Probing white dwarf interiors with LISA: periastron precession in double white dwarfs

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by  B. Willems

Probing white dwarf interiors with LISA: periastron precession in double white dwarfs
B. Willems1, A. Vecchio1,2, V. Kalogera1
1Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA
2School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
In globular clusters, dynamical interactions give rise to a population of eccentric double white
dwarfs detectable by the Laser Interferometer Space Antenna (LISA) up to the Large Magellanic
Cloud. In this Letter, we explore the detectability of periastron precession in these systems with
LISA. Unlike previous investigations, we consider contributions due to tidal and rotational dis-
tortionsof thebinarycomponentsin addition togeneral relativistic contributionstotheperiastron
precession. AtorbitalfrequenciesaboveafewmHz,wefindthattidesandstellarrotationdominate,
opening up a possibly uniquewindow to thestudy of theinterior and structureof white dwarfs.
8
PACSnumbers: 95.10.Ce,95.30.Sf,95.55.Ym,97.10.Cv,97.60.-s,97.80.-d,
0
0
2
Introduction.– Binary stars have long been recognized to the tidal distortions of the binary components. The

n as unique astrophysical laboratories for the study of apsidal motion depends primarily on the internal mass
a
physicsandcosmology[1]. When oneofthe binarycom- distributionofthe stars,andtakesplaceonconsiderably
J
ponents is a compact object, measurements of its inter- shorter time scales than dissipative spin-orbit coupling.
7
actions with the orbital companion provide a wealth of Theshortertimescalesfacilitatetheinferenceofapsidal-
1
information onthe compact object and the properties of motionratesfromelectromagneticandGWobservations.

] matter under extreme conditions.
h In addition to tides, rotational distortions of binary
The Laser Interferometer Space antenna (LISA; [2])
p willsurveythewholegalacticpopulationofbinariescon- stars and general relativity (GR) also contribute to the
-
o sisting of two white dwarfs (WDs) with periods < 6 hr apsidalmotionineccentricbinaries. Whiletherotational
r by monitoring their gravitational wave (GW) em∼ission. contribution depends on the internal mass distribution
st LISAwillindividuallyresolve 104 doublewhitedwarfs in a similar way as the tidal effect, the GR contribution
a ∼ depends only on the total system mass and orbital el-
(DWDs) [3] and will be particularly effective in discov-
[
ering short-period ( < 30 min) systems – in stark con- ements. For non-degenerate stars, comparisons between

2 trast to current and∼planned electromagnetic observato- theoreticallypredictedapsidal-motionratesandobserva-
v ries. LISA will therefore allow astrophysical studies of tionally inferred rates have long served as a critical test
0 of theories of stellar structure and evolution [7, 8, 9].
outstanding questions in compact object binary physics,
0
such as dynamically unstable mass transfer, accretion,
7 In this Letter, we examine the physics accessible by
3 and type Ia supernova formation [4]. measuring the apsidal motion of eccentric DWDs with
. So far, investigations of DWDs have focused on the
6 LISA. General relativistic apsidal motion has already
formationandpropertiesof binarieswith circularorbits,
0 been proposed as a tool to derive the total system mass
which dominate the Galactic DWD population. How-
7 ofeccentricneutronstar(NS)binaries[11]. Applications
0 ever, Willems et al. [5] recently predicted the existence
toWDbinarieshavesofarnotbeenconsidered,norhave
: of a sub-population of DWDs consisting of eccentric bi-
v the imprint on GWs of the periapse precession induced
naries formed through dynamical interactions in globu-
i by tides and stellar rotation.
X lar clusters. Unlike present and planned electromagnetic
r telescopes, LISA will be able to detect these systems to Apsidal motion.—Webrieflysummarizetheequations
a
distancesasfarastheLargeMagellanicCloud,providing governing the tidal, rotational, and GR contributions to
auniqueopportunitytostudydegeneratematterthrough theapsidalmotionineccentricbinaries. Forthispurpose,
the imprint of tidal effects in the detected GW signal. weconsiderabinaryconsistingoftwouniformlyrotating
Tidal interactions in close binaries couple the spins of starswithmassesM1,2,radiiR1,2,androtationalangular
the component stars to the orbital motion, driving bi- velocitiesΩ1,2.Thestellarrotationaxesareassumedtobe
naries to a state of minimum kinetic energy in which perpendicular to the orbital plane [22]. We furthermore
the orbit is circular, the stellar spin angular momenta let P be the orbital period, a the semi-major axis, e the
arealignedwith the orbitalangularmomentum, andthe orbitaleccentricity,andγtheargumentoftheperiastron.
