Tidal Spin-up of Subdwarf B Stars
Linhao Ma
(
马
林
昊
)
1
,
2
and Jim Fuller
1
1
TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA;
linhaoma@princeton.edu
2
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Received 2024 May 1; revised 2024 August 28; accepted 2024 September 3; published 2024 October 21
Abstract
Hot subdwarf B
(
sdB
)
stars are stripped helium-burning stars that are often found in close binaries, where they
experience strong tidal interactions. The dissipation of tidally excited gravity waves alters their rotational evolution
throughout the sdB lifetime. While many sdB binaries have well-measured rotational and orbital frequencies, there
have been few theoretical efforts to accurately calculate the tidal torque produced by gravity waves. In this work,
we directly calculate the tidal excitation of internal gravity waves in realistic sdB stellar models and integrate the
coupled spin
–
orbit evolution of sdB binaries. We
fi
nd that for canonical sdB
(
M
sdB
=
0.47
M
e
)
binaries, the
transitional orbital period below which they could reach tidal synchronization in the sdB lifetime is
∼
0.2 day, with
weak dependence on the companion masses. For low-mass sdBs
(
M
sdB
=
0.37
M
e
)
formed from more massive
progenitor stars, the transitional orbital period becomes
∼
0.15 day. These values are very similar to the tidal
synchronization boundary
(
∼
0.2 day
)
evident from observations. We discuss the dependence of tidal torques on
stellar radii, and we make predictions for the rapidly rotating white dwarfs formed from synchronized sdB binaries.
Uni
fi
ed Astronomy Thesaurus concepts:
B subdwarf stars
(
129
)
;
Stellar oscillations
(
1617
)
;
Stellar rotation
(
1629
)
;
Tidal interaction
(
1699
)
1. Introduction
Hot subdwarf B
(
sdB
)
stars,
fi
rst observed by M. L. Humason
&F.Zwicky
(
1947
)
, are compact and faint stars with surface
temperatures between 20,000 and 40,000 K and masses below
0.5
M
e
(
U. Heber
2009
,
2016
;X.Zhangetal.
2009
)
. These stars
have helium-burning cores and thin envelopes
(
Y. Götberg et al.
2018
)
, and they are thought to be stripped cores of helium-
burning red giants, whose envelopes were previously lost due to
some binary interactions
(
Z. Han et al.
2002
,
2003
; I. Pelisoli
et al.
2020
)
. About half of the observed sdB systems are found in
binaries
(
P. Maxted et al.
2001
;R.Napiwotzkietal.
2004
;
C. M. Copperwheat et al.
2011
; S. Geier et al.
2011
)
, with many
of them in close
(
P
orb
10 days
)
orbits. This suggests that a prior
common envelope phase might be responsible for the ejection of
their envelopes, as well as the inspirals of their companions to
their current orbital con
fi
guration
(
M. U. Kruckow et al.
2021
)
.
For binaries with short orbital pe
riods, tidal interactions can
shape both the migration of their orbits and the spin evolution of
individual stars. Historically, sdB binaries are sometimes assumed
to have reached tidal synchroni
zation, such that their orbital
parameters can be derived from measurements of sdB rotation
rates
(
see, e.g., R. P. Kudritzki & K. P. Simon
1978
; S. Geier et al.
2010
)
, even if the companions
(
typically white dwarfs, WDs, or
Mdwarfs,dMs
)
are too faint to be seen. However, this
assumption has been seriously challenged by observations from
the past decade, especially those with high-precision measure-
ments with TESS and Kepler
/
K2, where both spin and orbital
frequencies are available
(
see the summary of observation results
in Section
4.2
)
. These studies have found that sdB binaries with
orbital periods as short as
∼
7 hr are not always synchronized
(
R. Silvotti et al.
2022
)
. Nevertheless, these emerging new data
provide an excellent opportunity to test the theoretical modeling of
tidal interactions in sdB binaries.
For stars with convective cores
and radiative envelopes like
sdBs, tidally excited gravity wa
ves in their envelopes are thought
to be the most ef
fi
cient form of tidal interaction
(
J. P. Zahn
1977
)
.
These waves are excited by the tidal potential from the
companion, and when they propagate through the stellar interior,
the
fl
uid damps via radiative diffusion, exerting effective tidal
torques that transfer the angular momentum from the orbit to the
stellar spin
(
J. P. Zahn
1975
)
. This classical theory of dynamical
tides was originally proposed for massive stars, and several works
have calculated the tidal evolution of sdB binaries with this model
or its adaptations
(
S. Geier et al.
2010
; H. Pablo et al.
2012
;
H. P. Preece et al.
2018
)
.
However, L. Ma & J. Fuller
(
2023
)
recently pointed out that
an assumption in Zahn
’
s model may not be true for stripped
helium-burning stars, like Wolf
–
Rayet stars in the case of
massive stars and sdBs in the case of low-mass stars. While
J. P. Zahn
(
1975
)
assumed that the waves are all ef
fi
ciently
damped when they propagate to the stellar surface, in these
stripped stars, they may be otherwise re
fl
ected and form
standing waves. This is particularly true for high-frequency
gravity waves excited by short-period orbits, with less ef
fi
cient
radiative damping in stellar envelopes. By direct calculations of
tidally excited oscillations with radiative damping, L. Ma &
J. Fuller
(
2023
)
showed that real tidal torques should have a
more complicated frequency dependence than the simple
power-law relation derived from Zahn
’
s model. This new
approach brings concerns for the existing predictions of sdB
rotation rates based on Zahn
’
s tidal calculation.
In this paper, we build sdB models and calculate their tidal
evolution with the method in L. Ma & J. Fuller
(
2023
)
.Wecarry
out direct calculations of stellar oscillations and their tidal
response, and we
fi
nd that standing waves indeed exist in these
sdBs. The tidal torques are hence different from Zahn
’
s, and our
results for sdB rotation rates are consistent with the observed
trends of tidal synchronization for these systems. The manuscript
The Astrophysical Journal,
975:1
(
14pp
)
, 2024 November 1
https:
//
doi.org
/
10.3847
/
1538-4357
/
ad7788
© 2024. The Author
(
s
)
. Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the
Creative Commons Attribution 4.0 licence
. Any further
distribution of this work must maintain attribution to the author
(
s
)
and the title
of the work, journal citation and DOI.
1
is organized as follows. In Section
2
, we describe the physics of
sdB spin-up from tidally excited
g
-mode oscillations; in
Section
3.1
, we describe our model setup; and in Sections
3.2
and
3.3
, we describe how we calculate the oscillation modes and
the binary evolution. We show the results for tidal torque
calculations in Section
4.1
and the spin
–
orbit evolution of sdB
binaries in Section
4.2
. We discuss the limitations of our models
and the various related physical processes in Section
5
.We
fi
nally conclude in Section
6
.
