The Milky WayPart 3 — Stellar kinematicsPhysics of Galaxies 2011

part 8

1

Stellar motions in the MW disk

Let’s continue with the rotation of the Galaxy, this time from the point of view of the stars

First, we need to introduce the concept of the Local Standard of Rest (LSR)

The LSR is the frame rotating with the Galaxy that has zero deviation from the Galactic velocity of circular rotation at the radius of the Solar Neighborhood

2

As you saw when looking at the motion of gas in the MW disk, the Galaxy rotates differentially:

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

How can we tell?

3

Let’s start with the equation we derived last time for the motion of gas in circular orbits with velocity vc at some point R away from the GC:

Here, ω is the angular velocity around the center

At the Sun,

vc = ω ×R

v0 = ω0 ×R0

4

Recall that the line-of-sight (or radial) velocity when seen from the LSR (~the Sun) is then

Using

we can write

Now we need the cosine rule:

To first order in distance,

and

vlos = [ω − ω0]R0 sin l

ω − ω0 =dω

dR

����R0

(R−R0)

vlos =dω

dR

����R0

(R−R0)R0 sin l

R2 = R20 + d2 − 2R0d cos l

R2 ≈ R20 − 2R0d cos l

R2 −R20 = (R−R0)(R + R0) ≈ (R−R0)× 2R0

...at least near the Sun!

5

And so

Therefore we can write

or

where Oort’s A constant is

or, in terms of the magnitude of the circular velocity,

We’ll come back to its physical meaning shortly!

(R−R0) ≈ −d cos l

vlos = − dω

dR

����R0

R0d sin l cos l

vlos = −12

dω

dR

����R0

R0d sin 2l = Ad sin 2l

A = −12R

dω

dR

����R0

A =12

�vc

R− dvc

dR

�

R0

6

Now let’s look at the transverse velocity:

or

Using the same approximation for (R–R0) as before and noting that , we have

vt = [ω − ω0]R0 cos l − ωd

vt =dω

dR

����R0

(R−R0)R0 cos l − ωd

cos2 l = 12 (cos 2l + 1)

vt = d

�−ω − 1

2R0

dω

dR

����R0

(1 + cos 2l)

�

= d[B + A cos 2l]

7

where Oort’s B constant is

Note that the proper motion in the l direction is , so

B = −�

ω +12R

dω

dR

�����R0

= −12

�vc

R+

dvc

dR

�����R0

µl = vt/d µl = B + A cos 2l

8

Combining A and B, we see that the magnitude of the circular velocity at R0 (~Sun) is

and the slope of the rotation curve at R0 is

vc = R0(A−B)

dvc

dR

����R0

= −(A + B)

9

So what are A and B physically?

A is the local (azimuthal) shear in the MW’s rotation curve: the deviation from solid-body rotation

B is the local voriticity, 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

So the rotation curve appears to be gently falling and vc = 218(R0/8 kpc) km s−1

10

Variation of radial and tangential velocity for stars on circular orbits

!"#$"%$&'(&)(%*+(,$'+(&)(-$.*%("'/(%"'.+'%$",(0+,&1$%$+-("-()2'1%$&'(&)(3","1%$1(,&'.$%2/+()&#(/$-4(-%"#-(5&0$'.(&'(1$#12,"#(&#6$%-(

11

So the MW is in differential rotation

Question: why did we go through all this, if we can measure the MW’s rotation curve from the gas?

Historically, the gas velocities couldn’t be measured, and the A and B coefficients are relatively easy to measure if you have enough stars with accurate radial velocities and proper motions — note that near the Sun, distances are not really needed, just velocities as a function of longitude

The interpretation of A and B in terms of differential rotation was a great breakthrough by Oort in understanding the MW

12

Stars in the MW disk do not move in perfectly circular orbits

Although this is no surprise, let’s examine how we know this...

The motion of the Sun with respect to the LSR is called the Solar motion

The Solar motion is defined relative to the mean motion of stars with the same stellar type (gK, dM, etc.)

