Physics of Galaxies 2014 Lecture Six: The Milky Way: Kinematics There will be a guest lecture on Tuesday 20th May (11-13) - Peter Barthel will give the lecture on AGN http://www.astro.rug.nl/~etolstoy/pog14 (his area of research interest)
Sparke & Gallagher, chapter 2 6th May 2014
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Milky Way Disc Rotation in a Disc
Solid-Body
3 4 Differential Rotation Differential Rotation
The Galaxy rotates differentially, which means that stars closer to the center take less time to complete their orbits around the Galaxy than those farther out
If the Galaxy rotated as a solid (or rigid) body, then all stars in circular motion would take the same time to complete their orbits around the Galaxy
Solid-Body
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Stellar motions in the Milky Way disc: Stellar motions in the Milky Way disc: Galactic Rotation radial velocities To a good approximation stars and gas in the disc of the Milky Way move in nearly circular paths about the Galactic centre. We can calculate the radial velocity of a star (or a gas cloud), V0 cos(90 l) Stars closer to the Galactic centre complete their orbits in assuming that it follows a circular orbit. less time than than those further out At radius R0 the Sun orbits with speed V0, while a star at P at DIFFERENTIAL ROTATION radius R has orbital speed V(R). The star moves away from us at speed V = V cos ↵ V sin l r 0 sin A sin B sin C This was first discovered from the proper motions of nearby = = stars: we typically see stars orbiting in the same direction a b c but those closer to the Galactic centre pass us in their using the sine rule, we have sin l/R = sin(90 + ↵)/R0 R V cos ↵ orbits; those further away fall behind. cos ↵ = 0 sin l R V V0 and so V = R sin l This effect was already noticed in around 1900, but it wasn't until r 0 R R 1927 that Jan Oort explained this as an effect of Galactic rotation. ✓ 0 ◆
If the Milky Way rotated rigidly, the distance between stars We can take advantage of this orderly motion to map out the distribution from measured velocities. would not change, and Vr would always be 0. In fact stars further from the centre take longer to complete their orbits; the angular speed V/R drops with radius R.
7 8 V V0 V/R decreases Vr = R0 sin l tells us R R for larger R ✓ 0 ◆
o o Vr is positive for nearby For 180 < l < 270 Vr is objects where 0 < l < 90o always positive.
o o For 90 < l < 180 Vr is pattern of first quadrant always negative (R>R0). repeated with sign reversed
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Stellar motions in the Milky Way disc: Stellar motions in the Milky Way disc:
radial velocities a2 = b2 + c2 2bc cos(A) proper motions When the star or gas cloud P is very close to the Sun S, then The proper motion of a star at P relative to the Sun, S can be V0 cos l 2 d ≪ R and R ≈ R0 and we can neglect terms in d , using the calculated in a similar way. The tangential velocity is: cosine rule. 2 2 V0 cos(90 l) R = R 2R0d cos l V = V sin ↵ V cos l 0 t 0 R0 cos l = R sin ↵ + d gives R R0 d cos l ⇡ V V0 d Vt = R0 cos l V R R0 R V cos(90 ↵) V V0 then becomes, for small ✓ ◆ Vr = R0 sin l R R difference in V/R ✓ 0 ◆ Close to the Sun, R0 - R ≈ d cos l, and so Vt varies almost linearly with d(V/R) (V/R) (V/R) distance, d. = R0 sin l (V/R) = = dR R R R0 R d(V/R) d 1 d(RV ) V cos ↵ Vt d cos(2l) d(V/R) ⇡ 2 dR 2 R dR substitute R R = d cos l ✓ ◆ R0 R0 sin l (R R0) 0 ⇡ dR remembering sin 2✓ =2sin✓ cos ✓ d [A cos(2l)+B] R d(V/R) ⌘ d sin(2l) dAsin(2l) B is the second of Oort's constants, and is measured to be ⇡ 2 dR ⌘ ✓ ◆ R0 -12.4±0.6 km/s/kpc
A is one of Oort's constants, and is measured to be 14.8±0.8 km/s/kpc it measures the local vorticity, or angular momentum gradient in the disc. it measures local shear, or deviation from rigid rotation
11 12 Differential Rotation Stellar motions in the Milky Way disc: the Oort constants
