MMNRAS , 1–10 (2020) Preprint 17 August 2020 Compiled using MNRAS L A TEX style file v3.0
A magnetar parallax
H. Ding, , (cid:63) A. T. Deller, , M. E. Lower, , C. Flynn, , S. Chatterjee, W. Brisken, N. Hurley-Walker, F. Camilo, J. Sarkissian, V. Gupta , Centre for Astrophysics and Supercomputing, Swinburne University of TechnologyJohn St, Hawthorn, VIC 3122, Australia ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav) CSIRO Astronomy and Space Science, Australia Telescope National Facility, Epping, NSW 1710, Australia Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA National Radio Astronomy Observatory, P.O. Box O, Socorro NM 87801, USA International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia South African Radio Astronomy Observatory, 2 Fir Street, Observatory 7925, South Africa CSIRO Astronomy and Space Science, Parkes Observatory, PO Box 276, Parkes NSW 2870, Australia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
XTE J1810 −
197 was the first magnetar identified to emit radio pulses, and has been extensivelystudied during a radio-bright phase in 2003–2008. It is estimated to be relatively nearbycompared to other Galactic magnetars, and provides a useful prototype for the physics ofhigh magnetic fields, magnetar velocities, and the plausible connection to extragalactic fastradio bursts. Upon the re-brightening of the magnetar at radio wavelengths in late 2018, weresumed an astrometric campaign on XTE J1810 −
197 with the
Very Long Baseline Array ,and sampled 14 new positions of XTE J1810 −
197 over 1.3 years. The phase calibration forthe new observations was performed with two phase calibrators that are quasi-colinear onthe sky with XTE J1810 − . ± .
05 mas corresponds to a most probable distance 2 . + . − . kpc for XTE J1810 − ≈
200 km s − for XTE J1810 − ≈
70 kyr ago, but a direct association is disfavored by theestimated SNR age of ∼ Key words: radio continuum: transients – pulsars: individual: XTE J1810 −
197 – parallaxes– proper motions
Magnetars are a class of highly magnetized, slowly rotating neutron stars (NSs) with surface magnetic field strengths typically inferredin the range 10 –10 G, making them the most magnetic objects in the known universe. They have been observed to emit high energyelectromagnetic radiation, and to undergo powerful X-ray and gamma-ray outbursts. The high energy emission from these objects is thoughtto be powered by the decay of their magnetic fields (Thompson & Duncan 1995) as opposed to dipole radiation for classical pulsars. Todate, 29 magnetars and 6 magnetar candidates have been discovered (Olausen & Kaspi 2014) ; however, only 6 magnetars have ever beenobserved to emit radio pulsations, partly due to a small birthrate for the class (Gill & Heyl 2007). SGR J1935 + . ± . (cid:63) E-mail: [email protected] Catalogue: © 2020 The Authors a r X i v : . [ a s t r o - ph . I M ] A ug H. Ding et al. the Galaxy and confirms magnetars are plausible sources of extragalactic Fast Radio Bursts (FRBs). However, the mechanism by which suchstrong radio pulses are produced from magnetars is poorly understood (e.g. Margalit et al. 2020), as is the birth mechanism of magnetars.Multi-wavelength observations of Galactic magnetars, including long-term timing and
Neutron Star Interior Composition Explorer (NICER) observations, allow us to study the morphology and evolution of their magnetic fields, and potentially probe their internal structure(Kaspi & Beloborodov 2017). However, such studies are usually limited by the uncertainties in the underlying magnetar distances (anduncertain proper motions as well, in some cases). For instance, the X-ray spectrum fitting technique that has recently been made possible byobservations with NICER requires a well-constrained, pre-determined distance to the target (which can be a magnetar) in order to infer itsradius along with its mass (Bogdanov et al. 2019). Besides, owing to the enormous instability in spin-down rates (period derivative (cid:219) P ) ofmagnetars (e.g. Camilo et al. 2007; Archibald et al. 2015; Scholz et al. 2017), measuring the proper motion (not to mention parallax) viatiming is difficult for magnetars; as such, using an accurate, a priori proper motion and parallax in the timing analysis of a magnetar canimprove the reliability of the timing model, thus facilitating the study of long-term (cid:219) P evolution. Furthermore, an accurate distance wouldenable unbiased estimation of the absolute flux of X-ray flares or the absolute fluence of giant radio pulses. On top of the studies focusingon the magnetars, accurate distance and proper motion for a magnetar also enables constraints to be placed on the distance to the dominantforeground (scattering) interstellar-medium (ISM) screen (Putney & Stinebring 2006; see Bower et al. 2014, 2015 for an example).Proper motion measurements for magnetars are significant in their own right. Both the space velocities of neutron stars and their surfacemagnetic field strengths have been connected to the progenitor stellar masses and the processes of core-collapse supernovae. Duncan &Thompson (1992) suggested that the high magnetic fields of magnetars could be associated with very high space velocities of 10 km s − through e.g., asymmetric mass loss during core collapse or in the form of an anisotropic magnetized wind, or through a neutrino and/orphoton rocket effect. Such processes would be ineffective for “ordinary” neutron star field strengths ( (cid:46) G), leading to great interestin magnetar velocity estimates as diagnostics of their natal processes. So far the transverse velocity measurements that have been made(modulo large uncertainties) do not support a higher-than-average kick velocity for magnetars, with transverse velocities around the range of200 km s − inferred for XTE J1810 − − − −
197 (hereafter J1810) was discovered in 2003 due to an outburst at X-ray wavelengths (Ibrahim et al. 2004), and wassubsequently seen to be pulsating at radio wavelengths with a period of 5.4 seconds (Camilo et al. 2006), the first time radio pulsations hadbeen detected for a magnetar. During its brief period of radio brightness over a decade ago, two VLBA observations separated by 106 dayswere made, allowing the measurement of a proper motion of 13 . ± . − – also the first for a magnetar (Helfand et al. 2007). Theimplied transverse velocity was 212 ±
35 km s − (assuming a distance of 3 . ± . After the re-activation of J1810 at radio wavelengths in December 2018, we observed the magnetar with the
Very Long Baseline Array (VLBA) from January 2019 to November 2019 on a monthly basis (project code BD223). We re-visited J1810 with three consecutive VLBAobservations in the same observing setup on 28 March, 6 April and 13 April 2020 (when J1810 was at its parallax maximum; project codeBD231). Altogether there are 14 new VLBA observation epochs.All the observations were carried out at around 5.7 GHz in astrometric phase referencing mode (where the pointing of the array alternatesbetween the target and phase calibrator throughout the observation). Unlike typical astrometric observations, J1810 was phase-referencedto two phase calibrators: ICRF J175309.0-184338 (4 . ◦ . ◦ MNRAS , 1–10 (2020)
LBA Astrometry of XTE J1810 − J1753 J1819 J1810 virtual calibrator
J1819 virtual calibrator
Figure 1. a)
The diagram shows the positions of the magnetar (marked with red rectangle), the primary phase calibrator J1753 and secondary phase calibratorJ1819. The virtual calibrator is chosen as the closest point (12 . (cid:48) b) Yellow arrows are overlaid on the calibrator plan to illustrate the discussion in Sections 4.2 and 4.4: if the position for J1819 was in error(greatly exaggerated here for visual effect – any position offset is in reality only at the ∼ hereafter J1819), which are almost colinear with J1810 (see Figure 1). The purpose of this non-standard astrometric setup is explained inSection 3.1. ICRF J173302.7 − DiFX software correlator (Deller et al. 2011) in two passes, standard (ungated) and gated. For the gatedpass, the off-pulse durations were excluded from the correlation in order to improve the signal-to-noise ratio (S/N) on the magnetar. For allthe 14 observations, pulsar gating was applied based on the pulsar ephemerides obtained with our monitoring observations of the magnetar atthe Parkes and Molonglo telescopes. The timing observations at Parkes were carried out under the project code P885; the relevant observingsetup and data reduction are described in Lower et al. (in preparation). The timing observations at Molonglo were fulfilled as part of theUTMOST project (Jankowski et al. 2019; Lower et al. 2020).Apart from the 14 new VLBA observations, we re-visited two archival VLBA observations of J1810 taken in 2006 under the projectcode BH142 and BH145A (Helfand et al. 2007). An overview of observation dates is provided in Figure 2. The two observations in 2006 werecarried out at both 5 GHz and 8.4 GHz, using J1753 as the only phase calibrator. More details of the observing setup for the two observationsin 2006 can be found in Helfand et al. (2007). Hereafter, where unambiguous, positions obtained from the two epochs in 2006 and the 14 newepochs are equally referred to, respectively, as the “year-2006 positions" and the “recent positions".
All VLBI data were reduced with the psrvlbireduce ( https://github.com/dingswin/psrvlbireduce ) pipeline written in python-based ParselTongue (Kettenis et al. 2006).