2. Tidal Physics
For sdB binaries, the tidal dissipation inside the sdB star is
thought to be responsible for its tidal evolution. In this picture,
the tides are excited by the tidal gravity potential from the
companion star, which is usually a dM or a WD. When the
binary orbit is faster than the stellar spin, the tidal dissipation
transfers energy and angular momentum from the orbit to
the star.
For rotating sdB stars with convective cores and radiative
envelopes, three possible tidal dissipation mechanisms could be
at work, namely,
(
1
)
the turbulent viscous dissipation of
equilibrium tidal bulges in the stellar core
(
P. P. Eggleton et al.
1998
; P. Eggleton
2006
)
,
(
2
)
the turbulent viscous damping of
tidally excited inertial waves restored by the Coriolis force
(
G. I. Ogilvie
2013
; S. C. Wu et al.
2024
)
, and
(
3
)
the radiative
damping of tidally excited gravity waves in the envelope
(
J. P. Zahn
1975
,
1977
)
. Studies have found that the
fi
rst
scenario is usually inef
fi
cient for close-in subdwarf binaries, as
the orbital periods might be shorter than the convective
turnover time in the core, such that convective viscous
dissipation is suppressed
(
H. P. Preece et al.
2019
)
. The
second scenario is likely unimportant for the same reason, and
the small convective core sizes of sdB stars further suppress
inertial wave dissipation. We hence focus on the last scenario
to be the dominant process for tidal evolution.
We sketch the physics picture of radiative dissipation of
tidally excited gravity waves in Figure
1
. Gravity waves,
propagating in the radiative envelope of the star, can be tidally
excited by the orbit of the companion. In the outer envelope
with large thermal diffusion, these waves damp partially or
fully by radiative diffusion, releasing their energy and angular
momentum, and hence exert a tidal torque on the star. The
orbital angular momentum is thus transferred to the stellar spin.
In previous studies, radiative dissipation is often assumed to be
ef
fi
cient so that these gravity waves damp completely in the
radiative envelope
(
J. P. Zahn
1975
)
, while in principle, they
might re
fl
ect back and instead form standing waves, i.e.,
oscillation modes
(
L. Ma & J. Fuller
2023
)
. Hence, a realistic
estimate of tidal torques requires calculation of individual
stellar oscillation modes.
For an aligned and circular orbit, the tidal torque for a tidally
excited oscillation mode
α
is given by
(
L. Ma & J. Fuller
2023
)
∣∣
()
()
()
⎛
⎝
⎞
⎠
mqMRWQ
R
a
,1
lm
l
2
1
1
2
2
f
2
f
2
2
1
21
t
wg
w
ww g
=-
-+
a
aa
a
a
a
+
where
ω
α
and
γ
α
are the mode frequency and growth rate
(
with
γ
α
<
0 for damped modes and the corresponding
τ
α
>
0
)
,
ω
f
=
m
(
Ω
orb
−
Ω
spin
)
is the tidal forcing frequency
(
measured in
the frame corotating with the sdB
)
,and
Ω
spin
is the sdB
’
s angular
rotation frequency.
M
1
and
R
1
are the mass and radius of the sdB,
q
=
M
2
/
M
1
is the mass ratio of the companion to the sdB, and
a
and
Ω
orb
are the semimajor axis and the angular frequency of the
orbit.
l
and
m
are the mode
’
s angular and azimuthal wavenumbers,
and
W
lm
is an expansion coef
fi
cient of the tidal potential.
()∣()
Q
GR
rY
l
l
lm
1
32
xw
ºáñ
a
a
a
+
is the dimensionless overlap
integral describing the spatial c
oupling between the mode and the
tidal potential, where
ξ
α
is the displacement vector of mode
α
,
normalized by
∣·
*
dV M R
star
1
1
2
ò
xx x xr
á
ñº
=
aa
a
a
. In practice,
Q
α
is calculated by the relation
()(
)
Q
lR
21 4
2
1
2
dpw
=- + F
aa
a
(
J. Fuller
2017
)
,where
δ
Φ
α
is the surface gravity potential
perturbation. For weak damping
(
γ
α
<
ω
α
)
, the excitation of
individual oscillation modes is independent from each other, such
that the total tidal torque can be expressed as
()
.2
tide
å
tt
=
a
a
Hence, by solving for the internal oscillation modes
(
with
ω
α
,
γ
α
, and
a
)
inside the sdB, we are able to calculate the
torque and the angular momentum transfer rate, given a
companion mass and orbit. We stress that there are
no
free
parameters in estimating the strength of tidal torques with this
method.
Figure 1.
A sketch of the physics of sdB tidal spin-up. Gravity waves, propagating in the radiative envelope, can be tidally excited by the gravity from the orbit
ing
companion. In the hydrogen-rich outer envelope, the waves damp
(
either partially or fully
)
and deposit their angular momentum into the star, transferring angular
momentum from the orbit to the stellar spin. The color scale shows the hydrogen fraction in the radiative envelope.
2
The Astrophysical Journal,
975:1
(
14pp
)
, 2024 November 1
Ma & Fuller
3. Methods
We calculate the tidal evolution of sdB binaries with a
similar method developed for Wolf
–
Rayet black hole binaries
in L. Ma & J. Fuller
(
2023
)
.We
fi
rst build realistic single
evolving sdB models throughout their helium-burning lifetime
(
Section
3.1
)
. We then solve for stellar oscillations based on
these models to estimate the tidal torques
(
Section
3.2
)
. Finally,
we numerically integrate the coupled spin
–
orbit evolution of
sdB binaries with interpolation between the previously
calculated sdB models and tidal torques, with different choices
of initial binary parameters, i.e., the initial orbital periods and
companion masses
(
Section
3.3
)
.
3.1. Stellar Models
We build single sdB models with the MESA stellar evolution
code
(
r12778; B. Paxton et al.
2011
,
2013
,
2015
,
2018
,
2019
;
A. S. Jermyn et al.
2023
)
. We build two sdB models to
represent the two types of sdBs formed from progenitors of
different masses, summarized in Sections
3.1.1
and
3.1.2
.We
turn on element diffusion for
1
H,
4
He,
12
C,
14
N, and
16
O in the
MESA models to account for correct treatments of gravitational
settling and radiative acceleration for these atoms.