stellar rotation rates are synchronized with the orbital
motion. The efficiency of this process depends strongly The contribution to the apsidal motion from the tidal
onthe mode ofenergydissipationinthe stellarinteriors, distortion of the binary components is most commonly
which is reasonably well understood for non-degenerate determined under the assumption that the orbital and
stars,but anopen questionfor WDs [6]. However,in ec- rotationalperiodsarelongcomparedtotheperiodsofthe
centricbinaries,tidesalsocauseanon-dissipativepreces- freeoscillationmodesofthecomponentstars[12,13,14].
sionoftheperiastronoftheorbit. This“apsidalmotion” Underthisassumption,therateofsecularapsidalmotion
is caused by perturbations in the gravitational field due due to the dominant quadrupole tides raised in star i
2
(i=1,2) is given by
γ˙tid,i = 3P0π (cid:18)Rai(cid:19)5 MM3−ii 1+(132e2e+2)581e4 ki, (1)
−
where k is the quadrupolar apsidal-motion constant of
i
stari. Whentheorbitalandrotationalperiodsareofthe
order of the free oscillation modes of the binary compo-
nents,deviationsfromEq.(1)ariseduetotheincreasing
role of stellar compressibility on the tidal displacement
field for higher tidal forcing frequencies and due to the
occurranceofresonancesbetweendynamictidesandnon-
radial stellar oscillations [13]. FIG. 1: Left: Tidal (short-dashed), rotational (dotted), GR
Theapsidal-motionconstantsk measurethedegreeto (long-dashed),andtotal(solid)apsidal-motionrateforDWDs
i
whichmassisconcentratedtowardsthestellarcenterand with M1 = M2 = 0.3M⊙. Right: Total apsidal-motion rate
are determined by numerical integration of the equation for DWDs with M1 =M2 =0.3M⊙ (solid) and M1 =M2 =
of Clairaut (for details see, e.g., [14, 15]). The constants 0.6M⊙ (short-dashed). Bothpanelsshowresultsfore=0.01
(black) and e=0.5 (grey), and k1 =k2 =0.1.
are unaffected by dissipative effects as long as the tidal
forcing is not in resonance with any of the WDs’ non-
radial stellar oscillation modes [13, 21]. In the limiting
cases where the stars are approximated by point masses angularvelocityatperiastron. Sincethetidaleffectsusu-
or equilibrium spheres with uniform mass density, the allydominatetherotationaleffects,thisassumptiondoes
constantski takethe values 0and0.75,respectively. For not affect the main conclusions of the calculation.
more realistic stellar models, the constants take values
It is evident that the total apsidal-motion rate is sub-
between these two extremes.
stantialthroughouttheentireLISAband,evenforeccen-
Rotationcontributestotheapsidalmotionthroughthe
tricities as low as e 0.01: at ν 0.5 mHz, periastron
rotational quadrupole distortion caused by the centrifu- ≃ ≃
precession already induces a phase shift in the GW sig-
gal force [14]. The corresponding rate of secular apsidal
nalofmorethan2π overanobservationtimeTobs =5yr
motion depends on R1,2 and k1,2 in a similar way as the (the currentminimum missionlifetime requirement)and
tidal contribution to the apsidal-motion rate, but has a
thereforebecomespotentiallydetectable. Thephaseshift
different dependence on M1,2 and e: isevenlargerathigherorbitalfrequencies. Equallystrik-
ing is the dominance of tides and stellar rotation at fre-
5 2
2π Ri M1+M2 (Ωi/Ω) quenciesaboveafew mHz. Willemsetal.[5]haveshown
γ˙rot,i = P (cid:18) a (cid:19) Mi (1 e2)2 ki, (2) eccentric DWDs in this frequency rangeto be detectable
−
by LISA up to distances as far as the Large Magellanic
where Ω=2π/P is the mean motion. Cloud, opening up new avenues for GW astrophysics of
TheGRcontributiontotheapsidal-motionratediffers DWDs. The tidal and rotational contributions to γ˙ fur-
from the tidal and rotational contributions in that it is thermore decrease with increasing mass of the WDs due
independent of the radii and internal structure of the to the smaller radii of more massive WDs. At low fre-
binary components. At the leading quadrupole order, quencies,whereGReffectsdominate,theapsidal-motion
the GR apsidal-motion rate is given by rate increases with increasing mass of the WDs.
LISA observations.— Periastron precession leaves a
2π 3G M1+M2
γ˙GR = P c2 a(1 e2), (3) signature in the GW forms of eccentric DWDs by mod-
− ifying the phase of the signal recorded by laser interfer-
ometers. LISA can therefore probe into the structure of
where G is the Newtonian gravitational constant, and c
DWDs byobservingthe apsidalmotiondue tothe above
the speed of light [17].
contributions. Here, we first show conceptually how one
In Fig. 1, the tidal, rotational, GR, and total apsidal-
can measure γ and the parameters that drive its evolu-
motion rate are shown as functions of the orbital fre-
tion; next we explore how accurately these parameters
quency ν = 1/P for different orbital eccentricities and
can be measured. Throughout this discussion we model
conservative WD component masses of 0.3 and 0.6M⊙.
