Correcting velocity dispersions of dwarf spheroidal galaxies for binary orbital motion

0.64 MB·English

by  Quinn E. Minor

Draftversion January11,2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
CORRECTING VELOCITY DISPERSIONS OF DWARF SPHEROIDAL GALAXIES FOR BINARY ORBITAL
MOTION
Quinn E. Minor, Greg Martinez, James Bullock, Manoj Kaplinghat
DepartmentofPhysicsandAstronomy,UniversityofCalifornia,IrvineCA92697, USA
Ryan Trainor
0 DepartmentofAstronomy,CaliforniaInstitute ofTechnology, PasadenaCA91125,USA
Draft version January 11, 2010
1
0
ABSTRACT
2
Weshowthatmeasuredvelocitydispersionsofdwarfspheroidalgalaxiesfromabout4to10km/sare
n
unlikely to be inflated by more than 20% due to the orbital motion of binary stars, and demonstrate
a
that the intrinsic velocity dispersions can be determined to within a few percent accuracy using
J
two-epoch observations with 1-2 years as the optimal time interval. The crucial observable is the
1
threshold fraction—the fraction of stars that show velocity changes larger than a given threshold
1
between measurements. The threshold fraction is tightly correlated with the dispersion introduced

by binaries,independent ofthe underlyingbinary fractionanddistributionoforbitalparameters. We
]
A outlineasimpleproceduretocorrectthevelocitydispersiontowithinafewpercentaccuracybyusing
thethresholdfractionandprovidefittingfunctionsforthismethod. Wealsodevelopamethodologyfor
G
constrainingpropertiesofbinarypopulations frombothsingle-andtwo-epochvelocitymeasurements
. by including the binary velocity distribution in a Bayesian analysis.
h
p Subject headings: binary stars, theory—galaxies: kinematics and dynamics
-
o
r 1. INTRODUCTION flate the first-order(Gaussian) component of the disper-
t
s sion as well. In previous studies, Olszewski et al. (1996)
In recent years, a large number of dwarf spheroidal
a and Hargreaves et al. (1996) used Monte Carlo simula-
[ satellite galaxies of the Milky Way have been discovered
usingtheSloanDigitalSkySurvey(Willman et al.2005, tionstoshowthatthedispersionestimatedbythesetech-

2 Zucker et al.2006). Thesegalaxiesaremuchfainterthan niques is inflated due to binaries by an amount which is
v previously known Milky Way satellites, having larger smallcomparedtothe statisticalerrorintheirdatasets.
0 Sincethattime,thenumberofstarsinthen-knowndwarf
mass-to-light ratios and velocity dispersions that range
6 spheroidalswithspectroscopicdatahasincreasedconsid-
from 7.6 km/s down to 3.3 km/s (Simon & Geha 2007).
1 erably,fromless than100atthe time ofOlszewski et al.
Estimatingtheamountofdarkmattercontainedinthese
1 (1996) to more than 1000 for Draco, Fornax, and Ca-
galaxies becomes more susceptible to error than in the
. rina (Walker et al. 2009). For such large samples, we
1 largerdwarfspheroidals. This is not only because of the
will show that the bias in the first-order dispersion due
0 statisticalerrorassociatedwithsmallstellarsamples,but
0 alsobecausesourcesofvelocitycontaminationconstitute to binaries is larger than the statistical error. More im-
1 a larger fractionalerror due to the galaxy’ssmall intrin- portantly,forgalaxieswhoseintrinsicdispersionissmall,
: sic velocity dispersion. Potential sources of contamina- the first-order dispersion may be inflated by somewhat
v
more than the statisical error even in data sets as small
i tioncomefromforegroundMilkyWaystars,atmospheric
X jitter in red giant stars (Pryor et al. 1988), and an in- as 100 stars.
To improve previous estimates of the intrinsic disper-
r flated velocity dispersion due to the orbital motion of
a binarystars(Olszewski et al.1996). Among these,bina- sion for these cases, it is necessary to model the velocity
distributionofthe binarypopulationandinvestigatethe
rieshavebeenthemostdifficulttocorrectforbecausethe
behavior of the binary dispersion as model parameters
binary distribution of velocities in environments beyond
such as binary fraction are varied. With this approach
the solarneighborhoodisnotwellknownanddifficultto
it is desirable to find the best possible constraints on
constrainwithoutalargenumberofhigh-precisionradial
the binary velocity distribution, and observations taken
velocity measurements.
at multiple epochs are very useful for this purpose. In
The most prominent signature of binary stars in these
addition, estimating the binary fraction and other bi-
galaxies is a high-velocity “tail” in the velocity distribu-
nary properties in galaxies and clusters beyond the so-
tion due to short-period binaries, which gives the distri-
lar neighborhood is useful in its own right, since this
butionahigherkurtosisthanthatexpectedfromaGaus-
has been difficult to predict accurately in simulations
sian. Because of this departure from Gaussianity, the
(Tohline 2002, Goodwin et al. 2007). Binaries also af-
intrinsic velocity dispersion is usually estimated by us-
fect higher-order moments of the velocity distribution,
ing a robust estimator such as the biweight (Beers et al.
e.g. kurtosis, which has been shown in principle to pro-
1990), by discarding velocity outlier stars from the data
videusefulinformationaboutthemassprofileofgalaxies
sample, or by a combination of both approaches (cf.
(L okas et al. 2005).
Mateo et al.1991). Whilethesetechniqueseliminatethe
In section 2 we derive an analytic formula for the
largest component of the binary dispersion, binaries in-
2
center-of-mass velocity distribution of binary stars. In
sfoerctbioonth3siwngeleu-saentdhtiwsod-iesptroicbhutvieolnoctitoyddeartiav.eWlikeedliehmooodns- vz = 2Pπa(1−e2)−12ge(θ,ψ,φ), (1)
stratein section4 howproperties ofa binarypopulation
can be constrained from multi-epoch data. Fig. 4 shows g (θ,ψ,φ) sinθ[cos(ψ φ)+ecos(φ)] (2)
e
that repeat measurements with a baseline of 1-2 years ≡ −
is sufficient to constrain the binary population. We will where a is the semi-major axis of the primary star’s or-
prove that over a range of velocities of a few km/s, the bit, and θ and ψ are the second and third Euler angles,
binary fraction is nearly degenerate with the parame- respectively. TheazimuthalEulerangledoesnotappear
terscharacterizingthedistributionofperiods,andinsec- explicitly because vz is invariant under rotations about
tion 5 we use this fact to develop a model-independent the z-axis.
