Constraining the Solar Galactic Reflex Velocity using Gaia Observations of the Sagittarius Stream

It has been over 30 years since the International Astronomical Union addressed disparities in derived values of the solar Galactocentric distance, R₀, and the Galactic circular rotation velocity at the Sun, Θ₀, of ±1 kpc and ±20 km s⁻¹, respectively, by recommending the adoption of R₀ = 8.5 kpc and Θ₀ = 220 km s⁻¹ "in cases where standardization on a common set of galactic parameters is desirable."³ Despite continued work to establish these parameters, it seems that the dispersions in determinations have not significantly diminished.

Recent estimates of Θ₀ span a wide range, 218–254 km s⁻¹, including those from radio interferometric proper motion measures of star-forming regions (Bovy et al. 2009; Reid et al. 2009) and Sagittarius (Sgr) A* (Reid & Brunthaler 2004), as well as explorations of stellar kinematics from the APO Galactic Evolution Experiment (APOGEE; Bovy et al. 2012) and the Sloan Extension for Galactic Understanding and Exploration (SEGUE) surveys (Schönrich 2012), or the kinematics of Cepheids (Kawata et al. 2018); however, the latest of these studies have tended toward the middle of that range.

One source of uncertainty is that there is a degeneracy between the inferred Θ₀ and inferred R₀, for most measurement methods. Assessments are further complicated by the fact that the Sun has a peculiar motion with respect to the Local Standard of Rest (LSR), while the LSR may itself be moving with respect to a simple circular orbit around the Galaxy. Thus, Θ₀ is typically entangled with the Θ-directional components of the solar peculiar motion and LSR motion. The latter are often simplified to the Galactic Cartesian counterparts ${V}_{\odot ,\mathrm{pec}}$ and ${V}_{\mathrm{LSR},\mathrm{pec}}$, respectively (as these velocities are often measured using an ensemble of nearby stars), so that, in cylindrical coordinates, the net revolutionary component of the Sun's motion is ${{\rm{\Theta }}}_{\odot }={{\rm{\Theta }}}_{0}+{V}_{\mathrm{LSR},\mathrm{pec}}+{V}_{\odot ,\mathrm{pec}}$. Values of ${V}_{\odot ,\mathrm{pec}}$ hover around 12 km s⁻¹, and Bovy et al. (2012) claim that ${V}_{\mathrm{LSR},\mathrm{pec}}$ may also be as high as 12 km s⁻¹.

In this Letter we focus on a new measurement of Θ_⊙. Majewski et al. (2006) demonstrated that the Sgr stream provides an effective means to measure Θ_⊙ (from which ${{\rm{\Theta }}}_{\mathrm{LSR}}\,={{\rm{\Theta }}}_{0}+{V}_{\mathrm{LSR},\mathrm{pec}}$ and Θ₀ can be inferred), independent of the assumed R₀ or the precise shape of the Galactic mass distribution, and that avoids having to observe sources in the heavily crowded and dust-extinguished Galactic Center. The method exploits the favorable orientation of the Sgr stream, which is in a nearly polar orbit that intersects the Galactic plane virtually along the line between the Sun and the Galactic Center. In this orientation, the Sgr plane provides a non-rotating reference against which the solar motion may be measured, with almost all of the reflex motion imprinted on the proper motions of Sgr stars perpendicular to the stream. At the time of Majewski et al. (2006), the proper motions for known Sgr stream stars were not good enough to apply the method rigorously. However, exploiting proper motions of ∼1–2 mas yr⁻¹ accuracy for ∼1–2 dozen spectroscopically confirmed Sgr stream stars in four fields along the Sgr trailing arm, Carlin et al. (2012) found that their proper motions were best reproduced by models similar to the (Law & Majewski 2010, hereafter LM10) Sgr destruction models but utilizing an LSR velocity of 264 ± 23 km s⁻¹.

The second data release (DR2) from the ESA-Gaia mission (Gaia Collaboration et al. 2018a) now provides the opportunity make this measurement with a significantly larger sample of stars having proper motions an order of magnitude more precise.

Red giants are ideal tracers of the Sgr stream, because they can be seen to large distances. Thus we initially selected stars from the Two Micron All Sky Survey (2MASS) catalog (Skrutskie et al. 2006) with J, H, and K_s magnitudes between 10 and 13.5, covering a range of colors and magnitudes where giant stars trace the Sgr stream (e.g., Majewski et al. 2003). Such stars were selected in a series of rectangular regions (selected to cover Sgr longitudes $| {B}_{\odot }| \lesssim 20^\circ$) on the sky around the trailing arm of the Sgr stream, which collectively span right ascensions from α = 320° to 75° and declinations −40° < δ < +40°, which corresponds to about 90° of the Sgr stream trailing arm. The selected stars were then cross-matched with the Gaia DR2 source catalog (Gaia Collaboration et al. 2018a) using the CDS X-match service⁴ adopting a 1'' positional tolerance.

