Accelerated tidal circularization via resonance locking in KIC 8164262

MNRAS

472,

L25–L29 (2017)

doi:10.1093/mnrasl/slx130

Advance Access publication 2017 August 18

Accelerated tidal circularization via resonance locking in KIC 8164262

Jim Fuller,

1,2

‹

Kelly Hambleton,

Avi Shporer,

Howard Isaacson

and Susan Thompson

Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA

TAPIR, Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA

Department of Astrophysics and Planetary Science, Villanova University, 800 East Lancaster Avenue, Villanova, PA 19085, USA

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125

Astronomy Department, University of California, Berkeley, CA 94720, USA

SETI Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA

Accepted 2017 August 15. Received 2017 August 14; in original form 2017 June 14

ABSTRACT

Tidal dissipation in binary star and planetary systems is poorly understood. Fortunately,

eccentric binaries known as heartbeat stars often exhibit tidally excited oscillations, providing

observable diagnostics of tidal circularization mechanisms and time-scales. We apply tidal

theories to observations of the heartbeat star KIC 8164262, which contains an F-type primary

in a very eccentric orbit that exhibits a prominent tidally excited oscillation. We demonstrate

that the prominent oscillation is unlikely to result from a chance resonance between tidal

forcing and a stellar oscillation mode. However, the oscillation has a frequency and amplitude

consistent with the prediction of resonance locking, a mechanism in which coupled stellar and

orbital evolution maintain a stable resonance between tidal forcing and a stellar oscillation

mode. The resonantly excited mode produces efficient tidal dissipation (corresponding to an

effective tidal quality factor

∼

), such that tidal orbital decay/circularization proceeds

on a stellar evolution time-scale.

Key words:

binaries: close – stars: evolution – stars: individual: KIC 8164262 – stars: kine-

matics and dynamics – stars: oscillations – stars: rotation.

1 INTRODUCTION

The mechanisms underlying tidal energy dissipation in stellar and

gaseous planetary interiors remain uncertain despite decades of

research. Observationally, tidal orbital evolution is challenging to

measure because it typically proceeds on very long time-scales.

Theoretically, tidal dissipation is difficult to calculate from first

principles because it is produced by weak friction effects that depend

on details of the stellar structure, convective turbulence, non-linear

mode coupling and other complex hydrodynamical processes.

Heartbeat stars present a new opportunity to constrain tidal dissi-

pation processes through observations of tidally excited oscillations

(TEOs). These eccentric binary stars experience tidal distortion near

periastron that produces ‘heartbeat’ signals in high-precision light

curves, and they have been studied in a number of recent works

(Welsh et al.

2011; Thompson et al.

2012; Hambleton et al.

2013;

Beck et al.

2014;Schmidetal.

2015; Smullen & Kobulnicky

2015;

Hambleton et al.

2016;Kirketal.

2016; Shporer et al.

2016;

Dimitrov, Kjurkchieva & Iliev

2017; Guo, Gies & Fuller

2017).

A fraction of heartbeat stars also exhibit TEOs that can be rec-

E-mail:

jfuller@caltech.edu

ognized because they occur at

exact

integer multiples of the or-

bital frequency (Kumar, Ao & Quataert

1995; Burkart et al.

2012;

Fuller & Lai

2012). The TEOs are produced by tidally forced stel-

lar oscillation modes, in most cases by gravity modes (g modes).

Given sufficiently accurate stellar properties, one can identify the g

modes responsible for the TEOs. From observed mode amplitudes,

one can then calculate mode energies, damping rates, tidal energy

dissipation rates and circularization/synchronization time-scales.

In this paper, we compare detailed tidal theory from a companion

paper (Fuller

2017), with observations of the heartbeat star KIC

8164262 (K81) analysed in another companion paper (Hambleton

et al.

2017). This heartbeat star contains an F-type primary star in

a highly eccentric (

0.89), long period (

87 d) orbit with

alowmass(

≈

0.36 M

) companion. The F-type primary has

1.7

0.1 M

2.4

0.1 R

eff

6900

100 K,

and is nearing the end of its main-sequence lifetime. Most impor-

tantly, K81 exhibits a large amplitude (relative flux variability of

∼

−

) TEO at exactly 229 times the orbital frequency. A

light curve and power spectrum are shown in Fig.

