Precision orbital dynamics from interstellar scintillation arcs for PSR J0437-4715
Daniel J. Reardon, William A. Coles, Matthew Bailes, N. D. Ramesh Bhat, Shi Dai, George B. Hobbs, Matthew Kerr, Richard N. Manchester, Stefan Oslowski, Aditya Parthasarathy, Christopher J. Russell, Ryan M. Shannon, Renee Spiewak, Lawrence Toomey, Artem V. Tuntsov, Willem van Straten, Mark A. Walker, Jingbo Wang, Lei Zhang, Xing-Jiang Zhu
DDraft version September 29, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Precision orbital dynamics from interstellar scintillation arcs for PSR J0437 − Daniel J. Reardon,
1, 2
William A. Coles, Matthew Bailes,
1, 2
N. D. Ramesh Bhat, Shi Dai, George B. Hobbs,
5, 2
Matthew Kerr, Richard N. Manchester, Stefan Os(cid:32)lowski, Aditya Parthasarathy,
1, 2
Christopher J. Russell, Ryan M. Shannon,
1, 2
Ren´ee Spiewak,
1, 2
Lawrence Toomey, Artem V. Tuntsov, Willem van Straten, Mark A. Walker, Jingbo Wang, Lei Zhang,
11, 12, 5 and Xing-Jiang Zhu
13, 2 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav) Electrical and Computer Engineering, University of California at San Diego, La Jolla, California, U.S.A. International Centre for Radio Astronomy Research, Curtin University, Bentley, Western Australia 6102, Australia Australia Telescope National Facility, CSIRO Astronomy & Space Science, P.O. Box 76, Epping, NSW 1710, Australia Space Science Division, Naval Research Laboratory, Washington, DC 20375-5352, USA CSIRO Scientific Computing Services, Australian Technology Park, Locked Bag 9013, Alexandria, NSW 1435, Australia Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia Institute for Radio Astronomy & Space Research, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi, Xinjiang 830011, China National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, China University of Chinese Academy of Sciences, Beijing 100049, China School of Physics and Astronomy, Monash University, Victoria 3800, Australia (Received 2020 March 2; Revised 2020 August 25; Accepted 2020 September 26)
Submitted to ApJABSTRACTIntensity scintillations of radio pulsars are known to originate from interference between waves scat-tered by the electron density irregularities of interstellar plasma, often leading to parabolic arcs inthe two-dimensional power spectrum of the recorded dynamic spectrum. The degree of arc curvaturedepends on the distance to the scattering plasma and its transverse velocity with respect to the line-of-sight. We report the observation of annual and orbital variations in the curvature of scintillationarcs over a period of 16 years for the bright millisecond pulsar, PSR J0437 − > D = 89 . ± . D = 124 ± i = 137 . ± . ◦ , and longitude of ascending node, Ω = 206 . ± . ◦ .Using scintillation arcs for precise astrometry and orbital dynamics can be superior to modelling vari-ations in the diffractive scintillation timescale, because the arc curvature is independent of variationsin the level of turbulence of interstellar plasma. This technique can be used in combination with pul-sar timing to determine the full three-dimensional orbital geometries of binary pulsars, and providesparameters essential for testing theories of gravity and constraining neutron star masses. Keywords: pulsars: general, pulsars: individual (PSR J0437 − Corresponding author: Daniel J. [email protected] a r X i v : . [ a s t r o - ph . H E ] S e p Reardon et al. INTRODUCTIONPSR J0437 − i , which is difficult to achieve throughtiming unless the orbit is observed nearly edge-on orhas extremely high precision arrival times (van Stratenet al. 2001) to allow measurement of the relativisticShapiro delay (Shapiro 1964). It can also be obtainedfrom a subtle kinematic effect that depends on theEarth’s orbit and pulsar proper motion (Kopeikin 1995,1996; van Straten et al. 2001), but this is only possi-ble for the most precisely-timed millisecond pulsars, andPSR J0437 − i and Ω is the study of pulsar scintillation,which in contrast to timing is only sensitive to trans-verse motions (e.g. Lyne 1984; Ord et al. 2002; Ransomet al. 2004).Interstellar scintillation originates from spatial fluctu-ations in the electron density of the ionized interstellarmedium (IISM), which have a power-law distribution ofsizes and densities originating from turbulence (Rickett1990). These density fluctuations scatter incident wave-fronts by diffraction and produce an interference patternof intensity variations (Rickett 1969) that varies withfrequency, and with time because of the relative motionsof source, scattering media, and observer. Scintillationis observed in radio observations of compact sources atcentimetre to metre wavelengths and is captured in thedynamic spectrum (see left panel of Figure 1), which wedescribe in Section 2.The apparent quasi-periodic structures in the dynamicspectrum become more ordered in the secondary spec-trum; the two-dimensional power spectrum of the dy-namic spectrum (Figure 1, right panel). This may also be referred to as a delay-Doppler distribution becausethe Fourier conjugate variables on the axes correspondto the differential time delay f ν and differential Dopplershift f t between pairs of interfering waves (Cordes et al.2006). Parabolic arcs in a secondary spectrum were firstidentified by Stinebring et al. (2001), and the origin oftheir shape was explained by Walker et al. (2004).The arcs shown here would be called “forward” arcs be-cause the apex is at the origin and they can be describedby f ν = η ν f t . Their parabolic form can be understoodin terms of the interference of two plane waves scatteredat angles θ and θ . The Doppler shift, f t ∝ θ − θ andthe delay, f ν ∝ θ − θ . If one of θ or θ is near the ori-gin a forward arc arises naturally. In strong scintillation,particularly when the angular scattering is highly asym-metric, one often sees a large number of inverted arcswith their apexes distributed along or near the primaryarc (Brisken et al. 2010). We do not see any evidenceof inverted arcs in our observations, which are in muchweaker scintillation.The Doppler shift f t depends on the effective velocity, V eff of the line-of-sight relative to the medium (whichis a linear combination of the transverse velocities ofthe Earth, pulsar, and IISM). With multiple measure-ments of parabolic arcs at different epochs, the arc cur-vature will show cyclical variations due to the orbitalmotions of the Earth and the pulsar. However earlyanalyses of scintillation arcs have primarily involved soli-tary pulsars without binary motions (e.g. Stinebringet al. 2001), and individual epochs of observations (e.g.Brisken et al. 2010; Bhat et al. 2016). One previous ex-ample of annual and orbital velocity modulations to arccurvatures has been reported, which was for an analysisof PSR J0737 − − − − D = 156 . ± .
