THE PROPER MOTION OF THE GALACTIC CENTER PULSAR RELATIVE TO SAGITTARIUS A*

As shown in Table 4, inclusion of constant acceleration (due to Sgr A*) or constant core shift terms do not return detections of either of those parameters or significantly alter the quality of the fits. Accordingly, we consider here the proper-motion-only fits in order to assess the presence of systematic errors in our measured PSR J1745-2900 positions. Using all of the data and the formal positional errors estimated from JMFIT, we calculate χ² values for each fit in right ascension and declination. None of the χ² values are consistent with a good fit and accurately estimated errors. In particular, errors in declination are underestimated by ∼25%, and errors in right ascension are underestimated by a factor of approximately three. As shown in Figure 1, there is an apparent systematic error in the right ascension position associated with the first four epochs that accounts for most of the increase in χ². A number of potential effects could cause this discrepancy, and we examine each of them in turn.

The first three possibilities concern calibration. As noted in Bower et al. (2014b), it is difficult to generate an accurate model of Sgr A* to use for calibration because scatter broadening causes it to be heavily resolved on most VLBA baselines. This problem is doubly severe in right ascension compared to declination because the shorter VLBA baselines are predominantly east–west (meaning higher resolution in the right ascension coordinate) and the major axis of Sgr A* scattering is almost aligned with the right ascension axis. An incorrect model of Sgr A* will lead to different positional offsets between epochs if the (u, v) coverage differs (which it does because of different failed antennas), as the calibration changes the data in order to match the incorrect model. In this sense the first four epochs are not especially poor, nor should this type of error lead to a right ascension error that apparently varies linearly with time. It would affect the 8 GHz observations more severely than the 15 GHz because of the more severe effects of scattering, and a higher proportion of the first four epochs are at 8 GHz, so this could simply be random errors giving the appearance of linear motion in the first four epochs.

Second, the calibration of the phased VLA could be worse in the first four epochs. During this time, we used the source NRAO 530 to determine the real-time solutions to phase up the VLA, and after this time we used Sgr A* itself. The angular offset from our target field to NRAO 530 is 16°, and so it is likely that the phased VLA gain is slightly lower and more time-variable in these epochs. Because the VLA heavily affects the calibration solution, being the most sensitive antenna, errors in its gain calibration could propagate through to larger errors in the other telescopes gain calibrations and hence positional offsets. There is a direct link to the first four epochs; however, there is once again no reason to expect an apparently linear motion with time.

Third, ionospheric effects can introduce wavelength-dependent astrometric errors. In the case of our experiment, the small separation, δr, between calibrator and target leads to a correspondingly small error in relative positions (Reid & Honma 2014). After the application of total electron content (TEC) models as part of our calibration, the residual ionospheric delay could correspond to ≲ 10 cm at 3.5 cm. This corresponds to an error of approximately three beam widths, or δθ ≲ 10 mas at 3.5 cm. This leads to an astrometric error of δrδθ ≲ 1 μas for the 3.5 cm relative to a shorter wavelength (e.g., 7 mm) position where ionospheric effects are very small. Thus, ionospheric effects do not affect our result.

Fourth, the apparent structure of Sgr A* itself could change with time. Because we use a constant model at each frequency (derived from the concatenated data sets at that frequency), there is no way to detect or account for such a change, which could be due to intrinsic source effects (such as a time-variable core shift due to material propagating outward along a jet) or time-variable scattering or scintillation. By forcing Sgr A* back to the model position during calibration, we would impose an equivalent shift on the target magnetar. Unlike the previous two explanations, evolution of Sgr A* could lead to a linear position change with time, and it would likely be along the right ascension axis, which is the major axis for both the scattering and for the intrinsic structure of Sgr A*. However, the effect we see seems too large for this to be a likely explanation. Various estimates of the scatter-broadened size of Sgr A* (Bower et al. 2006, 2014b) show it to be constant over time within the error bars, and the intrinsic size of Sgr A* is likely much smaller than the required shifts here. The specific case of a time-variable core shift is considered in more detail in Section 4.