1. TEOs can be

excited to large amplitude if a stellar g-mode frequency happens to

be nearly equal to a tidal forcing frequency (i.e. a multiple of the

orbital frequency). We demonstrate that the prominent TEO in K81

requires a resonance so finely tuned that it is unlikely to occur by

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Figure 1.

Top left-hand panel: phased light curve of K81, in units of parts per thousand, with periastron near phase zero. The sharp variation near periastron

is the ‘heartbeat’ signal produced by the equilibrium tidal distortion, reflection and Doppler boosting. The rapid variability away from periastron

is due to the

prominent TEO at 229 times the orbital frequency. Top right-hand panel: same as top left-hand panel panel, zoomed in near periastron. Bottomleft-han

d panel:

Fourier transform of the light curve of K81. The large amplitude spike at

2.6 d

−

is produced by the dominant TEO and extends above the scale of the plot

to 1 ppt. The fuzz at lower amplitudes is composed of peaks at integer multiples of the orbital frequency. Bottom right-hand panel: Same as bottom left-

hand

panel, zoomed in to frequencies near 1.5 d

−

. Peaks at orbital harmonics are produced by TEOs (dynamical tide).

chance. Instead, the oscillation presents a compelling case for res-

onance locking (Witte & Savonije

1999, 2001; Fuller & Lai

2012;

Burkart, Quataert & Arras

2014), where tidal orbital decay natu-

rally maintains a finely tuned resonance with a g mode, resulting in

a large amplitude TEO.

2 TIDAL MODELS OF K81

TEOs are produced by tidally forced stellar oscillation modes. Com-

puting expectations for TEO frequencies, amplitudes, and phases

are discussed in detail in Fuller (

2017). In many cases such as in

K81, only the highest amplitude TEOs are detectable, and these

TEOs are produced by near-resonances between stellar g-mode fre-

quencies and multiples of the orbital frequency.

When a single resonant oscillation mode (labelled by

) domi-

nates the tidal response at a multiple

of the orbital frequency, it

produces a sinusoidal luminosity fluctuation of form

sin(

)

(1)

where

π

is the angular orbital frequency. The amplitude

√

(

−

)

(2)

The pulsation phase

also contains useful information

(O’Leary & Burkart

2014), but the longitude of periastron mea-

surement in K81 from Hambleton et al. (

2017) was not precise

enough to measure the phases of its TEOs. Each term in equation

(2) is defined in Fuller (

2017) and can be calculated relatively eas-

ily. The tidal forcing amplitude

has a value of

−

for K81.

describes the visibility of the modes based on viewing angle,

and is typically of the order of unity. Other quantities are a function

of frequency and are plotted in Fig.

2. The Hansen coefficient

is the strength of tidal forcing at each forcing frequency

and

peaks near

∼

40 for

2 for K81 (see Fig.

2). It has a long

tail to higher frequencies (due to the high eccentricity) allowing

oscillations up to

∼

300 to be observable.

The other terms in equation (2) are properties of stellar oscilla-

tion modes. To calculate them, we construct a stellar model with

parameters nearly equal to those found by Hambleton et al. (

2017)

for the primary star, using the stellar evolution code

MESA

(Paxton

et al.

2011, 2013, 2015). Our model has the same stellar/orbital

parameters quoted above, but with a 1

smaller stellar radius of

2.3 R

that better matched the observed TEOs. We calculate the

non-adiabatic stellar oscillation modes of our model using the

GYRE

oscillation code (Townsend & Teitler

2013).

MESA

GYRE

inlists and

a discussion of the process are provided in Fuller (

2017). Next,

we calculate tidal overlap integrals

that measure gravitational

coupling between oscillation modes and the tidal potential, and

bolometric luminosity perturbations

produced by normalized

modes at the stellar surface. Fig.

2 shows

and

for

modes (we show below that most of the observed oscillations are

likely generated by quadrupolar,

1 and 2 oscillation modes).

Although the value of

peaks for high-frequency (low radial or-

der) g modes, the value of

peaks for modes with

∼

100. Fig.

also shows mode damping rates

and frequency spacings

Damping rates are much larger for low-frequency (high-radial or-

der) modes, where the mode spectrum is very dense. Modes behave

like travelling waves when

∼

, which occurs at frequencies

100.

