2411.00088v1.pdf

Version November 4, 2024

Preprint typeset using L

X style openjournal v. 09/06/15

A GENERATIVE MODEL FOR

GAIA

ASTROMETRIC ORBIT CATALOGS: SELECTION FUNCTIONS FOR

BINARY STARS, GIANT PLANETS, AND COMPACT OBJECT COMPANIONS

Kareem El-Badry

, Casey Lam

, Berry Holl

, Jean-Louis Halbwachs

, Hans-Walter Rix

, Tsevi

Mazeh

, and Sahar Shahaf

Department of Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA

Max-Planck Institute for Astronomy, K ̈onigstuhl 17, D-69117 Heidelberg, Germany

Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA

Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland

Department of Astronomy, University of Geneva, Chemin d’Ecogia 16, 1290 Versoix, Switzerland

Universit ́e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, 11 rue de l’Universit ́e, Strasbourg, France

School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 6997801, Israel and

Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel

Version November 4, 2024

ABSTRACT

Astrometry from

Gaia

DR3 has produced a sample of

∼

170,000 Keplerian orbital solutions, with

many more anticipated in the next few years. These data have enormous potential to constrain the

population of binary stars, giant planets, and compact objects in the Solar neighborhood. But in order

to use the published orbit catalogs for statistical inference, it is necessary to understand their selection

function: what is the probability that a binary with a given set of properties ends up in a catalog?

We show that such a selection function for the

Gaia

DR3 astrometric binary catalog can be forward-

modeled from the

Gaia

scanning law, including individual 1D astrometric measurements, the fitting of

a cascade of astrometric models, and quality cuts applied in post-processing. We populate a synthetic

Milky Way model with binary stars and generate a mock catalog of astrometric orbits. The mock

catalog is quite similar to the DR3 astrometric binary sample, suggesting that our selection function is

a sensible approximation of reality. Our fitting also produces a sample of spurious astrometric orbits

similar to those found in DR3; these are mainly the result of scan angle-dependent astrometric biases

in marginally resolved wide binaries. We show that

Gaia’s

sensitivity to astrometric binaries falls off

rapidly at high eccentricities, but only weakly at high inclinations. We predict that DR4 will yield

∼

million astrometric orbits, mostly for bright (

≲

15) systems with long periods (

orb

≳

1000 d). We

provide code to simulate and fit realistic

Gaia

epoch astrometry for any data release and determine

whether any hypothetical binary would receive a cataloged orbital solution.

Subject headings:

astrometry – catalogues – methods: statistical – binaries: general

INTRODUCTION

The third data release of the

Gaia

mission (DR3; Gaia

Collaboration et al. 2016, 2023a) included astrometric or-

bital solutions for

∼

168

000 binary systems, represent-

ing a vast improvement in sample size and completeness

over all previous work (Gaia Collaboration et al. 2023b).

These data have already enabled discovery of several as-

trophysically interesting objects, including a sample of

compact objects in au-scale orbits with stellar compan-

ions (e.g. Shahaf et al. 2023) and a sample of giant plan-

ets orbiting nearby stars (Holl et al. 2023a). For a review

of the

Gaia

binary star sample, see El-Badry (2024).

Epoch-level astrometric data was not published in

DR3, and several quality cuts were imposed on the pub-

lished orbital solutions. These cuts depend on a variety of

quantities, such as the signal-to-noise ratio of the photo-

center semi-major axis, the orbital period, and the paral-

lax and eccentricity uncertainties. These cuts – combined

with the fact that

Gaia’s

sensitivity to astrometric orbits

Corresponding author: kelbadry@caltech.edu

depends on quantities such as the eccentricity and orien-

tation in ways that are difficult to predict analytically –

make it a nontrivial problem to use the

Gaia

astromet-

ric binary sample for population interference, and few

attempts at modeling the sample have been made so far.