Fifth, refractive wander could induce variations in the position of the size on an angular scale smaller than the size of the scattered image. The motion of ∼1 mas in right ascension is much less than the image size of ∼15 mas. The short timescale of the fluctuation of ∼10² days, however, appears inconsistent with the refractive timescale of τ_R  ∼  Dθ/v ≳ 8 × 10² days at 3.4 cm, where D ∼ 3 kpc is the Earth-to-scattering screen distance, θ is the scatter-broadened angular size, and v  ∼  100 km s⁻¹ is the relative velocity of the screen perpendicular to the line of sight. The variable τ_R is a minimum timescale for significant refractive changes because the turbulent medium may be uniform on scales larger than Dθ in the scattering screen. As noted above, the apparent size of Sgr A* has remained constant over the course of ≳ 10 yr, suggesting that the refractive timescale is likely quite large. In addition, the similarity between the pulsar and Sgr A* images, discussed below, suggests that the timescale for refractive changes could be as large as R/v ≈ 1000 yr, where R ∼ 0.1 pc is the separation between the two sources. Further, previous attempts to detect positional wander in Sgr A* relative to background quasars at a separation of 05 have not shown any effect on scales larger than 400 μas on timescales ranging from approximately one hour to one month (Reid et al. 2008). Longer timescales were probed by observations of water masers in Sgr B2, which has a degree of scatter broadening similar to Sgr A* (Gwinn et al. 1988). No positional wander was detected on a timescale of six months to a limit of 18 μas rms for maser spots spread over a region 4'' in scale.

On the other hand, the astrometric offsets we see are only a fraction of the total potential refractive image wander and thus could potentially occur on (proportionately) shorter timescales than τ_R. In particular, the recent discovery of substructure in the image of Sgr A* at the 1.3 cm wavelength (Gwinn et al. 2014) suggests that refractive effects could affect the determination of the centroid of PSR J1745-2900 and the correct model for Sgr A*. Point-like emission with a flux density ∼1% of the total flux density was found, effectively demonstrating that Sgr A* resides in the "average" image regime (Narayan & Goodman 1989; Goodman & Narayan 1989). The characteristic timescale for the average image regime ranges from the diffractive timescale of ∼1 s to the time for an interferometric resolution element to move by a single beam ∼10 yr, i.e., comparable to the refractive timescale. The amplitude of refractive image wander relative to the scattered size in the case of a shallow turbulent spectrum (α < 2) will scale as the ratio of the substructure to peak flux densities. That ratio is ∼100 at 1.3 cm in the results from Gwinn et al. (2014), a factor of 10 smaller than the actual imaging errors at 3.4 and 2.0 cm. Steeper turbulent spectra can lead to a more widely varying refractive image wander. We also do not know whether refractive image wander would be independent between Sgr A* and PSR J1745-2900 if the image wander is common between the two sources, then there would be no astrometric effect in our data. For a single epoch of our data at the Ku band, we did fit a Gaussian scattering model plus point source and found no significant change in the resultant position for PSR J1745-2900. More detailed study of the wavelength characteristics and time variability of this substructure can determine whether refractive image wander is a significant limitation for the astrometry of PSR J1745-2900.

Finally, the change in proper motion in right ascension could be due to real acceleration of the magnetar during a close encounter with a massive star (a wide companion in a highly elliptical orbit or a chance encounter). Producing the apparent change in proper motion between the first four and the last four epochs would require acceleration of ∼1.5  cm s⁻², equivalent to the effect of a massive star at a distance of ∼10⁻³ pc or 25 mas. No star is known to be this close to PSR J1745-2900  (see the discussion below). We also consider this implausible because the radial acceleration from such an encounter would also affect the pulsar timing, and no large deviations from the long-term timing trends are seen around MJD 56500. In particular, for an edge-on orbit, we require a change in the period derivative of ∼10⁻¹⁰, an order of magnitude larger than the period derivative $\dot{P}=6.12 \times 10^{-12}$ measured by Kaspi et al. (2014) at this epoch. In fact, timing observations detected an abrupt change in the period derivative at MJD = 56450, in the middle of the apparently linear proper motion and, therefore, in conflict with orbital motion. If we take the conservative estimate of the measured period derivative as being entirely due to acceleration, we set an upper limit that the inclination angle must be ${\lesssim } 4\deg$. A more detailed analysis of the timing data will set even stronger constraints. Thus, the encounter would have to have been seen almost face on, an unlikely but not impossible scenario.

