Chapter

15

Kinematics

of

the

Solar

Neighborhood

Unlik

e

a

n

elliptical

galaxy

,

the

Milk

y

W

ay

rotates

with

a

speed

much

lar

ger

than

the

random

v

eloci-

ties

of

typical

stars.

Our

position

inside

the

disk

of

the

Milk

y

W

ay

is

both

a

blessing

and

a

curse;

we

can

obtain

v

ery

detailed

data

on

stellar

motions,

b

u

t

must

tak

e

our

o

w

n

motion

into

account

when

interpreting

the

observ

ations.

15.1

Standards

of

Rest

Motions

in

the

Milk

y

W

ay

are

commonly

e

xpressed

with

respect

to

the

•

FSR

:

the

‘fundamental’

standard

of

rest

with

respect

to

the

galactic

center

,

o

r

•

LSR

:

the

‘local’

standard

of

rest

with

respect

to

a

circular

orbit

at

the

Sun’

s

radius.

The

FSR

is

used

when

describing

the

MW

as

a

whole,

while

the

LSR

is

more

useful

for

describing

motions

near

the

Sun.

Since

the

MW

is

approximately

rotationally

symmetric,

we

adopt

a

c

ylindrical

coordinate

sys-

tem;

let

z

be

distance

abo

v

e

the

plane

of

the

MW

,

R

be

the

distance

from

the

galactic

center

in

the

plane

of

the

MW

,

and

!

be

the

azimuthal

coordinate,

measured

in

the

direction

of

the

MW’

s

rotation.

The

LSR

mo

v

e

s

about

the

galactic

center

at

the

local

v

alue

of

the

circular

v

elocity

.

V

elocities

with

respect

to

the

LSR

are

represented

by

v

ertical,

radial,

and

azimuthal

components

w

!

dz

dt

,

u

!"

dR

dt

,

v

!

R

d

!

dt

"

V

(

R

0

)

,

(15.1)

respecti

v

ely

,

where

V

(

R

)

is

the

circular

v

elocity

at

radius

R

,

and

R

0

is

the

Sun’

s

galactic

radius.

Thus

a

star

with

w

>

0

i

s

climbing

abo

v

e

the

galactic

plane;

a

star

with

u

>

0

i

s

f

alling

inw

ard

to

w

ard

the

galactic

center

,

and

a

star

with

v

>

0

i

s

m

o

ving

in

the

direction

of

galactic

rotation

at

greater

then

the

local

circular

v

elocity

.

The

determination

of

the

LSR

is

a

t

w

o-step

process.

Basically

,

w

e

assume

that

o

n

a

v

e

rage

stars

in

the

Sun’

s

vicinity

are

mo

ving

in

the

!

direction

b

u

t

h

a

v

e

n

o

net

motion

in

the

R

or

z

directions.

It

might

also

seem

natural

to

assume

that

on

a

v

erage

the

stars

are

also

mo

ving

in

the

!

direction

with

97

98

CHAPTER

15.

KINEMA

TICS

OF

THE

SOLAR

NEIGHBORHOOD

the

local

circular

v

elocity

,

b

ut

this

is

not

true

because

of

the

asymmetric

drift

(see

belo

w);

the

greater

the

random

v

elocities

of

the

stars,

the

more

the

net

motion

lags

behind

the

local

circular

v

elocity

.

Thus

the

steps

are

1.

measure

the

solar

motion

with

respect

to

an

ensemble

of

nearby

stars

selected

in

a

kinemati-

cally

unbiased

manner

,

and

2.

correct

for

the

asymmetric

drift

due

to

random

motions

of

stars.

Step

#2

may

be

performed

empirically

by

in

v

estigating

ho

w

the

asymmetric

drift

v

elocity

v

a

depends

on

the

radial

v

elocity

dispersion

u

2

for

dif

ferent

sets

of

stars,

and

e

xtrapolating

to

u

2

=

0.

The

data

are

well-

ﬁ

t

b

y

a

linear

relation:

v

a

#

u

2

80

±

5k

ms

"

1

(15.2)

(BM98,

eq.

(10.12)).

Results

for

dif

ferent

ensembles

of

stars

may

be

combined

to

obtain

the

motion

of

the

Sun

with

respect

to

the

LSR:

(

u

,

v

,

w

)

$

=(

10

.

0

±

0

.

4

,

5

.

2

±

0

.

6

,

7

.

2

±

0

.

4

)

km

s

"

1

(15.3)

(BM98,

eq.

(10.11)).

T

o

relate

the

LSR

to

the

FSR,

we

need

to

kno

w

the

local

circular

v

elocity

,

V

0

=

V

(

R

0

)

.