ParselTongue serves as an interface to interact with
AIPS (Greisen 2003) and
DIFMAP (Shepherdet al. 1994). The pipeline has been incrementally developed for VLBI pulsar observations over the last decade, notably for the large VLBAprograms PSR π (Deller et al. 2019) and MSPSR π (e.g. Ding et al. 2020).For the two year-2006 epochs, only data taken at 5 GHz were reduced and analyzed, in order to avoid potential position uncertainties dueto any frequency-dependent core shift in the phase calibrators (e.g. Bartel et al. 1986; Lobanov 1998). All positions of J1810 were obtainedfrom the gated J1810 data, the S/N of which exceed the ungated J1810 data by 40% on average.Due to multi-path scattering caused by the turbulent ISM along the line of sight to J1810 or J1819, the deconvolved angular size ofJ1810 is mildly broadened by 0 . ± . ∼
10 mas. Accordingly, J1819 is heavily resolved by the longestbaselines of the VLBA, with a flux density of at most a couple of mJy on baselines longer than 50 M λ (mega-wavelength). The three stationsfurthest from the geographic centre of VLBA are MK (Mauna Kea), SC (St. Croix) and HN (Hancock), each of which only has 0–1 baselinesshorter than 50 M λ . Unsurprisingly, we found that while valid solutions could still often be found, the phase solutions for HN, MK, and SCare much noisier than for the remainder of the array. On top of this, atmospheric fluctuation causes larger phase variation at longer baselines,which, combined with the worse phase solutions, makes phase wraps (to be explained in Section 3.1) at MK, SC and HN hard to determine. MNRAS , 1–10 (2020)
H. Ding et al. r e l a t i v e D e c l . ( m a s ) R A . o ff s e t ( m a s ) D e c l . o ff s e t ( m a s ) virtual-cal frameJ1819 frame Figure 2. Left panel: position evolution of J1810 relative to 18 h m . s , − ◦ (cid:48) . (cid:48)(cid:48) Right panel: recent positions of J1810with proper motion subtracted measured in the virtual-calibrator frame (blue) and J1819 frame (yellow; see Section 4.1 for explanations of different referenceframes); to convenience visual comparison, the two sets of positions adopt the same systematic uncertainties at each epoch as obtained in the virtual-calibratorframe. Overlaid are the best-fit models for 500 bootstrap draws from the positions measured in the virtual-calibrator frame. Here, the systematic uncertaintiesfor the positions measured in the virtual-calibrator frame are still under-estimated, which is addressed in Section 4.3 using a bootstrap technique. Obviously,the positions measured in the virtual-calibrator frame provide tighter constraints to the model.
We found that the image S/N for J1810 generally improves when MK, SC and HN are excluded from the target field data, and as a result,taking the three stations out of the J1810 data leads to a statistical positional error comparable to that obtained when using the full array.Therefore, we consistently flagged the three stations from the final J1810 data for 14 recent epochs. However, we did not remove the threestations from any calibration steps, because we found the participation of the three stations allows better performance of self-calibration onJ1819 at other stations.The applied calibration steps in this work are largely the same as the PSR π project (Deller et al. 2019) except for the phase calibration.For the two year-2006 epochs, the phase solutions obtained with J1753 were directly transferred to J1810; whereas, for the recent 14 epochs,phase solutions were corrected based on the positions of J1753 and J1819, before being applied to J1810. Such a technique, though previouslyadopted by other researchers (e.g. Fomalont & Kopeikin 2003), is applied to a pulsar for the first time. During the phase calibration of VLBI data, the calibration step k ( k = , , ... ) provides an increment of the phase difference ∆ φ ( k ) n ( t ) betweenthe station n and the reference station in the VLBI array at a given time t (for simplicity, frequency-dependency is not accounted for),which is then added to the previous sum of the phase difference φ ( k − ) n ( t ) when the new solutions are applied. The phase calibration of theprimary phase calibrator (J1753 for this work) is followed by the self-calibration of the secondary phase calibrator (J1819 for this work), thephase solutions of which are predominantly limited by anisotropic atmospheric (including ionosphere and troposphere) propagation effect.Accordingly, the position-dependent phase difference should be formulated as ∆ φ n ( (cid:174) x , t ) , where (cid:174) x represents the 2-D sky position. In thenormal phase calibration, the solutions ∆ φ n ( (cid:174) x S , t ) obtained with the self-calibration of the secondary phase calibrator at (cid:174) x S are directlygiven to the target. The closer (cid:174) x S is to the target field, the better ∆ φ n ( (cid:174) x S , t ) can approximate the unknown ∆ φ n ( (cid:174) x T , t ) at the target field.In cases where (cid:174) x S is (cid:38) ◦ away from the