MNRAS
496,
5482–5502 (2020)
doi:10.1093/mnras/staa1858
Advance Access publication 2020 June 26
Non-linear dynamical tides in white dwarf binaries
Hang Yu
,
1
,
2
‹
Nevin N. Weinberg
1
and Jim Fuller
2
1
Department of Physics, and MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2
TAPIR, Walter Burke Institute for Theoretical Physics, California Institute of Technology, Mailcode 350-17, Pasadena, CA 91125, USA
Accepted 2020 June 22. Received 2020 June 22; in original form 2020 May 5
ABSTRACT
Compact white dwarf (WD) binaries are important sources for space-based gravitational-wave (GW) observatories, and an
increasing number of them are being identified by surveys like Extremely Low Mass (ELM) and Zwicky Transient Facility
(ZTF). We study the effects of non-linear dynamical tides in such binaries. We focus on the global three-mode parametric
instability and show that it has a much lower threshold energy than the local wave-breaking condition studied previously. By
integrating networks of coupled modes, we calculate the tidal dissipation rate as a function of orbital period. We construct
phenomenological models that match these numerical results and use them to evaluate the spin and luminosity evolution of a
WD binary. While in linear theory the WD’s spin frequency can lock to the orbital frequency, we find that such a lock cannot
be maintained when non-linear effects are taken into account. Instead, as the orbit decays, the spin and orbit go in and out of
synchronization. Each time they go out of synchronization, there is a brief but significant dip in the tidal heating rate. While most
WDs in compact binaries should have luminosities that are similar to previous traveling-wave estimates, a few per cent should
be about 10 times dimmer because they reside in heating rate dips. This offers a potential explanation for the low luminosity of
the CO WD in J0651. Lastly, we consider the impact of tides on the GW signal and show that the
Laser Interferometer Space
Antenna
(
LISA
) and
TianGO
can constrain the WD’s moment of inertia to better than 1 per cent for centi-Hz systems.
Key words:
gravitational waves – instabilities – (
stars
:) binaries (
including multiple
): close – stars: oscillations – white dwarfs.
1 INTRODUCTION
As binary white dwarfs (WDs) with short orbital periods inspiral due
to the emission of gravitational waves (GWs), they can evolve into a
variety of interesting systems, including AM CVn stars (Nelemans
et al.
2001
), R Cor Bor stars (Clayton
2012
), and rapidly rotating
magnetic WDs (Ferrario, de Martino & G
̈
ansicke
2015
). Merging
WDs may also explode as Type Ia supernovae (Iben & Tutukov
1984
; Webbink
1984
; Toonen, Nelemans & Portegies Zwart
2012
;
Polin, Nugent & Kasen
2019b
) or in other types of luminous
thermonuclear events (Shen et al.
2018
; Polin, Nugent & Kasen
2019a
). Compact WD binaries emit GWs with frequencies of
≈
1–
100 mHz, which makes them prominent sources for proposed space-
based GW observatories such as the
Laser Interferometer Space
Antenna
(
LISA
; Amaro-Seoane et al.
2017
),
TianQin
(Luo et al.
2016
), and
TianGO
(Kuns et al.
2019
).
The tidal interaction between the binary components spins them
up and heats their interiors. As they inspiral, the tide becomes pro-
gressively stronger and eventually their spin frequency nearly equals
the orbital frequency. However, they never become perfectly syn-
chronous because of the continual GW-induced orbital decay. The de-
gree of spin asynchronicity affects the tidal heating rate and luminos-
ity of the WDs (Iben, Tutukov & Fedorova
1998
; Fuller & Lai
2012a
,
2013
; Piro
2019
) and the outcome of their potential merger (Raskin
et al.
2012
; Dan et al.
2014
; Fenn, Plewa & Gawryszczak
2016
).
E-mail:
hangyu@caltech.edu
The dominant mechanism of tidal dissipation is most likely the
excitation of internal gravity waves, either in the form of standing
waves (i.e. g-modes; Fuller & Lai
2011
; Burkart et al.
2013
), or
traveling waves (Fuller & Lai
2012a
,
b
,
2013
,
2014
). As we will
show, for orbital periods between approximately 10 and 150 min,
which describe many of the observed WD binaries, the resonant g-
modes excited by the tide have such large amplitudes that they cannot
be considered small, linear perturbations to the background star. On
the other hand, the amplitudes are not so large that the modes break
due to strong non-linearities. The tidal dynamics and dissipation
in this intermediate, weakly non-linear regime are complicated and
depend on details of the non-linear coupling between g-modes driven
directly by the tide and the sea of secondary modes they excite.
In this paper, we apply the weakly non-linear tidal formalism
developed in Weinberg et al. (
2012
) to study tides in WD binaries.
Our study fills the gap between those that assume the excited modes
are linear standing waves (e.g. Fuller & Lai
2011
; Burkart et al.
2013
) and those that assume they break and form strongly non-linear
traveling waves (Fuller & Lai
2012a
,
b
,
2013
,
2014
). In Section 2,
we present the background WD model we use throughout much of
our analysis. In Section 3, we describe the mode coupling and tidal
driving equations that govern the mode dynamics, and in Section 4,
we describe our numerical method for solving these equations. In
Section 5, we present our solutions of the mode dynamics and show
how tidal dissipation and synchronization varies with orbital period
in the weakly non-linear regime. We also compare our results with
the previous studies that assumed the tide was either linear or strongly
non-linear. In Section 6, we describe the observable electromagnetic
C
2020 The Author(s)
Published by Oxford University Press on behalf of the Royal Astronomical Society
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Non-linear tides in white dwarf binaries
5483
Table 1.