I

n

1985,

IA

U

Commission

33

adopted

V

0

=

220

km

s

"

1

and

R

0

=

8

.

5

kpc

(K

err

&

L

ynden-Bell

1986).

More

recent

estimates

f

a

v

o

r

slightly

smaller

v

alues

of

R

0

;

BM98

adopt

R

0

=

8

.

0

kpc

.

15.2

Effects

of

Galactic

Rotation

The

rotation

of

the

galaxy

gi

v

e

s

rise

to

an

or

ganized

pattern

of

stellar

motions

in

the

vicinity

of

the

Sun.

In

this

section,

these

motions

are

described

under

the

assumption

that

random

v

elocities

are

zero.

15.2.1

Intuiti

v

e

pictur

e

A

physical

understanding

of

the

kinematic

consequences

of

galactic

rotation

may

be

gained

by

considering

separately

the

local

ef

fects

of

solid-body

and

dif

fer

ential

rotation

(MB81,

Chapter

8-1).

If

the

MW

rotated

as

a

solid

body

,

with

angular

v

elocity

"

independent

of

R

,

then

distances

between

stars

w

ould

not

change,

and

all

radial

v

elocities

w

ould

be

zero.

Ho

we

v

e

r

,

stars

w

ould

still

sho

w

proper

motions

with

respect

to

an

e

xternal

frame

of

reference;

the

transv

erse

v

elocity

of

a

star

at

a

distance

r

from

the

Sun

w

ould

be

v

t

=

"

"

r

.

(15.4)

Of

course,

the

MW

does

not

rotate

as

a

solid

body;

the

orbital

period

is

an

increasing

function

of

R

in

the

vicinity

of

the

Sun.

Stars

at

radii

R

<

R

0

therefore

catch

up

with

and

pass

us,

while

we

catch

up

with

and

pass

stars

at

radii

R

>

R

0

.

This

results

in

non-zero

radial

v

elocities,

v

r

#

r

sin

(

2

!

)

,

(15.5)

where

!

is

the

galactic

longitude

of

the

star

under

observ

ation.

15.3.

RANDOM

VELOCITIES

IN

THE

SOLAR

NEIGHBORHOOD

99

15.2.2

Global

f

ormulae

Consider

a

star

at

galactic

radius

R

mo

ving

at

the

circular

v

elocity

appropriate

to

that

radius,

V

(

R

) =

R

"

(

R

)

.

W

ith

respect

to

the

LSR

at

the

Sun’

s

galactic

radius

R

0

,

the

radial

and

transv

erse

components

of

the

star’

s

motion

are

v

r

=(

"

"

"

0

)

R

0

sin

(

!

)

,

v

t

=(

"

"

"

0

)

R

0

cos

(

!

)

"

"

r

,

(15.6)

where

"

0

=

"

(

R

0

)

.

15.2.3

Local

appr

oximations

In

the

local

limit,

where

r

/

R

0

is

a

small

parameter

,

the

abo

v

e

e

xpressions

become

v

r

#

Ar

sin

(

2

!

)

,

v

t

#

(

A

cos

(

2

!

) +

B

)

r

,

(15.7)

where

A

and

B

are

the

Oort

constants,

gi

v

e

n

b

y

A

!

1

2

!

V

R

"

dV

dR

"

0

,

B

!"

1

2

!

V

R

+

dV

dR

"

0

,

(15.8)

where

the

subscript

0

indicates

that

the

e

xpressions

in

parenthesis

are

e

v

aluated

at

the

solar

radius,

R

0

.

I

n

brief,

A

is

a

measure

of

the

shear

of

the

MW’

s

rotation,

while

B

is

a

measure

of

the

v

orticity

.

Observ

ations

of

local

stellar

motions

allo

w

a

direct

estimate

of

the

Oort

constants.

In

practice

these

quantities

are

subject

to

a

number

of

constraints;

a

recent

determination

from

Hipparcos

data

yields

A

#

14

.

8

±

0

.

8k

ms

"

1

kpc

"

1

,

B

#"

12

.

4

±

0

.

6k

ms

"

1

kpc

"

1

(15.9)

(Feast

&

Whitelock

1997).

15.3

Random

V

elocities

in

the

Solar

Neighborhood

Lik

e

the

Sun,

other

stars

ha

v

e

random

v

elocities

with

respect

to

the

LSR.

Complementing

the

dis-

cussion

abo

v

e

,

this

section

will

discuss

random

motions

of

stars

in

our

immediate

neighborhood,

while

ne

glecting

the

lar

ger

-scale

ef

fects

of

rotation.

15.3.1

Theor

etical

expectations

Consider

an

ensemble

of

stars

with

orbits

passing

through

the

vicinity

of

the

Sun.

