THE NANOGRAV NINE-YEAR DATA SET: MONITORING INTERSTELLAR SCATTERING DELAYS

The interstellar medium (ISM) consists partly of ionized plasma, which interacts with pulsar radio emission. This has several effects on pulsar signals, and will impact the times of arrival (TOAs) of the pulses at Earth. One such effect is dispersion, which occurs when the radio wave propagates through a column of free electrons in the ISM, and is characterized by a frequency-dependent time delay. The delays are proportional to DM × ${\nu }^{-2}$, where the dispersion measure, DM, is the integrated column density of free electrons and ν is the observing frequency. Since the pulsar and the ISM have different relative velocities, the DM of a pulsar is not constant in time. By observing pulsars at two or more separate frequencies, the DM variations can be tracked and corrected for in the data (e.g., Demorest et al. 2013; Keith et al. 2013; Lee et al. 2014).

In contrast to dispersion, which would be present in a completely homogeneous medium, scattering arises when radio waves travel through an inhomogeneous medium. Multi-path scattering manifests itself in several ways, including diffractive intensity scintillations and pulse broadening. Diffractive scintillation effects were first observed in pulsars by Lyne & Rickett (1968), and are the focus of this paper. The basic model usually used to describe diffractive scintillation assumes that the ISM is a thin screen of plasma, located between the pulsar and the observer (Scheuer 1968). As the signal propagates through the screen, inhomogeneities in the plasma introduce phase perturbations that are correlated over a scintillation bandwidth, which is inversely proportional to the scattering timescale. These perturbations are also known as scattering delays. The scintillation pattern, and hence the scattering timescale, of a pulsar can change drastically over time (Hemberger & Stinebring 2008). Similar to the case of DM variations, the relative velocities of the pulsar and the ISM give rise to the time variable scattering delays. The scaling of the scintillation parameters with frequency is often described as that of a Kolmogorov medium, with the scintillation bandwidth, ${\rm{\Delta }}{\nu }_{{\rm{d}}}\propto {\nu }^{4.4}$, and the scintillation timescale, ${\rm{\Delta }}{t}_{{\rm{d}}}\propto {\nu }^{1.2}$ (Cordes et al. 1985), although it has been shown that some sources have scaling indices that deviate from these (e.g., Bhat et al. 2004; Löhmer et al. 2004). These observed deviations may not necessarily be indicative of a non-Kolmogorov spectrum, but may be due to the size of the dominant scattering region transverse to the line of sight (Cordes & Lazio 2001).

In a pulsar timing array (PTA), millisecond pulsars (MSPs) are observed in an effort to detect nanohertz gravitational waves. The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) uses the 100 m Green Bank Telescope (GBT) and the 300 m Arecibo Observatory (AO) to observe ∼40 MSPs every 7−28 days. To succeed in detecting gravitational waves, it is necessary to obtain as high a timing precision as possible, over a long time span. The achievable timing precision of MSPs is continually increasing with longer data sets and improved instrumentation. This makes it ever more important to understand all non-gravitational wave effects, to improve the sensitivity of PTAs to gravitational waves.

As described above, the perturbations caused by interstellar scintillation, as well as the perturbations from DM variations, limit the timing precision for all pulsars. Some pulsars will be more affected than others, however, depending on the properties of the ISM along their line of sight. Most MSPs in PTAs have been chosen partly due to having a low DM, and hence it is expected that scattering will only contribute to a small part of the total timing error for these pulsars. However, accurately correcting for ISM perturbations needs to be done carefully, so as not to introduce additional errors. To correct for DM variations, nearly simultaneous observations with at least two separate frequency bands at widely spaced center frequencies are used. Because of the different frequency scaling of dispersion and interstellar scattering, by only correcting for DM variations systematic errors are introduced into the timing procedure (see the Appendix). One important caveat to this correction procedure is that measurements at different observing frequencies sample different parts of the ISM (Cordes et al. 2015). This significantly affects DM measurements and could also affect interstellar scintillation measurements for some pulsars.

A number of authors have previously addressed the issues of measuring scintillation parameters and mitigating ISM effects. A few examples include Coles et al. (2010) and Keith et al. (2013), who analyzed DM variations and scintillation parameters for pulsars in the Parkes Pulsar Timing Array (PPTA). Their observations were carried out at three different frequencies: a 64 MHz band centered at 685 MHz, a 256 MHz band centered at 1369 MHz, and a 1024 MHz band centered at 3100 MHz (Manchester et al. 2013). Some of the pulsars in the PPTA sample are also observed by NANOGrav and a comparison of the results is included in this work. In a different paper, Gupta et al. (1994) studied the scintillation properties of 8 pulsars over a 16 month period with the Lovell telescope at Jodrell Bank, using a 5 MHz band centered at 408 MHz. They found that the observed fluctuations in the spectra could be explained as refractive modulation of the diffractive scintillation parameters. Bhat et al. (1998) analyzed scintillation parameters for 20 slow pulsars with low DM (<35 pc cm⁻³), using observations at a 9 MHz frequency band centered at 327 MHz at 10−90 epochs spanning ∼100 days. They report large fluctuations in both diffractive scintillation timescale and bandwidth, with variations of a factor of 3−5 for most pulsars in their sample.

In this paper, we analyze the magnitude and variation of interstellar scattering delays in regular NANOGrav timing observations to investigate the effect of interstellar scintillation and its contribution to the total noise budget for this important set of pulsars.

This paper makes use of data from regular NANOGrav timing observations. The data are all included in the latest NANOGrav data release (9 year data set; Arzoumanian et al. 2015), and here we use a sub-set spanning ∼3.7 years for data from GBT and ∼1.7 years for data from AO. The exact MJD range of the data used for each pulsar is shown in Table 1, together with observational properties of the pulsar as well as of the ISM along the line of sight to the pulsar. We have focused on observations carried out at a center frequency of ∼1500 MHz for 20 pulsars at GBT and 19 pulsars at AO. Two of the pulsars (J1713+0747 and B1937+21) are observed with both telescopes.