The Solar motion and the LSR

13

This means that the LSR is defined to be the mean motion of stars in the Solar Neighborhood

In the LSR frame, the velocity components of a star are

(U, V,W ) = (vR, vφ −ΘLSR, vz)

14

Let’s examine the motions of nearby disk stars, using Hipparcos proper motions and measured radial velocities

Their motions should be a reflection of the Solar motion:

The U and W components are independent of color (spectral type):

�v� = −�v⊙�

Stromberg’s asymmetric drift equation [e.g. equation (4-34) of

Binney & Tremaine (1987, hereafter BT)]. That is, V increases

systematically with S2 because the larger a stellar group’s velocity

dispersion is, the more slowly it rotates about the Galactic Centre

and the faster the Sun moves with respect to its lagging frame.

For very early-type stars with B ! V ! 0:1 mag and/or S!

15 km s!1, the V-component of vh i decreases with increasing S,

colour, and hence age contradicting the explanation given in the last

paragraphs. However, the stars concerned are very young, and there

are several possibilities for them not to follow the general trend.

First, because of their youth these stars are unlikely to constitute a

kinematically well-mixed sample; rather, they move close to the

orbit of their parent cloud; many will belong to a handful of moving

groups. Secondly, Stromberg’s asymmetric drift relation predicts a

linear relation between V and S2 only if both the shape of the

velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the

radial density gradient are independent of S. Young stars probably

violate these assumptions, especially the latter one.

The radial and vertical components U0 and W0 of the velocity of

the Sun with respect to the LSR (the velocity of the closed orbit in

the plane that passes through the location of the Sun) can be derived

from hvi estimated for all stars together. The component in the

direction of rotation, V0, may be read off from Fig. 4 by linearly

extrapolating back to S " 0. Ignoring stars blueward of

B ! V " 0 mag we find

U0 " 10:00 " 0:36 #"0:08$ km s!1;

V0 " 5:25 " 0:62 #"0:03$ km s!1;

W0 " 7:17 " 0:38 #"0:09$ km s!1;

#20$

where the possible effect arising from a systematic error of 0:1 mas

in the parallax is given in brackets. When we use this value of the

Local stellar kinematics 391

! 1998 RAS, MNRAS 298, 387–394

Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ! V. Also shown is the variation of the dispersion S with

colour.

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 ! V " 0:

U⊙ = 10.0± 0.4 km s−1

W⊙ = 7.2± 0.4 km s−1

15

On the other hand, the V motion is strongly dependent on color for B–V<0.6

It turns out that the V motion is coupled to the random motions in the disk, due to scattering processes

So we need to find the V motion under the condition of no random motions:

Stromberg’s asymmetric drift equation [e.g. equation (4-34) of

Binney & Tremaine (1987, hereafter BT)]. That is, V increases

systematically with S2 because the larger a stellar group’s velocity

dispersion is, the more slowly it rotates about the Galactic Centre

and the faster the Sun moves with respect to its lagging frame.

For very early-type stars with B ! V ! 0:1 mag and/or S!

15 km s!1, the V-component of vh i decreases with increasing S,

colour, and hence age contradicting the explanation given in the last

paragraphs. However, the stars concerned are very young, and there

are several possibilities for them not to follow the general trend.

First, because of their youth these stars are unlikely to constitute a

kinematically well-mixed sample; rather, they move close to the

orbit of their parent cloud; many will belong to a handful of moving

groups. Secondly, Stromberg’s asymmetric drift relation predicts a

linear relation between V and S2 only if both the shape of the

velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the

radial density gradient are independent of S. Young stars probably

violate these assumptions, especially the latter one.

The radial and vertical components U0 and W0 of the velocity of

the Sun with respect to the LSR (the velocity of the closed orbit in

the plane that passes through the location of the Sun) can be derived

from hvi estimated for all stars together. The component in the

direction of rotation, V0, may be read off from Fig. 4 by linearly

extrapolating back to S " 0. Ignoring stars blueward of

B ! V " 0 mag we find

U0 " 10:00 " 0:36 #"0:08$ km s!1;

V0 " 5:25 " 0:62 #"0:03$ km s!1;

W0 " 7:17 " 0:38 #"0:09$ km s!1;

#20$

where the possible effect arising from a systematic error of 0:1 mas

in the parallax is given in brackets. When we use this value of the

Local stellar kinematics 391

! 1998 RAS, MNRAS 298, 387–394

Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ! V. Also shown is the variation of the dispersion S with

colour.