25 pc(Reardon et al. 2016). This makes the system anideal candidate for modelling of the transverse motionprobed by scintillation, and allows us to measure prop-erties of the scattering screens with unprecedented pre-cision. Scintillation arcs have previously been observedfor PSR J0437 − odelling scintillation arcs of PSR J0437 − Figure 1.
Dynamic spectrum S ( t, ν ) (left) and corresponding secondary spectrum P ( f t , f λ ) (right) for a long track observationof PSR J0437 − ∼ tions of the brightest arc, Bhat et al. (2016) could onlyestimate the distance to one screen.In this paper we show that measuring the scatter-ing screen distance from individual observations in thisway, with the necessarily restrictive assumptions of astationary IISM, can result in significantly biased mea-surements. Long-term modelling provides a robust wayto measure the screen distance, which is constrained bythe relative amplitudes of the arc curvature modulationdue to the orbits of the pulsar and the Earth. It can alsoprovide precise measurements of the IISM velocity andanisotropy angle, and determine binary orbital parame-ters such as i and Ω. These pulsar parameters are impor-tant for constraining neutron star masses and for testsof general relativity using relativistic binaries, but aredifficult to measure through pulsar timing alone. Mod-elling of the long-term changes to the diffractive scin-tillations (using the characteristic time- and frequency-scales of the dynamic spectrum) can also be used to pre-cisely measure these parameters (e.g. Rickett et al. 2014;Reardon et al. 2019), but the work in this paper is a newapproach that can work well even in the regimes of weakand/or time-varying levels of interstellar turbulence.Our observations make use of the second data releaseof the PPTA (Kerr et al. 2020), and are briefly describedin Section 2. In Section 3 we provide theoretical mo-tivation for the arcs in the secondary spectra, discussanisotropic scattering in the context of our observations, and describe our method for fitting the arc curvature.Section 4 details the model for the effective velocity andarc curvature variations, and the results are then pre-sented in Section 5. We identify curvature variationsfor two arcs corresponding to two discrete scatteringscreens, and we are able to model the long-term modu-lations for both. In the discussion in Section 6 we givesuggestions for candidate structures in the IISM respon-sible for the scattering and predict how our techniqueswill be extended and used for interpreting more sensi-tive observations from new telescopes such as MeerKAT(Bailes et al. 2018) and instruments such as the Parkesultra-wideband receiver (Hobbs et al. 2019). OBSERVATIONS AND DATAOur observations of PSR J0437 − f c ∼
685 MHz and f c ∼
732 MHz respectively), 20-cm( f c ∼ f c ∼ Reardon et al. serving systems are described in Manchester et al. (2013)and of the data processing are in Kerr et al. (2020).PSR J0437 − ∼
10 hours from rise to set forthe purpose of polarisation calibration and instrumentcommissioning. These long observations provide longdynamic spectra and improve the signal-to-noise ratioof any scintillation arcs in the secondary spectra.A dynamic spectrum from one of these long tracksis shown in the left panel of Figure 1, with a well-defined scintillation arc apparent in the secondary spec-trum (right panel of Figure 1). While these longer ob-servations show particularly clear scintillation arcs, wealso detect the primary arc in all available observationsin the 20-cm and 50-cm bands, provided they are nottoo contaminated with radio-frequency interference.2.1.
Computing dynamic and secondary spectra
The dynamic spectra, S ( t, ν ), are computed as part ofthe data processing pipeline that has been developed forthe second data release of the PPTA, which uses psr-flux from the psrchive package (Hotan et al. 2004; vanStraten et al. 2012). This pipeline has already been usedto study the relativistic binary pulsar PSR J1141 − B c ∼ . t sub ∼
30 s in time.Data flagged as RFI are replaced using linear inter-polation. Removing the RFI reduces artifacts in thesecondary spectrum (particularly along the axes) butdoes not affect curvature measurements. For the longtracks where we concatenate multiple dynamic spec-tra, the gaps (during which time a noise diode is ob-served) are filled with the mean flux. Before calculatingthe secondary spectrum we first re-sample the dynamicspectrum uniformly in wavelength rather than frequency(using cubic interpolation onto a grid with wavelengthstep size equal to the difference in the lowest two fre-quency channels), S ( t, λ ), as has been done previously(Fallows et al. 2014). This has the effect of removing thefrequency-dependence of the arc curvature and thereforesharpens the features of the arc to improve the curvaturemeasurements.We also apply a Hamming window function to theouter 10% of each dynamic spectrum to reduce side-lobe response that adds power along the secondary spec-trum axes. We then subtract the mean flux before com-puting the secondary spectrum, P ( f t , f λ ), which is itstwo-dimensional Fourier transform. This is computed by first pre-whitening the dynamic spectrum (using thefirst-difference method) before it is Fourier transformedwith zero padding. We take the squared magnitude ofthe transform, and shift and crop it to show only valuesfor f λ >
0. The spectrum is then “post-darkened” (thereverse process of pre-whitening Coles et al. 2011) andgiven in units of dB. So in summary we have P ( f t , f λ ) = 10 log ( | ˜ S ( t, λ ) | ) , (1)where the tilde denotes the two-dimensional Fouriertransform, f t and f λ are the Fourier conjugates of t and λ respectively, and ˜ S ( t, λ ) is the mean-subtracted andwindowed dynamic spectrum with wavelength. A sec-ondary spectrum computed in this way is shown in theright panel of Figure 1.2.2. The normalized secondary spectrum
We introduce a novel way to search for forward arcs insecondary spectra and analyse their power distributions,by re-sampling the spectrum in Doppler to transformany parabolas into vertical lines. This is done by ad-justing the sampling for each row of the spectrum, withlinear interpolation, such that the number of samplesdecreases with f λ . We refer to this a “normalized” sec-ondary spectrum, P ( f t /f arc , f λ ), with respect to somereference arc curvature η , when the transformation isdone such that the units on the x-axis are f tn = f t /f arc ,the fractional distance from the f t = 0 axis to the arcat f arc , for a given f λ . In this way, any arc at η will be-come a vertical line of power at the normalized f tn = 1,and for example, a second vertical line of power at nor-malized f tn = β would correspond to a second arc withcurvature η β = η/β . The normalized secondary spec-trum for the data shown in Figure 1, is given in the leftpanel of Figure 2.Taking cuts across P ( f tn , f λ ) along the f λ axis, showsthat the shape of the power distribution is approxi-mately constant with f λ , however the amplitude decayssteeply as ∼ f − / λ (see Appendix B). This constant pro-file shape means that P ( f tn , f λ ) can be averaged over f λ (with appropriate weighting) to increase the signal-to-noise ratio of the power distribution across and insidethe arc. This Doppler profile D t ( f tn ) is shown in thecenter panel of Figure 2. It is useful for analysing theanisotropy of the scattering and for fitting arc curva-tures. The Delay profile D λ ( f λ ) is obtained by averag-ing over Doppler. In weak scintillation the Delay profileis power-law with an exponent of α/ − α isthe spectral exponent of the turbulence (see AppendixB). Accordingly we display the Delay profile in the rightpanel of Figure 2 scaled by f / λ and plotted vs f / λ .The resulting plot is directly proportional to the phase odelling scintillation arcs of PSR J0437 − Figure 2.
