Radial Velocities in the Outermost Disk toward the Anticenter

The three components of the velocity derived from proper motions and radial velocities were already analyzed for Gaia data (G18; LS18). Here, however, we will only use the radial velocities. APOGEE provides independent information with a determination of distances different from parallax in Gaia that is more precise for the largest heliocentric distances, so we can reach greater distances with APOGEE than with Gaia. Moreover, given that our analyses will be restricted to the anticenter, proper motions make a negligible contribution to the radial component of the Galactocentric velocity. With only the radial heliocentric velocity v_r, we can obtain the Galactocentric radial velocity (v_R) through (LG16, Equation (4))

$$\begin{eqnarray}\begin{array}{rcl}{v}_{R} & = & -\displaystyle \frac{{v}_{r}}{\cos (\phi +{\ell })\cos b}-\displaystyle \frac{\cos {\ell }}{\cos (\phi +{\ell })}{U}_{\odot }\\ & & -\ \displaystyle \frac{\sin {\ell }}{\cos (\phi +{\ell })}{V}_{g,\odot }+\displaystyle \frac{\tan b}{\cos (\phi +{\ell })}(W-{W}_{\odot })\\ & & +\tan (\phi +{\ell }){V}_{\phi },\end{array}\end{eqnarray} \tag{ 1 }$$

where (ℓ, b) are the Galactic coordinates, ϕ is the Galactocentric azimuth (such that $x=R\cos \phi$, $y=R\sin \phi$, ${\phi }_{\odot }=0$), (U_⊙, V_⊙, W_⊙) is the velocity of the Sun with respect to the local standard of rest (LSR), ${V}_{g,\odot }={V}_{\phi }({R}_{\odot },z=0)+{V}_{\odot };$ ${V}_{\phi }(R,z)$ is the azimuthal velocity, and $W(R,z)$ is the vertical motion. In the calculation of the average or median of v_R, we neglect the possible asymmetries of random peculiar motions.

The disadvantage of Equation (1) is that it is model-dependent, since we need to know the values of V_ϕ and W, but we can make an appropriate selection of regions in which the dependence on them is very small. In particular, as we will see next, toward the anticenter, the dependence on these two functions is negligible.

Here, we adopt the values ${R}_{\odot }=8\,{\rm{kpc}}$, ${V}_{g,\odot }$ = 244 ± 10 km s⁻¹, ${U}_{\odot }=10\pm 1$ km s⁻¹ (Bovy et al. 2012), and ${W}_{\odot }=7.2\pm 0.4$ km s⁻¹ (Schönrich et al. 2010). The azimuthal velocity ${V}_{\phi }(R,z=0)={V}_{c,0}(R)-{V}_{a,0}(R)$ where ${V}_{c,0}$ is the rotation curve, taken from Sofue (2017, Figure 4), and ${V}_{a,0}$ is the asymmetric drift at z = 0. The contribution from the asymmetric drift ${V}_{a,0}$ is negligible, especially for the anticenter region; we include its small contribution as

$$\begin{eqnarray}&&{V}_{a,0}(R)=\displaystyle \frac{{f}_{B}(R){\sigma }_{R}{\left(R\right)}^{2}}{{V}_{c}(R)},\end{eqnarray} \tag{ 2 }$$

with ${f}_{B}(R)\approx 0.27\,R(\mathrm{kpc})$ (Bovy et al. 2012, Figure 4/top). ${V}_{a,0}$ has a value ∼10 km s⁻¹ for R ≲ 15 kpc and somewhat larger for larger R. The generalized expression with dependence on z that takes into account the lower rotation speed for $z\ne 0$ is taken as ${V}_{\phi }(R,z)={V}_{\phi }(R,z=0)-19.2$ km s⁻¹${\left(\tfrac{| z| }{\mathrm{kpc}}\right)}^{1.25}$ (Bond et al. 2010). As the systematic error we take ${\rm{\Delta }}{V}_{\phi }(R,z)={\rm{\Delta }}{V}_{g,\odot }=10$ km s⁻¹ for $R\lt 20\,\mathrm{kpc}$ and ${\rm{\Delta }}{V}_{\phi }(R,z)=(R(\mathrm{kpc})-10)$ km s⁻¹ for R ≥ 20 kpc, in agreement with the errors given in Sofue (2017, Figure 4). We set W = 0, considering negligible the contribution of possible vertical motions; but we take into account the range of possible values in the calculation of systematic errors using ${\rm{\Delta }}W\equiv {W}_{\mathrm{best}\mathrm{fit}}$ of Equation (15) of López-Corredoira et al. (2014).