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 ! V " 0:Stromberg’s asymmetric drift equation [e.g. equation (4-34) of

Binney & Tremaine (1987, hereafter BT)]. That is, V increases

systematically with S2 because the larger a stellar group’s velocity

dispersion is, the more slowly it rotates about the Galactic Centre

and the faster the Sun moves with respect to its lagging frame.

For very early-type stars with B ! V ! 0:1 mag and/or S!

15 km s!1, the V-component of vh i decreases with increasing S,

colour, and hence age contradicting the explanation given in the last

paragraphs. However, the stars concerned are very young, and there

are several possibilities for them not to follow the general trend.

First, because of their youth these stars are unlikely to constitute a

kinematically well-mixed sample; rather, they move close to the

orbit of their parent cloud; many will belong to a handful of moving

groups. Secondly, Stromberg’s asymmetric drift relation predicts a

linear relation between V and S2 only if both the shape of the

velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the

radial density gradient are independent of S. Young stars probably

violate these assumptions, especially the latter one.

The radial and vertical components U0 and W0 of the velocity of

the Sun with respect to the LSR (the velocity of the closed orbit in

the plane that passes through the location of the Sun) can be derived

from hvi estimated for all stars together. The component in the

direction of rotation, V0, may be read off from Fig. 4 by linearly

extrapolating back to S " 0. Ignoring stars blueward of

B ! V " 0 mag we find

U0 " 10:00 " 0:36 #"0:08$ km s!1;

V0 " 5:25 " 0:62 #"0:03$ km s!1;

W0 " 7:17 " 0:38 #"0:09$ km s!1;

#20$

where the possible effect arising from a systematic error of 0:1 mas

in the parallax is given in brackets. When we use this value of the

Local stellar kinematics 391

! 1998 RAS, MNRAS 298, 387–394

Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ! V. Also shown is the variation of the dispersion S with

colour.

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 ! V " 0:

V⊙ = 5.2± 0.6 km s−1

16

On the other hand, the V motion is strongly dependent on color for B–V<0.6

It turns out that the V motion is coupled to the random motions in the disk, due to scattering processes

So we need to find the V motion under the condition of no random motions:

Stromberg’s asymmetric drift equation [e.g. equation (4-34) of

Binney & Tremaine (1987, hereafter BT)]. That is, V increases

systematically with S2 because the larger a stellar group’s velocity

dispersion is, the more slowly it rotates about the Galactic Centre

and the faster the Sun moves with respect to its lagging frame.

For very early-type stars with B ! V ! 0:1 mag and/or S!

15 km s!1, the V-component of vh i decreases with increasing S,

colour, and hence age contradicting the explanation given in the last

paragraphs. However, the stars concerned are very young, and there

are several possibilities for them not to follow the general trend.

First, because of their youth these stars are unlikely to constitute a

kinematically well-mixed sample; rather, they move close to the

orbit of their parent cloud; many will belong to a handful of moving

groups. Secondly, Stromberg’s asymmetric drift relation predicts a

linear relation between V and S2 only if both the shape of the

velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the

radial density gradient are independent of S. Young stars probably

violate these assumptions, especially the latter one.

The radial and vertical components U0 and W0 of the velocity of

the Sun with respect to the LSR (the velocity of the closed orbit in

the plane that passes through the location of the Sun) can be derived

from hvi estimated for all stars together. The component in the

direction of rotation, V0, may be read off from Fig. 4 by linearly

extrapolating back to S " 0. Ignoring stars blueward of

B ! V " 0 mag we find

U0 " 10:00 " 0:36 #"0:08$ km s!1;

V0 " 5:25 " 0:62 #"0:03$ km s!1;

W0 " 7:17 " 0:38 #"0:09$ km s!1;

#20$

where the possible effect arising from a systematic error of 0:1 mas

in the parallax is given in brackets. When we use this value of the

Local stellar kinematics 391

! 1998 RAS, MNRAS 298, 387–394

Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ! V. Also shown is the variation of the dispersion S with

colour.

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 ! V " 0:Stromberg’s asymmetric drift equation [e.g. equation (4-34) of

Binney & Tremaine (1987, hereafter BT)]. That is, V increases

systematically with S2 because the larger a stellar group’s velocity

dispersion is, the more slowly it rotates about the Galactic Centre

and the faster the Sun moves with respect to its lagging frame.

For very early-type stars with B ! V ! 0:1 mag and/or S!