The normalized secondary spectrum P ( f t /f arc , f λ ) (left) of the observations shown in Figure 1. The normalizing f arc is found by a best fit for the curvature of the primary arc. We refer to the marginal distributions as the Doppler profile(center) and the Delay profile (right). The Doppler profile D t ( f t /f arc ) shows how the power decays inside the arc, which isrelated to the strength of scintillation and the anisotropy. The secondary arc can be seen in these spectra, at normalised f t , f tn ∼ . f / λ D λ ( f / λ ), follows the phase spectrum. Theorange curve on the marginal distributions shows the weak scintillation approximation for an isotropic image from Kolmogorovturbulence (Equation B5). spectrum in weak scintillation, if Kolmogorov the spec-tral exponent would be -11/3. INTERPRETING AND FITTING THESECONDARY SPECTRAIn this Section we give a brief overview of scintillationarcs, with a focus on the curvature parameter and adiscussion relevant to the expected Doppler profiles ofthe secondary spectra. Detailed reviews of the theoryof scintillation arcs can be found in Walker et al. (2004)and Cordes et al. (2006).A power-law distribution of density irregularities inthe IISM scatters incident radiation, by means of diffrac-tion, into a spectrum of angles relative to the direct line-of-sight to the source. The interference of waves arrivingat the observatory from two small angles in this spec-trum, (cid:126)θ and (cid:126)θ , produces a single frequency-dependentinterference fringe pattern, which is observed in time andfrequency as a sinusoid in the dynamic spectrum (Cordeset al. 2006). For a geometrically thin (in the radial di-rection) screen at some fractional position s along theline of sight from the source (where s = 0 is at the sourceand s = 1 is at observatory), the axes of the wavelength-resampled secondary spectrum P ( f t , f λ ) are related tothese scattering angles with f λ = D (1 − s )2 sλ c ( θ − θ ) (2) f t = 1 sλ c (cid:126)V eff · ( (cid:126)θ − (cid:126)θ ) , (3)where D is the distance to the source from the obser-vatory, (cid:126)V eff is the effective velocity of the line-of-sight through the screen (Equation 5), λ c is the central wave-length of the observation, and c is the speed of light. TheFourier variable conjugate to λ , f λ = cτ del /λ c , where c is the speed of light, τ del = f ν is the differential geomet-ric time delay between the paths taken to arrive fromthe two angles, and f t is their differential Doppler shift.Each Fourier component in the secondary spectrum cor-responds to a particular sinusoidal fringe pattern andthus to the summation of all pairs of components of theangular spectrum with the same values of f λ and f t .The form of the secondary spectra depends stronglyon the strength of scintillation. This is defined by thephase structure function D φ ( s ) = ( s/s ) α = (cid:104) ( φ ( r ) − φ ( r + s )) (cid:105) . The intensity variance in weak scintillation m b is the accepted measure of the strength of scintilla-tion. For an isotropic Kolmogorov spectrum (Equation 3of Coles et al. 2010) m b = 0 . D φ ( r F ) where r F is theFresnel scale. This is most easily measured by the frac-tional bandwidth of the diffractive scintillations, whichis given by ∆ ν d /ν c = ( s /r F ) (Rickett 1990). For aKolmogorov structure function, D φ ( r F ) = ( r F /s ) / ,and using this we can estimate the Born variance as m b = 0 .
773 ( ν c / ∆ ν d ) / . In this way we find m b ≈ m b ≈
22 at 40-cm. Thesevalues are appropriate for simulating scintillation withour observed ∆ ν d /ν c under the assumption of isotropicscattering. However if the scattering is anisotropic then∆ ν d /ν c will be reduced (Rickett et al. 2014). To ac-count for this in simulations, we iteratively decrease m b until the simulated ∆ ν d /ν c is within 5% of the isotropicsimulation. Reardon et al.
Figure 3.
Wavelength-resampled dynamic spectra (left panels), secondary spectra (middle panels), and Doppler profiles (rightpanels, with x-axis normalised with respect to the primary arc) for a typical 20-cm observation (on MJD 55832; top) and 40-cmobservation (on MJD 56319; bottom). The 20-cm observation was included in our dataset of curvature measurements for thesecondary arc, and the measured value is indicated by the black dashed lines. The orange curve on the 20-cm Doppler profileshows the expected Doppler profile for an isotropic image in the weak scintillation regime (Equation B5).
