Gravitational waves from in-spirals of compact objects in binary common-envelope evolution
Yonadav Barry Ginat Hila Glanz Hagai B. Perets Evgeni Grishin Vincent Desjacques


firstpage–lastpage
Transiting exocomets detected in broadband light by TESS 

   in the β Pictoris system
    S. Zieba 1
         K. Zwintz1
          M. A. Kenworthy2
          G. M. Kennedy 3
          
    Received 26 March 2019; accepted 
==================================================================================================





Detection of gravitational-wave (GW) sources enables the characterisation
of binary compact objects and of their in-spiral. However,
other dissipative processes can affect the in-spiral. Here we show
that the in-spiral of compact objects through a gaseous common-envelope
(CE) arising from an evolved stellar companion produces a novel type
of GW-sources, whose evolution is dominated by the dissipative gas
dynamical friction effects from the CE, rather than the GW-emission
itself. The evolution and properties of the GW-signals differ from those of isolated gas-poor mergers significantly. We find characteristic
strains of ∼10^-23-10^-21 (10 kpc/D) for such
sources – observable by next-generation space-based GW-detectors.
The evolution of the GW-signal can serve as a probe of the interior regions
of the evolved star, and the final stages of CE-evolution, otherwise
inaccessible through other observational means. Moreover, such
CE-mergers are frequently followed by observable explosive electromagnetic
counterparts and/or the formation of exotic stars.



Gravitational waves — (stars:) binaries (including multiple): close





§ INTRODUCTION

Two gravitating bodies emit gravitational
waves (GWs) when orbiting each other, whose amplitude and frequency depend on their relative
acceleration, which, in turn, depends on their masses
and separation. Generally, larger masses and smaller separations induce stronger accelerations and stronger GW emission.
Current and upcoming GW-detectors cannot detect mergers of stellar
binaries, whose radii – and separation at merger – are far
below the amplitude sensitivity of the detectors at the relevant frequencies. Indeed, only sufficiently compact (and massive) stars
could serve as detectable GW sources; current studies typically focus
on the mergers of white dwarfs (WDs) <cit.>, neutron stars
(NSs), and black holes (BHs) <cit.>.
However, the stellar cores of evolved stars could serve as a novel
type of compact object (CO) to produce GW sources, once they
merge with another compact object. Such mergers can occur following
a binary common-envelope (CE) evolution of evolved stars with compact companions, leading to the in-spiral of the companion and to its
merger with the core of the evolved star <cit.>.


During the evolution of stars beyond their main-sequence evolutionary
stage, they fuse the hydrogen in their cores into helium and heavier
elements, that accumulate in compact cores. The stellar envelopes
of the stars then expand to large radii – tens, or even up to hundreds of
solar radii (during the red-giant branch, and asymptotic giant branch,
phases of their evolution, as well as the Wolf-Rayet stage for very
massive stars). When such an evolved star has a close stellar companion
the expanded envelope engulfs the companion and, under appropriate
conditions, creates a shared envelope of gas, covering both the core of the
evolved star and the companion <cit.>. The stellar companion
is then thought to spiral inside the envelope due to its gravitational
interaction with the gas (see figure <ref> for depictions of such in-spirals). The CE phase
could result in either an ejection of the gaseous envelope before
the companion arrives at the core, or in a final merging
of the companion with the compact core. The former occurs
when the evolved stellar envelope is not sufficiently massive and
dense compared with the stellar companion, and vice versa.
Here we show that when the stellar companion is compact, and the core is also sufficiently compact, CE
evolution leads to various possible CO-core mergers with
detectable GW signals (see 
for an initial study for neutron star companions). Note that significant fractions of the envelope may be ejected on much longer non-dynamical timescales <cit.>; this, however is always preceded by the dynamical phase.