GW radiation at the leading Newtonian quadrupole or-
The tidal and rotational contributions are determined
der(post-Newtoniancorrectionsarenegligibleinthisfre-
using Nauenberg’s [20] zero-temperature mass-radiusre-
quency and mass range),andmodel LISA following [16].
lation, and setting k1 = k2 = 0.1. These k1 and k2 are
The signal from an eccentric DWD in the LISA frame
appropriate for cool WD models of 0.3M⊙ and 0.6M⊙;
can be schematically written as
the dependence of k1 and k2 on the WD mass and tem-
perature will be explored in more detail in a separate
√3
ifnuvrtehsteirgmatoiroena.sTsuhmeeWdtDobroetasytinocnharloannizgeudlawritvheltohceitoiersbiatrael h(t)= 2 F+(t)h+n(t)+F×(t)h×n(t) , (4)
Xn (cid:2) (cid:3)
3
whereF+,× arethe antennabeampatterns(thatdepend ν e
on the source right ascension α and declination δ, and (mHz) 0.01 0.1 0.3 0.5 0.7
the wave polarization ψ at a constant reference time), 0.1 0.03 0.03 0.03 0.02 0.03
and h(∆e)2i1/2 1 0.02 0.02 0.03 0.03 0.03
3 0.05 0.05 0.04 0.04 0.03
h+n(t) = An−(1+cos2ι)un(e)cos[nφ(t)+2γ(t)] h(∆γ˙/π)2i1/2/nHz 01.1 1174..3490 11..9853 11..0288 01..8037 10..7809
(1+cos2ι)v (e)cos[nφ(t) 2γ(t)]
− n − 3 32.30 3.28 0.12 0.86 0.69
+sin2ιw (e)cos[nφ(t)] , (5) h(∆ν˙)2i1/2/nHz2 1 8.56 7.62 5.27 3.64 2.12
n
o 3 8.57 8.36 6.89 4.82 2.82
h×(t) = 2Acosι u (e)sin[nφ(t)+2γ(t)]
n n
n
TABLEI:Measurementaccuracyofe,γ˙/π,andν˙ forselected
+v (e)sin([nφ(t) 2γ(t)]) . (6)
n
− o values of e and ν. The errors are for Tobs = 5yr and for a
source detected with optimal SNR = 10. No assumption is
Here, φ(t) is the Doppler shifted orbital phase φorb =
2πνt+πν˙t2+O(t3)+φ0,whereν˙ istheorbitalfrequency mγ˙ aadned/oonrtfhreeqpuheynsciycadlrmiftecν˙h.anTishme edrrriovrinogntνh˙eisaprespidoarltemdootniolny
derivative and φ0 an arbitraryinitial phase; ι is the con- forsystemswithν ≥1mHz,wheretheGRcontributiontoν˙
stant source inclination angle, A = (2πν)2/3 5/3/d the
becomes observable.
M
GW amplitude, the chirp mass, and d the distance
M
to the source; u (e), v (e), and w (e) are linear combi-
n n n ν e
nations of the Bessel functions of the first kind J (ne),
n (mHz) 0.01 0.1 0.3 0.5 0.7
Jn±1(ne) and Jn±2(ne); explicit expressions can be de- 0.1 19.86 2.20 1.12 0.72 0.99
rived using Eqs. (7) and (10) in [16]. h(∆M)2i1/2/M 1 0.35 0.05 0.04 0.07 0.11
In the absence of periastron precession, radiation is
3 0.13 0.02 0.04 0.07 0.11
emitted at multiples n of the orbital frequency ν, but h(∆M)2i1/2/M 1 0.52 0.43 0.20 0.30 1.64
when periastron precession is present, each of these har-
3 0.01 0.04 0.12 0.32 1.62
monicsissplitintoatripletwithfrequenciesnν γ˙/πand
±
nν, andamplitudes u (e), v (e)andw (e), respectively.
n n n
As already noted by [11], the observation of any two TABLEII:Relativeerrorontotalsystemmassandchirpmass
“emission lines”, allows us to derive the orbital and the measurements assuming only GR contributes to γ˙ and ν˙, for
apsidal-motion frequency. In practice, for typical DWD a5-yrobservationofaM1=M2=0.6M⊙ DWDatSNR=10.
eccentricities(e < 0.5;see[5]), u (e) v (e) , w (e),
n n n
so that LISA wi∼ll primarily rely| on o|b≫ser|vation|s|of GW|s
at frequencies nν+γ˙/π for at least two values of n. eccentricities e 0.01 [24]. At these frequencies most
∼
We can compute whether and how accurately perias- systems are also expected to exhibit a detectable change
tron advance can be measured by computing the Fisher ofthe orbitalfrequency. Assuming thatonly generalrel-
information matrix (see e.g. [16]) associated with the ativity affects γ˙ and ν˙, the combined measurement of γ˙
measurement. The signal depends on α, δ, ι, ψ, A, e, and ν˙ allows us to determine the total system mass M
ν, ν˙, γ˙, φ0 and γ0 (the argument of periastron at an ar- and chirp mass M, as shown in Table II, and therefore
bitrary reference time). Due to the fact that α, δ, ι and the individual WD masses M1 and M2.
