# Features of Gaia DR3 Spectroscopic Binaries
I. Tidal circularization of Main-Sequence Stars

Dolev Bashi, Tsevi Mazeh and Simchon Faigler
School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 6997801,
Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Previous studies pointed out that many observed samples of short-period
binaries display a cutoff period, $P_{\rm cut}$, such that almost all binaries
with periods shorter than $P_{\rm cut}$ have circular orbits. This feature is
probably due to long-term circularization processes induced by tidal
interaction between the two stars of each binary. It seemed as if coeval main-
sequence (MS) samples of open clusters display $P_{\rm cut}$ that depends on
the sample age. Using the unprecedentedly large sample of MS spectroscopic
orbits recently released by Gaia we have found that the $P_{\rm cut}$ does not
depend on the stellar age but, instead, varies with stellar temperature,
decreasing linearly from $6.5$ day at $T_{\rm eff}\sim 5700$ K to $\sim 2.5$
day at $6800$ K. $P_{\rm cut}$ was derived by a new algorithm that relied on
clear upper envelopes displayed in the period-eccentricity diagrams. Our
$P_{\rm cut}$ determines both the border between the circular and eccentric
binaries and the location of the upper envelope. The results are inconsistent
with the theory which assumes circularization occurs during the stellar MS
phase, a theory that was adopted by many studies. The circularization has
probably taken place at the pre-main-sequence phase, as suggested already in
1989 by Zahn and Bouchet, and later by Khaluillin and Khaluillina in 2011. Our
results suggest that the weak dependence of $P_{\rm cut}$ on the cluster age
is not significant, and/or might be due to the different temperatures of the
samples. If indeed true, this has far-reaching implications for the theory of
binary and exoplanet circularization, synchronization, and alignment.

###### keywords:

binaries: close – binaries: spectroscopic – methods: statistical

††pubyear: 2015††pagerange: Features of Gaia DR3 Spectroscopic Binaries  I.
Tidal circularization of Main-Sequence Stars–Features of Gaia DR3
Spectroscopic Binaries  I. Tidal circularization of Main-Sequence Stars

##  1 Introduction

The Gaia latest release of Non-Single Star catalogs (Gaia Collaboration et
al., 2022, hereafter NSS) includes the orbits of $181\,327$ single-lined
spectroscopic binaries (SB1), based on the radial velocities obtained by the
space-mission RVS spectrograph (Recio-Blanco et al., 2022; Blomme et al.,
2022; Katz et al., 2022). The NSS SB1 catalog is substantially larger than any
previously-known catalog, and therefore is a gold mine for investigating the
statistical features of short-period binaries (e.g., Duquennoy & Mayor, 1991;
Duchêne & Kraus, 2013; Torres et al., 2021), like the frequency of binaries as
a function of the primary mass (Raghavan et al., 2010; Troup et al., 2016; Moe
& Di Stefano, 2017) and mass-ratio distributions (e.g., Mazeh & Goldberg,
1992; Boffin, 2012, 2015; Shahaf et al., 2017).

This work utilizes the NSS catalog to follow the tidal circularization of
short-period binaries. Based on theoretical work, we expect short-period
binaries ($\lesssim 3$ – $10$ day) to be circularized (e.g., Kopal, 1956;
Mayor & Mermilliod, 1984; Giuricin et al., 1984; Zahn, 2008) by the tidal
interaction between the two components, whose strength is a strong function of
the binary separation and primary radius (e.g., Zahn, 1975, 1977, 1989). This
expectation caused Gaia Collaboration et al. (2022) to question the validity
of the very short-period binaries, with periods shorter than one day and large
eccentricities. We, therefore, rely on the work of Bashi et al. (2022), who
confronted the NSS SB1 orbits with the LAMOST and GALAH RV databases, and
constructed a clean sample of $91\,740$ Gaia SB1 orbits. This sample is the
basis of our period-eccentricity relation.

Observational evidence for tidal circularization, based on small samples of
spectroscopic binaries, was already pointed out by Mayor & Mermilliod (1984)
and Mathieu & Mazeh (1988). They considered samples of coeval spectroscopic
binaries and showed that each sample displays a circularization "cutoff
period", $P_{\rm cut}$, out to which most binaries have circular orbits, while
most binaries of longer periods have eccentric orbits (see also Mathieu et
al., 2004). The idea behind the cutoff period is the strong dependence of the
tidal interaction on the binary separation and therefore on the binary period.
Binaries with periods longer than $P_{\rm cut}$ were not circularized, keeping
their primordial eccentricity. In this model, coeval binary samples should
display a discontinuity jump of the eccentricity at $P_{\rm cut}$, which
should depend on the sample age — the older the sample the longer $P_{\rm
cut}$ (e.g., Mathieu & Mazeh, 1988; Mathieu et al., 2004; Geller et al., 2008,
2010; Geller & Mathieu, 2012; Nine et al., 2020; Geller et al., 2021; Zanazzi,
2022).

Another avenue to observationally study the tidal circularization was taken by
North & Zahn (2003) and Mazeh et al. (2006), and recently by the seminal works
of Van Eylen et al. (2016) and Justesen & Albrecht (2021); see also Zanazzi
(2022). They considered samples of eclipsing binaries (EB) — those of OGLE LMC
(Wyrzykowski et al., 2003), Kepler (Slawson et al., 2011) and TESS (Ricker et
al., 2015), obtaining the projected eccentricity of those binaries from the
timing of the secondary eclipse (Sterne, 1940). These studies used the fact
that a light curve of an EB allows for deriving the stellar radii relative to
the binary separation, and therefore the ratio between the sum of radii of the
two components and the binary separation, a ratio that directly determines the
tidal circularization effectiveness (North & Zahn, 2003). Thus, the analysis
of EB samples provides a cutoff ratio that divides circular from eccentric
orbits. Justesen & Albrecht (2021) derived the cutoff ratio for binaries with
different temperatures. Such a ratio is not available for spectroscopic
binaries, and therefore one can study only the cutoff period.