15 km s!1, the V-component of vh i decreases with increasing S,

colour, and hence age contradicting the explanation given in the last

paragraphs. However, the stars concerned are very young, and there

are several possibilities for them not to follow the general trend.

First, because of their youth these stars are unlikely to constitute a

kinematically well-mixed sample; rather, they move close to the

orbit of their parent cloud; many will belong to a handful of moving

groups. Secondly, Stromberg’s asymmetric drift relation predicts a

linear relation between V and S2 only if both the shape of the

velocity ellipsoid, i.e. the ratios of the eigenvalues of j, and the

radial density gradient are independent of S. Young stars probably

violate these assumptions, especially the latter one.

The radial and vertical components U0 and W0 of the velocity of

the Sun with respect to the LSR (the velocity of the closed orbit in

the plane that passes through the location of the Sun) can be derived

from hvi estimated for all stars together. The component in the

direction of rotation, V0, may be read off from Fig. 4 by linearly

extrapolating back to S " 0. Ignoring stars blueward of

B ! V " 0 mag we find

U0 " 10:00 " 0:36 #"0:08$ km s!1;

V0 " 5:25 " 0:62 #"0:03$ km s!1;

W0 " 7:17 " 0:38 #"0:09$ km s!1;

#20$

where the possible effect arising from a systematic error of 0:1 mas

in the parallax is given in brackets. When we use this value of the

Local stellar kinematics 391

! 1998 RAS, MNRAS 298, 387–394

Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ! V. Also shown is the variation of the dispersion S with

colour.

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 ! V " 0:

V⊙ = 5.2± 0.6 km s−1

16

So the magnitude of the Solar motion is |v⊙|=13.4 km s–1, in towards the Galactic Center, up towards the North Galactic Pole, and away from the plane

The trend of V⊙ with S2 is called asymmetric drift, the tendency of the mean rotation velocity (of some population of stars) to lag behind that of the LSR more and more with increasing random motion of the population

Here we define the magnitude of any random motion in direction i as σi = �(vi − �vi�)2�1/2

17

All velocity dispersions of MW disk stars in the Solar Neighborhood increase with color until B–V~0.6

Why this color? Stars with B–V<0.6 all have ages<10 Gyr

Random motions are clearly driven by processes that take time

scattering by GMCs, spiral arms, graininess in MW potential, etc.

solar motion to calculate values of hvyi relative to the LSR for

our stellar groups, Stromberg’s asymmetric drift equation is found

to be

vy

! "

! "j2xx=k; #21$

where k ! 80 ! 5 km s"1 and we have used S2! 1:14j2

xx.

4.3 The velocity dispersion tensor

We divided the 11 865 stars of our sample into nine bins in B " V

with equal numbers of stars in each bin, and determined j2 as

described in Sections 3.3 and 3.4.3. The results are displayed in

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

sampling assuming a multivariate Gaussian distribution in the j2ij

with variance evaluated via equation (15). In the upper panel of

Fig. 5 the three diagonal velocity dispersions and hvfi " hvyi % V0

are plotted versus B " 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-

nents of j2 is the same for all colour bins: jxx > jyy > jzz.

The lower three panels of Fig. 5 show

j0ij " sign#j2

ij$########

jj2ijj

$

#22$

for the mixed components of the velocity dispersion tensor j2.

Evidently, the mixed moments involving vertical motions vanish

within their errors. This is to be expected for essentially all possible

dynamical configurations of the Milky Way. On the other hand, the

mixed dispersion in the plane, j2xy, 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

coordinate frame. We diagonalized the tensor j2 to obtain its

eigenvalues j2i . The square root of the largest of these as well as

the ratios to the smaller ones are given in Table 1. The ratio

j1=j2 ! 1:6, whereas j1=j3 ! 2:2 with a trend for smaller values

at redder colours.

Also given in Table 1 is the ‘vertex deviation’, commonly used to

parametrize the deviation from dynamical symmetry. This is

392 W. Dehnen and J. J. Binney

! 1998 RAS, MNRAS 298, 387–394

Figure 5. Velocity dispersions for stars in different colour bins. The top

panel shows the mean rotation velocity (negative values imply lagging with

respect to the LSR) and the three main velocity dispersions. In the three

bottom panels j0ij " sign#j2

ij$ jj2ijj

1=2 is plotted for the mixed components of

the tensor j2ij.

