# The merger of hard binaries in globular clusters as the primary channel for
the formation of second generation stars.

Valery Kravtsov1, Sami Dib2, Francisco A. Calderón3
1Sternberg Astronomical Institute, Lomonosov Moscow State University,
University Avenue 13, 119899 Moscow, Russia
2Max Planck Institute for Astronomy, Königstuhl 17, D-69117, Heidelberg,
Germany
3Departamento de Física, Universidad Católica del Norte, Av. Angamos 0610,
Antofagasta, Chile
E-mail, VK<EMAIL_ADDRESS>
###### Abstract

We have recently presented observational evidence which suggests that the
origin of the second generation (G2) stars in globular clusters (GCs) is due
to the binary-mediated collision of primordial (G1) low-mass main-sequence
(MS) stars. This mechanism avoids both the mass budget problem and the need of
external gas for dilution. Here, we report on another piece of evidence
supporting this scenario: (1) the fraction of MS binaries is proportional to
the fraction of G1 stars in GCs and, at the same time, (2) the smaller the
fraction of G1 stars is, the more deficient binaries of higher mass ratio
(q$>0.7$) are. They are, on average, harder than their smaller mass-ratio
counterparts due to higher binding energy at a given primary mass. Then (2)
implies that (1) is due to the merging/collisions of hard binaries rather than
to their disruption. These new results complemented by the present-day data on
binaries lead to the following conclusions: (i) the mass-ratio distribution of
binaries, particularly short-period ones, with low-mass primaries, $M_{\rm
P}<1.5$ M☉, is strongly peaked close to q$=1.0$, whereas (ii) dynamical
processes at high stellar density tend to destroy softer binaries and make
hard (nearly) twin binaries to become even harder and favor their mergers and
collisions. G2 stars formed this way gain mass that virtually doubles the
primary one, $2M_{\rm P}$, at which the number of G1 stars is $\sim 5$ times
smaller than at $M_{\rm P}$ according to the slope of a Milky Way-like IMF at
$M_{\rm MS}<1.0$ M☉.

###### keywords:

globular clusters: general - (stars:) binaries (including multiple): close

††pagerange: The merger of hard binaries in globular clusters as the primary
channel for the formation of second generation stars.–References††pubyear:
2016

Accepted XXX. Received XXX

##  1 INTRODUCTION

Globular clusters (GCs) with a present-day mass exceeding a threshold value of
about $M_{\rm GC}\sim 10^{4}$ M☉ (Carretta et al., 2010; Bragaglia et al.,
2017; Tang et al., 2021) are known to be typically composed of two sub-
populations or generations of stars, a primordial generation (G1) and a
secondary one (G2). Whereas the surface abundance patterns of G1 red giant
branch (RGB) stars have been spectroscopically established to be similar to
those of the field counterparts of the same iron abundance, G2 stars have been
found (e.g., Carretta et al., 2009, and references therein) to have distinct
abundances of light elements (C, N, O, Na, Mg, Al) and key (anti)correlations
between them. Namely, the abundances of N, Na, Al are increased, while those
of C, O, Mg are decreased. However, there are important nuances. Specifically,
Pancino et al. (2017) and Nataf et al. (2019) found that the Mg–Al anti-
correlation tend to disappear in less massive or most metal-rich GCs. Nataf et
al. (2019) argue that this suggests more than one chemical polluter
contributed to the chemical manifestations of the multiplicity in GCs. The
presence of sub-populations of G2 stars with such chemical characteristics
depending on cluster parameters is specific to GCs with no evident
counterparts in open star clusters (see, however, Pancino, 2018, for possible
exceptions) nor in the field. An exception is the Galactic bulge (Schiavon et
al., 2017) which is characterized by a high stellar density. The presence of
G2 stars in the Galactic halo is not excluded (Martell & Grebel, 2010), in
particular, due to their escape from GCs.

Outside of the Milky Way, the manifestations typical of multiple stellar
populations in Galactic GCs were revealed in massive star clusters older than
2 Gyr in the Large and Small Magellanic Clouds (e.g., Hollyhead et al., 2017;
Niederhofer et al., 2017; Hollyhead et al., 2018; Milone et al., 2020; Li et
al., 2021; Martocchia et al., 2021). More details on this particular subject
and on the obtained results can be found in Salgado et al. (2022). Moreover,
young and intermediate-age massive star clusters in the the Magellanic Clouds
exhibit a complexity of their stellar populations in the form of an extended
MS turnoff (Mackey & Broby Nielsen, 2007; Mackey et al., 2008; Milone et al.,
2009), which is believed to be another manifestation of the multiplicity at
the respective age.

Abundant knowledge about multiple stellar populations in GCs have been
obtained thanks to observational and theoretical investigations carried out to
date. The key facts and various aspects concerning multiple populations in
GCs, including massive star clusters in the Magellanic Clouds, have been
reviewed in dedicated papers, for instance, by Bastian & Lardo (2018), Gratton
et al. (2019), and Milone & Marino (2022). In spite of gaining a deeper
insight, the origin of G2 stars paradoxically remains a controversial issue.
New findings make the problem even more complex and confusing. Very special
assumptions are sometimes required to explain some observations. This could be
attributed to the shared idea between virtually all of the proposed scenarios
about the nature of the majority of G2 stars, namely the canonical view that
G2 stars are formed from gas lost by intermediate- and high-mass G1 stars.

We suggest that the collision/merging of G1 stars might be a plausible
alternative for explaining the origin of G2 stars, by relying on observational
evidence. Kravtsov et al. (2022) have recently examined available data on both
the slope of the present-day mass function ($\alpha_{\rm pd}$, MF) and the
fraction of G1 RGB stars ($N_{\rm G1}/N_{\rm tot}$) for a sample of 35
Galactic GCs. They found that an anti-correlation between the two quantities
is statistically significant for GCs of higher mass ($M_{\rm GC}>10^{5.3}$
M☉). Also, a very significant anti-correlation has been found between ($N_{\rm
G1}/N_{\rm tot}$) and the stellar encounter rate for a sample of 54 GCs. Taken
together, these relationships imply that the origin of G2 stars presently
observed in GCs [i.e., with their (progenitor) mass $M_{\rm MS}<0.9$ M☉] could
be due to binary-mediated collision/merging of MS G1 stars originally
belonging to the low-mass end ($[0.1-0.5]$ M☉) of the initial mass function
(IMF). We also considered a generic model (Kravtsov et al., 2022; Dib et al.,
2022) that showed the feasibility of such a mechanism to produce (1) a
sufficient fraction of G2 stars and (2) the general trend of the dependence of
($N_{\rm G1}/N_{\rm tot}$) on $\alpha_{\rm pd}$. An increased dispersion of
points in the latter dependence can likely be caused by variations in the IMF
(Dib et al., 2022) and a higher mass loss from lower-mass GCs.

Our approach highlights the potentially important or even decisive
contribution to the formation of G2 stars via this collisional mode of star
(trans)formation. While none or negligible (trans)formation of stars from the
coalescence of other stars occurs in low to intermediate density environments,
this mode can play a much more prominent role at high stellar densities such
as those that are thought to be prevailing in GCs at their infancy (Dib,
2023). Moreover, it has been argued since long ago (e.g., Meylan & Heggie,
1997, and references therein) that binary mediated stellar encounters are more
probable among other possible channels of stellar collision. More recently, by
modeling stellar encounters relevant to the conditions in the Galactic nuclear
star cluster, Mastrobuono-Battisti et al. (2021) found that collisions between
pairs of MS stars are the most common stellar encounters leading to mergers.

In the context of the nature of multiple stellar populations, a particular
attention is given to the established difference of the occurrence of G2 stars
in the three above-mentioned types of environments, i.e. GCs, open star
clusters, and Galactic field. Interestingly, this is not the only
dissimilarity between stellar populations in GCs, on the one hand, and the
field and open clusters, on the other. The fraction of the presently observed
binaries in GCs is notably smaller than in both the field and open star
clusters of similar metallicity (e.g., Bica & Bonatto, 2005; Sollima et al.,
2007; Milone et al., 2012; Raghavan et al., 2010; Tokovinin, 2014b; Albrow &
Ulusele, 2022; Cordoni et al., 2023, among others). Oddly enough, this
important fact is not normally associated with the origin of G2 stars. We
believe that the reason for this is twofold. First, it is naturally believed
that the (much) smaller fractions of binary stars in GCs is fully due to their
destruction in the high-density environment. Second, the scenarios of the
formation of G2 stars from G1 stars ejecta do not make any particular room for
the issue of binarity. On the contrary, in the framework of the
collision/merger-based approach, the smaller fraction of MS binaries in GCs is
very relevant, particularly by comparing some details of the present-day data,
both theoretical and observational, on binary stars in different environment
with the results obtained here. Note, however, that the issue of binaries and
their occurrence in different environment is really complicated.

