Kt and clemens rcurves
THE ROTATION CURVE OF THE MILKY WAY AND SPATIALLY COHERENT STREAMING MOTIONS In this section, we analyze the KT-derived $\vlos$ field in the Galactic plane. The primary purpose of this analysis is to establish that on a coarse scale, we basically agree with the Milky Way kinematics literature. This serves as a complement to the previous section, where we established that we agree with fine-grained measurements of dense HMSFRs. We first decompose the in-plane $\vlos$ field into a rotation curve and residual velocity field under different assumptions about the form of the rotation curve and compare the results to results from the Milky Way kinematics literature. We find that the magnitude and spatial scale of spatially coherent streaming motions is such that we cannot reliably fit a rotation curve without a model for these streaming motions. When we fit a non-flat rotation curve to the in-plane $\vlos$ field, these streaming motions are incorporated into the rotation curve. This non-flat rotation curve is consistent with the rotation curve, which also describes large-scale streaming motions. When we instead attempt to fit a flat rotation curve while assuming residuals from flat rotation are spatially uncorrelated, we get results that change dramatically depending on which subset of the $\vlos$ field we use and which are inconsistent with estimates of the Galactic rotation rate from the literature. Next, we discuss the residual velocity field. We compare it to the two-dimensional residual velocity field of and find that the two agree. We associate some of the clear large-scale streaming motions with features discussed in , , and . Finally, we give a rough estimate of the spatial scale of the strongest fluctuations and compare this to power spectra of velocity fluctuations derived from measurements of gas and stellar kinematics . The rotation curve We assume that the global motion of the Milky Way ISM can be entirely described by a rotation curve, $V(\RGC)$. In particular, we assume that there are no global radial flows and that azimuthal and radial departures from the rotation curve average to zero over the Galaxy. We allow for local, spatially independent departures from the rotation curve and model these departures as a Gaussian perturbation superimposed on the global motion. We assume that the azimuthal and radial components of the velocity perturbation are uncorrelated and have standard deviations σϕ and σR. In addition to the motion of the ISM, we need to describe the location and motion of our observing site, the Sun. We use $$ for the Sun’s Galactocentric radius and Vϕ, ⊙ and VR, ⊙ for its azimuthal and radial motion relative to the Galactic center. Because we will only be using the $\glat \approx 0 \deg$ portion of the $\vlos(\glon, \glat, d)$ cube, we can ignore the Sun’s motion perpendicular to the Galactic plane. To compute the likelihood of a parameter set $V_\phi (\RGC)$, σϕ, σR, R⊙, Vϕ, ⊙, and VR, ⊙, we first need to convert these parameters to a model $(\glon, d)$. The Galactocentric radius of a given $\glon$ and d at $\glat = 0 \deg$ is \RGC = + d^2 - 2 d \cos \glon}. The mean line-of-sight velocity at a $\glon$ and d is (\glon, d) = \sin\glon}{\RGC} - V_{\phi, \odot} \sin \glon + V_{R, \odot} \cos \glon. The standard deviation of the distribution of line-of-sight velocities about this mean is \sigma^2_ = \sin \glon}{\RGC} \right)^2 + \left(\sigma_R \cos \glon + d (1 - 2 \cos^2 \glon)}{\RGC} \right)^2}. To these model $(\glon, d)$ values we compare the mass-weighted $\glat$-average of the KT-derived $\vlos(\glon, \glat, d)$ cube for $-2.5 < \glat < + 2.5$; we will call this $\glat-$averaged quantity the $\vlos(\glon, 0 \deg, d)$ _map_. Our likelihood function for a single $(\glon, 0\deg, d)$ voxel is a Gaussian with mean $(\glon, d)$ and standard deviation $\sigma_(\glon, d)$. The full likelihood function is the product of the single-voxel likelihood functions over all $(\glon, 0\deg, d)$ voxels. We fit two sets of models, one in which we assume a flat rotation curve and let the Solar parameters vary and one in which we assume a piecewise-linear rotation curve with breaks every 0.25 kpc and fix the solar parameters to $\RGC=8.1$ kpc CITEP E.G. BOVY, vϕ, ⊙ = TK, and vR, ⊙ = TK (CITE THAT STANDARDS PAPER THAT’S CITED IN IBID). We use different portions of the $\vlos(\glon, 0\deg, d)$ map for the different models. There are three competing considerations for choosing what portion of the map to use — the quality of the KT solution drops beyond some maximum heliocentric distance, model parameters become less degenerate as the $\glon$ and $\RGC$ ranges of the data widen, and parts of the Galaxy are known to deviate from flat rotation (e.g. CITEALT WRONG ROTATION CURVE). Motivated by the third consideration, our fiducial ranges for the flat rotation curve case are TK and TK. In the piecewise-linear rotation curve case, the first and second considerations are more important, so we use DRANGE TK and RGCRANGE TK, which translates to a non-separable $\glon$-d constraint. For the fiducial $\glon$- and d-ranges, our results for the flat rotation curve model are Vϕ = TK ± TK, σϕ = TK ± TK, σR = TK ± TK, R⊙ = TK ± TK, vϕ, ⊙ = TK ± TK, and vR, ⊙ = TK ± TK. The Vϕ and R⊙ values are in tension with the IAU-recommended values, as well as the results of CITET BOVY and CITET REID, if we take our quoted uncertanties to be accurate. If we vary the outer limit of the d-range used in the fit from TK to TK, Vϕ varies from TK to TK and R⊙ varies from TK to TK. This variation is smooth as a function of the outer limit of the d-range, indicating that the variation is not driven by a small problem region. We believe that the cause of this variation is global model misspecification — we have assumed that deviations from the rotation curve are spatially uncorrelated when they are in fact correlated, and on spatial scales that are of order the spatial extent of the $\vlos(\glon, 0\deg, d)$ map. We will discuss this issue further in Section [sec:rotation_discussion]. Our best-fit piecewise-linear rotation curve is shown, along with the