3
3.1.1. 0.47
M
e
Canonical sdB Model
This model represents the most abundant sdBs
(
“
canonical
”
sdBs
)
that are formed from low-mass
(
M
2
M
e
)
main-
sequence progenitor stars. When these stars evolve off the main
sequence, they start hydrogen shell burning, which deposits
helium into their helium core until they reach the tip of the red
giant branch
(
TRGB
)
when the helium core exceeds 0.46
M
e
.
At this moment, an off-center helium
fl
ash is triggered and the
helium burning propagates to the center of the core, while the
star loses most of its envelope through binary processes
(
e.g., a
common envelope event
)
, leaving a core-helium-burning sdB
star with a little of its envelope
(
∼
0.01
M
e
)
retained. sdB stars
formed this way are insensitive to the masses of their
progenitors and have universal masses of
≈
0.47
M
e
.We
establish this sdB model by evolving a 1.2
M
e
star from the
zero-age main-sequence
(
ZAMS
)
to the TRGB and then apply
an arti
fi
cial stellar wind to remove its envelope until the off-
center helium burning propagates to the stellar center. This
happens at the moment when the envelope mass reaches the
desired mass
(
see details in Section
3.1.3
)
due to the speci
fi
c
wind scaling factor we chose. The model then becomes a zero-
age canonical sdB model, and we evolve it until core helium
depletion.
3.1.2. 0.37
M
e
Low-mass sdB Model
Stars of
∼
2
–
3
M
e
can also ignite core helium burning
without forming a fully degenerate core at helium core masses
less than 0.46
M
e
. Hence, they usually form sdBs of lower
masses compared to canonical sdBs. To simulate this scenario,
we evolve a 2.7
M
e
star from the ZAMS to the TRGB and then
remove its envelope by a similar arti
fi
cial wind until its
envelope mass reaches the desired mass
(
see details in
Section
3.1.3
)
. The model then triggers central helium burning
as a zero-age sdB star. The sdB model we build this way has a
mass of 0.37
M
e
.
3.1.3. Envelope Mass Setup
sdBs are known to retain a small amount of hydrogen
envelope above their helium cores. The amount of hydrogen
can be constrained from their spectroscopic properties and is
found to be between 0.001 and 0.005
M
e
(
see, e.g., Figure 10
of T. Kupfer et al.
2015
)
. We hence adjust the arti
fi
cial winds
such that the stellar models start core helium burning
(
zero-age
sdB
)
when they have 10
−
3
M
e
hydrogen left. After that
moment, we turn off the stellar winds, as the envelope-stripping
phase is considered completed. We note that real sdB stars can
possibly retain more hydrogen than 10
−
3
M
e
. However, we
found many unstable stellar oscillations in sdB models with
more massive envelopes, and we are not able to calculate the
tidal dissipation of these modes with our current method. We
discuss the in
fl
uence of envelope masses and these unstable
modes in more detail in Section
5.1
.
3.1.4. Convective Core Boundary Setup
The excitation of gravity waves is sensitive to the size of the
convective core, which in turn can be sensitive to how its
boundary is treated in stellar evolution models. Unlike the
standard convective-overshooting paradigm that has been
established for main-sequence stars
(
see, e.g., the MIST
project; J. Choi et al.
2016
)
, overshooting parameters for stars
with a helium-burning convective core, like sdBs, are poorly
constrained. Nevertheless, asteroseismic measurements of core-
helium-burning stars suggest the existence of bigger cores
compared to theoretical modeling
(
T. Constantino et al.
2015
;
D. Bossini et al.
2017
; A. Noll et al.
2024
)
. We hence turn off
overshooting for our stellar models in the core-helium-burning
phase, instead applying the
“
predictive mixing
”
scheme for
convection
(
B. Paxton et al.
2018
)
. This allows for a steady
growth of the convective core during core helium burning,
more consistent with asteroseismic observations than other
choices for convective mixing. Furthermore, the predictive
mixing scheme helps prevent
“
breathing pulses
”
at late stages
of the core-helium-burning phase, which may split the
convective core to create small radiative zones, in which very
high-order gravity waves can be trapped. Breathing pulses have
been argued to be numerical artifacts
(
E. B. Bauer &
T. Kupfer
2021
)
, and we aim to avoid the associated dif
fi
culties
in computing gravity modes.
3.1.5. Rotation Setup
When a star evolves off the main sequence, its core contracts
and spins up, while its envelope expands and spins down. The
shear created between the core and the envelope could trigger
hydrodynamical and magnetohydrodynamical instabilities,
which transfer some of the core angular momentum to the
envelope, forming a slowly rotating stellar core
(
J. Fuller et al.
2019
; B. Tripathi et al.
2024
)
. Asteroseismic measurements of
red clump stars have shown that their core rotation periods are
typically
∼
100 days
(
B. Mosser et al.
2012
)
. Therefore, if these
stellar cores form sdBs, they should also be slowly rotating.
We applied the modi
fi
ed Taylor
–
Spruit torque as described
in J. Fuller et al.
(
2019
)
in our stellar models, and we found that
the stellar models at the end of the envelope-stripping phase
rotate slowly, with rotation rates insensitive to the initial
rotation at the ZAMS. The slow rotation rates are consistent
with the slow sdB rotation rates measured in wide binaries,
where tidal effects are not important
(
see, e.g., Section
4.2
)
.We
3
The MESA inlists are available on Zenodo under an open-source Creative
Commons Attribution license: doi:
10.5281
/
zenodo.13388526
.
3
The Astrophysical Journal,
975:1
(
14pp
)
, 2024 November 1
Ma & Fuller
hence set the sdB models to be nonrotating at the start of their
helium-burning phase. Since we only compute our spin
–
orbit
evolution by postprocessing of the stellar models, without
actually updating their rotation rates in MESA
(
see details in
Section
3.3
)
, the single sdB models remain nonrotating
throughout their lifetime.
3.2. Calculation of Oscillation Modes
We calculate the internal oscillation modes for the individual
snapshots of our sdB models with the GYRE stellar oscillation
code
(
R. H. D. Townsend & S. A. Teitler
2013
; R. H. D. Tow-
nsend et al.
2018
; J. Goldstein & R. H. D. Townsend
2020
)
.
We solve for nonadiabatic oscillations, which account for
radiative damping in the oscillation equations. We use the
second-order Magnus differential scheme, as it proves to be the
most reliable when dealing with low-frequency oscillations.
We specify our search to
l
=
m
=
2 modes, as this is the
dominant part of the tidal potential in aligned and circular
orbits, with the corresponding
W
310
22
p
=
.