Combining A and B, we see that the magnitude of the circular velocity at R0 (~Sun) is: v = R (A B) c 0 and the slope of the rotation curve at R0 is: dv c = (A + B) dR R0 Note that the proper motion in the l direction is µ l = v t /d , so µl = B + A cos 2l A is the local (azimuthal) shear in the Milky Way’s rotation curve: the deviation from solid- body rotation B is the local vorticity, the gradient in the angular momentum in the MW disk locally — the tendency for stars to circulate around a point Recent results: A=14.8±0.8 km s-1 kpc-1 B=–12.4±0.6 km s-1 kpc-1 1 So the rotation curve appears to be gently falling and vc = 218(R0/8 kpc) km s
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Stellar motions in the Milky Way disc: Measuring rotation speeds in the the rotation curve outer Galaxy is harder. Need to find the parallax distances to If we can measure Vr for stars of known distance through out the disc of the Milky Way we can work backward to find V(R), the rotation curve of the Milky Way. associations of young stars. Their Vr is then measured from the Unfortunately, visible light from disc stars and clusters is blocked by dust. Radio waves can travel emission lines of gas around the through dust, and we receive emission in the 21 cm hyperfine transition of atomic hydrogen (HI) stars. The stellar distances are from almost everywhere in the Galaxy. But distances are not straight forward to determine. often not very well determined, but they are good enough to show For the inner galaxy (R < R0), use Tangent-Point method. The tangent point method fails that the rotation speed V(R) does V V not decline much in the outer The angular speed, V/R, drops with radius, so V = R sin l 0 for R ≤ R0, because this is gas r 0 R R Galaxy and may even rise. ✓ 0 ◆ in the bar, which follows oval orbits. tells us that when we look out in the disc along a fixed direction with The gravitational pull of extra o 0 < l < 90 the radial speed Vr(l,R) is greatest at the tangent point (T). mass in spiral arms can easily This is where the line of sight passes closest to the Galactic centre. change the velocity of gas passing through them by 10-20km/s. If a tangent point R = R0 sin l V (R)=Vr + V0 sin l falls close to an arm, then the rotation speed found will different from the average so if there is emitting gas everywhere in the disc, we can find V(R) by speed of an orbit at that measuring the largest velocity seen in emission for each longitude, l. Galactocentric radius.
15 16 Milky Way Rotation Curve
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2 2 2 NGP v = U + V + W Local stellar kinematics 391 | | G.C. The Galactic Disc in 3D 2 2 2 Non-Circular motions v = 10p +5.2 +7.2 =Local 13. stellar4 kinematics 391 U W Stars in the Milky Way disc do not move in perfectly circular orbits. We | | know this by carefully studying the solar motion, which is the motion of Rot. So we need to find the V motion under p V the Sun with respect to the Local Standard of Rest (LSR). the condition of no random motions: 1 The Local Standard of Rest (LSR) is defined as the frame rotating with the Galaxy that has V =5.2 0.6kms ± zero deviation from the circular rotation at the radius of the Solar Neighborhood. So the magnitude of the Solar motion is In the LSR frame, the velocity components of a star are: (U, V, W )=(vR,v LSR,vz) –1 Local stellar kinematics 391 |v⊙|=13.4 km s , in towards the Galactic Center, up towards the North Galactic Pole, Looking at the motions of nearby disk and away from the plane. stars, using Hipparcos proper motions Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ¹ V. Also shown is the variation of the dispersion S with and measured radial velocities, we colour. should see properties similar to the Sun. Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ¹ V. Also shown is the variation of the dispersion S with colour.