While Gaia DR2 does not measure parallaxes to a precision sufficient enough to reach the distances of the Sgr stream, we can still identify and remove nearby Milky Way (MW) contamination using Gaia DR2 parallaxes. Stars with relative parallax uncertainties ${\sigma }_{\pi }/\pi \leqslant 0.2$ can provide relatively reliable distance measurements, and almost all such stars in our sample have 1/π < 10 kpc, which is considerably closer than the trailing arm of the Sgr stream (which ranges from about 20–40 kpc at different stream longitudes; Majewski et al. 2003; Koposov et al. 2012). We therefore remove all of these stars from our sample, and, although it seems counter-intuitive, we keep the stars with poor parallaxes, σ_π/π > 0.2, which are primarily distant stars that we refer to as our "distant Gaia sample." To reduce the MW contamination in this distant Gaia sample further, we also removed stars at α > 50° and within 25° of the Galactic plane because these sky regions lie near the MW disk where contamination washes out the signature of the Sgr trailing arm.

The signature of the Sgr trailing arm is apparent as an arcing overdensity in the Gaia DR2 proper motion vector point diagram (PMVPD) for our distant Gaia sample (Figure 1, lower-left panel), so we trimmed in the PMVPD around the overdensity using −4 mas yr⁻¹ < μ_α cosδ < 2 mas yr⁻¹ and −6 mas yr⁻¹ < μ_δ < 1 mas yr⁻¹. When restricting to these low proper motions, the stars in the Sgr trailing arm stand out with proper motion position angles (ϕ_μ) that are coherent and nearly linear with right ascension (Figure 1, top-left panel). The small scatter around this linear trend in ϕ_μ arises because the Sgr stream is kinematically cold, and provides an opportunity to greatly refine our selection of Sgr trailing arm candidates. However, elevated MW contamination at some α makes a simple linear fit to the data challenging, even with σ-clipping to reduce the noise. To avoid this problem, we apply the DBSCAN clustering algorithm (a density-based clustering algorithm that builds clusters of a given density; Ester et al. 1996) to the proper motion cut stars between −20° < α < 20°, where the Sgr trailing arm feature is most free of MW contamination. This allows us to select the Sgr trailing arm feature in a reproducible manner, and use it to extract the feature at right ascensions where the contamination is more considerable.

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Setting DBSCAN input parameters  = 5 and N = 100, with no normalization of ϕ_μ or α, the DBSCAN algorithm appears clearly to identify a cluster associated with the Sgr trailing arm feature (blue points in the top-right panel of Figure 1). We fit a line to this DBSCAN-identified cluster and calculated the dispersion (σ) in the residuals to this fit. We extend this Sgr stream candidate selection outside of the −20° < α < 20° range by extrapolating the fitted line and select stars within 3σ using this calculated dispersion. To measure the remnant contamination, in the same parameter space we count the number of stars in a region of the same shape/size just above and below our Sgr stream selection, which we average and compare to the number of selected stream candidates to estimate a contamination of about 16%. This selection therefore includes the DBSCAN selected stars (blue) and the extended sample (red) shown in the right panels of Figure 1.

Finally, we trim the selected stars to Sgr orbital longitudes Λ_⊙ = 30°–115° (so that the selection box cuts at right angles across the stream) and reject stars with proper motion errors in ${\mu }_{\alpha }\cos \delta$ or μ_δ greater than 0.2 mas yr⁻¹. This results in a final sample of 1963 candidate Sgr Stream stars, with an estimated contamination of 16%.

The bottom-right panel of Figure 1 shows that the arcing overdensity seen by eye in the PMVPD is due to the Sgr trailing arm as traced by these candidates. However, as seen in the top panels of Figure 1, there is still a higher level of contamination at either end of the α-range of our sample, particularly at α ≳ 30°. Moreover, in the μ_δ distribution of our Sgr trailing arm candidates there appears to be a tail toward values below ∼−4 mas yr⁻¹. This low-density, low μ_δ tail is not found in the Sgr stream models discussed below and is likely remnant MW contamination.

If the Sgr stream were on a perfectly polar orbit in the Galactic X − Z plane⁵ with no orbital precession along the stream, any systematic motion of the stream out of its orbital plane would be entirely due to the Sun's own reflex motion, which could then be measured directly from the data. In reality, Sgr is not on a purely polar orbit, the orbital plane is not precisely aligned with the Galactic X − Z axis, and the observed tidal streams precess slightly with increasing distance from the Sgr core. Therefore, we must compare the observations against N-body models that properly incorporate these various effects.

