dispersion measure (DM) accounts for the total electron content between a pulsar and us. High-precision observations for projects of pulsar timing arrays show temporal DM variations of millisecond pulsars. The aim of this paper is to decompose the DM variations of 30 millisecond pulsars using the Hilbert-Huang transform (HHT) method, so that we can curves resulting from the solar wind, interplanetary medium and/or ionosphere. We find that and double-component features of different origins. The amplitudes and phases of the curve peaks are related to the ecliptic latitude and longitude of pulsars, respectively.

Key words:pulsars: general-ISM: general.

1 INTRODUCTION

Pulsar signals propagate through the ionized medium between a pulsar and us, and are subject to a dispersive delay caused by the medium of t =e 2

4πm

e c? psr us n e (l)dl 2 ,(1) m e andeare the mass and charge of an electron,n e (l)represents the number density of electrons along the sight line, and dlis the in units of cm -3 pc) is defined to account for the total electron column density between a pulsar and us, and expressed as DM=? psr us n e (l)dl.(2) The observed DM of a pulsar includes contributions from the ion- ized interstellar medium, the inter-planetary medium in the Solar system, and the ionosphere around the Earth. For the DMs of most pulsars, the ionized interstellar medium is predominant, and the contributions from the ionosphere and the inter-planetary medium are often negligible but are known to affect the DMs of a few pul- sars located at low ecliptic latitudes in some seasons (e.g. You et al.

2007a).

sars (e.g. Lyne, Pritchard & Smith1988; Petroff et al.2013), which may reflect the drifting of interstellar clouds into or away from our line of sight to a pulsar, or the changes of electron density distri- bution in the interplanetary medium or even in the ionosphere. Re- cent observations with wide-band receivers or quasi-simultaneous multiple-band observations can determine pulsar DMs accurately (e.g. Keith et al.2013; Reardon et al.2016), up to an accuracy of 10 -4 cm -3 pc depending on the steepness of pulse profiles and the frequency range of observations. If temporal DM variations are not ond pulsars (e.g. Keith et al.2013; Lee et al.2014; Arzoumanian et al.2018), and are therefore one of the primary sources of low- frequency ‘noise" in measurements of the time of arrival (TOA) for pulses. Much effort has been expended in eliminating the ‘DM noise" in the pursuit of gravitational wave detection, for example

2016), the European Pulsar Timing Array (Caballero et al.2016;

Desvignes et al.2016), the North American Nanohertz Observatory for Gravitational Waves (Demorest et al.2013; Arzoumanian et al.

2015,2018), and their combination, i.e. the International Pulsar

Timing Array (Lentati et al.2016).

Using Bayesian methodology (e.g. Arzoumanian et al.2015; Lentati et al.2016), DM variations represented by a series of dis- can be analysed for yearly variations, for non-stationary DM events and for the spherically symmetric solar wind term (e.g. Lentati et al.2016). A number of methods have been developed to explore the temporal DM features and their power spectrum. For example, linear and periodic functions and their combinations have been fit- ted to DM time series to obtain the temporal scales and the trends for DM variations (Jones et al.2017). Power spectral analyses of DM time series have been conducted to search for periodic DM modulations (e.g. Keith et al.2013). Structure functions have been C?

2018 The Author(s)

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3394P. F. Wang and J. L. Han

employed to estimate the power for the stochastic, white noise and periodic components of the DM time series, and to verify the Kol- Keith et al.2013; Reardon et al.2016; Lam et al.2016;Jonesetal.

2017). The trajectories of sight lines to pulsars have sometimes

been plotted to help to interpret the trends and annual variations (Keith et al.2013;Jonesetal.2017). In order to understand the origins of DM variations, Lam et al. (2016) carried out a detailed modelling. They attributed the linear DM variations to the persis- tent gradient of the interstellar medium transverse to the lines of sight and/or the parallel motion between the pulsar and observer. Periodic DM variations were attributed to the combined effects of (You et al.2007b;Opheretal.2015) and also by the heliosphere and plasma lens in the interstellar medium (Lam et al.2016). These variations generally have a period of one year, while the periodicity Dupr ´e2006). Stochastic DM variations were generally attributed to interstellar turbulence, perhaps in the form of Kolmogorov turbulence. The long-term DM variations, often referred as the DM trends, have to be identified by decomposing the entire data series. Al- though the linear trends are often fitted to the data, the real DM variations generally exhibit structures much more complicated than a monotonic decrease or increase (e.g. Arzoumanian et al.2015,