2 The U and W components are The trend of V⊙ with S is called independent of color (spectral type): asymmetric drift, the tendency of the mean rotation velocity (of some 1 U = 10.0 0.4kms population of stars) to lag behind that of ± 1 W =7.2 0.4kms the LSR more and more with increasing ± random motion of the population
Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ¹ V. Also shown is the variation of the dispersion S with colour. On the other hand, the V motion is strongly dependent on color. V motion is coupled to the random motions in the disk, due to scattering processes. Figure 4. The dependence of U, V and W on S2. The dotted lines correspond to the linear relation fitted (V) or the mean values (U and W) for stars bluer than B ¹ V ¼ 0:
Figure 4. The dependence of U, V and W on S2. The dotted lines correspond to the linear relation fitted (V) or the mean values (U and W) for stars bluer than 19 StroB ¹¨mberg’sV ¼ 0: asymmetric drift equation [e.g. equation (4-34) of radial density gradient are independent of S20. Young stars probably Binney & Tremaine (1987, hereafter BT)]. That is, V increases violate these assumptions, especially the latter one. systematically with S2 because the larger a stellar group’s velocity The radial and vertical components U and W of the velocity of Stro¨mberg’s asymmetric drift equation [e.g. equation (4-34) of radial density gradient are independent of0 S. Young0 stars probably dispersion is, the more slowly it rotates about the Galactic Centre the Sun with respect to the LSR (the velocity of the closed orbit in Binney & Tremaine (1987, hereafter BT)]. That is, V increases violate these assumptions, especially the latter one. and the faster the Sun moves with respect to its lagging frame. the plane that passes through the location of the Sun) can be derived systematically with S2 because the larger a stellar group’s velocity The radial and vertical components U and W of the velocity of For very early-type stars with B ¹ V Շ 0:1 mag and/or SՇ from hvi estimated for all stars together.0 The0 component in the dispersion is, the more slowly it rotates about the Galactic Centre the Sun with respect to the LSR (the velocity of the closed orbit in 15 km s¹1, the V-component of hiv decreases with increasing S, direction of rotation, V , may be read off from Fig. 4 by linearly and the faster the Sun moves with respect to its lagging frame. the plane that passes through0 the location of the Sun) can be derived colour, and hence age contradicting the explanation given in the last extrapolating back to S ¼ 0. Ignoring stars blueward of For very early-type stars with B ¹ V Շ 0:1 mag and/or SՇ from hvi estimated for all stars together. The component in the paragraphs.¹1 However, the stars concerned are very young, and there B ¹ V ¼ 0 mag we find 15 km s , the V-component of hiv decreases with increasing S, direction of rotation, V , may be read off from Fig. 4 by linearly are several possibilities for them not to follow the general trend. 0 colour, and hence age contradicting the explanation given in the last extrapolating back to S ¼ 0. Ignoring¹1 stars blueward of U0 ¼ 10:00 Ϯ 0:36 ðϮ0:08Þ km s ; First,paragraphs. because However, of their youththe stars these concerned stars are are unlikely very young, to constitute and there a B ¹ V ¼ 0 mag we find V ¼ 5:25 Ϯ 0:62 ðϮ0:03Þ km s¹1; ð20Þ kinematicallyare several possibilities well-mixed for sample; them not rather, to follow they move the general close to trend. the 0 orbit of their parent cloud; many will belong to a handful of moving : ϮϮ : Ϯ : ¹¹11; First, because of their youth these stars are unlikely to constitute a UW00 ¼ 107:1700 00:3836 ðð 00:0809ÞÞ kmkm s s ; groups.kinematically Secondly, well-mixed Stro¨mberg’s sample; asymmetric rather, they drift moverelation close predicts to the a V ¼ 5:25 Ϯ 0:62 ðϮ0:03Þ km s¹1; ð20Þ 2 0 : linearorbit of relation their parent between cloud;V manyand S willonly belong if both to a handful the shape of moving of the where the possible effect arising from¹ a1 systematic error of 0 1 mas W0 ¼ 7:17 Ϯ 0:38 ðϮ0:09Þ km s ; velocitygroups. Secondly, ellipsoid, Stroi.e.¨mberg’s the ratios asymmetric of the eigenvalues drift relation of j predicts, and the a in the parallax is given in brackets. When we use this value of the linear relation between V and S2 only if both the shape of the where the possible effect arising from a systematic error of 0:1 mas ᭧ velocity1998 RAS, ellipsoid, MNRAS i.e.298 the, 387–394 ratios of the eigenvalues of j, and the in the parallax is given in brackets. When we use this value of the Figure 4. The dependence of U, V and W on S2. The dotted lines correspond to the linear relation fitted (V) or the mean values (U and W) for stars bluer than B ¹ V ¼ 0: ᭧ 1998 RAS, MNRAS 298, 387–394
