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
ostep
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
yndenBell
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
solidbody
and
dif
fer
ential
rotation
(MB81,
Chapter
81).
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
nonzero
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
wellmix
ed;
that
is,
slight
dif
ferences
in
orbital
period
will
ha
v
e
had
enough
time
to
spread
out
initiallycorrelated
groups
of
stars,
in
ef
fect
assigning
stellar
orbits
randomlychosen
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
steadystate
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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