# Gravitational Waves from Stellar Black Hole Binaries and the Impact on Nearby Sun-like Stars

Ilídio Lopes, Joseph Silk

► To cite this version:

Ilídio Lopes, Joseph Silk. Gravitational Waves from Stellar Black Hole Binaries and the Impact on Nearby Sun-like Stars. The Astrophysical Journal, 2017, 844 (1), pp.39. 10.3847/1538-4357/aa7758. hal-01582801

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# Gravitational Waves from Stellar Black Hole Binaries and the Impact on Nearby Sun-like Stars

Ilídio Lopes¹ and Joseph Silk²,³

¹Centro Multidisciplinar de Astrofísica, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal; ilidio.lopes@tecnico.ulisboa.pt

²Institut d'Astrophysique de Paris, UMR 7095 CNRS, Université Pierre et Marie Curie, 98 bis Boulevard Arago, Paris F-75014, France; silk@astro.ox.ac.uk

³Department of Physics and Astronomy, 3701 San Martin Drive, The Johns Hopkins University, Baltimore MD 21218, USA

Received 2016 September 10; revised 2017 June 2; accepted 2017 June 2; published 2017 July 20

## Abstract

We investigate the impact of resonant gravitational waves on quadrupole acoustic modes of Sun-like stars located nearby stellar black hole binary systems (such as GW150914 and GW151226). We find that the stimulation of the low-overtone modes by gravitational radiation can lead to sizeable photometric amplitude variations, much larger than the predictions for amplitudes driven by turbulent convection, which in turn are consistent with the photometric amplitudes observed in most Sun-like stars. For accurate stellar evolution models, using up-to-date stellar physics, we predict photometric amplitude variations of 1–10³ ppm for a solar mass star located at a distance between 1 au and 10 au from the black hole binary and belonging to the same multi-star system. The observation of such a phenomenon will be within the reach of the PLATO mission because the telescope will observe several portions of the Milky Way, many of which are regions of high stellar density with a substantial mixed population of Sun-like stars and black hole binaries.

**Key words:** asteroseismology – binaries: general – black hole physics – globular clusters: general – stars: general – X-rays: binaries

## 1. Introduction

The ground-breaking observations of gravitational waves from the coalescence of two black hole binaries (Abbott et al. 2016a, 2016b) swiftly initiated the age of gravitational-wave astronomy. These discoveries suggest that massive stellar black holes are more common than previously thought. As a consequence, the idea that intrinsic oscillations of stars may be stimulated by gravitational radiation from nearby stellar black hole binaries becomes a real possibility (i.e., Lopes & Silk 2015).

A compelling reason for our study comes from the fact that stellar oscillations stimulated by nearby stellar black hole binaries might have already been observed unintentionally by current missions. At the very least, such observations are now within the reach of future asteroseismology missions. During the last decade, the *Corot* (launched 2006, Baglin et al. 2006) and *Kepler* (launched 2009, Gilliland et al. 2010; Howell et al. 2014) satellites, despite their restricted fields of view, have discovered non-radial oscillations in approximately 700 main-sequence and sub-giant stars and more than 17,000 red-giant stars out of the 400 billion stars expected to exist in the Milky Way (i.e., Miglio et al. 2012; Chaplin & Miglio 2013; Stello et al. 2015). In these stars, known as Sun-like stars, such non-radial oscillations are driven stochastically by the turbulence of their external convective envelope. The *Kepler* mission alone has measured oscillations in stars located at distances up to 15 kpc, almost two times the distance of the Sun to the Galactic center. This mission has observed such stars uninterruptedly through long periods of time, typically with a duration varying from 3.5 to 4 years. In many of the cases, astronomers were able to determine the high-precision spectrum of their stellar non-radial oscillations. Such stars are located in contrasting space environments of the Milky Way, within the galactic disk and bulge, and in halo globular clusters. Many of these

locations are dense stellar environments, such as globular clusters with mixed populations of Sun-like stars and black hole binaries (Rodriguez et al. 2015). Therefore, it is to be expected that some of these Sun-like stars are located near these binaries and close enough for the incoming gravitational waves to excite some of their non-radial modes (Lopes & Silk 2015). In agreement with our previous work, such Sun-like stars are referred to as star detectors.

Most likely the Sun-like star and the stellar black hole binary will be part of the same gravitationally bound multi-body system. There is theoretical and observational evidence pointing to the formation of multi-star systems (Lopes & Silk 2015). Furthermore, the current theory of star formation predicts the existence of many binary, triple, and higher multiplicity stellar systems. Tokovinin (2014) studied the multiplicity of multi-body systems within 67 pc of the Sun and found that 13% of the stellar systems have three or more stars. Raghavan et al. (2010) estimated that as much as 8% are three-body systems, and Rappaport et al. (2013), using *Kepler* data, estimated that 20% of close binaries have tertiary companions. The *Kepler* mission alone has discovered more than 220 triple systems. In addition to the large number of triple-star systems, a number of higher-order multiple star systems have also been discovered by the *Kepler* mission and ground-based observations (Borkovits et al. 2015). For example, Rappaport et al. (2016) have discovered a quintuple star system that contains two small eclipsing binaries orbiting each other. Each of the small binaries have an orbital radius of 2 au, one of which has a third star. The large binary formed by the small binaries has an orbital radius of 12 au. Therefore, in such multi-star systems, it is very likely that triple systems can also be formed between a stellar black hole binary and a nearby Sun-like star, or even in higher-order star systems (Kobayashi et al. 2012). This is more likely to occur in dense stellar environments such as the core of the Milky Way or in the cores of globular clusters.
Nevertheless, as the amplitudes of oscillations for main-sequence stars are a few orders of magnitude smaller than the amplitudes of red-giant stars, and because the brightness of main-sequence stars are much smaller, consequently in order to observe the impact of gravitational radiation in main-sequence stars it will be more efficient to look for possible targets within the solar neighborhood. Alternatively, because red-giant stars are much more bright, it should be possible to look for potential red-giant star targets at much larger distances. Nevertheless, as we will discuss in this work, the most important limitation on detection of the impact of gravitational waves on the pulsation spectrum of Sun-like stars is related to the magnitude of gravitational radiation striking the star.