target field, considerable offsets are expected between ∆ φ n ( (cid:174) x S , t ) and ∆ φ n ( (cid:174) x T , t ) (especially atlower frequencies), which are commonly treated as systematic errors when using the normal phase calibration. However, if the primary phasecalibrator, secondary phase calibrator and the target happen to be quasi-colinear on the sky, ∆ φ n ( (cid:174) x T , t ) can be well approximated by phasesolutions corrected from ∆ φ n ( (cid:174) x S , t ) (Fomalont & Kopeikin 2003).It is easy to prove that for three arbitrary different co-linear positions (cid:174) x , (cid:174) x and (cid:174) x , φ n ( (cid:174) x , t ) = (cid:174) x − (cid:174) x (cid:174) x − (cid:174) x φ n ( (cid:174) x , t ) + (cid:174) x − (cid:174) x (cid:174) x − (cid:174) x φ n ( (cid:174) x , t ) , (1)assuming higher-than-first-order terms are negligible (as supported by Chatterjee et al. 2004; Kirsten et al. 2015). Specific to the self-calibration MNRAS000
We found that the image S/N for J1810 generally improves when MK, SC and HN are excluded from the target field data, and as a result,taking the three stations out of the J1810 data leads to a statistical positional error comparable to that obtained when using the full array.Therefore, we consistently flagged the three stations from the final J1810 data for 14 recent epochs. However, we did not remove the threestations from any calibration steps, because we found the participation of the three stations allows better performance of self-calibration onJ1819 at other stations.The applied calibration steps in this work are largely the same as the PSR π project (Deller et al. 2019) except for the phase calibration.For the two year-2006 epochs, the phase solutions obtained with J1753 were directly transferred to J1810; whereas, for the recent 14 epochs,phase solutions were corrected based on the positions of J1753 and J1819, before being applied to J1810. Such a technique, though previouslyadopted by other researchers (e.g. Fomalont & Kopeikin 2003), is applied to a pulsar for the first time. During the phase calibration of VLBI data, the calibration step k ( k = , , ... ) provides an increment of the phase difference ∆ φ ( k ) n ( t ) betweenthe station n and the reference station in the VLBI array at a given time t (for simplicity, frequency-dependency is not accounted for),which is then added to the previous sum of the phase difference φ ( k − ) n ( t ) when the new solutions are applied. The phase calibration of theprimary phase calibrator (J1753 for this work) is followed by the self-calibration of the secondary phase calibrator (J1819 for this work), thephase solutions of which are predominantly limited by anisotropic atmospheric (including ionosphere and troposphere) propagation effect.Accordingly, the position-dependent phase difference should be formulated as ∆ φ n ( (cid:174) x , t ) , where (cid:174) x represents the 2-D sky position. In thenormal phase calibration, the solutions ∆ φ n ( (cid:174) x S , t ) obtained with the self-calibration of the secondary phase calibrator at (cid:174) x S are directlygiven to the target. The closer (cid:174) x S is to the target field, the better ∆ φ n ( (cid:174) x S , t ) can approximate the unknown ∆ φ n ( (cid:174) x T , t ) at the target field.In cases where (cid:174) x S is (cid:38) ◦ away from the target field, considerable offsets are expected between ∆ φ n ( (cid:174) x S , t ) and ∆ φ n ( (cid:174) x T , t ) (especially atlower frequencies), which are commonly treated as systematic errors when using the normal phase calibration. However, if the primary phasecalibrator, secondary phase calibrator and the target happen to be quasi-colinear on the sky, ∆ φ n ( (cid:174) x T , t ) can be well approximated by phasesolutions corrected from ∆ φ n ( (cid:174) x S , t ) (Fomalont & Kopeikin 2003).It is easy to prove that for three arbitrary different co-linear positions (cid:174) x , (cid:174) x and (cid:174) x , φ n ( (cid:174) x , t ) = (cid:174) x − (cid:174) x (cid:174) x − (cid:174) x φ n ( (cid:174) x , t ) + (cid:174) x − (cid:174) x (cid:174) x − (cid:174) x φ n ( (cid:174) x , t ) , (1)assuming higher-than-first-order terms are negligible (as supported by Chatterjee et al. 2004; Kirsten et al. 2015). Specific to the self-calibration MNRAS000 , 1–10 (2020)