In this paper, we use a model of the solar neighbor-

hood’s binary population to build a forward-model for

the

Gaia

DR3 astrometric binary catalog. This entails

modeling the

Gaia

observations at the level of individ-

ual epochs, predicting the observation times, scan an-

gles, and simulated one-dimensional astrometry for each

simulated source from the

Gaia

scanning law. We in-

clude common false positives and reproduce the cascade

of single-star, accelerating, and full orbital astrometric

models employed in producing the observed catalog. Be-

cause our modeling results in a mock catalog of orbital

solutions that resembles the one actually published in

DR3, we believe that it captures the most important se-

lection effects and false positives of the real catalog. It

can thus be used both to interpret the DR3 binary sam-

ple and to forecast what will be discoverable in future

data releases.

arXiv:2411.00088v1 [astro-ph.SR] 31 Oct 2024

The remainder of this paper is organized as follows. In

Section 2, we summarize the basics of

Gaia

observations

and the astrometric signal caused by a binary. Section 3

describes the Galactic model, simulated binary popula-

tion, and the assumptions we make to predict 1D epoch

astrometry. We discuss the astrometric model cascade

in Section 4 and present the resulting mock catalog in

Section 5. Finally, Section 6 presents a Monte Carlo se-

lection function that can be used to model and fit epoch

astrometry for arbitrary binaries. Section 7 summarizes

our main results.

This paper aims to develop a relatively detailed model

Gaia

orbit catalogs by forward-modeling epoch as-

trometry and the astrometric model cascade. A com-

panion paper, Lam et al. (2024), develops a model that

is more approximate but significantly less computation-

ally expensive.

HOW GAIA OBSERVES

How

Gaia

observes has been summarized extensively

in other work, and we refer to Gaia Collaboration et al.

(2016) for a detailed description. We summarize the most

important aspects here.

Gaia

is equipped with two telescopes whose fields of

view are separated by 106.5 degrees. The satellite rotates

with a period of 6 hours, such that the two telescopes

sweep out an annulus with a width of about 0.7 deg on

the sky. The rotation axis precesses with a period of

63 days and a fixed tilt angle of 45 degrees with respect

to the Sun direction, causing this annulus to rotate on

the sky. At the same time, the spacecraft orbits the

Sun with a period of a year. The combination of these

three rotations results in full-sky coverage, though with

the observing cadence and distribution of scan angles

varying significantly across the sky (for details, see Holl

et al. 2023b). For stars brighter than

= 15, the median

number of visibility periods used in DR3 is 20, with a

(16-84)% range of 16-27. Here a visibility period refers

to a group of observations separated by other groups of

observations by at least 4 days. Data contributing to

DR3 solutions were obtained over a period of about 1000

As a source moves across the astrometric field of view

(FOV), it is independently observed by 8-9 different

CCDs. Each CCD observation results in an independent

measurement of the source’s position in the along-scan

(AL) direction relative to a reference position assigned

to the source at a reference time. Crucially,

only

a 1D

measurement in the AL direction is made with high pre-

cision and used in the astrometric solution. At

≲

– which is the regime relevant to most astrometric bi-

naries published so far – the per-CCD observations cur-

rently reach an AL precision of order 0.12 mas. All the

CCD transits in a given FOV transit are spread over

a period of less than a minute, which is much shorter

than the orbital periods of astrophysically plausible bi-

naries to which

Gaia

is sensitive. The 8-9 CCD transits

can thus be regarded as essentially independent measure-

ments of the same quantity, and can be averaged to ob-

tain smaller uncertainties in the CCD-averaged measure-

ments. Our modeling in this work suggest that these

measurements are indeed nearly independent and not

systematics-dominated, such that the per-FOV transit

uncertainties are

∼

3 times smaller than the per-CCD

uncertainties (Appendix C).

Gaia

astrometry is built upon a global astrometric so-

lution, whose calculation requires the simultaneous opti-

mization of millions of attitude and calibration parame-

ters, and astrometric parameters for

∼

100 million stars

(Lindegren et al. 2012). Most of these parameters are of

little interest for the analysis of one source. It is pos-

sible to transform the astrometric measurements for a

single source to “local plane coordinates” in a tangent

plane to the unit sphere in the vicinity of each source,

as described by Lindegren & Bastian (2022). The as-

trometric measurements required to describe the motion

of a source can then be represented with just a small

file containing the AL displacements, their uncertainties,

and associated metadata (scan angles, parallax factors,

transit times, etc.) for each source. Thus far, such epoch

astrometry has been published for only one source: the

binary Gaia BH3 (Gaia Collaboration et al. 2024b).

GALACTIC MODEL AND BINARY

POPULATION

We now describe the Galactic model and binary pop-

ulation from which we forward-model

Gaia

observations

and construct a mock astrometric orbit catalog. This will

allow us to validate our selection function by comparing

the mock catalog to the DR3 data.

3.1.