The interest in investigating the excitation of stellar oscillation modes by gravitational-wave radiation started in the 1960s, when Joseph Weber published the first article about the experimental detection of gravitational waves (i.e., Weber 1961). Use of the Earth was proposed as a laboratory mass to detect gravitational radiation (Forward et al. 1961). A few years later, Dyson (1969) studied the response of the Sun and Earth to incoming gravitational radiation. Other publications have followed (i.e., Boughn & Kuhn 1984; Siegel & Roth 2011; McKernan et al. 2014). With the advance of the observations, this idea was pursued with the gravitational techniques of helioseismology and asteroseismology, either by studying the impact of gravitational radiation on solar oscillations (i.e., Lopes & Silk 2014, and references therein) or by extending this approach to other stars (i.e., Lopes & Silk 2015, and references therein).

In this article, we predict that stellar black hole binaries located at short distances from Sun-like stars will excite a few of their non-radial oscillations well above the photometric amplitude variations caused by stochastic excitation.

We review some of the more current channels for the formation of black hole binaries, and we discuss the dynamic evolution of black hole binaries during the inspiral phase. Finally, we evaluate the excitation of stellar oscillations by gravitational waves.

## 2. Formation of Stellar Black Hole Binaries

The recent discovery of the GW150914 (Abbott et al. 2016a) and the GW151226 (Abbott et al. 2016b) black hole binaries<fnref n="4" /> is an indisputable confirmation that massive stellar black hole binaries are very likely quite common. Such compact binary systems can form, either from isolated binary stellar systems in galactic fields, or as a result of dynamic interactions of stars in dense stellar environments such as the cores of stellar clusters or galaxies.

In the first case, known as a standard scenario, the formation of black hole binaries follows through a complex number of stages (i.e., Postnov & Yungelson 2014). First, two massive stars form in a common stellar envelope, and a stable mass transfer is established between the two stars. By the action of gravity, the core of the largest star collapses into a black hole. Then, a second mass transfer starts from the external layers of the remaining star to the black hole. During this phase the binary continuously loses mass and angular momentum. The compact binary forms when the core of the second star also

collapses into a black hole (O'Shaughnessy et al. 2005). Although there are several stellar models to explain the formation of black hole binaries, the vast majority follow this evolutionary path (Belczynski et al. 2008; Belczynski et al. 2002). For instance, Mandel & de Mink (2016) and Marchant et al. (2016) recently proposed that massive stars, which throughout their lifetimes remain in a close compact binary, maintaining a rapid rotation and a chemically homogeneous interior, rapidly end by forming a massive stellar black hole binary.

In the second case, the black hole binaries form in massive stellar clusters, due to the drift of single or binary black holes to the core of the cluster by dynamical friction (Rodriguez et al. 2016b). In these dense stellar environments, the gravitational interaction of stars and black holes leads to the formation of multi-stellar binary systems, some of which are quite often ejected from the cluster (Sigurdsson & Hernquist 1993). Such interactions preferentially keep the heaviest objects in binaries and eject the lightest ones leading to a concentration of heavier black hole binaries in the cluster core (Heggie 1975). In some cases, such processes leads to the formation of triple compact star systems. Meiron et al. (2017) suggest that the presence of a nearby third star can be determined by the perturbation caused to the gravitational wave emitted during the merger of the two black holes of the inner binary.

It is well accepted that both evolution scenarios predict the formation of black hole binaries with masses comparable to those recently discovered (Abbott et al. 2016b, 2016c), though further improvements are required to fine-tune their evolutionary paths. Nevertheless, all scenarios predict that the more massive binaries are formed in stellar environments of low metallicity (e.g., Dominik et al. 2013). Equally, the formation of binaries by dynamical friction is intensified in the cores of globular clusters, where there is a high density of stars with low metallicity (Rodriguez et al. 2016a).

One test of all these formation scenarios is that the black holes of such binaries must merge on a timescale smaller than the age of the universe (~13.7 Gyr). For this to happen, the orbital separation of the black holes must be sufficiently small to allow rapid binary contraction by energy loss to gravitational radiation, culminating in the coalescence of the two black holes. Although this property is verified for all binaries, the binary formation timescales and their coalescence timescales are very different. For instance, in the standard scenario, compact binaries form on a timescale of 10 Gyr (Postnov & Yungelson 2014). In a variation of this scenario, a close binary formed by two massive stars coalesces into a black hole binary with a total mass of 80 $M_{\odot}$ in a record time of 2.6 Gyr (Rodriguez et al. 2016b). In this case, black hole binaries created by stellar encounters in globular clusters have a typical waiting time to coalescence of 6.3 Gyr.