We use the technique described in (Coles et al. 2010)to simulate dynamic spectra with the same fractionalbandwidth of our observations. In this we implicitlyassume that the scintillation is dominated by the thinscreen which provides the primary arc. We have re-produced this technique in Python and have made itpublicly available .In weak scintillation, where m b <
1, the parabolicarc in the secondary spectrum can be interpreted as theinterference between the unscattered image of the pulsarand the surrounding scattered image. In this case wehave Equations 2 and 3 with θ = 0 and θ = θ , where θ is the angular separation between a component of thescattered image and the line-of-sight. The arc curvaturethen comes from the quadratic relationship between f λ and f t through their dependence on θ . We introducethe curvature parameter, η , such that f λ = ηf t , andthis is then given by η = Ds (1 − s )2 V cos ψ (4)where ψ is the angle between V eff and the position vectoralong the anisotropy in the scattered image and comesfrom the dot product in Equation 3. For an isotropically- From https://github.com/danielreardon/scintools. See also Ap-pendix A distributed image, such as a ring or halo, the equation isthe same, but with cos ψ = 1 describing the outer-edgeof the arc (Cordes et al. 2006). We give the equationfor Doppler profiles in weak scintillation in Appendix B,Equation B5.In the case of strong scintillation (such as in 40/50-cmobservations of PSR J0437 − ψ with respect to the position angle of (cid:126)V eff . In the case of bright, discrete scattered images be-yond the rms scattering angle, these arclets can be re-solved individually (as in Brisken et al. 2010). However,more continuous anisotropy at high scattering angles isexpected to produce a forest of these arclets that can to-gether appear as a broadened scintillation arc (see exam-ples in Cordes et al. 2006). In this regime, the curvaturedescribed by Equation 4 is the line through the centresof the arclets, rather than the outer edge of power.Arcs may also show asymmetries in their total powerabout f t = 0 (Doppler asymmetries) formed by asym-metric scattering about the line-of-sight in the direction odelling scintillation arcs of PSR J0437 − (cid:126)V eff (e.g. Cordes et al. 2006). This can be caused by aphase (and electron density) gradient across the line-of-sight, or physically asymmetric structures in the scatter-ing medium. We do not observe any clearly asymmetricarcs at any epoch or orientation of the (cid:126)V eff vector. Thismay indicate that the scattered image is symmetric, suchas a linear structure (for the case of highly anisotropicscattering), an ellipse (for a moderately anisotropic im-age), or a circularly-symmetric halo (for isotropic scat-tering). 3.1. Fitting arc curvature
To measure the arc curvature and estimate its uncer-tainty from the secondary spectrum of each observation,we use the Doppler profiles described in Section 2.2. Were-scale the x-axis of the normalized secondary spectruminto physical units of the curvature, η . The transforma-tion to this Doppler profile is computed only once foreach observation and displays the mean power as a func-tion of arc curvature P ( η ), so curvature measurementsare found simply by detecting peaks in this distribution.The P ( η ) curve for our long 20-cm observation fromFigures 1 and 2 is shown in Figure 4. The primaryand secondary arcs are clearly seen as peaks in P ( η ).For most of our observations, the signal-to-noise of thearc peak is not as high as in this figure, because mostobservations have a 64 min duration. Some more typ-ical examples from 20-cm and 40-cm observations areshown in Figure 3. The curvature measurements forpeaks in this distribution are found by first smoothing the data, and then fitting a parabola to the un-smootheddata in a region from − − . σ η isdetermined from the noise level in the secondary spec-trum, far from the power in the arc σ s (as in Bhat et al.2016). We construct a standard error confidence re-gion around the curvature measurement, correspondingto the change in η required for the power to drop by σ s , from the peak in the smoothed data. The curva-ture measurement and the uncertainty regions for theprimary and secondary arcs are shown in Figure 4.Using this method, the secondary spectrum must becropped (or truncated) at a τ del (or its corresponding Using a first-order Savitzky-Golay filter.
Figure 4.
Result of arc curvature fitting to the secondaryspectrum Doppler profile shown in Figure 2. Here the posi-tive and negative sides of the Doppler profile have been aver-aged to increase the signal-to-noise ratio. The measurementand uncertainty regions for the primary and secondary arcsare shown with the green and purple bands respectively. Theorange line is the smoothed data that were used to select thefitting regions, and the inverted parabolas in black show theresult of these fits. The arc curvature axis is displayed on alogarithmic scale for visualization only. f λ after wavelength re-sampling) value that is just be-yond where the arc power becomes lower than the noise.This is difficult because the power in the arcs decayswith increasing τ del . We chose to crop each secondaryspectrum at a fixed maximum time delay, τ del , max be-yond which most observations show no evidence of theprimary arc. This delay depends on the observing fre-quency for the observation, and we have therefore de-fined τ del , max = 0 . µs × (1400 MHz /f ) for the primaryarc and τ del , max = 0 . µs × (1400 MHz /f ) for fitting thesecondary. In general this is a conservative figure thatwill include noise in our data and our estimated uncer-tainties for the arc curvature are expected to be slightlyoverestimated as a result.Figure 4 also shows that there are potentially lowsignal-to-noise arcs at η ≈
13 m − mHz − , and η ≈
200 m − mHz − . We do not analyse these arcs in de-tail because we cannot reliably measure their curvaturesfor a significant number of epochs. They appeared atmultiple epochs only in the long track observations, andalways with a low signal-to-noise ratio. MODELLING ARC CURVATURE VARIATIONSThe curvature of arcs in the secondary spectrum de-pends on the distance to the scattering region s (as-sumed to be a thin screen), and the velocity of the line- Reardon et al. of-sight with respect to the medium, (cid:126)V eff , as given inEquation 4. We therefore expect the curvature to betime-dependent as (cid:126)V eff changes because of the changingtransverse components of the Earth’s velocity ( (cid:126)V E ) andthe pulsar’s orbital velocity ( (cid:126)V p ). The effective velocityis a linear combination of these velocities and the ve-locity of the medium itself ( (cid:126)V IISM ) (Cordes & Rickett1998), (cid:126)V eff = (1 − s )( (cid:126)V p + (cid:126)V µ ) + s(cid:126)V E − (cid:126)V IISM , (5)where (cid:126)V µ is the constant transverse velocity of the pulsarsystem (corresponding to its proper motion).We model the variations in η ( t ) with (cid:126)V eff componentsin right ascension ( α ) and declination ( δ ) v eff ,α =(1 − s )( v p ,α + v µ,α ) + sv E ,α − v IISM ,α v eff ,δ =(1 − s )( v p ,δ + v µ,δ ) + sv E ,δ − v IISM ,δ (cid:126)V eff = (cid:113) v ,α + v ,δ (6)The distance and proper motion for PSR J0437 − v µ,α = 90 . ± .