Table 1. Eigenvalues of j2 for the nine colour bins and all stars beyond Parenago’s discontinuity

(last row).

bin #B " V$ min; max j1 j1=j2 j1=j3 !v

1 "0:238 0.139 14:35%0:49#0:20$"0:40#0:15$ 1:52%0:16

"0:14 2:63%0:94"0:28 30:2%4:7

"5:3

2 0.139 0.309 20:17%0:50#0:20$"0:43#0:20$ 2:11%0:13

"0:28 2:51%0:83"0:10 22:8%2:8

"3:0

3 0.309 0.412 22:32%0:56#0:15$"0:47#0:18$ 1:89%0:14

"0:20 2:40%0:66"0:14 19:8%3:2

"3:4

4 0.412 0.472 26:26%0:80#0:21$"0:59#0:01$ 1:66%0:12

"0:15 2:16%0:52"0:15 10:2%5:0

"5:4

5 0.472 0.525 30:37%0:96#0:22$"0:70#0:21$ 1:66%0:13

"0:15 2:28%0:77"0:18 6:9%5:1

"5:3

6 0.525 0.582 32:93%1:09#0:22$"0:75#0:21$ 1:51%0:13

"0:12 2:19%0:64"0:19 1:9%6:0

"6:1

7 0.582 0.641 37:64%1:37#0:23$"0:94#0:23$ 1:61%0:07

"0:18 1:78%0:48"0:04 10:2%5:6

"6:0

8 0.641 0.719 38:13%0:71#0:23$"0:31#0:22$ 1:60%0:10

"0:15 1:84%0:42"0:08 7:6%5:9

"6:0

9 0.719 1.543 37:20%1:41#0:16$"0:93#0:15$ 1:44%0:12

"0:12 2:04%0:61"0:16 13:1%6:7

"7:6

— 0.610 1.543 37:91%0:79#0:20$"0:63#0:20$ 1:54%0:07

"0:09 1:86%0:21"0:08 10:3%3:9

"3:9

j1, j2, j3 are the roots of the largest, middle and smallest eigenvalues of the velocity dispersion tensor

j2. !v is the vertex deviation (equation 23). Units are mag, km s"1 and degrees for B " V, ji and !v

respectively. The errors 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 brackets (only for

j1; for all other quantities listed the deviations are neglible compared with the uncertainties).

18

The Schwarzschild distribution

Consider a population of, say, oxygen or nitrogen molecules in air at room temperature

Each component of their velocity distribution will have a Gaussian probability distribution, independent of direction

Schwarzschild (1907) proposed that the same might be true for stars, if the Gaussians were dependent on direction: the probability of a star having a velocity in the volumed3v = dv1dv2dv3 is

P (v)d3v =d3v

(2π)3/2σ1σ2σ3exp

�−

3�

i=1

v2i

2σ2i

�

19

The Schwarzschild distribution implies that stars move on velocity ellipsoids

f(v) is constant on ellipsoidal surfaces in velocity space

20

When examining the velocity distributions of old stars (dK,dM) it is clear that the U and W velocities are well-described by the Schwarzschild distribution, but the V velocities are not

The V distribution is skewed towards negative velocities, with a sharp cutoff on the positive side and a long tail to the negative side

21

Why is this?

Stars with V<0 have orbits at smaller radii

Consider a star on an elliptical orbit inside the Solar circle, with pericenter R1<R0

At R1, its velocity is purely tangential and larger than the circular velocity at that point, because it needs to reach larger radii:

Center

R0

R0

R1

Θ(R1) > Θc(R1)

22

At apocenter R0, its velocity is again purely tangential, but now it must be smaller than the circular velocity, because it needs to reach a smaller radius:

There are more stars with V<0 than V>0, because there are more exponentially more stars as you move closer to the center of the Galaxy

The probability that a star from R<R0 visits the Solar neighborhood is higher than those from R>R0, because the velocity dispersion grows exponentially towards the center

Center

R0

R0

R1

Θ(R0 = Rmax) < Θc(R0)

23

Star streamsHowever, the distribution of stars in velocity space is not smooth

projections in U and V space show clear substructure: moving groups/streams

Groups of stars born together

Dynamical perturbations (due to bar, spiral structure)