Table 1.  Scattering Properties of NANOGrav MSPs

Pulsar	period	DM	${S}_{{\rm{1400}}}$	${T}_{{\rm{ref}}}^{{\rm{NE2001}}}$	${\tau }_{{\rm{d}}}^{{\rm{NE2001}}}$	${\bar{\tau }}_{{\rm{d}}}$	${\rm{\Delta }}\bar{{\nu }_{{\rm{d}}}}$	SM	${N}_{{\rm{scint}}}^{{\rm{med}}}$	${N}_{{\rm{m}}}$	${N}_{{\rm{obs}}}$	${\rm{\Delta }}{\tau }_{{\rm{d}}}$	${\sigma }_{\ \mathrm{TOA}}^{{\rm{med}}}$	MJD range
	(ms)	(pc cm−3)	(mJy)	(days)	(ns)	(ns)	(MHz)	(kpc ${{\rm{m}}}^{-20/3}$)				(ns)	(ns)	(days)
J0023+0923	3.05	14.3	0.48	0.28	2.5	6.2 ± 2.1	20.5 ± 6.6	2.9 $\times \quad {10}^{-4}$	7	5	24	6.3	72	55989–56599
J0030+0451	4.87	4.3	1.1	0.029	0.055	...	...	...	...	...	27	...	104	55989–56599
J0340+4130	3.29	49.6	0.31	2.0	49	14.7 ± 5.5	9.1 ± 3.3	2.6 $\times \quad {10}^{-4}$	16	15	28	23.6	638	55972–56586
J0613–0200	3.06	38.8	1.7	1.1	16	11.7 ± 3.9	11.1 ± 4.3	2.2 $\times \quad {10}^{-4}$	12	25	49	20.7	180	55275–56586
J0645+5158	8.85	18.2	0.29	0.45	6.2	...	...	...	...	...	33	...	394	55700–56586
J0931–1902	4.64	41.5	0.42	1.5	24	3.2	50.3	6.4 $\times \quad {10}^{-3}$	4	1	11	...	622	56351–56586
J1012+5307	5.26	9.0	3.2	0.15	1.2	2.5 ± 0.1	66.4 ± 5.6	1.8 $\times \quad {10}^{-4}$	3	4	51	0.5	207	55275–56586
J1024–0719	5.16	6.5	1.5	0.055	0.17	2.8 ± 1.3	46.8 ± 17.8	2.4 $\times \quad {10}^{-4}$	7	5	52	3.8	338	55275–56586
J1455–3330	7.99	13.6	0.65	0.15	0.94	4.0 ± 1.1	69.6 ± 17.5	1.6 $\times \quad {10}^{-4}$	3	4	51	3.1	706	55275–56584
J1600–3053	3.60	52.3	2.2	2.7	93	...	...	...	...	...	50	...	81	55275–56584
J1614–2230	3.15	34.5	0.70	1.3	29	16.2 ± 4.3	9.0 ± 2.6	3.5 $\times \quad {10}^{-4}$	18	42	64	18.1	170	55265–56584
J1640+2224	3.16	18.4	0.70	0.56	5.8	2.6 ± 1.1	56.3 ± 14.6	8.8 $\times \quad {10}^{-3}$	4	9	32	2.8	33	56023–56598
J1643–1224	4.62	62.4	3.8	3.2	90	...	...	...	...	...	51	...	192	55275–56584
J1713+0747	4.57	16.0	8.5	0.42	4.1	7.1 ± 2.4	21.1 ± 8.6	2.3 $\times \quad {10}^{-4}$	8	62	110	11.9	20	55275–56598
J1738+0333	5.85	33.8	0.67	1.2	21	9.4 ± 2.3	16.8 ± 7.7	1.9 $\times \quad {10}^{-4}$	11	12	20	11.1	126	56018–56591
J1741+1351	3.75	24.0	0.37	0.19	0.86	5.0 ± 4.4	17.1 ± 17.3	2.2 $\times \quad {10}^{-4}$	4	10	24	20.7	93	55996–56593
J1744–1134	4.08	3.1	2.8	0.020	0.020	3.8 ± 1.3	42.1 ± 9.1	2.7 $\times \quad {10}^{-4}$	5	12	47	4.1	81	55275–56584
J1747–4036	1.65	152.9	1.0	19	2 300	...	...	...	...	...	25	...	445	55977–56584
J1832–0836	2.72	28.2	0.79	0.97	18	...	...	...	...	...	10	...	169	56367–56584
J1853+1303	4.09	30.6	0.45	0.72	5.3	11.7 ± 4.5	12.7 ± 5.1	1.7 $\times \quad {10}^{-4}$	11	4	24	12.0	148	55989–56598
B1855+09	5.36	13.3	3.8	0.47	4.0	21.3 ± 9.9	5.2 ± 2.4	5.4 $\times \quad {10}^{-4}$	26	26	33	45.3	59	55989–56598
J1903+0327	2.15	297.5	0.61	260	230 000	...	...	...	...	...	21	...	178	55997–56591
J1909–3744	2.95	10.4	1.6	0.18	1.5	4.9 ± 1.8	39.0 ± 14.7	2.6 $\times \quad {10}^{-4}$	5	17	59	7.0	41	55275–56598
J1910+1256	4.98	38.1	0.50	0.96	8.4	57.8 ± 16.6	2.3 ± 0.9	6.3 $\times \quad {10}^{-4}$	63	24	25	83.0	96	55989–56598
J1918–0642	7.65	26.6	1.4	0.78	10	9.7 ± 3.0	14.9 ± 5.1	2.3 $\times \quad {10}^{-4}$	11	32	49	11.8	219	55275–56584
J1923+2515	3.88	18.9	0.22	0.36	1.7	6.1 ± 0.8	21.6 ± 9.5	1.1 $\times \quad {10}^{-4}$	6	4	23	3.6	214	55996–56593
B1937+21	1.56	71.0	13.3	4.7	130	44.3 ± 21.4	2.8 ± 1.3	3.6 $\times \quad {10}^{-4}$	43	11	67	88.5	5	55275–56593
J1944+0907	5.19	24.3	2.6	0.50	2.9	10.1 ± 5.6	10.6 ± 5.1	2.1 $\times \quad {10}^{-4}$	12	16	21	23.2	161	55997–56591
J1949+3106	13.14	164.1	0.12	14	610	...	...	...	...	...	17	...	566	56139–56593
B1953+29	6.13	104.5	0.82	7.3	240	55.3	2.9	3.2 $\times \quad {10}^{-4}$	63	3	21	...	223	56018–56591
J2010–1323	5.22	22.2	0.60	0.57	6.6	19.3 ± 6.4	6.9 ± 2.3	5.1 $\times \quad {10}^{-4}$	21	30	48	26.5	293	55275–56584
J2017+0603	2.90	23.9	0.28	0.53	3.8	8.4 ± 3.4	21.6 ± 9.5	1.4 $\times \quad {10}^{-4}$	8	5	29	8.7	95	55989–56598
J2043+1711	2.38	20.7	0.094	0.41	2.0	1.8	86.3	4.2 $\times \quad {10}^{-3}$	3	1	33	...	68	55997–56591
J2145–0750	16.05	9.0	5.5	0.12	0.53	2.8 ± 0.7	47.8 ± 13.3	1.7 $\times \quad {10}^{-4}$	4	9	48	3.2	214	55275–56584
J2214+3000	3.12	22.6	0.53	0.48	3.2	3.1 ± 3.4	23.0 ± 17.9	1.2 $\times \quad {10}^{-4}$	6	7	25	10.1	160	55989–56598
J2302+4442	5.19	13.8	0.90	0.22	0.84	14.1 ± 2.7	9.9 ± 2.4	3.5 $\times \quad {10}^{-4}$	15	10	26	12.2	637	55972–56586
J2317+1439	3.45	21.9	0.80	0.27	1.9	3.0 ± 1.0	41.8 ± 11.8	1.4 $\times \quad {10}^{-4}$	6	3	19	2.1	81	56100–56599