4
When solving
for modes, we
fi
nd that the Brunt
–
Väisälä frequency
(
V. Väisälä
1925
)
pro
fi
les solved from MESA are sometimes not consistent
with the density and pressure pro
fi
les from the same model,
which may lead to inaccurate mode solutions. We hence slightly
adjust the stellar pro
fi
les with the process described in the
Appendix
for our stellar models. We checked that the change of
stellar structure due to this process is negligible.
In principle, we need to sum over
all
modes to get the total
tidal torque through Equation
(
2
)
. This is not practically
possible, as there are an in
fi
nite number of modes that could be
excited at all frequencies. Nevertheless, we note from
Equation
(
1
)
that for a given tidal forcing frequency
ω
f
,
typically only the few nearly resonant modes with
ω
α
close to
ω
f
contribute signi
fi
cant torques. Torques from other non-
resonant modes are usually negligible due to the
(
)
f
2
ww
-
a
term in the denominator of Equation
(
1
)
. We hence restricted
our mode solutions to a
fi
nite period range, namely, from 0.005
day to 0.5 day, and we hence found a
fi
nite number of modes.
We can then calculate the total tidal torques as long as the
forcing frequency
ω
f
is between 2
π
/
(
0.5 day
)
=
12.57 day
−
1
and 2
π
/
(
0.005 day
)
=
1257 day
−
1
. The
ω
f
calculated from our
spin
–
orbit evolution usually lies well within this range, except
for some systems that reach tidal synchronization, whose
ω
f
should approach 0. We hence stop the evolution when
ω
f
reaches the minimum mode frequency 12.57 day
−
1
.We
checked that our binary models reaching this condition are at
least at 80% synchronization, so we
de
fi
ne
all systems with
Ω
spin
0.8
Ω
orb
as tidally synchronized.
3.3. Spin
–
Orbit Evolution
We integrate the spin
–
orbit evolution of the sdB binaries
from the stellar models and tidal torques we computed. For
simplicity, we ignore the tidal dissipation in the companion star
(
see discussion in Section
5.7
)
. Assuming circular orbits, the
orbital angular momentum of the system is lost due to
gravitational-wave
(
GW
)
radiation and tides, while the sdB
receives spin from the tidal torque:
()
J
,3
orb
GW
tide
tt
=- -
()
J
,4
spin
tide
t
=
where
()()
Ga c MM M M
32 5
GW
72 5
1
2
2
2
12
t
=+
-
is the
effective torque by GW radiation
(
P. C. Peters
1964
)
. The
GW orbital decay timescale is then given by
() (()
)
()
()
TcqP
GMq
P
51
644
180 Myr
1 h
5
GW
513
orb
83
243 53
1
53
orb
83
p
=+
»
for an equal-mass sdB binary
(
q
=
1
)
with a 0.46
M
e
canonical
sdB star, comparable to the sdB lifetime of
∼
150 Myr for very
short-period systems. This means that GW orbital decay needs
to be included in the spin
–
orbit evolution.
As we expect ef
fi
cient angular momentum transport during
the core-helium-burning phase
(
J. Fuller et al.
2019
; J. Fuller &
W. Lu
2022
; see discussions in Section
3.1.5
)
, we assume rigid
rotation for the sdB star, with a uniform rotational frequency
Ω
spin
. We discuss the case of differential rotation in Section
5.3
.
The coupled spin
–
orbit evolution can then be integrated by
()
J
I
I
I
,6
spin
spin
spin
spin
spin
spin
W= -W
()
J
I
I
I
J
I
3,
7
orb
orb
orb
orb
orb
orb
orb
orb
W = -W =-
where
I
spin
is the moment of inertia of the sdB star,
I
orb
=
μ
a
2
is
the moment of inertia of the orbit, and we made use of Kepler
’
s
third law to simplify Equation
(
7
)
. This means that the spin of
the sdB star may also change due to the changes of its internal
structure and hence moment of inertia.
We make use of the integration machinery constructed in
L. Ma & J. Fuller
(
2023
)
, with the same interpolation method.
As sdB lifetime is typically longer than the Wolf
–
Rayet stars in
L. Ma & J. Fuller
(
2023
)
, we choose the integration time step to
be 0.1 times the values derived from the time step control
method described in L. Ma & J. Fuller
(
2023
)
. We integrate the
evolution from 1 yr after the start of the sdB helium-burning
phase, and we stop when the model depletes its core helium
(
de
fi
ned by the time when the core helium fraction drops below
1%
)
or when the system reaches
ω
f
12.57 day
−
1
(
see
Section
3.2
)
. The initial rotational period of sdBs is set to be
60 days, to match the observed values from single sdB stars
(
R. Silvotti et al.
2022
)
. While single sdBs may not represent a
fair sample of binary sdBs at birth, this assumed initial
rotational frequency is very low and never important for
systems that reach synchronization. As there are only limited
theoretical constraints on the initial binary parameters of sdBs,
we vary the companion masses uniformly between 0.1
M
e
and
0.8
M
e
and choose the initial orbital periods to be between 1
and 18 hr, aiming to cover the observed parameter space of
close-in sdB binaries
(
V. Schaffenroth et al.
2022
)
. We checked
that our results are robust against different choices of time-step
resolution.
4. Results
In this section, we show the results of our tidal torque
calculation and spin
–
orbit evolution.
4.1. Tidal Torque
In Figure
2
, we show our calculated tidal torque magni-
tude with its dependence on the period of tidal forcing
4
Example GYRE inputs are available on Zenodo under an open-source
Creative Commons Attribution license: doi:
10.5281
/
zenodo.13388526
.
4
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(
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)
, 2024 November 1
Ma & Fuller
(
P
f
≡
2
π
/
ω
f
)
. The calculation is based on the 0.47
M
e
sdB model
with a companion of 0.4
M
e
, when its central helium fraction
drops to 60%, and we assume that the sdB is nonrotating such that
ω
f
=
m
Ω
orb
. The torque generally has a complicated dependence
on the forcing period. By plotting the torque contributions from
each individual oscillation mode
J
α
, we see that this dependence is
caused by summing over the resonance peaks of many modes with
different frequencies. When the tidal forcing frequency gets close
to one of the mode frequencies, the
(
)
f
2
ww
-
a
term in
Equation
(
1
)
vanishes, and the total torque becomes dominated
by the strong resonance peak of that mode. Therefore, the
frequencies
/
periods of these peaks are the frequencies
/
periods of individual oscillation modes inside the star. These
peaks have a nearly uniform period spacing, a feature
expected for gravity
(
g
)
modes. In the right panel of
Figure
2
, we show some example eigenfunctions of these
modes, and we can see that they are indeed
g
-modes that
propagate inside the radiative envelope of the star.