15 km s − and v µ,δ = − . ± .
09 km s − (Reardon et al. 2016). The precise timingmodel also allows us to derive the mean orbital velocity V = 2 πxc sin iP b (cid:112) (1 − e ) = 18 . ± .
015 km s − (7)from the projected semi-major axis x (in light-seconds),orbital period P b , eccentricity e , and the inclination an-gle i . The orbital transverse velocity in components par-allel and perpendicular to the line of nodes ( v p , (cid:107) , and v p , ⊥ respectively) is then defined in terms of the trueorbital anomaly θv p , (cid:107) = − V ( e sin ω + sin ( θ + ω )) v p , ⊥ = V cos i ( e cos ω + cos ( θ + ω )) , (8)where ω is the longitude of periastron. These compo-nents are rotated into right ascension α and declination δ with the longitude of the ascending node Ω, definedEast of North. This is the most uncertain parameterin the pulsar timing model, with Ω = 207 . ± . ◦ , be-cause it is measured through a subtle kinematic effectcaused by the combination of the Earth’s orbital mo-tion and pulsar’s transverse velocity, which changes theprojection of the orbit (Kopeikin 1995).Remarkably, by modelling the annual and orbitalmodulation of diffractive scintillations (e.g. Rickett et al.2014; Reardon et al. 2019) or arc curvature, we are ableto measure Ω and other parameters (such as i ) often with higher precision than through pulsar timing, be-cause these parameters have a strong influence on thetransverse velocity variations. From Equations 6 and 8we see that the only parameters required for modellingthe arc curvatures for PSR J0437 − s , v IISM ,α ,and v IISM ,δ , although we also fit for Ω and i , as consis-tency checks for our model. Our measurements of theseparameters therefore demonstrate the precision and ac-curacy achievable from the scintillation arcs, which maybe useful for other systems that do not have such precisetiming.The models are fitted to the data using the em-cee (Foreman-Mackey et al. 2013) Markov chain MonteCarlo algorithm within the lmfit python package(Newville et al. 2014). This allows us to probe the fullposterior probability distribution of our model, deriverobust measurement of the parameter uncertainty, andvisualize any parameter correlations. RESULTSWe have measured the curvature of scintillation arcsin a set of observations across two observing bands forPSR J0437 − t obs , and observing bandwidth B . In observations withthe highest signal-to-noise ratio, we see a fainter, sec-ondary arc at a lower curvature.The time series of η for each measurement of the pri-mary scintillation arc is shown in Figure 5. There is aclear annual modulation to the curvatures, as well as a ∼ . χ values for the models of each of these arcs, and wedescribe these models in detail in the following sections.5.1. Testing for anisotropy
Extreme anisotropy of the scattered image cannot beassumed a priori. Axial ratios of order 2 have been ob-served in the two relativistic binaries whose scatteringgeometry and three-dimensional orbits have been com-pletely solved (Rickett et al. 2014; Reardon et al. 2019).However the solitary source PSR B0834+06, which hasalso been solved with the help of VLBI scintillations(Brisken et al. 2010), shows extremely anisotropic scat-tering. For this work we fit models for both isotropicand anisotropic scattering to both of our measured arcs. odelling scintillation arcs of PSR J0437 − Figure 5.
Arc curvature measurements for the primary scintillation arc (top panel) with the best-fit model shown as the orangeline. The annual cycle component of this model was subtracted from the data and model to show the orbital modulation with ∼ . In addition to a goodness-of-fit assessment, we use ex-isting theory for the expected power distribution in thesecondary spectra to give us some additional insight intothe validity of these models. Fortunately we have both20-cm and 40-cm observations. The latter have a scin-tillation bandwidth of ∆ ν d ∼
25 MHz, which translatesto ∼
400 MHz at 20-cm, following a ∆ ν d ∝ f − relation-ship. While the 40-cm observations are in the strongscintillation regime, the 20-cm observations are in thetransition between weak and strong. For this transitionscattering the theory is not well-developed, but we cansimulate the scattering using the techniques discussedby Coles et al. (2010). The effect of anisotropy on weakscintillation is best illustrated in the Doppler profiles.Examples of such simulated profiles, for a strength ofscintillation that matches our observations in the 20-cm band, are shown in Figure 6. The full secondaryspectrum corresponding to these profiles, as well as theanalytical profile for the case of weak scintillation, aregiven in Appendix B.The Doppler profile in Figure 2 (middle panel) showsa sharp outer edge and a decay in power inside the arcof ∼ ∼ ∼
25 dB at A r ∼
5. Theobserved level of power inside the arc is actually slightlyhigher than that expected from a single isotropic scat-tering screen, which suggests that there are potentiallycontributions from other screens. One screen inside thisarc is resolved in a few epochs (Section 5.3), and theremay be more unresolved screens.For the near-weak scintillation at 20-cm, the arcs re-main sharp for moderate axial ratios, but at A r (cid:38) A r (cid:46) Reardon et al.
Table 1.