First, we choose the lines of sight symmetrically around the anticenter, which minimizes the systematic errors. We choose $| b| \lt 10^\circ$, which makes the contribution of W or W_⊙ very low, and $| {\ell }-180^\circ | \lt 20^\circ$, to make $\cos (\phi +{\ell })$ close to one for all of the stars and $\langle \tan (\phi +{\ell })\rangle$ close to zero, which reduces the impact of the error in V_ϕ. The average Galactocentric velocities are shown in Figure 2 for $R-{R}_{\odot }\lt 20\,\mathrm{kpc}$, together with other results of previous studies for $R\lt 18\,\mathrm{kpc}$ in fair agreement with our data. Note that the selected regions in the different surveys are not exactly the same, although all are centered in the anticenter region, so no perfect agreement in the results is expected. The median of V_R(R) is also plotted for the first sample, with results very similar to the average; thus, we will use the "average" hereafter. In Table 1 we give the numerical values. Bins are divided into ${\rm{\Delta }}\mathrm{log}R(\mathrm{kpc})=0.03$, so as to have N ≥ 6 stars in each of them. We plot the average velocity for all the stars, and separately for those at [Fe/H] > −1. The second subsample, which is restricted in metallicity, is almost pure disk stars with a very small contamination of the halo population (López-Corredoira et al. 2018). Both of them show compatible results for $\langle {V}_{R}\rangle$ because, even for the first sample, disk stars dominate. The outcome is an increase in positive velocity (expansion of the disk) from R ≈ 9 kpc to R ≈ 13 kpc, reaching a maximum of ≈6 km s⁻¹, and a decrease outwards, reaching ≈ −10 kpc for R > 17 kpc (negative means contraction of the disk). The signal is significantly larger than the statistical errors (dispersion in the measurements of v_r) and systematic errors (due to the uncertainties in the parameters used in Equation (1)).

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If we divide the sample with [Fe/H] > −1 into two subsamples corresponding to the northern and southern Galactic hemispheres, the results show some differences (Figure 3) but within similar trends. The systematic errors are larger due to the ΔW term when $\langle z\rangle$ is significantly different from zero. If the division is carried out into ℓ < 180° and ℓ > 180°, however, the results are significantly different (Figure 4), indicating also that there is an important dependence of the radial velocity on the azimuth.

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The errors in the determination of heliocentric distance for any given star (around 10% as stated; Figure 5 shows the average error as a function of Galactocentric distance for this line of sight) might produce a smoothing in the function $\langle {v}_{R}(R)\rangle$, but, given that the width of the bins is of the order of these errors, the variation of the function should not be important. The positive or negative signal cannot be due to errors in distance estimates.

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We wonder whether this feature of non-zero Galactocentric radial velocity can also be observed in neutral hydrogen. The anticenter is a region that is usually avoided for the analysis of the 21 cm line, because distances should be derived from heliocentric velocities assuming circular orbits, and thus one would expect zero velocity (except for the corrections for the motion of the Sun with respect to the LSR) at any distance, so there is no way to disentangle the integrated flux. Nonetheless, there might be some net redshift/blueshift or some distortion that could make us suspect that the orbits are not perfectly circular in the gas in the outer disk.

A simple exercise can illustrate this. Assuming that gas and stars have a similar mean velocity, the integrated flux of the antenna temperature in H i in the anticenter should be (corrected to Galactocentric coordinates)

$$\begin{eqnarray}&&T(v)\propto {\int }_{{R}_{\odot }}^{\infty }{\rho }_{{\rm{H}}{\rm{I}}}(R)\exp \left(-\displaystyle \frac{{\left(v-{v}_{R}(R)\right)}^{2}}{2{\sigma }_{R}{\left(R\right)}^{2}}\right){dR},\end{eqnarray} \tag{ 3 }$$

where ${\rho }_{{\rm{H}}{\rm{I}}}(R)$ is the H i density at distance R and σ_R is the dispersion of v_R. We assume a constant velocity dispersion for the H i gas. The observational evidence for a constant dispersion is indirect. Dickey et al. (2009) have shown that in the outer part of the Milky Way up to the observational limit at radii of ∼25 kpc the ratio of the emission to the absorption, which determines the mean spin temperature of the gas, stays nearly constant. This implies on average a constant mixture between cold and warm neutral medium, hence a constant velocity dispersion.