The in-spiral of a compact object proceeds on a dynamical
timescale when the dynamical friction due to the CE is sufficiently
strong. The closest approach accessible for a CO-core binary
occurs when the CO reaches the tidal radius of
the core r_t∼(M_CO/M_core)^1/3 R_core,
at which point the latter is disrupted tidally. To characterise the GW-detectability
of such binary mergers, one needs to derive the strain and frequency
of the GW emission of such binaries close to the final radius. One also needs to account for the evolution due to the interaction
with the gaseous envelope. This process is special, in-so-far as the
driving force behind the in-spiral is not energy-loss to gravitational
waves, but rather interaction with an environment. As such the gravitational-wave
signal should reflect the properties of the environment, and allow
for potentially direct observations of
such otherwise-inaccessible environments/processes <cit.>.

To study such GW sources we model
the evolution as two point masses orbiting inside an envelope with
a given constant density profile, and we neglect the back-reaction
of the motion of the system on the envelope itself. We model
the in-spiral through the effects of gas dynamical friction (GDF) using an approach originally presented by <cit.>, rather than the approach taken previously
by <cit.>, and include 2.5 post-Newtonian corrections
to the gravitational interaction of the CO and the core. The resultant GW signal would depend
on the density profile of the CE, thereby enabling the utilization of gravitational waves to probe the properties of the interior of the red-giant/CE.

The in-spiral is driven by GDF and not by aerodynamic drag, due to the
high ratio between the mass of the in-spiralling CO and its geometric
cross-section. <cit.> found that the critical
transition radius between GDF-dominated evolution and aerodynamic-drag
dominated evolution is R_c∝ v_ rel/√(Gρ_m),
where v_ rel is the relative velocity between the CO and
the gas, and ρ_m is the density of the CO. The proportionality
constant depends on the dimensionless Reynolds and Mach numbers of
the flow, and is of order unity for most plausible situations.

In our case, we take an initial relative velocity of ∼100 km s^-1
(similar to the Keplerian velocity around a massive star at 1  AU).
For a typical WD density of ρ_ WD≈10^6  g cm^-3,
we get a critical radius of a few percent of the WD radius,
R_c≈0.05R_ WD, with R_ WD=0.01R_⊙.
For typical NS densities of ρ_ NS=10^14  g cm^-3,
the critical radius is around 4×10^3 cm, much less
than a NS radius. Therefore, for compact objects embedded in envelopes of giant stars, GDF dominates over
aerodynamic drag.

We begin by describing the set-up
of our models and the equations of motion, which we numerically integrate
to calculate the GW signatures of the CE gravitational-wave
(CEGW) sources, exploring various configurations of CE-binaries and
COs. We then discuss the properties of such CEGW sources
and the possibility of detecting them, and then we summarise.




§ MODEL DESCRIPTION


We work in the centre-of-mass frame of the binary, and neglect any back-reaction of the companion
on the envelope – we take the density of the envelope
to be constant in time, and `glued' to the core of the star, which
we model as a point mass [This approximation is more consistent for cases where the envelope
is significantly more massive than the companion (see below). In practice, the
CE evolution changes the structure of the envelope even when the envelope
is more massive.]. This system may be described as an effective one body, whose position
𝐫 describes the relative separation between the companion
and the core. Its equations of motion are

    𝐫̈=-G(M+m)/r^3𝐫-GM_env(r)/r^3𝐫-F(𝐫,𝐯)𝐯+,

where M_env(r) is the mass of the envelope inside a
sphere of radius r (excluding the core), M=M_core
is the core mass, and m=M_CO is the companion mass,
and `P.N.' denotes post-Newtonian terms. We further define M_tot=M+M_env(R),
where R is the radius of the envelope. F describes the effects
due to GDF, which model like <cit.>:

    F(𝐫,𝐯)=2π G^2mρ(𝐫)/v^3ln(1+ℳ/1-ℳe^-2ℳ),    ℳ<1
    ln(Λ^2-Λ^2/ℳ^2),    ℳ>1.
    ,

where ℳ=v/c_s is the Mach number, and c_s is
the local sound-speed of the envelope. The Coulomb logarithm lnΛ
is given by Λ=b_max/b_min <cit.>,
where b_min=maxGm/v^2,r_coll; r_coll
is the radius at which the compact object collides with the core. We take b_max = 2r, as in <cit.> rather than b_max = R_env, because the density outside r is a lot smaller than the density inside r.