method for correcting the dispersion due to binaries by Now taking the log of eq. 1 and subsituting Kepler’s
using multi-epoch data. Fig. 10 encapsulates our main third law, we find
result, from which we conclude that the velocity disper-
sions of dwarf spheroidals are unlikely to be inflated by logP = k 3log vz , (3)
− | |
more than 20% by binaries. The procedure to correct
the dispersion for binaries is summarized at the end of
g (θ,ψ,φ) 3 (2πq)3m
section 5. In section 6 we outline a method to combine k log | e | (4)
single- and multi-epoch data in a Bayesiananalysis,and ≡ ((cid:18) √1−e2 (cid:19) (1+q)2)
discuss the issue of foreground contamination and how
this affects the apparent binary population. In section whereP is inyears,m is the massofthe primarystarin
7 we find a fitting function for the velocity distribution solarmasses,qistheratioofsecondary-to-primarymass,
of binary stars and use it to derive an analytic formula and vz is in units of AU/year.
forthebinarydispersion. Finally,insection8wefurther As an aside, we note that eqs. 3 and 4 can be used
explore the degeneracy of model parameters and discuss to make a back-of-the-envelope estimate of the velocity
prospects for constraining the period distribution of bi- scale associated with a given orbital period. By averag-
nary systems from multi-epoch velocity data. ing overorientations,mass ratio andeccentricity (whose
PDF’s are given below), we find
2. DISTRIBUTIONOFVELOCITIESINTHE
CENTER-OF-MASSFRAMEOFBINARYSYSTEMS 1
M/M 3
⊙
Spectroscopic velocity measurements of binary stars vz (5.7 km/s) (5)
| |≈ P/year
are typically dominated by the primary star due to (cid:18) (cid:19)
the difference in luminosities and spectra between the where M is the mass of the primary star. In dwarf
two stars. This is especially true for stars that lie spheroidalgalaxiesandglobularclusterswhereredgiants
on the red giant branch, for which the luminosity- havemassesM 0.8M ,the aboveestimateshowsthat
⊙
≈
mass relation steepens drastically. By way of com- periodslongerthanafewdecadeswillyieldvelocitiesless
parison, Duquennoy & Mayor (1991) found the ratio of than 2 km/s. Meanwhile, velocities larger than 10 km/s
secondary-to-primary masses of binary stars within the will be dominated by binaries with periods shorter than
solar neighborhood to peak at m/M 0.23 with only 1 month. The above estimate is somewhat sensitive to
≈
20% of the stars having mass ratios larger than 0.6. the mass ratio; whereas we used the approximate mean
≈
For systemswith a mass ratioof0.6 andthe primary ly- value q 0.4 in deriving eq. 5, if a system has a mass
≈
ing nearthe base ofthe redgiantbranch,the luminosity ratioq 0.8,thecoefficientinfrontbecomes 10km/s.
ratiowouldbeatmostl (m/M)2.3 0.3;thisratiobe- To fin≈d a distribution in the center-of-ma≈ss velocity
≈ ≈
comesmuchsmalleriftheprimarystarisfurtheralongon v , the distribution of orbital periods must be averaged
z
the red giant branch. Thus, although the cross correla- over all the parameters in eq. 4. We must also average
tionfunctionofthestellarspectramaybedouble-peaked over the time taken to traverse one orbital cycle, with
(Tonry & Davis 1979), the spectrum of the primary star all times being weighted equally. We express the orbital
will likely dominate the signal unless the mass ratio is angle φ in terms of the eccentric anomaly parameter η,
close to 1, in which case the velocities of the two stars so that a uniform distribution in time corresponds to a
willbenearlyequal. Inviewoftheprecedingarguments, distribution f(η)=(1 ecosη)/2π.
−
fortheremainderofthispaperwewillusethevelocityof TheorbitalperiodsofG-dwarfstarsinthesolarneigh-
theprimarystarinmodelingthe spectroscopicvelocities borhood were found by Duquennoy & Mayor (1991) to
of binary systems. follow a log-normal distribution with a mean period of
To model the velocity distribution of the primary star 180 years. Fischer & Marcy (1992) also found a log-
in a population of binary systems, first we must find its normal period distribution for M-dwarfs in the solar
velocity distribution in the center-of-mass frame of the neighborhood with a mean period similar to that of the
binary. The motion of two stars orbiting each other can G-dwarfs. In terms of logarithm of the period P, the
be simply expressed in terms of four parameters: the distribution found by Duquennoy & Mayor (1991) has
semi-major axis a, eccentricity e, period P, and orbital mean µ =2.23 and dispersion σ =2.3 (where P
logP logP
angle φ with respect to the center of mass. If we also is in years, and the logarithm is base 10). We shall use
specify the orientation of the orbital plane in terms of this as the fiducial binary model in this paper, but will
Euler angles, with the z-axis pointing along the line of also allow µ and σ to take on other values.
logP logP
sight,the line-of-sightvelocityofthe primarystar inthe Aswiththeperioddistribution,weusedistributionsof
center-of-mass frame is given by the mass ratio q and eccentricity e observed in G-dwarf
3
stars in the solar neighborhood (Duquennoy & Mayor eventually merge. This effect is not included in the dis-
1991). When comparing different empirically derived tributions of Duquennoy & Mayor (1991) because their
mass functions, Duquennoy & Mayor (1991) showed the sample consisted entirely of dwarf stars. Therefore, we
distribution of mass ratios q was best fit by the Gaus- maketheapproximationofexcludingsystemswhosepri-
sian mass function considered in Kroupa et al. (1990) mary star is larger than its Roche lobe, assuming the
with mean q¯ = 0.23 and dispersion σ = 0.32. This binary to be destroyed over a timescale much less than
q
is somewhat misleading, since for q > 0.5 the distribu- 1 Gyr. We use an approximation to the radius r (a,q)
L
tion is in fact consistent with the power-law initial mass of the Roche lobe given by Eggleton (1983). While this
functionsofSalpeterandKroupa(Salpeter1955,Kroupa radius is not exactly correct for eccentric orbits, a re-
2001). At smaller mass ratios the distribution decreases cent smoothed-particle hydrodynamics simulation of an
sharply compared to the well-established Kroupa initial eccentric binary of mass ratio q = 0.6 by Church et al.
mass function of single stars, which implies that small (2009) found the Roche lobe radius to decrease only
mass ratios are strongly affected by interaction between slightlywitheccentricity. Theyalsoderiveafittingfunc-
primary and secondary during the formation of the bi- tion for the Roche lobe radius with an eccentricity e,
nary system. given by r (e) = r (e = 0)(1 0.16e). We find using
L L
−
To further complicate matters, Mazeh et al. (1992) this formula that the velocity distribution changes only
foundthatshort-periodbinariestendtohavehighermass by a small amount (less than 2% at 5 km/s) compared
ratios than those found by Duquennoy & Mayor (1991). to when using the Roche lobe radius evaluated at peri-
By analyzing spectroscopic binaries, they found that bi- center, given by r (e = 0) above. For the following cal-
L
narieswithperiodsshorterthan3000dayshavemassra- culations, we therefore adopt the Roche lobe evaluated
tios consistentwith a uniform distribution, althoughthe at pericenter for an eccentric orbit.
