Inferring the Presence of Tides in Detached White Dwarf Binaries
aa r X i v : . [ a s t r o - ph . S R ] S e p D RAFT VERSION S EPTEMBER
18, 2019Typeset using L A TEX twocolumn style in AASTeX62
Inferring the Presence of Tides in Detached White Dwarf Binaries A NTHONY
L. P
IRO The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA; [email protected]
ABSTRACTTidal interactions can play an important role as compact white dwarf (WD) binaries are driven togetherby gravitational waves (GWs). This will modify the strain evolution measured by future space-based GWdetectors and impact the potential outcome of the mergers. Surveys now and in the near future will generatean unprecedented population of detached WD binaries to constrain tidal interactions. Motivated by this, Isummarize the deviations between a binary evolving under the influence of only GW emission and a binary thatis also experiencing some degree of tidal locking. I present analytic relations for the first and second derivativeof the orbital period and braking index. Measurements of these quantities will allow the inference of tidalinteractions, even when the masses of the component WDs are not well constrained. Finally, I discuss tidalheating and how it can provide complimentary information.
Keywords: binaries: close — binaries: eclipsing — gravitational waves — white dwarfs INTRODUCTIONThe merger of compact white dwarf (WD) binarieshas been hypothesized to generate an incredible rangeof astrophysical systems and/or events. This includessdB/O stars (Saio & Jeffery 2000, 2002), R Cor Bor stars(Webbink 1984), fast radio bursts (Kashiyama et al. 2013),magnetic and DQ WDs (García-Berro et al. 2012), mil-lisecond pulsars (Bhattacharya & van den Heuvel 1991),magnetars (Usov 1992), a subset of gamma-ray bursts(Dar et al. 1992; Metzger et al. 2008), Type Ia supernovae(Iben & Tutukov 1984), a source of ultra-high energy cos-mic rays (Piro & Kollmeier 2016), and AM CVn binaries(Postnov & Yungelson 2006). An important uncertainty inconnecting specific binaries to these outcomes is the role oftidal interactions (Marsh et al. 2004) . Furthermore, thesebinaries are among the strongest gravitational wave (GW)sources in our Galaxy and will be prime targets for futurespace-based GW detectors (e.g., Nelemans 2009; Marsh2011; Nissanke et al. 2012; Tauris 2018). Tides will againbe key in determining the orbital evolution detected by theseobservations.Many double WD binaries are transferring mass, whichmakes it difficult to isolate the impact of tidal interactions.Detached WD binaries provide the perfect laboratory formeasuring these effects. The sample has grown with sur-veys such as ELM (Kilic et al. 2012), SPY (Napiwotzki et al.2004), and ZTF (Bellm et al. 2019), and will only accelerate Although see the arguments by Shen (2015) that the vast majority ofthese binaries may merge. in size over the next decade with SDSS-V (Kollmeier et al.2017) and LSST (LSST Science Collaboration et al. 2009;Korol et al. 2017). In particular, there is now a binary witha period of 12 .
75 min, SDSS J065133.338+284423.37 (here-after J0651, Brown et al. 2011), and another with a periodof 6 .
91 min, ZTF J153932.16+502738.8 (hereafter J1539Burdge et al. 2019), which should be especially helpful forstudying the role of tides.Motivated by these issues, in the following I provide ana-lytic relations that summarize how a binary evolving due toGW emission changes with the degree of tidal locking. Thesewill simplify the interpretation and isolate the impact of tidesin future binary WD observations. This is similar to discus-sions in Piro (2011), but here I focus on relations that willhelp constrain tidal interactions empirically rather than fol-lowing the time dependent orbital evolution. Some similarrelations are also provided in Shah & Nelemans (2014), butthese use approximations from Benacquista (2011). Here Ipresent exact relations that can be expressed analytically.In Section 2, I summarize the main equations determin-ing the binary evolution