 §.§ Time-Scales

During the most of the in-spiral v>c_s, and the dynamical friction force is well approximated
by

    F(𝐫,𝐯)∼ NG^2m^2ρ(r)/v^3,

where N is a dimensionless constant, proportional to the Coulomb
logarithm lnΛ. Let us try to get an order-of-magnitude estimate
on the decay time: as F_DF=-Fv∼ v^-2, we can treat the orbit
as approximately circular. Taking v^2∼ G(M+m+M_env(r))/r, gives a collapse time-scale of 
    τ_DF=μ v^3/NG^2m^2ρ̅∼μ(M+m+M_env(r))^3/2/Nr^3/2√(G)m^2ρ̅.

If we take ρ=ρ̅, we need to take r as the radius
at which the density is equal to the average density, which we denote
by r=CR, where C<1 describes the concentration of the system.
Setting M_env(r)≈ M_env, we have

    τ_DF    =4πμ(M+m+M_env)^3/2a^3/2/3C^3/2N√(G)m^2M_env
        =0.1 year(M+m+M_env/2M_⊙)^3/2(R/100R_⊙)^3/2
       ×(M_⊙/m)^2(M_⊙/M_env)(0.1/C)^3/2(4ln10/N).





 §.§ Wave-Forms

During the in-spiral the binary emits gravitational waves due
to a changing quadrupole moment. As r decreases, the orbital
acceleration increases, and so does the GW-amplitude, right up to
the final merger. The leading-order GW-strain h_ab(t) is given by the quadrupole
formula:

    h_ab^TT(t)=2G/Dc^4Q̈_ab^TT(t_ret),

where TT denotes the transverse-traceless gauge, and an upper TT
on Q implies a projection on the direction of observation; Q_ab=M^ab-δ_abM^kk/3
is the quadrupole moment, and t_ret=t-D/c is the retarded
time, with D being source-to-detector distance.
Even though one of the bodies is not point-like, it is spherical,
so the mass moment is

    M^ij=mx_1^i(t)x_1^j(t)+M_totx_2^i(t)x_2^j(t)+1/3δ^ijC,

where C is a higher moment of the density ρ, and 𝐱_1,2
is the position of the centre of the CO/giant. The second term does not contribute to Q_ab, so we can ignore it. The contribution
of the evolved star is thus equal to that of a point mass, and we may therefore
use the formula for M^ij of a binary system in the centre-of-mass frame (e.g. given by equation (3.72) of ). The
solution to equation (<ref>) defines the wave-form, so
the problem of calculating the wave-form reduces to solving the
equations of motion.



§ NUMERICAL CALCULATIONS


To calculate the CEGW signatures, we integrate equation (<ref>)
numerically using a Runge-Kutta integrator (MATLAB's ode113). We added relativistic corrections to the motion of the core
and the companion up to 2.5PN (to account for GW dissipation)[These have significant contributions only when they are extremely
close to each other, so the post-Newtonian terms do not include the
gravity of the gas in the envelope.], as given by <cit.>. The orbit can still be described
by a point particle with mass μ=mM_tot/(m+M_tot),
moving under the influence of the gravitational field a mass M+m and inside the envelope's potential. Here
we considered a few combinations of m,M and M_env
(see table <ref> and figure <ref>).

The density profiles of the stars were obtained from stellar
evolution models, produced by the MESA code <cit.>, evolved from initial masses
of 15, 8 and 5 M_⊙, until they reached 110R_⊙; the stars retained almost all of their original mass. Examples of orbits are shown in figure <ref>.