The present paper is organized in the following way. In the next section
(Section 2), we describe the data used for our analysis. In Section 3, we show
the existence of a tight correlation between the fraction of binary stars and
the fraction of G1 RGB stars in GCs, ($N_{\rm G1}/N_{\rm tot}$). A supporting
evidence that this correlation is not due to the disruption of binaries but
rather to their merger is also presented. It is twofold: (i) a trend shown
between the same parameter ($N_{\rm G1}/N_{\rm tot}$) and the relative
fraction of higher mass-ratio binaries, and (ii) the present-day data
characterizing them as high-mass ratio hard binaries with fate as mergers. We
discuss our findings and their implications in Section 4, along with our
conclusions.

##  2 DATA

In the present study, we rely on publicly available sets of data on Galactic
GCs and their stellar populations (SPs). One of these is the atlas composed by
Milone et al. (2017) on multiple SPs in 57 GCs with previously obtained HST
photometry in different passbands (Sarajedini et al., 2007; Piotto et al.,
2015) in the central parts of the GCs of this sample. The authors isolated RGB
stars, separated them into subpopulations belonging to different generations,
and deduced a set of parameters characterizing each generation. Specifically,
Milone et al. (2017) quantified the relative number of G1 red giants ($N_{\rm
G1}/N_{\rm tot}$) as the ratio between the number of G1 ($N_{\rm G1}$) and the
total number of red giants in the observed area of each GC. Milone et al.
(2017) have been able to estimate the ($N_{\rm G1}/N_{\rm tot}$) parameter in
54 of the 57 GCs.

Another set of data is on the fraction of binaries derived by Milone et al.
(2012) in 59 Galactic GCs. The list of these GCs essentially overlaps with the
one for which the parameter ($N_{\rm G1}/N_{\rm tot}$) is measured. However,
the final list of GCs in common was defined by the following details
concerning the data on binaries. The fraction of binaries were estimated and
listed for three projected GC areas confined between radial distances from the
cluster centers corresponding to the GC structural parameters: within the core
radius ($R_{\rm C}$), between the core and the half-mass radius ($R_{\rm
C-HM}$), and outside the latter ($R_{\rm oHM}$). Moreover, the binary
fraction, $f$, measured in any cluster region was separately estimated in
three ranges of the binary mass ratio: q$>0.5$ [f(q$>0.5$)], q$>0.6$
[f(q$>0.6$)], and q$>0.7$ [f(q$>0.7$)]. Formally, the total binary fraction
f(tot), which takes the expected (not measured) fraction of binaries with
q$\leq 0.5$, is given, too. However, this parameter was calculated by the
authors from f(q$>0.5$) by accepting one of the existing versions of the mass-
ratio distribution of binary stars. So, the former is simply equal to the
latter multiplied by a numerical coefficient. Therefore, f(q$>0.5$) may be
used as f(tot) whenever the absolute value of f(tot) is unimportant.
Unfortunately, there is a lack of binary measurements within the $R_{\rm C}$
region in a notable number of the sample GCs, namely in 16 clusters being
primarily more massive and concentrated globulars, such as NGC 104 (47 Tuc),
NGC 1851, NGC 5904 (M 5), NGC 6093 (M 80) NGC 6388, NGC 7078 (M 15) among
others. Fortunately, 11 of these GCs have measurements in the $R_{\rm C-HM}$
region where the populations of binaries still remains large. Milone et al.
(2012) find that the fraction of binaries, on average, decreases by a factor
of $\sim 2$ from the center to about two core radii. For this reason, in order
to obtain both a uniform and large sample size as possible, we used a larger
set of GCs where binary star fractions were measured just in the $R_{\rm
C-HM}$ region. Overall, 45 GCs forming this data set turned out to be in
common between the lists of Milone et al. (2017) and Milone et al. (2012). The
GCs of this sample fall in a large range of the key GC parameters, such as
metallicity, mass, and stellar density.

##  3 RESULTS

Due to high stellar densities, the central regions of GCs are characterized by
unique conditions, which favor frequent close encounters between stars and
important effects caused by such encounters. In particular, in relation with
the subject of the present paper, these events dynamically affect hard and
soft binaries in such a way that may favor them to finally merge/collide or
disrupt, respectively (see more detail in next Subsection 3.2 and Section 4).
We analyze the above-described data on GCs and their multiple stellar
populations to further examine one of the key aspects of the suggested
collisional/merging formation of G2 stars presently observed in GCs, i.e. with
mass $M_{MS}<0.9$ M☉, of primordial low-mass MS (G1) stars. Namely, the
present paper looks into the role of (initial) binaries and respective
implications.

###  3.1 The fraction of G1 stars as a function of binary star fraction

The crucial role of binary-mediated collision/merging of G1 stars in the
formation of (the bulk of) G2 stars in GCs implies that it is a binary
consuming process. Hence, one may expect a relationship between the observed
binary fraction and the population of G1 stars. Figure 1 displays the
dependence of the parameter ($N_{\rm G1}/N_{\rm tot}$), the fraction of G1
stars, on f(q$>0.7$), the fraction of binaries with the mass ratio q$>0.7$ in
GCs of the sample. In the middle panel of Figure 1, the parameter ($N_{\rm
G1}/N_{\rm tot}$) is plotted as a function of the logarithm of f(q$>0.7$). The
continuous line in the middle panel is a linear fit to the data. The plot in
the right panel of Figure 1 shows the parameter ($N_{\rm G1}/N_{\rm tot}$) as
a function of the ratio between the fractions of binaries with different mass
ratios, f(q$>0.7$) and f(q$>0.5$). This dependence is discussed in detail in
Subsection 3.2 along with a summary of closely related relevant results
obtained and accumulated to date on binary stars.

We estimated the level of statistical significance of the dependencies shown
in Figure 1. The results of the Spearman’s rank correlation test done to
assess the correlation between the data sets plotted in Figure 1 are listed in
Table 1. As can be seen, the correlation is statistically significant at very
high confidence level for the same dependence shown in two forms (left and
middle panels). We made separate estimates for both forms of the dependence in
order to take care of a formality. The significance of the trend in the right
panel was estimated for two its versions, the total and reduced. The reduced
version is coded by magenta color in Figure 1. It is obtained by reducing from
the total set of data (45 units) 10 GCs (22% of the sample) with the largest
errors of the ratio f(q$>0.7$)$/$f(q$>0.5$). These points make the trend more
diffuse and somewhat less significant according to the test. The reduced
version of the trend is surely significant at a high confidence level of
99.91% as compared to somewhat lower confidence level of 97.07% for the total
data set.

These dependencies provide another piece of evidence supporting our scenario.
First, the fraction of MS binaries is proportional to the fraction of G1 stars
in GCs or, in other words, this means that the smaller fraction of binary
stars in GCs, the larger, on average, the fraction of G2 stars. A
systematically decreasing fraction of binaries in GCs with systematically
increasing fraction of G2 stars may be explained as due to the formation of G2
stars from binaries. However, it is known (Milone et al., 2012) that GCs with
larger fraction of binaries are also systematically more massive. So, the
discussed relation may alternatively be interpreted as being due to more
efficient disruption of binaries in more massive GCs. The trend presented in
the right panel of Figure 1 does not support the latter interpretation. The
trend shows that the smaller the fraction of G1 stars is, the more deficient
binaries of higher mass ratio (q$>0.7$) are. These binaries of higher mass
ratio are, on average, harder than their smaller mass-ratio counterparts due
to higher binding energy at a given primary mass. In the next Subsection, we
justify this in detail.

Figure 1: For a sample of 45 Galactic GCs: The dependencies of the fraction of
G1 stars ($N_{\rm G1}/N_{\rm tot}$) on (a, left panel) the fraction of binary
stars with the mass ratio q$>0.7$, f(q$>0.7$), isolated and quantified by
Milone et al. (2012) in the cluster area between the core and the half-mass
radius ($R_{\rm C}<R<R_{\rm HM}$); (b, middle panel) the logarithm of
f(q$>0.7$); (c, right panel) the ratio between the fractions of binary stars
with the mass ratio q$>0.7$ and q$>0.5$, f(q$>0.7$)$/$f(q$>0.5$). The
continuous line in middle panel is a linear fit to the plot. Black points in
the right panel are 10 GCs (22% of the sample) with the biggest errors of the
ratio f(q$>0.7$)$/$f(q$>0.5$) (see more details in the text). Table 1:
Spearman’s rank correlation test.