The last term in equation (2) is the resonant deturning for with

. The detuning factor can be very large for nearly resonant

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Resonance locking in KIC 8164262

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Figure 2.

Top panel: properties of normalized

2 g modes in our

model of the primary star of K81 (see the text). All values have been cal-

culated at multiples

of the orbittal frequency by interpolating between

neighbouring g modes. Modes within the grey shaded region have very

large radial orders and have not been calculated. The top axis shows the cor-

responding observed oscillation frequency

. Bottom panel: Mode damping

rate

, and mode frequency separation

. Note that most mode damp-

ing times are longer than 10

d so that TEOs maintain a nearly constant

amplitude throughout each orbit.

modes but is very sensitive to the degree of detuning. Because of this

extreme sensitivity, it is difficult to reliably calculate this quantity.

However, we can still predict mode amplitudes as a function of

frequency in a statistical fashion, as outlined in Fuller (

2017). The

median luminosity fluctuation is

med

∣

L

∣

(3)

while the maximum possible fluctuation amplitude is

max

∣

L

∣

(4)

where

L

With the K81 parameters from Hambleton et al. (

2017) and the

oscillation modes computed above, we can compute expected fre-

quencies and amplitudes of TEOs in K81. Fig.

3 shows both the

observed TEOs and our model results. We plot the median lumi-

nosity fluctuation at each orbital harmonic

(equation 3) and the

maximum luminosity fluctuation (equation 4). In addition, we plot

the luminosity fluctuations at each harmonic for our representative

stellar model. Finally, the background colour indicates the probabil-

ity density dlog

of observing a TEO in that part of the plot.

Most of the observed TEOs in K81 are likely to be produced by

quadrupolar

2 and 1 modes, with amplitudes above the median

amplitude of equation (3) due to chance resonances between mode

frequencies and forcing frequencies. In our model,

0 modes

are less visible due to the viewing angle relative to the stellar spin

axis.

Hambleton et al. (

2017) found evidence for spin-orbit misalign-

ment in K81, measuring an orbital inclination of

◦

and

a spin inclination of

◦

, implying a minimum obliq-

uity of 30

◦

. Such spin-orbit misalignment allows for excitation of

quadrupolar

1 modes. The model shown in Fig.

3 has

◦

and an actual (non-projected) obliquity of

◦

. We find mod-

els with lower values of

or higher obliquity provide a worse

fit to the data because they produce an excess of low-frequency

(

110)

1, 0 TEOs that are not observed. Another mod-

elling feature that improves agreement is the use of small amounts

of convective core overshoot and diffusive mixing. Our model uses

0.01 and

mix

1cm

−

, on the low side of values inferred

from aseteroseismology (Moravveji et al.

2015, 2016; Deheuvels

et al.

2016), which are typically in the range 0.01

0.03

and 1

mix

100 cm

−

. Larger overshoot tends to overpredict

TEO amplitudes. Our models use the Schwarzschild criterion for

convective boundaries, appropriate for main-sequence stars of this

mass (Deheuvels et al.

2016; Moore & Garaud

2016).

3 RESONANCE LOCKING IN K81

The primary TEO at

229 is not easily explained as a chance

resonance. Fig.

3 shows that this TEO lies in a region of parame-

ter space unlikely to contain TEOs, in contrast to TEOs at lower

frequencies and amplitudes. A very close resonance is required to

produce the primary TEO, which is possible but unlikely.

To quantify this statement, we compute the cumulative distribu-

tion of the expected number of TEOs to exist above a given am-

plitude

, and compare with the distribution of observed TEO

amplitudes. This calculation is described in Fuller (

2017), and is

performed by integrating the probability distribution in Fig.

3 over

all

and up to a chosen amplitude

. Fig.

4 shows the results.

Our model slightly overpredicts the number of low-amplitude TEOs

with

−

but the agreement is fairly good. However, our

model indicates the expected number of TEOs with

≥

−

is about 0.05. In other words, only 5 per cent of systems with prop-

erties nearly identical to K81 would be expected to exhibit such a

large amplitude TEO. We thus find it unlikely that the primary TEO

in K81 is caused by a chance resonance.