and derive the tidal corrections tothe first derivative of the orbital period, while in Section 3,I focus on the second derivative. In Section 4, I discuss thebraking index n , defined by ˙ Ω ∝ Ω n where Ω is the binary or-bital frequency. In Section 5, I discuss how the tidal lockingmay change with time, and what impact this has on the sys-tem’s evolution. Simple estimates for these expressions arepresented in Section 6. In Section 7, I discuss the durationof observations needed to measure deviations in the inspiralfrom just GW emission. The role of tidal heating in con-straining the degree of tidal locking is described in Section 8,and I conclude in Section 9 with a summary. FIRST DERIVATIVES OF THE ORBITAL PERIODConsider a binary with orbital separation a composed ofWDs with masses M and M and orbital frequency Ω = GM / a where M = M + M . The orbital angular momen-tum is J orb = ( Ga / M ) / M M . The angular momentum ofthe system decreases due to GW emission at a rate ˙ J gw = − G c M M Ma J orb . (1)Taking the time derivative of J orb and setting it equal to ˙ J gw results in a period derivative of ˙ P gw = − G c M M MPa = 3 ˙ J gw J orb P , (2)if GW emission is acting alone without any tides.The total angular momentum of the binary system is J tot = J orb + J wd = ( Ga / M ) / M M + I Ω + I Ω , (3)where I i and Ω i are the moments of inertia and spins of theWDs, respectively. To simply include the impact of tidal in-teractions, I introduce the variable η , the tidal locking factor,and set Ω = Ω = η Ω . When η = 0, there is no tidal effects,and η = 1 corresponds to being completely tidally locked. Indetail, there should be a separate η for each WD, since theycould have different degrees of tidal locking. Given the cur-rent level of measurements, it is simpler to consider a single η that represents the whole binary system. The total angularmomentum becomes J tot = ( Ga / M ) / M M + η ( I + I )( GM / a ) / . (4)Taking the time derivative of this and setting ˙ J tot = ˙ J gw , ˙ J gw = 12 ˙ aa J orb − ˙ aa J wd + ˙ ηη J wd (5)Using the relation that ˙ P / P = (3 / ˙ a / a , I find that when tidesare included the period derivative is ˙ P tide = ˙ P gw − ˙ η/η )( J wd / J orb ) P − J wd / J orb . (6)Note that I use the subscript “tide” to denote when both tidesand GWs are acting on the binary. Also, since J wd ∝ η , thesecond term in the numerator is well-behaved even for η = 0.Equation (6) demonstrates how tides cause the period todecrease more rapidly than with GW emission alone becauseangular momentum is being taken from the WD orbits and put into the individual WDs. If the tidal locking is increas-ing as the orbital period shrinks ( ˙ η > η = 1 with ˙ η = 0. Note in the former work (which used thevariable ∆ I = J wd / J orb ) their expression for the change in thefrequency derivative is only correct to order J wd / J orb . Equa-tion (6) ignores deviations due to the energy it takes to makea tidal bulge, since this effect is small (Benacquista 2011).The angular momentum ratio J wd / J orb continually comesup when considering tidal corrections. Useful ways to ex-press this include J wd J orb = η ( I + I ) MM M a = η ( I + I ) M / Ω / G / M M , (7)which is used for some of the estimates below. SECOND DERIVATIVES OF THE ORBITAL PERIODI next consider the impact of tidal interactions on the sec-ond derivative of the orbital period. For GW emission alone,taking the derivative of Equation (2) results in ¨ P gw = − ˙ aa ˙ P gw + ˙ PP ˙ P gw = − ˙ PP ˙ P gw , (8)where I have been careful to distinguish between ˙ P (the ex-act derivative of P ) and ˙ P gw (the derivative when just GWsare considered) because this difference will be important forsubsequent discussions. Since in this case ˙ P = ˙ P gw , then ¨ P gw = − ˙ P P . (9)Given that ¨ P will be difficult to measure, I focus on the sim-pler scenario where η = 1 with ˙ η = 0. To find the secondderivative including tides, it is helpful to first rewrite Equa-tion (6) as ˙ PJ orb − ˙ PJ wd = ˙ P gw J orb . (10)Taking