ψ are only weakly correlatedwith the remaining param- Astrophysical implications.—Wehaveinvestigatedthe
eters for an observation lasting several years [18], we do impact of tides, rotation, and GR on the apsidal motion
not include them inour analysis andcompute the angle- ofeccentricDWDsanditssignatureontheemittedGWs.
averaged Fisher information matrix for Tobs = 5 yr. We Basedon our present understanding of the astrophysical
conservatively consider the first 10 harmonics in Eq. (4) scenarios [5], we conclude that LISA will be able to ob-
and normalize the results to signals detected at an opti- servetheperiastronadvanceforthevastmajorityofsuch
mal signal-to-noise ratio (SNR) of 10 (the mean-square sources detected. These observations provide a new and
errorsscaleas 1/SNR).The resultsaresummarizedin unique probe into the internal structure of WDs.
≈
Table I for the parameters relevant to this investigation. Tides and stellar rotation strongly dominate the
For ν < 1mHz, the apsidal motion becomes progres- apsidal-motion rate at orbital frequencies above 1
sively ha∼rder to measure due to the smaller and smaller mHz, inducing phase shifts much larger than those≈esti-
phaseshift(cf.Fig.1). Atν 0.1mHztheestimateder- mated using only the GR contribution. In GW searches
≈
ror on γ˙/π is greater than γ˙/π itself and apsidal motion for eccentric binaries, it is therefore essential to include
becomes undetectable (the details of course depend on γ˙ in the signal templates as a phenomenological param-
theactualvaluesofe,ν,M1,M2 andSNRforthesource eter not bound by γ˙GR in order to not bias LISA surveys
at hand) [23]. However, for ν > 1mHz, LISA can detect against eccentric DWDs. In the interpretation of the
periastron precession and me∼asure γ˙/π with a relative data,neglectingtidalandrotationalcontributionswould
error of 1% 10%, depending on the mass and ec- lead to an overestimate of the total system mass derived
∼ −
centricity of the binary. This result holds even for small fromγ˙. This will likely induce a misclassificationof GW
4
sources as NS rather than DWD binaries, thus affecting nation of the total system mass [11]. For ν > 0.5mHz
the ratios of different populations of compact object bi- [5], the radiation reaction may also cause a m∼easurable
naries that hold essential signatures of stellar evolution drift in the orbital frequency. Assuming that there is
and binary formation mechanisms. On the other hand, no significant contribution from tidal and/or magnetic
the dependence of γ˙ on R1,2, M1,2, and k1,2 provides a spin-orbit coupling, measuring ν˙ with LISA yields the
new window into the internal structure of WDs and the sourcedistance aswellas the chirpmass[e.g.19]. In the
astrophysicsofsuchstars,but poses a severedegeneracy (small)regimewheregeneralrelativitydominatestheap-
problem for the extraction of astrophysical information sidalmotionandν˙ isdetectable,thecombinedknowledge
from measured apsidal-motion rates. More refined theo- ofthechirpmassandtotalsystemmassyieldsthemasses
reticalmodelingisthereforeneededinpreparationofthe of the individual WD components. For eccentric NS-NS
LISA mission to fully characterize the dependence of γ˙ binaries – which have negligible tidal/rotational distor-
on the WD physical parameters and to identify routes tions and tidal/magnetic spin-orbit coupling throughout
to untangle them. Even though we have here focused on the LISA band and a strongerν˙ than WD binaries – the
DWDs, our results also apply to WDs with NS compan- measurement of the individual masses at 10% level
≈
ions. These sources are in fact much “cleaner”probes of should be routine.
WD physics since the tidal and rotational distortions of
the NS contribute negligibly to the apsidal motion. In Acknowledgments.— This work is partially supported
this case, the apsidal-motion rate therefore carries the by a Packard Foundation Fellowship, a NASA BEFS
unique signature of only one WD rather than two. grant (NNG06GH87G), and a NSF CAREER grant
Atfrequencies < 1mHz,GReffectsdominatethepre- (AST-0449558) to VK. We are grateful to Brad Hansen
cession rate and m∼easurements of γ˙ allow the determi- and Chris Deloye for providing theoretical WD models.
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2
[7] Claret, A., & Gimenez, A. 1993, Astron. & Astrophys., is ≃61(1.3)/(1−e ) nHzfor ν =1(0.1) mHz.
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