Ultimately, we cannot confirm or rule out any of the explanations listed above, with the exception of differential ionospheric errors. Additional astrometric epochs may eventually provide enough information to favor one of these explanations, but the additional uncertainty imposed on the right ascension proper motion due to the scatter in the first four epochs does not alter any of the conclusions that follow in this paper. Removing this systematic uncertainty and approaching the S/N limited uncertainty in proper motion, however, will be crucial in detecting the acceleration of PSR J1745-2900 due to Sgr A* over a long period (see Section 5).

Due to the relatively poor quality of the right ascension least-squares fit, for the case of proper motion only, we also performed a bootstrap resampling method to estimate solutions and errors more accurately. The bootstrap method resampled the data with replacement 10⁴ times. We did not group the data by date, which may permit some small amount of correlated error to propagate into our solutions, i.e., as the result of same-day tropospheric corrections. In practice, we found a negligible difference in the solutions (i.e., δμ_α = 0.015 mas yr⁻¹) if we averaged the data by date before including it in the bootstrap algorithm. We adopt the bootstrap result, which is consistent with the least-squares fit but with larger errors, as our best result.

We consider initially the astrometric fits to proper motion alone. The acceleration constraints are weak, and the proper-motion-only fits provide the highest accuracy. The best-fit proper motion is μ_α = 2.44 ± 0.33 mas yr⁻¹ and μ_δ = 5.89 ± 0.11 mas yr⁻¹ (67% confidence limits). The empirical probability distribution of μ_δ is strongly peaked and resembles a Gaussian (Figure 2). The probability distribution of μ_α, on the other hand, is asymmetric with a longer tail to smaller values. The 95% confidence intervals for the best-fit parameters are 1.6 < μ_α < 3.0 mas yr⁻¹ and 5.7 < μ_δ < 6.1 mas yr⁻¹.

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The best-fit proper motion is the opposite sign and within 25% of the magnitude of the proper motion of Sgr A* relative to the ICRF (caused by Galactic rotation). This produces the result that the pulsar is nearly at rest relative to the ICRF. Nevertheless, it is the motion of the pulsar relative to Sgr A* that is relevant for understanding its origin and future. The transverse velocity for the pulsar relative to Sgr A* is v = 236 ± 11 km s⁻¹ in position angle 22 ± 2 deg east of north. From our fit, the positional offset of PSR J1745-2900 from Sgr A* at MJD 56406 (when it was first detected) was 2402 mas, corresponding to a distance of 0.097 pc. The observed separation of the pulsar is consistent with lower precision X-ray estimates (Rea et al. 2013).

The proper motion of the pulsar is similar to the motion of stars near Sgr A* studied through near-infrared adaptive optics (Figure 3; Yelda et al. 2014). The stars S2–4 and S2–6 are within 500 mas of PSR J1745-2900 and have proper motions of (7.9, 3.1) mas yr⁻¹ and (7.9, 2.3) mas yr⁻¹, respectively. The magnitude of the velocity vector is similar to that of these nearby Galactic center stars, but the vector orientation differs by ∼45° from that of the nearest stars. This variation in the orientation is comparable to the variations among the stars. Thus, the motion of the pulsar appears to be consistent with clockwise (CW) rotation around Sgr A*. In the next section, we explore the probability of an origin in the CW stellar disk through Monte Carlo simulations and conclude that an origin in that disk is likely, although we cannot rule out an origin from the isotropic distribution of stars.