Galactic model

Our modeling uses

Galaxia

(Sharma et al. 2011),

which generates synthetic resolved-star surveys from the

Besan ̧con model of the Milky Way (Robin et al. 2003).

We use a modified version of the code described by Lam

et al. (2020). We only attempt to model sources within

2 kpc of the Sun, because 99% of all astrometric orbits

published in DR3 are found within 2 kpc.

Galaxia

predicts a total of 1.1 billion sources (all rep-

resenting single stars) within 2 kpc of the Sun. Com-

paring to the

Gaia

DR3

gaia

source

catalog, we find

that

Galaxia

predicts 1.4 times more sources than are

observed (with

parallax

over

error > 10

) within both

50 and 100 pc. Since

Gaia

is expected to be nearly com-

plete within these volumes for sources above the hydro-

gen burning limit (Gaia Collaboration et al. 2021), we

conclude that

Galaxia

predicts 40% too many sources

in the Solar neighborhood. We thus discard a random

subset of predicted sources, chosen independent of dis-

tance and apparent magnitude, to reduce source counts

by a factor of 1.4

3.2.

Binary population

Galaxia

does not include binary stars. We thus use

COSMIC

(Breivik et al. 2020) to generate a zero-age binary

population according to the model from Moe & Di Ste-

fano (2017). We assume a Kroupa (2001) primary mass

function

and draw mass ratios, periods, and eccentrici-

ties from the covariant, mass-dependent distributions in-

ferred by Moe & Di Stefano (2017). The assumed binary

fraction is

≈

41% for solar-type primaries,

increasing

Note that this is different from the more top-heavy mass func-

tion assumed in COSMIC by default.

This is lower than

mult

≈

5, the mean number of com-

panions per solar-type primary, because higher-order multiples

contribute more than one companion.

Only companions with

log (

orb

8 are considered.

to 57% at

= 3

⊙

and 80% at

= 6

⊙

. Below

⊙

, the binary fraction is assumed to decrease lin-

early with log

, from 40% at 0

⊙

to 0 at 0

⊙

Almost all of the binaries predicted to be detectable in

DR3 have primary masses between 0.7 and 1

⊙

, so

it is the binary population in this mass range – which

is constrained primarily by the Raghavan et al. (2010)

survey of nearby G stars – that is most important for

our modeling.

COSMIC

predicts populations of single and binary stars.

We match both binaries and singles to the

Galaxia

predicted sources, generating sources until the total

number of stellar systems (singles plus binaries, with

each binary representing a single system) in the zero-

age

COSMIC

-predicted population exceeds the number of

Galaxia

sources within 2 kpc by 10%. This accounts for

the fact that the

Galaxia

population only contains stars

that are still alive; for a Kroupa IMF and constant star

formation history over 12 Gyr, about

≈

10% of all stars

that have been born will have already died.

Massive stars are preferentially found in the Galac-

tic disk and thus are subject to higher extinction than

typical lower-mass stars. To account for this, we place

each simulated binary at the position of the

Galaxia

star whose mass is closest to that of the primary. We

assign inclinations by drawing from a sin(

) distribution.

We draw longitudes of the ascending node, Ω, and ar-

guments of periastron,

, from

). We define the

reference phase as

πT

orb

, where

is the epoch of

periastron, and draw

from

1).

We assign

−

band absolute magnitudes to the pri-

mary and secondary components using the

isochrones

package (Morton 2015), using

MIST

models (Choi et al.

2016) and assuming a uniform age distribution between

0 and 12 Gyr. We remove binaries with initial masses

⊙

whose primaries have died, assuming that most

would-be black hole or neutron star binaries are de-

stroyed or dramatically tightened by common envelope

evolution and supernova kicks. We similarly assume that

most binaries containing white dwarfs (WDs; i.e., pri-

maries with initial masses

⊙

that have terminated

their evolution) shrink to short periods and are not de-

tectable astrometrically. However, motivated by the re-

sults of Shahaf et al. (2024) and Yamaguchi et al. (2024),

we assume that 10% of WD + luminous star binaries with

initial separations of (2

−

6) au form wide post-common

envelope binaries with separations that are 50% of their

initial separations. This prescription is quite simplified

and is not expected to capture all aspects of the observed

population. The 10% fraction is chosen to approximately

match the number of binaries with large mass functions

in the observed catalog (Section 3.4). We assign WD

masses using the initial-final mass relation of Weidemann

(2000). We remove binaries in which either star currently

fills its Roche lobe or would have filled it previously (e.g.,

red clump stars that would have filled their Roche lobes

at the tip of the first giant branch). Because most bi-

naries that are astrometrically detectable contain main-

sequence stars in au-scale orbits, the treatment of evolu-

tionary effects has minor effects on the overall properties

of the observable population.