Dependence | $N_{\rm GC}$ | $\rho^{a}$ | P(%)b
---|---|---|---
($N_{\rm G1}/N_{\rm tot}$) vs f(q$>0.7$) | 45 | 0.674 | $>99.9999$
($N_{\rm G1}/N_{\rm tot}$) vs log f(q$>0.7$) | 45 | 0.674 | $>99.9999$
($N_{\rm G1}/N_{\rm tot}$) vs f(q$>0.7$)$/$f(q$>0.5$) | 45 | 0.325 | 97.07
($N_{\rm G1}/N_{\rm tot}$) vs f(q$>0.7$)$/$f(q$>0.5$) | 35 | 0.537 | 99.91

00footnotetext: a

a Spearman’s $\rho$ coefficient of correlation between ($N_{\rm G1}/N_{\rm
tot}$) and other quantities characterizing binary stars in GCs.
00footnotetext: bb The probability that two data sets are associated.

###  3.2 The fraction of G1 stars as a function of the relative fraction of
higher mass-ratio binaries

This subsection deals with the subject of the most suitable and probable kind
of binaries that could primarily be involved in the (trans)formation of (G1
into) G2 stars. The dependence of ($N_{\rm G1}/N_{\rm tot}$) on the binary
fraction ratio (right panel, Figure 1) implies that it is their fraction with
q$>0.7$ of the mass-ratio distribution. Indeed, this dependence unambiguously
shows that the relative fraction of the higher mass-ratio binaries decreases
with decreasing fraction of G1 RGB stars in GCs. Virtually the same binaries
are expected to be the hardest in GCs at a given primarily mass. Strictly
speaking, the data on the fractions of MS binary stars of different mass ratio
in GCs used in our analysis do not allow one to distinguish between soft and
hard binaries. So, one cannot directly refer to the kind of these binaries, as
soft or hard. Then, do we have any ground to relate the obtained dependencies
with our approach? In other words, may we interpret the formally decreasing
amount of binaries with decreasing fraction of G1 stars in GCs as probably due
to the transformation of the binaries into G2 stars rather than to the
disruption of a larger number of binaries in GCs with smaller ($N_{\rm
G1}/N_{\rm tot}$)? Of course, the issue is really complex. Nevertheless the
trend in the right panel of Figure 1 supports such an interpretation, given a
body of both observational and theoretical present-day data about binaries and
multiple systems. In order to explain this trend, we briefly summarize key
actual data on binary stars and their characteristics appropriate to the case,
particularly on hard (close) binaries, their probable mass-ratio distribution
and their fate in GCs.

According to the present-day understanding of the issue, the dynamical
evolution of binaries (triples) in GCs develops in two opposite ways, namely:
dynamically hard and soft binaries become even harder and softer, respectively
(e.g., Spera et al., 2022; Rozner & Perets, 2022, and references therein).
Thus, dynamically soft binaries can hardly survive for a long time in the
high-density environment of GCs. In particular, both tidal disruption by the
gravitational potential of GCs and dynamical evaporation tend to soften and
finally disrupt these binaries (Rozner & Perets, 2022). Therefore, a fraction
of the initially formed binaries, primarily soft ones, are expected to be
essentially disrupted, while on the other hand, hard binaries increase their
binding energy. There is a well known complexity of the effects on hard
binaries in the condition of high-density environment due to their
interactions with other GC constituents, such as single stars, other binary
stars, and black holes. A part of their increasing binding energy is lost to
kinetic energy of escaping single stars. Also, one cannot exclude that hard
binaries achieving a threshold binding energy can eject themselves before
merging. Nevertheless, despite these effects, the most probable development
and the fate of the dynamical evolution of hard binaries in GCs is expected to
be merger or collision (Goodman & Hut, 1993). Supporting evidence of a similar
effect with a lower efficiency is observed among field stars. Tokovinin et al.
(2006) established that close spectroscopic binaries with and without tertiary
companion have significantly different period distributions whereas their
mass-ratio distributions are similar. So, the dynamical effect of such
tertiary companions leads to shorter periods of respective binaries. It is now
evident that virtually all short-period binary stars in the field have a
tertiary companion. In the more general context of the observed relation
between close binaries and solar-type hierarchies, Tokovinin (2023) finds that
the inner subsystems in these hierarchies show strong excess of periods
shorter than 30 days. Interestingly enough, the eccentricities of the same
subsystems tend to be zero, and a larger fraction of twins belongs to these
short-period binaries, too.

In addition to the dynamical effects making hard binaries become harder in the
dense environment of GCs, with their probable fate being a merger, there may
be other potential effects favoring the merging of binaries. For instance,
Rozner & Perets (2022) argue that the properties of binaries in the early GCs
could be affected by the gas produced by the evolved G1 stars. The presence of
gas can lead to an efficient merger of binaries, on the one hand, and to the
formation of additional population of binaries, on the other.

It is now widely accepted that the mass-ratio distributions of MS binaries in
both the field and clusters are in obvious disagreement with random pairing of
stars taking their proportions from the IMF into account. Moreover, a body of
observational evidence converge towards a particular feature in the mass-ratio
distribution of solar-type binaries in the form of a prominent peak (excess)
of twin binaries, i.e. with mass-ratio close to unity. We primarily refer just
to low-mass, so-called solar-type stars, with the primary mass $0.5-1.5$ M☉,
which are relevant to this study. Fisher et al. (2005) studied a sample of
field spectroscopic MS binaries with the primary mass greater than $1.1$ M☉
and found an evident peak in the mass-ratio distribution near q=1. Using a
sample of 454 solar-like (F6–K3) stars from the Solar vicinity, Raghavan et
al. (2010) revealed that the mass-ratio distribution shows ”a preference for
like-mass pairs”, which are more frequent in relatively close pairs. Tokovinin
& Moe (2020) compared their model with observational data on binary stars,
namely with a large sample of solar-type binaries obtained by Tokovinin
(2014a) within 67 pc from the Sun. They noted that the model reproduces well
the large excess of twin binaries (with q$>0.95$), the fraction of which is
larger among close binaries (Tokovinin, 2014b). In the old open cluster M67,
Albrow & Ulusele (2022) estimated the fraction of photometric binaries with
q$>0.8$ to be $\sim$14% (and $\sim$26% with q$>0.5$). From their overview of a
variety of observational studies on binary stars including low-mass ones,
Pinsonneault & Stanek (2006) argued that there is clear evidence of a
typically large fraction of twin binaries (or at least with q$>0.8$) revealed
in these works (up to 35% of such binaries).

By proceeding from the observational evidence that close binaries typically
have mass ratios close to unity, the so-called twin binaries, Adams et al.
(2020) theoretically explained this as being due to the fact that this
corresponds to the lowest energy state of such binary systems.

##  4 DISCUSSION AND CONCLUSION

The initial star formation and the dynamical processes acting in the dense
stellar environment of GCs objectively define both the fraction and mass-ratio
distribution of the hard binaries which are suggested to be the direct
progenitors of G2 stars. It cannot be excluded that the mass-ratio
distribution of binaries in this particular environment of GCs can deviate
from its version in the Galactic field in the sense that the former
distribution is likely more weighted toward higher mass ratio so that the
majority of hard binaries fall between $0.8<$ q $<1.0$.

###  4.1 The formation of G2 stars and quantitative estimate of the fraction
of G1 stars at a given mass

We qualitatively formulate our approach in a more specific form defined by the
transformation of high mass-ratio G1 binaries into G2 stars of approximately
two times higher mass than the primary one. This means that lower-mass G1
stars forming such twin or almost twin binaries are transformed into a more
massive G2 stars, which replace less numerous (in absolute value, but at the
same or comparable fraction) G1 binaries at this higher mass, since these
latter stars are in turn transformed into even more massive G2 stars. This
way, a successive shifting of more numerous (in absolute value) lower-mass
stars, transformed into G2 stars, towards two times more massive (and
therefore many times less numerous at that mass) G1 stars is a mechanism that
can naturally and uniformly operate and lead to similar outcome in different
GCs.