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Our fits are consistent with no acceleration with 3σ upper limits of 3 mas yr⁻² and 1.5 mas yr⁻² in each coordinate, respectively. This does not provide a strong constraint because the maximal acceleration that can be obtained at this separation from Sgr A* is 0.04 mas yr⁻². The magnitude of the acceleration is dependent on the distance z along the line of sight of the pulsar from Sgr A*. If the pulsar remains bright at high frequencies for ∼3–10 yr, future astrometric observations will have the sensitivity to detect the acceleration and, therefore, determine z. Constraints on acceleration will grow rapidly with increasing observing time.

In the absence of an acceleration, we cannot conclusively determine whether the pulsar is bound to Sgr A*. We can set limits on the velocity along the line of sight, v_z, as a function of the line-of-sight distance from the pulsar to Sgr A*, z, for which the pulsar is bound using the constraint that E = T + U < 0 (Figure 4). For |z| < 0.1(1.0) pc, velocities up to 550 (170) km s⁻¹ are bound to Sgr A*. Faucher-Giguère & Kaspi (2006) estimate the mean natal three-dimensional velocity for pulsars at 380 km s⁻¹, corresponding to a single coordinate mean of 220 km s⁻¹. Therefore, to |z| ≲ 1.0 pc, the pulsar is likely to be bound. For z = 0 and v_z = 0, the orbital period is ∼700 yr, perigee is ∼0.01 pc, and apogee is ∼0.1 pc. For higher velocities and values of z, the orbital period increases.

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Two other pieces of evidence support a hypothesis that the pulsar is bound to Sgr A*. One, the chance of random superposition of an unbound object within 3'' of Sgr A* in the 1400''-square Swift field of view is ∼10⁻⁵. Two, the relatively high transverse proper motion gives a short window of ∼400 yr in which the pulsar will be within 0.1 pc of Sgr A*. This timescale is shorter than the likely lifetime of the pulsar of ∼10³–10⁴ yr. Thus, the observed position and velocity point to an object that is likely bound to Sgr A*.

We also set an upper limit to the parallax with a fit to the ICRF position. We find π < 0.6 mas at 95% confidence. The limit is poor because of the relatively limited temporal coverage and because of the systematic errors in the right ascension position. Future observations will significantly improve this limit.

The observed proper motion of PSR J1745-2900 is CW, consistent with the young disk of massive stars orbiting Sgr A* (Paumard et al. 2006; Lu et al. 2009; Yelda et al. 2014); however, the angle of its orbit is offset by ≈45°. We assess whether a natal kick can move a pulsar onto this orbit using a suite of Monte Carlo simulations of potential orbits of pulsars in the Galactic Center, assuming it originated in the stellar disk. We compare these results to Monte Carlo simulations of an alternative model of the pulsar origin, where the pulsars are kicked from an isotropic distribution of stars with a thermal distribution of eccentricities.

For the stars originating in the disk, we initially generate orbits consistent with the best-fit parameters of the observed CW disk (Ω ≈ 96°  ±  3°, i ≈ 130°  ±  8°, e ≈ 0.27  ±  0.07; Yelda et al. 2014). We then kick each star from its orbit by selecting a random velocity from the double-exponential distribution of Faucher-Giguère & Kaspi (2006), which has a median kick velocity of ≈380 km s⁻¹. Other distributions of pulsar kick velocities have been created with similar broad properties (e.g., Arzoumanian et al. 2002). Although this distribution was derived to fit the overall pulsar (not magnetar) population of the Milky Way, it is consistent with the observed distances of magnetars that lie near massive star-forming regions and the two measured magnetar transverse velocities (Helfand et al. 2007; Deller et al. 2012). We can compare these results with stars selected from an isotropic distribution of orbits with a thermal eccentricity distribution and a three-dimensional density profile n∝r⁻² consistent with the stars not located in the disk (Paumard et al. 2006; Lu et al. 2013; Yelda et al. 2014). In both models, ∼10⁷ orbits are each integrated for a uniform random duration up to 10⁴ yr. The longest integration time is comparable to the inferred spin-down age of PSR J1745-2900 and approximately 10 orbital periods at 0.1 pc from Sgr A*; it is also slightly less than the Newtonian precession timescale at that distance (Kocsis & Tremaine 2011). Because the Newtonian precession only acts in the plane of the orbit (i.e., keeps the inclination fixed), it has little impact on the currently observable properties of the magnetar, so longer integration times should not be necessary. Shorter integration times are needed to assess unbound orbits that spend little time within 0.1 pc of Sgr A*. We found no noticeable difference in the resultant distributions when using a fixed integration time of 10⁴ or 10⁵ yr. These integrations are carried out using the galpy¹² code with a static potential that replicates the Milky Way rotation curve from the Galactic center through the halo.

