Publications
of
the
Astronomical
Society
of
the
Pacific
Vol.
104
1992
November
No.
691
Publications
of
the
Astronomical
Society
of
the
Pacific
104:
981-1034,
1992
November
Binaries
in
Globular
Clusters
1
Piet
Hut
Institute
for
Advanced
Study,
Princeton,
New
Jersey
08540
Electronic
mail:
piet@guinness.ias.edu
Steve
McMillan
Department
of
Physics
and
Atmospheric
Science,
Drexel
University,
Philadelphia,
Pennsylvania
19104
Electronic
mail:
steve@zonker.drexel.edu
Jeremy
Goodman
2
Princeton
University
Observatory,
Princeton,
New
Jersey
08544
Electronic
mail:
jeremy@astro.princeton.edu
Mario
Mateo
3
The
Observatories
of
the
Carnegie
Institution
of
Washington,
Pasadena,
California
91101
Electronic
mail:
mateo@ociwl.caltech.edu
E.
S.
Phinney
2,4
Theoretical
Astrophysics,
151-33,
California
Institute
of
Technology,
Pasadena,
California
91125
Electronic
mail:
esp@tapir.caltech.edu
Carlton
Pryor
5-7
Department
of
Physics
and
Astronomy,
Rutgers
University,
Piscataway,
New
Jersey
08855
Electronic
mail:
pryor@pryor.rutgers.edu
Harvey
B.
Richer
5
Department
of
Geophysics
and
Astronomy,
University
of
British
Columbia,
Vancouver,
BC
V6T
1Z4,
Canada
Electronic
mail:
h.richer@mtsg.ubc.ca
Frank
Verbunt
Astronomical
Institute,
University
of
Utrecht,
P.O.
Box
80000,
3508
TA
Utrecht,
The
Netherlands
Electronic
mail:
verbunt@fys.ruu.nl
Martin
Weinberg
Department
of
Physics
and
Astronomy,
University
of
Massachusetts,
Amherst,
Massachusetts
01003
Electronic
mail:
weinberg@owl.phast.umass.edu
Received
1992
August
9;
accepted
1992
August
19
ABSTRACT.
Recent
observations
have
shown
that
globular
clusters
contain
a
significant
binary
pop-
ulation.
This
is
a
dramatic
change
from
the
conventional
view
of
even
a
decade
ago,
which
held
that
globular
clusters
formed
without
any
binaries
at
all,
since
the
observed
X-ray
binaries
were
understood
to
be
formed
through
dynamical
capture.
Over
the
last
few
years,
a
number
of
different
observational
techniques
have
resulted
in
the
detection
of
a
substantial
number
of
binaries
most
of
which
are
believed
to
be
primordial.
When
the
many
selection
effects
are
taken
into
account,
these
detections
translate
into
a
binary
abundance
in
globular
clusters
that
may
be
somewhat
smaller
than
those
in
the
Galactic
disk
and
halo,
but
not
by
a
large
factor.
Within
the
current
uncertainties,
it
is
even
possible
that
the
primordial
binary
abundance
in
globular
clusters
is
comparable
to
that
in
the
Galactic
disk.
We
discuss
different
successful
optical
search
techniques,
based
on
radial-velocity
variables,
photometric
variables,
and
the
positions
of
stars
in
the
color-magnitude
diagram.
In
addition,
we
review
searches
in
other
wavelengths,
which
have
turned
up
low-mass
X-ray
binaries
and
more
recently
a
variety
of
radio
981
©
1992.
Astronomical
Society
of
the
Pacific
982
HUT
ET
AL.
pulsars.
On
the
theoretical
side,
we
give
an
overview
of
the
different
physical
mechanisms
through
which
individual
binaries
evolve.
We
discuss
the
various
simulation
techniques
which
recently
have
been
employed
to
study
the
effects
of
a
primordial
binary
population,
and
the
fascinating
interplay
between
stellar
evolution
and
stellar
dynamics
which
drives
globular-cluster
evolution.