LBA Astrometry of XTE J1810 − of the secondary phase calibrator, we have ∆ φ n ( (cid:174) x , t ) = ∆ φ n ( (cid:174) x S , t ) · ( (cid:174) x P − (cid:174) x )/( (cid:174) x P − (cid:174) x S ) , where (cid:174) x P and (cid:174) x S refer to the position of the primaryphase calibrator and secondary phase calibrator, respectively; this relation allows us to extrapolate to ∆ φ n ( (cid:174) x , t ) at any position (cid:174) x colinear withJ1753 and J1819 based on ∆ φ n ( (cid:174) x J1819 , t ) .We calculated the closest position to J1810 on the J1753-to-J1819 geodesic line (an arc), where it can best approximate ∆ φ n ( (cid:174) x J1810 , t ) .Since it acts like a phase calibrator for J1810, hereafter we term it the “virtual calibrator” for J1810. The position of the virtual calibratoris (cid:174) x v = . · (cid:174) x J1753 + . · (cid:174) x J1819 , which is 12 . (cid:48) ∆ φ n ( (cid:174) x v , t ) = . · ∆ φ n ( (cid:174) x J1819 , t ) ; using this relation, we corrected the phase solutions of J1819 obtained with its self-calibration. The correction was implemented with a dedicated module called calibrate_target_phase_with_two_colinear_phscals that was newly added to the vlbatasks.py , as a part of the psrvlbireduce package. One of the functions of the module is to solve thephase ambiguity of ∆ φ n ( (cid:174) x J1819 , t ) prior to multiplying ∆ φ n ( (cid:174) x J1819 , t ) by 0.62.The biggest challenge of the dual-calibrator phase calibration is the phase ambiguity of ∆ φ n ( (cid:174) x J1819 , t ) , which can be equivalentlyexpressed as ∆ φ n ( (cid:174) x J1819 , t ) ± i π ( i = , , , ... ) . This phase ambiguity will not change the quality of solutions for the normal phasecalibration, but will cause trouble for the dual-calibrator phase calibration, as the periodicity of the phase is broken when multiplied by afactor. The degree of phase ambiguity depends on observing frequency and angular distance between the main and secondary phase calibrator;for this work (5.7 GHz, 6.5 ◦ ), we run into mild phase ambiguity, mainly at the longest baselines (that we do not use anyway as mentionedearlier in this section). Among the recent epochs, the smallest size of the synthesized beam excluding MK, SC and HN is 2 . × . ∆ φ n ( (cid:174) x J1819 , t ) less likely to turn more than one wrap, and impossible to turn more than two wraps (i.e. | i | ≤ ∆ φ n ( (cid:174) x J1819 , t ) in a semi-automatic and iterative manner. The pipeline would go though all values of ∆ φ n ( (cid:174) x J1819 , t ) at each station. If | ∆ φ n ( (cid:174) x J1819 , t )| < π / ∆ φ n ( (cid:174) x J1819 , t ) are deemedphase-unambiguous, and no human intervention is needed for the station n . Otherwise, ∆ φ n ( (cid:174) x J1819 , t ) solutions are plotted out for inspectionand interactive correction. In most cases, no interactive correction is necessary after the inspection of the ∆ φ n ( (cid:174) x J1819 , t ) plot, as the solutionslook continuous, oscillating around 0 within a reasonable range (e.g. between ± π / π radians to the solution) with interactivecorrection. For every possibility, we implemented the dual-calibrator phase calibration using the resultant solution and ran through thecomplete (data-reduction and imaging) pipeline. The correct ∆ φ n ( (cid:174) x v , t ) should outperform other possibilities in terms of the image S/N forJ1810 (and the S/N difference is normally significant), as it better approximates the ∆ φ n ( (cid:174) x J1810 , t ) . This S/N criterium helps us find the “real” ∆ φ n ( (cid:174) x J1819 , t ) , thus the right ∆ φ n ( (cid:174) x v , t ) . The obtained solutions ∆ φ n ( (cid:174) x v , t ) were then transferred to J1810.More generally, if no phase-calibrator pair quasi-colinear with the target is found, one can extrapolate to ∆ φ n ( (cid:174) x , t ) at any position (cid:174) x withthree non-colinear phase calibrators (also known as 2-D interpolation, Fomalont & Kopeikin 2003; Rioja et al. 2017). Despite the longerobserving cycle and hence sparser time-domain sampling (unless using the multi-view observing setup, Rioja et al. 2017), the tri-calibratorphase calibration can in principle remove all the first-order position-dependent systematics. Similar to the way a reference frame is normally defined in non-relativistic (Cartesian) contexts, a reference frame in the context of relativeVLBI astrometry (hereafter reference frame or frame) generally refers to a system of an infinite amount of sky positions that are tied to aphase calibrator (not necessarily a real one), in which positions are measured relative to the phase calibrator. In this work there are threedifferent reference frames where we can measure the positions of J1810: the J1753 frame, the J1819 frame, and the virtual-calibrator frame.To be more specific, in the J1753/J1819 frame, the positions are measured relative to the brightest spot of the model image for J1753/J1819respectively. By applying an identical model of J1753/J1819 during the fringe fitting and self-calibration steps, the J1753/J1819 images atdifferent epochs are aligned, respectively, to 17 h m . s , − ◦ (cid:48) . (cid:48)(cid:48)
520 and 18 h m . s , − ◦ (cid:48) . (cid:48)(cid:48) h m . s , − ◦ (cid:48) . (cid:48)(cid:48) (cid:174) x v = . · (cid:174) x J1753 + . · (cid:174) x J1819 . For the 14 recentepochs, the final positions of J1810 were measured in the virtual-calibrator frame, though the positions of J1810 were also measured in theother two frames for various purposes (see Section 4.2 and Figure 2). The two year-2006 positions were merely measured in the J1753 frame,as J1753 is the only available phase calibrator for these observations.