The state of the H i is determined by gas-phase reactions, in equilibrium situations with isotropic velocity dispersions for individual gas clouds. The velocity dispersion of the gas is not related to the dynamical state of the stellar population, but depends on the star formation rate (SFR). Dib et al. (2006) investigated the relationship between the velocity dispersion of the gas, the supernova rate, and feedback efficiency with three-dimensional numerical simulations of supernovae-driven turbulence in the interstellar medium and explained the approximate constancy of observed velocity dispersion profiles in the outer parts of galactic disks. Thermal instabilities in the H i gas, driven by supernovae, determine spin temperatures, composition, and velocity dispersion of the H i.

Tamburro et al. (2009) studied a sample of galaxies with a resolved systematic radial decline in the H i line width, implying a radial decline in kinetic energy density of H i. They find a correlation between the kinetic energy of gas and SFR with a slope close to unity and a similar proportionality constant in all objects.

Observations of velocity dispersions in external galaxies are faced with difficulties in disentangling the radial H i surface density, the rotation curve, and the H i velocity dispersion. O'Brien et al. (2010) observed the H i in edge-on galaxies, where it is in principle possible to measure the force field in the halo vertically and radially from gas layer flaring and rotation curve decomposition respectively. They found (their Figure 21) radial fluctuations of the velocity dispersions, but on average the dispersions are constant in the outer parts of the disks. In all cases a distinct flaring is observed (O'Brien et al. 2010, Figure 25). In fact, for a constant velocity dispersion the gradient of the force field in the vertical direction implies exponential flaring for the gas layer. Such a dependence is also observed in the Milky Way (Kalberla et al. 2007).

In Figure 6 we plot the expected profile of the H i line in observations toward the anticenter for ${\sigma }_{R}(R)=12$ km s⁻¹, ${\rho }_{{\rm{H}}{\rm{I}}}(R)\propto \exp (-R/{h}_{R,{\rm{H}}{\rm{I}}})$ with a gas scale length of ${h}_{R,{\rm{H}}{\rm{I}}}=3.75\,\mathrm{kpc}$ (Kalberla & Dedes 2008), and mean radial velocities null or equal to the observed APOGEE data for all sources (second column in Table 1). The result shows a small asymmetry with respect to perfect circular orbits with slightly larger dispersion for negative velocities: $T(-v)\gt T(v)$. The effect is not very large, but it might be significant. Note, however, that, with these parameters, 40% of the temperature comes from gas at R < 10 kpc and 80% from gas at R < 15 kpc, so the outermost disk features are not dominant in the maps of gas flux.

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As a matter of fact, the results of Liu (2008, Figure 6) or those from the EBHIS survey (Winkel et al. 2016),⁷ reproduced in our Figure 6, show an even larger asymmetry, which would confirm this trend; the larger asymmetry might be due to ${h}_{R,{\rm{H}}{\rm{I}}}\gt 3.75\,\mathrm{kpc}$ or to a higher amplitude of negative v_R for R < 10 kpc (which is affected by very large errors in our APOGEE sample) or because the stars in our region (160° < ℓ < 200°, $| b| \lt 10^\circ$) have on average lower radial velocities than stars in the region of the anticenter alone, or simply because gas presents higher asymmetries than the stars in the velocity distribution, as observed in other galaxies (Pizzella et al. 2008).

The velocity distribution of the 21 cm line emission is usually affected by turbulent motions (Kalberla & Kerp 2009). In Figure 6, we compare observations with significantly different FWHM beam sizes of ∼6° and ∼02. It appears implausible that the asymmetries in both cases are caused by random turbulent motions, but we cannot exclude this possibility.

