# Kinematics of the Solar

Mechanics

Nov 13, 2013 (4 years and 8 months ago)

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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
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
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
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,
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
R
,
and
R
0
is
the
Sun’
s
galactic
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
v
elocity
dispersion
u
2
for
dif
ferent
sets
of
stars,
and
e
xtrapolating
to
u
2
=
0.
The
data
are
well-

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
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
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
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
R
<
R
0
therefore
catch
up
with
and
pass
us,
while
we
catch
up
with
and
pass
stars
at
R
>
R
0
.
This
results
in
non-zero
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
R
mo
ving
at
the
circular
v
elocity
appropriate
to
that
V
(
R
) =
R
"
(
R
)
.
W
ith
respect
to
the
LSR
at
the
Sun’
s
galactic
R
0
,
the
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
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
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
enough
time
to
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

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
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

cant
term
to
be
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

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

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

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

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
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
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

nition
of
the
asymmetric
drift
v
elocity
v
a
.
F
or
the
motion,
v
2
R
=
\$
2
R
,
(15.17)
since
there
is
no
net
streaming
motion
in
the
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
component
of
the
v
elocity
dispersion.
If
we
compare
ensembles
of
stars
which
ha
v
e
the
same
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

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;
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
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