TABLE
OF
CONTENTS
1.
2.
Introduction
Observations
2.1
Radial-Velocity
Variables
2.1.1
Radial-Velocity
Searches
for
Binary
Stars
in
Globular
Clusters
2.1.2
Implementing
a
Survey
2.1.3
Current
Results
2.1.4
Future
Prospects
2.2
Photometric
Variables
2.2.1
Introduction
2.2.2
Eclipsing
Binaries
in
Globular
Clusters
2.2.3
Cataclysmic
Variables
2.2.4
Blue
Stragglers
2.3
Binaries
in
the
Color-Magnitude
Diagram
2.3.1
The
“Second
Sequence”
2.3.2
Effects
of
Crowding
on
Estimates
of
Binary
Frequency
2.3.3
Maximum-Likelihood
Estimates
2.3.4
Testing
Assumptions
about
the
Mass-
Ratio
Distribution
2.3.5
Other
Evidence
for
Binaries
2.3.6
Summary
2.4
X-ray
Binaries
2.4.1
Observations
of
the
Bright
Sources
2.4.2
Individual
Systems
2.4.3
They
are
Neutron
Stars
...
2.4.4
...
but
are
they
Binaries?
2.4.5
Lifetime
of
the
Sources
2.4.6
Dim
X-ray
Sources
2.4.7
The
Nucleus
of
M33
2.5
Pulsars
2.5.1
Formation
Mechanisms
2.5.2
Radio
Searches
3.
Theory
3.1
Globular-Cluster
Evolution
3.1.1
Core
Collapse
3.1.2
Post-Collapse
Evolution
of
Globular
Clusters
3.1.3
The
Central
Energy
Source
3.1.4
Core
Oscillations
3.2
Binary-Star
Evolution
3.2.1
Physical
Mechanisms
3.2.2
Point-Mass
Dynamics
3.2.3
Tidal
Capture
3.2.4
Stellar
Evolution
3.3
A-Body
Simulations
3.3.1
Direct
Integration
Methods
3.3.2
Binary
Formation
3.3.3
Primordial
Binaries
3.4
Fokker-Planck
Simulations
3.4.1
Fokker-Planck
Formalism
3.4.2
Three-body
binaries
3.4.3
Tidal-capture
binaries
3.4.4
Primordial
Binaries
3.5
Stochastic
Simulations
3.5.1
A
Simple
Model
with
Binary-Binary
Scattering
3.5.2
A
More
Realistic
Treatment
of
Binary/Single-Star
Encounters
4.
Summary
and
Outlook
4.1
Radial-Velocity
Variables
4.2
Photometric
Variables
4.3
Binaries
in
the
Color-Magnitude
Diagram
4.4
X-ray
Binaries
4.5
Pulsars
4.6
A-Body
Simulations
4.7
Fokker-Planck
Simulations
4.8
Stochastic
Simulations
1.
INTRODUCTION
Only
a
dozen
years
ago,
there
was
no
evidence
for
a
substantial
population
of
binaries
in
any
globular
cluster.
invited
review
paper.
2
A.
P.
Sloan
Foundation
Fellow.
3
Hubble
Fellow.
Presidential
Young
Investigator.
5
Visiting
Astronomer,
Canada-France-Hawaii
Telescope,
operated
by
the
National
Research
Council
of
Canada,
the
Centre
National
de
la
Recherche
Scientifique
of
France,
and
the
University
of
Hawaii.
6
Work
based
partly
on
observations
made
at
the
Multiple
Mirror
Tele-
scope,
a
joint
facility
of
the
Smithsonian
Astrophysical
Observatory
and
the
University
of
Arizona.
7
Visiting
Astronomer,
Cerro
Tololo
Inter-American
Observatory
and
Kitt
Peak
National
Observatory,
National
Optical
Astronomy
Observa-
tories,
which
are
operated
by
the
Association
of
Universities
for
Re-
search
in
Astronomy,
Inc.
(AURA),
under
cooperative
agreement
with
the
National
Science
Foundation.