We calculate extinctions to each binary using the

combined19

dust map in the

mwdust

package (Bovy et al.

[mag]

[mas]

per CCD

per FOV transit

Fig. 1.—

Assumed AL displacement uncertainties. Black line

shows the uncertainty per CCD transit, adopted from Holl et al.

(2023a). Tan line shows our assumed uncertainty per FOV transit

and is smaller than the black line by a factor of

√

8, representing the

result of averaging measurements from an average of 8 uncorrelated

CCD measurements per FOV transit. The uncertainties are small-

est at

= 8

−

14 and are dominated by calibration systematics

in this magnitude range, but we find (Appendix C) that measure-

ments from individual CCDs can still be usefully combined to yield

FOV transit-averaged uncertainties a factor of

≈

√

8 smaller. The

uncertainties at

≳

14 are photon-limited and thus rise rapidly

toward faint magnitudes. Calibration problems worsen at

≲

2016), which combines the maps from Drimmel et al.

(2003), Marshall et al. (2006), and Green et al. (2019).

We assume

= 2

(

−

), where

(

−

) is on

the Schlegel et al. (1998) scale. We then calculate each

binary’s total

−

band apparent magnitude, angular sep-

aration,

, and magnitude difference, ∆

−

Binaries that are resolved or marginally resolved are re-

moved from the sample. As we show in Appendix B,

the transition between resolved and unresolved binaries

depends on both

and ∆

, and in DR3 can be approx-

imated as ∆

[mag] =

mas

−

200

, where binaries

with ∆

larger than this value are unresolved. We re-

move resolved binaries from the sample, as well as un-

resolved binaries with

G >

19, which were not fit with

binary solutions in DR3. We only generate mock as-

trometry for binaries, discarding single stars and neglect-

ing both higher-order multiples and chance alignments of

physically unassociated stars. We are left with 46 million

unresolved binaries with

G <

19 within 2 kpc.

As a consistency check, we calculated the total number

of predicted binaries with absolute magnitude

= 3

−

(since solar-type binaries dominate the astrometric bi-

nary sample in DR3), finding 9

and 1

within 200 pc and 500 pc, respectively. These fractions

represent 44% and 45%, respectively, of the total num-

ber of

Gaia

sources with

= 3

−

6 within the same

volumes. The good agreement between these fractions

and the 41% assumed binary fraction for solar-type stars

suggests that the

Galaxia

model is reasonably accurate.

3.3.

Epoch astrometry

We mock-observe each binary using the scan times and

scan angles predicted by the

Gaia

Observation Forecast

Tool (

GOST

)

. Because querying

GOST

involves a relatively

slow web query, we first pre-computed the scanning law

https://gaia.esac.esa.int/gost/

at 49152 uniformly spaced points on the sky (healpix

level 6), and then used the saved value at the point closest

to each simulated source. This results in an effective

resolution of about 0.9 degrees, which is comparable to

the 0.7 degree across-scan FOV.

When modeling astrometry for

Gaia

DR3, we only in-

clude observations between JD 2456891 and JD 2457902

(Halbwachs et al. 2023). For DR4 observations, we in-

clude scans taken between JD 2456891 and JD 2458868.

Due to a variety of issues (see Lindegren et al. 2021b),

about 10% of FOV transits do not result in usable data.

To account for this, we reject FOV transits for each

source with 10% probability.

3.3.1.

Noise model

We add Gaussian noise to the predicted AL displace-

ments according to the empirical median CCD AL ab-

scissa uncertainty in DR3 (Holl et al. 2023a, their Fig-

ure 3, “EDR3 adjusted” model). At

≲

13, these un-

certainties are on average a factor of

≈

2 larger than

the formal uncertainties calculated by the image param-

eter determination (IPD) pipeline (e.g. Lindegren et al.