Parameters from models of the curvature variations in the primary and secondary arcs for PSR J0437 − ◦ and Ω = 207 . ◦ (Reardon et al. 2016). For the secondary arc, we also present models with theinclination angle fixed at this timing value, which significantly improves the precision for s . The goodness of fit is quantifiedwith the χ value for each model. Primary SecondaryIsotropic Anisotropic Isotropic AnisotropicFitted i Fixed i Fitted i Fixed is . . . . . . i ( ◦ ) 136 . . · · · · · · Ω ( ◦ , N → E) 205 . . v IISM ,α (km s − ) − . · · · − − · · · · · · v IISM ,δ (km s − ) 31 . · · · · · · · · · ξ ( ◦ , N → E) · · · . · · · · · · v IISM ,ξ (km s − ) · · · − . · · · · · · − − χ
162 163
If there is any persistent anisotropy, then the cur-vature of the arcs (measured as the peak power) willbe modulated by the 1 / cos ψ term in Equation 4 asthe velocity vector moves with respect to the majoraxis of the anisotropy. This modulation is not exclu-sive to extreme anisotropy, as demonstrated in the sim-ulations with A r = 3 . ψ in the right panel of Figure 6. The transverse velocityfor PSR J0437 − ◦ on the sky for the primary arc, because itis dominated by the pulsar’s proper motion. This is notsufficient to estimate the direction of the anisotropy ac-curately, but the observed sharpness of the arcs impliesthat the major axis of any moderate anisotropy must bealigned roughly with the velocity vector.5.2. Primary arc
The primary arc appears in all observations in the20-cm and 40/50-cm bands that are not too contami-nated with RFI, giving us 2645 unique measurements of η . The maximum likelihood model for the variations in η is shown in Figure 5.This model gives a precise measurement of the inclina-tion angle, i = 137 . ± . ◦ , which differs from the timingsolution of i = 137 . ± . ◦ (Reardon et al. 2016) by ∼ . σ . Similarly, our measurement of the longitude ofascending node Ω = 206 . ± . ◦ , is within ∼ σ of thetiming measurement Ω = 207 . ± . ◦ , and actually sur-passes its precision despite this being one of the mostprecisely timed millisecond pulsars. The precision (andpotential accuracy) of these measurements is impressivefor scintillation studies, which can often be complicatedby changes to properties of the scattering with time. The IISM velocity and anisotropy in the direction ofPSR J0437 − ∼
16 years ofour observations, meaning that the scattering geometryand kinematics of the screen are stable over a spatialscale of hundreds of AU. Since the proper motion forthis pulsar is known to high precision and included inour model, the measured constant components of thevelocity are only due to the velocity of the IISM, withmagnitude | V IISM | (cid:38)
32 km s − . We know very littleabout the IISM features causing scintillation. This ve-locity is high compared with the expected thermal orAlfv´en velocity of the IISM (Goldreich & Sridhar 1995), ∼
10 kms − , but an entire cloud or outflow could bemoving at this rate without any internal disturbance.This best-fit model is of anisotropic scattering, inwhich the scattered image is elongated on the sky withaxial ratio A r (cid:38)
2. In this model we fit for the orienta-tion of the image on the sky, ξ (defined East of North)and then derive for each observation the angle, ψ , be-tween the image and the effective velocity vector. Wealso fit for the component of the IISM velocity alongthe image, but not the perpendicular component be-cause the data are completely insensitive to any motionperpendicular to the image. As discussed in the pre-vious Section, any anisotropy must be roughly alignedwith the velocity to produce arcs that are sharp atall epochs. We find v IISM ,ξ = − . ± . − and ξ = 134 . ± . ◦ , which is indeed close to direction ofproper motion.The measured parameters from the isotropic modelare also shown in Table 1, and demonstrate that theorbital parameters i and Ω are nearly independent ofthe choice of scattering model (isotropic or anisotropic).The distance to the screen s is also nearly model- odelling scintillation arcs of PSR J0437 − Figure 6.
Simulated Doppler profiles for secondary spectra with m b selected to match the observed fractional scintillationbandwidth for 20-cm observations of PSR J0437 − A r = 3 . independent because it is constrained by the relative am-plitudes of the annual and orbital modulations. There-fore this technique can be a powerful tool for solving pul-sar orbits and accurately determining screen distances.5.3. Secondary and additional arcs
We measure the secondary arc using 165 of the high-est signal-to-noise ratio observations from the PDFB4signal processing system. As with the primary arc, wehave fitted both isotropic and anisotropic models, withmeasured parameters given in Table 1. In this casethe isotropic model provides a slightly better fit to thedata. The orbital and annual components of the veloc-ity model for this secondary screen are shown in Figure7. Using this secondary screen alone, we are able to mea-sure the inclination angle i = 138 ± ◦ and longitude ofascending node Ω = 211 ± ◦ for the isotropic scatter-ing model (with i = 137 ± ◦ and Ω = 214 ± ◦ for theanisotropic model). By fixing i at the superior measure-ment from pulsar timing, we determine the fractionalscreen distance to be s = 0 . ± . | V IISM | ∼
50 km s − .In approximately half of the long track observa-tions we see evidence for additional arcs. The twomost prominent appear in Figure 4 at curvatures η ≈
13 m − mHz − , and η ≈
190 m − mHz − , correspondingto fractional screen distances of approximately s = 0 . s = 0 . DISCUSSION6.1.
Advantages of scintillation arcs
The analysis of variations in the timescale of intensityscintillations for a binary pulsar was originally suggestedand attempted by Lyne (1984), but it did not providethe desired accuracy because the plasma turbulence wasinhomogeneous over the orbit of the first pulsars tested.The method had more success with relativistic binariesbecause their compact orbits appear entirely within theregion of scattering on the sky, meaning the scintilla-tion timescale is primarily controlled by the velocity ofthe line of sight. This was first demonstrated by Ordet al. (2002) with PSR J1141 − − − − Reardon et al.
Figure 7.
Orbital (left panel) and annual (right panel) variations in arc curvature measurements for the secondary arc (as inFigure 5 for the primary arc). The isotropic model is shown as an orange line. independent of the strength of scintillation, where themain effect of inhomogeneities appears. Thus includingthe orbital motion of the Earth can be done with preci-sion comparable with fitting the timescale over a binaryorbit.Scintillation arcs are particularly useful in weak scin-tillation where they appear sharper than in strong scin-tillation. In weak scintillation both the scintillationbandwidth and timescale increase and are typically com-parable with the observing bandwidth and the observ-ing time respectively. Thus there may be only a few“scintles” in a dynamic spectrum, which makes any mea-surement of a correlation function difficult. Howeverarcs originate from a finer-scale pattern caused by muchhigher scattering angles. Therefore, there are manymore degrees of freedom in a secondary spectrum andthe accuracy is correspondingly increased.From our observations, we see that the apparent prop-erties of the IISM sampled by our line-of-sight (velocity,distance, and anisotropy) change slowly enough for thecurvature to remain stable over many years, which givesus clean annual variations for precise transverse velocitymodelling.6.2.