CITET CLEMENS rotation curve for comparison, in Figure [fig:rotation_curves]. Both curves can be qualitatively described as 3 “bumps” or “wiggles” about a mean circular velocity. Pointwise, the two curves agree to within about 10 km/sec. In terms of the qualitative description, the curves are in even better agreement — the mean circular velocities and the locations of the bumps in the two curves are essentially the same, while the amplitudes of the bumps are greater by the same factor of ≈10 km/sec for the KT-derived rotation curve. The differences that are present can most likely be explained by differences in the sets of $\vlos$ measurements that go in to curves’ derivations. Spatially coherent streaming motions The Milky Way velocity field has structure on four main spatial scales — the size of the Galaxy i.e. the rotation curve; ≈2 kpc (e.g. CITEALT CLEMENS for atomic and molecular gas, ) We are looking for motions with spatial scales larger than of order 100 parsecs. [studies such as Clemens85, B&B93, Bovy+15, McG+16 have found high-amplitude motions on scales of about 200 pc, 2 kpc, and the Galaxy. there’s also cloud-cloud motions, seen in high-density tracers (Clemens85). vertical averaging may have washed some stuff out; also most voxels have a long side length > 200pc. so, we don’t expect to cleanly detect the 200 pc motions or cloud-cloud motions.] In Figure [fig:six_pies], we show residuals between the $\vlos$ measurements and (panel b) the flat rotation curve derived from the nearest 4 kpc and (panel c) the piecewise-linear rotation curve. For comparison, we also show residuals between the $\vlos$ measurements and (panel a) our fiducial flat rotation curve and Galactic parameters, (panel d) the non-flat rotation curve, (panel e) the flat rotation curve, (panel f) and the A5 model flat rotation curve. In every panel, there are obvious, multi-kpc regions with consistent non-zero velocity residuals. Most of these regions persist across all six panels, though there is some panel-to-panel variation in the exact bounds and residual magnitude of each region. For example, there is always a blue-shifted feature just outside the solar circle at $90^\circ \lesssim \glon \lesssim 180^\circ$, a smaller red-shifted feature ≈12 kpc from the Galactic center at the high-$\glon$ boundary of the map, and another red-shifted feature just inside the solar circle. Persistence across different sets of rotation curves and Galactic parameters provides evidence that the spatially coherent residuals are not caused by a misspecification of the global Galactic motion and geometry. We can argue for the accuracy of most of the residual regions by taking the HMSFR measurements as truth and appealing to the regions’ strong spatial coherence. The argument goes as follows – if the $\vlos$ map accurately predicts the velocities of HMSFRs that are spatially coincident with a specific residual region, it is reasonable to assume that the $\vlos$ values assigned to the rest of that region are also likely to be accurate. Based on the discussion in Section [sec:KT-validation], we can claim that the HMSFR velocities are accurately predicted everywhere outside of the inner 4-5 kpc of the Galaxy or so. A visual inspection of Figure [fig:maser_pie] shows that there are HMSFRs in every residual region within about 6 kpc of the Sun. We interpret the combination of these two facts as evidence that all of the residual regions shown in Figures [fig:maser_pie] and [fig:six_pies] outside of the inner 4-5 kpc of the galaxy are, in fact, actually present in the velocity field of the Milky Way ISM. Once we have established that the spatially-structured residuals are real, we can ask what they are. In particular, we can ask what fraction of the residuals can be ascribed to radial rotation curve variations and what fraction should instead be interpreted as streaming motions. If we compare the residuals from the two radially-varying rotation curves in Figure [fig:six_pies] (panels c and d) to the residuals from the four flat rotation curves (panels a, b, e, and f), the typical magnitudes of the former are clearly lower than typical magnitudes of the latter. While we could interpret this difference in residual magnitudes as evidence that the residuals are mostly attributable to radial rotation curve variations, that interpretation would only be true for a very empirical definition of a rotation curve. We can consider two definitions of a rotation curve: the azimuthal average of the azimuthal component of the Galactic velocity field (empirical) and the circular velocity in the Galactic gravitational potential (theoretical), both of which may be functions of Galactocentric radius. If azimuthal departures from the theoretical rotation curve, i.e. azimuthal streaming motions, at some Galactocentric radius do not average to zero, then the empirical and theoretical rotation curves at that radius will not be equal. We are particularly prone to cross-talk between spatially coherent streaming motions and radial variations in the rotation curve because of the limited azimuthal coverage of our $\vlos$ measurements. At some radii, a streaming motion that is coherent over a few kiloparsecs is enough to fill most of the azimuth range in our map at those radii. In the inner Galaxy, particularly near the tip of the Galactic bar, this danger has been pointed out by e.g. . Those authors argue that most of the radial variations in the inner Galaxy in , in particular, are actually streaming motions. While our azimuth coverage is wider than that of , the similarity in the rotation curve shapes (Figure rotation curves) suggests that the point still applies. While assume that deriving the rotation curve in the outer Galaxy should be relatively free of hassle, extragalactic studies such as and (some of those Halpha ones) have shown that strongly spiral galaxies can have quite strong streaming motions far away from the bar. So, the advice seems to be to just steer clear of assuming fluctuations in azimuthal velocities have much to do with the theoretical rotation curve – it’s probably all streaming motions.