the derivative of both sides of this expression, ¨ PJ orb + ˙ aa ˙ PJ orb − ¨ PJ wd − ˙ ΩΩ ˙ PJ wd = − ˙ PP ˙ P gw J orb + ˙ aa ˙ P gw J orb . (11)Rewriting all the first derivatives in terms of ˙ P , dividing by J orb , and then collecting similar terms, the final result is ¨ P tide (cid:18) − J wd J orb (cid:19) = − ˙ P tide ˙ P gw P − ˙ P P (cid:18) + J wd J orb (cid:19) . (12)When combined with Equation (6) for ˙ P tide (using ˙ η = 0 forconsistency), this expression provides an analytic relationsfor ¨ P tide . Comparing Equations (9) and (12) shows that tidalinteractions make the second derivative even more negativethan the GW only case. BRAKING INDEXBeyond the values of the first and second derivative, an-other way to think about tides is in terms of a braking index.The basic idea is analogous to pulsars (Shapiro & Teukolsky1983) where one wishes to measure n such that ˙ Ω ∝ Ω n . (13)The main difference is that here Ω refers to the orbital fre-quency of the binary rather than the pulsar spin frequency(and perhaps it should be referred to as an “acceleration in-dex” here), although also see discussions of the braking in-dex by Nelemans et al. (2004) and Stroeer et al. (2005) in thecontext of AM CVn systems. It is straightforward to show n = Ω ¨ Ω ˙ Ω = 2 − P ¨ P ˙ P . (14)Using the relations from Sections 2 and 3, for GWs only(Webbink & Han 1998), n gw = 11 / , (15)while when tides are included n tide = 103 + / + J wd / J orb − J wd / J orb . (16)Thus fitting for the power law ˙ Ω ∝ Ω n in real systems couldbe another way to infer the presence of tides. Although theexact value of n depends on the specific parameters of thebinary and the degree of tidal locking, simply showing that n > / THE RATE OF CHANGE OF TIDAL LOCKINGAn additional factor in ˙ P tide that is often not considered is ˙ η , the rate of change of the tidal locking. This will depend indetail on the model for the tidal interaction. In Piro (2011),the tide is treated using a parameterization with a standardquality factor Q . When this tidal interaction is integrated for-ward in time, it is found that the binary reaches an equilib-rium spin at any given point where the ratio of the tidal forc-ing frequency to the orbital frequency is roughly the ratio ofthe tidal synchronization time τ tide to the GW inspiral time τ gw = − (3 / P / ˙ P , i.e., 1 − η ≈ τ tide τ gw . (17)Furthermore, this work finds that for constant Q , τ tide /τ gw ∝ P / . This ratio thus gets smaller at shorter orbital period,resulting in a more tidally locked binary. In reality, this willdepend on the details of how the tides act, and so I considera general form of τ tide /τ gw ∝ P β with β >
0. Assuming a model of this form and taking the derivativeresults in, ˙ η ≈ − β ˙ PP τ tide τ gw ≈ − β ˙ PP (1 − η ) . (18)This particular model has a couple of important conse-quences for ˙ P . First, it makes ˙ P even more negative. Second,it adds corrections to ˙ P even when η ≈ η , while for η ≈ ˙ η ≈ ˙ η might befor more physical models, such as dynamical tides when theWDs are near a resonance (e.g., Fuller & Lai 2011). ESTIMATES FOR REAL SYSTEMSEquations (6) and (12) summarize the analytic expressionsfor the first and second period derivatives with tidal interac-tions, and Equation (16) the braking index. It is helpful toestimate what values are implied by these expressions.Low mass WD binaries are typical composed of a He andC/O WD. Taking typical values of M = 0 . M ⊙ and M =0 . M ⊙ , for GW emission alone, ˙ P gw = − . × − M . M . M − / . P − / s s − , (19)where M . = M / . M ⊙ , M . = M / . M ⊙ , M . = M / . M ⊙ , and P = P /