2018). When the DM trend cannot be properly identified through

periodic and stochastic DM variation components remain unclear, which makes it difficult to understand their origins. The trends of the complicated DM variations have been analysed by cutting data into discrete pieces, and then fitting each section by a combination of the triangle function and a linear term (e.g. Jones et al.2017). The DM variations caused by the interplanetary medium should be correlated among pulsars (Lam et al.2016), depending on their the interstellar medium should be uncorrelated. Previous efforts prior to this paper. In this paper, we employ the Hilbert-Huang transform (HHT, Huang et al.1998; Huang, Shen & Long1999;Wu&Huang2009) method to decompose the temporal DM variations of pulsars into components from different physical origins. This recently devel- oped signal-processing method provides a new tool to identify the various contributions by decomposing the data series. Because this is the first time that the HHT has been applied in pulsar astronomy, the HHT algorithm is briefly introduced in Section 2. Its applica- tion to the DM time series of 30 pulsars in order to identify the general trends, annual and stochastic components is then presented in Section 3. A discussion of the decomposed components and the conclusions are given in Sections 4 and 5, respectively.

2 THE HILBERT-HUANG TRANSFORM

The HHT was developed to process non-linear and non-stationary signals (Huang et al.1998,1999). It has been used in many ar- eas of research, for example in geophysics and meteorology to decompose the ionospheric scintillation effects from global navi- gation satellite system signals (e.g. Sivavaraprasad, Sree Padmaja & Venkata Ratnam2017) and to retrieve the wind direction from rain-contaminated X-band nautical radar sea-surface images (e.g.

Liu, Huang & Gill2017), in solar physics to analyse sunspot num-bers (e.g. Gao2016), and in astrophysics to analyse gravitational

waves from the late inspiral, merger, and post-merger phases of binary neutron star coalescence (e.g. Kaneyama et al.2016). It has not, however, heretofore been used for any analysis of real pulsar data. The HHT consists of ‘empirical mode decomposition" (EMD) and the well-known Hilbert transform. EMD can decompose any methods. These IMFs are generally in agreement with physical sig- the Hilbert spectrum in order to obtain the energy-frequency-time distribution of the signal. To calculate the EMD of a given signal,x(t), the signal"s local maxima and minima are first identified, and then the envelopes for the two types of extremes are constructed. A mean curve is cal- culated by averaging the two envelopes, which is then subtracted from the signal. This procedure is called ‘sifting", and is performed iteratively many times until the remaining signal meets the fol- lowing criteria: (1) the number of extremes and the number of zero-crossings are equal or differ by one; (2) the mean for the en- velopes is zero. After this iterative process, the finest component of the signal, namely the first intrinsic mode function (IMF1),c 1 (t), is decomposed, which shows very fast variations depending on the sampling cadence.

After this, the first residual,r

1 (t)=x(t)-c 1 (t), is computed and serves as the input signal for re-doing the iterative sifting processes toobtainIMF2,c 2 (t).Then,thesecondresidual,r 2 (t)=r 1 (t)-c 2 (t), is obtained as the input signal forc 3 (t), etc. Such a decomposition procedure is performed iteratively to obtain IMF1 to IMFn, until the residual,r n (t), becomes monotonic or has only one extremum, which is called the ‘trend" term. The original signal thus can be expressed as x(t)= n j=1 c j (t)+r n (t).(3) The upper panels of Fig.1illustrate the EMD of a simulated signal. The EMD has proved to be very useful in geophysics, solar physics and other scientific fields, as noted above. It does, how- ever, have some drawbacks. The most serious problem is the mode- mixing; that is, the fact that a signal of similar scales and fre- quencies appears in different IMFs. The ensemble empirical mode decomposition (EEMD), a noise-assisted data analysis method, wasdevelopedbyWu&Huang(2009) to solve the problem, in which independent white noise realizations are made and added to the original data and then an ensemble of EMDs are performed. The IMFs of an ensemble of EMDs are then averaged to elimi- nate the added white noise, which enables the mixed modes to be separated. For each of the decomposed IMFs, the Hilbert transform can be applied to obtain their instantaneous frequencies and amplitudes. The Hilbert transformation of a signal,x(t), can be written as: y(t)=P x(τ) t-τdτ.(4) Here,Pis the Cauchy principal value of the signal integration. The