Similar to the way in which systematic positional errors for pulsars in the PSR π project were evaluated using Eqn 1 of Deller et al. (2019) toaccount for both differential ionospheric propagation effects and thermal noise at secondary phase calibrators, the estimation of systematic available at https://data-portal.hpc.swin.edu.au/dataset/calibrator-models-used-for-vlba-astrometry-of-xte-j1810-197 MNRAS , 1–10 (2020)
H. Ding et al. uncertainty for the measured positions of J1810 is based on the mathematical formalism ∆ sys2 = (cid:16) A · s · csc (cid:15) (cid:17) + ( B / S ) , (2)where ∆ sys is the ratio of the systematic error to the synthesized beam size, (cid:15) stands for elevation angle, csc (cid:15) is the average csc (cid:15) for a givenobservation (over time and antennas), s is the angular separation between J1810 and the calibrator of the frame, S represents the image S/Nof the calibrator of the frame, and A and B are coefficients to be determined. However, unlike Eqn 1 of Deller et al. (2019), in Eqn 2 the twocontributing terms on the right side are added in quadrature. Given that they should be uncorrelated, this is more appropriate than the linearsummation in Eqn 1 of Deller et al. (2019).For this work, the second term of Eqn 2 is negligible for any of the three frames, as both J1753 and J1819 are strong sources, with abrightness ≥
24 mJy beam − at our typical resolution (after MK, SC and HN have been removed from the array). In order to find a reasonableestimate of A for this work, we measured the positions of J1810 consistently in the J1753 frame for all 16 epochs, and determined the valueof A that renders an astrometric fit (see “direct fitting” in Section 4.3) with unity reduced chi-square, or χ ν =
1. We obtained A = × − .We note that, in principle, A is invariant with respect to different reference frames. As is mentioned in Section 4.1, the final positions ofJ1810 were measured in the virtual-calibrator frame for the 14 recent epochs and in the J1753 frame for the two year-2006 epochs. Using A = × − and Eqn 2 without the second term, we acquired systematic errors for each epoch, which was then added in quadrature to therandom errors.After the determination of the systematic errors, the next step is to transform positions into the same reference frame. Since J1753 andJ1819 are remote quasars almost static in the sky, the frame transformation is simply translational. We translated the two year-2006 positionsmeasured from the J1753 to the virtual-calibrator frame. The translation is equivalent, but in the reverse direction, to translate from thevirtual-calibrator frame to the J1753 frame, which is easier to comprehend. In order to translate the virtual-calibrator frame to the J1753frame, the position of the virtual calibrator needs to be measured in the J1753 frame, which can be accomplished by measuring the positionof J1819 in the J1753 frame. The method to estimate the position of J1819 (cid:174) x (cid:48) J1819 and its uncertainty (cid:174) σ (cid:48) J1819 in the J1753 frame is detailed inSection 3.2 of Ding et al. (2020).As is shown in Figure 1, once (cid:174) x (cid:48) J1819 is measured in the J1753 frame, the new position of the virtual calibrator (cid:174) x (cid:48) v in the J1753 frame isalso determined, the uncertainty (cid:174) σ (cid:48) v of which is 0.62 times (cid:174) σ (cid:48) J1819 . The difference between (cid:174) x (cid:48) v and (cid:174) x v (or 0.62 times the difference between (cid:174) x (cid:48) J1819 and (cid:174) x J1819 ) was used to translate the two year-2006 positions of J1810 from the J1753 frame to the virtual-calibrator frame. The (cid:174) σ (cid:48) v was added in quadrature to the error budget (already including systematic and random errors) of the two year-2006 positions. After including the systematic errors and unifying to the virtual-calibrator frame, the 16 positions of J1810 can be used for astrometric fitting.Astrometric fitting was performed using pmpar . The median among the 16 epochs, MJD 58645, was adopted as the reference epoch. Theresults out of direct fitting are reproduced in Table 1, the χ ν of which is 10.6. The large χ ν suggests the systematic errors for the recent14 positions are probably under-estimated, and the actual uncertainty for either parallax or proper motion is about 3 times larger than theuncertainty from direct fitting.Applying a bootstrap technique to astrometry can generally provide more conservative uncertainties, compared to direct fitting (e.g.Deller et al. 2019). In the same way as is described in Section 3.1 on Ding et al. (2020), we bootstrapped 100000 times, from which weassembled 100000 fitted parallaxes, proper motions and reference positions for J1810. The marginalized histograms for parallax and propermotion as well as their paired error “ellipses” are displayed in Figure 3. We reported the most probable value at the peak of each histogramas the measured value; the most compact interval containing 68% of the sample was taken as the 68% uncertainty range of the measuredvalue (see Figure 3). The parallax and proper motion estimated with bootstrap are listed in Table 1, which are highly consistent with directfitting while over 3 times more conservative (as is expected from the χ ν of direct fitting). Thus, the precision achieved for parallax and propermotion gauged with bootstrap can be deemed reasonable. The parallax corresponds to the distance 2 . + . − . kpc. Such a precision in distancewould not be achieved with normal phase calibration using the same data (see Figure 2). Along with proper motion and parallax, a reference position 18 h m . s ± .