On
the
contrary,
a
systematic
search
for
radial-velocity
variables
by
Gunn
and
Griffin
(
1979)
did
not
discover
any
spectroscopic
binaries.
This
was
interpreted
by
them
to
imply
that
globular
clusters
are
significantly
deficient
in
binaries
with
respect
to
a
younger
Galactic
population.
Because
we
still
have
only
a
rudimentary
understanding
of
the
process
of
binary
formation,
it
is
not
possible
to
predict
theoretically
what
fraction
of
stars
should
have
been
formed
in
binaries
in
globular
clusters,
even
in
the
idealized
case
where
we
knew
the
initial
conditions
under
which
the
clusters
formed.
Therefore,
it
even
seemed
pos-
sible,
if
perhaps
not
plausible,
that
globular
clusters
could
have
been
born
without
any
binaries
whatsoever.
In
this
case,
the
12
observed
X-ray
sources
and
the
2
novae
might
have
formed
through
the
dynamical
capture
by
main-
sequence
stars
of
neutron
stars
and
white
dwarfs,
respec-
tively.
BINARIES
IN
GLOBULAR
CLUSTERS
983
As
a
consequence,
nearly
all
theorists
happily
modeled
globular
clusters
as
if
indeed
all
stars
had
started
off
single.
This
assumption
was
made
partly
for
reasons
of
simplicity
and
economy.
However,
another
reason
was
that
theorists
were
occupied
in
the
early
1980s
with
the
then
unresolved
question
concerning
the
fate
of
globular
clusters
during
and
after
core
collapse.
As
long
as
this
problem
was
not
solved
for
a
population
of
single
stars,
and
as
long
as
there
were
no
compelling
observational
reasons,
it
did
not
seem
necessary
to
worry
about
primordial
binaries.
After
a
coherent
picture
of
post-collapse
evolution
be-
gan
to
emerge
in
the
mid-1980s,
simulations
of
cluster
ev-
olution
became
increasingly
refined
and
detailed.
Around
the
same
time,
a
number
of
different
observational
tech-
niques
began
to
offer
a
more
bountiful
picture
of
the
globular-cluster
binary
population.
The
last
few
years,
es-
pecially,
have
witnessed
a
rich
harvest
of
direct
and
indi-
rect
binary
detections.
In
Sec.
2
we
discuss
the
main
observational
techniques
that
have
recently
been
successful
in
identifying
and
ana-
lyzing
binaries
in
globular
clusters.
In
Sec.
2.1
we
review
how
red-giant
radial-velocity
measurements
have
led
to
the
discovery
of
a
number
of
relatively
wide
binaries.
In
Sec.
2.2
we
discuss
the
identification
of
(much
shorter
period)
photometrically
variable
binaries.
In
Sec.
2.3
we
describe
how
the
presence
of
a
binary
population
has
been
deduced
and
quantified
from
analyses
of
color-magnitude
diagrams.
Finally,
in
Secs.
2.4
and
2.5
we
review
the
X-ray
binary
and
binary
pulsar
content
of
globular
clusters.
In
response
to
all
these
detections,
theorists
finally
be-
gan
to
include
primordial
binaries
in
their
simulations.
As
expected
on
the
grounds
of
earlier
analytic
estimates,
even
a
modest
fraction
of
primordial
binaries
turned
out
to
be
sufficient
to
dramatically
alter
the
evolution,
as
well
as
some
of
the
observational
properties,
of
globular
clusters
after
core
collapse.
In
Sec.
3
we
first
summarize
the
current
physical
picture
of
cluster
evolution
(Sec.
3.1),
and
of
the
evolution
of
individual
binaries
(Sec.
3.2).
We
then
review
the
different
types
of
recent
simulations
of
globular-cluster
evolution,
using
direct
A-body
techniques
(Sec.
3.3),
Fokker-Planck
methods
(Sec.
3.4),
and
stochastic
(Monte
Carlo)
models
(Sec.
3.5).
Finally,
Sec.