We now attempt to obtain more quantitative estimates of how the binary star
fraction can relate to the fraction of G1 stars. We use the following
arguments. Suppose that the fraction of hard binaries of G1 stars, $f_{\rm
B,M}$, corresponding to the number $N_{\rm B,M}$, with both the primarily mass
around $M$ and high mass ratio (virtually twins), are transformed into the
same number of G2 stars with mass around 2$M$. Similarly, due to the same
process, $b$ times larger number of hard binaries with the primary mass around
$0.5M$ are transformed into single G2 stars, $N_{\rm S,M}({\rm G2})$, with
mass around $M$, while the number of G1 stars left around the same mass $M$ is
$N_{\rm S,M}({\rm G1})$. In other words, the transformed $N_{\rm B,M}$ G1
binaries were substituted with $b$ times larger number of G2 stars, $N_{\rm
S,M}({\rm G2})$. Then, $N_{\rm S,M}({\rm G2})=bN_{\rm B,M}$. Note that the
fraction of hard binaries $f_{\rm B,0.5M}$ is not necessary equal to $f_{\rm
B,M}$ in general case, since the fraction of hard MS binaries in a GC may vary
along the MS. One can easily qualitatively express the binary fraction $f_{\rm
B,M}$ the following way:

$f_{\rm B,M}=\frac{N_{\rm B,M}}{N_{\rm S,M}({\rm G1})+N_{\rm B,M}},$ (1)

Our goal is to approximately estimate a typical realistic value of the
parameter $g_{\rm 1}$, the fraction of G1 MS stars around mass $M$, which was
defined in Kravtsov et al. (2022) as the analog of the fraction of G1 RGB
stars [i.e., $N_{\rm 1}({\rm G1})/N_{\rm tot}=N_{\rm 1}({\rm G1})/(N_{\rm
1}({\rm G1})+N_{\rm 2}({\rm G2}))$]. Taking the above-mentioned expressions
into account, $g_{\rm 1}$ is expressed as:

$g_{\rm 1}=\frac{N_{\rm S,M}({\rm G1})}{N_{\rm S,M}({\rm G1})+N_{\rm S,M}({\rm
G2})}=\frac{N_{\rm S,M}({\rm G1})}{N_{\rm S,M}({\rm G1})+bN_{\rm B,M}},$ (2)

From here and using Equation 1 we easily come to the following expression:

$g_{\rm 1}=\frac{1-f_{\rm B,M}}{1+f_{\rm B,M}(b-1)}$ (3)

For estimating typical value of $g_{1}$, the value of $b$ may be around
$b\approx 5$, i.e. equal to the ratio of the number of single stars, with the
given ratio of their mass, defined by the canonical Kroupa IMF (Kroupa, 2001,
2002)111This value of $b$ can be different if the shape of the IMF is
calibrated according to the IMF slope versus stellar surface density of the
cluster found by Dib (2023).. Namely, we accept the ratio of their mass is
equal to $\approx 2$ and the slope of the IMF is $\alpha=2.3$ in the range of
low-mass stars, at $M_{\rm MS}<1.0$ M☉. Since the fraction of hard, high-mass
ratio (virtually twin) binaries that have finally merged in GCs is an obvious
variable, we consider two distinct values for it, namely $f_{\rm B,M}=0.10$
and $f_{\rm B,M}=0.30$. They presumably bracket the plausible range for this
variable in GCs. Each of the two values is accepted to be constant in the
considered stellar mass range, although, in the general case, the fraction of
hard (close) binaries may vary with stellar mass as we noted earlier. Then, by
applying Equation 3, we obtain the values of $g_{\rm 1}$ equal to 0.64 and
0.32, respectively. This is a reasonable range in comparison with the
fractions of the G1 RGB stars, ($N_{\rm G1}/N_{\rm tot}$) that are estimated
for most GCs in the Milky Way. However, massive GCs have lower values of
($N_{\rm G1}/N_{\rm tot}$), typically between 0.20 and 0.25. Such values of
$g_{\rm 1}$ can be obtained for some useful and interesting cases. In
particular, it is worth noting the aforementioned case of $f_{\rm B,M}$
varying along MS, for example, as an increasing function of decreasing stellar
mass. This tendency seems to be real at least for the field stars at solar
metallicity. For instance, Tokovinin & Moe (2020) note an interesting detail,
namely that ”the excess twin fraction is substantially reduced for close
early-B binaries and quickly diminishes with increasing separation” in
contrast with the same characteristic estimated by the same authors for solar-
type stars. In such a case, suppose that the value of $b$ increases, say, up
to $b=7$ between the same masses, $0.5M$ and $M$, within $M_{\rm MS}<1.0$ M☉.
Then the value of $g_{\rm 1}$ decreases up to $g_{\rm 1}=0.25$ for $f_{\rm
B,M}=0.30$. Note, however, that a detailed analysis of this issue is outside
the scope of the present paper.

###  4.2 The compatibility of a merger-induced formation of G2 stars with
their surface abundance of light elements

The mechanism described above for the formation of G2 stars directly suggests
that the abundance of light elements (primarily CNO-Na) distinguishing G2
stars from their G1 counterparts, were synthesized in the stellar interiors
and transported to the stellar surface. However, this point is very disputable
and it suffers from high grade of uncertainty. The formation of a merger-
induced G2 MS star as a result of the merger of two G1 MS stars has yet to be
investigated in detail. Its particularly challenging step is the formation of
one whole hydrogen-burning nucleus from the two merging ones. The merging of
G1 stars is accompanied with the transformation of the system angular momentum
and with an initially strongly perturbed non-equilibrium state of the merger
product. The coalescence of the hydrogen-burning nuclei of G1 stars into one
whole nucleus should temporally boost the central temperature and pressure in
the forming nucleus as compared to the same parameters in the nucleus of a G1
star of the same mass. As of now, there is unfortunately significant
uncertainty about this process and a lack of reliable knowledge about the
temporal variations of the central temperature, its maximum possible value
achievable at the initial stage of the formation of the merger product. This
would have a strong impact on the yields of relevant nuclear reactions.
Whether the difference of the CNO-Na surface abundances between G1 and G2
stars can be explained in the framework of our scenario due to the processes
generated by low-mass mergers in the interiors of their products is a key
question that should be clarified by dedicated numerical simulations. The
results obtained so far on the modeling and simulation of the processes
related to the mergers of low-mass stars do not exclude or even partially
support the ant-correlated variation of the surface abundances of at least
some elements. Such a variation of N and C may take place, increasing with
time, but emerging at best $\sim 2$ Gyr after the event occurred, depending on
the model (Sills et al., 2005).

According to the models of Prantzos et al. (2007), the core H-burning
temperature in stars increases towards the end of the MS evolution and it may
reach up to $\sim 27\times 10^{6}$K even in low-mass stars with $M_{\rm
MS}=0.9$ M☉ (at [Fe/H]=-1.6). The authors note that this temperature is yet
somewhat lower than what is typically required for the NeNa-chain to operate
(it becomes sufficient at the RGB, particularly near its tip, where the
temperature can exceed $\sim 60\times 10^{6}$K). In their Figure 4 (left
panel), Prantzos et al. (2017) indicate the temperature range as a function of
the produced He amount ($\Delta Y$, which is used as a proxy for time) where
each of the relevant chemical elements is either overproduced (including Na)
or depleted with respect to its initial value by the factors corresponding to
the extreme abundances observed in the GC NGC 2808 where [Na/Na0]$>0.7$. The
low temperature limit necessary for synthesizing the required amount of Na
exactly for this particular case is around $40\times 10^{6}$K at $\Delta
Y<0.02$. The lower the value of [Na/Na0] is, the lower is the temperature
limit.

We only briefly note that in the framework of the merger/collision formation
of G2 stars, a qualitative interpretation of the disappearing Mg-Al anti-
correlation in less massive or most metal-rich GCs can be considered in terms
of the expected impact of decreasing velocity dispersion and increasing
viscosity of the stellar matter, respectively. Indeed, the kinetic energy of
colliding/merging H-burning nuclei apparently depends on the two factors at a
given stellar mass.

In this context, it is important to point out to another interesting advantage
of the merger-induced scenario for the formation of G2 stars as compared with
their canonical formation from H-processed material ejected by polluters (and
diluted with pristine gas with composition of G1 stars). The former mechanism
requires an order of magnitude less amount of the enriching material to form
the same amount (i.e., the same total mass) of G2 stars with a given chemical
composition. The main reason is that this difference is defined by the much
lower efficiency of star formation of the canonical mechanism. It can hardly
be more than 20% as compared with the almost 100% efficiency ($\sim$90%) of
the merger-induced mechanism. In other words, this mechanism requires to
synthesize, say, at least 5 times less amount of the required chemical
elements. This directly implies an apparently lower required rate of the
involved nuclear reactions corresponding to a lower temperature limit. Whether
the central temperature could temporally achieve or even exceed the minimum
required level in a forming low-mass merger/collision products of $M_{\rm
MS}=0.8-0.9$ M☉ (the bulk of the available spectroscopic data on the O, Na,
Mg, Al abundances in GCs were obtained in GCs for RGB stars) is one of the key
question that have yet to be answered.