Poisson errors in their analysis were quite large. Sub- The radius of each star is found by estimating its ef-
sequent studies (Goldberg et al. 2003, Halbwachs et al. fective temperature from an isochrone of given age t in
g
2004) have shown that the mass ratio distribution in the stellar population synthesis model of Girardi et al.
short-period binaries is bimodal (also seen in previous (2004). This, together with its magnitude, provides an
samples; cf. Trimble 1990), with a peak at low mass estimate of the stellar radius. We denote M (m;t )
V g
ratios similar to that of the long-period binaries but and R(m;t ) as the absolute V-band magnitude and
g
with another peak near q 1. In the sample ana- radius (respectively) of a star of mass m assigned by
≈
lyzed by Goldberg et al. (2003), the peak at high mass an isochrone of age t . If the star lies on the horizon-
g
ratios is smaller for primaries with masses larger than tal branch or asymptotic giant branch, instead of using
0.6 M , but larger for halo stars. In view of lingering its present radius (which may be small) we compare its
⊙
uncertainties in the nature of the mass ratio distribu- Roche lobe to the largest radius previously attained by
tion, for simplicity we will adopt the Gaussian distri- the star at the end of its red giant phase.
bution fromDuquennoy & Mayor(1991) for long-period WeshallexpressourformulaintermsofanabsoluteV-
binaries (P >3000 days) and a uniform distribution for banduppermagnitudelimitM andaget . Assuming
lim g
short-period binaries (P <3000 days). For the distribu- the lower magnitude limit to be near the tip of the red
tionoftheprimarymassf(m)wewillusetheKroupaini- giant branch, we find the velocity distribution is quite
tial mass function corrected for binaries (Kroupa 2002). insensitivetotheexactvalueofthelowermagnitudelimit
The distributionofeccentricities f(e logP)wasfound because the suppression of binaries due to Roche lobe
|
by Duquennoy & Mayor (1991) to have three different overflow dominates the high-luminosity end of the red
regimesdepending onthe period. For periods of11days giant branch.
or shorter, the orbits are circularized due to tidal forces Averaging over distributions for all the model param-
and are therefore approximated to have e = 0. For pe- eters and dropping the z subscript for readability, we
riods between 11 days and 1000 days, f(e) can be ap- obtain
proximated by a Gaussian with mean e¯= 0.25 and dis-
persion σ = 0.12. For periods longer than 1000 days,
e f (log v ;M ,t ) = (6)
highereccentricitiesaremoreprevalentandthedistribu- b lim g
| |
tion approximately follows f(e) = 3e1/2. Among these, 3 1 2π 1 2π
2 d(cosθ) dψ f(e logP)de f(η)dη
the Gaussian regime (11 days < P < 1000 days) has 8π2 |
the greatest impact on the velocity distribution at ve- Z−1 Z0 Z0 Z0
1 ∞
locities of order km/s. While the adopted distributions
f(q logP)dq f(m)dm Θ[M M (m;t )]
lim V g
of mass ratios and eccentricities are undoubtedly only a × | · −
Z0 Z0
rough approximation to the true distributions, our cen- exp −[3log|v|−k+µlogP]2
tralresults(presentedinsection5)willprovetobequite Θ[r (a,q) R(m;t )] 2σl2ogP ,
insensitive to the nature of the adopted distributions. × L − g n 2πσ2 o
An important effect that must be taken into account logP
is the effect ofmass transferbetweenthe stars ifthe pri- q
mary star is a red giant whose size is larger than the whereΘ[x]istheHeavisidestepfunction. Thevariable
radius of its Roche lobe (Paczyn´ski 1971). In such a k is a function of all the other parameters according to
case, matter from the surface of the giant will accrete (4), with φ φ(η). We use a Monte Carlo simulation to
≡
onto the smaller star and the separation between the perform the integration over a grid of log v values and
| |
stars will decrease. The end result is that either the interpolate to find fb(log v ). In fig. 1 we plot fb(log v )
| | | |
otherstarwillexplodeinasupernovaIa,orthestarswill for different ages and absolute magnitude limits. This
figureshowsthatforvelocities&10km/s,suppressionof
4
where v′ is the line-of-sightcomponentof the velocity in
the center-of-mass frame of the binary system. To find
the binary likelihood we therefore average the velocity
distribution in v over the distribution of the binary
nb
component v′ given in eq. 6:
L (v σ ,v¯) = ∞ e−(v−v′−v¯)2/2σ02 fb(log|v′|)dv′ (8)
b | 0 Z−∞ 2πσ02 2|v′|ln10
Thefactorof2inthedenpominatorarisesfromthefact
that f (log v ) is normalized in log v , whereas the like-
b
| | | |
lihood is normalized in v (allowing for positive and neg-
ative velocities).
By taking the second moment of the velocity distribu-
tion in eq. 7, one obtains the result that
σ2 =σ2+Bσ2 (9)
0 b
Fig.1.— Distribution of velocities in the center-of-mass frame
of binary systems, plotted for different absolute magnitude limits where σ is the measured dispersion and σ is the binary
b
Mlim = 0, 3 and stellar ages tg = 1 Gyr, 10 Gyr. The suppres- dispersionfoundbytaking the secondmomentofthe bi-
sionofbinariesduetoRoche-lobeoverflowbecomesimportantfor
nary velocity distribution f (v) = f (log v )/2v ln10.
velocities & 10 km/s. Except at the turnover point v ≈10 km/s, b b
| | | |
the distribution behaves locally as a log-normal to good approxi- As in the usual case, it can be shown that given a nor-
mation. mallydistributedmeasurementerrorwithdispersionσ ,
m
binaries due to Roche-lobe overflow becomes important. one need only make the replacement σ2 σ2+σ2 in
0 −→ 0 m
Asymptotically for large velocities, the velocity distri- the above formulas.