4
sums
up,
and
presents
an
outlook
for
developments
in
the
near
future.
It
was
this
summary
and
outlook,
arrived
at
during
an
informal
gathering
at
the
Institute
for
Advanced
Study
in
the
Summer
of
1991,
which
stimulated
us
to
write
the
present
review.
Originally,
we
had
planned
that
small
round-table
meeting
as
a
one-
day
consulting
session,
to
let
the
observers
tell
the
theorists
what
is
out
there
to
be
explained,
and
to
let
the
theorists
tell
the
observers
what
they
thought
would
still
be
out
there
waiting
to
be
discovered.
Since
all
of
us
learned
so
much
from
this
exchange,
we
decided
by
the
end
of
the
day
to
bundle
our
contributions
into
a
review.
The
authorship
of
the
following
sections
and
subsec-
tions
cannot
be
disentangled
exactly,
since
the
manuscript
has
gone
through
a
large
number
of
revisions
with
mutual
comments,
and
has
seen
whole
blocks
of
text
being
shuffled
around
to
increase
readability.
However,
the
main
authors
of
the
subsections
of
Secs.
2
and
3
can
be
listed
as
follows:
Sec.
2.1
by
C.P.;
Sec.
2.2
by
M.M.;
Sec.
2.3
by
H.B.R.
and
M.W.;
Sec.
2.4
by
F.V.;
Sec.
2.5
by
E.S.P.;
Sec.
3.1
by
P.H.;
Sec.
3.2
by
S.M.,
E.S.P.,
and
F.V.;
Sec.
3.3
by
P.H.
and
S.M.;
Sec.
3.4
by
J.G.;
Sec.
3.5
by
S.M.
and
E.S.P.
2.
OBSERVATIONS
Recently,
a
variety
of
techniques
has
revolutionized
our
view
of
the
stellar
populations
of
globular
clusters.
Be-
tween
them,
they
cover
a
wide
range
of
binary
types
and
orbital
parameters.
New
CCD
cameras
and
fiber-fed
spec-
trographs
have
greatly
improved
our
capabilities
in
the
optical,
while
radio
observations
have
complemented
X-
ray
observations
in
giving
us
detailed
insight
into
the
neu-
tron
star
population,
both
as
single
stars
and
as
members
of
binaries.
In
this
section,
we
describe
these
observational
techniques.
For
each
we
present
the
most
recent
results
concerning
the
binary
population
in
globular
clusters.
2.1
Radial-Velocity
Variables
2.1.1
Radial-Velocity
Searches
For
Binary
Stars
In
Globular
Clusters
What
is
the
frequency
of
primordial
binaries
in
globular
clusters
today?
Does
the
frequency
correlate
with
other
cluster
properties?
What
is
the
radial
distribution
of
bina-
ries
in
clusters?
Have
primordial
binaries
been
destroyed
in
globular
clusters
by
dynamical
processes?
One
tool
that
can
address
these
questions
is
a
survey
for
radial-velocity
variables.
The
most
luminous
giants
in
well-studied
globular
clus-
ters
typically
have
V=
12-13,
which
is
already
faint
for
making
radial-velocity
measurements
with
a
precision
of
1
kms
-
1
.
As
a
result,
most
surveys
have
examined
stars
within
several
magnitudes
of
the
tip
of
the
giant
branch.
Such
surveys
have
three
useful
features:
complete
radial
coverage,
the
ability
to
derive
some
properties
of
the
binary
orbits,
and
selection
effects
that
are
relatively
easy
to
cal-
culate
on
the
probability
of
discovering
a
binary.
Even
at
ground-based
observatories,
the
brighter
cluster
stars
can
be
observed
anywhere
in
the
cluster,
so
complete,
magnitude-limited
samples
are
obtainable.
Such
samples
determine
the
binary
frequency
without
uncertainties
about
the
radial
distribution
of
binaries
and
allow
that
distribution
to
be
explicitly
determined.