10 min. The fractional changewhen tides are included to first order in J wd / J orb when ˙ η = 0is ˙ P tide − ˙ P gw ˙ P gw ≈ J wd J orb ≈ . η I M − . M − . M / . P − / , (20)where I = ( I + I ) / g cm , which is estimated using amoment of inertia of I i ≈ . M i R i (Marsh et al. 2004). If oneinstead assumes ˙ η follows the model described in Section 5, ˙ P tide − ˙ P gw ˙ P gw ≈ [ β + (1 − β ) η ] 3 J wd ( η = 1) J orb . (21)Even for η = 0, this model gives a deviation in the periodderivative.The second derivative due to GW emission alone is calcu-lated from Equation (9) to be ¨ P gw = − . × − M . M . M − / . P − / s s − . (22)To estimate the second derivative with tides, consider Equa-tion (12) in the limit J wd ≪ J orb , ¨ P tide ≈ − ˙ P tide ˙ P gw P (cid:18) + J wd J orb (cid:19) − ˙ P P (cid:18) + J wd J orb (cid:19) (cid:18) + J wd J orb (cid:19) . (23)Substituting for ˙ P tide using Equation (6), and collecting termsfirst order in J wd / J orb results in ¨ P tide ≈ − ˙ P P (cid:18) + J wd J orb (cid:19) , (24)or a fractional change in the second derivative of the orbitalperiod of ¨ P tide − ¨ P gw ¨ P gw ≈ J wd J orb ≈ . η I M − . M − . M / . P − / . (25)Thus the deviation in the second derivative should be morepronounced (by a factor of a few) than the first derivative.The braking index estimated to first order in J wd / J orb is n tide ≈ + J wd J orb ≈ / + . η I M − . M − . M / . P − / . (26)Changes in n are at a level of around ten percent. Unlikecomparing ˙ P gw and ˙ P tide , n gw is independent of the massesof the WDs, so any deviation from 11 / ∼
50% larger than theestimate in Shah & Nelemans (2014). MEASURING THE BRAKING INDEXThe previous sections show that measuring n (or equiva-lently ¨ P ) provides a way to infer the presence of tides with-out having to know the WD masses. It is natural to ask howlong it will take to measure n with sufficient accuracy. Taylorexpanding the orbital phase of the binary (assuming φ = 0 at t = 0) φ = Ω t + ˙ Ω t + ¨ Ω t + · · · . (27)If the uncertainty in the eclipse timing of a binary is δ t , thenthe uncertainty in the phase is δφ = Ω δ t . The fractional un-certainty in constraining the frequency derivative can be readoff from the second term of the Taylor expansion to be δ ˙ Ω ˙ Ω ≈ δφ ˙ Ω t , (28)where t the length of time of the observation, while the un-certainty in the braking index is δ nn ≈ δ ¨ Ω ¨ Ω ≈ δφ ¨ Ω t . (29)Using the ¨ Ω for GW emission alone, this results in δ nn ≈ . M − . M − . M / . P / δ t (cid:18) t
10 yrs (cid:19) − , (30) where δ t = δ t /
10 ms for the eclipse timing accuracy. Thestrong scalings with P and t mean that n will be much easierto measure for short orbital period systems that are observedfor a long time. In particular, for J1539 a ≈
10% constrainton n should be possible after ≈ δ t / yrs. CONSTRAINING TIDES WITH TIDAL HEATINGAlthough direct measurements of the first and second pe-riod derivative are the cleanest way to infer tidal interactions,such measurements require observations over a long timebaseline. It is therefore useful to have other complementarymethods such as tidal heating.The total energy of the WD binary is composed of orbitaland spin components, E tot = E orb + E wd = − GM M a +
12 ( I + I ) η Ω . (31)Taking the time derivative of this, − ˙ aa E orb + ˙ ΩΩ E wd + ˙ ηη E wd = ˙ E gw − L tide , (32)where L tide is energy lost to tidally heating the WDs (definedto be positive here) and ˙ E gw = − G c M M Ma , (33)is the energy lost to GWs. Combining Equations (5) and (32),results in L tide = 2 (cid:18) ˙ ηη − ˙ PP (cid:19) (cid:18) E wd + E orb J wd J orb (cid:19) = (cid:18) ˙ ηη − ˙ PP (cid:19) ( I + I ) Ω η (1 − η ) , (34)for the tidal heating rate. A rough estimate can be made fromjust considering ˙ P from GWs (Iben et al. 1998), L tide ≈ − ˙ P gw P ( I + I ) Ω ≈ . × I M . M . M − / . P − / erg s − , (35)but the exact amount depends on the degree of tidal lockingand the model for how the tidal locking changes with time.In the case of ˙ η = 0, the heating rate scales as L tide ∝ η (1 − η ) , (36)Thus, in the limit of extremely weak tidal locking ( η ≈
0) theenergy losses go all into GWs and for strong tidal locking( η ≈
1) the energy losses go into spinning up the WDs, andin each case the tidal heating is small (also see the discussionin Fuller & Lai 2012). For η = 1 /
2, the heating is maximum.If the WD spins are allowed to vary independently (so that
Figure 1.