When studying spectroscopic samples, Meibom & Mathieu (2005) suggested a more
composite approach for determining the cutoff period of a binary sample. They
considered the eccentricity as a piece-wise function of the period, $P$, with
zero eccentricity for $P<P_{\rm cut}$ and a smooth rising asymptotic function
that goes up to eccentricity close to unity for $P>P_{\rm cut}$. The idea
behind the Meibom & Mathieu (2005) approach is that even binaries with
$P>P_{\rm cut}$ are subject to the circularization processes that decreased
the primordial eccentricity, but were not strong enough to make the orbit
circular. The reduced eccentricities of those binaries depend on the
primordial orbital period and eccentricity and the strength of the tidal
interaction. The asymptotic function reflects the present characteristics of
the whole binary sample and not only the very short-period circularized
binaries (see this approach applied, for example, by Nine et al. (2020) and
with a different version by Zanazzi (2022)).

Following Mazeh (2008) (see also, for example, Pourbaix et al., 2007) we
consider here a piece-wise asymptotic function that presents an upper envelope
for the eccentricity as a function of the period for a given sample. The
function marks the edge of the populated part of the period-eccentricity
plane. For a given period, there are almost no binaries with eccentricity
larger than the asymptotic-function value at this period, while there is a
high probability of finding binaries with eccentricities smaller than that
value. In short, we view the period-eccentricity diagram as divided into two
regimes, separated by an upper-envelope asymptotic function. We use Mazeh et
al. (2016) algorithm to find the best upper envelope by a maximum-likelihood
approach, given a sample of spectroscopic binaries. Similar to Meibom &
Mathieu (2005), we define the cutoff period, $P_{\rm cut}$, as the period at
which the upper envelope cuts the eccentricity axis, i.e., $e=0$. In this
approach, $P_{\rm cut}$ denotes both the border between the circular and
eccentric binaries and the location of the upper-envelope base.

The large cleaned Gaia sample of Bashi et al. (2022) allows us to apply our
technique to a few sub-samples of the Gaia SB1s. We concentrate on the
binaries with main-sequence (MS) A-, F- and G-primaries, identified by their
position on the Gaia Colour-Magnitude Diagram (CMD). Similar to the Justesen &
Albrecht (2021) analysis, we divided the sample into temperature bins, and
derived their corresponding $P_{\rm cut}$ independently, finding a clear
dependence on the stellar temperature of each bin.

Section 2 details the Gaia sample used, including removing binaries with
unconstrained eccentricities. Section 3 presents our algorithm to obtain the
best parameters of the upper envelope, given a binary sample, and Section 4
tests our algorithm for six simulated binary samples. Section 5 presents our
main result — the dependence of $P_{\rm cut}$ on the primary temperature of
each sub-sample. Section 6 shows that the results are inconsistent with the
theory which assumes circularization occurs during the stellar MS phase.
Section 7 summarizes our results and discusses their possible implications.

##  2 The sample

Figure 1: Colour-Magnitude Diagram (CMD) of the clean Gaia sample of Bashi et
al. (2022) binaries. The colour scale represents the number of binaries
falling into each hexagonal bin. The dashed line (see text) delineates our
border of the MS binaries.

We wish to study the eccentricity distribution of the Gaia binaries, as a
function of the stellar temperature in particular. We, therefore, consider a
uniform sample of binaries with unevolved primaries, of derived effective
temperature, and orbits with well-constrained eccentricity. Such a reduced
sample is composed in this section.

###  2.1 The main-sequence binaries

We first cross-matched the clean sample of Bashi et al. (2022) with the TESS
Input Catalog111https://tess.mit.edu/science/tess-input-catalogue/ (TIC;
Stassun et al., 2018; Paegert et al., 2021), yielding $89\,742$ SB1s. Fig. 1
shows these binaries on the Gaia Color-Magnitude Diagram (CMD), clearly
displaying the difference between the MS and the evolved stars. To identify
the MS binaries we used the limits

$\mathrm{MS}:\begin{cases}-5.5+10.5(G_{\mathrm{BP}}-G_{\mathrm{RP}})<G,&\mathrm{if~{}}G_{\mathrm{BP}}-G_{\mathrm{RP}}\leq
1\ ,\\\
3.1+2(G_{\mathrm{BP}}-G_{\mathrm{RP}})<G,&\mathrm{if~{}}G_{\mathrm{BP}}-G_{\mathrm{RP}}>1\
,\end{cases}$ (1)

which left us with $17\,474$ binaries.

###  2.2 Removing orbits with unconstrained eccentricity

Figure 2: Period-eccentricity diagram for the cleaned MS SB1 sample.

Fig. 2 shows the period-eccentricity ($P$-$e$) diagram for the MS-reduced
sample. Two features are clearly seen:

  * –

An upper envelope that rises from $e\sim 0.1$ at period of $P\sim 3$ day
towards $e\sim 0.8$ at period of $P\sim 100$ day.

  * –

A high concentration of binaries centered at $P\sim 5$ day and low
eccentricity of $\sim 0.03$.

To check which of these low eccentricities are real (see also a discussion by
Bashi et al., 2022), we used here the Lucy & Sweeney (1971) approach (see also
Hara et al., 2019, in the context of the eccentricity of planetary orbits),
who suggested that the actual eccentricity uncertainty is larger than the one
derived by the regular analysis. Thus, Lucy & Sweeney (1971) suggested that
many derived eccentricities of their time might be spurious.

Figure 3: Ratio between the derived eccentricities and their corresponding
uncertainties as a function of the orbital period. Red dashed horizontal line
marks the $e/\Delta e=2.45$ threshold of Lucy & Sweeney (1971).