Note. Flux density values at 1400 MHz (S${}_{{\rm{1400}}}$) are average values from the calibrated timing observations. Scattering delays from the NE2001 model (${\tau }_{{\rm{d}}}^{{\rm{NE2001}}}$) have been scaled to an observing frequency of 1500 MHz, and the refractive timescales (T${}_{{\rm{ref}}}^{{\rm{NE2001}}}$) are calculated from the scaled values of the scintillation bandwidth and timescale estimated within the model. The weighted average of the scattering delay is given as ${\bar{\tau }}_{{\rm{d}}}$ and the weighted average of the measured scintillation bandwidth is given as ${\rm{\Delta }}{\bar{\nu }}_{{\rm{d}}}$, where the error is represented by the standard deviation of the weighted mean in both cases. The maximum variation of the scattering delay is calculated as the maximum measured scattering delay minus the minimum measured scattering delay and is here given as ${\rm{\Delta }}{\tau }_{{\rm{d}}}$. The scattering measure (SM) is calculated through Equation (9), using distances estimated from the NE2001 model. The number of epochs included in ${\bar{\tau }}_{{\rm{d}}}$ is given as ${N}_{{\rm{m}}}$, while the total number of epochs analyzed is given as ${N}_{{\rm{obs}}}$. The number of scintles in each of the ${N}_{{\rm{m}}}$ epochs are calculated through the expression for ${N}_{{\rm{scint}}}$ given in Equation (3) and the median number of scintles is given as ${N}_{{\rm{scint}}}^{{\rm{med}}}$. The median TOA uncertainty (${\sigma }_{{\rm{TOA}}}^{{\rm{med}}}$) is given as the averaged residual error for the combined band, with values from Arzoumanian et al. (2015).

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At GBT, the data are collected with the FPGA-based spectrometer GUPPI (Green Bank Ultimate Pulsar Processing Instrument) using coherent dedispersion techniques. The observations are carried out over a frequency band of 800 MHz centered at 1500 MHz, divided into 1.5625 MHz wide frequency channels. All observations are ∼30 minutes in length and are folded in real time with 15 s subintegrations.

At AO, we are using data collected and coherently dedispersed with the Puerto Rico Ultimate Pulsar Processing Instrument (PUPPI) backend. Here the observations are conducted over a 700 MHz bandwidth centered near 1500 MHz, divided in 1.5625 MHz wide channels. The data are recorded in 1 s subintegrations (or 10 s subintegrations for observations before MJD∼56540) for ∼30 minutes per pulsar and epoch.

At the start of each observation, a polarization calibration scan is performed by injecting a 25 Hz noise diode for both polarizations. Once during each epoch and for each observing frequency, a flux calibrator (B1442+101) is observed. For the analysis in this paper, total intensity profiles have been used, by summing the polarizations of the calibrated data (Arzoumanian et al. 2015).

The GUPPI and PUPPI backends provide substantially larger observation bandwidths compared to previously used backends Astronomical Signal Processor (ASP) and Green Bank Astronomical Signal Processor (GASP), which both had 64 MHz bandwidth capacity (e.g., Demorest et al. 2013). The wider bandwidths not only result in higher timing precision due to a higher signal-to-noise value for the pulsar signal overall, but they also provide a larger number of scintillation maxima and minima over the observed band. The wide-band observations carried out with GUPPI and PUPPI prompted a need to investigate the effect of interstellar scattering delays, and are essential for the analysis in this paper.

We have created and analyzed two-dimensional dynamic spectra of each 1500 MHz observation for each of the NANOGrav pulsars, following a procedure similar to that described in Cordes (1986) unless otherwise stated (e.g., for part of the delay uncertainties as described in Equation (3) below). A dynamic spectrum displays how the intensity of the pulsar signal varies with time, t, and observing frequency, ν. Each point in the spectrum is calculated by

$$\begin{eqnarray}&&S(\nu ,t)=\displaystyle \frac{{P}_{{\rm{on}}}(\nu ,t)-{P}_{{\rm{off}}}(\nu ,t)}{{P}_{{\rm{bandpass}}}(\nu )},\end{eqnarray} \tag{ 1 }$$

where ${P}_{{\rm{bandpass}}}$ is the total power of the observation as a function of observing frequency, and ${P}_{{\rm{on}}}$ and ${P}_{{\rm{off}}}$ are the power in the on- and off-pulse part of the pulse profile, respectively. The on-pulse part is here defined as all bins in the summed pulse profile that have an intensity $\gt 5\%$ of the maximum intensity, after smoothing the original 2048 pulse profile bins down to 64 bins. This is done for each observation of each pulsar. An example of a dynamic spectrum can be found in the top panel of Figure 1, where lighter pixels indicate greater interference between radio waves and each local maxima is known as a scintle. To calculate the sizes of the scintles and hence analyze the interference pattern, we compute a 2D autocorrelation function (ACF). The ACF is then summed over time and frequency separately and a Gaussian function, centered at zero lag, is fitted to the two 1D ACFs to determine the scintillation parameters. The scintillation timescale, ${\rm{\Delta }}{t}_{{\rm{d}}}$, is defined as the half-width at ${e}^{-1}$ of the summed frequency lag and the scintillation bandwidth, ${\rm{\Delta }}{\nu }_{{\rm{d}}}$, is the half-width at half-maximum of the summed time lag. Most of the observations have total integration time $T\lt {\rm{\Delta }}{t}_{{\rm{d}}}$ at our observing frequencies, and hence we cannot calculate values for ${\rm{\Delta }}{t}_{{\rm{d}}}$.