We extend our analysis using EBHIS observations for ℓ = 180° and latitude −10° < b < 10°. Figure 7 shows a position–velocity diagram for this case. The observed H i emission is shown color-coded and overplotted with model data. The model is based on a global fit of the 3D H i distribution in the Milky Way, including flaring and warping of the H i disk (Kalberla et al. 2007 with updates from Kalberla & Dedes 2008 ⁸ ). Clearly, from the equilibrium model of the H i distribution no asymmetries in velocity are expected for all latitudes at ℓ = 180°. The observed H i emission shows, as usual, significant turbulent fluctuations in intensity and velocity that cannot be modeled. But for all latitudes there is a common trend that the emission is shifted to negative velocities. To quantify this velocity shift we determine velocity centroids (${v}_{{\rm{c}}}=\tfrac{\sum {Tv}}{\sum T};$ Kerp et al. 2016), the first moment of the velocity field, shown by the black line in Figure 7. The average centroid velocity for $| b| \lt 10^\circ$ is −4 km s⁻¹; however, the H i emission for $| b| \gt 5^\circ$ is affected by emission at negative velocities. For latitudes b ≳ 5° there are several clouds at −25 km s⁻¹ ≲ v_LSR ≲ −10 km s⁻¹ with peak brightness temperatures T_B ∼ 50 K. At b ≲ 5° this emission at intermediate negative velocities is weaker, 20 K ≲ T_B ≲ 40 K, but more extended in velocity, −35 km s⁻¹ ≲ v_LSR ≲ −10 km s⁻¹. Emission at positive velocities v_LSR ≳ 15 km s⁻¹ is almost completely absent. Thus the velocity centroid for $| b| \gt 5^\circ$ is biased toward negative velocities. The observed emission at negative velocities is from part of a population of intermediate-velocity clouds, which are most likely located in the transition region between the Galactic disk and halo (Stanimirović et al. 2006). Emission from similar clouds in the negative-velocity wings is insignificant close to the disk at $| b| \lt 5^\circ$ (Figure 7): we determine for this range a centroid velocity of −3.5 km s⁻¹, which we consider characteristic for the H i in the anticenter region.

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Variations and systematic changes in the velocity dispersion of the H i along the line of sight and uncertainties in the density distribution imply uncertainties in the shape of calculated model profiles in Figures 6 and 7, but they do not affect the velocity centroids. The center velocities of individual gas clouds along the line of sight at ℓ = 180° depend merely on the dynamics of the system.

The same analysis may be carried out in regions away from the anticenter, but in these cases the systematic error is much larger because the uncertainties in the rotation curve have an important effect on the conversion of heliocentric to Galactocentric coordinates.

The results are plotted in Figure 8, where we can see that the average velocities are of the order of the systematic errors, rendering these measurements unuseful. We need the proper motions to reduce these systematics in these directions, work that was already done with Gaia-DR2 data in G18 up to R ≈ 13 kpc or in LS18 up to R ≈ 20 kpc.

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In the case of elliptical orbits, ${v}_{R}\ne 0$ would indicate an eccentricity, e, different from zero. We have no information on the dependence of v_R on ϕ, so we cannot derive the exact orbits, but we can statistically derive the most likely values of e for the mean orbit.⁹ In the approximation of low e, the probability of having a given value of the eccentricity e when we measure a radial Galactocentric velocity¹⁰ v_R with rms σ, at a particular point of the orbit, is (LG16, Equation (11))

$$\begin{eqnarray}\begin{array}{rcl}P(e){de} & \approx & \displaystyle \frac{{2}^{1/2}{de}}{{\pi }^{3/2}\sigma \,{V}_{\phi }\,{e}^{2}}{\int }_{-{V}_{\phi }\,e}^{{V}_{\phi }\,e}\displaystyle \frac{| x| }{\sqrt{1-\tfrac{{x}^{2}}{{V}_{\phi }^{2}{e}^{2}}}}\\ & & \times \,\exp \left[-\displaystyle \frac{{\left(x-{v}_{R}\right)}^{2}}{2{\sigma }^{2}}\right]{dx}\end{array}\end{eqnarray} \tag{ 4 }$$

where, as mentioned above, V_ϕ is the azimuthal velocity (as usual, we take the rotation curve data of Sofue 2017, included in their Figure 4, and we subtract the asymmetric drift as above; we neglect here the variation with z). Using this expression, we can evaluate the most likely values of the eccentricity as a function of R derived from our results in the previous section. We carry out the calculation for the anticenter data with [Fe/H] > −1, without taking into account the systematic errors (we take σ equal to the statistical error), which are plotted in Figure 9. We can observe that values of e ≈ 0.07 are the most likely for R ≈ 18 kpc, and lower values for other Galactocentric radii.