We, therefore, plot in Fig. 3 the ratio between the derived Gaia
eccentricities and their corresponding uncertainties as a function of the
orbital periods. Two emerging clusters can be seen for binaries with orbital
periods shorter than $100$ days. Using K-means clustering (Pedregosa et al.,
2011)222https://scikit-
learn.org/stable/modules/generated/sklearn.cluster.KMeans.html, we found the
center of the bottom cluster at ($P,e/\Delta e$) = ($6.37$, $1.32$), while the
upper-cluster center is at ($19.38$, $10.15$). Using the Lucy & Sweeney (1971)
$5\%$ level of significance for testing the hypothesis that $e=0$, we found
most binaries in the lower cluster to be under the threshold of $e/\Delta
e=2.45$. To define a sample of binaries with well-constrained eccentricities,
we drew a line perpendicular to the line connecting the two cluster centres
through the middle of the gap between the two clusters. This has left us with
a sample of $10\,325$ binaries with at least $95\%$ confidence for non-
circular eccentricity, and $7\,149$ binaries with unconstrained
eccentricities.

Fig. 4 presents the two sub-samples. As can be clearly seen, the dense area of
near-circular binaries is populated by the unconstrained eccentricity sources
only.

Figure 4: Period-eccentricity diagram for the cleaned SB1 sample of MS
unconstrained eccentricities (left) and non-circular (right) binaries. Figure
5: Stellar Age-effective temperature scatter plot of the unconstrained
eccentricities sample (left) and non-circular (right) samples. A similar trend
between the two parameters is evident in both panels.

Next, we wish to test whether the clear distinction between the two sub-
samples of Fig. 4 is a result of an age difference, as one might argue that
the nearly circular binaries are old MS binaries that had already completed
their tidal circularization. We, therefore, plot in Fig. 5 the age-effective
temperature dependence of each of the two sub-sample. The left panel includes
a sample of $4\,530$ binaries with unconstrained eccentricities for which Gaia
FLAME stellar effective temperature and age (age_flame) are available.
Similarly, the right panel includes a sample of $6\,389$ sources with non-
circular solutions. Overall, a strong correlation between effective
temperature and age is evident in both panels, as expected for MS stars, with
no apparent age difference.

In what follows, we concentrate on analysing the sub-sample of $10\,325$
binaries with well-constrained eccentricities, deriving the dependence of
$P_{\rm cut}$ on the stellar temperature. We will also show that $P_{\rm cut}$
does not display a dependence on stellar age.

##  3 Fitting an Upper Envelope to the period-eccentricity diagram

The right panel of Fig. 4 clearly shows that the binary eccentricities do not
present a clear cutoff period that separates the circular and the eccentric
binaries, but, instead, show a gradual rise toward high values. Furthermore,
the eccentricities are not concentrated around an upper envelope but are
spread out between almost zero eccentricity and the upper bound. The figure
suggests that the base of the envelope can serve as a better definition for
the cutoff period. Any circularization analysis should try to account for this
feature.

To better characterize the $P$-$e$ diagram we define an upper envelope as

$f(P)=\begin{cases}1-\bigg{(}\frac{P_{\mathrm{cut}}}{P}\bigg{)}^{1/\tau}&\quad\quad\quad
P_{\mathrm{cut}}<P\ ,\\\ 0&\quad\quad\quad P_{\mathrm{cut}}\geq P\
,\end{cases}$ (2)

where $P_{\rm cut}$ is the cutoff period (see also Zanazzi, 2022) and $\tau$
is a dimensionless parameter that determines the slope of the envelope. The
function gets a value of zero at $P_{\rm cut}$ for any value of $\tau$, and
asymptotically rises to unity as $P$ gets substantially larger than $P_{\rm
cut}$. Note that the function is simple with only two free parameters —
$P_{\rm cut}$ and $\tau$. $P_{\rm cut}$ defines both the separation between
the circular and eccentric orbits and the location of the upper envelope.

To get a sense of the role of $\tau$, one can plot (see the bottom panel of
Fig. 6) the analysed sample in the ($\log P$, $\log(1-e)$) plane. In this
plane, the upper envelope turns into a straight-line lower envelope with a
slope of $-1/\tau$. Another way to follow the meaning of $\tau$ is to
consider, for example, $P_{0.8}(P_{\rm cut},\tau)$ — the period for which the
upper envelope gets the value of $e=0.8$. It is easy to show that
$P_{0.8}=5^{\tau}P_{\rm cut}.$ In other words, $\tau$ measures by how much we
have to increase $P$, relative to $P_{\rm cut}$, for the upper envelope to get
a value of $e=0.8$.

We then follow Mazeh et al. (2016) and use a modified Fermi function to
describe an assumed probability density function (PDF) of binaries above and
below the upper envelope in the ($\log P$, $e$) plane. The PDF converges to
zero above the envelope, and to a positive constant below it. The transition
region between the two parts is along the envelope, with a width characterized
by a parameter $\delta$. The probability density at a point $(\log P,e)$ is a
function of its distance $d(\log P,e;\,P_{\mathrm{cut}},\tau)$ to the envelope
curve along the $\log P$ axis and can be expressed as:

$d=\log P-\log
f^{-1}(e)=\log\Bigg{[}\frac{P}{P_{\mathrm{cut}}}(1+e)^{\tau}\Bigg{]}\ ,$ (3)

where $f^{-1}(e)$ is the inverse function of $f(P)$ defined in equation 2. The
distance $d$ is positive for points to the right and below the envelope and
negative on the other side. The PDF is

$\mathcal{F}_{\mathrm{PDF}}(P,e;\,{P_{\mathrm{cut}}},\tau,\delta)=\mathcal{A}\frac{1}{1+\mathrm{exp}(-d/\delta)}\
,$ (4)

where the constant $\mathcal{A}$, in units of $1/[\log(P/1\,{\rm day)}\,e]$,
is defined such that the 2D integral over the ($\log P$, $e$) plane, using the
three parameters — ${P}_{\mathrm{cut}},\tau,\delta$, equals unity. To get a
sense of this function, one can see that when $d=0$ the PDF value is simply
$\mathcal{A}/2$, while at $d=\pm\delta$ it changes between $0.27\mathcal{A}$
and $0.73\mathcal{A}$.