bution in eq. 6 behaves as a log-normal with dispersion Next,itisdesirabletohavealikelihoodfunctionforve-
similartothatoftheperioddistribution,σ . Thiscan locities measured at two different epochs. Since velocity
logP
beseenasfollows: first,atlargevelocities,thelog-normal changesoforderkm/soveratimescaleofyearsisentirely
in the integrand is far from its maximum and therefore negligiblefornonbinarystars,themostfruitfulapproach
varies slowly in the model parameters (eq. 4) compared istousealikelihoodinthedifference∆vbetweenthetwo
to their respective probability distributions, provided k velocities. Keeping in mind the log-normal behavior of
isnotlargeandnegative. Thereforeasafirstapproxima- the velocitydistribution, we write the binary partof the
tion we can apply the method of steepest descents and likelihood as gb(log ∆v ;∆t). As with the single-epoch
| |
find the resulting distribution to be log-normal with a velocity distribution fb(log v ), we use a Monte Carlo
pmreoaxnimgaivteionnboyflsolgow|v¯l|y=va31ryk¯in−g lµolgog-nPor.mHaolwisevneort,sthtreicatply- isnimthuelastiimonultaoticoanlcwuelafitnedg∆b(vlo|bgy||∆evvo|;lv∆int)g.tFhoerorebaicthalbpinhaarsye
true, since k becomes larg(cid:0)e and nega(cid:1)tive if the mass ra- to its value after a time ∆t and calculating the resulting
tio q and direction function g (θ,ψ,φ) are close to zero. changeinvelocity. In the absenceof measurementerror,
e
Thismeansthatthemeank¯isinfactafunctionofv. We the nonbinaries will have zero change in velocity, so the
find, however, that locally k¯ is linear in log v to good total likelihood can be written as
| |
approximation, with the result that the velocity distri-
bution still behaves locally as a log-normal but with a g (log ∆v ;∆t)
b
somewhat different dispersion from σ . We will use L(∆v ∆t,B)=(1 B)δ(∆v)+B | | (10)
logP | − 2∆v ln10
thistoconstructafittingfunctionforthebinaryvelocity | |
distribution in section 7. If there is a normally distributed measurement error,
the likelihood must be averaged over two Gaussians of
3. VELOCITYDISTRIBUTION OFAPOPULATIONOF widths σ and σ for the first and second velocity
BINARYSTARS m,1 m,2
errors, respectively. A little calculation shows this to
3.1. Single- and two-epoch likelihood functions be equivalent to averaging over a single Gaussian with
Suppose that among a population of stars, a fraction dispersionσ2e, which is the equivalent 2-epochmeasure-
B of them are in binary systems. Further suppose that ment error:
the velocity distribution for stars not in binary systems
is Gaussian with dispersion σ0 and systemic velocity v¯. σ22e =σm2,1+σm2,2, (11)
The velocity likelihood function will have the following
form:
e−∆v2/2σ22e
L(∆v ∆t,B,σ )=(1 B) (12)
2e
e−(v−v¯)2/2σ02 | − 2πσ22e
L(v B,σ ,v¯) = (1 B) + BL (v σ ,v¯)
| 0 − 2πσ02 b | 0 +B ∞ e−(∆v−∆v′)2/2σ22e gbp(log|∆v′|;∆t)d(∆v′)
where L (v σ ,v¯) is the likelpihood for binary stars. (T7o) Z−∞ 2πσ22e 2|∆v′|ln10
b 0
|
derivethebinarylikelihood,wenotethatthecomponent Note that the plikelihood is identical in form to that
ofthevelocitynot duetothebinaryorbitisv =v v′, of eqs. 7 and 8, since in both cases we are averaging the
nb
−
5
distributionoveraGaussian. Boththesingle-andmulti- eq. 11. Note that in the limit as σ 0, the first term
2e
→
epoch likelihoods will be put to use in later sections. goes to zero while the complementary error function in
the integrand reduces to a step function 2Θ(∆v′ ∆v),
3.2. Threshold fraction of a binary population so that eq. 14 reduces to eq. 13 as expected. −
A convenient observable quantity for characterizing a 4. CONSTRAININGPROPERTIESOFABINARY
binarypopulationisthethresholdfraction,definedasthe POPULATIONBYMULTI-EPOCHOBSERVATIONS
fraction F of stars in a sample which exhibit a change
In this section we investigate how properties of a pop-
in radial velocity greater than a threshold ∆v after a
ulation of binary stars affect the observed velocity dis-
time ∆t between measurements. For ∆v > 1 km/s, this
tribution measured at two or more epochs. Specifically,
fraction is typically smaller than 0.2, so the threshold
we explore how our proposed observable, the threshold
number (given by n = NF where N is the number of
factionF (section3.2),willbeaffectedbychangesinthe
stars)followsaPoissondistributionwithmeann¯ =NF¯.
underlying binary fraction B, absolute magnitude limit,
Thereforethe distributionofF ischaracterizedbyasin-
stellar age, size of the measurement error, and time in-
gle number, the mean threshold fraction F¯, and the ex-
terval between measurements. We will also demonstrate
pectederrorcanbeestimated. Inparticular,theerrorin
how the binary fraction B can be inferred by a likeli-
F is approximately F¯/N (appendix 2). Fornotational hood analysis, and show how this leads to a better de-
simplicity, for the remainder of this paper we will refer termination of the threshold fraction F. We first con-
to the mean threshopld fraction F¯ as simply the thresh- sider binarymodels with our fiducialperiod distribution
oldfractionF (withoutthebar),withtheunderstanding (inferred from the solar neighborhood)and then explore
that the observed threshold fraction will have a Poisson howtheinferredbinaryfractionisaffectediftheassumed
scatter about this value. period distribution parameters are incorrect. Unfortu-
Despiteitsstraightforwarddefinition,therearetwodif- nately, and as we discuss more fully in the next section,
ficulties in measuring the threshold fraction from actual the effect of changing the binary fraction B on the ob-
data sets. First, often there does not exist a common served binary velocities can be mimicked closely by al-
time interval ∆t between measurements in the sample, tering the assumed distribution of orbital periods (i.e.
but rather several time intervals for various subsets of changing the parameters µ , σ ). While this is
logP logP
stars. Furthermore,differentvelocitymeasurementshave badnewsforanyattemptatconstrainingtheunderlying
theirownassociatedmeasurementerrorsandthisinturn properties of a galaxy’s binary population in full gen-
affects the measured value of F. The latter issue can erality, it turns out to be good news for correcting the
be dealt with in an approximate way by using the me- observed velocity dispersion for the effects of binary or-
dian(orotherrobustlocationestimator)ofthemeasure- bital motion, as we will show in section 5.
menterrorofthe sample,in termsofwhichσ¯ =σ¯ √2 First, let us make the rather optimistic assumption
2e m
(eq. 11). However, both problems can be surmounted that the distribution of binary orbitalperiods is approx-
more rigorously by estimating the error-free threshold imately universal, so that it follows our fiducial choice
fraction F via a Bayesian or maximum-likelihood ap- µ = 2.23, σ = 2.3 (section 2). Before launch-
0 logP logP
proach. By using the likelihood in ∆v defined in eq. 12, ing into the full-fledged calculation, one would like to
the thresholdfractionata particularthresholdandtime estimate how well the fiducial binary fraction B can be
interval can be estimated even if measurements were constrainedfor a givensample, or conversely,how many
taken at various epochs—moreover, the inferred thresh- starsarerequiredtoconstrainBbyacertainamount. To
oldfractionF isfreeofmeasurementerror. Thismethod simplifymatters,letusassumewehaveadatasetwhere
0
will be demonstrated in section 4. the two epochs have the same time interval ∆t between
The threshold fraction without measurement error, them and the same measurement error σ . The equiv-
m
whichwe denote by F0, canbe expressedinterms of the alent two-epoch measurement error is then σ2e =σm√2
binary two-epoch velocity distribution gb(log ∆v ;∆t) (eq. 11).