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The scattering delay, ${\tau }_{{\rm{d}}}$, that arises as a result of scintillation can be calculated from

$$\begin{eqnarray}&&2\pi {\rm{\Delta }}{\nu }_{{\rm{d}}}{\tau }_{{\rm{d}}}={C}_{{\rm{1}}}\end{eqnarray} \tag{ 2 }$$

where ${C}_{{\rm{1}}}$ is a constant with value ranging 0.6−1.5 (Lambert & Rickett 1999) depending on the geometry and spectrum of the electron density fluctuations. Here we assume ${C}_{{\rm{1}}}=1$.

The uncertainties of the scattering delay measurements consist partly of a "finite scintle error" and partly of the uncertainty of the least-square fit of a Gaussian to the ACF, which are added in quadrature to get the total uncertainty. The finite scintle error is calculated as

$$\begin{eqnarray}\epsilon & \approx & {\tau }_{{\rm{d}}}{N}_{{\rm{scint}}}^{-1/2}\\ & \approx & {\tau }_{{\rm{d}}}{[(1+{\eta }_{{\rm{t}}}T/{\rm{\Delta }}{t}_{{\rm{d}}})(1+{\eta }_{\nu }B/{\rm{\Delta }}{\nu }_{{\rm{d}}})]}^{-1/2}\end{eqnarray} \tag{ 3 }$$

where ${N}_{{\rm{scint}}}$ is the number of scintles, T and B are total integration time and total bandwidth, respectively, and ${\eta }_{{\rm{t}}}$ and ${\eta }_{\nu }$ are filling factors in the range 0.1–0.3, here set to 0.2 (Cordes & Shannon 2010). Since commonly for our observations $T\ll {\rm{\Delta }}{t}_{{\rm{d}}}$, the first term in Equation (3) is approximately equal to 1 and hence depends only on values related to the bandwidth.¹⁸

A positive consequence of the wide bandwidth observations is the larger numbers of scintles observed compared to observations with narrower bandwidths. However, it also implies large differences in the scintillation bandwidths measured at the lowest part of the band compared with the higher part of the band, which in turn causes problems when creating ACFs. This is only problematic if the collected data span a large range in frequency. Current methods to calculate scintillation parameters were developed for narrower bands, and do not provide a direct solution to the scaling issue. To resolve this, we have developed a method to "stretch" the spectra to a reference frequency based on how the scintillation bandwidth scales with observing frequency.

To investigate the scaling, we divided the wide bandwidth observations into four equally sized sub-bands. The scattering delays inferred from the measured scintillation bandwidths of the four separate bands were then plotted against the center frequency of the bands, and a function of the form ${\tau }_{{\rm{d}}}={\nu }^{-\zeta }$ was fitted to the data to calculate the scaling index, ζ. An example of a scaling index measurement is given in Figure 2. Noting that the number of scintles in each sub-band drastically changes with observing frequency, which could potentially affect this analysis, we also tried dividing the band up into four parts with the same number of scintles per band instead of a set bandwidth. The result of this exercise agreed well with the set bandwidth method, and hence the values reported here are all calculated from 200 MHz sub-bands.

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This method was used to measure scaling indices for 10 of the NANOGrav pulsars. The remaining pulsars all have either scintillation bandwidths that are too wide to get reliable measurements in a 200 MHz band, or scintles that are too narrow to get resolved scintillation bandwidths with the 1.5625 MHz frequency channels. However, in an effort to keep the number of variables to a minimum, we choose to use a scaling index as predicted in a Kolmogorov medium (${\zeta }_{{\rm{stretch}}}$ = 4.4) in the stretching even for the sources with a measured ζ-value. This choice is also based on the observation of possible variable scaling indices between epochs (see Section 5.4), and hence using a measured scaling index from one epoch in the stretching of another epoch may bias the dynamic spectrum analysis. In addition, comparisons show that the consequences of using a fixed ${\zeta }_{{\rm{stretch}}}$ are smaller than the other statistical uncertainties on the scattering delay measurements.

Hence, each dynamic spectrum was stretched using a scaling index ${\zeta }_{{\rm{stretch}}}$ = 4.4, by rescaling the frequency axis and setting the reference frequency, ${\nu }_{{\rm{ref}}}$, to the center of the band. The result is an 800 MHz wide dynamic spectrum with evenly sized scintles that all refer to the same observing frequency. We calculated the ACFs of these stretched spectra and derived from them the scintillation bandwidths that are used in the remainder of this paper. See the bottom panel of Figure 1 for an example of a stretched spectrum.

4.1. Scaling Over the Observed Frequency Band

Using the sub-banding method described in Section 3, we have successfully measured scaling indices for 10 of the 37 NANOGrav MSPs. These values are given in Table 2. For most of the pulsars, the scaling index is lower than that for a Kolmogorov medium ($\zeta =4.4$). In addition to the variation between different pulsars, the scaling index for a particular pulsar may not necessarily be constant in time and ideally also this variation should be monitored. However, it has proven difficult to measure scaling indices at most epochs because scintles are not visible in all of the sub-bands, or because radio frequency interference is severely affecting parts of the band. Only a few of the pulsars have multiple scaling index measurements (for further discussion, see Section 5.4).

Table 2.  Measured Scattering Delay Scaling Indices

Pulsar	ζ	MJD	${N}_{{\rm{scint}}}^{{\rm{subband}}}$
J0613–0200	2.8(2)	55275	11
	1.2(8)	56130	8
J1614–2230	2.6(9)	55269	6
	6.3(4)	55304	9
	2.5(4)	55892	13
	2(2)	56288	13
J1713+0747	1.1(5)	55949	5
	4.1(2)	56299	4
	3.7(5)	56391	3
B1855+09	3.8(3)	56294	10
J1910+1256	2.3(6)	56319	29
J1918–0642	2.4(6)	55463	5
	3(1)	56066	5
B1937+21	1.3(3)	56131	11
	2.7(5)	56403	13
	3(1)	56451	10
	3(1)	56492	24
	1.8(1)	56548	8
	4.9(8)	56593	26
J1944+0907	3(2)	56018	7
J2010–1323	3.9(4)	56095	10
	4.4(1)	56250	10
	1.9(4)	56352	9
	1(1)	56472	9
	3.1(5)	56523	23
J2145–0750	3.1(2)	56195	6

Note. Scattering delay scaling over frequency band, where ${\tau }_{{\rm{d}}}\propto {\nu }^{-\zeta }$. The indices are measured at the given observing epoch, and the numbers in parentheses are the errors on the last given digit. ${N}_{{\rm{scint}}}^{{\rm{subband}}}$ represents the average number of scintles in each subband, calculated through the expression given in Equation (3).