## 3. Stellar Black Hole Binaries in the Inspiral Phase

The final evolutionary stage of a black hole binary, as recently validated by the discovery of GW150914, follows a well-defined process of three phases (Blanchet 2014; Buonanno & Damour 2000): it begins with the black holes being far apart and rotating in near-perfect circular orbits (inspiral), entering into a stage of rapid orbital shrinking (speed-up), and terminating by merging into a single black hole (ringdown). Here our study focuses on an approximate description of the first two phases in the sprit of Dvorkin et al. (2016).

<footnote n="4">⁴ GW150914: located at a distance of 410 Mpc, with black hole masses of 36 $M_{\odot}$ and 29 $M_{\odot}$; and GW151226: located at a distance of 440 Mpc, with black hole masses of 14.2 $M_{\odot}$ and 7.5 $M_{\odot}$.</footnote>
We start by assuming that the two black holes in the binary are in a circular orbit. The detector star is considered to be sufficiently far away from the binary, such that the incoming gravitational radiation is described by a plane wave field, but near enough that the redshift corrections are ignored in the gravitational-wave spectrum. During the two initial phases of the binary contraction, the frequency and amplitude of gravitational-wave strain vary as follows.

(a) As the gravitational frequency $\nu$ is two times the orbital frequency, Kepler's laws show that $\nu$ increases with the reduction of the orbital radius of the binary. As usual, we define the angular frequency $\omega$ to be equal to $2\pi\nu$. The smaller black hole slowly enters in an adiabatic inspiral process, by going through a succession of quasi-circular orbits during which it loses energy by gravitational radiation. Consequently, the time to coalescence $\tau$ decreases as $\nu$ increases,

$$ \nu/\nu_c = 0.0728(\nu_c\tau)^{-3/8}, \qquad (1) $$

where $\nu_c$ is a characteristic frequency of the binary system. $\nu_c$ is equal to $(c^3/(GM_c))$, where c and G are the speed of light and Newton constant, and $M_c$ is the chirp mass of the binary system, $M_c \equiv (m_1m_2)^{3/5}/M_t^{1/5}$, where $m_1$ and $m_2$ are the masses of the two black holes and $M_t = m_1 + m_2$. Similarly, the orbital radius a, which is equal to $(GM_t/(\pi\nu))^{1/3}$, decreases as $\nu$ increases.

The inspiral phase ends when the radial distance between the two black holes is shorter than the last stable circular orbit, also known as the innermost stable circular orbit (ISCO). The frequency of this orbit $\nu_{ISCO}$ is approximately equal to 2.2 kHz ($M_\odot/M_t$). When this orbit is passed, the two black holes merge and coalesce. Hence, $\nu_{max}$, the frequency of the gravitational wave emitted on the ISCO orbit is equal to twice the $\nu_{ISCO}$ (Maggiore 2008). $\nu_{max}$ also corresponds to the maximum frequency of the gravitational radiation emitted in the inspiral phase.

(b) The strain $h$ increases as the binary system approaches coalescence. The nature of gravitational radiation determines that $h$ has two polarized components, $h_+$ (plus polarization) and $h_-$ (cross polarization). These quantities are written in a condensed form: from the start of the inspiral phase until coalescence, $h_k(\tau)$ is equal to $A_* g_k(\varphi) C_k[\Phi(\tau)]$, where $k$ is equal to $+$ or $-$, $A_*$ is the amplitude, $g_k$ are geometrical functions depending of the directional angle $\varphi$, and $C_k$ are circular functions depending of the phase $\Phi(\tau)$ (Maggiore 2008). The strain amplitude $A_*$ reads

$$ A_* = c/(d_*\nu_c) (5/(\nu_c\tau))^{1/4}, \qquad (2) $$

where $d_*$ is the distance of the black hole binary system to the detector star.

Figure 1 shows the time variation of two binary parameters—orbital radius and time to coalescence, as a function of $\nu$ for the two recently discovered black hole binaries: GW150914 (Abbott et al. 2016a) and GW151226 (Abbott et al. 2016b). Since their formation (the upper limit is the age of the universe), these binaries have emitted gravitational radiation in the frequency range of $10^{-7}-10^{-2}$ Hz. During this time, the orbital radius reduces to a tenth of the solar radius (~10¹⁰ cm), being only 50 years (~1.5 10⁹ s) away from the binary coalescence. As illustrated in Figure 1 during most of the inspiral phase, the gravitational radiation emitted can stimulate non-radial oscillations of Sun-like stars located nearby these binaries.

<figure><img src="image_3.png" /><figcaption>Figure 1. Frequency time variation of the gravitational wave emitted by the GW150914 (continuous curves) and GW151226 (dashed curves) black hole binaries: the blue (left axis) and red (right axis) lines correspond to the variation of the orbital radius (a[cm]) and time to coalescence (τ[sec]) as the binary proceeds toward coalescence. The vertical black lines correspond to the frequency $\nu_{max}$, for GW150914 (dashed line) and GW151226 (dotted-dashed line). The pink area defines the frequency range of non-radial oscillations observed in Sun-like stars: main-sequence and sub-giant stars from $10^{-4}$ Hz to $10^{-2}$ Hz (Kjeldsen et al. 2008); and red-giant stars from $10^{-7}$ Hz to $10^{-3}$ Hz (Mosser et al. 2013). The orange area corresponds to the frequency oscillation range of the star detector shown in Figure 2. The red, green, and yellow circles correspond to the orbital radius 1 au, 50 R$_\odot$ and 5 R$_\odot$ solar radius. The red, green, and yellow diamonds correspond to the time to coalescence of $10^6$ year, $10^4$ year, and 50 year, respectively.</figcaption></figure>

## 4. Impact of Gravitational Waves in Non-radial Oscillating Stars

The response of the Sun to resonant gravitational radiation has been discussed by several authors, following the detailed study made originally by Boughn & Kuhn (1984). Most aspects of the analysis for other stars are similar. We follow the standard discussion about gravitational radiation emission (Maggiore 2008).