!"#$%&"$'#(&

) *+'%,-&"$-./"-01%02%&"#$&%-1%3'405-"6%&7#5'%-&%10"%&(00"+8%

) 9%3& :;%:%3& <%740"&%&+0=%&/.&"$/5"/$'&>%(03-1?%?$0/7&>%&"$'#(&

@$-?-1A

)B$0/7&%02%&"#$&%.0$1%"0?'"+'$A%07'1%54/&"'$&%>%#&&05-#"-01&%>%#4&0%&7#"-#4%&"$/5"/$'

)C61#(-5#4%0$-?-1%D4-E'%,/'%7'$"/$.#"-01&%.6%&7-$#4%#$(&;%.#$F

!"#$%$& &%$!"#$%$&'(

&'(%$!"#$%$&') &'*%$!"#

LSR SiriusHyades

Pleiades

24

Kinematics of the thick disk

If the thick disk is selected by abundances, the thick disk appears to lag the LSR by ~40–50 km s-1, with higher velocity dispersions but lower rotational velocity

Thick and Thin Disk 5

Figure 4. Top panel: The open histogram shows the distribu-tion of the rotation velocity (V component) for all stars in thesample. (Units are number of stars per bin.) The shaded his-togram corresponds to those in the thin disk, as defined in thebottom panel of Fig. 2. The remainder are shown by the openhistogram. Bottom panel: V-distribution of stars in the metal-poor tail, i.e., [Fe/H]< !1.5. This illustrates the velocity distri-bution of the slowly-rotating, dynamically-hot stellar halo. Theblue curve illustrates the best fitting gaussian, with "V # = !60km/s and !V = 144 km/s. Middle panel: V-distribution of starswith [Fe/H]> !1.5 but excluding the thin disk. The distributionis strongly non-gaussian, and shows two well-defined peaks; oneat V $ 0 km/s and another at V $ 160 km/s. The latter is welltraced by stars in the “thick disk” component identified in thetop panel of Fig. 2 (with Eu; filled green histogram). Includingstars without Eu measurements but of comparable ["/Fe] ratios(the “Thick” region in the bottom panel of Fig. 2) results in theshaded green histogram. Note that stars near the V $ 0 peakare almost exclusively those in the debris (“D”) region of Fig. 2.They define a non-rotating, dynamically-cold component distinctfrom either the thick disk and the stellar halo. Both the debrisand thick disk components are relatively cold dynamically; theV-distribution can be well approximated by the sum of two gaus-sians with similar velocity dispersion, of order $ 40 km/s (seegreen and magenta solid curves).

3.2 A chemical definition for the thick disk

If our interpretation is correct, then the familiar increasein velocity dispersion with decreasing metallicity for starsin the vicinity of the Sun must result from the increasedprevalence of the thick disk at low metallicity: as mentionedin Sec. 1, the thick disk is metal-poor, lags the thin disk inrotation speed, and has a higher velocity dispersion. Thisis shown in the top panel of Fig. 4, where we compare thedistribution of the rotation speed (V component) of all starsin our sample with that of the thin disk.

The V distribution of stars not in the thin disk is com-plex, and hints at the presence of distinct dynamical compo-nents. It shows two well defined peaks, one at V! 160 km/sand another at V ! 0 km/s, as well as a tail of fast coun-

terrotating stars at highly-negative values of V. The firstpeak corresponds to a rotationally-supported structure: thetraditional thick disk. Stars belonging to this rotating com-ponent seem to disappear from our sample when only metal-poor stars with [Fe/H] < "1.5 are considered (bottom panelof Fig. 4). The latter trace the classical, kinematically-hotstellar halo: the V distribution is consistent with a gaus-sian with velocity dispersion !V ! 144 km/s (shown in thebottom panel with a blue curve).

Interestingly, the V distribution of non-thin-disk starswith [Fe/H] > "1.5 (middle panel of Fig. 4) shows evenmore clearly the double-peak structure noted above. Thepeak at V ! 160 km/s is well traced by the stars identifiedwith the thick disk in the "+Eu panel of Fig. 2, shown bythe solid-shaded histogram in the middle panel of Fig. 4.

Indeed, the peak is traced almost exclusively by starswith high values of ["/Fe]. This is shown by the shadedgreen histogram, which corresponds to all stars in the re-gion labelled “Thick” in the bottom panel of Fig. 2. The Vdistribution of these stars is well approximated by a gaus-sian with #V$ = 145 km/s and !V = 40 km/s that accountsfor nearly all stars with V > 100 km/s.