03 mas , − ◦ (cid:48) . (cid:48)(cid:48) ± . (cid:174) x v = . · (cid:174) x J1753 + . · (cid:174) x J1819 , change in (cid:174) x J1753 or (cid:174) x J1819 would cause the position shift of the virtual calibrator ∆ (cid:174) x v (hence the position of J1810), following the relation ∆ (cid:174) x v = . · ∆ (cid:174) x J1753 + . · ∆ (cid:174) x J1819 . (3) https://github.com/walterfb/pmpar MNRAS000
03 mas , − ◦ (cid:48) . (cid:48)(cid:48) ± . (cid:174) x v = . · (cid:174) x J1753 + . · (cid:174) x J1819 , change in (cid:174) x J1753 or (cid:174) x J1819 would cause the position shift of the virtual calibrator ∆ (cid:174) x v (hence the position of J1810), following the relation ∆ (cid:174) x v = . · ∆ (cid:174) x J1753 + . · ∆ (cid:174) x J1819 . (3) https://github.com/walterfb/pmpar MNRAS000 , 1–10 (2020)
LBA Astrometry of XTE J1810 − Table 1.
Proper motion and distance measurements for J1810method µ α ≡ (cid:219) α cos δ µ δ (cid:36) D References(mas yr − ) (mas yr − ) (mas) (kpc)direct fitting − . ± . − . ± .
03 0 . ± .
01 2 . ± . − . + . − . − . ± . . ± .
05 2 . + . − . this workPrevious VLBI astrometry − . ± . − . ± . − − Helfand et al. (2007)red clump stars − − − . ± . − − − − − . − . − . − . − . µ α ( m a s y r − ) . . . . parallax (mas) − . − . − . − . µ δ ( m a s y r − ) − . − . − . − . − . µ α (mas yr − ) − . − . − . − . µ δ (mas yr − ) Figure 3.
Error “ellipses” and marginalized histograms for parallax and proper motion. In each histogram, the dashed line marks the measured value; the shadestands for the 68% confidence interval. In each error “ellipse”, the dark and bright contour enclose, respectively, 68% and 95% of the bootstrapped data points.MNRAS , 1–10 (2020)
H. Ding et al.
Using Eqn 3 and the method outlined in Section 3.2 on Ding et al. (2020), the reference position was shifted to align with the latest positionsof J1753 and J1819 . The shifted reference position 18 h m . s ± . , − ◦ (cid:48) . (cid:48)(cid:48) ± . (cid:46) . As is shown in Table 1, our new proper motion significantly improves on the previous value inferred from the two year-2006 positions; thenew distance D = . + . − . kpc is consistent with 3 . ± . − v t = + − km s − .Using the Galactic geometric parameters provided by Reid et al. (2019) and assuming a flat rotation curve between J1810 and the Sun, thepeculiar velocity (with respect to the neighbourhood of J1810) perpendicular to the line of sight was calculated to be v b = − ± − and v l = − ±
26 km s − . Our refined astrometric results consolidate the conclusion by Helfand et al. (2007) that J1810 has a peculiarvelocity typically seen in “normal” pulsars, unless its radial velocity is several times larger than the transverse velocity. The closest cataloged SNR to J1810 is G11.0 − , a partial-shell SNR 9 (cid:48) × (cid:48) in size (Brogan et al. 2004, 2006). The positionof its geometric center is 18 h m s , − ◦ (cid:48) , 19 (cid:48) away from J1810. The latest distance estimate of G11.0 − . ± . h m . s , − ◦ (cid:48) (cid:48)(cid:48) , about 1 (cid:48) east to the geometric center of G11.0 − − − (cid:219) P is erratic for J1810 (Camilo et al. 2007) as well as other magnetars (Archibald et al. 2015; Scholz et al. 2017), making the characteristic age τ c an unreliable estimate of the true age for J1810. Over the course of a decade, the changing value of (cid:219) P for J1810 has led to the τ c ( ∝ / (cid:219) P )increasing from 11 kyr (Camilo et al. 2007) to 31 kyr (Pintore et al. 2018). While the characteristic age is currently less than the tentativekinematic age τ ∗ k that the tentative association would imply, the unreliability of the τ c estimator in the case of magnetars suggests that theassociation cannot be ruled out on this basis.From the perspective of G11.0 − R SNR and its age τ SNR can be rewritten from Sedov (1959) as R SNR ≈ (cid:18) E erg (cid:19) / (cid:18) n
30 cm − (cid:19) − / (cid:18) τ SNR (cid:19) / pc , (4)where the injected energy E is expressed in a value typical of spherical SNRs expanding into the Galactic ISM, and the ambient ISM density n ∼