Obviously, the probability density function is of a statistical nature. One
can find binary systems above the upper envelope, either due to erroneous
measurements or invalid orbits or because of some special astrophysical
circumstances, like a high primordial eccentricity or a distant faint
companion that pumped eccentricity into the spectroscopic orbit (e.g., Mazeh &
Shaham, 1979; Mazeh, 1990; Jha et al., 2000). The advantage of our approach is
that for large enough samples the derivation of the envelope is only slightly
sensitive to these ’outlier’ binaries, as the fit is to the population as a
whole. On the other hand, the results are not biased by the number of binaries
in the sample.

For each set of parameters of the PDF — ${P}_{\rm cut}$, $\tau$ and $\delta$,
we derive the likelihood of a given sample of $N$ binaries, as

$\mathcal{L}=\prod_{i=1}^{N}\mathcal{F}_{\mathrm{PDF}}({P}_{i},e_{i};{P}_{\rm
cut},\tau,\delta)\ .$ (5)

where $e_{i}$ and $P_{i}$ are the eccentricity and period of the $i-$th
binary.

We applied our algorithm to the sample of the Gaia well-constrained
eccentricities discussed above, using only a reduced sub-sample of $4\,430$
binaries with orbital periods shorter than 30 days. This was done because the
density of the binaries with periods larger than 30 days is clearly not
constant, contrary to the assumption of our algorithm. To find the upper
envelope of the reduced MS sample of Fig. 4 we ran an MCMC routine with $50$
walkers and $10^{4}$ steps, with uninformative priors on the free parameters:
$\log P_{\mathrm{cut}}\sim\mathcal{U}(-1,1)$; $\tau\sim\mathcal{U}(0,4)$ and
$\delta\sim\mathcal{U}(0.01,0.3)$. We used the Python emcee package (Foreman-
Mackey et al., 2013) to find the parameter values (and their uncertainties)
that maximize the sample likelihood.

Fig. 6 shows our best-fit upper envelope with
$P_{\mathrm{cut}}=4.67~{}\mathrm{day}$, $\tau=1.70$ and $\delta=0.09$. The
bottom panel of the figure displays the analyzed sample in the ($\log P$,
$\log(1-e)$) plane. In this plane, the upper envelope turns into a straight-
line lower envelope with a slope of $-1/\tau$, showing how $\tau$ controls the
slope of the upper (lower) envelope.

Figure 6: Top: Log period – eccentricity diagram for the cleaned SB1 sample of
MS binaries, after removing orbits with unconstrained eccentricities (see
text). The red dashed line marks our upper envelope best-fit model, based on
binaries with an orbital period shorter than $30$ days, marked by a vertical
dotted line (see Section 3 for more details). Bottom: The same sample plotted
on the ($\log P$, $\log(1-e)$) plane. The same upper envelope is now a
straight-line lower envelope, with a slope of $\tau$.

##  4 Testing our algorithm on simulated binary samples

To test our upper-envelope algorithm, we performed a simplified simulation of
six binary samples with different circularization efficacy coefficients, all
starting with the same flat distribution of eccentricity and log-period.

The binaries were numerically evolved with Hut (1981) (see also Terquem &
Martin, 2021) equations (Zahn, 1975, 1977; Zahn & Bouchet, 1989; Zahn, 1989)
for stars with convective envelopes:

$\displaystyle\frac{de}{dt}=-\frac{18}{7}B\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}^{-16/3}\frac{e}{(1-e^{2})^{13/2}}\Bigg{[}1+\frac{15}{4}e^{2}+\frac{15}{8}e^{4}+\frac{5}{64}e^{6}$
(6) $\displaystyle\ \ \ \ \ \ \ \ \
-\frac{11}{18}(1-e^{2})^{3/2}\Big{(}1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}\Big{)}\Bigg{]},\
\ {\rm and}$
$\displaystyle\frac{d}{dt}\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}=-\frac{6}{7}B\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}^{-13/3}\frac{1}{(1-e^{2})^{15/2}}\Bigg{[}1+\frac{31}{2}e^{2}+\frac{255}{8}e^{4}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{185}{16}e^{6}+\frac{25}{64}e^{8}-(1-e^{2})^{3/2}\Big{(}1+\frac{15}{2}e^{2}+\frac{45}{8}e^{4}+\frac{5}{16}e^{6}\Big{)}\Bigg{]}\
.$

We chose the constant $B$ to control the circularization efficacy over the
binary lifetime, which we define as going from $t=0$ to $t=1$, where $t$ is
unitless.

Figure 7: Simulated binary samples of $500$ binaries each that were integrated
through the circularization process following equations 8. The red lines show
the upper envelopes, while the red area marks the transition region of
$\pm\delta$ width along the envelope.

Note that we ignore the changes in the stellar parameters, like radius and
structure, and the stellar rotation, which must account for the varying
angular momentum of the orbital motion. Furthermore, equations 6 are a
simplified version of the actual tidal interaction, neglecting the dependence
on the binary mass ratio, for example. Nevertheless, they are good enough for
our purpose here (see, for example, the discussion of Terquem, 2021).

We note that if we keep only the leading order of the eccentricity in
equations 6 (i.e., $e<<1$), we get:

$\displaystyle\frac{de}{dt}=-eB\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}^{-16/3},\
\ {\rm and}$ (7)
$\displaystyle\frac{d}{dt}\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}=-\frac{57}{7}e^{2}B\Big{(}\frac{P}{1~{}\rm{day}}\Big{)}^{-13/3}.$

The approximated equations allow one to derive the eccentricity as a function
of the period for a given binary

$\displaystyle\ln(P/P_{0})=\frac{57}{14}(e^{2}-e^{2}_{0})\ ,$ (8)

where $P_{0}$ and $e_{0}$ are the period and eccentricity at the starting
point of the circularization track. This equation does not predict the amount
of change the eccentricity and period will go through in a given timespan but
instead draws an evolutionary trajectory in the ($\log P$, $e$) plane.