For each of these models we calculate the emitted waves, with m=0.6M_⊙ representing a WD companion,
and m=1.4M_⊙ – a NS. The detector is assumed to be at a distance of
10 kpc. The resulting wave-forms are shown
in figure <ref>. All in-spirals show a characteristic
evolution beginning with regular low-amplitude oscillations at the early phases, which then gradually increase
in frequency and amplitude down to the final plunge accompanied by
a high amplitude burst.





§ DETECTABILITY


Let us proceed to consider the detectability of such CEGWs: the size of the core
is comparable to (or larger than) that of WDs, and the orbital frequency
of the binary before its final merger is therefore lower than the
detection range of aLIGO. However, as we show below, such
GW sources are detectable by next-generation space-based GW-detectors,
such as LISA.

Whether the predicted signal is detected or not is determined by its characteristic strain, which is related to the signal-to-noise
ratio (SNR). If S_n(f) is the noise power-spectrum density of
a detector, and h is the signal (without noise) then the
SNR is given by <cit.>

    (S/N)^2=4∫_0^∞h̃(f)^2/S_n(f)df=∫_0^∞h_c^2(f)/h_n^2(f)df,

where h̃(f)=∫_-∞^∞e^-2πifth(t)dt.
The characteristic strain is h_c(f)=2fh̃(f); so,
if h_n=√(fS_n(f)) signifies the sensitivity of the detector,
the area between the characteristic strain and the sensitivity curve
in logarithmic scale is the SNR. These are calculated using a discrete
fast Fourier transform, applying a Tukey window <cit.> to alleviate spectral leakage.

The characteristic strains of the cases we considered here are presented at the top row of figure <ref>. SNRs for the three detectors: LISA, DECIGO and BBO, are presented
in table <ref>. Note, that contrary to a gas-free GW in-spiral, h_c(f) is not proportional to a power of f – it is not a straight line in logarithmic scale.







§ EXTENDED ENVELOPES



As mentioned above, the models described
here are simplified and neglect the evolution of the envelope
itself due to the in-spiral. In reality, gravitational energy is deposited into the envelope by gas dynamical friction, which inflates the envelope and results in its partial
or full unbinding <cit.>. The in-spiral is therefore
expected to occur in more extended density profiles than the initial
one, and therefore the density close to
the core should be lower.
In this scenario the magnitude of the GDF force on the companion should be smaller, and the amount of time it spends
very close to the core, emitting the strongest GWs, is increased.
In other words, our SNR calculations serve as lower limits to the
expected GW signatures for such CE-in-spirals, whilst being bounded
by an upper limit – the expected signal from a gas-free in-spiral of two COs of similar masses (see the violet line in figure <ref>).
As a simplified approach we gauge the
magnitude of the extended-envelope effect using a simple, inflated
envelope model, where we again integrate equation (<ref>),
but we now consider a density profile which is linearly dilated, with

    ρ'(r) = ρ(r-R_core/λ+R_core)λ^-3
       
      c_s'(r)=c_s(r-R_core/λ+R_core)λ^-1/2,

etc. A plausible scale-factor given results from hydrodynamical simulations
of CE-evolution from the literature <cit.> is λ≈ 10;
the characteristic strains we find for the simplified extended envelopes
are shown in the second row of figure <ref>
and the total SNRs are given in table <ref>.


As is evident from figure <ref> and table
<ref> CE-in-spirals occurring in the Galaxy could potentially
be observed by LISA, DECIGO or BBO, where the
latter might even be sensitive to extra-Galactic sources up to
distances of a few Mpc.