by taking the integral of eq. 10 with respect|to |∆v′ First, consider the mean threshold fraction of
| |
from a threshold ∆v to : the binaries without measurement error, denoted by
∞
F (∆v ∆t,B = 1) (eq. 13). In that case the threshold
0
|
∞ g (log ∆v′ ;∆t) fraction scales with the binary fraction B, i.e. is given
F (∆v ∆t,B)=B b | | d∆v′ (13) by F (∆v ∆t,B) = BF (∆v ∆t,B = 1). Now consider
0 | ∆v′ ln10 | | 0 | 0 |
Z∆v | | the threshold fraction without binaries, but with a mea-
Note that in the absence of measurement error, the surement error σm. The two-epoch measurement error
thresholdfractionF scaleslinearlywiththebinaryfrac- is then σ √2σ (eq. 11), and the threshold fraction
0 2e m
≈
tion B. The threshold fraction with measurement error F(∆v ∆t,B =0,σ ) is given by the complementary er-
2e
|
is likewise obtained by taking the integralof eq. 12 from ror function (first term in eq. 14 with B = 0). In fig. 2
∆v to , with the result weplotthethresholdfractionF (∆v ∆t,B)producedby
0
∞ |
a Monte Carlo simulation with an absolute magnitude
limit M =1 and stellar age t = 10 Gyr. The thresh-
∆v lim g
F(∆v ∆t,B,σ )=(1 B)erfc (14) old fraction is plotted for different binary fractions, and
2e
| − (cid:20)√2σ2e(cid:21) we also plot the threshold fraction from measurement
+B ∞ erfc ∆v−∆v′ gb(log|∆v′|;∆t)d(∆v′) etirornor∆wvith σwmhe=re F2 k(∆mv/s. N∆eta,Br )th=e Fpo(i∆ntvof i∆ntte,rBse=c-
Z−∞ (cid:20) √2σ2e (cid:21) 2|∆v′|ln10 0,σ ),tthaieleffectofb0inartyaislt|arsbecomesnottiaciel|ableover
2e
where σ is the 2-epoch measurement error given by the measurement error. Since the Poisson errors are
2e
6
Fig.2.— Threshold fraction F(∆v|∆t,B,σm), defined as the Fig.3.— Fractional constraint ǫb on the binary fraction B, de-
fraction of stars with observed change in velocity greater than fined by eq. 15. In this plot we a measurement error σm = 2.0
a threshold ∆v after a time interval ∆t between measurements. km/s. For comparison we also plot the 95% confidence interval
Thesolidcurvehasnobinaries(B=0)andameasurement error obtainedbyaBayesiananalysisofsimulatedtwo-epochdatafrom
σm = 2 km/s. The other curves are plotted from a Monte Carlo arandomsampleofN stars,averagedoverahundredrealizations.
simulation for binary fractions B = 1 and 0.5, with ∆t = 1 year AuniformpriorisassumedforB.
and no measurement error. The stellar population has an age tg
= 10 Gyr and the absolute magnitude limit Mlim = 1. Given a
measurementerrorσm,thebinaryfractioncanbeconstrainedfor
thresholds ∆v &∆vtail, where ∆vtail is the point of intersection
where F(∆v|∆t,B =0,σm)=F(∆v|∆t,B,σm =0). For agiven
binary fraction B, the total threshold fraction without measure-
menterrorisgivenbyB×F(∆v|∆t,B=1,σm=0).
larger at higher velocity thresholds, to first approxima-
tion we can say that the error-free threshold fraction F
0
is best constrained at thresholds near ∆v . It follows
tail
that the fiducial binary fraction will be constrained by
thestarswith∆v &∆v . (Foraroughapproximation,
tail
one can also use ∆v 2σ 2√2σ .) A little alge-
tail 2e m
≈ ≈
bra (see appendix 2) shows that to constrain the binary
fractiontowithinafractionalaccuracyofǫ ,thenumber
b
of stars required is approximately
1 2B 2
N(ǫ ) (15)
b ≈ F¯(∆v ) ǫ
tail (cid:18) b (cid:19) Fig.4.— Threshold fraction F(∆v|∆t,B,σm), defined as the
Infig.3wegraphtheapproximationformulafordiffer- fraction of stars with observed change in velocity greater than
a threshold ∆v after a time interval ∆t between measurements.
entvaluesofB andcomparetothe95%confidenceinter-
Thesolidcurve hasnobinaries(B=0)andameasurementerror
valinthe binaryfractioninferredby aBayesiananalysis σm = 2 km/s. The other curves are plotted from a Monte Carlo
of the simulated data (described later in this section). simulation for different time intervals ∆t, with no measurement
As is evident for the B = 0.7 curve, the approximation error and a binary fraction B = 1. The stellar population has
formuladiffersforhighbinaryfractionsbecauseB >1is an age tg = 10 Gyr and the absolute magnitude limitMlim = 1.
Given a measurement error σm, the binary fraction can be con-
notallowedintheBayesiananalysis. Theapproximation strained for thresholds ∆v &∆vtail, where ∆vtail is the point of
formula is discussed further in appendix 2. intersection where F(∆v|∆t,B =0,σm)=F(∆v|∆t,B,σm =0).
In fig. 4 we plot the threshold fraction F¯(∆v ∆t,B = ForagivenbinaryfractionB,thetotalthresholdfractionwithout
1,σ = 0) produced by a Monte Carlo simulat|ion with measurementerrorisgivenbyB×F(∆v|∆t,B=1,σm=0).
2e
an absolute magnitude limit M = 1 and stellar age
lim
t =10Gyr for differenttime intervals∆t. We find that 2km/s,littleisgainedbyextendingtheintervalfrom1-2
g
for a measurement error σ = 2 km/s, the observable years to 5 or more years.
m
threshold fraction steadily increases as ∆t is increased, Infig.5 weplotthe thresholdfractionfor differentab-
untilroughly∆t=1year. Thisresultdependssomewhat solute magnitude limits M = 0, 3 and stellar ages t
lim g
on the mass ratio distribution, since higher mass ratios = 1 Gyr and 10 Gyr. Extending the magnitude limit to
resultinhighervelocitiesforagivenorbitalperiod. Ifthe fainter magnitudes increases the threshold fraction be-
mass ratio distribution in Duquennoy & Mayor (1991) cause there is a greater contribution from smaller stars
is assumed for all periods (as opposed to the uniform withless binary suppressiondue to Roche-lobeoverflow.