With
accurate
color-magnitude
diagrams,
radial
velocities,
and,
in
some
cases,
proper-motion
studies,
the
cluster
membership
of
the
binaries
discovered
is
relatively
secure.
Continuing
radial-velocity
observations
can
determine
each
system’s
orbital
period
and
eccentricity.
The
distribu-
tion
of
periods
is
itself
interesting,
since
dynamical
pro-
cesses
preferentially
eliminate
the
wider
systems
(cf.
Heg-
gie
1975;
Hut
1983).
When
the
period
is
known,
it
is
also
possible
to
discriminate
between
tidal
capture
and
primor-
dial
binaries
and
to
place
some
constraints
on
the
system’s
mass.
Some
of
these
points
and
the
selection
effects
are
discussed
in
more
detail
in
Sec.
2.1.2.
984
HUT
ET
AL.
Velocity
surveys
of
luminous
giants
also
have
problems.
The
large
radii
of
these
stars,
0.1-0.4
AU,
impose
a
bias
on
the
periods
detectable.
Binary
systems
with
periods
shorter
than
about
40
days
(separations
less
than
about
0.25
AU)
will
not
reach
the
luminosities
required
to
be
included
in
magnitude-limited
samples
because
they
will
have
been
in-
volved
in
mass
transfer
that
either
truncates
the
evolution
of
the
giant
or
leads
to
coalescence
through
a
common-
envelope
stage
(e.g.,
see
the
discussion
in
Pryor
et
al.
1988,
hereafter
PLH,
and
in
Sec.
3.2.4).
This
bias
towards
long
periods
and
low
orbital
velocities
also
means
that
higher
velocity
precisions
and
longer
time
baselines
are
needed
to
discover
those
binaries
that
do
make
it
into
the
sample.
A
binary
containing
0.8
and
0.4
M
Q
stars
separated
by
0.25
AU
has
a
period
of
42
days
and,
if
the
orbit
is
circular,
the
more
massive
star
has
an
orbital
velocity
of
22
km
s“
1
.
Increasing
the
period
by
a
factor
of
10
increases
the
sepa-
ration
to
1.2
AU
and
decreases
the
velocity
to
10
km
s
_
1
.
Thus
detecting
binaries
over
a
decade
or
larger
range
in
period
requires
a
study
lasting
years
and
velocities
accurate
to
1
kms
-
1
or
better.
Velocities
with
this
precision
have
been
attainable
for
globular-cluster
stars
only
for
about
the
last
20
years
and
many
surveys
for
radial-velocity
variabil-
ity
have
extended
for
less
than
5
years.
Today,
velocity
precisions
of
1
km
s
-
1
can
be
achieved
for
giants
in
the
closest
clusters
with
efficient
instruments
on
small
telescopes
(CORAVEL
on
the
1.5-m
Danish
tele-
scope
at
ESO,
the
radial-velocity
scanner
on
the
Dominion
Astrophysical
Observatory
1.2-m
telescope,
and
the
inten-
sified
Reticon
system
on
the
1.5-m
Whipple
Observatory
telescope
of
the
Smithsonian
Astrophysical
Observatory),
but
rapidly
surveying
large
samples
in
most
clusters
re-
quires
4-m-class
telescopes.
Since
access
to
large
telescopes
is
extremely
competitive,
the
studies
that
have
been
done
to
date
have
been
limited
to
modest
numbers
(3-6)
of
obser-
vations
per
star.
Also
acting
to
limit
the
number
of
obser-
vations
per
star
is
the
rarity
of
spectroscopic
binary
stars
in
globular
clusters,
discussed
in
Sec.
2.1.3,
which
has
forced
the
surveys
to
include
large
numbers
of
stars
in
order
to
get
statistically
significant
results.
The
complete
characterization
of
the
population
of
bi-
naries
in
clusters
is
unfortunately
many
years
off.
What
questions
can
and
should
be
addressed
first?
A
survey
of
a
complete
sample
of
globular-cluster
stars
for
binaries
can
test
two
key
predictions
of
the
simulations
of
cluster
dy-
namical
evolution
discussed
in
Secs.