Instead, we suggest that the prominent TEO in K81 is generated

via resonance locking (see Witte & Savonije

1999, 2001; Burkart

et al.

2012, 2014; Fuller & Lai

2012). Because the g-mode frequen-

cies change as the star evolves, they pass through resonances with

tidal forcing frequencies. A resonance lock occurs when tidal dissi-

pation from a resonant mode causes orbital decay such that the tidal

forcing frequency increases at the same rate as the mode frequency,

and the mode remains resonant. Resonance locking configurations

can be stable and last for long periods of time (Burkart et al.

2014),

and they would create a single high-amplitude TEO like that

in K81.

To test the resonance locking hypothesis, we generate stellar mod-

els slightly younger and older than our model for K81, and compute

their g-mode spectra. We then compute the rates at which the mode

frequencies evolve,

,where

is the mode

frequency in the inertial frame. We assume no angular momentum

loss and rigid stellar rotation. In contrast to previous assumptions

(Burkart et al.

2012; Fuller & Lai

2012), g-mode frequencies typ-

ically increase with age in intermediate-mass stars because they

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Figure 3.

TEOs in the K81 system. Observed TEOs are shown as purple squares, with the prominent TEO denoted by a magenta star. Black dots are TEOs

from our representative model. Black lines are the median amplitudes of the

1 and 2 TEOs in our model, while the blue line is the maximum possible

amplitude of

1 or 2 TEOs in our model. Background colour denotes the probability density for

1 and 2 TEOs, which are more likely to exist in red

regions of the plot. Note the primary TEO lies in a very unlikely region of parameter space. Blue–green and light-green circles are expected amplitude

sfor

resonantly locked

2and1modes.

Figure 4.

Cumulative distribution of number of TEOs above an amplitude

as a function of

. We plot both observed numbers and expected

numbers from our representative model. At low amplitudes, the observed

and expected distributions overlap, indicating the model adequately explains

most of the low-amplitude TEOs. The primary TEO at

−

(magenta

star) is unexpected from the model and is unlikely to occur by a chance

resonance (the probability is

≈

5 per cent), and instead could be the result of

resonance locking.

become more highly stratified as they evolve. This creates increas-

ing Brunt–V

ais

a frequencies and g-mode frequencies, allowing

resonance locking to occur with modes of any value of

After calculating the mode frequency evolution rates, we calcu-

late corresponding resonance locking mode amplitudes and lumi-

nosity fluctuations,

ResLock

[

]

(5)

Here,

is a dimensionless factor of the order of 10

−

–10

−

in our

models, which is derived in Fuller (

2017). We plot these predictions

for

1 and 2 modes in Fig.

3.Wehaveaddedan

∼

3 per cent

uncertainty in frequency and

∼

30 per cent uncertainty in luminos-

ity fluctuation to account for the uncertainty in the stellar/orbital

parameters. The primary TEO has an amplitude and frequency con-

sistent with being a resonantly locked

1 oscillation mode.

A resonantly locked

2 mode may be possible but the pre-

dicted amplitude is slightly too low. Addtionally, using the system

parameters, we evaluate equation (53) of Burkart et al. (

2014)to

find that resonance locks will be stable for both

1 and 2 modes.

In the resonance locking scenario, both the resonantly locked

mode frequency

and tidal forcing frequency

are increasing

at the same rate, set by the mode frequency evolution time-scale

∼

4Gyrfor

1 g modes in K81. We compute a corresponding

tidal orbital evolution time-scale of

orb

tide

orb

tide

∼

6Gyr,

which is somewhat longer than the stellar age in this particular case.

This tidal dissipation rate can be translated into an effective tidal

quality factor Q (Goldreich & Soter

1966) by defining

orb

tide

orb

(

peri

)

(6)

where

peri

−

) is the periastron orbital separation and

−

is the Love number for our model. We calculate

∼

, much lower than might be naively expected for an

F-type star. Indeed, in our model without a resonantly locked mode,

the combined energy dissipation of all TEOs leads to

∼

2.5

meaning the resonantly locked mode increases the tidal energy

dissipation rate by a factor of

∼

500.

Equation (53) of Burkart et al. (

2014

) has a typo, the

sign should be

sign.