Here, to compute the stimulus of non-radial oscillations of a Sun-like star struck by gravitational waves from a nearby black hole binary, we follow the study and notation of Lopes & Silk (2015, and references therein). Accordingly, the stimulation of gravitational waves only excites stellar modes of even degree $l$. Of these, we focus our study on the quadrupole modes ($l=2$) for which the gravitational forcing is the greatest. Because our goal is to compute the amplitude of these modes, the problem is simplified by focusing our attention on the quadrupole acoustic modes of frequency $\omega_n$ and order $n$ (overtone).

In general, because a black hole binary radiates gravitational waves, it loses energy and thus spins up. This process is accompanied by a reduction of the binary size and an increase of the orbital frequency. Accordingly to Lopes & Silk (2014), we have computed the maximum impact of gravitational waves on photospheric velocity amplitudes of resonant stellar quadrupole modes; for convenience, we choose here to write as the variation of photometric amplitudes

$$ \left( \frac{A_n}{\text{ppm}} \right) = \frac{A_{\odot}}{\nu_{\odot}} f_s \frac{A_*}{\text{cm s}^{-1}} \frac{L_n}{\alpha_s} \frac{\omega_n^2}{\eta_n}, \qquad (3) $$

where $A_*$ is the strain amplitude of the gravitational wave (see Equation (2)), and $\omega_n$ and $\eta_n$ are the angular frequency and
damping rate of the star's mode. $\alpha_s$ is a constant equal to $2\sqrt{2}$, for which the exact value is fixed by observations. $f_s$ is equal to $\sqrt{T_\odot/T_\star}$, where $T_\odot$ and $T_\star$ are the effective temperatures of the Sun and the star. $A_\odot$ is the maximum observed bolometric amplitude and $v_\odot$ is the maximum surface velocity of the mode in the Sun's surface. The values of $A_\odot$, $v_\odot$ and $T_\odot$ are equal to $2.53 \pm 0.11$ ppm (parts-per-million), $18.5 \pm 1.5$ cm s$^{-1}$, and 5777 K. All the previous quantities are evaluated at the solar photosphere (Michel et al. 2009). In the determination of $A_n$ (Equation (3)), we use the photospheric velocity amplitude expression $V_n = A_*L_n/\alpha_s \omega_n^2/\eta_n$ from Lopes & Silk (2014). The conversion from photospheric velocity to photometric amplitude variations is made using the relation obtained by Michel et al. (2009). We note that the photometric amplitude given by Equation (3) can be easily modified for stars in other stages of evolution. In particular, for red-giant stars, this can be done by using a scale relation like the ones obtained by Huber et al. (2011) and Barban et al. (2011). These scale relations were obtained empirically from large data photometric sets by the Kepler Mission. These observational data sets comprise photometric time series for stars in quite distinct stages of evolution, like the main-sequence and red-giant phases. $L_n$ is the effective modal length that depends on the properties of the quadrupole mode of order $n$: $L_n = 1/2 R_* |\chi_n|$, where $R_*$ is the star's radius and $\chi_n$ is a coefficient that depends on the eigenfunction $\xi_n$:

$$ \chi_n = \frac{3}{4\pi\bar{\rho}_*} \int_0^1 \rho(r) [\xi_{r,n}(r) + 3\xi_{h,n}(r)] r^3 dr, \quad (4) $$

where $\rho$ and $\bar{\rho}_*$ are the density profile and mean density of the star, and $\xi_{r,n}$ and $\xi_{h,n}$ are the radial and horizontal components $\xi_n$ (Lopes & Silk 2014).

If the strain of the incoming gravitational wave is considered to be constant, Equation (3) becomes identical to the previous amplitude predictions (for generic gravitational-wave sources), like the ones obtained by Siegel & Roth (2011) and Lopes & Silk (2014). Equation (3) differs from the previous ones because in this case it takes into account the variation of $A_n$ due to the orbital contraction of the black hole binary (see Equation (2)). Accordingly, from the previous Equations (2) and (3), the new expression reads

$$ \left( \frac{A_n}{\text{ppm}} \right) = \frac{A_\odot}{v_\odot} f_s \frac{h_*}{\text{cm s}^{-1}} \left( \frac{5}{\nu_c \tau} \right)^{1/4} \frac{L_n}{\alpha_s} \frac{\omega_n^2}{\eta_n}, \quad (5) $$

where $h_* = c/(d_*v_c)$. The above expression is valid provided that the frequency of the gravitational wave is in resonance with the frequency of the stellar mode and its variation can be neglected. This equation includes the term $A_\odot/v_\odot f_s \times (h_*/\text{cm s}^{-1}) \times (5(\nu_c\tau))^{1/4}$ that converts velocity amplitudes to photometric amplitudes, and also takes into account the strain amplitude time variation emitted by the compact binary.