Stro¨mberg’s asymmetric drift equation [e.g. equation (4-34) of radial density gradient are independent of S. Young stars probably Binney & Tremaine (1987, hereafter BT)]. That is, V increases violate these assumptions, especially the latter one. 2 systematically with S because the larger a stellar group’s velocity The radial and vertical components U0 and W0 of the velocity of dispersion is, the more slowly it rotates about the Galactic Centre the Sun with respect to the LSR (the velocity of the closed orbit in and the faster the Sun moves with respect to its lagging frame. the plane that passes through the location of the Sun) can be derived For very early-type stars with B ¹ V Շ 0:1 mag and/or SՇ from hvi estimated for all stars together. The component in the ¹1 15 km s , the V-component of hiv decreases with increasing S, direction of rotation, V0, may be read off from Fig. 4 by linearly colour, and hence age contradicting the explanation given in the last extrapolating back to S ¼ 0. Ignoring stars blueward of paragraphs. However, the stars concerned are very young, and there B ¹ V ¼ 0 mag we find are several possibilities for them not to follow the general trend. Ϯ Ϯ ¹1 First, because of their youth these stars are unlikely to constitute a U0 ¼ 10:00 0:36 ð 0:08Þ km s ; ¹1 kinematically well-mixed sample; rather, they move close to the V0 ¼ 5:25 Ϯ 0:62 ðϮ0:03Þ km s ; ð20Þ orbit of their parent cloud; many will belong to a handful of moving ¹1 W0 ¼ 7:17 Ϯ 0:38 ðϮ0:09Þ km s ; groups. Secondly, Stro¨mberg’s asymmetric drift relation predicts a linear relation between V and S2 only if both the shape of the where the possible effect arising from a systematic error of 0:1 mas velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the in the parallax is given in brackets. When we use this value of the
᭧ 1998 RAS, MNRAS 298, 387–394 392 W. Dehnen and J. J. Binney
solar motion to calculate values of hvyi relative to the LSR for The Galactic Disc in 3D our stellar groups, Stro¨mberg’s asymmetricStellar drift equation ismotions found in the Milky Way disc: to be 2 = ; All velocity dispersions of MW disk stars in vy ¼¹jxx k ð21Þ Asymmetric Drift Ϯ ¹1 2 Ӎ 2 the Solar Neighborhood increase with color where k ¼ 80 5 km s and we have used S 1:14jxx. until B–V~0.6, and all stars with B–V<0.6 have ages<10 Gyr. 4.3 The velocity dispersion tensorasymmetric drift - the net lag of a given ensemble of stars with respect to the LSR is We divided the 11 865 stars of ourapproximated sample into nine by bins in B ¹ V Random motions are clearly driven by with equal numbers of stars in each bin, and determined j2 as 2 processes that take time. described in Sections 3.3 and 3.4.3. The results are displayedv iny = R/(110km/sec) Fig. 5 and Table 1. The errors correspond to the 15.7 and 84.3 percentiles (1j error) and have been evaluated via Monte Carlo e.g., scattering by GMCs, spiral arms, 2 Velocity distribution of stars in the plane of the Milky Way w.r.t. the sun sampling assuming a multivariateThis Gaussian relationship distribution is in a theconsequencejij of the collisionless Boltzmann equation. Stars with larger graininess ino MW potential, etc. o [ in the direction l = 90 (v) and in the direction l = 180 (u) ] : with variance evaluated via equationvelocity (15). dispersions In the upper panel lag ofmore strongly. Fig. 5 the three diagonal velocity dispersions and hvfi ϵ hvyi þ V0 are plotted versus B ¹ V. Parenago’s discontinuity is visible in all three jii and hvfi, although, owing to the larger bin size, less clearly than in Fig. 3 above. The ordering between the diagonal compo- 2 Relative to the LSR, the Sun is moving inwards at 10km/s and travels faster than the nents of j is the same for all colour bins: jxx > jyy > jzz. The lower three panels of Fig.direction 5 show of rotation by 5km/s, with an upward speed of 7km/s.