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4.2. Scattering Delay Variation

Interstellar scattering delays (${\bar{\tau }}_{{\rm{d}}}$) for all NANOGrav pulsars are given as the weighted average over all observing epochs in Table 1. For comparison, values for the predicted scattering delay and refractive timescales given by the NE2001 model for free electrons in the ISM (Cordes & Lazio 2002) are listed for all the pulsars. In addition, the table includes the maximum variation measured (${\rm{\Delta }}{\tau }_{{\rm{d}}}={\tau }_{{\rm{d;max}}}-{\tau }_{{\rm{d;min}}}$), together with the median TOA uncertainty resulting from timing. Comparing these values can give insight into the feasibility of correcting the timing residuals for scattering delays for a particular pulsar (see Section 5.5 for more details).

Two typical observations are shown in Figure 3, with the top dynamic spectrum displaying narrow scintles for PSR J1910+1256 with ${\rm{\Delta }}{\nu }_{{\rm{d}}}\approx 3\;{\rm{MHz}}$ and the bottom dynamic spectrum showing wide scintles for PSR J2317+1439 with ${\rm{\Delta }}{\nu }_{{\rm{d}}}\approx 34\;{\rm{MHz}}$. Examples of scattering delay variability can be found in Figure 4, where all NANOGrav pulsars with at least ten ${\tau }_{{\rm{d}}}$ measurements are plotted along with their DM variations.

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Previous work on scintillation properties of pulsars has been carried out at various radio frequencies. To compare the scattering delays in this paper with previously published values, we need to scale all values to a common observing frequency. This introduces large uncertainties, since the comparison will be largely dependent on the chosen scaling law, and as shown in this paper, the measured scaling indices are not only different for different pulsars and hence observing directions, but may also vary with time (as discussed in more detail in Section 5.4). Nevertheless, a comparison with previously published values is listed in Table 3, by scaling the published scattering delays to their value at an observing frequency of 1500 MHz, using a scaling index $\zeta =4.4$.

Table 3.  Previously Published Scintillation Parameters

	This work	NE2001	Previously published values
Pulsar	${\bar{\tau }}_{{\rm{d}}}$	${\tau }_{{\rm{d}}}^{{\rm{NE2001}}}$	${\tau }_{\text{d;1500 MHz}}$	ν	References
	(ns)	(ns)	(ns)	(MHz)
J0030+0451	...	0.055	0.12	435	Nicastro et al. (2001)
J0613–0200	11.7 ± 3.9	16	61.2	1369	Coles et al. (2010)
∥	∥	∥	97.0a	1500a	Keith et al. (2013)
J1024–0719	2.8 ± 1.3	0.17	0.34	685	Coles et al. (2010)
∥	∥	∥	0.59a	1500a	Keith et al. (2013)
J1455–3330	4.0 ± 1.1	0.94	0.51	436	Johnston et al. (1998)
J1600–3053	...	93	1768a	1500a	Keith et al. (2013)
∥	∥	∥	1072	3100	Coles et al. (2010)
J1640+2224	2.6 ± 1.1	5.8	3.3	430	Bogdanov et al. (2002)
J1643–1224	...	90	7234a	1500a	Keith et al. (2013)
∥	∥	∥	2918	3100	Coles et al. (2010)
J1713+0747	7.1 ± 2.4	4.1	1.1	430	Bogdanov et al. (2002)
∥	∥	∥	0.48	436	Johnston et al. (1998)
∥	∥	∥	1.1	685	Coles et al. (2010)
∥	∥	∥	6.6	1500a	Keith et al. (2013)
J1744–1144	3.8 ± 1.3	0.020	0.52	436	Johnston et al. (1998)
∥	∥	∥	1.87	660	Johnston et al. (1998)
∥	∥	∥	0.40	685	Coles et al. (2010)
∥	∥	∥	2.7a	1500a	Keith et al. (2013)
B1855+09	21.3 ± 9.9	4.0	6.5	430	Dewey et al. (1988)
∥	∥	∥	1.1	685	Coles et al. (2010)
∥	∥	∥	13.0	1369	Coles et al. (2010)
∥	∥	∥	28.9a	1500a	Keith et al. (2013)
J1903+0327	...	$2.3\times {10}^{5}$	$9.3\times {10}^{4}$	1400	Champion et al. (2008)
J1909–3744	4.9 ± 1.8	1.5	0.68	685	Coles et al. (2010)
∥	∥	∥	4.3a	1500a	Keith et al. (2013)
B1937+21	44.3 ± 21.4	130	127	320	Cordes et al. (1990)
∥	∥	∥	155	430	Cordes et al. (1990)
∥	∥	∥	48.4	1369	Coles et al. (2010)
∥	∥	∥	127	1400	Cordes et al. (1990)
∥	∥	∥	132a	1500a	Keith et al. (2013)
J2145–0750	2.8 ± 0.7	0.53	0.47	436	Johnston et al. (1998)
∥	∥	∥	0.45	685	Coles et al. (2010)
∥	∥	∥	0.82a	1500a	Keith et al. (2013)
J2317+1439	3.0 ± 1.0	1.9	1.4	436	Johnston et al. (1998)

Note. The published values are reported at the given observing frequency (ν) and here are converted to scattering delays (${\tau }_{\text{d;1500 MHz}}$) at 1500 MHz using a scaling index of $\zeta =4.4$. For reference, scattering delay values from this work are included as ${\bar{\tau }}_{{\rm{d}}}$ and scaled values from the NE2001 are given as ${\tau }_{{\rm{d}}}^{{\rm{NE2001}}}$.

^aOnly an already scaled value is reported in the original publication.

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In some cases, the previously published values show large discrepancies with the values in this work. Coles et al. (2010) and Keith et al. (2013) both list scintillation parameters for MSPs in the PPTA, some of which are also included in the NANOGrav array. The values in Keith et al. have already been scaled to a common observing frequency of 1500 MHz, while Coles et al. list the actual frequency of each observation. Some of the values from Coles et al. were measured at an observing frequency ${\nu }_{{\rm{obs}}}$ = 1369 MHz, and these values agree better with our measurements than the rest of the pulsars in Coles et al., which suggests that the choice of scaling index may be an issue. Another explanation for the discrepancy is the variation of scattering delay over time (see Figure 4), which is also observed by Coles et al. Similarly, Bhat et al. (1998) observed variations of the scintillation bandwidth of a factor of 3−5 when analyzing scintillation parameters of slow low-DM pulsars. In the data presented here, we observe delays with a variation of up to an order of magnitude.