Figure 2 shows the photometric amplitude variations of quadrupole acoustic modes for resonant gravitational waves emitted by GW150914 and GW151226 black hole binaries. In this calculation, the values of $\omega_n$ and $\chi_n$ are obtained for the present Sun (i.e., Turck-Chièze & Lopes 2012; Lopes & Silk 2014), and $\eta_n$ corresponds to damping rates for realistic solar

<figure><img src="image_4.png" /><figcaption>Figure 2. Maximum intensity amplitude of quadrupole modes computed for a solar main-sequence star excited by a nearby GW150914 (red curve) and GW151226 (blue curve) black hole binary. The detector star is located at a distance of 10 au (square symbol) or 1 au (circle symbol). The orbital radius of these binaries varies from 3 $R_\odot$ to 0.66 $R_\odot$, during which the frequency of the gravitational waves emitted varies from $3 \times 10^{-4}$ Hz to $3 \times 10^{-3}$ Hz (corresponds to $\log_{10}(\nu[\text{Hz}])$ varies from -3.4 to -2.8, see the orange area in Figure 1).</figcaption></figure>

envelope models (Houdek et al. 1999; Grigahcène et al. 2005). The result shows that these black hole binaries when in the vicinity of typical Sun-like stars of a solar mass are able to strongly excite the lowest order overtones of these modes, producing photometric amplitude variations well above 100 ppm. In some extreme cases, the amplitude variation is above $10^3$ ppm. These values are much larger than the photometric amplitudes of just a few ppm observed for low-degree acoustic modes in main-sequence stars (Campante et al. 2011), as predicted to be stochastically excited by turbulent convection (Houdek et al. 1999). In a recent study, Huber et al. (2011) found that main-sequence and red-giant stars show photometric amplitudes increasing from a few ppm up to 1000 ppm.

Among the possible physical processes capable of suppressing the impact of a gravitational wave in a stellar mode (see Equation (5)), the damping rate $\eta_n$ is the most significant one. It is true that until now, no low-order quadrupole modes have been successfully observed in Sun-like stars, possibly because the fluid motions of the upper layers of stars are suppressing the stochastic excitation of the modes due to turbulence in the layers beneath. Indeed, the energy exchange between convection pulsations could affect the amplitude of such modes. This specific point was discussed by several authors, including Chaplin et al. (1997) and Dupret et al. (2006), among others. Nevertheless, we note that the excitation provided by gravitational radiation is external to the star and therefore the prediction of the mode excitation is simpler than the case of modes excited by the turbulent processes of the convection zone, which depends of many local physics processes such as magnetic fields and differential rotation. We believe that the predictions made in this study are realistic since the theoretical damping rates used in this study follow from the same theory of stellar pulsation that successfully predicted the excitation and damping of stellar oscillations for the Sun and many Sun-like stars (Houdek et al. 1999; Grigahcène et al. 2005; Turck-Chièze & Lopes 2012; Belkacem et al. 2013). This theory successfully describes the amplitude of the observed acoustic
modes of low-degree *l* and high-order *n*. Therefore, it is
reasonable to assume that at lower frequencies such a
theoretical model of stellar pulsations is still valid. On the
unlikely possibility that lower-order modes have damping rates
comparable to the observed value of $10^{-3}$ µHz (and predicted
by theoretical pulsation models) for stellar modes with
$\omega/2\pi \sim 1000$ µHz the amplitude of such modes will be below
the PLATO threshold of detection. Nevertheless, as explained
by Lopes & Silk (2014), the current theoretical predictions
suggest that such stellar modes with $\omega/2\pi < 1000$ µHz have
much smaller values of damping rates.

The next generation of satellites, like PLATO (Rauer 2013),
should be able to measure photometric amplitudes with the
necessary precision to detect such an impact of gravitational
waves in low-order acoustic quadrupole modes. The current
estimate for the PLATO mission predicts that the telescope will
be able to measure photometric amplitude variations above
3.2 ppm after 5 days (or 1.3 ppm after 1 month) of continuous
observation.

**5. Summary and Conclusions**

The recent detection of gravitational waves from two distant
black hole binaries of intermediate mass implies that such
binary systems are quite common in the universe. This also
hints at the existence of very efficient mechanisms for the
formation of these binaries. In this work, we mention two of
these processes: as the end product of the evolution of binaries
of massive stars and as resulting from the gravitational
encounters of stars in high density stellar environments, as
found in the cores of stellar clusters. Moreover, the formation
of black hole binaries from progenitor stellar binaries is
facilitated in low-metallicity stellar environments (e.g., Abbott
et al. 2016c), most notably in the cores of globular clusters
(Rodriguez et al. 2016a). These facts substantiate the existence
of regions in the Milky Way with a mixed population of stellar
black hole binaries and Sun-like stars.