Previous studies involving the tidal dissipation of gravity
waves in sdBs usually use Zahn
’
s model for dynamical tides
(
S. Geier et al.
2010
; H. Pablo et al.
2012
; H. P. Preece et al.
2018
)
, which assumes that these waves are ef
fi
ciently damped
as they reach the stellar surface
(
“
traveling wave limit
”
)
.We
see from the right panel of Figure
2
that this is clearly not
always the case for individual resolved stellar oscillations. The
blue line shows the eigenfunction of an example oscillation at a
short
(
P
α
0.14 day
)
period. We see that instead of ef
fi
ciently
damping near the stellar surface, the wave re
fl
ects back at the
stellar surface and forms a standing wave with nodes. This
means that Zahn
’
s picture may overestimate the mode damping
rate, hence the tidal dissipation.
For comparison, we show the tidal torques calculated based
on Zahn
’
s formalism with a modi
fi
ed formula given by
D. Kushnir et al.
(
2017
)
:
̄ ̄
()
⎜⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
GM
r
r
a
s
1, 8
Zahn
2
2
2
c
c
6
c
83 c
c
c
c
2
tb
r
r
r
r
=-
where
(
̄
)∣
∣
s
G
3
c
c
orb
spin
pr
=W-W
, while
r
c
,
ρ
c
, and
̄
c
r
are
the convective core radius, the density at the core boundary,
and the average density of the core, respectively.
β
2
is a
dimensionless coef
fi
cient solved from stellar structures, and
different main-sequence and Wolf
–
Rayet stellar models have
β
2
≈
1
(
D. Kushnir et al.
2017
)
. Its dependence on the tidal
forcing period is mostly the power-law term in
s
a
c
83
6
-
, as seen
in Figure
2
. This is clearly different from the resonance peak
dependence we
fi
nd from realistic mode calculations. We see
that at short periods, if the binary orbit has an off-resonance
tidal forcing period
(
i.e., not close to any stellar oscillation
modes
)
, the real tidal torque can be orders of magnitude lower
than Zahn
’
s prediction. On the other hand, if the orbit is on
resonance, the torque might be signi
fi
cantly larger than Zahn
’
s
prediction.
However, when the tidal forcing is on resonance with one of
the oscillation modes, the mode amplitude becomes so large
that it can trigger nonlinear wave dissipation. In this scenario,
the oscillation mode excites a number of nearby daughter and
granddaughter modes, and the overall damping rate by this sea
of coupled modes could be much larger than the radiative
damping of individual modes
(
A. J. Barker & G. I. Ogilvie
2011
; N. N. Weinberg et al.
2012
)
. The level of nonlinearity
can be estimated by the second-order nonlinear term in the
momentum equation
ξ
·
∇
ξ
∼
(
d
ξ
r
/
dr
)
ξ
: when
d
ξ
r
/
dr
>
1,
nonlinear effects become very strong.
With the above criterion, we showed where tidally excited
modes are strongly nonlinear in Figure
2
with dashed gray
Figure 2.
Left: the tidal torque calculated for a 0.47
M
e
sdB model with a companion of 0.4
M
e
, when the central helium fraction is 60%, assuming the sdB is
nonrotating. The thick lines show the total torque calculated by summing over contributions from individual tidally excited
g
-modes
(
thin lines
)
. We also show the
torque calculated from Zahn
’
s formalism for comparison. At short periods, the total tidal torque is dominated by resonance peaks from individual weakly damped
g
-modes, different from the power-law dependence on forcing period of Zahn
’
s formalism. However, on-resonance torques at short periods
(
thick dashed gray lines
)
may trigger nonlinear dissipation, and the real torques may not be as large as seen in the plot. At longer periods,
g
-modes are more ef
fi
ciently damped, and the torque
agrees better with Zahn
’
s model. Right: the mode eigenfunctions for an example weakly damped standing mode
(
blue line
)
and an example strongly damped mode
(
red line
)
in the left panel. The weakly damped mode has nodes in its eigenfunction, while the strongly damped mode ef
fi
ciently damps near the surface.
5
The Astrophysical Journal,
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Ma & Fuller
lines. We see that nonlinear effects mostly affect on-resonance
torques at short
(
P
tide
<
0.07 day
)
periods. For these torques,
the resonance peaks will be smoothed out when additional
nonlinear damping is present, and their actual magnitude may
not be as large as seen in Figure
2
.
As the star and the orbit evolve, both the oscillation mode
period
(
hence the location of the resonance peaks
)
and the tidal
forcing period change over time, so the system can quickly pass
through resonances
(
as long as resonance locking does not
happen; see discussions in Section
5.2
)
. Since the resonances
are narrow, the system spends more time out of resonance than
in resonance
(
i.e., with torques much weaker than Zahn
’
s
prediction
)
, and the accumulated angular momentum received
by the sdB star can still be less than the predictions from
Zahn
’
s theory.
At longer periods, gravity waves have larger wavenumbers
and are expected to damp more ef
fi
ciently with radiative
damping. With larger
γ
α
, the
2
g
a
term becomes more important
in the denominator of Equation
(
1
)
, smoothing out the
resonance peaks. We see in Figure
2
that this is exactly the
case for the tidal torques at long periods
(
P
f
>
0.14 day
)
, when
the individual modes damp so much that the resonance
structure gets smoothed out. In addition, the shape of the
example eigenfunction
(
red line
)
shown in the right panel of
Figure
2
becomes closer to a traveling wave that ef
fi
ciently
damps near the stellar surface, as Zahn
’
s formalism assumes.
The tidal torque
’
s dependence on tidal forcing period also gets
closer to Zahn
’
s power-law dependence, as expected. This
further shows that Zahn
’
s traveling wave picture is a limit case
of realistic tidal torques at long periods.
4.2. Tidal Synchronization
With the tidal torques calculated, we are able to integrate the
coupled spin
–
orbit evolution of our models. In Figure
3
,we
show the calculation for our 0.47
M
e
canonical sdB model and
0.37
M
e
low-mass sdB model with a
fi
xed companion mass of
0.4
M
e
, with different initial orbital periods. The rotational and
orbital periods are evaluated at the end of the spin
–
orbit
evolution
(
de
fi
ned in Section
3.3
)
, and some ultra-short-orbit
synchronized systems that reach
P
orb
<
0.02 day due to GW
orbital decay are not shown. We see that for the 0.47
M
e
sdB
model, all systems with orbital periods less than
∼
0.2 day reach
tidal synchronization, while for the 0.37
M
e
sdB model, the
synchronization period becomes
∼
0.15 day.