In the cases where the published values are ${\tau }_{{\rm{d}}}$ ≲ 1 ns, the values measured in this work are always larger. This is likely due to the limited bandwidth of the data, which allows only the observations with higher scattering delay values in the distribution to be resolved. Many of the values published before 2001 are very similar to the corresponding value from the NE2001 model. This is expected, since in the creation of the electron density model, as many scattering delay measurements as possible were included (Cordes & Lazio 2002) and hence these early measurements were likely used as input to the model.

5.1. Comparison with the NE2001 Model

The most comprehensive and frequently used model of the free electrons in the Galaxy is the NE2001 model (Cordes & Lazio 2002). In addition to predicting distances to pulsars from their DM values, this model also estimates the scintillation properties from a position and a DM value. The scattering delays predicted for the NANOGrav pulsars by the model are listed in Table 1.

These values have also been plotted against the average measured scattering delays in Figure 5. The horizontal dotted lines show the minimum and maximum measurable delays in the data, while the diagonal line shows equality of the measurement to the model. In general, the measured values are slightly higher than the predictions. It is known that the model predictions have large associated uncertainties, in particular at higher Galactic latitudes, where the numbers of sources with known distances and DM are low.

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For some of the pulsars, no measurement was possible, e.g., due to too narrow or too wide scintles. All but two of these non-detections are predicted to lie close to the detection limits of the data, and hence it is not surprising that no scattering delays have been measured for these pulsars. One exception is PSR J0645+5158, which has ${\tau }_{{\rm{d;NE2001}}}=6.2$ ns at 1500 MHz. Manual inspection of the dynamic spectra obtained for this pulsar shows that in the few observations with scintles strong enough to be detected, radio interference contamination caused errors in the ACF calculation. By eye, we estimate a scintillation bandwidth of ${\nu }_{{\rm{d}}}\approx 100\;{\rm{MHz}}$, which corresponds to ${\tau }_{d}\approx 1.5$ ns. The other exception is PSR J1832–0836, with ${\tau }_{{\rm{d;NE2001}}}=18$ ns at 1500 MHz. This pulsar is a recent addition to the NANOGrav array, and only 10 observations at 1500 MHz are included in the 9 year data release. Even by manually inspecting the dynamic spectra for PSR J1832−0836, no scintles are detected, which is likely due to the low flux density of the signal.

For the pulsars with ${\tau }_{{\rm{d;}}{\rm{NE2001}}}\ll 1$ ns, the measured scattering delay is much larger than the model delay. It is possible that there is an underlying structure of wide scintles that are too wide to detect in this data set, and what is measured here are small modulations on top of the larger structure. Another possibility is that the NE2001 model largely underestimates the amount of scattering in the direction of these pulsars, which are all at high Galactic latitudes where the model is known to have large uncertainties (e.g., Gaensler et al. 2008; Chatterjee et al. 2009). For the cases where the ${\tau }_{{\rm{d;}}{\rm{NE2001}}}$-values are close to 1 ns, we may only be detecting the highest values in the distribution of delays. Since the bandwidth of the observations is limited, the true mean ${\tau }_{{\rm{d}}}$-values may be lower than what we can measure. A similar argument can be applied to the ${\tau }_{{\rm{d;}}{\rm{NE2001}}}$-values close to the maximum delay limit, where the limited frequency resolution would permit the detection of only the lower parts of the delay distributions. In addition, the delays in particular observations may be underestimated for pulsars close to the maximum limit, since a strong scintle with true ${\rm{\Delta }}{\nu }_{{\rm{d}}}$ smaller than the channel bandwidth would still be measured as being one channel wide.

5.2. ISM Variations

The measured values of ${\tau }_{{\rm{d}}}$ are plotted in Figure 4 as a function of observing day for all pulsars with at least 10 measured scattering delay values. To analyze the delay variability we have calculated a modulation index for the scintillation bandwidth for each pulsar, defined as the RMS fluctuations of ${\rm{\Delta }}{\nu }_{{\rm{d}}}$ divided by its average value (Bhat et al. 1999) and given by

$$\begin{eqnarray}&&{m}_{{\rm{b;measured}}}=\displaystyle \frac{1}{\langle {\rm{\Delta }}{\nu }_{{\rm{d}}}\rangle }{\left(\displaystyle \frac{1}{{N}_{{\rm{obs}}}-1}\displaystyle \sum _{i=1}^{{N}_{{\rm{obs}}}}{({\rm{\Delta }}{\nu }_{{\rm{d}},i}-\langle {\rm{\Delta }}{\nu }_{{\rm{d}}}\rangle )}^{2}\right)}^{1/2},\end{eqnarray} \tag{ 4 }$$

where $\langle {\rm{\Delta }}{\nu }_{{\rm{d}}}\rangle$ is the average scintillation bandwidth, ${N}_{{\rm{obs}}}$ is the number of measurements, and ${\rm{\Delta }}{\nu }_{{\rm{d}},i}$ is the scintillation bandwidth at the ith epoch. The measurement error on ${\rm{\Delta }}\nu$ will induce an apparent increase in the modulation indices, that needs to be corrected for. Following Bhat et al. (1999), we calculate corrected modulation indices as

$$\begin{eqnarray}&&{m}_{{\rm{b;corrected}}}^{2}={m}_{{\rm{b;measured}}}^{2}-{m}_{{\rm{b;noise}}}^{2}\end{eqnarray} \tag{ 5 }$$

where the error induced modulation index, ${m}_{{\rm{b;noise}}}$, is calculated as the typical uncertainty in ${\rm{\Delta }}{\nu }_{{\rm{d}}}$. The corrected values of ${m}_{{\rm{b}}}$ are given in the upper panel for each pulsar in Figure 4. For two of the pulsars (J1744–1124 and J1909–3744), ${m}_{{\rm{b;noise}}}\approx {m}_{{\rm{b;measured}}}$. Both of these pulsars have only a small numbers of scintles in each observation, which contributes to the large uncertainty in the scattering delay measurements, as seen in Figure 4, and hence are excluded from further modulation index analyses. For the remaining pulsars, ${m}_{{\rm{b;noise}}}\ll {m}_{{\rm{b;measured}}}$, so the corrected modulation is approximately equal to the measured modulation.