This study suggests that the quadrupole acoustic modes of
stars located nearby black hole binaries in some cases can have
relatively large amplitudes well above the experimental
threshold of the current asteroseismic satellites. Nevertheless,
there are important caveats that need to be addressed if one
wishes to successfully find such stars. In particular, these
amplitude predictions do not take into account the visibility of
stellar modes, which depends on the photometric properties
of the star, as well as the degree and azimuthal order of the
mode. This important point was discussed in the context of
gravitational radiation for the first time by Siegel & Roth
(2014, 2015) for the case of high-precision radial velocity of
solar gravity modes. These authors have shown that visibility
effects lifted the degeneracy in the azimuthal order of the mode
leading to the reduction of amplitudes by a few orders of
magnitude. However, as pointed out by Berthomieu & Provost
(1990), these visibility corrections depend on the properties of
stellar modes and on the detailed structure of the upper layers
and atmosphere of the star. Unfortunately, these stellar regions
are still poorly described by the current stellar models (i.e.,
Lopes & Silk 2013, and references therein). Nevertheless, this
difficulty should be overcome since the large quantity and
high-quality data made available by the current asteroseismic
satellites should allow astronomers to accurately model the
upper layers of Sun-like stars in the near future.

The possibility of observing the imprint of gravitational
radiation on the stellar pulsation spectrum of a Sun-like star by
a future asteroseismology mission is limited by the existence of
many unknowns related to the dynamic properties of the stellar
systems being observed, as well as by the experimental
specifications of the telescopes being used for the observations.
Nevertheless, there are a few points that we can make that
could help the discussion to successfully discover such stars.

Since it is now known that stars show Sun-like oscillations in
many stages of the stellar evolution including main-sequence,
sub-giant, and red-giant phases. It is possible that some of these
stars are located sufficiently near such black hole stellar
systems. As discussed previously, black hole binaries can form
in two types of scenarios: in stellar multiple systems where the
black holes form as the end stage of stellar evolution, or the
binaries are created due to the dynamical interactions between
stars. Both processes are more likely to occur in dense stellar
regions of the Milky Way, like the ones found in the core of
stellar clusters and the regions of intense stellar formation
near the galactic center. In any case, for the spectrum of a Sun-
like star to be affected by gravitational radiation, it is
paramount that the black hole binary be located relatively near
the Sun-like star, possibly for even the black hole binary and
the Sun-like star to be part of the same multiple stellar system
(Toonen et al. 2016). There is strong evidence to support that a
significant fraction of stars resides in a multi-stellar system
including binaries (Raghavan et al. 2010). Whitworth & Lomax
(2015) estimate that 54% of Sun-like stars are in binaries or
multiple stellar systems. Rappaport et al. (2013), using a
limited range of orbital periods for the triple-star systems,
estimate that at least 20% of all close binaries have tertiary
companions. Recently, Roberts et al. (2015) have observed a
few triple systems with solar-type stars in the optical as well as
in the near-infrared (Roberts et al. 2017) and Rappaport et al.
(2017) discovered a quadruple stellar system and quintuple
stellar system (Rappaport et al. 2016). Moreover, as much as
70% of massive stars are observed to have companions
(Kobulnicky & Fryer 2007). This is significant, since some
of these massive stars could lead to the formation of black
holes, in which the instantaneous supernova expulsion shell
will not affect the companion stars (Toonen et al. 2016).
However, the explosion will modify the dynamics of the
multiple stellar system (Pijloo et al. 2012).

In spite of not knowing how many stellar black hole binaries
exist in the Milk Way that are located nearby Sun-like stars, at
least we can make an estimation of the total number of stellar
black hole binaries in our Galaxy. Recently, Christian & Loeb
(2017) using the merger rates of LIGO found that the number
of black hole binaries in the Milky Way *N* with semimajor
axis *a* smaller than *a*<sub>bh</sub> to be given by $N(a \le a_{bh}) \approx K_{bh}R_{100}(a_{bh}/10R_{\odot})^4$, where $K_{bh}$ is a rate factor related to the
black hole binary mass and $R_{100}$ is the rate in units of
100 Gpc<sup>-3</sup>yr<sup>-1</sup>. If only binaries with black hole mass of 30 $M_{\odot}$
are taken into account, then $K_{bh} = 3 \times 10^2$. However, if the
same calculation is repeated assuming a power law for the mass
function of massive stars, i.e., taking into account all binaries
(with black holes of different masses) with the same orbital
radius, then $K_{bh} = 3 \times 10^4$. For instance, *N* is equal to 20 in
the case of binaries formed by two black holes with a mass of
30 $M_{\odot}$ each and with a semimajor *a* axis smaller than
$a_{bh} \sim 5 R_{\odot}$. In the case in which the black hole masses in
the binary can take other values, *N* will increase to 2000. The
same expression predicts the existence of 19 × 10⁶ black hole binary systems with $a_{bh} \sim 50 R_⊙$. This type of binaries will emit gravitational radiation that will excite the quadrupole modes of nearby Sun-like stars, black hole binaries with a $a_{bh} \sim 50 R_⊙$ and $\sim 5 R_⊙$ will emit waves with a frequency greater than $10^{-6}$ Hz and $10^{-4}$ Hz (see Figure 1).

The important point is that the Sun-like stars to be observed by the asteroseismic satellite must be located at a short distance of the black hole binary. Therefore, we will increase the chances to make such detections if the telescope is able to observe stars at large distances. Among the star targets observed by the *Corot* and *Kepler* satellites (Mosser et al. 2010; Huber et al. 2011), a few tens of thousands of stars have been discovered to show Sun-like oscillations.