The tidal heating rate given by Equation (34) for thecases of J0651 (Hermes et al. 2012) and J1539 (Burdge et al. 2019)in the upper and lower panels, respectively. In each case, the bluecurves assume ˙ η = 0, while the red and purple curves use the modelfrom Section 5 for ˙ η with β = 1 / β = 1, respectively. Thedashed horizontal lines show the currently observed luminosities ofthe brighter component of each binary, while the dimmer compo-nent would be below the lowest plotted luminosities. Ω = η Ω and Ω = η Ω ), then the heating simply scales with I η (1 − η ) + I η (1 − η ) instead, and heating is maximum for η = η = 1 / ˙ η from Section 5, the heatingscales instead scales as L tide ∝ [ β + (1 − β ) η ](1 − η ) . (37)This again goes to zero for η = 1 because all of the extra en-ergy goes into spinning up the WDs, but now at η = 0 there isa non-zero amount of heating because the tidal locking factoris changing rapidly at small η . In addition, L tide will be largerfor the ˙ η = 0 case because ˙ P is more negative as well.Figure 1 shows how the tidal heating changes with η us-ing Equation (34). I consider cases where ˙ η = 0 (blue curves)and when ˙ η = 0 with either β = 1 / β = 1(purple curves). The β = 1 / Q model (Piro 2011), while β = 1 is chosen because it resultsin L tide ∝ − η , similar to the most extreme tidal heating ex-pected (Fuller & Lai 2012). The upper and lower panels useparameters (masses, radii, and orbital periods) appropriatefor J0651 and J1539, respectively. In each case, the brighterWD could be consistent with tidal heating, but it depends onthe details of the tidal dissipation. If ˙ η = 0 or if β is too small,then the tidal heating is insufficient to explain the observed luminosities. The similarity of the top and bottom panelspotentially indicates that similar physical processes are oc-curring to dissipate the tides in each of the systems. It iscurious that in this picture the C/O WD rather than the HeWD would be more tidally heated in J1539, but perhaps thiscan be accomplished via resonances with a dynamical tide.Alternatively, the large luminosities of the brighter WDsmay simply mean that the observed emission cannot be dom-inated by tides. Instead, the relative luminosities of theseWDs could reflect their age (Istrate et al. 2014, 2016). Itmay seem somewhat paradoxical that the C/O WD wouldbe younger in J1539, but there are binary scenarios wherethis can occur (Toonen et al. 2012). These binaries wouldthen have to be generated with periods fairly close what isobserved now, perhaps explaining how surveys were able tofind them so close to merger (one would expect many morelong period WD binaries if they start these longer periods,although there may be selection effects that favor short peri-ods). Another possibility discussed by Burdge et al. (2019)is that the C/O WD has undergone recent accretion.For both J0651 and J1539, the dimmer WDs argue thatthere is little tidal heating (they are both so dim to be belowthe luminosities plotted in Figure 1). Given the symmetri-cal dependence on η in Equation (36), they would have toeither have η . .
005 or η & . CONCLUSIONS AND DISCUSSIONIn this work, I have presented analytic relations for the im-pact of tidal interactions on ˙ P , ¨ P , the braking index, and heat-ing for binaries driven together by GWs. The ˙ P measured forJ0651 and J1539 cannot currently answer whether tidal inter-actions are occurring because the masses of the WDs are notknown to sufficient accuracy. The braking index n , definedby ˙ Ω ∝ Ω n , is 11 / n = 11 / n improves withtime in Equation (30) shows that it will take on the order of ≈
10 yrs to make such measurements for the shortest knowndetached eclipsing binaries. These simple prescriptions areuseful for assessing the ability of space-based GW detectorsto infer the presence of tides (e.g., Shah & Nelemans 2014;Littenberg & Cornish 2019). Measuring tidal heating is another way to constrain tidalinteractions. The brighter WDs in both J0651 and J1539 haveluminosities that could be explained by tidal heating, but thisdepends on the tidal dissipation model used. This makes itdifficult to directly constrain the influence of tides from theluminosities alone. The dimmer WDs must either be nearlytidally locked or not locked at all under the assumption thattheir tidal heating can be readily radiated. Having a widersample of detached WDs with different orbital periods in the future will hopefully help determine how well WD luminosi-ties can be used to constrain tidal interactions.I thank Jim Fuller, J. J. Hermes, Thomas Kupfer, ThomasMarsh, Thomas Tauris, and Silvia Toonen for helpful discus-sions and feedback. I also thank the organizers of The Begin-ning and Ends of Double White Dwarfs meeting in Copen-hagen, Denmark (July 1-5, 2019), where some of this workwas inspired.REFERENCES10 yrs to make such measurements for the shortest knowndetached eclipsing binaries. These simple prescriptions areuseful for assessing the ability of space-based GW detectorsto infer the presence of tides (e.g., Shah & Nelemans 2014;Littenberg & Cornish 2019). Measuring tidal heating is another way to constrain tidalinteractions. The brighter WDs in both J0651 and J1539 haveluminosities that could be explained by tidal heating, but thisdepends on the tidal dissipation model used. This makes itdifficult to directly constrain the influence of tides from theluminosities alone. The dimmer WDs must either be nearlytidally locked or not locked at all under the assumption thattheir tidal heating can be readily radiated. Having a widersample of detached WDs with different orbital periods in the future will hopefully help determine how well WD luminosi-ties can be used to constrain tidal interactions.I thank Jim Fuller, J. J. Hermes, Thomas Kupfer, ThomasMarsh, Thomas Tauris, and Silvia Toonen for helpful discus-sions and feedback. I also thank the organizers of The Begin-ning and Ends of Double White Dwarfs meeting in Copen-hagen, Denmark (July 1-5, 2019), where some of this workwas inspired.REFERENCES