In contrast to the high degree of uncertainty of the possibility for the
necessary amount of material to be processed at least through the CNO cycle
and the NeNa chain in low-mass MS collision products, the availability of a
mechanism capable of transporting such a material to the external parts of
these products (MS G2 stars) looks very likely. Indeed, the effects of
rotation in the interiors of collision products are theoretically known more
reliably, with lower degree of uncertainty. Collision products are expected to
be fast rotators at least in the initial period after their formation.
Lombardi et al. (1996) used smoothed particle hydrodynamics simulations of
stellar collisions between MS stars in GCs and noted that any off-axis
collisions result in rapidly, differentially rotating merger remnants. Later,
Sills et al. (2005) confirmed that these off-axis collision products have
substantial angular momentum from their initial configuration (even assuming
initially non-rotating stars). Moreover, Sills (2015) noted that very high
initial rotation rates of the collision products must be reduced by an unknown
process to allow these products to collapse to the MS. So, rotational mixing
is the very likely mechanism that would transport the yields of the assumed
CNO cycle to the stellar surface. This process and its impact on the variation
of the surface elemental abundance of collision products is considered, in
particular, in the models of low-mass collision products by Sills et al.
(2005). Their models show, for example, essential variation of the surface
abundance of N (in excess) and C (in depletion). Add also that one of the main
effects of tides in close binaries is that they tend to spin-up the companions
and this tidal spin-up establishes rotational mixing in the stellar interiors
of the companions before their merging (Spera et al., 2022).

###  4.3 A possible effect of merging on the surface Li abundance

The proposed mechanism of the formation of G2 stars not only avoids the
problematic mass budget problem common to other alternatives, but also it
apparently does not need an external primordial gas for dilution. We have to
note, however, that according to Prantzos et al. (2007), in the case of the
dilution of Li-free gas processed through the CNO cycle at high temperature in
the polluters (which are massive or intermediate-mass stars in the scenarios
relying on the canonical formation of G2 stars, though), the situation is
different, i.e. ”the unprocessed material has to be of pristine nature (i.e.
pure ISM) rather than from the stellar envelope of the polluter itself”.

On the other hand, rotational mixing, as the main mechanism responsible for
the variation of the surface elemental abundances of merger products, should
not lead to an essential depletion in the surface lithium abundance of the
merger products (G2 stars). Strassmeier et al. (2012) find that binaries
exhibit, on average, approximately 0.25 dex less surface Li than single stars
of comparable temperature, as expected if the depletion is due to the
rotation. In turn, Aguilera-Gómez et al. (2023) find no evident impact of
binarity on Li abundance among field RGB stars of their sample. Even more so
Sayeed et al. (2023) find a larger fractions of fast rotators among Li-rich
stars. The issue of Li abundance in low-mass stars, enriched by numerous
observational results for the last years, is contradictory and even
surprising. Recent studies of low mass-stars sometimes show paradoxical
results on Li abundance. Sayeed et al. (2023) find that Li-rich stars at the
base of the RGB have somewhat higher mean s-process abundances and confirm a
prevalence of Li-rich stars on the red clump, i.e. among core helium-burning
red giants. Surprising enough that this enrichment with Li, not predicted by a
standard evolution theory, is somehow related with the helium flash, since it
is observed among stars with mass $M<2$ M☉ (Kumar et al., 2020). According to
Li et al. (2018), Li-rich stars are found on the MS. They argue that this
indicates the presence of an unknown efficient process that affects the
surface Li abundances before the RGB stage. In addition, Li abundance is a
function of various parameters. For instance, Martos et al. (2023) reveal
robust anti-correlations between Li abundance and both metallicity and age
among solar twins.

Perhaps the most relevant result to mergers between non-compact stars has been
recently obtained by Kamiński et al. (2023) on the presence of lithium in
three Galactic red nova remnants, which are probably produced by mergers of MS
stars. Super-solar lithium abundances they find in these objects ”may suggest
that at least some merger products activate mixing mechanisms capable of
producing lithium”. V1309 Sco is among these red novae. Its outburst was
detected and studied photometrically. It has been established to be caused by
the merged stars of a contact binary with the mass of the main companion $\sim
1$ M☉ (Tylenda et al., 2011) or perhaps somewhat higher, $\sim 1.5$ M☉. This
is now the least massive binary system known to give rise to a red novae
(Howitt et al., 2020). This object and studies on its nature were discussed in
more detail in Kravtsov et al. (2022).

###  4.4 Concluding remarks

We obtained new results and complemented them with both a body of
observational evidence and theoretical results on binary stars in different
environments including GCs. This allowed us to tentatively identify the
probable progenitors of G2 stars in the framework of our approach that
suggests the binary-mediated merging/collision formation of G2 stars in GCs.
These progenitors are high mass-ratio (virtually twin) hard binaries in GCs.
They form in any environment, but only in the condition of the high density
environment in GCs, and possibly in some other objects, such as ultra-compact
dwarf galaxies, the bulge of the Galaxy, etc., these ”seeds” are converted
into G2 stars.

The consumption of G1 binaries for building G2 stars means that G1 and G2 sub-
populations should be deficient and virtually free of primordial binaries,
respectively. This is in agreement with observations that show that the
fractions of binaries among G1 sub-populations in GCs are normally an order of
magnitude higher than among G2 ones (see Gratton et al., 2019, and references
therein).

Primordial (G1) MS stars composing low-mass binaries also undergo an essential
transformation due to mass transfer between the components, particularly the
accretor. The transferred gas not only increases the accretor’s mass but it
can also modify the surface elemental abundances under certain conditions. For
example, by using the MESA code, Wei et al. (2020) carried out model studies
of surface chemical anomalies of low-mass MS stars due to mass transfer. Their
results show that in some low-mass binaries, the accretors can exhibit
elemental abundances similar to those of G2 stars, i.e. C and O are in
depletion whereas Na and N are in excess (at constant Mg and Al, however),
while still remaining on the MS. Note, however, that the donor is considered
to be at least as massive as $1$ M☉ and the final result depends on a few
other key parameters including the mass ratio that is typically adopted to be
q$=0.5$. When the accretor evolves and is on the RGB, these abundance
anomalies can become weaker or disappear entirely. Therefore, given these
model results, one can note two useful points. First, the mass-transfer
binaries that recently were at the stage of blue stragglers and now are (their
accretors) at the RGB, are expected to originally be of systematically lower
mass ratio as compared to their counterparts, hard binaries with q$>0.8$ that
merged a long ago (G2 stars) and actually are also on the RGB. Second, how
many of the actually observed RGB stars in GCs are the evolved blue
stragglers? Wei et al. (2021) estimated that their fraction is about 10%. Our
estimate (Kravtsov & Calderón, 2021) made for the GC NGC 3201 taking the
observed number of blue stragglers in this GC cluster into account is of the
same order of magnitude, possibly slightly smaller. However, strictly
speaking, this fraction depends on the specific frequency of blue stragglers
recently ($\sim 1$ Gyr ago) populated GCs. Normally, the specific frequency of
the presently observed blue stragglers is higher in GCs with higher fractions
of binary stars (see Milone et al., 2012; Chatterjee et al., 2013, and
references therein).

By considering the particular example of NGC 3201, Kravtsov & Calderón (2021)
paid attention that the present-day RGB stars in GCs are not uniform in terms
of their mass, since not all of them are the progenies of the primordial MS
stars. One (minor) fraction are the relatively recently evolved blue
stragglers of any nature (with mass typically $M_{\rm BS}>1.1$ M☉) and another
fraction of RGB stars are expected to be the progenies of merger/collision
products of the primordial MS stars (latter, in Kravtsov et al. (2022) and in
the present paper we argued that the bulk of G2 stars can be such products).
By relying on the models of Sills et al. (2009), Kravtsov & Calderón (2021)
argued also that such products should achieve the RGB and evolve on it
together with the progenies of the primordial stars of (typically) lower mass.
In this context, it is worth noting recent results on estimating the masses of
stars belonging to the advanced stages of evolution, mainly the RGB, by using
asteroseismic observations of the GCs M4 (Howell et al., 2022) and M80 (Howell
et al., 2023). Several outliers with masses exceeding $M_{\rm RGB}>1$ M☉ were
detected as candidates evolved blue stragglers. Moreover, an increased
dispersion of the derived masses of RGB stars, excluding the outliers, at and
below the RGB bump allows to assume a real variation of mass of the order of
$\sim 0.05$ M☉.