To compare with observations, we also plot the measured
rotational and orbital periods fo
r short-period sdB binaries in
Figure
3
. The different colors show the sdB rotation rates derived
from spectral line measurements
(
HS 0705
+
6700, H. Drechsel
et al.
2001
; CD-30 11223, S. Vennes et al.
2012
;SDSS
J162256.66
+
473051.1, V. Schaffenroth et al.
2014
; PTF1 J0823
+
0819, T. Kupfer et al.
2017
; PTF1 J011339.09
+
225739.1,
M. Wolz et al.
2018
;ZTFJ2130
+
4420, T. Kupfer et al.
2020a
;
ZTF J2055
+
4651, T. Kupfer et al.
2020b
; SDSS J082053.53
+
000843.4, V. Schaffenroth et al.
2021
;HWVir,E.M.Esmer
et al.
2021
; and EPIC 216747137, R. Silvotti et al.
2021
)
,
asteroseismic
p
-mode frequency splitting
(
NY Vir, S. Charpinet
et al.
2008
;Feige48,V.VanGrooteletal.
2008
; V1405 Ori,
M. D. Reed et al.
2020
; and HD 265435, I. Pelisoli et al.
2021
)
,or
g
-mode frequency splitting
(
KIC 11179657 and KIC 2991403,
H. Pablo et al.
2012
; FBS 1903
+
432, J. H. Telting et al.
2014
;
KIC 7664467, A. S. Baran et al.
2016
; EQ Psc and PHL 457,
A. S. Baran et al.
2019
; KIC 2438324, S. Sanjayan et al.
2022
;
TYC1 4544-2658-1, R. Silvotti et al.
2022
; and PG 0101
+
039,
X. Y. Ma et al.
2023
)
.
We see that all the observed systems with
P
orb
0.2 day are
close to tidal synchronization, while all but two systems
5
above
this period are not synchronized. This matches strikingly well
with the theoretical prediction from our sdB models. In
addition, the models with 0.3 day
P
orb
0.6 day are tidally
spun up to a rotational period of a few days, also consistent
with the observed partially synchronized systems in that period
range. Hence, our theoretical calculations agree with the
observation data.
Note that in Figure
3
, the spin and orbital frequencies are
shown at the end of the spin
–
orbit evolution, while measure-
ments for realistic systems usually occur when the sdB still
undergoes core helium burning. Therefore, it is illustrative to
show the tidal synchronization timescales calculated for our
models and to compare them with the sdB lifetime
T
EHB
. If the
synchronization time is shorter, then we expect that those
systems are likely to reach tidal synchronization to be
observed. As we consider systems with 80% synchronization
as synchronized
(
see discussions in Section
3.2
)
,wede
fi
ne the
tidal synchronization timescale throughout the whole evolution
as
̄
()
⎜⎟
⎧
⎨
⎪
⎩
⎪
⎛
⎝
⎞
⎠
T
t
T
, if synchronized
0.8
, if not
,9
sync
0.8
EHB
spin
orb
final
1
spin orb
º
W
W
WW=
-
where
t
is the stellar age since the start of sdB core helium
burning. We then run a grid of spin
–
orbit evolution for both of
our 0.47
M
e
and 0.37
M
e
sdB models, with initial orbital
periods of
(
1, 2, 3, 4, 5, 6, 7, 8
)
hr and companion masses of
(
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8
)
M
e
, and calculate their
tidal synchronization timescale de
fi
ned by Equation
(
9
)
.
We show the interpolated results for the ratio
̄
TT
sync EHB
based
on this grid in the whole parameter space in Figure
4
.Onthe
y
-axis, we also label the typical companion masses for sdB
+
dM
and sdB
+
WD binaries
(
V. Schaffenroth et al.
2022
)
.Weseethat
this ratio ranges from 10
−
1.5
to 10
1.5
, with weak dependence on
the masses of the companion star. This is an expected result from
our tidal torque formula
(
Equation
(
1
))
: the torque depends
on the mass ratio
(
hence the secondary mass
)
as
τ
α
∝
q
2
a
−
6
,where
((
)
)
(
)
a
GM M
q
1
12
orb
2
13
13
=+Wμ+
, hence
τ
α
∝
q
2
/
(
1
+
q
)
2
. For typical sdB binaries,
q
ranges from 0.3 to
1.5, and the corresponding torque scaling is maximally different
by only a factor of
∼
7. In contrast,
̄
TT
sync EHB
varies by a factor of
∼
10
3
over the period range shown in Figure
4
due to its strong
dependence on semimajor axis.
The blue colored regions in Figure
4
show the parameter
space where
̄
TT
1
sync EHB
<
, or where the binaries are expected
to be synchronized. We see that for 0.47
M
e
sdB binaries,
systems with initial orbital periods less than
∼
0.15
–
0.22 day
have synchronization timescales shorter than the 164 Myr sdB
lifetime, while for 0.37
M
e
sdBs, this period becomes
∼
0.10
–
0.17 day for its 444 Myr lifetime, depending on the
companion masses. As binaries at these orbital periods produce
5
The two exceptional systems are Feige 48 and V1405 Ori. For Feige 48,
there are some discrepancies in its measured orbital and rotational periods
(
see,
e.g., G. Fontaine et al.
2014
; H. P. Preece et al.
2018
; A. S. Baran et al.
2024
)
.
For V1405 Ori, there is some evidence that it might be a differentially rotating
sdB
(
M. D. Reed et al.
2020
)
, such that its envelope can be synchronized at
longer periods while its interior is not
(
see discussions in Section
5.3
)
.
6
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Ma & Fuller
weak GW emission, their orbital periods are nearly constant,
and these results con
fi
rm the critical orbital periods for
synchronization shown in Figure
3
.
We note from Figure
4
that, even though systems below the
synchronization periods can reach tidal synchronization in the
sdB lifetime, in most of the parameter space, their synchroniza-
tion timescales are not less than the corresponding sdB lifetime
by 1 order of magnitude. This is especially true for sdB
+
dM
binaries with
M
companion
0.3
M
e
, and we can see that
̄
TT
0.1
sync
EHB
<
is only achieved for those binaries in
P
orb
0.05 day
≈
1 hr orbits. This is consistent with the
fi
ndings
that sdB binaries with small companions
(
dMs or brown dwarfs
)
can be slightly subsynchronized e
ven at orbital periods less than
∼
2.5 hr
(
e.g., SDSS J162256.66
+
473051.1, a 64% synchronized
system with
P
orb
=
1.67 hr, V. Schaffenroth et al.
2014
;and
SDSS J082053.53
+
000843.4, a 65% synchronized system with
P
orb
=
2.3 hr, V. Schaffenroth et al.