Table 1:  Best-fitted values for $P_{\rm cut}$, $\tau$, $\delta$ of our simulated binary sample of $500$ binaries for six bins of circularization parameter $B$ B |  $P_{\rm cut}$ [day] | $\tau$ | $\delta$
---|---|---|---
$100$ | $1.42\pm 0.05$ | $1.61\pm 0.05$ | $0.020\pm 0.007$
$500$ | $2.04\pm 0.05$ | $1.62\pm 0.04$ | $0.020\pm 0.005$
$1500$ | $2.57\pm 0.07$ | $1.66\pm 0.04$ | $0.022\pm 0.005$
$6000$ | $3.24\pm 0.08$ | $1.70\pm 0.05$ | $0.018\pm 0.005$
$25000$ | $4.35\pm 0.08$ | $1.67\pm 0.04$ | $0.013\pm 0.003$
$100000$ | $5.75\pm 0.12$ | $1.72\pm 0.05$ | $0.015\pm 0.004$

Table 1, as well as the six panels of Fig. 7, present the results of the
integration through the "lifetime" of the samples, for six different values of
the circularization parameter, or alternatively for different lifetimes; these
values span a range of three orders of magnitudes. Each sample contains $500$
binaries of the same lifetime and circularization effectiveness $B$. Other
simulations, not shown in the figure, showed that the best-fit obtained
parameters do not depend on the number of binaries in the sample; only the
uncertainties of the parameters do.

The panels clearly show that the separation between the circular and eccentric
orbit is not a straight line but has a curved shape, supporting our approach
that uses an upper envelope to characterize the period-eccentricity relation.
Those envelopes are included in the figure, together with their corresponding
cutoff periods.

The relation between the cutoff period $P_{\rm cut}$ and the circularization
parameter $B$ is presented in Fig. 8 by a tight linear relation, $\log P_{\rm
cut}=(0.198\pm 0.011)\,\log(B)-(0.231\pm 0.042)$, implying that

$P_{\rm cut}\varpropto B^{3/16}\,.$ (9)

This can be traced back to equations 7 by noting that for small eccentricity
the logarithmic derivative of $e$ is approximated by
$\frac{1}{e}\frac{de}{dt}\propto BP^{-16/3}$. $P_{\rm cut}$ is the period for
which $\frac{\Delta e}{e}\sim 1$ during the "lifetime" of the binary, so that
$BP_{\rm cut}^{-16/3}\sim 1$.

Figure 8: $P_{\rm cut}$ as function tidal dissipation effectiveness $B$ for
six simulated samples. Red-dotted line, $\log P_{\rm cut}=(0.198\pm
0.011)\,\log(B)-(0.231\pm 0.042)$, marks the best-fit linear model.

##  5 Cutoff period vs. effective temperature in MS binaries

The large number of Gaia MS binaries at hand enables us to look for the
dependence of the cutoff period on the stellar temperature, which is readily
available for most binaries. Fig. 9 shows a histogram of stellar temperatures
of the $4\,376$ orbits we have at hand (see above). We proceeded with the
$3\,959$ binaries in the range $5300$ – $7700$ K.

Figure 9: Stellar effective temperature histograms of our non-circular MS,
$P<30$ day, binary sample. Inside the range marked by red vertical dotted
lines ($T_{\mathrm{eff}}=5300-7700$) there are $3\,959$ binaries.

We divided those binaries into $18$ temperature bins and fitted an upper
envelope to each sub-sample with the our algorithm. We show in Fig. 10 the
upper envelope best-fit model (red curve) and transition region of $\pm\delta$
width along the envelope (red area) of these binary samples. The results are
summarized in Table 2 and Fig. 11, where the points and error bars mark our
median and $16\%,~{}84\%$ percentiles.

Table 2:  Best-fitted values of $P_{\rm cut}$, $\tau$, $\delta$ for $18$ temperature bins of the Gaia MS spectroscopic binaries. $T_{\rm eff}$ [k] | N |  $P_{\rm cut}$ [day] | $\tau$ | $\delta$
---|---|---|---|---
$5300-5500$ | $148$ | $7.00\pm 0.77$ | $1.42\pm 0.16$ | $0.08\pm 0.01$
$5500-5600$ | $137$ | $7.19\pm 0.86$ | $1.37\pm 0.17$ | $0.10\pm 0.02$
$5600-5700$ | $198$ | $6.57\pm 0.68$ | $1.52\pm 0.14$ | $0.08\pm 0.01$
$5700-5800$ | $237$ | $6.16\pm 0.49$ | $1.40\pm 0.11$ | $0.07\pm 0.01$
$5800-5900$ | $280$ | $6.52\pm 0.63$ | $1.52\pm 0.12$ | $0.09\pm 0.01$
$5900-6000$ | $310$ | $5.90\pm 0.47$ | $1.50\pm 0.11$ | $0.07\pm 0.01$
$6000-6100$ | $307$ | $5.44\pm 0.41$ | $1.49\pm 0.11$ | $0.08\pm 0.01$
$6100-6200$ | $334$ | $4.58\pm 0.32$ | $1.76\pm 0.11$ | $0.07\pm 0.01$
$6200-6300$ | $348$ | $4.52\pm 0.34$ | $1.67\pm 0.11$ | $0.07\pm 0.01$
$6300-6400$ | $336$ | $3.95\pm 0.40$ | $1.83\pm 0.13$ | $0.09\pm 0.01$
$6400-6500$ | $261$ | $4.22\pm 0.43$ | $1.77\pm 0.14$ | $0.09\pm 0.01$
$6500-6600$ | $222$ | $3.39\pm 0.44$ | $1.91\pm 0.17$ | $0.10\pm 0.02$
$6600-6700$ | $165$ | $3.19\pm 0.46$ | $1.91\pm 0.22$ | $0.09\pm 0.02$
$6700-6800$ | $123$ | $2.55\pm 0.41$ | $2.26\pm 0.23$ | $0.08\pm 0.03$
$6800-6900$ | $104$ | $2.90\pm 0.63$ | $1.98\pm 0.24$ | $0.11\pm 0.03$
$6900-7100$ | $136$ | $2.94\pm 0.38$ | $1.95\pm 0.20$ | $0.08\pm 0.02$
$7100-7300$ | $126$ | $3.46\pm 0.43$ | $1.84\pm 0.20$ | $0.08\pm 0.02$
$7300-7700$ | $187$ | $3.28\pm 0.38$ | $1.87\pm 0.16$ | $0.09\pm 0.01$
Figure 10: Period-eccentricity diagram with fitted upper envelope (see Table 2
for details) in red dashed line for the $18$ sub-samples of the MS Gaia
binaries. The red area marks the transition region of $\pm\delta$ along the
envelope.  Figure 11:  $P_{\rm cut}$ of $18$ Gaia sub-samples as a function of
their corresponding temperature. Red-dotted line, $P_{\rm cut}=-3.69\,(\pm
0.39)\,({T_{\rm eff}/1000{\rm K}})+27.6\,(\pm 2.5){\rm\ day}$, marks the best-
fit linear model ignoring the rightmost and four leftmost bins.