2021a). This discrepancy is a result of imperfect treat-

ment of systematics for bright sources. In the astromet-

ric non-single star (NSS) pipeline, epoch uncertainties

were correspondingly inflated, with the intention that the

residuals for typical good solutions should be consistent

with the uncertainties (see Holl et al. 2023a). While this

uncertainty inflation was only partially successful (i.e.,

some trends in goodness-of-fit statistics with apparent

magnitude still exist), it is important to remember that

different uncertainties were assumed in calculating bi-

nary and single-star astrometric solutions. This implies

that their goodness-of-fit metrics should not be directly

compared, and that the two kinds of solutions are likely

affected by different types of systematics (e.g. Nagarajan

& El-Badry 2024).

Each FOV transit contains 8 or 9 individual CCD tran-

sits, each separated by a few seconds. We bin these

into a single CCD-averaged value per transit, with un-

certainty

√

bin

times smaller than the per-CCD uncer-

tainty. Here

bin

represents the number of measure-

ments from individual CCDs that are being averaged;

we adopt

bin

= 8 to account for the fact that some

CCD measurements flagged as outliers are discarded. We

show the assumed per-CCD and per-FOV transit un-

certainties in Figure 1. We verified that applying this

method of single sources fit with 5-parameter astromet-

ric solutions results in astrometric uncertainties that are

in good agreement with those published in

Gaia

DR3

(see Appendix C).

For sources brighter than

= 13, we add unmodeled

Gaussian noise to account for underestimated uncertain-

ties due to various systematics, which have been shown

to increase at

G <

13 (e.g. Lindegren et al. 2021b,a; El-

Badry et al. 2021). A sharp increase in

goodness

of

fit

of the DR3 astrometric binary solutions at

G <

13 (e.g.

El-Badry et al. 2023c) shows that this effect was not fully

corrected by error inflation applied to the epoch astrom-

etry in the NSS astrometric pipeline. We draw the per-

FOV transit “

” of the unmodeled noise from

04)

mas. This is motivated by our fit to the epoch astrometry

for Gaia BH3 (Gaia Collaboration et al. 2024b), where

we find that adding 0

04 mas in quadrature to the per-

FOV transit uncertainties results in a reduced

of 1.

3.4.

Generating mock observations

We now consider how an unresolved or marginally re-

solved binary affects the epoch astrometry. Our model-

ing follows Lindegren (2022). In a given scan, a source

is observed with scan angle

, defined as degrees east of

north. We define

as the sky-projected, instantaneous

angular separation of the two stars, and

as the binary’s

position angle. Then ∆

cos(

−

) is the angular

separation projected onto the scan angle. Finally, we de-

fine

= ∆

η/u

, where

quantifies the resolution of the

Gaia

line spread function. In detail,

will depend on

the flux ratio and other quantities, but Lindegren (2022)

and Holl et al. (2023b) find that

= 90 mas provides

a reasonable approximation for modeling astrometry of

binaries, and we adopt this value throughout.

We assume that the two stars each contribute flux

along the AL direction as 1D Gaussians with unit vari-

ance separated by

, and with amplitudes scaled accord-

ing to the flux ratio,

= 10

(

−

)

, where

and

are the magnitudes of the primary and secondary. We

further assume that the measured AL displacement will

be the

peak

of the resulting flux profile. At sufficiently

large angular separations, the two peaks will be resolved,

and in this case we assume the IPD pipeline will correctly

centroid the peak of the primary. We predict the AL dis-

placement relative to the binary’s center of mass in these

two cases as

δη

(

f,ξ

)

−

∆

if 0

|≤

−

∆

if 3

−

f <

(1)

Here

(

f,ξ

) is obtained by iteratively solving the equa-

tion

fξ

+ exp(

−

ξB

)

(2)

for

, beginning at

= 0. This yields the AL coordi-

nate of the peak of the total flux profile (see Lindegren

2022, for details). Inspection of Equation 1 reveals that

in the limit of small

(i.e., close separations), it reduces

δη

−

∆

. In this case, the observations

exactly trace the photocenter. As

increases, the peak

of the flux profile no longer exactly traces the photo-

center, but is displaced toward the primary. For typical

flux ratios, the difference between the first case in Equa-

tion 14 and the photocenter displacement becomes sig-

nificant (

≳

05 mas) only for separations ∆

≳

30 mas.

This means that

Gaia

effectively observes the photocen-

ter for the vast majority of binaries with periods short

enough to be astrometrically constrained. On the other

hand, for marginally resolved wide binaries,

Gaia

does

not

observe the photocenter. The difference between the

observed AL displacements and the photocenter will vary

with scan angle, and this can lead to spurious orbits with

a range of periods related to the scanning law, as we ex-

plore in Section 5.2 and Appendix E.