0 ϵ 2 2 jij signðjijÞ jjijj ð22Þ q IAU recommended value for the Sun's distance from the centre of the Milky Way is for the mixed components of the velocity dispersion tensor j2. R0=8.5kpc and with a rotation velocity of V0=220km/s. These are often used even though Evidently, the mixed moments involving vertical motions vanish
U within their errors. This is to be expectedwe now for believe essentially the all possible true values are more like R0=8kpc and V0=200km/s. dynamical configurations of the Milky Way. On the other hand, the 2 mixed dispersion in the plane, jxy, differs significantly from zero, which is not allowed in a well-mixed axisymmetric Milky Way. Thus the principal axes of j2 are not aligned with our Cartesian V coordinate frame. We diagonalized the tensor j2 to obtain its 2 Figure 5. Velocity dispersions for stars in different colour bins. The top eigenvalues ji . The square root of the largest of these as well as panel shows the mean rotation velocity (negative values imply lagging with the ratios to the smaller ones are given in Table 1. The ratio Conclusions: - different stellar populations have different distributions Ӎ Ӎ respect to the LSR) and the three main velocity dispersions. In the three j1=j2 1:6, whereas j1=j3 2:2 with a trend for smaller values - young stars have a uniform distribution 0 2 2 1=2 21 22 bottom panels jij ϵ signðjijÞ jjijj is plotted for the mixed components of at redder colours. 2 - the “weighted central point” can be used as a velocitythe tensor jij. Also given in Table 1 is the ‘vertex deviation’, commonly used to reference, the so called Local Standard of Rest (LSR) parametrize the deviation from dynamical symmetry. This is
The Schwarzschild distribution 2 The Schwarzschild distribution Table 1. Eigenvalues of j for the nine colour bins and all stars beyond Parenago’s discontinuity (last row). Consider a population of, say, oxygen or nitrogen molecules in air at room temperature. bin ðB ¹ VÞ j j =j j =j ᐉ Each component of their velocity distribution will have a Gaussian probability distribution,min; max 1 1 2 1 3 v 1 ¹0:238 0.139 14:35þ0:49ð0:20Þ 1:52þ0:16 2:63þ0:94 30:2þ4:7 independent of direction. ¹0:40ð0:15Þ ¹0:14 ¹0:28 ¹5:3 þ0:50ð0:20Þ þ0:13 þ0:83 þ2:8 2 0.139 0.309 20:17¹0:43ð0:20Þ 2:11¹0:28 2:51¹0:10 22:8¹3:0 þ0:56ð0:15Þ þ0:14 þ0:66 þ3:2 cutoff sharp 3 0.309 0.412 22:32¹0:47ð0:18Þ 1:89¹0:20 2:40¹0:14 19:8¹3:4 Schwarzschild (1907) proposed that the same might be true for stars in galaxies, with the tail long 4 0.412 0.472 26:26þ0:80ð0:21Þ 1:66þ0:12 2:16þ0:52 10:2þ5:0 Gaussians dependent on direction: the probability of a star having a velocity in the volume ¹0:59ð0:01Þ ¹0:15 ¹0:15 ¹5:4 þ0:96ð0:22Þ þ0:13 þ0:77 þ5:1 5 0.472 0.525 30:37¹0:70ð0:21Þ 1:66¹0:15 2:28¹0:18 6:9¹5:3 is the sum of these three probability distributions. þ1:09ð0:22Þ þ0:13 þ0:64 þ6:0 6 0.525 0.582 32:93¹0:75ð0:21Þ 1:51¹0:12 2:19¹0:19 1:9¹6:1 þ1:37ð0:23Þ þ0:07 þ0:48 þ5:6 7 0.582 0.641 37:64¹0:94ð0:23Þ 1:61¹0:18 1:78¹0:04 10:2¹6:0 3 3 2 d v 8 0.641v 0.719 38:13þ0:71ð0:23Þ 1:60þ0:10 1:84þ0:42 7:6þ5:9 3 3 i ¹0:31ð0:22Þ ¹0:15 ¹0:08 ¹6:0 is P (v)d v = exp þ1:41ð0:16Þ þ0:12 þ0:61 þ6:7 d v = dv1dv2dv3 3/2 9 0.7192 1.543 37:20 1:44 2:04 13:1 (2 ) 1 2 3 2 i ¹0:93ð0:15Þ ¹0:12 ¹0:16 ¹7:6 i=1 þ0:79ð0:20Þ þ0:07 þ0:21 þ3:9 — 0.610 1.543 37:91¹0:63ð0:20Þ 1:54¹0:09 1:86¹0:08 10:3¹3:9
where we define the magnitude of any random motion in directionj i1 ,as:j2, j3 are the roots of the largest, middle and smallest eigenvalues of the velocity dispersion tensor j2. ᐉ is the vertex deviation (equation 23). Units are mag, km s¹1 and degrees for B ¹ V, j and ᐉ v 2 1/2 i v i = respectively.(vi Thev errorsi ) given correspond to the 15.7 and 84.3 percentiles, i.e. 1j error. The maximum deviations resulting from a systematic error of 0:1 mas in the parallaxes are given in bracketsWhen (only for examining the velocity distributions of old stars (dK,dM) it is clear that the U and W j1; for all other quantities listed the deviations are neglible compared with the uncertainties).velocities are well-described by the Schwarzschild distribution, but the V velocities are not. ᭧ 1998 RAS, MNRAS 298, 387–394 The V distribution is skewed towards negative velocities, with a sharp cutoff on the positive side and a long tail to the negative side. NGP