Screen distances and IISM velocity
We have measured | V IISM | (cid:38)
32 km s − for the pri-mary screen and | V IISM | ∼
50 km s − for the secondary.We know so little about the origin for this plasma thatit is difficult to say whether this velocity is unusual. Itwould be large compared with the thermal or Alfv´enspeed of the interstellar plasma (Goldreich & Sridhar1995), but an entire cloud could be moving with thisvelocity without causing any internal disturbance. Any-thing fast moving would cause shocks, which we may bepreferentially seeing in the data since the density andlevel of turbulence increases, which then results in morescattering. Taking these velocities into account, we were ableto make robust screen distance measurements of s =0 . ± .
002 and s = 0 . ± .
016 for the primary andsecondary arcs respectively. Using the precise measure-ment of the distance to PSR J0437 − D = 156 . ± .
25 pc, the absolutedistances to these screens are D = 89 . ± . D = 124 ± − ∼
33 pc. This similarity is rather improbable,but with only two screen measurements we cannot deter-mine if this is coincidence or suggestive of an associationbetween these two screens. Future high signal-to-noiseobservations could permit measurements of more arcsand clarify the relative velocities of the population ofscreens. 6.3.
Object candidates
The structures in the IISM that cause scintillation arepoorly understood because they are difficult to study.Compact, turbulent, and over-dense regions of electrondensity in the IISM are known to have a high scatteringefficiency that can dominate the scattering of the entireline-of-sight (Coles et al. 2015), meaning that the scin-tillation can often be described by a single thin screenscattering model. However the origin of such compactregions, including extreme scattering events (ESEs; e.g. odelling scintillation arcs of PSR J0437 − − ∼ −
120 pc (e.g. Spangler 2009). Our updateddistance for the primary screen is significantly less thanthe measurement of Bhat et al. (2016), but either ofthe screens (at 89 . ± . ± Future work
We have identified additional faint arcs; one at a lowercurvature and one at a higher curvature in a few obser-vations. However we were unable to reliably measurethe curvature for these arcs in multiple observations tofind curvature modulations. This is because the arc withsmaller curvature is generally faint and near to the poweron the leading-edge of secondary and/or primary arc.The arc with higher curvature is hidden mostly withinthe power inside the primary arc. Future observations can be optimized for fitting anyarcs with higher curvature by taking long observations,and lower curvatures can be probed by using wide ob-serving bandwidths or detecting flux in shorter sub-integration times. These additional arcs (and poten-tially more) will be analysed using more sensitive pulsarobservations from the MeerKAT (Bailes et al. 2018) ra-dio telescope, and the ultra wide-band receiver of theParkes radio telescope (Hobbs et al. 2019).In further studies on the Doppler profiles it will bepossible to develop templates for matched filtering toimprove arc fitting measurements and to determineanisotropy and orientation simultaneously with a veloc-ity model. This technique will become increasingly im-portant for high signal-to-noise and wide-bandwidth ob-servations, however it requires an estimate of the scatter-ing strength and currently assumes that the scatteringoriginates from a thin screen with a power spectrum ofdensity irregularities (such as Kolmogorov turbulence). CONCLUSIONSMeasuring the curvature of scintillation arcs as theychange with the velocity of the line-of-sight through thescattering medium is a powerful way of precisely mea-suring properties of the scattering and the orbit of abinary pulsar. We have measured annual and orbitalmodulations in the curvature of two separate arcs forthe millisecond pulsar, PSR J0437 − D = 89 . ± . D = 124 ± v IISM ,α and v IISM ,δ for isotropic scattering, or ξ and v IISM ,ξ for anisotropic)across the whole ∼
16 years of observations, meaningthat these properties of the interstellar plasma remainstable over a spatial scale of at least ∼
400 AU.Our precise velocity model has provided a mea-surement of the longitude of the ascending node forPSR J0437 − . ± .
4, which is more precise thanthat obtained from the timing model of Reardon et al.(2016). Our data are also sensitive to the orbital incli-nation angle, and we have measured i = 137 . ± . ◦ .This shows that modelling these variations gives an al-ternate means for obtaining precise measurements of i for pulsars that are not observed edge-on. When applied4 Reardon et al. to other pulsars that have only one post-Keplerian pa-rameter measured from timing, this could lead to moremeasurements of neutron star masses, depending on thepulsar’s flux and scintillation properties (from the pul-sar catalogue (Manchester et al. 2005), approximately10 pulsars are currently suitable for this application).Measurements of i and Ω can also improve tests of gen-eral relativity by allowing the correction of kinematiceffects in relativistic parameters (Kopeikin 1995, 1996),or by enabling tests of gravitational symmetries (e.g.Zhu et al. 2019). Similarly to the screen distance, thesebinary parameters are nearly model-independent.If the proper motion of a pulsar is not known frompulsar timing, this method could be used to estimate itby assuming that its velocity is much larger than anyIISM velocity. However we have shown that care needsto be taken when estimating parameters of the screenunder the assumption of a stationary IISM, since it mayhave a substantial (of order a few tens of km s − ) mag-nitude. Arc curvature modelling is promising for pulsars observed in the weak scintillation regime, where the scin-tillation bandwidth and timescale is too unstable fromobservation-to-observation to reliably measure the prop-erties of diffractive scintillation and their change withtime. ACKNOWLEDGEMENTSParkes radio telescope is part of the Australia Tele-scope, which is funded by the Commonwealth Gov-ernment for operation as a National Facility managedby CSIRO. M.B., S.O., R.M.S., and R.S. acknowledgeAustralian Research Council grant FL150100148. Partsof this research were conducted by the Australian Re-search Council Centre of Excellence for GravitationalWave Discovery (OzGrav), through project numberCE170100004. Work at NRL is supposed by NASA.This research has made use of NASA’s AstrophysicsData System.REFERENCES Bailes, M., Barr, E., Bhat, N. D. R., et al. 2018, arXive-prints, arXiv:1803.07424Bannister, K. W., Stevens, J., Tuntsov, A. V., et al. 2016,Science, 351, 354Bhat, N. D. R., Ord, S. M., Tremblay, S. E., McSweeney,S. J., & Tingay, S. J. 2016, ApJ, 818, 86Bhat, N. D. R., Tremblay, S. E., Kirsten, F., et al. 2018,ApJS, 238, 1Brisken, W. F., Macquart, J.