The figure and the table offer a few features.

  * –

$P_{\rm cut}$ drops linearity from $\sim 6.5$ day to $\sim 2.5$ day when
moving from $T_{\rm eff}$ of $5500$ to $6800$.

  * –

The linear model for the $13$ bins is

$\frac{P_{\rm cut}}{\rm day}=A+B\left(\frac{T_{\rm eff}}{1000K}\right)\,,$
(10)

where $A=27.6\pm 2.5$ and $B=-3.69\pm 0.39$.

  * –

The linear fit is quite tight, suggesting that the uncertainties in $P_{\rm
cut}$ are overestimated by a factor of $\sim 1.7$. This could be expected, as
the typical uncertainty in $T_{\rm eff}$ is of the same order as the bin width
we used, inserting extra noise into the bins.

  * –

$P_{\rm cut}$ is flat at the $7700$ – $6800$ K range ($\sim 3$ day), and might
be also flat at $5600$ – $5300$ K.

  * –

A possible small jump of $P_{\rm cut}$ might be seen at the Kraft (1967) break
of $6100$ K.

  * –

There is a strong correlation between $P_{\rm cut}$ and $\tau$ of the $18$
bins — the shorter $P_{\rm cut}$ the larger $\tau$ is. This reflects the fact
that the shape of the different upper envelopes is such that all reach the
same eccentricity at a period of $\sim 30$ day. This can be explained by the
fact that binaries with such relatively long periods retain their primordial
eccentricities and therefore do not show any dependence on $T_{\rm eff}$.

###  5.1 Cutoff period dependence on stellar age?

As shown in Fig. 5, the MS Gaia binaries show a strong correlation between
stellar temperature and age. In this sub-section we examine the dependence of
$P_{\rm cut}$ on both stellar parameters and show that the dependence on age,
is weak if existing at all.

We divided the age-temperature plane into evenly spaced bins of $250$ K and
$2$ Gyr, and derived $P_{\rm cut}$ for the $14$ bins that contain more than
$50$ binaries. These $P_{\rm cut}$ values are colour coded in Fig. 12.

We fitted to the $14$ bins a two-dimensional function of

$\frac{P_{\rm cut}}{\rm day}=A_{\scriptscriptstyle\rm
2D}+B_{\scriptscriptstyle\rm 2D}\left(\frac{T_{\rm
eff}}{1000K}\right)+C_{\scriptscriptstyle\rm
2D}\left(\frac{\mathrm{Age}}{\mathrm{Gyr}}\right)\,,$ (11)

and ran an MCMC procedure to find the best values. We obtained
$A_{\scriptscriptstyle\rm 2D}=25.5\pm 6.9$, $B_{\scriptscriptstyle\rm
2D}=-3.46\pm 1.01$ and $C_{\scriptscriptstyle\rm 2D}=0.08\pm 0.11$.

This shows that $P_{\rm cut}$ strongly depends on stellar temperature and does
not depend on the stellar age.

Figure 12: 2D histogram in evenly spaced bins of effective temperature and
age. Colour represents $P_{\rm cut}$ of Gaia binaries in bins with more than
$50$ sources (otherwise white). Grey crosses mark median and $16\%,~{}84\%$
percentiles in each bin. The number of sources in each bin is displayed on the
bin’s top right side.

##  6 Approximated scaling law of circularization

In this section, we try to scale the tidal interaction with the stellar mass
and radius and the orbital period, in order to derive the theoretical
expectation for the dependence of $P_{\rm cut}$ on the stellar temperature and
compare it to our observational results.

According to Zahn (1975, 1977); Zahn & Bouchet (1989); Zahn (1989) and Zahn
(2008), the timescale of the circularization is determined by the turbulent
dissipation, in cool stars with convective envelopes, and radiative damping,
in hotter stars with radiative envelopes. We consider here the circularization
of stars with radiative envelopes only, as at least half of the Gaia binaries
analyzed here are of that type.

In a very simplified way, we consider only the eccentricity decay and ignore
the period change associated with it. This can be close to reality only when
the eccentricity is relatively small. Nevertheless, this approximation is
probably good enough for the purpose of this section. The approximation of the
eccentricity derivative, to first order in $e$, was already presented in
equation 7

$\displaystyle-\frac{1}{e}\frac{de}{dt}=B_{\rm rad}{P^{-7}},\ \ $ (7)

where we follow the convention of Section 4. As emphasized there, changing the
lifetime of a binary while keeping the circularization effectiveness is
equivalent in our formulation to varying instead the $B_{\rm rad}$ parameter
by the same factor.

Using Claret & Cunha (1997) equation 18 (see also Van Eylen et al., 2016,
equation 3), we write

$B_{\rm rad}\propto\frac{R_{*}^{9}}{M_{*}^{3}}\mathcal{F}(q)E_{2}\,T_{\rm
MS}\,,$ (12)

where $\mathcal{F}(q)=q/(1+q)^{5/3}$ is a function of the mass ratio, $E_{2}$
is a tidal constant evaluated numerically and $T_{\rm MS}$ is the MS lifetime.
Adopting Van Eylen et al. (2016) scaling of $R_{*}\propto M_{*}^{0.8}$ and
$T_{\rm MS}\propto M_{*}^{-2.9}$, assuming the observed-stars age is a
substantial fraction of their MS lifetime, we get

$B_{\rm rad}\propto{M_{*}^{1.3}}\mathcal{F}(q)E_{2}\,,$ (13)

Using equation 43 of Hurley et al. (2002), $E_{2}\propto{M_{*}^{2.8}}$ , we
finally get

$B_{\rm rad}\propto{M_{*}^{4.1}}\mathcal{F}(q)\,,$ (14)

indicating that $B_{\rm rad}$ is a monotonic increasing function of stellar
mass, and consequently of stellar temperature.