G.C. U W Rot. V 23 24 R0 LSR Sirius Velocities in the plane of the disc Stellar StreamsStar streams Hyades Consider a star on an elliptical orbit inside the Solar The distribution of stars in velocity circle, with pericenter R1 At apocenter R0, its velocity is again purely tangential, U (km/s) but now it must be smaller than the circular velocity, R1 Groups of stars born together and/or because it needs to reach a smaller radius. dynamical perturbations (due to a bar R0 ) U vs V, V vs Wor spiralplots structure) show Stars with V<0 have orbits at smaller radii. substructures/ moving groups/ 0.4< B-V < 0.8 0.6< B-V There are more stars with V<0 than V>0, because there streams are exponentially more stars as you move closer to the V (km/s) center of the Galaxy. The probability that a star from R If the thick disk is selected by abundances, the thick disk appears to lag the LSR by The halo stars (selected by abundance) show a -1 0 The velocity distribution of halo stars ~40–50 km s , with higher velocity dispersions but lower rotational velocity. very-nearly Schwarzschild distribution in Thickis close and to Thin Schwarzschild Disk 5 distribution velocities. 1 R = 61 km s terrotating stars at highly-negativeprincipal axes values closely of V.aligned The firstto (R, Θ, Z) peak corresponds to a rotationally-supported structure: the 1 directions.0 The principal axes are closely aligned = 58 km s traditional thick disk. Stars belongingto the ( to this, ,Z) rotating directions, com- and the halo ponent seem to disappear from our sample when only metal- 1 does not rotate z = 39 km s poor stars with [Fe/H]The< − 1halo.5areconsidered(bottompanel does not appear to rotate and has of Fig. 4). The latter trace the classical, kinematically-hot -1 1 velocity dispersions (140, 105, 95) km s V 160 km s stellar halo: the V distribution is consistent with a gaus- sian with velocity dispersionHalo0 starsTσVhe∼ movevelocity144 km/s on (shownverydispersions eccentric in the areorbits. bottom panel with a blue curve). Interestingly, the V distribution(140, 105, of non-thin-disk 95) km/s stars with [Fe/H] > −1.5(middlepanelofFig.4)showsevenLarge speeds means they must travel very far more clearly the double-peakaway from structure the Galactic noted above. Center. The peak at V ∼ 160 km/s is well traced by the stars identified with the thick disk inCan the α be+Eu used panel to of estimate Fig. 2, shown escape by velocity and thus Halo stars have very distinct motions from disk the solid-shaded histogrammass in of the Milky middle Way panel of Fig. 4. stars (known at least since Oort 1926) and are Indeed, the peak is traced almost exclusively by stars easy to pick out in proper motion surveys: very with high values of [α/Fe]. This is shown by the shaded large velocities with respect to the Sun. green histogram, which corresponds to all stars in the re- gion labelled “Thick” in the bottom panel of Fig. 2. The V NGP distribution of these stars is well approximated by a gaus- sian with #V$ = 145 km/s and σV = 40 km/s that accounts G.C. for nearly all stars with V>100 km/s. U W The association between the thick disk and “high-α” Rot. Figure 4. Top panel: The open histogram shows the distribu- stars is reinforced by inspecting the location of counterro- V tion of the rotation velocity (V component) for all stars in the tating stars in Fig. 2 (shown in cyan). These stars, which sample. (Units are number of stars per bin.) The shaded his- 27 clearly do not belong to a rotationally-supported structure 28 togram corresponds to those in the thin disk, as defined in the like the thick (or thin) disk, are evenly distributed among bottom panel of Fig. 2. The remainder are shown by the open stars with [Fe/H] < −1.5butshunthe“high-α”regionin histogram. Bottom panel: V-distribution of stars in the metal- the range −1.5 < [Fe/H] < −0.7. Of the 38 counterrotating poor tail, i.e., [Fe/H]< −1.5. This illustrates the velocity distri- bution of the slowly-rotating, dynamically-hot stellar halo. The stars in our sample