distributionweadoptforP <3000days),theobservable The threshold fraction is also higher for a younger stel-
threshold fraction increases until roughly ∆t = 2 years. lar population because of their larger mass at a given
Inanycase,unlessthemeasurementerrorissmallerthan stage of stellar evolution, which produces higher orbital
7
Fig.5.— Threshold fraction F(∆v|∆t,B,σm), defined as the Fig.7.—Posteriorprobabilitydistributionofthethresholdfrac-
fraction of stars with observed change in velocity greater than a tionF ofasimulatedgalaxywithbinaryfractionB=0.5andwith
threshold∆vafteratimeinterval∆tyearbetweenmeasurements. aperioddistributioncharacterized by µlogP =1.5, σlogP =2 (P
Here we have picked ∆t = 1 year, binary fraction B =1, and no
inyears). ThethresholdfractionF(∆v|∆t)isdefined asthefrac-
measurement error (σm = 0). We plot the threshold fraction for tionofstarswithobservedchangeinvelocitygreaterthanathresh-
differentabsolutemagnitudelimitsMlim=0,3andstellaragestg old∆v afteratimeinterval∆tyearbetween measurements. Here
=1Gyr,10Gyr. ForagivenbinaryfractionB,thetotalthreshold
wechoseathreshold∆v =5km/sandtimeinterval∆t=1year;
fraction is given by F(∆v|∆t,B,σm = 0) = B×F(∆v|∆t,B = thecorrectthresholdfractionforthisgalaxyisF ≈0.05. Thesim-
1,σm=0). ulated data sample consisted of 300 stars, each with two velocity
measurements taken ∆tdata = 2 years apart, and a measurement
error of 2 km/s. Solid curve is the posterior calculated assuming
thefiducial(solarneighborhood)model,whichisincorrectforthis
galaxy. Dashedcurveusesthecorrectmodel,withthesameperiod
distributionparameters µlogP, σlogP given above. Note that the
correctthresholdfractioncanberecoveredevenifthewrongmodel
isassumed(inthiscase,thefiducialmodel).
measurementstaken ∆t = 2 yearsapart, anda mea-
data
surement error of 2 km/s. First we assume the fiducial
model (with µ = 2.23, σ = 2.3, which is incor-
logP logP
rect for this galaxy) and, assuming a uniform prior in
the binary fractionB, generate a posteriorin the binary
fraction. We then repeat this procedure using the cor-
rect period distribution parameters µ , σ in our
logP logP
model, whose values are givenabove. The resulting pos-
teriors are plotted in fig. 6. This figure shows that the
binary fraction B is a highly model-dependent quantity,
andgiventhe unknownnatureofthe perioddistribution
of binaries outside the solar neighborhood, the inferred
Fig.6.—Posteriorprobabilitydistributionofthebinaryfraction
binaryfractionmust be takenwith a grainof salt. How-
B of a simulated galaxy with binary fraction B =0.5 and with a
period distribution characterized by µlogP = 1.5, σlogP = 2 (P ever, the fiducial binary fraction can still be used as a
inyears). The simulated data sampleconsisted of 300 stars, each relative indicator of the fraction of observable binaries,
with two velocity measurements taken ∆tdata = 2 years apart. aslongasitisinterpretedinreferencetothefractionob-
Solidcurveistheposteriorcalculatedassumingthefiducial(solar
servedinabinarypopulationfollowingthefiducial(solar
neighborhood) model, which is incorrect for this galaxy. Dashed
curve uses the correct model, with the same period distribution neighborhood) distributions of orbital parameters.
parametersµlogP,σlogP givenabove. Althoughthe inferredbinaryfractionB isverymodel-
dependent,thisanalysisisstillusefulinthatitleadstoa
velocities. We find that the threshold fraction changes better determination of the threshold fraction F, which
little for ages between 2-10 Gyr; as the stellar age t is more directly observable than the binary fraction. To
g
is reduced from 2 Gyr, however, the threshold fraction see this, we use the Monte Carlo to generate the binary
steadily increases. thresholdfractionF ofeachmodel,forathreshold∆v=
b
Toestimatethebinaryfractioninatwo-epochsample, 5km/s,timeinterval∆t=1year,andzeromeasurement
weuse the likelihoodfunction L(∆v ∆t,B,σ ) (eq.12). error, i.e. σ =0 (see eq. 14). We then transform each
2e 2e
|
Forthesakeofillustration,weanalyzeasimulatedgalaxy posterior in fig. 6 from B to the threshold fraction F =
with binary fraction B =0.5 but with a different period BF . The renormalized posteriors P(F) are plotted in
b
distribution from that of the solar neighborhood. We fig. 7; the correct threshold fraction for this galaxy is
choose the period distribution parameters µ = 1.5, F 0.05. Note that the correct threshold fraction can
logP
≈
σ = 2 (P in years) for this galaxy. The simulated be recoveredevenif the wrongmodelis assumed(in this
logP
data sample consists of 300stars,eachwith two velocity case, the fiducial model).
8
threshold fraction, we therefore find it a more profitable
strategy to make two-epoch measurements over a larger
sample of stars, as opposed to adding more repeat mea-
surements over an existing sample (assuming a similar
number of overall measurements in either case).
A possible complicating factor in the above analysis is
that selection criteria for making repeat measurements
can bias the inferred threshold fraction and binary frac-
tion. If stars whose spectra yield multiple peaks in the
crosscorrelationfunctionaresingledoutforrepeatmea-
surements, the multi-epoch sample may have an inordi-
nately high binary fraction comparedto the overallstel-
larpopulation. Thisselectionbiasisprobablynotsignif-
icant in red giant stars due to the typically large differ-
ence in luminosity between primary and secondary star.
However,insamplesthatcontainasignificantfractionof
main sequence stars the bias may be more problematic,
although an upper bound on the binary dispersion can
Fig.8.—Posteriorprobabilitydistributionforbinaryfractionofa
still be obtained.
simulatedgalaxywith500starsandbinaryfractionB=0.5. Solid
curveiscalculatedfromthreevelocitymeasurements,whereasdot-
ted curve uses only the first two velocity measurements. v1 and 5. CORRECTINGTHEOBSERVEDVELOCITY
v2 were taken 1 year apart, while v2 and v3 were taken 10 years DISPERSIONFROMMULTI-EPOCHDATA
apart.
In the previous section we demonstrated how uncer-
tainties in the underlying period distribution can ad-
Since even the stars with velocities smaller than the versely affect our ability to constrain the underlying bi-
threshold ∆v are used in the likelihood analysis, the er- nary fraction from multi-epoch data. Here we demon-
ror in the threshold fraction F is smaller than if F is strate that if our goal is to correct the observed velocity
measured directly, especially for higher velocity thresh- dispersion for the effects of binary stars, the degeneracy
olds. The approximate error in the threshold fraction between the period distribution parameters and binary
F estimated by this technique is derived in appendix 2 fractionisquiteuseful: regardlessoftheprecisenatureof
andgivenbyeq.43. Furthermore,the thresholdfraction thebinarypopulation,itseffectontheobservablethresh-
at ∆t = 1 year is recovered even though the data was old fraction F can be directly related to the associated
taken with a time interval of ∆t = 2 years. More correctionintheobservedvelocitydispersioninamodel-
data
generally, the threshold fraction for a specific time in- independent way.