2021
)
.
V. Schaffenroth et al.
(
2021
)
further point out that synchronized
binaries locate further away from the zero-age extreme horizontal
branch on the
gT
log
eff
-
diagram compared to these subsyn-
chronized systems, suggesting that those synchronized binaries
might be older. Our
fi
ndings that the tidal
synchronization
timescales at small orbital periods are shorter than the sdB lifetime
(
but not by orders of magnitude
)
agrees with this explanation.
Historically, the orbital inclinations and companion masses
are hard to acquire for noneclipsing sdB binaries. Some works
hence assume tidal synchronization for short-period binaries
and derive the orbital parameters from the orbital periods by
setting
P
orb
=
P
rot
(
e.g., S. Geier et al.
2010
)
. However, if we
can assume tidal synchronization for systems with
̄
TT
0.1
sync
EHB
<
, we see that this method should only apply
to binaries with
P
orb
1 hr. This is much shorter than the
synchronization period
(
P
orb, sync
=
1.2 days
)
assumed in
S. Geier et al.
(
2010
)
, meaning that in their work, the
companion masses
/
inclinations might be over
/
underestimated.
Additionally, binaries with
P
orb
1 hr can undergo signi
fi
-
cant orbital decay due to GW radiation, and it is questionable
whether these systems can
ever
reach 100% tidal synchroniza-
tion, as tides at subsynchronization may not be strong enough
for
Ω
spin
to fully catch up with
Ω
orb
(
as in the case of WD
binaries; see, e.g., P. Scherbak & J. Fuller
2024
)
. Mass transfer
may also happen for these binaries, making their evolution
more complicated
(
E. B. Bauer & T. Kupfer
2021
)
.
5. Discussion
5.1. Tidal Torque Scaling with Stellar Radii
We saw in Section
4.2
that the binary orbital period required
to reach tidal synchronization is shorter for the 0.37
M
e
low-
mass sdB compared to the 0.47
M
e
canonical sdB. This means
the tidal torque must be weaker for low-mass sdBs. To explain
the reason, we consider an equal-mass binary
(
q
=
1
)
and
Figure 3.
The observed trend of tidal synchronization
(
de
fi
ned by 0.8
Ω
spin
/
Ω
orb
1; shaded gray region
)
for short-period sdB binaries
(
crosses
)
vs. the calculation
results from binary spin
–
orbit evolution
(
dots
)
. The red, purple, and green crosses indicate rotational measurements from spectral line broadening,
p
-mode frequency
splitting, and
g
-mode frequency splitting. The red and blue dots indicate the modeled sdB binaries with the 0.47
M
e
and the 0.37
M
e
sdB primary, respectively. All
modeled binaries have a 0.4
M
e
companion and initial orbital periods ranging from 1 to 18 hr. For 0.47
M
e
sdB binaries, we see that all systems with orbital periods
less than
∼
0.2 day reach tidal synchronization, while for 0.37
M
e
sdB binaries, this synchronization period becomes
∼
0.15 day due to weaker torques on these
smaller sdBs. These results match the observed trends of sdB tidal synchronization. In addition, most systems with 0.3 day
P
orb
0.6 day are spun up to rotational
periods of a few days, which also agrees with the observed period range of partially synchronized binaries.
7
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)
, 2024 November 1
Ma & Fuller
rewrite the mode torque of Equation
(
1
)
as
()
()
()
fJCSR
,10
f1
twg
=-
aaaa
where
()
()
()
f
11
f
f
2
f
2
2
w
w
ww g
º
-+
a
a
is a dimensionless function describing the resonance depend-
ence of the torque on
ω
f
, whose scaling should be of similar
order for different stellar models. The quantity
∣()
Jm
12
wxx
ºáñ
aa
aa
is the angular momentum of the oscillation mode
α
with
azimuthal wavenumber
m
, which roughly scales as
JMR
1
1
2
μ
a
for stars of similar structure at the same mode frequency. For
dissipating modes, the rate for mode
α
to dissipate its angular
momentum
(
i.e., exerting a torque
)
is
γ
α
J
α
. The quantity
∣∣
∣
()
C
WQ
MR
13
lm
2
1
1
2
xx
º
áñ
a
a
aa
describes the dimensionless coupling of the tidal potential and
the oscillation mode, and
() (
)
()
S
RRa
l
11
21
º
+
is the scaling of
the tidal forcing strength.
With this notation, we can see that the quantity
γ
α
J
α
C
α
represents the rate at which mode
α
deposits its angular
momentum into the star
(
i.e., the tidal torque
)
per tidal forcing
strength. We plot this quantity for the 0.47
M
e
and 0.37
M
e
sdB models in Figure
5
, and we see that they have similar
orders of magnitude without a clear dependence on the
different stellar models. This is expected, as these sdB stars
have very similar internal structures.
Therefore, the main scaling of the physical tidal torque
comes from the forcing strength
S
(
R
1
)
. As shown in the right
panel of Figure
5
, the 0.37
M
e
sdB is more compact than the
0.47
M
e
sdB due to its lower mass, and its radius
R
1
is smaller
by a factor of
∼
2. For
l
=
2 modes, this produces a difference
in the tidal torque by
(
)
RR
164
0.37 0.47
6
~
. Since the 0.37
M
e
sdB has a smaller moment of inertia by a factor of a few, its
synchronization timescale is about 10 times longer. We can
con
fi
rm this result from Figure
4
: the tidal synchronization
timescale at
P
∼
0.2 day for 0.47
M
e
sdB binaries is
T
EHB,0.47
=
164 Myr, roughly 10 times shorter than the time-
scale of 0.37
M
e
sdB binaries at the same period
(
a few times
T
EHB,0.37
=
444 Myr
)
.
The above analysis can also provide insight into the tidal
torques for sdBs with different hydrogen envelope masses.
Detailed sdB modeling has shown that hydrogen envelope
masses range from 0.001 to 0.005
M
e
(
T. Kupfer et al.
2015
)
.
This small amount of hydrogen never affects the core structure
of the helium-burning sdBs, but it can greatly change the stellar
radius.
In the right panel of Figure
5
, we show the stellar radii for
some 0.47
M
e
sdB models that retained more hydrogen than
10
−
3
M
e
. Compared to the original 10
−
3
M
e
hydrogen model,
we can see that even a slight increase of hydrogen could
increase the sdB radius by a factor of
∼
1.5
–
2. We further plot
the
γ
α
J
α
C
α
calculated for these models in the left panel of
Figure
5
, and we see that despite some scatter, they are similar
to the 10
−
3
M
e
hydrogen model. We hence expect that the
τ
∝
R
6
scaling roughly holds for these models, and the tidal
torque for these larger sdBs could be larger by a factor of
∼
10
–
50, which will increase the synchronization transitional
period.