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For a particular line of sight in the anticenter, the relative variation of stellar mass within radii between R and R + dR is (LG16, Equation (13))

$$\begin{eqnarray}&&\displaystyle \frac{\dot{M}}{M}(R)=\displaystyle \frac{-1}{R\sigma (R)}\displaystyle \frac{d[\langle {v}_{R}\rangle (R)R{\rm{\Sigma }}(R)]}{{dR}},\end{eqnarray} \tag{ 5 }$$

where Σ(R) is the stellar surface density, assuming a constant average mass/luminosity ratio throughout the disk. With an exponential disk ${\rm{\Sigma }}(R)\propto {e}^{-R/{h}_{R}}$, we obtain that the relative gain of stellar mass toward the anticenter at radius R is

$$\begin{eqnarray}&&\displaystyle \frac{\dot{M}}{M}(R)=\langle {v}_{R}\rangle (R)\left(\displaystyle \frac{1}{{h}_{r}}-\displaystyle \frac{1}{R}\right)-\displaystyle \frac{d\langle {v}_{R}\rangle (R)}{{dR}}.\end{eqnarray} \tag{ 6 }$$

Assuming ${h}_{R}=2.0\pm 0.4\,\mathrm{kpc}$ for a thin disk (López-Corredoira & Molgó 2014), we get the relative mass gain/loss given in Figure 10. It is indeed positive (gain) with a maximum amplitude at R ≈ 15 kpc and negative (loss) with a maximum amplitude at R ≈ 19 kpc.

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The values of Figure 10 indicate a very fast evolution of the mass distribution. Most of the mass will be concentrated in the central regions without much change, but the outer disk has a trend at present to lose stars at R ≲ 11 kpc and R ≳ 16 kpc and gain them at R between 11 and 16 kpc. Assuming a surface density of stars in the solar neighborhood of 27 M_⊙ pc⁻² (McKee et al. 2015), the total variation of stellar mass within ${\rm{\Delta }}\phi =40^\circ =\tfrac{2\pi }{9}$ in our line of sight toward the anticenter is −0.48 ± 0.13 M_⊙ yr⁻¹ between R = 9 and 11 kpc, +0.18 ± 0.06 M_⊙ yr⁻¹ between R = 11 and 16 kpc, and −0.03 ± 0.02 M_⊙ yr⁻¹ between R = 16 and 23 kpc. This is not possible if the Galaxy is in a steady state, since the life of the Galaxy is much longer than the time ${\rm{\Delta }}T={\left(\tfrac{\dot{M}}{M}\right)}^{-1}$ and the size of the Galaxy cannot change so fast. This means that either there is a different trend for other azimuths, or these velocities v_R(R) are not constant with time, so we live now in a period of fast expansion/contraction but this period will be short. This mechanism may contribute to the variation of the disk size over cosmological times. Also some stars will escape the Galaxy and will be part of the halo once they abandon the disk.

This net migration of stars might be related to the hypothesis known as stellar radial migrations, which has indeed been useful to explain breaks in surface brightness (Sánchez-Blázquez et al. 2009), the formation of the thick disk (Sales et al. 2009), disk flaring (Vera-Ciro et al. 2014) or metallicity distributions (Casagrande et al. 2011; Grand et al. 2015), and age–metallicity distributions (Frankel et al. 2018).

Minor mergers may raise vertical waves (Gómez et al. 2013), but these perturbations may not intensively affect the in-plane velocity (Tian et al. 2017). A local stream cannot be the explanation for radial velocities along a wide range of ∼20 kpc, but it could be a large-scale stream associated with the Galaxy in the Sun–Galactic center line. Since we do not have evidence of such a huge structure embedded in our Galaxy, we think this is not very likely. Moreover, there is a very small asymmetry between the northern and southern Galactic hemispheres (Figure 3), thus we think it is very unlikely that the same stream independent of the Galaxy would be so symmetric with respect to the plane.

In principle, the migrations of stars are theoretically expected as a consequence of the resonances of the bar and transient spiral arms in the disk (Sellwood & Binney 2002; Roskar & Debattista 2015; Halle et al. 2018; Hunt et al. 2018). No gradient in the radial velocity is expected from a bar effect for the observed distances (Monari et al. 2014), but it can be produced by spiral arms (Faure et al. 2014).