The association between the thick disk and “high-"”stars is reinforced by inspecting the location of counterro-tating stars in Fig. 2 (shown in cyan). These stars, whichclearly do not belong to a rotationally-supported structurelike the thick (or thin) disk, are evenly distributed amongstars with [Fe/H] < "1.5 but shun the “high-"” region inthe range "1.5 < [Fe/H] < "0.7. Of the 38 counterrotatingstars in our sample with "1.5 < [Fe/H] < "0.7, only 3 lieabove the dotted line that delineates the “Thick” region inthe bottom panel of Fig 2.

It is tempting therefore to adopt a purely chemical def-inition of the thick disk in terms of [Fe/H] and ["/Fe]:

• (i) [Fe/H] > "1.5;• (iii) ["/Fe] > 0.2"([Fe/H]+0.7)/4 for "1.5 < [Fe/H] <

"0.7;• (ii) ["/Fe] > 0.2 for [Fe/H] > "0.7.

If correct, then the thick disk emerges from this analysisas a chemically and kinematically coherent component thatspans a wide range in metallicity ("1.5 <[Fe/H]< "0.3) andcontains mainly stars highly enriched in " elements. Starsin this component have (#U$,#V$,#W$)= (10, 133,"2) km/s,and (!U ,!V ,!W )= (95, 61, 61) km/s.

The average V lags below the ! 160 km/s peak be-cause of the inclusion of a few stars with discrepant veloci-ties (some with negative V) more closely associated with theV ! 0 peak than with a rotationally supported structurelike a thick disk. This suggests that the chemical definitionof the thick disk proposed above is not perfect, and that itincludes spurious stars belonging to a di!erent, non-rotatingcomponent. We turn our attention to that component next.

3.3 Tidal debris in the stellar halo

The last feature of note in the top and middle panels ofFig. 4 is the peak centered at V ! 0. These are stars withno net sense of rotation around the Galaxy and with a sur-prisingly low velocity dispersion. (The gaussian fit to thelow-V tail shown in magenta in the middle panel of Fig. 4has a dispersion of just 40 km/s.) Their vertical (W) velocity

σR = 61 km s−1

σφ = 58 km s−1

σz = 39 km s−1

Vφ ∼ 160 km s−1

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Kinematics of halo starsThe halo stars (selected by abundance) show a very-nearly Schwarzschild distribution in velocities

principal axes closely aligned to (R, Θ, Z) directions

The halo does not rotate (at least the majority of the nearby halo) and has velocity dispersions (140, 105, 95) km s-1

!"#$%&'"()*+,*-&.+*)'&/)

0 1-$*2$.+("'3*4")'/"56'"+#*+,*-&.+*)'&/)*")*(.+)$*'+*7(-8&/9)(-".4*4")'/"56'"+#

0 1-$*:/"#(":&.*&;$)*&/$*(.+)$.3*&."<#$4*'+*'-$*= > >?@*4"/$('"+#)>*&#4*'-$*-&.+*4+$)*#+'*/+'&'$

0 1-$*2$.+("'3*4"):$/)"+#)*&/$*=ABC>*ACD>*ED@*F%G)

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The velocity ellipsoid is aligned along the radial direction

Halo stars move on very eccentric orbits

Halo stars have very distinct motions from disk stars (known at least since Oort 1926) and are easy to pick out in proper motion surveys: very large velocities with respect to the Sun

Large speeds means they must travel very far away from the Galactic Center

Can be used to estimate escape velocity and thus mass of MW

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Kinematically, the bulge is not an extension of the halo

...bulge stars also have higher metallicities, closer to the disk, although the abundance pattern is similar to the halo

The bulge rotates at ~100 km s-1

Velocity dispersion is lower than the disk:

Kinematics of the Galactic Bulge

σlos ∼ 60− 110 km s−1

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How do we select components of the MW in surveys?

Often, we are interested in the properties of a single component of the Galaxy

We can select from a number of criteria:

Spatial distribution

Color

Kinematics

Abundances

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How do we select components of the MW in surveys?

However, no criterion is perfect

Overlap

Contamination

...and 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

selecting by abundance makes this more difficult!

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