terval can be recovered by the likelihood analysis even The important degeneracy arises from the log-normal
if the data is taken at various different epochs and with behavior of the binary velocity distribution fb(log v )
| |
different measurement errors. As we will show in sec- (eq. 6). Binary orbital motion along the line of sight of
tion 5, the threshold fraction can be used to correct the orderkm/sisthemostimportantfortheintrinsicdisper-
measuredvelocitydispersionofasampleforthe effectof sions we are interested in. For these velocities,the value
binary motion. oflog v isfarfromthe meanofthelog-normal,whichis
| |
Finally, one may naturally wonder: how much are approximately 1 for a magnitude limit Mlim = 1 and
−
the constraints improved by including more than two age tg = 10 Gyr. The exponent of the log-normal is ap-
epochs in the analysis? To address this question, we proximatelylinearoverthescaleofkm/s,sowecanwrite
do a similar calculation on simulated 3-epoch data us- itas β αln v . Thereforethebinarypartoftheveloc-
− − | |
ing the Monte Carlo to generate a 3-epoch likelihood, ity distribution can be written as f(v) Be−β v −1−α,
∝ | |
L(∆v ,∆v ∆t ,∆t ,B,σ ), where the indices refer where B is the binary fraction. If the mean binary pe-
21 32 21 32 3e
|
to three velocity measurements v , v , and v , and σ riod µ is varied, the log-normalis offset in log-space
1 2 3 3e logP
is the 3-epochmeasurementerrordefinedanalogouslyto so to good approximation only β changes; therefore the
eq. 11. We generate a data sample of 500 stars from a velocity distribution f(v) changes by a constant factor
simulated galaxy with binary fraction B = 0.5 and the over the scale of km/s. If the dispersion of the period
period distribution parameters taking their fiducial val- distribution σ is varied, both the offset β and the
logP
ues. For comparison,we generatea posteriorP(B) from slope α change; however, the slope changes by a rela-
the two-epoch calculation which ignores v . The results tivelysmallamountforσ rangingfrom1-3(itsviable
3 logP
are plotted in fig. 8. The velocity measurements v and range of values; see section 8), so again the velocity dis-
1
v weretakenoneyearapart,whilev andv weretaken tribution changes by an approximately constant factor.
2 2 3
tenyearsapart. Whilethemostprobableinferredbinary The important point is if that the parametersσ and
logP
fractionB didimproveinthiscase,the95%confidence µ are varied, they change the velocity distribution
fit logP
limits are only improved by 25%. The fractional im- by an amount which is nearly the same over the scale
≈
provementintheconfidencelimitsisevenlessforsmaller of several km/s—in other words, they behave similarly
data sets; this is because in a sample of a few hundred as if the binary fraction were changed. This is also true
stars, there is significant scatter in the binary fraction, of the magnitude limit and stellar age, which effectively
andthe inferredbinaryfractionB has infact a signif- changethe mean of the log-normaland therefore behave
fit
icant probability of becoming worse when more epochs similarly to µ . We therefore conclude that the pa-
logP
areaddedtothesample. Toconstrainthebinaryand/or rametersµ , σ , magnitudelimit M andstellar
logP logP lim
9
aget areallnearlydegeneratewithbinaryfractionover
g
the scale of km/s.
The degeneracy of the period distribution parameters
withbinaryfractionalsoholdsforthetwo-epochvelocity
distribution g (log ∆v ;∆t), since this also has a log-
b
| |
normalformforkm/svelocities. Byeq.14,therefore,the
samedegeneracyholdsforthethresholdfractionF. The
effect of this degeneracy on the threshold fraction and
its implications for constraining the binary distribution
of periods will be explored in further detail in section 8.
In this section we will consider the threshold fraction
F with a fixed time intervalof 1 yearand without mea-
0
surement error, i.e. F = F(∆v ∆t =1 year,σ = 0)
0 2e
|
(section 3.2). There is no loss of generality in this; as
we demonstrated in section 4, the threshold fraction for
any given time interval ∆t can be estimated by a likeli-
hood analysis even if measurements are taken at various
different epochs and with various different measurement
Fig.9.—Ratioofmeasuredvelocitydispersionσovertheintrin-
errors. However,if the threshold fraction F is measured
sicdispersionσ0,plottedwithrespecttothresholdfractionF0 for
directly for a fixed time interval, it is necessary to ac- differentbinaryfractionsingalaxiesofintrinsicdispersionsσ0=5,
count for the effect of measurement error on F; we will 7,and10km/s. Themeasureddispersionswerecalculatedbyanin-
address this later in the section. terative3σ-clippingroutine,andthethresholdfractionF0 denotes
thefractionofstarswithobservedchangeinvelocitygreaterthan
In the absence of measurement error, by definition
a threshold ∆v =5 km/s after a time ∆t=1 year between mea-
F0 scales linearly with the binary fraction B (eq. 13). surements, assumingzero measurement error. Thefiducial period
Furthermore, because of the near-degeneracy of the pe- distribution (µlogP = 2.23, σlogP = 2.3, P in years) is assumed,
rioddistributionparameterswithbinaryfraction,F also andthebinaryfractionisvariedbetween 0.1and1.
0
scaleslinearly with µ and σ to goodapproxima-
logP logP
tion over their viable range of values (roughly 1-3 with Next we repeat the procedure over a grid of values for
P in years; see section 8). The essential point is that a the parameters B, µ , and σ , and for each point
logP logP
similar relationship holds for the velocity dispersion if a weplottheratioσ/σ withrespecttothethresholdfrac-
0
high-velocity cutoff is used, e.g. at v = 3σ, since the tionF . Theresultsareplottedinfig.10,againforgalax-
c 0
degeneracy approximately holds for velocities v <v . It ies with intrinsic dispersions of 4, 7, and 10 km/s. We
c
follows that if velocity outlier stars are excluded in de- see that for each group, the graph forms a tight relation
terminingvelocitydispersion,theextradispersiondueto for all but the most extreme values of the period distri-
binaries can be determined from the threshold fraction bution parameters. In plotting these points we varied B
F with reasonable confidence even if the parameters B, from 0.2 to 1, µ from -1 to 4, and σ from 0.5 to
0 logP logP
µ , and σ are entirely unknown. 4(with P in years). The lowermostpoints ofeachgroup
logP logP
We demonstrate this by simulating galaxies with var- are the points for with σ has its smallest value of
logP
ious intrinsic dispersions and characterized by different 0.5, producing only a very small number of short-period
binarypopulations. The dispersionσ iscalculatedbyit- binaries. The uppermost points are the points for which
eratively discarding stars with velocities larger than 3σ; µ hasits smallestvalue,sothe perioddistributionis
logP
on the first iteration the biweight is used to estimate shifted toward short periods. For these extreme values,
the dispersion, and the dispersion is then calculated on the velocity distribution becomes distorted into a dis-
every subsequent iteration until all the remaining stars tinctly non-Gaussian shape so these can be considered
have velocities that lie within 3σ. In order to make the highly improbable configurations. We have also varied
statisticalerrornegligible,weusedaverylarge“sample” the ellipticity distribution parameters e¯, σ (section 2)
e
of100,000stars. Wealsocalculatethethresholdfraction andfindthatthetightcorrelationinfig.10isunchanged,
F forthesamedataset,forwhichwepickedathreshold andalthoughtheamountofscatterincreasesslightly,the
0
∆v =5 km/s and time interval ∆t=1 year. correctionstill holds to within a few percent accuracy.