There might be accelerations due to spiral arms, which pull the stars toward the arm. In such a case, we would see fluctuations around zero in the positions where we cross a spiral arm: around R = 15 kpc (the value at which $\langle {v}_{R}\rangle$ crosses zero in Figure 2), and this is roughly coincident with the position of the Outer (Norma–Cygnus) spiral arm (Camargo et al. 2015; Vallée 2017). Therefore, the influence of the spiral arm looks like a plausible candidate for driving our asymmetry. This mechanism, with compression where the stars enter the spiral arm and expansion where they exit, was also proposed as a tentative explanation for the variations of more local radial velocities by Siebert et al. (2012), Monari et al. (2016) and G18. The association of spiral structure potential with some ripples and ridges in the kinematic data is also proposed (Faure et al. 2014; Hunt et al. 2018; Quillen et al. 2018) without the need to invoke an external perturbation, such as from the passage of the Sagittarius dwarf galaxy (Antoja et al. 2018). In this case, one thing is remarkable from our data in Figure 2: the transition from the highest velocity to the lowest velocity is quite smooth, over 4–5 kpc (although we must also bear in mind that the errors in the distance determination of the order of 10% might also produce this smoothing), much larger than the typical observable width in spiral arms of ∼0.8 kpc (Vallée 2014). Nonetheless, the transition from positive to negative velocities (or vice versa) is also smooth and short in the case of spiral arms and it is not directly linked to the inter-arm distance (Faure et al. 2014, Figure 6). In the simulations by Faure et al., the value of $\langle {v}_{R}\rangle$ is expected to be quite stable in the inter-arm region.

Before we discuss this question, we want to comment briefly on large-scale deviations from circular motions as observed in the H i gas. The spiral density wave theory is based on the hypothesis of quasi-stationary spiral structures (for a recent review see Shu 2016). Accordingly the motion of the gas is assumed to be forced by the background gravitational potential of spiral arms. As the H i gas streams through the spiral pattern it gets compressed and shocked, leading to phase transitions from the warm to the cold neutral medium. Hydrodynamic instabilities may cause the growth of knots in the spiral arms, eventually leading to molecular clouds and star formation. Spiral structure in the H i distribution is observed predominantly from velocity crowding along the line of sight (Burton 1972).

These assumptions have been used by Roberts (1972) and Simonson (1976) (see also the review by Shu 2016) to model spiral structures in the gaseous Milky Way disk. From Figures 5 and 6 of Roberts (1972), we estimate at ℓ = 180° an expected velocity shift of the H i line of −13 to −10 km s⁻¹ for the Perseus arm, approximately three times the centroid velocity measured by us. The plots by Simonson (1976) show also some expected velocity shift, but it is hard to read off numerical values. In both cases the Outer arm is not part of the model. In any case, these numbers should depend on R.

Let us now consider this question: how much mass do we need in the spiral arm overdensity to produce the observed effects? In order to estimate it, let us assume a simple model in which a spiral arm is a ring with radius R_sp, width Δ_sp, and total overdensity mass equal to M_sp. Of course, we know arms are spiral-shaped but for a simple calculation of the local effect of the gravitational force this approximation is good enough. The radial component of the acceleration of a star at Galactocentric radius R is (Feng & Gallo 2011)

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{a}_{R} & = & 2G\sigma {\displaystyle \int }_{{R}_{\mathrm{sp}}-{{\rm{\Delta }}}_{\mathrm{sp}}/2}^{{R}_{\mathrm{sp}}+{{\rm{\Delta }}}_{\mathrm{sp}}/2}S\left[\displaystyle \frac{E(m)}{R(S-R)}-\displaystyle \frac{K(m)}{R(S+R)}\right]\,{dS},\\ m & = & \displaystyle \frac{4{RS}}{{\left(R+S\right)}^{2}};\sigma =\displaystyle \frac{{M}_{\mathrm{sp}}}{2\pi {R}_{\mathrm{sp}}{{\rm{\Delta }}}_{\mathrm{sp}}}.\end{array}\end{eqnarray} \tag{ 7 }$$

K(m) and E(m) are respectively the first and second complete elliptical integrals. The radial velocity of a star follows from ${v}_{R}{{dv}}_{R}={a}_{R}{dR}$. The distribution of mean velocities that we get in our Figure 2 might be a combination of particles with different initial conditions, some of them coming from the inner parts of the Galaxy with initial v_R = 0 and others coming from the fall-off of particles from very high values of R. A simple estimation of the order of magnitude by integrating the previous motion equation numerically gives us that

$$\begin{eqnarray}&&{\rm{Max}}[{v}_{R}]\sim 100\sqrt{\displaystyle \frac{{M}_{\mathrm{sp}}}{2\times {10}^{11}\,{M}_{\odot }}}\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 8 }$$

with a slight dependence on the width of the spiral arms Δ_sp. If we compare this number with the value of +6 km s⁻¹ that we obtained in Figure 2, we get that the mass of the whole spiral arm should be of the order of ∼7 × 10⁸ M_⊙.