We note, however, that there is a reason that we did not
actually compute the tidal torques for these more extensive sdB
models. We see in Figure
5
that for sdB models with
M
H
>
10
−
3
M
e
, there exists a period range where the mode
growth rate
γ
α
is positive, i.e., where the modes are
unstable
.
This is caused by the so-called
κ
-mechanism in these stars,
where the partial ionization of iron creates an opacity bump,
generating self-excited oscillations
(
S. Charpinet et al.
1996
,
1997
)
. Our torque in Equation
(
1
)
only holds for
damped oscillation modes, and it is unclear how these self-
excited oscillations would interact with tidal forcing
(
see, e.g.,
J. Fuller
2021
)
. These unstable modes have periods of
Figure 4.
The ratio between the tidal synchronization timescale
̄
T
syn
c
and the sdB lifetime
T
EHB
interpolated between different choices of companion masses and initial
orbital periods. The blue regions show the parameter space where
̄
TT
sync
EHB
<
. The brackets on the
y
-axis indicate the typical companion masses for sdB
+
WD or sdB
+
dM systems
(
V. Schaffenroth et al.
2022
)
. Left: the results for the 0.47
M
e
sdB model, with
T
EHB
=
164 Myr. We see that for systems in orbits less than
∼
0.15
–
0.22 day, the synchronization timescale is less than the stellar lifetime, meaning these systems are likely to be observed as tidally synchronized. T
he results
have a weak dependence on companion masses. Right: the results for the 0.37
M
e
sdB model. The critical orbital period below which systems become synchronized
now becomes
∼
0.10
–
0.17 day.
8
The Astrophysical Journal,
975:1
(
14pp
)
, 2024 November 1
Ma & Fuller
∼
0.05
–
0.1 day, so they could be very important for the tidal
evolution of sdBs in
∼
0.1
–
0.2 day orbits. Future works should
investigate how these modes will behave under tidal excitation.
The above analysis also explains the discrepancies between
the tidal synchronization period we calculated and those
estimated by H. P. Preece et al.
(
2018
)
, who applied Zahn
’
s
traveling wave limit. The synchronization periods we found for
the 0.47
M
e
canonical sdB model are longer than the periods
from their work, which means the tidal torque in our cases is
stronger. This might be because H. P. Preece et al.
(
2018
)
used
an sdB model with only 10
−
4
M
e
hydrogen left, whose radius
is smaller than our model. Hence, even though the gravity
waves are more damped with Zahn
’
s traveling wave limit in
their models, the tidal torque can still be weaker due to its
strong dependence on the stellar radius.
5.2. Resonance Locking
In binary systems, if the tidal torque consists of many
resonance peaks from individual modes, a process called
resonance locking may occur
(
M. G. Witte & G. J. Savon-
ije
1999
,
2001
)
. In this scenario, the forcing frequency of the
binary enters a resonance with one of the oscillation modes
α
and stays as
()
()
m
14
forbspin
ww
ºW-W
a
throughout the binary lifetime. As
ω
α
evolves on its own
timescale, which is independent of the binary separation, this
scenario may result in a very different binary evolution history
compared to other tidal theories.
To see whether resonance locking can happen for sdB
binaries, we write the evolution of the forcing frequency as
()
()
⎜⎟
⎛
⎝
⎞
⎠
m
II
3
,15
f
tide
GW
orb
tide
spin
w
tt t
=
+
-
where we substitute Equations
(
3
)
,
(
4
)
,
(
6
)
, and
(
7
)
and neglect
the
I
spi
n
term, as the stellar structure and moment of inertia only
change slowly during the evolution. To maintain a resonance
lock, we must have
f
w
w
=
a
. For helium-burning subdwarfs,
their
g
-mode frequency increases over time; hence, the
necessary
(
but not suf
fi
cient
)
condition for resonance locking
to occur is
0
f
w
>
,or
()
I
I
1
3
.16
GW
tide
orb
spin
t
t
+>
For binaries of order-of-unity mass ratios,
I
a
orb
2
m
=
~
Ma MR I
1
2
1
1
2
spin
>
. The above relation hence never holds
for realistic sdB binaries unless
τ
GW
?
τ
tide
. This can happen
either for already close-to-synchronization binaries or for wide
binaries, where
τ
tide
becomes very small in both cases. In the
former case, the low-frequency oscillating
g
-modes that
contribute most to the tidal torque should be very ef
fi
ciently
damped
(
see Section
4.1
)
, which prevents resonance peaks
from forming. In the latter case, tidal evolution is not important,
because it would occur on a GW inspiral time that is very long
for wide binaries. We hence do not expect resonance locking to
happen for sdB binaries, which is con
fi
rmed with our numerical
spin
–
orbit evolution calculations.
5.3. Differential Rotation
As we expect very ef
fi
cient angular momentum transport
inside the sdBs, we assume that they are rigidly rotating in our
spin
–
orbit evolution calculations. Observationally, asteroseis-
mology can measure the internal rotation of stars via frequency
splittings of
g
-mode and
p
-mode oscillations
(
C. Aerts et al.
2010
)
. Since
g
-modes mainly probe the deeper region of the
star, while
p
-modes probe the outer layers, a difference
between the rotational rates derived from their frequency
splittings may suggest the level of differential rotation between
the stellar core and the outer layers.
With Kepler
/
K2, there have now been a handful of pulsating
sdBs with both
p
-mode and
g
-mode frequency splitting
measured. J. W. Kern et al.
(
2018
)
report that for the sdB
+
WD system KIC 11558725, the rotational rate derived from
Figure 5.
Left: the magnitude of
γ
α
J
α
C
α
(
de
fi
ned in the main text
)
for different sdB models. This quantity determines the tidal torque of each mode per unit tidal
forcing strength, and we see that they are of similar orders of magnitude without clear dependence on the different stellar models. Hence, the physica
l torque should be
roughly proportional to the tidal forcing strength, which scales as
R
6
. Note that sdB models with hydrogen masses greater than 10
−
3
M
e
have some unstable modes,
and their tidal excitation cannot be treated with our current method. Right: stellar radii as a function of age for different sdB models. More massive s
ystems with more
hydrogen left in the envelope have larger radii, and the tidal torques are expected to be stronger based on the
R
6
scaling.
9
The Astrophysical Journal,
975:1
(
14pp
)
, 2024 November 1
Ma & Fuller