3.3,
3.4,
and
3.5:
(1)
that
binaries
should
be
more
concentrated
towards
the
cen-
ter
of
the
cluster
than
single
stars
and
(2)
that
many
bi-
naries
should
be
destroyed
over
a
Hubble
time
in
a
typical
dense
cluster.
A
simple
test
of
the
first
prediction
is
to
compare
the
radial
distribution
of
binaries
containing
a
giant
with
that
of
apparently
single
giant
stars.
A
test
of
the
second
is
that
there
should
be
fewer
primordial
binaries
today
in
those
clusters
in
which
the
binary
hardening
and
destruction
processes
(described
in
Sec.
3.2)
have
been
the
most
im-
portant.
The
time
scale
for
these
processes
is
approxi-
mately
the
time
for
another
star
to
approach
to
within
the
binary’s
semimajor
axis,
a
(Heggie
1980;
Hills
1983;
Hut
1984):
7W2.5X10V)
/10
4
M
Q
p
c-
3
\
/I
AU\
(
(v
2
)
w
l
\
\
p
a
j\10
kmsj’
(1)
where
p
is
the
density
of
perturbing
stars,
and
(
v
2
)
1
/
2
is
the
rms
velocity
of
these
stars.
This
time
is
less
than
a
Hubble
time
at
the
centers
of
many
globular
clusters
for
binaries
with
a
^
1
AU.
Thus
the
models
predict
lower
binary
fre-
quencies
in
those
clusters
where
the
binary
destruction
time
T
h
evaluated,
say,
at
the
center
of
the
cluster
is
shorter.
If
clusters
with
present-day
destruction
times
much
longer
than
a
Hubble
time
are
included
in
the
sam-
ple,
such
a
study
will
also
produce
at
least
a
lower
limit
on
the
initial
frequency
of
binaries
in
globular
clusters.
2.1.2
Implementing
A
Survey
The
two
questions
that
must
be
faced
in
implementing
a
search
for
binaries
by
radial
velocities
are
(
1
)
how
to
iden-
tify
binaries
with
only
a
few
observations
per
star
and
(2)
how
to
turn
the
number
of
binaries
discovered
into
the
true
binary
frequency.
The
standard
test
for
significant
variabil-
ity
in
a
series
of
observations
is
the
x
¿
statistic.
Since
ve-
locity
measurement
uncertainties
are
usually
determined
from
the
data
themselves,
the
proper
statistic
is
actually
the
F
(variance
ratio)
statistic,
but,
for
samples
with
more
than
—200
degrees
of
freedom,
the
difference
is
slight.
The
problem
with
the
x
2
statistic
is
that
it
is
sensitive
to
the
assumption
of
normally
distributed
errors.
Large
errors
can
result
from
blunders
in
identifying
the
stars
observed,
but
subtler
effects
may
present
a
more
serious
problem.
For
example,
velocity
errors
due
to
miscentering
the
star
on
the
slit
(guiding
errors)
tend
to
be
worse
during
times
of
good
seeing,
which
come
and
go
during
a
night.
Another
potential
problem
is
the
blending
of
stellar
images
in
crowded
cluster
cores.
Careful
observational
procedures
can
minimize
some
of
these
effects,
as
can
estimating
the
uncertainties
directly
from
the
data,
but
adopting
stringent
limits
for
accepting
the
reality
of
variability
is
the
wisest
course.
This,
of
course,
reduces
the
number
of
binaries
that
are
“detected.”
A
different
sort
of
problem
is
real
velocity
variability
that
arises
from
causes
other
than
orbital
motion.
Gunn
and
Griffin
(1979,
hereafter
referred
to
as
GG),
Mayor
et
al.
(1984),
Lupton
et
al.
(1987),
and
PLH
all
reported
that
the
most
luminous
globular-cluster
giants
show
more
velocity
variability
than
fainter
stars,
which
is
not
the
trend
expected
from
binaries.