If we have four spiral arms in our Galaxy with the same mass as this one, the total mass of the spiral arms would be ∼3 × 10⁹ M_⊙, which is about 3% of the Galactic disk mass (Pérez-Villegas et al. 2015). This is precisely the mass that is attributed to the spiral arms in the models (Pérez-Villegas et al. 2015), in agreement with other observations.

Kalberla et al. (2007) also posited the existence of a dark ring at 13 kpc < R < 18.5 kpc, in order to reproduce the observed flaring in the H i distribution; however, they attribute a mass to this dark ring of (2.2–2.8) × 10¹⁰ M_⊙, which is much larger than our estimation of the necessary mass of a ring. The general assumption of Kalberla et al. (2007) was that the H i distribution is isothermal, which means velocity dispersions that cause the observed scale heights are constant with R; possibly this was a wrong assumption. Anyway, apart from the calculation of the mass, apparently something is taking place at R ≈ 15 kpc that produces important kinematical effects.

In this section we discuss the kinematic properties of two different kinds of system that are caused by different dynamical processes and that can be compared, only from a qualitative point of view, with the observed velocity field. On the one hand we consider a mock galaxy catalog that was built from a cosmological N-body simulation, and on the other hand we discuss the properties of some systems created by the gravitational collapse of isolated overdensities. The signature of the dynamical processes at work in the two cases is imprinted in the amplitude of radial motions: let us start from the latter case.

In B17, it was shown that the purely self-gravitating systems evolving from quite simple initial configurations can give rise easily to a quasi-planar spiral structure surrounding a virialized core and qualitatively resembling a spiral galaxy. In particular, for a broad range of non-spherical and non-uniform rotating initial conditions, gravitational relaxation gives rise quite generically to a rich variety of structure characterized by spiral-like arms; the main characteristic of these structures is that they are features of the intrinsically out-of-equilibrium nature of the system's collapse. That is, they represent long-lived non-stationary configurations characterized by predominantly radial motions in their outermost parts, but they also incorporate an extended flattened region that rotates coherently about a well virialized core of triaxial shape with an approximately isotropic velocity dispersion. Let us consider a few examples.

To characterize the velocity field, we consider the radial component of a particle's velocity ${\boldsymbol{v}}$,

$$\begin{eqnarray}&&{v}_{r}=\displaystyle \frac{{\boldsymbol{v}}\cdot {\boldsymbol{r}}}{| {\boldsymbol{r}}| }\end{eqnarray} \tag{ 9 }$$

and the vectorial "transverse velocity"

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{t}(r)=\displaystyle \frac{{\boldsymbol{r}}\times {\boldsymbol{v}}(r)}{| {\boldsymbol{r}}| },\end{eqnarray} \tag{ 10 }$$

i.e., the vector of which the magnitude is that of the non-radial component of the velocity, but oriented parallel to the particle's angular momentum relative to the origin. We will refer hereafter to the average of a quantity in a spherical shell with a radius r: coherent rotation of all the particles in the shell about the same axis then corresponds to $\langle | {{\boldsymbol{v}}}_{t}| \rangle$ = $| \langle {{\boldsymbol{v}}}_{t}\rangle | ={v}_{\phi }$. The asymmetric drift is defined as (Binney & Tremaine 2008)

$$\begin{eqnarray}&&{v}_{a}={v}_{c}-\overline{{v}_{\phi }}\end{eqnarray} \tag{ 11 }$$

where v_c is the circular speed that corresponds to perfectly circular orbits, i.e.,

$$\begin{eqnarray}&&{v}_{c}^{2}(r)=\displaystyle \frac{{GM}(r)}{r}.\end{eqnarray} \tag{ 12 }$$

To characterize the kinematics further, we also consider the anisotropy parameter $\beta (r)$,

$$\begin{eqnarray}&&\beta (r)=1-\displaystyle \frac{\langle {v}_{t}^{2}(r)\rangle }{2\langle {v}_{r}^{2}(r)\rangle },\end{eqnarray} \tag{ 13 }$$

where $\langle {v}_{t}^{2}(r)\rangle$ and $\langle {v}_{r}^{2}(r)\rangle$ are respectively the average square values of the transverse and radial velocities.