In Figure 5, we compare the observed proper motion of PSR J1745-2900 with the Monte Carlo simulations by selecting only pulsars within 1'' of the position of PSR J1745-2900. Stars that are kicked out of the stellar disk tend to remain on CW orbits. The observed proper motion of PSR J1745-2900 is completely consistent with this distribution. The proper motion distribution of the isotropic origin, on the other hand, is rather broad, which prevents us from excluding it as the origin at a robust level. Assuming the disk fraction of stars is 50 %, we find that 83 % of the pulsars with kinematic properties similar to PSR J1745-2900 were born in the CW disk. However, Yelda et al. (2014) has recently revised the estimated disk fraction down to 25%. Using this lower disk fraction as our prior lowers the likelihood of originating from the disk to ≈62%.

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Our simulations support the hypothesis that stars originating in the disk and isotropic distribution are likely to remain bound to Sgr A*. For stars originating in the disk, with the ordinary pulsar kick distribution, none of ∼720 stars with kinematic properties similar to the magnetar were unbound to Sgr A* and the central parsec of stars. For stars originating from the isotropic distribution, there was one star out of 149 with similar kinematic properties that was unbound to the central parsec. It received a kick of 675 km s⁻¹and was 172 yr old.

Unlike in isolated massive star clusters, only very large natal kicks are sufficient to unbind a pulsar from Sgr A*. In Figure 6, we show the natal kick velocity distribution assumed in this work to the distribution of kick velocities that resulted in magnetars with a position and proper motion similar to PSR J1745-2900. We find that the range of kicks allowed, without unbinding the magnetar, spans a large range of 100–500 km s⁻¹. These limits are consistent with the finding for the four other magnetars with observed kick velocities (∼130–350 km s⁻¹) that are comparable to that of ordinary pulsars (Helfand et al. 2007; Deller et al. 2012; Tendulkar et al. 2012). Early models predicted that magnetar birth kicks could be much larger than those of normal pulsars, enabled by exotic kick mechanisms that could not operate efficiently in the formation of normal pulsars (Duncan & Thompson 1992; Thompson & Duncan 1993); our results for PSR J1745-2900 add further evidence disfavoring these predictions. On its own, this one transverse velocity does not strongly constrain the natal kick velocity distribution of magnetars, but three measurements that each require unlikely viewing geometries to support a natal kick in excess of 500 km s⁻¹ strongly disfavor a distribution that peaks at high velocities (>500 km s⁻¹). On its own, this one kick measurement does not provide much constraint on the natal kick velocity, but three kick measurements in which the kick is either <500 km s⁻¹or the alignment is very fortuitous is unlikely. It is difficult to give a quantitative estimate because the width and functional form of the velocity distribution would also be free parameters, but a simple assumption like the scaling up of the Maxwellian distribution assumed for normal pulsars to give an approximately two times higher kick (and hence a mean three-dimensional kick velocity of 800 km s⁻¹versus 400 km s⁻¹) gives a probability of sampling five magnetars with transverse velocity measurements <300 km s⁻¹ of ≲ 0.1%.

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In Figure 7, we show the expected distance of PSR J1745-2900 relative to Sgr A* assuming it formed in a disk or from an isotropic distribution. At PSR J1745-2900's current position, stars in the CW stellar disk are nearest to us and have measured positive radial velocities. We find that if the magnetar originated in the disk, it has |z| ≲ 0.1 pc in 88 % of the simulations. If the magnetar originated from an isotropic distribution of stars, we find it still has |z| ≲ 0.1 pc and a proper motion comparable to the measured value in ≈49 % of our simulations. As noted above, future interferometric observations of PSR J1745-2900 will likely detect the acceleration of PSR J1745-2900 and determine |z|.