We do so using the LM10 model, which is well fit to the observed run of Sgr stream angular positions, radial velocities, and distances throughout the Λ_⊙ = 30°–115° range of the trailing arm. Despite the inability of this model to reproduce some key features of the broader Sgr–MW system (particularly, the bifurcation of the stream and the large apocentric distance for the trailing arm at Λ_⊙ ∼ 180°), it nonetheless remains the best-constrained model at the orbital longitudes considered here. Indeed, this particular section of the stream is easy to fit regardless of the assumptions made about the depth and detailed shape of the Galactic gravitational potential (in contrast to the leading arm for which a triaxial halo is necessary to fit both the angular positions and radial velocities simultaneously; Law et al. 2009). As discussed by LM10 (see also the discussion by Law et al. 2005), the dynamically young trailing arm has little power to constrain the shape of the Galactic dark matter halo and can be equally well fit in an oblate, spherical, prolate, or triaxial potential, and regardless of the exact distance to the Sgr core, the distance to the Galactic center, or the overall normalization of the depth of the Galactic gravitational potential via the local circular speed Θ₀.

The insensitivity to various factors is exactly what makes the trailing arm ideal for this study: for these stars a single and interesting model parameter, Θ₀, is strongly coupled to a single observational parameter, which is one dimension of their proper motion. In previous studies, for widely varying values of Θ₀, the tangential velocity of the Sgr core along its orbit—hitherto largely unconstrained given the uncertainties of past proper motion studies—could be dialed to optimize the match of the implied radial velocities along the trailing stream to available observations. Now, with the new proper motion constraints available from Gaia, we can discriminate between models having different values of Θ₀ and constrain the solar reflex velocity.

Carlin et al. (2012) reproduced the LM10 analysis (which assumed Θ₀ = 220 km s⁻¹), but for a range of different values of Θ₀ = 190–310 km s⁻¹ (sampled every 30 km s⁻¹, along with two "best-fit" cases where Θ₀ = 232 km s⁻¹ and 264 km s⁻¹). Because the dark matter halo contributes minimally within the solar circle, these models were realized by scaling the Galactic bulge/disk mass jointly to produce the desired local circular speed. The details of these models are discussed in LM10 and Carlin et al. (2012), and in brief, assume a three-component Galactic mass distribution consisting of a Hernquist spheroid, Miyamoto-Nagai disk, and a triaxial logarithmic dark matter halo with minor/major axis ratio ${(c/a)}_{{\rm{\Phi }}}=0.72$, intermediate/major axis ratio ${(b/a)}_{{\rm{\Phi }}}=0.99$, and the minor axis pointing toward $(l,b)=(7^\circ ,0^\circ )$. The Sgr dwarf within this Galactic gravitational potential is constrained to lie at $(l,b)=(5\buildrel{\circ}\over{.} 6,-14\buildrel{\circ}\over{.} 2)$, a distance of 28 kpc, and a heliocentric radial velocity ${v}_{\mathrm{hel}}=142.1$ km s⁻¹, with leading and trailing arms that match observed trends of angular position and radial velocity along orbital longitude of the stream. Each model assumed that the solar peculiar motion with respect to the local circular speed was given by $(U,V,W)=(-9,12,7)$ km s⁻¹ in a left-handed coordinate frame (Cox 2000), for a range in solar reflex velocities of Θ_⊙ = 202–322 km s⁻¹.

As illustrated in Figure 2, the LM10 model (middle column) continues to be a good match to observations of the Sgr stream in the Λ_⊙ = 30°–115° range of the trailing tail, despite the fact that no proper motions were used to constrain the model. This is perhaps unsurprising given that LM10 tuned the other four dimensions of phase space to reproduce all of the observational data available at the time. We note, however, that the run of proper motions in the declination direction is not quite a perfect match to the Gaia observations; furthermore, versions of the LM10 model in potentials with slower or faster circular speeds (left and right columns, respectively) result in linear shifts of the model in μ_δ while remaining almost unchanged in ${\mu }_{\alpha }\cos \delta$ and heliocentric radial velocity.

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This situation arises because of another fortuitous orientation of the Sgr stream with respect to celestial coordinates. In N-body models where the MW circular speed is larger, the proper motion of model stars in the direction perpendicular to the Sgr plane is larger due to the greater solar reflex motion. As the model Sgr dwarf must be made to move faster along its orbit to compensate for the deeper gravitational potential, these model stars also have faster motion within the Sgr plane. In the trailing stream, the vector addition of these components is such that the net change in proper motion for stream stars between different models happens to be almost entirely along μ_δ.