Following Fig. 8 and Equation 9, we conclude that for radiative-envelope
primaries $P_{\rm cut}\varpropto B_{\rm rad}^{1/7}$ and therefore

$P_{\rm cut}\varpropto M_{*}^{0.6}\ .$ (15)

This implies that $P_{\rm cut}$ should also increase with stellar temperature,
clearly in contrast with our results that show that $P_{\rm cut}$ is
decreasing with stellar temperature (or mass) in the range of $5500$ \- $6700$
K.

##  7 Discussion

We used the cleaned Gaia sample of spectroscopic binaries to study the
distribution of the orbital eccentricity as a function of the binary period.
The distribution is characterized by an upper envelope (Pourbaix et al., 2007;
Mazeh, 2008) that starts at $P_{\rm cut}$ and eccentricity zero and rises
monotonically toward high eccentricities for longer periods. The assumption is
that circularization processes induced by tidal interaction between the two
stars shaped the eccentricity distribution. The Gaia sample is a new
opportunity to confront the theory of circularization with fresh data.

We model the upper envelope by a simple function of two parameters, one of
which is $P_{\rm cut}$, while the other, $\tau$, determines the rise of the
envelope. We then follow the Mazeh et al. (2016) approach and use a modified
Fermi function to describe a probability density distribution of the binaries
above and below the upper envelope in the ($\log P$, $e$) plane. The
probability density function converges to zero above the envelope, and to a
positive constant below it, with a transition region of a derived width. We
use an MCMC routine to find the best parameters of the distribution, given a
sample of orbits.

The unprecedentedly large sample of Gaia orbits with MS A-, F- and G-type
primaries enables us to derive the dependence of the envelope, and $P_{\rm
cut}$ in particular, on the stellar temperature in the range of $5500$ –
$7500$ K. To do that we divide the binaries into sub-samples of different
temperatures, and fit the envelope of each sub-sample independently.

Our main finding is that $P_{\rm cut}$ presents a linearly decreasing function
of stellar temperature for the G and F stars — from $6.5$ day for $5700$ K to
$\sim 2.5$ day for $T_{\rm eff}\sim 6800$ K, at a rate of $-3.7$ days/1000 K.
The linear slope is at more than $9\sigma$ significance.

The uncertainties of the different $P_{\rm cut}$ values are larger than the
scatter around the linear fit. We suggest these are due to "imperfections" of
the samples, the result of different primordial eccentricity and period
distributions, binaries with different ages and mass ratios, and stellar
temperature spread, as the $T_{\rm eff}$ uncertainties are probably on the
order of 100 K. It is quite surprising that, nevertheless, the linear
dependence of $P_{\rm cut}$ is so pronounced and tight over a range of
temperatures that correspond to stellar envelopes of convective and radiative
nature alike.

In addition, $P_{\rm cut}$ is probably flat for $T_{\rm eff}$ at $5500$ –
$5700$ K, and $6800$ – $7500$ K. At the well-known stellar Kraft brake of
$\sim 6100$ K (see Kraft, 1967) we possibly see a small jump, yet this is
still barely significant.

We do not see any reason to believe our results are due to some Gaia
observational bias. For example, we could not find any dependence of the
quality score of Bashi et al. (2022), which estimates the degree of validity
of the orbits, on stellar temperatures or orbital periods and eccentricities.

As we have shown, the $P_{\rm cut}$ trend revealed for the G- and F-stars is
inconsistent with Zahn’s circularization theory, when we assume that
circularization took place during the stellar MS lifetime. This inconsistency
could indicate that

  * –

Zahn dynamical tide theory is not accurate enough to account for the
circularization processes. An assertion of studies suggested different
approaches (e.g., Alexander, 1973; Hut, 1981; Tassoul, 1987; Hut et al., 1992;
Dolginov & Smel’Chakova, 1992; Goldman & Mazeh, 1991, 1994; Goldman, 2008;
Witte & Savonije, 1999, 2001, 2002; Duguid et al., 2020; Zanazzi & Wu, 2021;
Terquem, 2021; Terquem & Martin, 2021; Koenigsberger et al., 2021; Zanazzi,
2022; Barker, 2022; Preece et al., 2022; Wei, 2022), in stars with radiative
envelopes in particular; see, for example, a heated discussion between Tassoul
& Tassoul (1997) and Rieutord (1992) and Rieutord & Zahn (1997), or

  * –

The eccentricity distribution of the binaries was determined during the pre-
main-sequence (PMS) phase of these stars, when the stars were much larger and
therefore the circularization processes much faster, as suggested by Mayor &
Mermilliod (1984), and worked out by Zahn & Bouchet (1989) and later by
Khaliullin & Khaliullina (2011, KK11), using updated PMS models; see also
Terquem & Martin (2021).

The latter conjecture is interesting, especially because it is coming from
Zahn himself. Zahn & Bouchet (1989) claimed that for masses of $0.5$ –
$1.25M_{\odot}$, the circularization took place during the PMS phase with
expected $P_{\rm cut}$ between $7.2$ and $8.5$ day (see some observational
evidence by Mathieu et al., 1992; Mathieu, 1994; Melo et al., 2001). In fact,
assuming the circularization occurred during the PMS phase, KK11 published a
series of expected $P_{\rm cut}$ for different stellar masses based on updated
PMS evolutionary tracks. Although the tracks are prone to a few uncertainties,
the table of KK11 displays a trend that might be similar to our results —
$P_{\rm cut}$ decreases as a function of the primary mass.