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We plot the observed size of the pulsar as a function of wavelength in Figure 8. The pulsar size shows excellent agreement from 3.6 cm to 0.7 cm with the scattering model for Sgr A* in both major and minor axes. This result further supports the angular broadening interpretation of Bower et al. (2014a) that the pulsar and Sgr A* share the same scattering medium along the line of sight. The similarity over all wavelengths is important because it demonstrates that the mean scattering properties of either source will not change for a timescale of τ ≳ R/v  ∼  1000 yr, where R ∼ 0.1 pc is the separation between the pulsar and Sgr A*, and v  ∼  100 km s⁻¹ is the velocity of the scattering screen across the line of sight. This timescale is much larger than the refractive timescale τ_R  ∼  0.05 yr at 7 mm, indicating that the scattering medium appears to be smooth over large scales.

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We also find new constraints on the size of Sgr A* from a closure amplitude analysis of visibilities. A closure amplitude analysis provides the most accurate method of determining the size of Sgr A* (Bower et al. 2014b). We provide new results for the size at 3.5 and 2.0 cm that are a substantial improvement on sizes previously measured (Bower et al. 2004). Improvements are due to the greater bandwidth and, therefore, improved sensitivity of VLBA observations and the inclusion of the VLA that provides shorter baselines. At both wavelengths, the new measured major axis sizes are more precise and differ by less than 2σ from the earlier measured sizes. Any differences could be due to changes in the intrinsic size or in the refractive properties of the medium. Including the new 3.5 cm major axis size with the earlier 6 cm VLBA size and L-band VLA sizes from Bower et al. (2006), we estimate the major axis scattering normalization of b_{maj, scatt} = 1.32 ± 0.02 mas cm⁻¹. This is consistent within the errors with the previous estimate of b_{maj, scatt}. The new minor axis sizes at 3.5 and 2.0 cm are a significant improvement over previous values. We combine these measurements with the previous 6 cm VLBA size but not the L-band VLA minor axis sizes, which were corrupted by the presence of a radio transient, and we find a new estimate of the minor axis scattering normalization, b_{min, scatt} = 0.67 ± 0.02 mas cm⁻¹. The error is a factor of two smaller than previous estimates. We also estimate the mean position angle from the 2 to 6 cm data to be 818 ± 02. These new estimates are more precise but within the errors of previous estimates and so do not significantly affect the conclusions on the intrinsic size of Sgr A* (e.g., Falcke et al. 2009; Bower et al. 2014b).

The core shift arises in jet sources because the optical depth changes as a function of wavelength (Blandford & Konigl 1979; Falcke & Biermann 1995). At shorter wavelengths, jets appear more compact, and the τ = 1 surface is closer to the origin of the jet and, therefore, the black hole. We phase-reference the pulsed emission from PSR J1745-2900 to Sgr A* in this experiment. The pulsed emission must be intrinsically point-like because it originates within the pulsar magnetosphere, which has an angular size smaller than 1μas. Thus, any wavelength dependence in the position of the pulsar must originate from wavelength-dependent structures in Sgr A*. The discovery of refractive substructure by Gwinn et al. (2014), however, does suggest that differential image wander could also affect this relative measurement. Many accretion models for Sgr A* suggest that the accretion flow is likely to be spherically symmetric and, therefore, produce zero core shift. Our data provides a powerful opportunity to probe the structure of Sgr A* at wavelengths where the source image is strongly dominated by interstellar scattering. We define the core shift Φ in Equation (2) for right ascension and declination as a wavelength scaling to the position.