The 2nd case of Equation 1 corresponds to cases where

two peaks are detected in the AL flux profile and the

IPD processing correctly identifies the centroid of the

primary. It is rarely relevant for binaries published in

Single star solution

9 parameter solution

7 parameter solution

Orbital solution

and

or resolved

and

Yes

and

Yes

Orbital solution

published

All sources within 2 kpc

Not detected by

Gaia

Yes

9 parameter solution

published

7 parameter solution

published

Yes

Fig. 2.—

Astrometric model cascade used in constructing the

mock catalog of orbital solutions. The cascade approximates the

one used in generating the DR3 binary catalog Halbwachs et al.

(2023, their Figure 1). The guiding principle is to only fit more

complex models (e.g. orbits) in cases where simpler models (single-

star and acceleration solutions) produce a manifestly poor fit. The

number of sources retained and excluded at each branch is indi-

cated.Many of the orbital solutions rejected at the last step of

filtering are spurious. Only a small fraction (

1%) of all binaries

detected by

Gaia

within 2 kpc ultimately receive astrometric or-

bital solutions in DR3.

DR3, because resolvable pairs are removed by the cut on

ipd

frac

multi

peak

used in selecting sources for pro-

cessing with orbital solutions (Appendix B).

For widely separated pairs with

ξ >

5, we assume

that the marginally-resolved nature of the source leads

to poor centroiding. We thus add unmodeled Gaussian

noise with

= 0

5 mas to the epoch astrometry in epochs

where

ξ >

5. This choice is somewhat ad-hoc but is mo-

tivated by the fact that the model otherwise predicts too

many spurious solutions with good formal fits, leading to

features in the recovered period distribution that are not

present in the DR3 data.

FITTING THE ASTROMETRIC DATA

We now describe how we fit the epoch astrometry

generated above. We attempt to reproduce the pro-

cedure used to generate the binary solutions published

in DR3 whenever possible, as described by Halbwachs

et al. (2023). The astrometric model cascade is illus-

trated in Figure 2. We emphasize that our main goal is

to reproduce the DR3 catalog of orbital solutions (i.e.,

nss

solution

type = Orbital

AstroSpectroSB1

so the cascade shown in Figure 2 is somewhat simplified

compared to the one actually used in DR3.

It is not known a priori which sources are single stars

and which are detectable binaries or higher-order multi-

ples. In

Gaia

DR3, the guiding principle was to publish

the simplest astrometric solution that could satisfactorily

explain the data, beginning with a single-star model.

4.1.

5-parameter single-star solution

All sources are first fit with a 5-parameter, single-star

astrometric model. It predicts AL displacements given

= [∆

∗

] sin

+ [∆

] cos

+ Π

π.

(3)

Here ∆

∗

and ∆

represent the source position at a

reference time

ref

, relative to the reference point (

,δ

)

assigned to each source.

∗

and

are proper motions,

and

is the source parallax. The scan angles,

, and

parallax factors, Π

, are precomputed for each source by

GOST

; the latter are defined such that multiplying the

true parallax by Π

gives the source’s parallactic motion

in the AL direction at the time of the relevant scan. Note

that

and Π

are functions of time.

Given a list of observation times,

, measured AL dis-

placements,

, their uncertainties,

η,i

, the scan angles

, and parallax factors, Π

η,i

, the maximum-likelihood

values of the astrometric parameters

{

∆

∗

,μ

∗

∆

δ,μ

,π

}

and their uncertainties can be obtained via linear regres-

sion. These parameters can then be used to calculate the

predicted AL displacements,

pred

, and a corresponding

statistic:

(

pred

−

)

η,i

(4)

At this stage it is necessary to correct for the fact that

we average (“bin”) measurements from different CCDs

within a FOV transit, reducing the number of astromet-

ric measurements by a factor of

bin

= 8. In general, nei-

ther

nor reduced

is conserved under binning. As-

suming that all CCDs yield independent measurements

of the same quantity, we derive an expression for the ex-

pected

of the unbinned data in terms of the

of the

binned data:

unbinned

binned

FOV transits

(

bin

−

(5)

In the subsequent text,

always refers to the value cal-

culated for the unbinned data via Equation 5.

For

Gaia

DR3,

ref

= J2016

0 = JD 2457389

0. For DR4, we

anticipate

ref

= J2017

5 = JD 2457936

875.