As we show in Figure 3 (left-hand panel), the difference between the (2σ-clipped) mean proper motion of the N-body models versus the observed stream is nearly constant with Λ_⊙, such that the model with reflex velocity Θ_⊙ = 202 km s⁻¹ is systematically offset by about +0.5 mas yr⁻¹ (purple curve), whereas the model with Θ_⊙ = 322 km s⁻¹ is offset by about −0.6 mas yr⁻¹ (red curve). The consistency of these systematic differences suggests that we can average these offsets over orbital longitude to obtain a single value $\langle {\rm{\Delta }}{\mu }_{\delta }\rangle$ describing the N-body model stream offset from the Gaia proper motions for different Θ_⊙ (Figure 3, right-hand panel). This relation is well described by a simple linear fit to within observational uncertainty. By taking this fit (and the 1σ uncertainties therein) we solve for the $\langle {\rm{\Delta }}{\mu }_{\delta }\rangle =0$ km s⁻¹ crossing point and determine that this occurs at Θ_⊙ = 253 ± 5 km s⁻¹. Limiting our analysis to the DBSCAN selection α-range where contamination is minimal, Λ_⊙ ∼ 52–80, we find Θ_⊙ = 247 ± 9 km s⁻¹, which is a slightly lower value but one that is more uncertain because the sample is three times smaller. This result has a larger uncertainty and results in less than a 1σ difference, therefore we proceed with Θ_⊙ derived from the full sample.

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An estimate of the possible systematic uncertainty in this measurement can be obtained by comparing the proper motion of the Sgr core in these N-body models with the Gaia observations. Because the models adopted an orbital pole defined by the path of the tidal streams (Majewski et al. 2003), varying Θ_⊙ in these models describes a linear relation in the PMVPD for the Sgr core (see Figure 2.8 of Law & Majewski 2016) according to the speed of Sgr along its orbit perpendicular to the line of sight (dialed up and down to compensate for the altered Galactic potential following from changes in Θ₀). The observed Gaia proper motion of the Sgr core ($({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta })=(-2.692,-1.359)$ mas yr⁻¹; Gaia Collaboration et al. 2018b) lies slightly off this relation by about 0.15 mas yr⁻¹, but is most consistent with a choice of Θ_⊙ = 256 km s⁻¹. We therefore adopt 3 km s⁻¹ (the difference between this value and the value derived from fitting the trailing stream) as our systematic uncertainty; by combining systematic and random uncertainty terms our final estimate of the solar reflex velocity is Θ_⊙ = 253 ± 6 km s⁻¹.

This measurement is consistent with both the values of ${{\rm{\Theta }}}_{\odot }={242}_{-3}^{+10}$ km s⁻¹ obtained by Bovy et al. (2012)⁶ and Θ_⊙ = 256 ± 17 km s⁻¹ obtained by Carlin et al. (2012, using all observable constraints applied to their highest-purity fields) to within 1σ. Meanwhile, combining the proper motion of Sgr A* in the Galactic Plane, μ_l = 6.379 ± 0.026 mas yr⁻¹ (Reid & Brunthaler 2004), with the recent, high-precision measure of R₀ = 8.122 ± 0.031 kpc from Gravity Collaboration et al. (2018) yields a value of Θ_⊙ = 245.6 ± 1.4 km s⁻¹. Combining the uncertainties on this R₀ dependent measure of Θ_⊙ and the Θ_⊙ reported here, there is a 1.2σ difference between the two results, which is still a reasonable agreement. While the reflex motion measured with respect to Sgr A* nominally provides higher precision, our estimate is an important, independent probe using a method that does not depend on the Galactocentric radius of the Sun. If we assume that ${V}_{\odot ,\mathrm{pec}}=12$ km s⁻¹ and likewise follow Bovy et al. (2012) in assuming that ${V}_{\mathrm{LSR},\mathrm{pec}}=12$ km s⁻¹, then our results imply a local MW circular velocity of Θ₀ = 229 ± 6 km s⁻¹.

We note that the overall 6 km s⁻¹ uncertainty in our estimate of the solar reflex velocity is driven primarily by the large intrinsic width of the stream. In both the observations and N-body models the 1σ width of the μ_δ distribution is about 0.5 mas yr⁻¹; with 10 longitude bins each containing about 200 stars, this translates to an uncertainty of about 11 μas yr⁻¹ in the mean. Similar efforts using dynamically colder streams may be able to obtain more precise results, and such generalizations of our method to arbitrary streams appear promising (Malhan & Ibata 2017).

We thank the referee for helpful comments. This research made use of topcat (Taylor 2005). C.R.H. acknowledges the NSF Graduate Research Fellowship through grant DGE-1315231.