These modulations of the scintillation bandwidths, and hence the scattering delays, could originate from refractive scintillation effects. The theoretical model of the observable modulation effects due to refractive scintillation is highly dependent on the form of the density spectrum, which can be written as

$$\begin{eqnarray}&&{P}_{\delta {n}_{{\rm{e}}}}(\kappa )={C}_{{\rm{n}}}^{2}{\kappa }^{-\beta }\end{eqnarray} \tag{ 6 }$$

for values of the spatial wavenumber, κ, between inner and outer cutoffs in spatial scale sizes of the density irregularities in the ISM (Armstrong et al. 1995). Here ${C}_{{\rm{n}}}^{2}$ represents the scattering strength and β is the density spectral index, which for a Kolmogorov spectrum is equal to 11/3. For $\beta \lt 4$, the corresponding scaling over the observed frequency band is

$$\begin{eqnarray}&&\zeta =\displaystyle \frac{2\beta }{\beta -2}\end{eqnarray} \tag{ 7 }$$

and hence for a Komogorov medium, $\zeta =4.4$ (Cordes & Shannon 2010). The scattering measure is defined as

$$\begin{eqnarray}&&{\rm{SM}}={\displaystyle \int }_{0}^{D}{C}_{n}^{2}{ds}\end{eqnarray} \tag{ 8 }$$

which can be simplified to

$$\begin{eqnarray}&&{\rm{SM}}=292{\left(\displaystyle \frac{{\tau }_{{\rm{d}}}}{D}\right)}^{5/6}{\nu }_{{\rm{GHz}}}^{11/3},\end{eqnarray} \tag{ 9 }$$

with units of kpc ${{\rm{m}}}^{-20/3}$ if the scattering delay, ${\tau }_{{\rm{d}}}$, is given in seconds and the distance, D, is given in kpc (Cordes et al. 1991).

Romani et al. (1986) presented a theory for the refractive effects in a few special cases using power-law forms of density spectra with different spectral indices. In the case of a Kolmogorov spectrum and considering scattering in a thin screen halfway between the observer and the pulsar, the modulation index for the scintillation bandwidth is given by

$$\begin{eqnarray}&&{m}_{{\rm{b;Kolmogorov}}}\approx 0.202{({C}_{{\rm{n}}}^{2})}^{-1/5}{\nu }_{{\rm{obs}}}^{3/5}{D}^{-2/5},\end{eqnarray} \tag{ 10 }$$

where ${\nu }_{{\rm{obs}}}$ is the observing frequency in GHz, D is the distance to the pulsar in kpc, and ${C}_{{\rm{n}}}^{2}$ is the scattering strength in units of 10⁻⁴m${}^{-20/3}$ (Bhat et al. 1999). For a Kolmogorov spectrum,

$$\begin{eqnarray}&&{C}_{{\rm{n}}}^{2}=0.002{\rm{\Delta }}{\nu }_{{\rm{d}}}^{-5/6}{D}^{-11/6}{\nu }_{{\rm{obs}}}^{11/3}{{\rm{m}}}^{-20/3},\end{eqnarray} \tag{ 11 }$$

where ${\rm{\Delta }}{\nu }_{{\rm{d}}}$ is given in MHz, D is given in kpc, and the observing frequency, ${\nu }_{{\rm{obs}}}$, is given in GHz (Rickett 1977; Cordes 1986). Calculating the theoretical modulation indices for the pulsars in Figure 4, using pulsar distances from parallax measurements where available¹⁹ and otherwise as given by the NE2001 model (Cordes & Lazio 2002), results in values ranging $0.15\leqslant {m}_{{\rm{b}}}\leqslant 0.20$. If the density spectrum is not Kolmogorov in shape, but rather has $\beta \gt 11/3$, the fluctuations are expected to be independent of the observing wavelength, but give larger modulation than a Kolmogorov spectrum. For $\beta =4.0$ the theoretical ${m}_{{\rm{b}}}\approx 0.35$ and for $\beta =4.3$ the theoretical ${m}_{{\rm{b}}}\approx 0.57$ (Romani et al. 1986; Bhat et al. 1999).

Comparing the corrected modulation indices with the theoretical ones, all of the measured ${m}_{{\rm{b}}}$-values are larger than those predicted for a Kolmogorov medium. However, this does not necessarily mean that the density spectrum is not Kolmogorov. Instead, if the ISM in these directions is not well described by a thin screen, but rather as an extended Kolmogorov medium, that would explain the larger observed modulation.

We can also approach this discussion from the other direction, by looking at the measured scattering delay frequency scaling. Many of the measured scaling indices in Table 2 are smaller than that expected for a Kolmogorov medium, even within the errors, and similar trends have been shown in previous work by other authors. In an effort to measure pulse scattering broadening of slow pulsars, Bhat et al. (2004) observed a number of pulsars at a minimum of two different frequencies with the Arecibo telescope. Indeed, for most of their sources, they measured a scaling index $\zeta \quad \lt$ 4.4. By using their results together with previously published values, Bhat et al. inferred an empirical frequency scaling index of 3.9 ± 0.2. Following the discussion above, the smaller observed scaling indices agree well with the larger modulation observed in our data.

5.3. Dispersion Measure Variations

5.4. Modulations of the Frequency Scaling Index

5.5. Possibility of Increased Timing Precision

We measure scintillation bandwidths for the NANOGrav pulsars at multiple epochs by creating dynamic spectra of wide-band GUPPI and PUPPI data. The scintillation bandwidths are then converted into scattering delays and analyzed for their variation over time. The delays are found to vary significantly. The large fluctuations of ${\rm{\Delta }}{\nu }_{{\rm{d}}}$ give modulation indices higher than expected for a Kolmogorov medium, suggesting that the density spectrum of the ISM is steeper than a Kolmogorov spectrum, or alternatively that the ISM is not well described by a thin screen in the direction of the sample pulsars. The mean values of the measured scattering delays are often found to be slightly higher than the corresponding values in the NE2001 electron density model. The wide bandwidths of the receivers have allowed for an analysis of the scaling of scattering delays with observing frequency. The scaling indices measured are in most cases smaller than that expected for a Kolmogorov medium and may also vary with time.