The
extensive
data
on
M3
stars
in
PLH
showed
that
the
variability
was
largest
within
0.5
mag
of
the
giant-branch
tip,
though
it
was
certainly
present
fainter
than
that
and
may
have
been
present
at
a
low
level
over
the
whole
1.75
mag
range
surveyed.
This
variability
is
attributed
to
motions
in
the
atmospheres
of
the
stars;
the
most
variable
are
low-level
photometric
variables
(PLH).
Excluding
the
large-amplitude
W
Virginis
and
long-period
BINARIES
IN
GLOBULAR
CLUSTERS
985
photometric
variables,
the
largest
velocity
ranges
observed
(always
for
stars
within
0.5
mag
of
the
tip),
are
about
8
km
s
_
1
.
Thus
a
criterion
that
will
identify
binaries
but
reject
“atmospheric”
variables
is
a
velocity
range
larger
than
10
km
s~
l
.
This
restrictive
limit
could
be
reduced
to
4
km
s~
1
(for
measurement
accuracies
of
about
1
km
s
-
1
)
for
stars
fainter
than
0.5
mag
below
the
tip
of
the
giant
branch.
However,
the
most
natural
discovery
criterion
is
a
lower
limit
on
the
probability
of
obtaining
a
x
2
larger
than
is
observed.
Another
commonly
used
variability
criterion,
the
ratio
of
the
external
to
the
internal
uncertainties,
is
closely
related
to
the
x
2
statistic.
These
criteria
can
be
employed
successfully
for
globular-cluster
stars
if
the
effect
of
atmospheric
variability
is
included
as
an
additional
ad-
ditive
uncertainty
(see,
e.g.,
GG
and
PLH)
and
if
velocity
ranges
greater
than
10
km
s“
1
are
required
for
the
stars
in
the
brightest
0.5
mag
of
the
giant
branch
to
be
identified
as
binaries.
Reasonable
limits
for
the
x
2
probability
are
0.01-
0.001.
Once
a
criterion
for
identifying
binaries
has
been
estab-
lished,
the
fraction
of
stars
in
a
sample
that
satisfy
the
criterion
is
the
discovery
fraction.
Any
criterion
will
miss
some
binaries
because
of
face-on
orbits,
long
periods,
low
velocity
amplitudes,
or
simple
bad
luck
in
the
timing
of
the
few
observations.
Each
criterion
has
a
corresponding
dis-
covery
efficiency:
the
fraction
of
binaries
that
will,
on
av-
erage,
be
found.
The
efficiency
depends
on
a
complex
way
on
the
properties
of
the
binaries
and
on
the
number
and
spacing
of
the
observations,
and
is
best
determined
by
Monte
Carlo
simulations
of
the
observations.
Figure
1
shows
the
results
of
such
a
simulation
using
the
criterion
that
the
velocity
range
be
larger
than
10
km
s
_
1
.
The
sample
consists
of
the
504
velocities
given
in
GG
and
PLH
for
110
M3
stars,
which
has
an
average
time
baseline
of
10.6
yr.
Binaries
were
chosen
with
periods
between
0.1
and
100
yr
and
given
a
primary
mass
of
0.8
The
four
solid
curves
are
the
discovery
efficiencies
for
four
equal
intervals
of
the
logarithm
of
the
mass
ratio
q
in
the
range
0.33
>
log(q)
>
—1.0.
These
curves
are
labeled
by
the
mean
mass
ratio
of
the
binaries
discovered.
The
efficiencies
were
averaged
in
four
bins
per
decade
of
period.
The
dashed
lines
show
the
bias
introduced
by
the
removal
of
binaries
from
this
magnitude-limited
sample
by
mass
trans-
fer
resulting
from
the
evolution
of
the
primary
into
a
red
giant.
This
effect
was
determined
by
calculating
the
radius
of
every
giant
in
the
sample
using
a
luminosity-radius
re-
lation
constructed
from
the
infrared
photometry
of
Cohen
et
al.
(1978).