Since

the

Milk

y

W

a

y

i

s

about

50

rotation

periods

old,

it

is

reasonable

to

assume

that

stars

are

well-mix

ed;

that

is,

slight

dif

ferences

in

orbital

period

will

ha

v

e

had

enough

time

to

spread

out

initially-correlated

groups

of

stars,

in

ef

fect

assigning

stellar

orbits

randomly-chosen

phases.

In

an

axisymmetric

galaxy

,

the

v

elocity

ellipsoid

at

z

=

0

should

then

ha

v

e

principal

ax

es

aligned

with

the

R

,

!

,

and

z

directions.

To

a

ﬁ

rst

approximation,

histograms

of

random

stellar

v

elocities

do

not

dif

fer

much

from

gaus-

sians,

so

a

con

v

enient

approximation

to

the

v

elocity

distrib

ution

with

respect

to

the

LSR

of

a

well-

mix

ed

stellar

ensemble

is

f

(

u

,

v

,

w

) =

f

0

ex

p

(

"

Q

(

u

,

v

"

v

a

,

w

))

,

(15.10)

100

CHAPTER

15.

KINEMA

TICS

OF

THE

SOLAR

NEIGHBORHOOD

where

v

a

is

the

asymmetric

drift

v

elocity

of

the

ensemble,

the

function

Q

(

u

,

v

%

,

w

) =(

u

,

v

%

,

w

)

∙

T

∙

(

u

,

v

%

,

w

)

,

(15.11)

and

the

symmetric

tensor

T

!

1

2

#

$

%

$

2

R

0 0

0

$

2

!

0

0 0

$

2

z

&

'

(

(15.12)

describes

the

shape

of

the

v

elocity

ellipsoid.

Note

that

T

is

proportional

to

the

r

andom

part

of

the

kinetic

ener

gy

tensor

of

the

stellar

ensemble;

it

is

diagonalized

as

a

consequence

of

the

mixing

assumption

made

abo

v

e

.

F

or

this

T

,

the

quantity

Q

(

u

,

v

"

v

a

,

w

)

is

Q

=

1

2

)

u

2

$

2

R

+

(

v

"

v

a

)

2

$

2

!

+

w

2

$

2

z

*

.

(15.13)

15.3.2

Obser

v

ational

r

esults

In

practice

the

v

elocity

ellipsoid

for

a

g

i

v

en

ensemble

of

stars

(

e.

g.

all

stars

of

a

g

i

v

en

stellar

type)

is

ne

v

e

r

prefectly

diagonalized.

The

most

signi

ﬁ

cant

term

to

be

added

to

(15.13)

is

proportional

to

u

(

v

"

v

a

)

,

indicating

that

the

v

elocity

ellipsoid

lies

in

the

plane

of

the

disk,

b

u

t

i

s

not

precisely

oriented

to

w

ard

the

galactic

center

.

The

angle

between

the

long

axis

of

the

v

elocity

ellipsoid

and

the

R

direction

is

called

the

longitude

of

verte

x

,

!

v.

Results

for

a

wide

range

of

stellar

types

are

listed

in

T

ables

10.2

and

10.3

of

BM98.

F

o

r

d

w

arf

stars,

the

v

arious

components

of

the

v

elocity

dispersion

become

greater

progressing

from

early

to

late

spectral

types.

This

is

an

age

ef

fect:

late

spectral

classes

include

more

old

stars,

and

the

random

v

elocities

of

stars

increase

with

time

(presumably

due

to

gra

vitational

scattering,

although

a

complete

theory

is

not

a

v

ailable).

F

o

r

giant

stars,

kinematic

parameters

re

ﬂ

ect

those

of

the

dw

arf

stars

the

y

e

v

olv

ed

from.

The

lar

gest

random

v

elocities

are

found

in

samples

of

white

dw

arfs

which

include

the

lar

gest

fractions

of

v

ery

old

(age

&

10

10

year

)

stars.

The

angle

!

v

depends

on

the

sample

of

stars

studied,

ranging

from

&

30

'

for

the

earliest

main-

sequence

samples

to

&

10

'

for

later

samples.

This

trend

with

spectral

type

is

probably

an

age

ef

fect;

older

stars

are

more

completely

mix

ed.

But

e

v

en

the

latest

samples

still

sho

w

a

signi

ﬁ

cant

de

viation;

spiral

structure

in

the

galactic

disk

may

be

to

blame.

While

the

o

v

erall

distrib

ution

of

random

v

elocities

is

roughly

consistent

with

(15.10),

signi

ﬁ

cant

structures

occur

in

the

stellar

v

elocity

distrib

ution.