The maximum variation of the scattering delays can be compared to the timing precision of each pulsar, to investigate whether the timing of a particular pulsar would benefit from correction for scattering delay variations. For most of the pulsars in the NANOGrav array, ${\sigma }_{{\rm{TOA}}}^{{\rm{med}}}\gg {\rm{\Delta }}{\tau }_{{\rm{d}}}$, implying that the scattering delay variation has only a small effect on the measured TOAs, and a correction analysis would likely not significantly improve the timing precision, as shown by a trial analysis with PSR J1910+1256. However, even if the scattering delays in the 1500 MHz band are small, the DM correction procedure may contribute to scattering contamination of the TOAs from lower frequency bands, increasing the timing error from scattering variations by up to a factor of ∼20 for the NANOGrav data.

With longer time spans and more sensitive instruments, pulsars are being timed with higher and higher precision. Over the next few years, noise budget constraints for high precision timing pulsars will be increasingly important (e.g., Dolch et al. 2014) and better correction for ISM contributions will improve sensitivity to gravitational waves. Implementation of frequency-dependent modeling of the pulse profile in wide-band timing (e.g., Pennucci et al. 2014) may account for part of the scattering delays discussed in this paper. An analysis of simultaneous removal of DM and scattering delays as well as inclusion of wide-band timing techniques should be carried out moving forward. In addition, with the development of a reliable procedure to correct for ISM delays, higher scattering pulsars may be considered as potential PTA pulsars, and would help in increasing the number of pulsars suitable for inclusion in PTAs.

The NANOGrav project receives support from the National Science Foundation (NSF) PIRE program award number 0968296. The National Radio Astronomy Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the NSF (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association. NANOGrav research at UBC is funded by an NSERC Discovery Grant and Discovery Accelerator Supplement and by the Canadian Institute for Advanced Research.

Author contributions. L.L. undertook the majority of the data analysis, software development, and calculations, as well as generated most of the text, figures, and tables. L.L., M.A.M., G.J., J.M.C., D.R.S., S.C., T.D., M.T.L., T.J.W.L., and N.P. all contributed to the ideas and discussion that led to the analysis carried out here. M.A.M., J.M.C. and D.R.S. assisted with guidance throughout the project and provided significant input about the contents of this paper. M.A.M. and G.J. provided some software and ideas that led to improvements in existing software. D.R.S. contributed with calculations leading to the analysis presented in the Appendix. Z.A., J.M.C., T.D., M.T.L., T.J.W.L., M.A.M., D.J.N., and T.T.P. commented on and suggested improvements to the manuscript text. This paper makes use of the NANOGrav nine-year data set (Arzoumanian et al. 2015). Z.A., K.C., P.B.D., T.D., R.D.F., E.F., M.E.G., G.J., M.L.J., M.T.L., L.L., M.A.M., D.J.N., T.T.P., S.M.R., I.H.S., K.S., J.K.S., and W.W.Z. all ran observations and analyzed timing models for this data set. Additional specific contributions to the data set are summarized in Arzoumanian et al. (2015). All authors reviewed the manuscript text and figures prior to the submission of the paper.

Most high-precision timing campaigns currently correct for DM variations, but ignore all contributions from scattering delays in the data. Scattering delays are then absorbed into the DM correction, resulting in systematic errors in the arrival time analysis. This is shown below for the case in which timing measurements have been made at two observing frequencies. Assuming that scattering scales with observing frequency as ${\nu }^{-4.4}$, we can write the observed TOA, $t(\nu )$, as

$$\begin{eqnarray}&&t(\nu )={t}_{\infty }+D{\nu }^{-2}+S{\nu }^{-4.4},\end{eqnarray} \tag{ 12 }$$

where t_∞ is the TOA at infinite frequency, D is the time dependent dispersive delay, and S is the time dependent scattering delay. By observing at two frequencies and ignoring scattering effects, we can solve for the dispersion constant ${D}^{\prime }$ as

$$\begin{eqnarray}&&{D}^{\prime }\equiv \displaystyle \frac{t({\nu }_{1})-t({\nu }_{2})}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}}.\end{eqnarray} \tag{ 13 }$$

However, since scattering is really present in the data, the dispersion constant will be overestimated:

$$\begin{eqnarray}{D}^{\prime } & = & \displaystyle \frac{({t}_{\infty }+D{\nu }_{1}^{-2}+S{\nu }_{1}^{-4.4})-({t}_{\infty }+D{\nu }_{2}^{-2}+S{\nu }_{2}^{-4.4})}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}}\\ & = & D+S\displaystyle \frac{{\nu }_{1}^{-4.4}-{\nu }_{2}^{-4.4}}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}}.\end{eqnarray} \tag{ 14 }$$

This overestimated ${D}^{\prime }$ will then be used when determining ${t}_{\infty }$ from the measured $t(\nu )$. Using $t({\nu }_{1})$ to calculate the systematic error in arrival time at infinite frequency yields

$$\begin{eqnarray}t{{}^{\prime }}_{\infty } & = & t({\nu }_{1})-{D}^{\prime }{\nu }_{1}^{-2}-S{\nu }_{1}^{-4.4}\\ & = & {t}_{\infty }-S{\nu }_{1}^{-2}\displaystyle \frac{{\nu }_{1}^{-4.4}-{\nu }_{2}^{-4.4}}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}},\end{eqnarray} \tag{ 15 }$$

hence the TOA will be overcorrected due to the overestimated DM and will cause an inferred infinite-frequency TOA that arrives too early, with an error of

$$\begin{eqnarray}&&{\epsilon }_{{\rm{TOA}}}=-S{\nu }_{1}^{-2}\displaystyle \frac{{\nu }_{1}^{-4.4}-{\nu }_{2}^{-4.4}}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}}.\end{eqnarray} \tag{ 16 }$$

Finally, the DM correction procedure will increase the scattering variation in the ${\nu }_{1}$-band with a factor of

$$\begin{eqnarray}&&{\epsilon }_{{\rm{scattering}}}=\displaystyle \frac{{\epsilon }_{{\rm{TOA}}}}{S{\nu }_{1}^{-4.4}}={\nu }_{1}^{2.4}\displaystyle \frac{{\nu }_{1}^{-4.4}-{\nu }_{2}^{-4.4}}{{\nu }_{1}^{-2}-{\nu }_{2}^{-2}}\end{eqnarray} \tag{ 17 }$$

compared to the intrinsic scattering variation in that band. For the frequency band considered in this paper, the scattering delay variation at the lowest part of the band (1100 MHz) will therefore increase the timing error from scattering by a factor of ∼3.4 at the reference frequency (1500 MHz), if scattering delays are ignored. For the pairs of frequency bands used in the full NANOGrav data set, the factors are 21.8 for (430, 1500 MHz), 5.7 for (820, 1500 MHz), and 3.5 for (1500, 2100 MHz).