As Table 4 demonstrates, our least-squares fit does not find a significant detection of a constant core shift. The 3σ upper limit is ∼0.3 mas cm⁻¹ in right ascension and ∼0.2 mas cm⁻¹ in declination. The fit to all data is heavily weighted by the high S/N detections at wavelengths of 1.3 and 0.7 cm. Giving equal weight to all epochs in the fit also does not lead to a statistically significant detection. We also consider the possibility that the core shift may be time variable. Sgr A* is known to have a time-variable flux density at all wavelengths, including the longer wavelengths examined here (Herrnstein et al. 2004; Macquart & Bower 2006). In five epochs (MJD 56658, 56710, 56750, 56772, and 56899), we obtain simultaneous observations at two wavelengths from which we can estimate a core shift (Figure 9). For three of these epochs, the two bands are Ku and X; in the fourth and fifth epoch, the bands are K and Q. In only one epoch (MJD 56772, Ku and X band) do we see a >3σ core shift with an amplitude ∼−0.4 mas cm⁻¹; for epoch MJD 56899, K and Q bands, the significance is ∼2.8σ with a value of 0.7 mas cm⁻¹. Errors in Φ are determined using the formal statistical errors on the measured positions rather than errors scaled to achieve $\chi ^2_\nu =1$ because we are testing the question of whether apparent systematic offsets may be due to the core shift. Given the large scatter in the core shift measurements and systematic error in the right ascension positions, we conclude that we have produced an upper bound on the amplitude of the core shift. Higher sensitivity measurements with three or more simultaneous frequencies are necessary to convincingly demonstrate the presence (or absence) of a core shift. Multifrequency measurements could also test against refractive effects that are due to large-scale inhomogeneities in the scattering screen, leading the positions for the two sources to wander independently.

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The sign and angle of the core shift is indicative of the direction of a jet. If our single-epoch detection of the core shift is accurate, this indicates a jet that extends primarily to the southwest in position angle 245° east of north. Bower et al. (2014b) found an extension of the source in position angle ∼95°, which is not statistically consistent with the direction of the core shift. There is also some suggestion in the data of an alignment of the individual epochs along the 245° axis, which could be consistent with a time-variable bipolar outflow.

We compare the measurements with the core shift predictions presented in Mościbrodzka et al. (2014). The theoretical models are based on the general relativistic magnetohydrodynamic (GRMHD) simulations of an accreting black hole in which jets are naturally produced by magnetic processes. These simulations combined with the ray-tracing radiative transfer models can predict the synthetic images of the jetdiskblack hole triad at various wavelengths (Mościbrodzka & Falcke 2013; Mościbrodzka et al. 2014). To compute the theoretical core shifts, the time-averaged images of the jet models are convolved with the scattering-screen Gaussian for various position angles (P.A.s) of the jet on the sky. Then, the theoretical core shift is Φ_theory(P.A.) = (ϕ₁(P.A.) − ϕ₂(P.A.))/(λ₁ − λ₂), where ϕ_{1, 2} are the centroids of the scattering-broadened images at λ_{1, 2} = 0.7 and 1.3 cm, respectively. The theoretical core shifts for a jet inclined at i = 60° and various P.A. are shown in Figure 9. We have chosen to show this model because its broadband synthetic spectrum and image size at millimeter wavelengths are consistent with all observations of Sgr A*. Other self-consistent models may show different profiles. We find that the model time-averaged core shifts are of the same order of magnitude (∼0.2 mas cm⁻¹) as those measured only during individual epochs of our observations. One can further test the time-dependent GRMHD models of jets by studying the core shift as a function of time. A variable core shift puts strong constraints on the jet particle heating, its speed, and the inclination angle, but this requires more detailed investigation.

We also consider a simple order-of-magnitude estimate of the amplitude of the core shift based on measured time lags in variability for Sgr A*. Yusef-Zadeh et al. (2006) observed delays of τ  ∼  30 min between variability at 1.3 and 0.7 cm. If these delays represent the size of the photosphere at different wavelengths and the underlying structure is a jet with velocity v_j and inclination angle of 90°, then the apparent core shift is Φ  ∼  v_jτ/D_GC/Δλ (Falcke et al. 2009), where D_GC = 8.3 kpc is the distance to the Galactic center. We find Φ ≈ 0.7(v_j/c) mas cm⁻¹, which is an order of magnitude that is comparable to the detailed model estimates and our measured values.