These

‘star

streams’

or

‘mo

ving

groups’

are

interpreted

as

the

remains

of

stellar

associations

and

clusters

which

ha

v

e

been

sheared

out

by

the

tidal

ﬁ

eld

of

the

Milk

y

W

ay

.

15.4

Asymmetric

Drift

Empirically

,

the

net

lag

of

a

g

i

v

en

ensemble

of

stars

with

respect

to

the

LSR

is

approximated

by

(15.2)

abo

v

e

.

This

relationship

is

a

consequence

of

the

collisionless

Boltzmann

equation.

In

c

ylin-

drical

coordinates,

the

CBE

for

a

steady-state

axisymmetric

system

is

(

e.

g.

BT87,

Chapter

4.1a)

0

=

v

R

%

f

%

R

+

v

z

%

f

%

z

+

)

v

2

!

R

"

%

&

%

R

*

%

f

%

v

R

"

v

R

v

!

R

%

f

%

v

!

"

%

&

%

z

%

f

%

v

z

.

(15.14)

15.4.

ASYMMETRIC

DRIFT

101

T

o

deri

v

e

the

relationship

for

the

asymmetric

drift

we

tak

e

the

radial

v

elocity

moment

of

(15.14);

multiplying

by

v

R

and

inte

grating

o

v

e

r

all

v

elocities,

the

result

is

a

Jeans

equation:

0

=

R

'

%

%

R

(

'

v

2

R

) +

R

%

%

z

(

v

R

v

z

) +

v

2

R

"

v

2

!

+

R

%

&

%

R

.

(15.15)

The

azimuthal

motion

can

be

di

vided

into

net

streaming

and

random

components:

v

2

!

=

v

!

2

+

$

2

!

=(

V

0

"

v

a

)

2

+

$

2

!

,

(15.16)

where

the

second

equality

follo

ws

from

the

de

ﬁ

nition

of

the

asymmetric

drift

v

elocity

v

a

.

F

or

the

radial

motion,

v

2

R

=

$

2

R

,

(15.17)

since

there

is

no

net

streaming

motion

in

the

radial

direction.

Combining

(15.15),

(15.16),

and

(15.17),

using

the

identity

V

2

0

=

Rd

&

/

dR

,

and

assuming

that

v

a

/

V

0

is

small,

we

obtain

v

a

#

$

2

R

2

V

0

+

$

2

!

$

2

R

"

1

"

%

ln

%

ln

R

(

'$

2

R

)

"

R

$

2

R

%

%

z

v

R

v

z

,

.

(15.18)

This

equation

relates

the

asymmetric

drift

v

elocity

to

the

radial

component

of

the

v

elocity

dispersion.

If

we

compare

ensembles

of

stars

which

ha

v

e

the

same

radial

distrib

ution

and

v

elocity

ellipsoids

of

similar

shapes,

the

e

xpression

inside

square

brack

ets

is

constant

and

we

reco

v

e

r

(15.2).

Se

v

eral

of

the

terms

in

(15.18)

may

be

further

simpli

ﬁ

ed.

By

multiplying

the

CBE

by

v

R

v

!

and

inte

grating

o

v

e

r

all

v

elocities

it

is

possible

to

sho

w

that

$

2

!

$

2

R

=

"

B

A

"

B

,

(15.19)

(BT87,

Ch.

4.2.1(c)).

Here

A

and

B

are

the

Oort

constants;

adopting

the

v

alues

and

uncertainties

in

(15.9),

we

e

xpect

$

2

!

/

$

2

R

#

0

.

46

±

0

.

05.

This

is

reasonably

consistent

with

observ

ational

data,

which

gi

v

e

$

2

!

/

$

2

R

#

0

.

3 t

o 0

.

5

(BM98,

Ch.

10.3.2).

The

most

uncertain

part

of

(15.18)

is

the

last

term

within

the

square

brack

ets.

This

term

rep-

resents

the

tilt

of

the

v

elocity

ellipsoid

at

points

abo

v

e

(and

belo

w)

the

galactic

midplane,

z

=

0.

W

e

do

not

presently

kno

w

v

ery

much

about

the

orientation

of

the

v

elocity

ellipsoid

a

w

a

y

from

the

midplane.

If

the

ellipsoid

remains

parallel

for

all

z

v

alues

then

this

term

is

identically

zero,

while

if

the

ellipsoid

tilts

to

al

w

ays

point

at

the

galactic

center

then

R

$

2

R

%

%

z

v

R

v

z

=

1

"

$

2

z

$

2

R

.

(15.20)

Numerical

e

xperiments

indicate

that

the

most

lik

ely

beha

vior

is

some

where

between

these

tw

o

e

x-

tremes.

102

CHAPTER

15.

KINEMA

TICS

OF

THE

SOLAR

NEIGHBORHOOD

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