30 cm − for the γ -ray-emitting region including G11.0 − γ -ray emission above 1 TeV at a distanceof 2.4 kpc (Castelletti et al. 2016). At an SNR distance D SNR = . R SNR ≤ τ SNR (cid:46) τ SNR of 70 kyr can be made possible with an injected energy 500 times smaller than the typically-assumed value of 10 erg, which is extremelyunlikely (Leahy 2017). Therefore, we conclude G11.0 − − − ≈
70 kyr ago and becameunbound from its original companion. The companion star continued evolving in isolation before itself undergoing a supernova explosionat (cid:46) ≈
67 kyr (compared to the typical supergiant age of (cid:38) http://astrogeo.org/vlbi/solutions/rfc_2020b/rfc_2020b_cat.html MNRAS000
67 kyr (compared to the typical supergiant age of (cid:38) http://astrogeo.org/vlbi/solutions/rfc_2020b/rfc_2020b_cat.html MNRAS000 , 1–10 (2020)
LBA Astrometry of XTE J1810 − Assuming no peculiar velocity of the progenitor binary system (with regard to its neighbourhood mean) as well as a flat rotation curvebetween the Sun and J1810, the expected proper motion of the barycentre of the supergiant binary as observed from the Earth is only1.2 mas yr − . The additional proper motion of the companion due to the orbital motion at the moment of unbinding is even smaller forsupergiant binaries. Thus, the accumulated position shift of the companion star after the unbinding is at the 1 . (cid:48) level across 67 kyr, whichdoes not violate the premise of the “companion SNR” scenario.In principle, given the large age scatter of supergiants, the “companion SNR” scenario allows J1810 to be indirectly associated with anSNR further away. For example, J1810 can be traced back to 17’ west to the centre of SNR G11.4 − ≈
144 kyr ago. However, the relatively small characteristic age τ c = − − − τ c monitoring with timing observations on J1810 will offer a more credible range of τ c to be compared with the tentativekinematic age τ ∗ k ≈
70 kyr suggested by the possible “indirect” association between G11.0 − − ACKNOWLEDGEMENTS
We thank Marten van Kerkwijk for his in-depth review and helpful comments on this paper. H.D. is supported by the ACAMAR (Australia-ChinA ConsortiuM for Astrophysical Research) scholarship, which is partly funded by the China Scholarship Council (CSC). A.T.D is therecipient of an ARC Future Fellowship (FT150100415). S.C. acknowledges support from the National Science Foundation (AAG 1815242).Parts of this research were conducted by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav),through project number CE170100004. This work is based on observations with the Very Long Baseline Array (VLBA), which is operated bythe National Radio Astronomy Observatory (NRAO). The NRAO is a facility of the National Science Foundation operated under cooperativeagreement by Associated Universities, Inc. Data reduction and analysis was performed on OzSTAR, the Swinburne-based supercomputer.This work made use of the Swinburne University of Technology software correlator, developed as part of the Australian Major NationalResearch Facilities Programme and operated under license.
DATA AND CODE AVAILABILITY
The pipeline for data reduction is available at https://github.com/dingswin/psrvlbireduce .All VLBA data used in this work can be found at https://archive.nrao.edu/archive/advquery.jsp under the project codes bd223,bd231, bh142 and bh145a. The calibrator models for J1753 and J1819 can be downloaded from https://data-portal.hpc.swin.edu.au/dataset/calibrator-models-used-for-vlba-astrometry-of-xte-j1810-197 . REFERENCES
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APPENDIX A: MEASURING SCATTER-BROADENED SIZE OF XTE J1810–197
The angular size of a radio source can be measured from its image deconvolved by the synthesized beam. As magnetars are point-like radiosources, a non-zero deconvolved angular size of J1810 can be attributed to scatter-broadening effect caused by ISM. For each epoch, thedeconvolved image of J1810 is obtained as an elliptical gaussian component; the mean of its major- and minor-axis lengths is used as thescatter-broadened size of J1810. Assuming the degree of scatter-broadening did not vary across the 14 recent epochs, the 14 measurementsof scatter-broadened sizes yield a scatter-broadened size of 0 . ± . This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000