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4 DISCUSSION AND CONCLUSIONS

We have demonstrated that resonance locking in K81 is an appealing

mechanism to account for its prominent TEO. Resonance locking

correctly predicts the amplitude of the TEO, whereas producing

the TEO by a chance resonance between tidal forcing and a stellar

oscillation mode is unlikely. However, we note that K81 was chosen

for detailed analysis because of its high-amplitude TEO, which is

larger amplitude than nearly all other TEOs in catalogued heartbeat

systems. Of the

∼

175 catalogued heartbeat stars,

roughly 30 show

evidence for TEOs visible by eye. Thus, the chances of finding

one system amongst these with an unexpectedly large TEO (at the

level) is of the order of unity. We therefore cannot exclude

the possibility the prominent TEO in K81 is simply an uncommon

occurrence that was selected for study due to its high amplitude.

K81 can be compared with the KOI-54 (Welsh et al.

2011) heart-

beat system. Resonance locking with

2 modes was proposed

by Fuller & Lai (

2012) to explain KOI-54’s large amplitude TEOs,

but O’Leary & Burkart (

2014) demonstrated that the TEO phases

identifies them as

0 modes. However, g-mode frequencies in-

crease as stars evolve off the main sequence and allows resonance

locking to occur with

0 modes, and hence the prominent TEOs

in KOI-54 could still be explained as a resonantly locked

TEO within each star. A preliminary calculation indicates this may

be possible, but the observed TEO amplitudes are a factor of a few

smaller than expected for resonance locking.

Other heartbeat systems may not exhibit resonance locking ef-

fects if they are not in an evolutionary stage where resonantly locked

modes are visible. Since TEOs are most visible in stars without thick

surface convective zones, we only expect to see large amplitude

TEOs in stars with

eff

6500 K. Moreover, resonance locking

amplitudes increase as the stellar evolution rate accelerates, i.e.

when stars evolve off the main sequence. Therefore, we expect to

see high-amplitude resonantly locked modes in heartbeat stars con-

taining somewhat massive (

1.5 M

) stars in the brief period

during which they are beginning to evolve off the main sequence,

but have not yet cooled to

eff

6500 K. A firm conclusion will re-

quire detailed analyses of a greater population of heartbeat systems.

If resonance locking does commonly occur, it can greatly enhance

tidal dissipation, causing orbital decay and spin synchronization to

proceed on a stellar evolution time-scale. During a resonance lock,

the tidal orbital energy dissipation rate is

orb

∼

Resonance locking is not necessarily limited to eccentric binary

star systems, and may operate in many astrophysical scenarios, in-

cluding circular (but non-synchronized) binary stars, exoplanetary

systems, inspiraling white dwarfs (Burkart et al.

2013) and out-

wardly migrating planetary moon systems Fuller, Luan & Quataert

(

2016). The observable feature of resonance locking is a larger-than-

expected variability at an integer multiple of the orbital frequency.

This could potentially be detected as ‘anomalous’ ellipsoidal varia-

tions (see e.g. Borkovits et al.

2014) or a perturbed gravity field. Res-

onance may not be able to operate in all scenarios, as it could some-

A list of catalogued heartbeat stars can be found at

http://keplerebs.

villanova.edu/search

, by filtering with the ‘HB’ flag checked.

Resonance locking cannot explain the pulsations in HAT-P-2 reported by

deWitetal.(

2017

) because those pulsations are larger than

,max

at the

observed pulsation frequency.

times be quenched by non-linear instabilities or be overwhelmed

by other tidal/orbital effects. When resonance locking can operate,

it generally accelerates tidal evolution to proceed on the relevant

evolutionary time-scale, for example, a stellar evolution time-scale,

magnetic braking time-scale, or gravitational radiation orbital decay

time-scale.

ACKNOWLEDGEMENTS

We thank the Planet Hunters and Dan Fabrycky for discovering

this system, and the anonymous referee for a thoughtful report. JF

acknowledges partial support from NSF under grant no. AST-

1205732 and through a Lee DuBridge Fellowship at Caltech. KH

acknowledges support through NASA ADAP grant (16-ADAP16-

0201). This research was supported by the National Science Foun-

dation under Grant No. NSF PHY11-25915, and by NASA under

grant 11-KEPLER11-0056 and the Gordon and Betty Moore Foun-

dation through grant GBMF5076.

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