The original *Kepler* mission alone has made such discoveries only by looking in a fixed direction of the sky; however, with its renewed observational program K2 mission (Huber et al. 2014), the survey is now extended to many other directions of the Milky Way, significantly increasing the chances to look for Sun-like stars near black hole binaries. Although there is a possibility that such phenomena will be found within the current asteroseismic data sets, it is more likely that it will happen in a future mission, such as PLATO (Rauer et al. 2014) and TESS (Ricker et al. 2014) satellites, or even more likely that it will happen in a post-PLATO asteroseismology mission for which specific observational targets could be chosen with this objective in mind. To succeed in this endeavor, it is necessary to understand which are the most likely stellar environments where the targets with Sun-like stars nearby black hole binaries could be found.

The PLATO satellite is expected to obtain 1,000,000 stellar photometric light curves for stars with a magnitude smaller 14 over the course of the full mission (Rauer et al. 2014). If we consider that a proportional of 10% of stars have Sun-like oscillations like it was found in the case of the *Kepler* mission, a total of 100,000 should also have Sun-like oscillations. In principle, this mission should be able to observe a typical star like the Sun up to a distance of 2 kpc and a red-giant star up to a distance of 20 kpc, if it was to the light obstruction by dust, the PLATO mission could observe red-giant stars in the center of the Milky Way (at a distance of 8 kpc). Therefore, it is reasonable to expect that some of these Sun-like stars are located near the black hole binaries.

Since some Sun-like stars that likely are located nearby black hole binary systems can be found in the dense stellar regions of the Milky Way obscured by dust, an asteroseismology mission dedicated to observe in the near-infrared band could be an excellent alternative to optical asteroseismology. As the star brightness and amplitude of stellar modes in this band is only a factor of five smaller than the optical measurements counterpart, this could significantly increase the possibility of finding such types of stars once it allows us to observe in the dense regions of the Milky Way. In spite of the asteroseismology of the near-infrared being very poorly understood at present, this technique could be particularly useful to look for stars in the dense regions of the galactic disk in the direction of the galactic center, where it is known to have large populations of compact stars.

We show here that if the generally accepted ideas about general relativity are correct, the low-order overtones of quadrupole acoustic modes of Sun-like stars could be driven by gravitational radiation of nearby stellar black hole binaries,

identical to the ones discovered by the Advanced LIGO collaboration (Abbott et al. 2016a, 2016b). Nonetheless, only if there is a sustained resonance between the incident gravitational radiation and the stellar quadrupole mode can there be a chance of an observable response from the star. In general, the energy loss by gravitational radiation causes the black hole binary to spin up, prohibiting the gravitational wave from remaining in resonance with a mode for a period long enough to have a perceptible effect. However, because the orbital frequency drift is negligible for many of these black hole binaries during most of the inspiral phase, oscillations in a solar mass star can be stimulated to an observable photometric amplitude by a binary at a distance of 1 au. In particular, we predict that the photometric amplitude variations of these modes are well above the experimental threshold of present and future asteroseismic satellite missions (Rauer 2013). Therefore, it is even possible that these observations could have already been made.

The author thanks the anonymous referees for the insightful comments and suggestions. In particular, the suggestion to include a discussion about the possibility of observing such phenomena by a future asteroseismology satellite was very helpful. The work of I.L. was supported by grants from “Fundação para a Ciência e Tecnologia” and “Fundação Calouste Gulbenkian.” We are grateful to the authors of ADIPLS and CESAM codes for having made their codes publicly available. The work of J.S. was supported by ERC Project No. 267117 (DARK) hosted by Université Pierre et Marie Curie (UPMC)—Paris 6.

## References

* Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, PhRvL, 116, 061102

* Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016b, PhRvL, 116, 241103

* Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016c, ApJL, 818, L22

* Baglin, A., Auvergne, M., Barge, P., et al. 2006, in Proc. ESA Special Publication 1306, The CoRoT Mission Pre-launch Status, ed. M. Fridlund, J. Lochard, & L. Conroy (Noordwijk: ESA), 33

* Ballot, J., Barban, C., & van't Veer-Mennertet, C. 2011, A&A, 531, A124

* Belczynski, K., Kalogera, V., & Bulik, T. 2002, ApJ, 572, 407

* Belczynski, K., Kalogera, V., Rasio, F. A., et al. 2008, ApJS, 174, 223

* Belkacem, K., Samadi, R., Mosser, B., Goupil, M. J., & Ludwig, H. G. 2013, in ASP Conf. Ser. 479, Progress in Physics of the Sun and Stars: A New Era in Helio- and Asteroseismology, ed. H. Shibahashi & A. E. Lynas-Gray (San Francisco, CA: ASP), 61

* Berthomieu, G., & Provost, J. 1990, A&A, 227, 563

* Blanchet, L. 2014, LRR, 17, 2

* Borkovits, T., Rappaport, S., Hajdu, T., & Sztakovics, J. 2015, MNRAS, 448, 946

* Boughn, S. P., & Kuhn, J. R. 1984, ApJ, 286, 387

* Buonanno, A., & Damour, T. 2000, PhRvD, 62, 064015

* Campante, T. L., Handberg, R., Mathur, S., et al. 2011, A&A, 534, A6

* Chaplin, W. J., Elsworth, Y., Isaak, G. R., et al. 1997, MNRAS, 288, 623

* Chaplin, W. J., & Miglio, A. 2013, ARA&A, 51, 353

* Christian, P., & Loeb, A. 2017, MNRAS, 469, 930

* Dominik, M., Belczynski, K., Fryer, C., et al. 2013, ApJ, 779, 72

* Dupret, M. A., Barban, C., Goupil, M.-J., et al. 2006, in ESA Special Publication 624, Proc. of SOHO 18/GONG 2006/HELAS I, Beyond the Spherical Sun (Noordwijk: ESA), 97