The present paper aims at developing further our recently proposed approach
(Kravtsov et al., 2022) on the merger/collision nature of G2222In the present
paper, we use the terms multiplicity, first and second generations (G1 and G2)
to follow the definitions adopted in the widely accepted paradigm of multiple
stellar populations in GCs, which unambiguously implies the canonical mode of
the formation of G2 stars from the gas produced by G1, (very) massive and
intermediate-mass stars. Perhaps, G1 and G1tr (transformed) instead of G1 and
G2 (or P1 and P2), respectively, would be more appropriate for our conceptual
framework. stars in GCs. The new results reported in the present paper are
added with a variety of important both observational evidence and theoretical
results obtained so far in various works on this and closely related subjects.
Thereby, we further argue a new way of understanding the origin of G2 stars.

##  5 Acknowledgments

The authors thank the anonymous referee for both the expressed interest to the
results obtained in the present paper and for useful comments that improved
its final version.

## Data Availability

The data underlying this article are available in tables from the papers
referred to here, at https://doi.org/10.1051/0004-6361/201016384 and
https://doi.org/10.1093/mnras/stw2531.

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# Toward Astrometric Constraints on a Supermassive Black Hole Binary
in the Early-Type Galaxy NGC 4472

J. M. Wrobel National Radio Astronomy Observatory, P.O. Box O, Socorro, NM
87801, USA  T. J. W. Lazio  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA J. M. Wrobel
<EMAIL_ADDRESS>
(Received 2022 February 8; Accepted 2022 April 17)

###### Abstract

The merger of two galaxies, each hosting a supermassive black hole (SMBH) of
mass $10^{6}$ M⊙ or more, could yield a bound SMBH binary. For the early-type
galaxy NGC 4472, we study how astrometry with a next-generation Very Large
Array (ngVLA) could be used to monitor the reflex motion of the primary SMBH
of mass $M_{\rm pri}$, as it is tugged on by the secondary SMBH of mass
$M_{\rm sec}$. Casting the orbit of the putative SMBH binary in terms of its
period $P$, semimajor axis $a_{\rm bin}$, and mass ratio $q=M_{\rm sec}/M_{\rm
pri}\leq 1$, we find the following: (1) Orbits with fiducial periods of $P=4$
yr and 40 yr could be spatially resolved and monitored. (2) For a 95% accuracy
of $2\,\mu$as per monitoring epoch, sub-parsec values of $a_{\rm bin}$ could
be accessed over a range of mass ratios notionally encompassing major
($q>\frac{1}{4}$) and minor ($q<\frac{1}{4}$) galaxy mergers. (3) If no reflex
motion is detected for $M_{\rm pri}$ after 1(10) yr of monitoring, a SMBH
binary with period $P=4(40)$ yr and mass ratio $q>0.01(0.003)$ could be
excluded. This would suggest no present-day evidence for a past major merger
like that recently simulated, where scouring by a $q\sim 1$ SMBH binary formed
a stellar core with kinematic traits like those of NGC 4472. (4) Astrometric
monitoring could independently check the upper limits on $q$ from searches for
continuous gravitational waves from NGC 4472.

Active galactic nuclei (16); Early-type galaxies (429); Supermassive black
holes (1663); Interferometry (808)

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2018), CARTA
(Wang et al., 2020)

##  1 Motivation

The merger of two galaxies, each hosting a supermassive black hole (SMBH) of
mass $10^{6}$ M⊙ or more, is expected to yield an SMBH binary. Eventually the
orbit of the SMBH binary will shrink due to gravitational wave (GW) emission
(Begelman et al., 1980). Such GW signatures are currently being sought with
the Parkes Pulsar Timing Array (PPTA; Manchester et al., 2013; Shannon et al.,
2015; Lasky et al., 2016), the European Pulsar Timing Array (Lentati et al.,
2015; Babak et al., 2016; Desvignes et al., 2016), and the North American
Nanohertz Observatory for Gravitational Waves (NANOGrav; Arzoumanian et al.,
2018). It is also anticipated that such GW signatures will be detected with
the future Laser Interferometer Space Array (Amaro-Seoane et al., 2017).

In addition, searches are underway for the electromagnetic (EM) signatures of
SMBH binaries. Decades of data seeking indirect EM signatures, such as
periodicities in the optical emission-line velocities or photometric
variability of active galactic nuclei (AGNs), have been used to identify
candidate SMBH binaries. See Burke-Spolaor et al. (2019) and De Rosa et al.
(2019) for recent reviews of all of the topics mentioned above.

One direct EM signature would be to spatially resolve and monitor the orbit of
one or both members of an SMBH binary. Bansal et al. (2017) report possible
evidence for orbital motion in the 7.3-pc binary in the radio galaxy 0402+379.
But the estimated period spans millennia, making it difficult to acquire
sufficient observables to solve for an orbit. EM strategies for studying
tighter and more easily measured orbits are only now being devised (D’Orazio &
Loeb, 2018; Safarzadeh et al., 2019; Dexter et al., 2020). These strategies
rely upon recent or projected advances in interferometric techniques at
millimeter (mm) or near-infrared wavelengths.

Here, we consider a strategy proposed by Safarzadeh et al. (2019) for
continuum targets at Jansky (Jy) levels observable at 230 GHz (1.3 mm) with
the Event Horizon Telescope (EHT; EHT Collaboration et al., 2019), and adapt
it for the fainter, more abundant targets at milliJansky (mJy) levels that
could be observed at 80 GHz (3.7 mm) with a next-generation Very Large
Array111https://ngvla.nrao.edu (ngVLA; Murphy et al., 2018).

Section 2 describes the selection of an example target, while Section 3
describes an observing strategy that capitalizes on a powerful, designed-in
capability of the ngVLA at its highest resolutions, namely paired antenna
calibration (Carilli & Holdaway, 1999; Carilli et al., 2021). Section 4
explores the implications of high-resolution ngVLA observations for
astrometric monitoring and for related tie-ins to galaxy evolution and to
multimessenger astronomy in the context of GW observations. We close in
Section 5 with a summary and conclusions. A preliminary version of this work
appeared in Wrobel & Lazio (2021).

##  2 Target Selection

We focus on the early-type galaxy NGC 4472, the dominant galaxy in Virgo
Subcluster B (e.g., Janowiecki et al., 2010; Arrigoni Battaia et al., 2012).
Throughout we assume a distance of 16.7 Mpc, where 81.0 pc subtends 1″
(Blakeslee et al., 2009).

NGC 4472 may have experienced one or more galaxy mergers, as suggested from
its overall slow rotation and its stellar core, which exhibits a substantial
mass deficit, counter rotation and tangentially-biased orbits (e.g., Bettoni
et al., 2001; Krajnovic et al., 2011; Emsellem et al., 2011; Thomas et al.,
2014). Such core traits are linked to stellar scouring by an SMBH binary, a
potential byproduct of a galaxy merger (e.g., Milosavljevic et al., 2002;
Gualandris & Merritt, 2008; Kormendy & Bender, 2009). Finally, though somewhat
fortuitous, Mingarelli et al. (2017) conduct a series of simulations to
construct realizations of the local GW landscape. They find that NGC 4472
plausibly could host a SMBH binary and illustrate some of their results by
considering it specifically.

High-resolution imaging of the low-luminosity AGN in NGC 4472 using the
technique of very long baseline interferometry (VLBI) is available only at 5
GHz (6.0 cm) and 8.4 GHz (3.6 cm) (Anderson & Ulvestad, 2005; Nagar et al.,
2005). The most constraining size information comes from the Very Long
Baseline Array image at 8.4 GHz showing a 4-mJy source (Anderson & Ulvestad,
2005). The diameter of the cm source has a full width at half maximum (FWHM)
of less than 0.73 mas (0.059 pc). Attempts to probe smaller size scales, via
time variability at 8.4 GHz and 15 GHz (2.0 cm) using the Very Large Array,
were inconclusive (Anderson & Ulvestad, 2005; Nagar et al., 2005).

Hydrodynamical simulations including radiative processes and SMBH feedback
suggest that NGC 4472’s low-luminosity AGN can be understood as arising from
jet-like outflows traced via synchroton emission and launched from a
radiatively-inefficient accretion inflow (e.g., Inayoshi et al., 2020).

Figure 1: ALMA archival image of the emission at 98 GHz (3.1 mm) from the low-
luminosity AGN in NGC 4472. The scale is 1″ = 81.0 pc (Blakeslee et al.,
2009). The ellipse in the southeast corner shows the synthesized beam
dimensions at FWHM of 0$\farcs$76 (62 pc) $\times$ 0$\farcs$59 (48 pc) with an
elongation position angle of 46°. The cross marks the location of the emission
peak, which has a value of 2.92 mJy beam-1. The white box in the northwest
corner highlights the label $I$ for the Stokes parameter.