Understanding the theory of tidal circularization is crucial for understanding
the evolution of short-period binaries (e.g., Hurley et al., 2002; Fragos et
al., 2023). This has deep impact on how we think short-period binaries (e.g.,
Fabrycky & Tremaine, 2007), close triple systems (Mazeh & Shaham, 1979; Naoz,
2016; Toonen et al., 2022), cataclysmic binaries (e.g., Patterson, 1984) and
X-ray binaries (e.g., Bildsten & Cutler, 1992; Podsiadlowski et al., 2002)
have evolved. Even some of the black-hole merger models depend on tidal
interaction (e.g., Antonini et al., 2017; Belczynski et al., 2002).

Finally, tidal interaction could have a crucial role in the formation and
evolution of exoplanets (Gu et al., 2003; Ida & Lin, 2004; Ogilvie & Lin,
2004; Terquem & Papaloizou, 2007; Jackson et al., 2008; Rao et al., 2018), hot
Jupiters (e.g., Rasio et al., 1996; Wu, 2005; Ferraz-Mello et al., 2008;
Leconte et al., 2010; Dawson & Johnson, 2018) and their orbit alignment with
the stellar spin (Dobbs-Dixon et al., 2004; Winn & Holman, 2005; Fabrycky &
Winn, 2009; Scherrer et al., 2010; Lai, 2012; Albrecht et al., 2012; Dawson,
2014; Mazeh et al., 2015; Albrecht et al., 2021; Albrecht et al., 2022) in
particular.

In the future, five observational avenues might be of use in order to deepen
our understanding of the tidal interaction in binary systems. First, one can
use other samples of spectroscopic binaries, to confirm our results and to
find the dependence of $P_{\rm cut}$ on other parameters, like the mass ratio,
stellar age and metallicity. Such samples include, for example, the next
planned Gaia release,333https://www.cosmos.esa.int/web/gaia/release and
results of multi-object spectrographs, like RAVE (Matijevič et al., 2011),
LAMOST (Cui et al., 2012), APOGEE (Price-Whelan et al., 2020) and the near-
future 4MOST (de Jong et al., 2019).

The second avenue, analyzing samples of eclipsing binaries, was already taken
by North & Zahn (2003); Mazeh et al. (2006); Van Eylen et al. (2016) and
Justesen & Albrecht (2021), as detailed in the introduction. The advantage of
using EBs is the capability to accurately derive small eccentricities, which
is not possible for SB1s. It would be interesting to use additional samples,
like that of Gaia (Mowlavi et al., 2022) and derive $P_{\rm cut}$ as a
function of $T_{\rm eff}$. The large photometric surveys at work, like ZTF
(Chen et al., 2020) and ASAS-SN (Paczyński et al., 2006; Rowan et al., 2022)
can yield large samples of additional eclipsing binaries.

The third avenue has to do with the obvious realization that the long-term
tidal interaction is not limited to circularization but acts to synchronize
(see Khaliullin & Khaliullina, 2010) and align the binary with the stellar
rotation (e.g., Hurley et al., 2002; Naoz & Fabrycky, 2014). Therefore, a
sample of binaries should also display synchronization and alignment cutoff
periods. It would be interesting if we could compare those periods with the
circularization period and their dependence on stellar parameters.

To follow the stellar rotation one can use the available large set of
photometric light curves, like those of OGLE (e.g., Soszyński et al., 2016),
TESS (Huang et al., 2020) and Gaia (Eyer et al., 2022), which can reveal the
stellar rotation periods (McQuillan et al., 2013, 2014; Avallone et al.,
2022). Synchronization can be assumed, for example, if one detects ellipsoidal
variability, which is modulated with the binary period (Faigler & Mazeh, 2011;
Faigler et al., 2012; Green et al., 2022).

The fourth avenue has to do with observed eccentric pseudosynchronized
binaries (Hut, 1981), discovered in the Kepler lightcurves (e.g., Zimmerman et
al., 2017; Saio & Kurtz, 2022), which are going through strong tidal
interaction. It would be interesting to compare their stellar rotation periods
with the tidal theory expectation. In one of these systems, an orbital-period
decay driven by tidal interaction has been probably observed (Ou et al.,
2021).

Finally, recently some observational evidence has been presented for the
orbital decay of very close hot Jupiters, like WASP-12 (e.g., Yee et al.,
2020; Turner et al., 2021; Wong et al., 2022); see also Harre et al. (2023),
Yang & Wei (2022) and Rosário et al. (2022), based on the precise timings of
transits that span over more than a decade. Following the planetary period
decay in real time (Yang et al., 2022) has the exciting potential to further
constrain the theory of tidal interaction.

When additional information is available, the observational evidence could be
compared with a more realistic tidal model that might be combined with some
conjecture about the primordial period-eccentricity distribution. Such a model
may include other evolutionary mechanisms, like magnetic breaking (e.g.,
Mestel, 1968; Fleming et al., 2019) and interaction with accretion discs
and/or third companions (e.g., Fabrycky & Tremaine, 2007; Naoz, 2016; Toonen
et al., 2022), so it can account for the observed statistical features of the
population of short-period binaries.

## Acknowledgements

We are deeply indebted to the NSS group and all the Gaia team for producing a
vast high-quality catalogue that enabled us to follow the tidal
circularizationof short-period binaries. We are extremely grateful to Josh N.
Winn and Robert D. Mathieu for their illuminating comments and wise
suggestions that have significantly improved this manuscript. The referee
contributed very helpful comments on a previous version of the paper, helping
us presenting our results in a clearer way. This research was supported by
Grant No. 2016069 of the United States-Israel Binational Science Foundation
(BSF) to TM, and Grant No. I-1498-303.7/2019 of the German-Israeli Foundation
for Scientific Research and Development (GIF) to TM. This work has also made
use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/
dpac/consortium). Funding for DPAC has been provided by national institutions,
in particular the institutions participating in the Gaia Multilateral
Agreement.

## Data Availability

Data used in this study are available upon request from the corresponding
author.

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