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The decomposition of temporal variations of pulsar DMs3395

Figure 1.Demonstration of the Hilbert-Huang transform for a simulated time series.The upper panels show the simulated data and intrinsic mode functions

(IMFs) from the ensemble empirical mode decomposition. The simulated time series is represented by the curve and 300 uniformly sampled data points in

the top panel. The signal is originally composed of four components: a linear term of 0.2-0.4t/90.0 (the same line as in the trend panel), a low-frequency

oscillation of 1.0 sin(2π/20.0t) (the curve in the panel for IMF3), a high-frequency oscillation of 0.3 sin(2π/4.0t) appearing between 30 and 60 s (the curve

in the panel for IMF2), and a normally distributed random noise withσ=0.1. The sampled signal is decomposed by the HHT into four IMFs and the trend,

namely Signal=? 4i=1

IMFi+Trend: here IMF1 is the noise, IMF2 is the high-frequency oscillation, IMF3 is the low-frequency oscillation, and IMF4 is

the residual which has a small amplitude.The bottom panels are the instantaneous frequencies and amplitudes from the Hilbert transform of the four IMFs.

Each point in a given IMF has a corresponding pair of amplitude and frequency. The bottom middle panel is an amplitude-frequency-time plot. The time

and frequency are represented by the horizontal and vertical axes. Each IMF has a dominating frequency interval, as shown by the histogram of instantaneous

frequencies in the bottom left panel, with the peak indicating the most probable frequency. By integrating the instantaneous amplitudes over time, we obtain

the distribution of integrated amplitudes over frequency for each IMF, namely the marginal spectrum, as shown in the bottom right panel.

complex signal then reads z(t)=x(t)+iy(t)=a(t)e iθ(t) ,(5)

The instantaneous frequency is defined as

ω(t)=dθ

dt.(6) The instantaneous frequencies and amplitudes of the IMFs demon- strate the energy-time-frequency distribution for the input sig- nalx(t), as shown in the bottom panels of Fig.1. The EMD or

EEMD and Hilbert spectral analysis are combined to form theHHT. The open-access tools of the HHT are available at the web-

page. 1

3 HHT ANALYSIS OF PULSAR DM

VARIATIONS

DM data from 30 pulsars, termed ‘DMX", denoting the offsets from the formal DM values in the ephemerides, are taken from Arzoumanian et al. (2018). These pulsars were observed as part 1

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3396P. F. Wang and J. L. Han

of the North American Nanohertz Observatory for Gravitational Waves project, and more than 48 DM measurements over more than 3.8 yr are available (see Table1) for each pulsar. We first decompose the DM time series directly by EEMD for the long- term ‘general trend" (not just the trend from the HHT, see below) and the short-term ‘random noise", and then obtain the annual DM variations by folding the trend- and noise-subtracted data.