##  3 ngVLA Observations

We adopt the ngVLA’s Long Baseline Array (LBA) component, which features ten
sites and three 18 m antennas per site (Carilli et al., 2021). We assume an
observing frequency of 80 GHz to match that assigned to notional continuum
studies in the highest frequency band (Wrobel et al., 2020). The point spread
function after tapering the natural weights is
modelled222https://gitlab.nrao.edu/vrosero/ngvla-sensitivity-calculator to
have a FWHM of $PSF=0.1$ mas (0.0081 pc).

To accrue a reasonable time on the target, one antenna per LBA site would
observe the phase calibrator while the other two continuously observed NGC
4472 (Carilli & Holdaway, 1999; Carilli et al., 2021). This capability is
generally not available with current VLBI arrays, but has been designed into
the ngVLA’s LBA component. After 4 hours on target with dual-polarization
receivers and a bandwidth of 20 GHz per polarization, the thermal noise is
modelled2 to have a root-mean-square value of about 3 $\mu$Jy beam-1.

The overall astrometric accuracy of the ngVLA observation will have
contributions from terms due to (i) the signal-to-noise ratio $S/N$ on the
target, $\sigma_{\rm s/n}$; and (ii) the relative accuracy achieved via phase
referencing, $\sigma_{\rm pr}$. Regarding term (i), Figure 1 shows an image of
NGC 4472 at 98 GHz (3.1 mm), obtained from the archives of the Atacama Large
Millimeter/submillimeter Array (ALMA) using the Cube Analysis and Rendering
Tool for Astronomy (CARTA).333https://doi.org/10.5281/zenodo.4905459 If the
peak signal in Figure 1 is available to the ngVLA at 80 GHz, it would have a
$S/N\sim 970$ and be localized with an associated accuracy of $\sigma_{\rm
s/n}=PSF/(1.665\times S/N)\sim 0.1\,\mu$as. This expression stems from
equation (25) in Condon et al. (1998) and applies to Gaussian-based fitting.

Figure 2: Parameter space for $a_{\rm bin}$ and $q$ for a putative SMBH binary
in the early-type galaxy NGC 4472. $M_{\rm bin}$ is from Rusli et al. (2013).
The region to the right of the navy diagonal line is accessible with ngVLA
astrometric monitoring at 80 GHz with the labelled accuracy. The associated
$PSF$ of the ngVLA Long Baseline Array (LBA) is marked for reference. The GW
constraints from the PPTA are tabulated in Schutz & Ma (2016), while that from
NANOGrav is derived from Arzoumanian et al. (2021). All quantities invoke the
Blakeslee et al. (2009) distance of 16.7 Mpc.

Regarding term (ii), the phase referencing will be imperfect due to the
different atmospheric conditions associated with two separations, that between
the sky locations of the target and the phase calibrator, and that between the
ground locations of the target antenna and the calibrator antenna (equation
13.128 of Thompson et al., 2017). To reach levels of $\sigma_{\rm pr}\sim
1\,\mu$as at mm wavelengths, it is desirable to employ multiple phase
calibrators whose sky locations are separated from the target by less than
1-2° (Broderick et al., 2011; Reid & Honma, 2014; Rioja & Dodson, 2020).
Several such calibrator candidates are already known for NGC 4472, one with a
sub-degree separation.444http://www.vlba.nrao.edu/astro/calib/ Some of the
calibrator candidates appear to be adequately strong according to the ALMA
Calibrator Source Catalog.555https://almascience.nrao.edu/sc/ Others could be
briefly observed to assess their suitability. It is also desirable to closely
pack the target and calibrator antennas per LBA site. Below, we assume that
the overall astrometric accuracy of the ngVLA observation will be dominated by
term (ii), with $\sigma_{\rm pr}\sim 1\,\mu$as.

##  4 Implications

###  4.1 Astrometric Monitoring

Following Safarzadeh et al. (2019), we examine how astrometric monitoring of
NGC 4472 could constrain the reflex motion of the low-luminosity AGN’s primary
SMBH of mass $M_{\rm pri}$, as it is tugged on by a putative secondary SMBH of
mass $M_{\rm sec}$. These masses define a binary mass $M_{\rm bin}=M_{\rm
pri}+M_{\rm sec}$ and mass ratio $q=M_{\rm sec}/M_{\rm pri}\leq 1$. We allow
$q$ to vary. We adopt $M_{\rm bin}=2.4^{+0.3}_{-0.1}\times 10^{9}$ M⊙ from
Schutz & Ma (2016), which is the dynamically derived mass estimate from Rusli
et al. (2013) but linearly scaled to an assumed distance of 16.7 Mpc
(Blakeslee et al., 2009). Because early-type galaxies with larger SMBH masses
are more likely to exhibit nuclear radio emission (e.g., Nyland et al., 2016),
we assume that the mm emission from NGC 4472 is associated with its primary
SMBH.

We assume a circular orbit for the SMBH binary and use Kepler’s third law to
link the binary’s semimajor axis $a_{\rm bin}$ to its mass $M_{\rm bin}$ and
orbital period $P$ (equation (1) of Dexter et al., 2020). The reflex motion of
$M_{\rm pri}$ as it orbits, with semimajor axis $a_{\rm pri}$, about the
binary’s center of mass is $a_{\rm pri}=a_{\rm bin}\times q/(1+q)$ (equation
(1) of Safarzadeh et al., 2019). We recast this as

$a_{\rm bin}=a_{\rm pri}\times(1+q)/q.$ (1)

Suppose that ngVLA astrometry of the low-luminosty AGN could achieve a 95%
accuracy of $2\,\mu$as for each epoch in a monitoring sequence. Then the
reflex constraint per epoch would become $a_{\rm pri}=2\,\mu{\rm as}\times
81.0\,{\rm pc}/10^{6}\,\mu{\rm as}=$ 0.00016 pc. Inserting this value into
equation (1) then defines how $a_{\rm bin}$ can be related to $q$. This
behavior is shown as the navy diagonal line in Figure 2. The parameter space
to the right of this line is accessible via ngVLA monitoring of NGC 4472 with
the adopted astrometric accuracy. The parameter space to the left of this line
is inaccessible.

A possible complication regarding the ngVLA monitoring deserves mention: Su et
al. (2019) suggest that NGC 4472 is moving northward, relative to its
surrounding X-ray-emitting gas, with a velocity $v_{\rm X-ray}\sim
450^{+192}_{-143}$ km s-1. A putative SMBH binary would share this velocity.
Projection effects are not known but if this velocity is purely in the plane
of the sky, it would have a secular proper motion $\mu_{\rm X-ray}\sim
5.7^{+2.4}_{-1.8}\,\mu$as yr-1. ngVLA monitoring may reveal such a secular
motion, upon which any reflex motion of $M_{\rm pri}$ would be superposed.
Such a complication can be handled as for Sgr A⋆, where proper motion studies
exclude any oscillatory reflex signals (Reid & Brunthaler, 2004).

In contrast to numerous prior studies that aim to detect the cm emission from
both SMBHs in the remnant of a galaxy merger (Burke-Spolaor et al., 2019; De
Rosa et al., 2019, and references therein), our strategy needs detectable mm
emission from only one SMBH. Also unlike those studies, our strategy can
leverage the frequency coverage, angular resolution, and sensitivity of the
ngVLA to conduct searches well into the regime in which a SMBH binary emits
GWs, a topic further developed in Section 4.3.

###  4.2 Tie-Ins to Galaxy Evolution

Figure 2 shows the values of $a_{\rm bin}$ associated with fiducial SMBH
binary periods of $P=4$ yr and $P=40$ yr. Astrometric monitoring through a
quarter of a period would be sufficient to constrain the range of mass ratios
$q$ allowed for the period. As is evident from Figure 2, orbits with these
periods could be spatially resolved by the ngVLA.

If no reflex motion is detected for $M_{\rm pri}$ after 1 yr of ngVLA
monitoring of NGC 4472, a SMBH binary with period $P=4$ yr and mass ratio
$q>0.01$ could be excluded. The darker blue shading in Figure 2 indicates
where SMBH binaries with shorter periods and higher mass ratios could also be
excluded. This exclusion is based on ngVLA astrometry and applies even for
orbits that are not spatially resolved with the ngVLA.