The mass
M
, radius
R
, effective temperature
T
eff
,
and moment of inertia
I
WD
of our WD model. We will often
express results in terms of the primary’s natural units of
energy
E
0
≡
GM
2
/
R
=
1.10
×
10
50
erg and frequency
ω
0
≡
√
GM/R
3
=
0
.
346 rad s
−
1
.
MRT
eff
I
WD
0.6 M
8.75
×
10
8
cm
9000 K
0.257
MR
2
Figure 1.
Propagation diagram (top panel) and composition profile (bottom
panel) of our WD model from the centre to the surface. Note that we have
oriented the bottom
x
-axis such that the radius increases to the right.
and GW signatures of the tidal interaction, including the tidal heating
luminosities, GW phase shifts, and projected constraints on the WD
moment of inertia. In Section 7, we summarize our key results and
conclude.
2 BACKGROUND MODEL
We u s e
MESA
(version 10398; Paxton et al.
2011
,
2013
,
2015
,
2018
)
to construct a WD model, whose key parameters are summarized
in Table
1
. To construct this model, we adopt parameters similar to
those used by Timmes et al. (
2018
). Specifically, we start with a pre-
main-sequence star with an initial mass of 2
.
8M
and metallicity
Z
=
0.02 and let it evolve to a CO WD with mass
M
=
0
.
6M
and
effective temperature
T
eff
=
9000 K. We include element diffusion,
semiconvection, and thermohaline mixing throughout the evolution.
We u s e
GYRE
(Townsend & Teitler
2013
; Townsend, Goldstein &
Zweibel
2018
) to compute the model’s eigenmodes and construct
our mode networks.
In the upper panel of Fig.
1
, we show the propagation diagram of
our WD model. The solid line is the buoyancy frequency
N
,where
N
2
=
g
2
1
c
2
e
−
1
c
2
s
,
(1)
c
2
e
=
d
P/
d
ρ
is the equilibrium sound speed squared,
c
2
s
=
1
P/ρ
is the adiabatic sound speed squared, and
1
is the adiabatic index.
All other quantities have their usual meaning. The dashed line is the
Lamb frequency
S
l
for
l
=
2, where
S
2
l
=
l
(
l
+
1)
c
2
s
r
2
.
(2)
For the short-wavelength g-modes that comprise the dynamical tide,
the square of the radial wavenumber,
k
2
r
=
ω
2
c
2
s
S
2
l
ω
2
−
1
N
2
ω
2
−
1
,
(3)
where
ω
is the angular eigenfrequency of the mode. A g-mode
propagates where
k
2
r
>
0, i.e. in regions where
ω<
N
and
ω<
S
l
, and is evanescent where
k
2
r
<
0.
The lower panel of Fig.
1
shows the composition profile of our
model. As is typical of stars supported by degeneracy pressure, the
buoyancy is due largely to composition gradients, with peaks in
N
associated with sharp transitions in the internal composition.
3 FORMALISM
3.1 Equation of motion
Consider a primary star of mass
M
and a secondary star of mass
M
and choose a coordinate system whose origin is at the centre of the
primary and corotates with it. We assume that the orbit is circular
and that the spin angular momentum of the primary is aligned with
the orbital angular momentum. For simplicity, we do not account
for the effect of rotation on the mode dynamics except through the
Doppler shift of the tidal driving frequency. The equation of motion
governing the Lagrangian displacement field
ξ
(
r
,t
) of a perturbed
fluid element at location
r
at time
t
is then (see e.g. Schenk et al.
2002
; Weinberg et al.
2012
, hereafter
WAQ B 1 2
)
ρ
̈
ξ
=
f
1
[
ξ
]
+
f
2
[
ξ
,
ξ
]
+
ρ
a
tide
,
(4)
where
f
1
and
f
2
represent the linear and leading-order non-linear
internal restoring forces, and
a
tide
=−∇
U
−
(
ξ
·∇
)
∇
U
(5)
is the tidal acceleration. The tidal potential can be expanded as
U
(
r
,t
)
=−
l
≥
2
,m
W
lm
GM
D
(
t
)
r
D
(
t
)
l
Y
lm
(
θ,φ
)e
−
i
m
(
orb
−
s
)
t
,
(6)
where
Y
lm
is the spherical harmonic function, and
D
,
orb
,and
s
are the orbital separation, the orbital angular frequency, and the spin
frequency of the primary, respectively. We focus on the leading-order
quadrupolar (
l
=
2) tide, whose non-vanishing
W
lm
coefficients are
W
2
±
2
=
√
3
π
/
10 and
W
20
=−
√
π
/
5. It is useful to define
=
M
M
R
D
3
=
y
1
+
y
orb
ω
0
2
,
(7)
where
y
=
M
/
M
is the mass ratio. The quantity
characterizes the
overall tidal strength and will be useful when we want to distinguish
the system’s dependence on the tidal strength from its dependence
on the driving frequency 2(
orb
−
s
).
In order to solve equation (4), we expand the six-dimensional
phase-space vector as
⎡
⎣
ξ
(
r
,t
)
̇
ξ
(
r
,t
)
⎤
⎦
=
q
a
(
t
)
⎡
⎣
ξ
a
(
r
)
−
i
ω
a
ξ
a
(
r
)
⎤
⎦
,
(8)
where
q
a
(
t
),
ω
a
,and
ξ
a
(
r
), are the amplitude, frequency, and
displacement of an eigenmode labelled by subscript
a
. The frequency
MNRAS
496,
5482–5502 (2020)
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