From this we calculate a “unit weight error”,

UWE =

(6)

where

is the number of degrees of freedom. In this case,

FOV transits

bin

−

5; i.e., the number of unbinned

datapoints minus the number of free parameters. UWE

is analogous to the

RUWE

statistic published in

Gaia

DR3

(Lindegren 2018; Lindegren et al. 2021b), where the lat-

ter is renormalized to correct for empirical trends in the

astrometric uncertainties with color and apparent mag-

nitude. It would

not

be appropriate to renormalize our

calculated UWE to match the

Gaia

RUWE

, because the

latter is calculated from underestimated epoch-level un-

certainties while the former is calculated from uncertain-

ties that have already been inflated to yield UWE

∼

for single sources (Section 3.3.1). We verify that our

UWE values are a good approximation of

Gaia’s

RUWE

Section 5.1.3.

Various authors have shown that many sources with

RUWE

4 are binaries (e.g. Belokurov et al. 2020;

Penoyre et al. 2022a). However, not all binaries are ex-

pected to have

RUWE

4, since in some cases the orbital

period is too long or the astrometric uncertainties are too

large for the companion to produce detectable deviations

from single-object motion. In

Gaia

DR3, only sources

with

RUWE

4 were processed with binary models. We

thus fit all 46 million simulated unresolved binaries satis-

fying

G <

19 with single-star solutions, discarding those

with UWE

4. Following the DR3 procedure, we also

discarded 3

sources that were observed in fewer

than 12 visibility periods.

4.2.

Acceleration solutions

The simplest extension of the single-star solution is a

7-parameter acceleration solution, which adds two free

parameters, ̇

∗

and ̇

, for acceleration in the right as-

cension and declination directions:

∆

∗

sin

∆

cos

+ Π

π.

(7)

This model is appropriate if the deviation from single-

object motion can be satisfactorily described by a con-

stant accelerations, as is expected for binaries with or-

bital periods much longer than the observing baseline.

A similar 9-parameter model can be defined for sources

with variable acceleration, as might be expected for bi-

naries with periods only a few times longer than the ob-

serving baseline:

∆

∗

sin

∆

cos

+ Π

π.

(8)

Here ̈

∗

and ̈

represent the acceleration derivatives.

Equations 7 and 8 are not identical to the models fit by Halb-

Adding additional free parameters will in general al-

ways lead to a better fit, so it is useful to quan-

tify whether the data significantly constrain the ad-

ditional free parameters.

For acceleration solutions,

“significance” is quantified as the modulus of the two-

dimensional vector of additional free parameters (relative

to the next-simplest model) divided by its uncertainty:

−

(9)

Here

and

represent the additional parameters of

the model,

and

their uncertainties, and

the cor-

relation coefficient between them. For the 7-parameter

model (Equation 7),

and

represent the accelera-

tion terms, ̇

∗

and ̇

; for the 9-parameter model, they

represent the acceleration derivatives, ̈

∗

and ̈

The goodness-of-fit of acceleration and orbital solution

is quantified by the

statistic (Wilson & Hilferty 1931),

which is a transformation of

−

(10)

For well-behaved solutions with reliably-estimated uncer-

tainties,

is expected to follow a normal distribution,

1). Values of

significantly above 1 suggest a poor

solution or underestimated uncertainties. To correct for

underestimated uncertainties, the uncertainties of the as-

trometric parameters are re-scaled by a constant,

−

(11)

such that sources with poor fits have inflated uncertain-

ties at fixed apparent magnitude.

Gaia

DR3, the 9-parameter variable acceleration so-

lution (Equation 8;

not

the simpler 7-parameter model)

was the first model tried for sources with

RUWE

the acceptance criteria applied before testing the orbital

model were:







(12)

where

is the significance of the 9-parameter solution

(Equation 9) and

and

are the parallax and parallax

uncertainty of the same solution.

Solutions satisfying these cuts were provisionally ac-

cepted with 9-parameter solutions (i.e., other models

were not tried, but only solutions passing additional cuts

were actually published). Sources not passing Equa-

tion 12 were fit with 7-parameter acceleration solutions

and (provisionally) accepted using similar criteria:







(13)

wachs et al. (2023, their Equations 6 and 7) because they include

an additional time offset to make the source positions at the refer-

ence epoch similar between accelerating and single-star solutions.

This does not affect the magnitude of the inferred accelerations,

the significance, or the goodness-of-fit.