To test our semi-analytical calculations, we perform a hydrodynamical
simulation for the case M+M_env = 8M_⊙, m = 0.6M_⊙. Our
simulation is run via the AMUSE framework <cit.>,
which allows us to combine different type of codes. We use the same density
profile for the 8M_⊙ red giant as above, and convert it to a
3-dimensional smoothed-particle-hydrodynamics (SPH) model with 250000 gas
particles. The core is represented by a point mass (dark-matter particle) to
avoid numerical issues and unnecessary calculation in this dense region,
whose internal changes are irrelevant for our purposes. Since the MESA model
is one-dimensional and is described by different evolution equations, we run
a relaxation phase of the SPH model alone first, for about 1000 days, while
we allow only small changes in the centre of mass velocity and position (a
restriction which is lifted later on). After we have a relaxed model, we use
it to simulate the actual CE phase with the companion which is also modelled
as a dark-matter particle. The simulation runs with the SPH code Gadget 2
<cit.> until the core and the companion become closer than the
smoothing radius. A more thorough explanation of the simulation is presented
in <cit.>.

Figure <ref> displays a
comparison between the core-companion distance obtained from the SPH model,
and with our approximation, taking λ = 11. Also shown is a plot of
the characteristic strains from both cases. The comparison between our
simplified approach and the SPH model is physically meaningful only down to
the smoothing parameter, as the SPH model breaks down at lesser radii; in
this range, the two agree, thereby justifying our model even at smaller
distances which the SPH simulation cannot probe, too. Figure <ref> displays the orbits in the two models.



§ SUMMARY

In this paper we explored gravitational-wave emission by the
in-spiral of a compact object in an evolved common-envelope binary,
leading to the final merger of the compact object with the stellar
core of the evolved star. As we show the in-spirals differ significantly from gravitational-wave in-spirals of binary isolated compact objects;
common-envelope in-spiral are dominated by a gas dynamical friction interaction
with the envelope and not by dissipation from the gravitational-wave emission
itself. Such an evolution changes the acceleration of the binary and
yields gravitational-wave signatures with a unique
frequency evolution. In particular such gravitational-wave sources could be potentially
observable by next-generation gravitational-wave detectors such as LISA/DECIGO/BBO,
and their unique frequency evolution could provide not only a smoking-gun
signature for their origin, but also enable the use of gravitational-wave detectors for mapping the interiors of evolved stars,
otherwise inaccessible through electromagnetic observations.

The common-envelope phase is also typically accompanied by explosive
electromagnetic transients. The in-spiral itself might give rise
to a luminous flare, possibly resembling stellar mergers such as V1309 Scorpii <cit.>. The merger of a white dwarf (or neutron star) and the degenerate core
might cause a thermonuclear explosion, potentially observable as
peculiar luminous (sub-luminous) type II/Ib supernova (given the large
amounts of hydrogen and helium in the envelopes). Finally,
the in-spiral of a neutron star into the center of the evolved star may produce
a Thorne-Żytkow object <cit.>. Systems with heavier compact companions, such as black holes, may generate even stronger signals. Even though there is expected to be little accretion onto such small compact objects as we consider here <cit.>, such accretion would generate gravitational waves at frequencies that are much higher than the characteristic ones for space-based detectors <cit.>, leading to a possibility of observing the same process in many detectors at the same time.

Estimates of the rates of mergers of compact objects with degenerate
cores of evolved stars leading to type Ia supernovæsuggest that they should
occur at rates of 10-100% of the observationally-inferred rates
of type Ia SNe <cit.>; the rates of CEGW sources could be higher. <cit.>
estimate a rate of 0.002 yr^-1 for the formation of
Thorne-Żytkow objects, based on <cit.>, implying
an overall rate of one CEGW event per a few centuries in our Galaxy. Although less likely to be observed by LISA given the expected Galactic rates, the DECIGO/BBO observatories would enable observing extra-Galactic CEGWs and potentially detect a few such events.



§ ACKNOWLEDGMENTS


We wish to thank Andei P. Igoshev and Yael Raveh for helpful discussions. YBG and EG acknowledge support from the Technion
Jacobs scholarship; and VD acknowledges support by the Israel Science Foundation (grant no. 1395/16).

mnras