Let us now consider three examples of this class of system (see Figures 11–13): the initial conditions were represented by uniform and isolated ellipsoids (with different ratio between the semi-axes) and with normalized spin parameter (Peebles 1969) λ = 0.1–0.3 (see B17 and Benhaiem et al. 2018 for more details). We note that at sufficiently small scales all simulations are characterized by a compact core with isotropic velocity dispersion, for which β(r) ≈ 0. At larger distances from the system's center, the simulation A1 is characterized by a regime where rotational motions dominate, i.e., v ≈ v_ϕ ≈ v_c and β(r) ≪ 0. In this case, the radial velocity remains small even in the outermost region of the system. On the other hand, the simulation A2, in an intermediate range of distances larger than the core's size but with r/r₀ ≤ 1, is dominated by rotational motions (i.e., v_ϕ > 0) and has negligible net radial motions, i.e., $\langle {v}_{r}\rangle \approx 0$; however, it shows a non-zero radial velocity dispersion so that 0 < β(r) < 1 and thus the asymmetric drift becomes large. Then at larger scales, the system is completely dominated by radial motion so that β(r) ≈ 1. Finally the simulation A3 shows an intermediate situation between A1 and A3: radial motions, in the very outermost region of the system, remain of the same order as the rotational ones.

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By a closer inspection we may note that, in all cases, the behavior of the average radial velocity, when different from zero, is first positive but close to zero, then negative, and then positive again. This is due to the fact that there is a stream of particles, spatially correlated, that are expanding outward and they produce an infall motion of the shells that lie in front of them (see Figure 14). The amplitude of the negative radial velocities reduces with time. Thus, a region where the radial velocity is negative is a natural outcome of these out-of-equilibrium models.

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The two cases where radial motions dominate in the outermost part of the systems, i.e., A2 and A3, are characterized by a noticeable break of axisymmetry, while in the case where radial motions are smaller (i.e., A1) the system remains close to axisymmetric. This is due to the fact that radial motions develop during the collapse because of the amplification of the initial anisotropy: indeed, the simulation A1 was started from an oblate ellipsoid and thus remains symmetric in the plane of the two largest semi-axes—while it contracts in the orthogonal direction parallel to the minor semi-axis. Thus, because of the correlation between the direction of the radial motion and the direction of the major semi-axis, one must take into account that a measurement of the radial velocities in the direction of the anticenter of a given observer (in the real case, the Sun) is not generally expected to give a strong radial signal in these simulations, unless such a direction is (by chance) aligned with the direction of the largest semi-axis or if the departure from axisymmetry is small enough. On the other hand, the more the system is axisymmetric, the higher is the probability that a random observer sees at its anticenter a growing radial velocity but the smaller is its amplitude.

Finally, let us now briefly discuss the difference between the models that we have presented, where the formation of a disk-like flat structure is caused solely by a gravitational, and thus dissipationless, dynamics and models in which instead the formation of a disk-like structure is driven by dissipational effects. Figure 15 shows the velocity components of a mock stellar catalog that was built from a cosmological simulation of the Auriga Project.¹¹ This provides a large suite of high-resolution magnetohydrodynamical simulations of galaxies simulated in a fully cosmological environment by means of the "zoom-in" technique (Grand et al. 2018). One may note in Figure 15 that the average radial component of the velocity is almost zero at all distances, while the velocity of stars is dominated by the circular component. In addition, the radial velocity dispersion is of the same order as the tangential one at large radii, while β(r) < 0 at smaller radii where the motion is predominantly circular.

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Figure 16 shows that the radial velocity field is not characterized by any coherent spatial structure at large enough distances from the center (i.e., r > 5 kpc), where the system is very close to axisymmetric. As mentioned above, the departure from axisymmetry is related to the presence of a complex velocity field, where radial motions become predominant at large distance: systems of this kind can be generated by a dynamics that involves globally the whole structure, i.e., a collapse, rather than a bottom-up aggregation process of the type at work in cosmological simulations.

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