If reflex motion remains undetected after a decade of ngVLA monitoring of NGC
4472, a SMBH binary with $P=40$ yr and $q>0.003$ could be excluded. Shorter
periods with higher mass ratios could also be excluded, as shown by the
lighter blue shading in Figure 2.

Below, we cast such constraints on $q$, the mass ratio of the SMBH binary, in
terms of traits traceable to galaxy merger events as NGC 4472 is assembled.
The close coupling between galaxy and SMBH masses makes it reasonable to
assume that $q$ for a SMBH binary will be similar to the mass ratio of the two
progenitor galaxies which, after merging, give rise to the SMBH binary. Major
galaxy mergers involve progenitors of comparable mass, whereas minor galaxy
mergers involve progenitors of dissimilar mass (e.g., Volonteri et al., 2003).
Following the common practice of separating major and minor mergers at a mass-
ratio boundary of about $\frac{1}{4}$, Figure 2 implies access to both types
of galaxy mergers.

####  4.2.1 Large q Values

We first examine whether large values for the mass ratio $q$ could be relevant
to NGC 4472. Rantala et al. (2019) use numerical simulations to show that the
dissipationless, major merger of two galaxies can yield a SMBH binary that
builds a stellar core, with tangentially biased orbits, that is counter
rotating relative to the surrounding merger remnant, a massive early-type
galaxy.

While not meant to model NGC 4472 specifically, the simulations of Rantala et
al. (2019) do appear to offer a potential explanation for the kinematics of
its stellar core (see Section 2). In this context, ngVLA astrometric
monitoring of NGC 4472 would be seeking present-day evidence for the SMBH
binary at high $q$ and on sub-parsec scales (Figure 2).

A dissipationless scouring process shrinks the binary orbit on a timescale
$t_{\rm sc}\propto a_{\rm bin}^{-1}$, slow compared to the timescale $t_{\rm
gw}\propto a_{\rm bin}^{+4}$ for orbit shrinkage driven by GW emission (e.g.,
Supplementary Figure 5; Mingarelli et al., 2017). Stated differently, the
residence time over which a SMBH binary might be detected via astrometry is
longer than the residence time over which it might be detected via GW
emission.

####  4.2.2 Small q Values

Small $q$ values could also be relevant to NGC 4472, given observational
evidence suggesting it has built up some of its size and mass by accreting
satellite galaxies (e.g., Janowiecki et al., 2010; Arrigoni Battaia et al.,
2012). As noted by Kormendy & Ho (2013), typical merger trees involve galaxies
that could transport their own SMBHs inward. For example, an in-coming SMBH
with a mass $M_{\rm sec}\sim 2\times 10^{7}$ M⊙ could lead to a SMBH binary
with a mass ratio $q\sim 0.01$, accessible astrometrically according to Figure
2.

###  4.3 Tie-Ins to GW Findings

NGC 4472 is sufficiently close that, if it hosts a SMBH binary, any GW
emissions from that binary might be detectable by current or near-future
pulsar timing arrays. As such, there is the possibility to obtain independent
checks on certain parameters or more comprehensive information about the
system than would be available from only electromagnetic (ngVLA) or
gravitational wave (pulsar timing) measurements.

ngVLA astrometric monitoring could independently check the GW-based upper
limits on $q$ plotted in Figure 2 for NGC 4472, provided the GW findings
invoke the same distance and SMBH binary mass adopted from Schutz & Ma (2016)
for the astrometry. For the PPTA, the $q$ values in Figure 2 are as tabulated
by, and thus consistent with, the Schutz & Ma (2016) distance and SMBH binary
mass.

For NANOGrav, we focus first on a GW frequency of 8 nanoHz (nHz), at which the
most constraining limits on GW strain can be set (Aggarwal et al., 2019;
Arzoumanian et al., 2021). From equation (1) of Schutz & Ma (2016), we set a
95% upper limit on the associated chirp mass of $M_{\rm chi}<0.44\times
10^{9}$ M⊙ for our adopted distance to NGC 4472. We then insert the ratio of
the chirp mass to our adopted binary mass into equation (5) of Schutz & Ma
(2016) to set the 95% upper limit of $q<0.064$ that we plot in Figure 2. These
upper limits on $M_{\rm chi}$ and $q$ are higher than those shown near 8 nHz
in Figure 2 of Arzoumanian et al. (2021); such differences arise because that
study adopts a cosmic-flow distance which is larger than the Schutz & Ma
(2016) distance.

The NANOGrav search spans a GW frequency range of 2.8 nHz to 317.8 nHz
(Aggarwal et al., 2019). This range corresponds to orbital periods of a
putative SMBH binary from $P=22.6$ yr to $P=0.2$ yr, equivalent to $a_{\rm
bin}=10600$ au to $a_{\rm bin}=453$ au for $M_{\rm bin}=2.4\times 10^{9}$ M⊙
(Schutz & Ma, 2016). GW-derived separations are so small that they are often
expressed not in pc, but in au. (An au axis is provided in Figure 2.) The
$PSF$ adopted for the ngVLA astrometry of NGC 4472 corresponds to 1670 au,
making it complementary to and midway within the range of separations
constrained by Aggarwal et al. (2019). Those GW constraints degrade
significantly below about 5 nHz, due to the 11 yr data span. Future
observations, potentially enhanced with ngVLA pulsar timing data (Chatterjee
et al., 2018), will improve the NANOGrav constraints, and extend them to lower
GW frequencies or longer orbital periods.

Finally, most GW analyses have been conducted assuming circular orbits. As
might be expected, for a SMBH binary on an elliptical orbit, GWs are emitted
not only at a frequency determined by the orbital period but also at higher
harmonics. The consequence is that there can be a penalty in signal-to-noise
ratio if a circular orbit is assumed, but the orbit is elliptical (Huerta et
al., 2015; Taylor et al., 2016). This penalty particularly applies if the
orbital period is shorter than $1/T$, where $T$ is the data span for the
pulsar timing. Thus, the limits quoted above for NGC 4472 become less
constraining if one considers elliptical orbits, particularly those at the
higher GW frequencies probed by NANOGrav. If ngVLA astrometry of NGC 4472 were
to suggest that it contained a SMBH binary on an elliptical orbit, improved
limits on the chirp mass $M_{\rm chi}$ or mass ratio $q$ could likely be
obtained.

##  5 Summary and Conclusions

A direct EM signature of an SMBH binary would be to spatially resolve and
monitor the orbit of one or both of its members. To that end, we adapted the
strategy of Safarzadeh et al. (2019) for EHT targets at 230 GHz to the more
abundant targets observable with the ngVLA at 80 GHz. The strategy involves
astrometric monitoring to trace the reflex motion of the binary’s primary SMBH
as it is tugged on by the secondary SMBH. Only one member of the binary needs
to have detectable mm emission.

Picking the early-type galaxy NGC 4472 as an example and casting the circular
orbit of a putative SMBH binary in terms of its period $P$, semimajor axis
$a_{\rm bin}$, and secondary-to-primary mass ratio $q$, we found the
following:

  1. 1.

Orbits with fiducial periods of $P=4$ yr and 40 yr could be spatially resolved
and monitored with the ngVLA.

  2. 2.

For a 95% accuracy of $2\,\mu$as per ngVLA monitoring epoch, sub-parsec values
of $a_{\rm bin}$ could be accessed over a range of mass ratios notionally
encompassing major ($q>\frac{1}{4}$) and minor ($q<\frac{1}{4}$) galaxy
mergers.

  3. 3.

If no reflex motion was detected for the primary after 1(10) yr of ngVLA
monitoring, a SMBH binary with period $P=4(40)$ yr and mass ratio
$q>0.01(0.003)$ could be excluded. This would suggest no present-day evidence
for a past major merger like one recently simulated, wherein scouring by a
$q\sim 1$ SMBH binary formed a stellar core with the kinematic properties of
NGC 4472’s stellar core.

  4. 4.

Astrometric monitoring with the ngVLA could independently check the upper
limits on $q$ for NGC 4472 from searches for continuous gravitational waves
with the PPTA and NANOGrav.

We thank the reviewer for their helpful and timely report. The NRAO is a
facility of the National Science Foundation (NSF), operated under cooperative
agreement by Associated Universities, Inc. (AUI). The ngVLA is a design and
development project of the NSF operated under cooperative agreement by AUI.
The NANOGrav project receives support from NSF Physics Frontiers Center award
number 1430284. Part of this research was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract with the
National Aeronautics and Space Administration. This paper makes use of the
following ALMA data: ADS/JAO.ALMA# 2015.1.00926.S. ALMA is a partnership of
ESO (representing its member states), NSF (USA) and NINS (Japan), together
with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.

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