-P., Gao, J. J., et al. 2010,ApJ, 708, 232Coles, W., Hobbs, G., Champion, D. J., Manchester, R. N.,& Verbiest, J. P. W. 2011, MNRAS, 418, 561Coles, W. A., Rickett, B. J., Gao, J. J., Hobbs, G., &Verbiest, J. P. W. 2010, ApJ, 717, 1206Coles, W. A., Kerr, M., Shannon, R. 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APPENDIX A. ACCESSING DATA AND REPRODUCING RESULTSThe raw pulsar observations are available for download from the CSIRO data access portal (DAP), with the majorityof these data being taken under the “P456” project code (https://data.csiro.au/dap/search?q=P456). The observationswere processed using the processing pipeline for the PPTA second data release, as described in Kerr et al. (2020).The pulsar ephemeris used to compute the pulsar’s transverse velocity is from Reardon et al. (2016) and available athttps://doi.org/10.4225/08/561EFD72D0409. Finally, all processed dynamic spectra of PSR J0437 − Scintools , which will be described and documented in detail ina future work. The package is available from https://github.com/danielreardon/scintools and includes some examplescripts that can be used to reproduce our results. These examples show the techniques for dynamic and secondaryspectrum processing, arc curvature measurement, and modelling curvature measurements with time. The currentversion as of this publication will be preserved as “pre-release version 0.2”. The code makes use of
Astropy (Price-Whelan et al. 2018) to calculate the transverse velocity of the Earth with respect to the pulsar, and the Romer delayto the Solar System Barycenter (SSB).We have also reproduced the dynamic spectrum simulation code of Coles et al. (2010), and include this in
Scintools . B. ADDITIONAL EQUATIONS AND FIGURESThe secondary spectrum in weak scintillation (equation D5 in Cordes et al. (2006)) can be written in terms of thespatial spectrum of the phase shift the wave experiences in traversing the scattering region P φ ( (cid:126)κ ) S ( f λ , f t ) = 8 π V eff D e κ y [ P φ ( κ x = 2 πf t /V eff , κ y ) + P φ ( κ x = 2 πf t /V eff , − κ y )] . (B1)Here the x-axis is in the direction of (cid:126)V eff , κ x and κ y are spatial wavenumbers, D e = Ds (1 − s ), κ y = (cid:112) κ − κ x , and κ = 8 π | f λ | /D e . We then define P φ as a power-law in a quadratic form P φ ( (cid:126)κ ) = C φ /Q ( (cid:126)κ ) α/ (B2)where the exponent α = 11 / A r and orientation ψ with respect to the (cid:126)V eff is (Reardon et al. 2019) Q ( (cid:126)κ ) = aκ x + bκ y + cκ x κ y (B3)where a = cos ψ/A r + A r sin ψ , b = A r cos ψ + sin ψ/A r and c = 2 sin ψ cos ψ (1 /A r − A r ). The secondary spectrumis then separable S ( f λ , f t ) = (cid:32) π C φ V D e (cid:33) κ − α +10 D t ( f t /f arc ) (B4)for | f t | < f arc and 0 elsewhere. The Doppler profile is given in normalized Doppler f tn = f t /f arc = κ x /κ so it isdimensionless D t ( f tn ) = [( af tn + b (1 − f tn ) + cf tn (1 − f tn ) − / ) − α/ + ( af tn + b (1 − f tn ) − cf tn (1 − f tn ) − / ) − α/ ](1 − f tn ) − / . (B5)If the medium is isotropic a = b = 1 and c = 0 so D t ( f tn ) = (1 − f tn ) − / .We use this Equation to plot the theoretical lines in Figures 2, 3, 9, and 10. For Figures 2 and 3 we show theexpectation for purely isotropic scattering to guide the eye and demonstrate that the data at these epochs is consistentwith near-isotropic scattering if the majority of power inside the arc does indeed originate from the primary scatteringscreen itself, rather than from other scattering material along the line-of-sight. In Figures 9, and 10 we overlay theweak scintillation approximation for our simulations of near-weak scintillation that match the properties of 20-cmobservations for PSR J0437 − A r (cid:46) odelling scintillation arcs of PSR J0437 − Figure 8.
Posterior probability distributions for the parameters in the isotropic (top) and anisotropic (bottom) models for theprimary arc. The contours in the 2D distributions mark the 68%, 95% and 99.7% confidence intervals and the blue lines markthe mean for each parameter. This figure was created with the corner package (Foreman-Mackey 2016). The most substantialcovariances are observed between s and v IISM ,ξ for the anisotropic model with a correlation coefficient of − .
96, and s and v IISM ,α for the isotropic model with a correlation coefficient of − . Reardon et al.
Figure 9.
Secondary spectra (left panels) and corresponding Doppler profiles (right panels) for a series of simulations withvarying degrees of anisotropy. The axial ratios ( A r ; labelled on the Doppler profiles in the right panels) for the simulationsare spaced log-uniformly from isotropy to a 10:1 anisotropy, each one aligned with the velocity. The strength of scintillationand the sampling characteristics for the simulations were chosen to approximately match our observations of PSR J0437 − A r = 10, the appearance of inverted arclets broadens the arc,which adds power internally to the arc, which is not expected in the weak scattering regime and therefore the model is a poorapproximation. odelling scintillation arcs of PSR J0437 − Figure 10.
As in 9, but with varying orientation of the anisotropy ψ with respect to the velocity, for a fixed axial ratio of A r = 3 ..