First, we assume the fiducial binary period distribu- Wefindthatthepointsplottedinfig.10arewellfitby
tion(µ andσ )andvarythebinaryfractionfrom alineplusanexponentialfunction,providedthatoutlier
logP logP
B =0.1to1. Infig.9weplottheratioσ/σ ofmeasured points are discarded. To define “outliers”, first we di-
0
dispersion over the intrinsic dispersion as a function of vide the domain F [0,0.2] into sections small enough
0
∈
threshold fraction, for galaxies with intrinsic dispersions sothe graphis approximatelylinearwithineachsection.
of 4, 7, and 10 km/s. We used an absolute magnitude We then further divide eachsectioninto two subsections
limit M =3,howeverthe graphremains virtuallyun- andcalculate the medianandmedianabsolutedeviation
lim
changed for other magnitude limits because of the near- (MAD) of the y-values of the points in each subsection.
degeneracy of magnitude limit with binary fraction dis- Next we draw lines through the two points defined by
cussed at the beginning of this section. We see that for the median twice the MAD of eachof the two subsec-
±
a given intrinsic dispersion, the observed threshold frac- tions,takingthecenterofthesubsectionastheirx-value.
tion can be mapped in a one-to-one way to the intrinsic The plotted points that lie outside the regiondefined by
dispersion. The relation shown in the graph also holds these lines represent extreme and highly improbable pe-
regardless of the age of the stellar population, again be- rioddistributions,andarethereforediscarded. Wefitthe
cause of the degeneracy of age with binary fraction. remaining points and repeat the procedure for galaxies
10
Fig.10.— Ratio of measured velocity dispersion σ over the in- Fig. 11.— Ratio of measured velocity dispersion σ over the in-
trinsicdispersionσ0,plottedwithrespecttothresholdfractionF0 trinsicdispersionσ0,plottedwithrespecttothresholdfractionF0
fordifferentbinarypopulations ingalaxies ofintrinsicdispersions forfixed measured dispersions σ. These curves werefound by fit-
σ0=5,7,and10km/s. Themeasureddispersionswerecalculated tinggraphs likethose showninfig.10,then transformingtofixed
by an interative 3σ-clipping routine, and the threshold fraction valuesofthemeasureddispersionσ. Themeasureddispersionsin
F0 denotes the fraction of stars with observed change in velocity fig. 10 were calculated by an interative 3σ-clipping routine, and
greater than a threshold ∆v = 5 km/s after a time ∆t = 1 year thethresholdfractionF0 denotesthefractionofstarswithchange
between measurements, assuming zero measurement error. Each in velocity greater than a threshold ∆v = 5 km/s after a time
pointrepresents adifferentbinarypopulationwithitsownbinary ∆t = 1 year between measurements, with zero measurement er-
fractionandperioddistribution;weplottedthepointsoveragrid ror. Weshowattheendofsection5how F0 canbeinferredfrom
of values, with binary fraction B ranging from 0.2 to 1, µlogP observationsaccounting formeasurementerrors.
from -1 to 4 (in log(P/year)) and σlogP from 0.5 to 4. We show
at the end of section 5 how F0 can be inferred from observations the value of F in eq. 16 must be scaled by the ratio
accounting formeasurementerrors. 0
F (∆v)/F (5 km/s). This ratio can be calculated by us-
0 0
of dispersions ranging from 3-12 km/s. ing Monte Carlo realizations to plot the threshold frac-
The plots in fig. 10 are not directly applicable to real tionasafunctionofthreshold,whichwewilldoinsection
data because eachgraphwas plotted for a fixed intrinsic 4 (figs. 2, 4). Again, the degeneracy of magnitude limit
dispersion σ , which is unknown (and is in fact what we and period distribution parameters with binary fraction
0
are attempting to calculate!). We therefore use our fits ensures that this ratio is virtually independent of the
togetherwitharoot-findingprocedure,interpolatingthe model parameters and magnitude limit, provided one
fitting parameters in σ , to draw similar graphs at fixed does not transform to thresholds that are too high (>
0
values of σ. A few resulting curvesare plotted in fig. 11. 10 km/s). We find that for thresholds in the range 4
Again, we find these curves are well fit by a line plus km/s < ∆v < 10 km/s, the ratio F0(∆v)/F0(5 km/s)
exponential, can be fit by the function
F (∆v)
σ 0 =a+be−∆v/∆vs (20)
σ =a(σ)+b(σ)F0+c(σ) eF0/0.1−1 (16) F0(5 km/s)
0
h i where the best-fit parameters are a=0.0725, b=1.897,
where F is the threshold fraction at 5 km/s. We also
0 and ∆v = 6.947 km/s. Thus to find the correction to
find fitting functions for the parameters a(σ), b(σ), and s
the dispersionin terms of a given velocity threshold∆v,
c(σ)whichfitwellfordispersionsσ rangingfrom4km/s
one substitutes eq. 20 into eq. 16 so that the fit is in
to 10 km/s. Defining ∆σ =σ 4 km/s, we find:
− terms of F0(∆v).
Thesamplingerrorintheintrinsicdispersionσ deter-
0
a(σ)=0.988e−0.0007∆σ (17) minedbythisprocedurecanbeestimatedbynotingthat
foratwo-epochsampleofN stars,the Poissonerrorin
b(σ)=0.576 0.08∆σ+0.772 1 e−0.1∆σ (18) 2e
− − the thresholdfractionis δF F0(∆v)/N2e. Letus as-
c(σ)=0.043e−0.247∆σ (cid:0) (cid:1) (19) sume the dispersion is measu≈redin a larger single-epoch
sample of N stars; then we canpmake the approximation
These formulas hold for any magnitude limit and stel-
thatthesamplingerrorsinσandF areweaklycorrelated
lar age, and the threshold fraction F refers here to a
0 so they add in quadrature. Propagating the error using
velocitythresholdof5km/s,time intervalof1year,and
eq.16givesthesamplingerrorintheintrinsicdispersion,
zero measurement error.
How are these formulas adjusted if a different veloc-
ity threshold is desired? Ideally, one should use the δσ 2 (σ/σ )2 σ ∂(σ/σ ) 2 F (∆v)
0 0 0 0 0
smallest possible threshold that is not significantly af- + (21)
σ ≈ 2N σ ∂F N
fected by measurement error–this will include the most (cid:18) 0 (cid:19) (cid:12) 0 (cid:12) 2e
(cid:12) (cid:12)
stars and therefore have a smaller scatter compared to where the second term in(cid:12)eq. 21 is the(cid:12) two-epoch sam-
(cid:12) (cid:12)
higher thresholds. To use a different threshold ∆v, plingerror. Forexample,ifthe single-epochsample con-