with −1.5 < [Fe/H] < −0.7, only 3 lie blue curve illustrates the best fitting gaussian, with "V # = −60 above the dotted line that delineates the “Thick” region in km/s and σV =144km/s.Middle panel: V-distribution of stars the bottom panel of Fig 2. with [Fe/H]> −1.5butexcluding the thin disk. The distribution It is tempting therefore to adopt a purely chemical def- is strongly non-gaussian, and shows two well-defined peaks; one inition of the thick disk in terms of [Fe/H] and [α/Fe]: at V ∼ 0km/sandanotheratV ∼ 160 km/s. The latter is well traced by stars in the “thick disk” component identified in the • (i) [Fe/H] > −1.5; top panel of Fig. 2 (with Eu; filled green histogram). Including • (iii) [α/Fe] > 0.2−([Fe/H]+0.7)/4for−1.5 < [Fe/H] < stars without Eu measurements but of comparable [α/Fe] ratios −0.7; (the “Thick” region in the bottom panel of Fig. 2) results in the • (ii) [α/Fe] > 0.2for[Fe/H] > −0.7. shaded green histogram. Note that stars near the V ∼ 0peak are almost exclusively those in the debris (“D”) region of Fig. 2. If correct, then the thick disk emerges from this analysis They define a non-rotating, dynamically-cold component distinct as a chemically and kinematically coherent component that from either the thick disk and the stellar halo. Both the debris spans a wide range in metallicity (−1.5 <[Fe/H]< −0.3) and and thick disk components are relatively cold dynamically; the contains mainly stars highly enriched in α elements. Stars V-distribution can be well approximated by the sum of two gaus- in this component have (#U$,#V$,#W$)= (10, 133, −2) km/s, sians with similar velocity dispersion, of order ∼ 40 km/s (see and (σU ,σV ,σW )= (95, 61, 61) km/s. green and magenta solid curves). The average V lags below the ∼ 160 km/s peak be- cause of the inclusion of a few stars with discrepant veloci- 3.2 A chemical definition for the thick disk ties (some with negative V) more closely associated with the V ∼ 0peakthanwitharotationallysupportedstructure If our interpretation is correct, then the familiar increase like a thick disk. This suggests that the chemical definition in velocity dispersion with decreasing metallicity for stars of the thick disk proposed above is not perfect, and that it in the vicinity of the Sun must result from the increased includes spurious stars belonging to a different, non-rotating prevalence of the thick disk at low metallicity: as mentioned component. We turn our attention to that component next. in Sec. 1, the thick disk is metal-poor, lags the thin disk in rotation speed, and has a higher velocity dispersion. This 3.3 Tidal debris in the stellar halo is shown in the top panel of Fig. 4, where we compare the distribution of the rotation speed (V component) of all stars The last feature of note in the top and middle panels of in our sample with that of the thin disk. Fig. 4 is the peak centered at V ∼ 0. These are stars with The V distribution of stars not in the thin disk is com- no net sense of rotation around the Galaxy and with a sur- plex, and hints at the presence of distinct dynamical compo- prisingly low velocity dispersion. (The gaussian fit to the nents. It shows two well defined peaks, one at V∼ 160 km/s low-V tail shown in magenta in the middle panel of Fig. 4 and another at V ∼ 0km/s,aswellasatailoffastcoun- has a dispersion of just 40 km/s.) Their vertical (W) velocity Kinematics of the Galactic Bulge How do we select components of the Milky Way? Often, we are interested in the properties of a single component of the Galaxy Kinematically, the bulge is not an extension of the halo We can select from a number of criteria: Bulge stars also have higher metallicities, closer to the disk. • Spatial distribution The bulge rotates at ~100 km s-1 • Color 1 Velocity dispersion is lower than the halo but higher than the disk: los 60 110 km s • Kinematics • Abundances However, no criterion is perfect, due to overlap, contamination etc. Selecting one criterion may bias the results that you’re seeking e.g., there is a huge fight in the literature right now about whether the thick disk is really a separate entity or just an extension of the thin disk 29 30