If
the
giant
in
a
binary
chosen
for
the
sim-
ulation
was
larger
than
its
Roche
radius,
then
that
binary
was
assumed
to
have
been
removed
from
the
sample
by
truncation
of
the
evolution
of
the
giant
or
by
coalescence
through
a
common-envelope
stage.
Figure
1(a)
shows
the
discovery
efficiency
for
binaries
on
circular
orbits.
Figure
1(b)
is
the
same
plot
for
a
pop-
ulation
of
binaries
with
a
“thermal”
(see
Hut
1985)
dis-
tribution
of
eccentricities,
e.
This
distribution
is
f(e)=
2e
and
thus
is
weighted
towards
large
eccentricities.
The
effect
of
mass
transfer
in
elliptical
orbits
is
approximated
by
O
1
1
1
1
1
n
1
—
1
1
1
1
'
T
i
—
1
1
—
l
J
0.1
1
10
100
Period
(YR)
O
i
-1—i
0.1
,
i
i'll!
—
I
1—i..i
i
I
0.1
1
10
100
Period
(YR)
Fig.
1—Binary
discovery
efficiencies
as
a
function
of
orbital
period
for
the
sample
of
504
velocities
from
GG
and
PLH
for
110
M3
stars.
The
discovery
criterion
is
a
velocity
range
larger
than
10
km
s
_
1
.
The
efficien-
cies
were
calculated
from
1000
simulated
samples
in
each
of
the
24
bins
produced
by
dividing
the
interval
between
—1
and
+2
in
the
logarithm
of
the
binary
period
(in
years)
uniformly
into
12
parts
and
the
interval
between
—
1
and
+0.33
in
the
logarithm
of
the
binary
mass
ratio
into
4.
The
four
solid
curves
are
for
the
individual
mass-ratio
bins
and
are
labeled
with
the
mean
ratio
of
the
binaries
discovered.
The
dashed
lines
show
the
effect
on
the
discovery
efficiency
of
the
removal
of
binaries
from
the
sample
by
the
mass
transfer
resulting
from
the
evolution
of
the
primaries
into
red
giants,
(a)
The
discovery
efficiencies
for
binaries
with
circular
orbits,
(b)
The
discovery
efficiencies
for
binaries
with
a
thermal
distribu-
tion
of
eccentricities.
eliminating
systems
whose
giants
overflow
their
Roche
ra-
dii
at
periastron.
The
results
in
Fig.
1
are
typical
of
the
best-observed
samples
of
luminous
globular-cluster
stars.
From
these
and
similar
diagrams
for
other
clusters
we
conclude
that:
(1)
Binaries
with
elliptical
orbits
are
generally
some-
what
harder
to
find
than
those
with
circular
orbits,
since
they
spend
much
of
their
orbit
moving
slowly
near
apas-
tron.
(2)
The
biasing
of
the
discovered
binary
orbits
by
the
size
of
the
giants
is
clearly
significant,
particularly
when
the
orbits
are
elliptical.
The
bias
can
be
reduced
by
observ-
ing
fainter
stars.
(3)
The
average
discovery
efficiency
is
high
(25%-
50%
)
for
binaries
with
periods
between
about
0.2
and
10
years
and
with
mass
ratios
larger
than
about
0.25.
Binaries
![Fig. 17—Cross sections for close approaches of a field star of mass 1.0 (in arbitrary units) to either of two stars (of mass 0.5 and 0.35) initially in a circular binary of semimajor axis a. To determine the cross section, field stars were fired with velocity at infinity v between 0.05 and 0.15i>c [vc defined by Eq. (8) at 1000 binaries with random orientations, phases, and impact parameters]. The cross section is cumulative (i.e., is for approach of the field star to less than the given distance from the other stars). Notice that the cross section for encounters within 10“2a is roughly equal to the area of the binary times the gravitational focusing factor ( vc/v) 2. This large cross section for close encounters is due to the many passes in resonant encounters.](/figures/fig-17-cross-sections-for-close-approaches-of-a-field-star-18dbqxt7.webp)