* Dvorkin, I., Uzan, J.-P., Vangioni, E., & Silk, J. 2016, PhRvD, 94, 103011

* Dyson, F. J. 1969, ApJ, 156, 529

* Forward, R. L., Zipoy, D., Weber, J., Smith, S., & Benioff, H. 1961, Natur, 189, 473

* Gilliland, R. L., Jenkins, J. M., Borucki, W. J., et al. 2010, ApJL, 713, L160

* Grigahcène, A., Dupret, M.-A., Gabriel, M., Garrido, R., & Scuflaire, R. 2005, A&A, 434, 1055
Heggie, D. C. 1975, MNRAS, 173, 729

Houdek, G., Balmforth, N. J., Christensen-Dalsgaard, J., & Gough, D. O. 1999, A&A, 351, 582

Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398

Huber, D., Bedding, T. R., Stello, D., et al. 2011, ApJ, 743, 143

Huber, D., Silva Aguirre, V., Matthews, J. M., et al. 2014, ApJS, 211, 2

Kjeldsen, H., Bedding, T. R., Arentoft, T., et al. 2008, ApJ, 682, 1370

Kobayashi, S., Hainick, Y., Sari, R., & Rossi, E. M. 2012, ApJ, 748, 105

Kobulnicky, H. A., & Fryer, C. L. 2007, ApJ, 670, 747

Lopes, I., & Silk, J. 2013, MNRAS, 435, 2109

Lopes, I., & Silk, J. 2014, ApJ, 794, 32

Lopes, I., & Silk, J. 2015, ApJ, 807, 135

Maggiore, M. 2008, Gravitational Waves, Vol. 1: Theory and Experiments (Oxford: Oxford Univ. Press)

Mandel, I., & de Mink, S. E. 2016, MNRAS, 458, 2634

Marchant, P., Langer, N., Podsiadlowski, P., Tauris, T. M., & Moriya, T. J. 2016, A&A, 588, A50

McKernan, B., Ford, K. E. S., Kocsis, B., & Haiman, Z. 2014, MNRAS, 445, 74

Meiron, Y., Kocsis, B., & Loeb, A. 2017, ApJ, 834, 200

Michel, E., Samadi, R., Baudin, F., et al. 2009, A&A, 495, 979

Miglio, A., Morel, T., Barbieri, M., et al. 2012, EPJWC, 19, 05012

Mosser, B., Belkacem, K., Goupil, M.-J., et al. 2010, A&A, 517, A22

Mosser, B., Samadi, R., & Belkacem, K. 2013, in SF2A-2013: Proc. Annual Meeting of the French Society of Astronomy and Astrophysics, ed. L. Cambresy, F. Martins, E. Nuss, & A. Palacios (Montpellier: Société Française d'Astronomie et d'Astrophysique), 25

O'Shaughnessy, R., Kaplan, J., Kalogera, V., & Belczynski, K. 2005, ApJ, 632, 1035

Pijloo, J. T., Caputo, D. P., & Portegies Zwart, S. F. 2012, MNRAS, 424, 2914

Postnov, K. A., & Yungelson, L. R. 2014, LRR, 17, 3

Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1

Rappaport, S., Deck, K., Levine, A., et al. 2013, ApJ, 768, 33

Rappaport, S., Lehmann, H., Kalomeni, B., et al. 2016, MNRAS, 462, 1812

Rappaport, S., Vanderburg, A., Borkovits, T., et al. 2017, MNRAS, 467, 2160

Rauer, H. 2013, in EGU General Assembly 2013 (Gottingen: Copernicus), EGU2013-2281

Rauer, H., Catala, C., Aerts, C., et al. 2014, ExA, 38, 249

Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, Proc. SPIE, 9143, 914320

Roberts, L. C., Jr., Tokovinin, A., Mason, B. D., Hartkopf, W. I., & Riddle, R. L. 2015, AJ, 150, 130

Roberts, L. C., Jr., Tokovinin, A., Mason, B. D., & Marinan, A. D. 2017, AJ, 153, 100

Rodriguez, C. L., Chatterjee, S., & Rasio, F. A. 2016a, PhRvD, 93, 084029

Rodriguez, C. L., Haster, C.-J., Chatterjee, S., Kalogera, V., & Rasio, F. A. 2016b, ApJL, 824, L8

Rodriguez, C. L., Morscher, M., Pattabiraman, B., et al. 2015, PhRvL, 115, 051101

Siegel, D. M., & Roth, M. 2011, ApJ, 729, 137

Siegel, D. M., & Roth, M. 2014, ApJ, 784, 88

Siegel, D. M., & Roth, M. 2015, ApJ, 810, 84

Sigurdsson, S., & Hernquist, L. 1993, Natur, 364, 423

Stello, D., Huber, D., Sharma, S., et al. 2015, ApJL, 809, L3

Tokovinin, A. 2014, AJ, 147, 87

Toonen, S., Hamers, A., & Portegies Zwart, S. 2016, ComAC, 3, 6

Turck-Chièze, S., & Lopes, I. 2012, RAA, 12, 1107

Weber, J. 1961, Interscience Tracts on Physics and Astronomy (New York: Interscience)

Whitworth, A. P., & Lomax, O. 2015, MNRAS, 448, 1761