3.1 EEMD for the general trend and noise

DMXs are very small deviations from the formal DM values of pulsars and are scaled to units of 10 -4 pc cm -3 . Through EEMD, the DMX time series can be decomposed: DMX=? N i=1 IMFi+ trend. Fig.2shows an example of the EEMD for IMFs. It can be seen that each DMX measurement has an uncertainty. The uncertainties for data obtained in the early years are much larger than those in recent years because of the lower sensitivity of the old observing systems with their limited observing bandwidth. Because the uncertainties are not considered in EEMD, we have to ‘clean" the DMX data before the EEMD process. We first omit those data with a very large uncertainty (larger than three times of the median uncertainty), which often deviate from the general significantly (larger than ten times of the median derivation) from their adjacent measurements, owing to the very sharp annual DM variations (e.g. J0023+0923) or an extreme scattering event (e.g. PSR J1713+0747) (Arzoumanian et al.2015; Coles et al.2015; Lentatietal.2016), and so are not the proper tracers for the trend. If these data points are included, the HHT analysis will introduce a number of oscillations with various amplitudes and frequencies to the IMFs around these data points. the trend, as shown for the DM time series for PSR J0613-0200 in Fig.2. The ‘noise" term is IMF1, as seen in the upper panels of Fig.2(a), which represents the finest structure of DM variation with a time-scale depending on the observational cadence. The Hilbert transform of IMF1 gives its instantaneous frequencies and amplitudes, as shown by the plus in the bottom middle panels of Fig.2(a). The median values of the instantaneous amplitudes and random noise component of DM variations, as listed in columns (9) and (10) in Table1. IMF3 and IMF2 are annual and semi-annual variations, which will be analysed in the next section. the rough rate of DM changes, namely dDM/dt, as also listed in column(8)ofTable1.Thegeneral trendterm for DM variations of PSR J0613-0200, however, is not just the EEMD trend term, but is the composite "Trend+IMF5+IMF4", which shows long-term variations with time-scales longer than 1 yr, as indicated by the line in the top panel. In order to demonstrate the effectiveness of the HHT in decom- posing DM variations, we simulate a DM time series by using the trend term of Fig.2(a) plus the white noise. The simulated data in Fig.2(b) have the same cadence as the real observations for PSR J0613-0200. The HHT decomposes the simulated data into five IMFs and a trend term. Each IMF has a dominant frequency range, but is not sharply peaked at any given frequency (e.g. 1 yr -1 ). The marginal spectra have comparable power among these IMFs, rather than a significant power excess for a given IMF. It is also apparent that the phases for the peaks of a given IMF (e.g. IMF2 and IMF3)

vary substantially, rather than being fixed at a given phase overyears. In other words, these IMFs of simulated data do not show

any significant periodic signal, which demonstrates that HHT does not artificially introduce any regular annual signals in the DM time series. As shown in Appendix A, we have also decomposed the noise terms and thegeneral trendsfor 30 pulsars (see Fig.A1). Thegeneral trendsare plotted together with the data in

Fig.3.

3.2 Annual variations from folding DMX data

There is no doubt that annual variations exist for pulsar DMs, as been shown by the clear frequency peaks for IMF2 and IMF3 of PSR J0613-0200 in Fig.2(a). Observations with higher cadence since about the year 2010 (see Fig.A1) exhibit clear semi-annual and annual variations. et al.2016). They refer to annual, semi-annual and other forms, as shown in Fig.2(a). Such an annual variation can be obtained by adding IMF2 and IMF3. The most effective approach for obtaining the annual term is folding the DMX curves with a 1-yr period, after the general trend and noise terms have been subtracted from the original data. Data from recent observations after the epochs indicated by the arrows in Fig.2(a) and Fig.A1are folded into 12 are weighting-averaged according to the measurement uncertainty. Among the 30 pulsars, seven show complicated structures in the trend- and noise-subtracted data, and no annual variations can be identified within short data spans. Another one, PSR J1640+2224, shows a complicated feature (see Fig.A1). The resulting annual curves for the remaining 22 pulsars are displayed in Fig.A1and

Fig.4.

AsshowninFig.3, data deviating from adjacent measure- ments by more than 10 times the median derivation were omit- ted for the trend analysis for five pulsars. Most of these ‘cleaned" data points are around the peaks for annual variations (see Fig. A1for PSRs J0023+0923, J0030+0451 and J1614-2230). These "cleaned" data indicated by asterisks near the curve peak are put back to form the annual variation curves, if they do not differ significantly from the data in other peaks. Although near each peak only one measurement is available, the recurrence of outlying data near the peak of annual variations for PSRs J0023+0923 and J0030+0451 indicate that the outlying data are part of the vari- ations and that the real peak should be much sharper than we see. For PSRs J1614-2230, J1853+1303 and J1923+2515, only one data point is abnormal near the peak, which is not taken back because no recurrence of the abnormal measurements near the other yearly peaks has been observed for confirmation. The data points for the extreme scattering event of PSR J1713+0747 are certainly not included in the folding to obtain the annual curve. To quantitatively describe the annual variations, we fitted vonquotesdbs_dbs46.pdfusesText_46

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