prospects for detecting exoplanets around double white
TRANSCRIPT
Draft version September 7, 2021Typeset using LATEX twocolumn style in AASTeX63
Prospects for detecting exoplanets around double white dwarfs with LISA and Taiji
Yacheng Kang ,1, 2 Chang Liu ,1, 2 and Lijing Shao 2, 3
1Department of Astronomy, School of Physics, Peking University, Beijing 100871, China2Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China3National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
(Received June 1, 2021; Revised June 1, 2021; Accepted June 1, 2021)
Submitted to AJ
ABSTRACT
Recently, Tamanini & Danielski (2019) discussed the possibility to detect circumbinary exoplanets(CBPs) orbiting double white dwarfs (DWDs) with the Laser Interferometer Space Antenna (LISA).Extending their methods and criteria, we discuss the prospects for detecting exoplanets around DWDsnot only by LISA, but also by Taiji, a Chinese space-borne gravitational-wave (GW) mission whichhas a slightly better sensitivity at low frequencies. We first explore how different binary masses andmass ratios affect the abilities of LISA and Taiji to detect CBPs. Second, for certain known detachedDWDs with high signal-to-noise ratios, we quantify the possibility of CBP detections around them.Third, based on the DWD population obtained from the Mock LISA Data Challenge, we presentbasic assessments of the CBP detections in our Galaxy during a 4-year mission time for LISA andTaiji. We discuss the constraints on the detectable zone of each system, as well as the distributionsof the inner/outer edge of the detectable zone. Based on the DWD population, we further inject twodifferent planet distributions with an occurrence rate of 50% and constrain the total detection rates.We finally briefly discuss the prospects for detecting habitable CBPs around DWDs with a simplifiedmodel. These results can provide helpful inputs for upcoming exoplanetary projects and help analyzeplanetary systems after the common envelope phase.
Keywords: Gravitational waves (678) — White dwarf stars (1799) — Exoplanet detection methods(489) — Habitable zone (696)
1. INTRODUCTION
So far, more than 4,300 exoplanets have been dis-covered using electromagnetic (EM) techniques, but weknow very little about planetary systems under extremeconditions, such as around white dwarfs (WDs). Theo-retical works suggest that a planet can survive the host-star evolution (Livio & Soker 1984; Duncan & Lissauer1998; Nelemans & Tauris 1998), and the observationalresults also confirm that P-type exoplanets (Dvorak1986) can exist around stars after one or two commonenvelope (CE) phases, for example, around system NNSer, which contains a WD and a low-mass star (Beuer-mann et al. 2010, 2011), and PSR B1620−26AB, whichcontains a WD and a millisecond pulsar (Sigurdsson1993; Thorsett et al. 1993). Nevertheless, no exoplan-ets have been discovered orbiting double WDs (DWDs)
Corresponding author: Lijing Shao
to date (Tamanini & Danielski 2019). Given that morethan ∼ 97% of stars will become WDs (Althaus et al.2010) and about 50% of Solar-type stars are not single(Raghavan et al. 2010; Duchene & Kraus 2013), thereshould be a considerable population of DWDs in ourGalaxy. If exoplanets do exist around DWDs, the de-tection of such a population in the future would be verypromising.
However, even if exoplanets can endure the CEphase(s), they may collide with each other or be ejectedfrom evolving systems due to the complex orbital evolu-tion (Debes & Sigurdsson 2002; Veras et al. 2011; Veras& Tout 2012; Veras 2016; Mustill et al. 2018). Strongtidal forces can crush the planetary cores during theirmigration or scattering processes (Farihi et al. 2018),which may be associated with the WD pollution effect(Jura et al. 2009; Farihi 2016; Brown et al. 2017; Small-wood et al. 2018). Therefore, as noted by Danielskiet al. (2019), the detection and study of these objectscan help analyze planetary systems after CE phases andthe planetary formation processes.
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2 Y. Kang, C. Liu, L. Shao
1 10 100 1000 10000Distance [pc]
10 3
10 2
10 1
100
101
Plan
etar
y m
ass [
MJ]
Imaging Microlensing RV Transit
Figure 1. Distribution of planetary mass and distance of the confirmed exoplanets. The black dotted line marks a distance of
3 kpc away from our Solar system. Different markers correspond to the currently known exoplanets using different EM detection
techniques. Note that we only plot exoplanets with masses below the deuterium burning limit, i.e. Mp = 13MJ (see Sec. 3.2).
The data were obtained from the NASA Exoplanet Archive.
Owing to the intrinsic faintness of DWDs and the sen-sitivity limits of the current EM methods, there are nomore than 200 known detached DWD systems (Brownet al. 2020). The amount of known interacting (AMCVn) DWD systems is even fewer (Ramsay et al. 2018).A more frustrating fact is that most detected exoplan-ets are restricted to the Solar neighborhood (∼ 3 kpc)and discovered successfully by EM detection methods(see Fig. 1), such as radial velocity (RV) and transitmeasurements.1 Gravitational microlensing is capableof detecting exoplanets farther away (∼ 8 kpc) towardsthe Galactic bulge, but scarcity and unrepeatability canbe two of the main restricting factors. All these showthat it is hard to discover exoplanets orbiting DWDs us-ing traditional EM techniques in the Milky Way (MW).
Differently, gravitational waves (GWs) can providea powerful tool in the detection of exoplanets beyondour Solar system without the above selection problem(Seto 2008; Wong et al. 2019). Recent studies have ex-plored the prospects for detecting new circumbinary ex-oplanets (CBPs) around DWDs in our Galaxy by usingthe Laser Interferometer Space Antenna (LISA) mission(Tamanini & Danielski 2019; Danielski et al. 2019). Themethod, measuring the perturbation on the GW signalsdue to CBPs, is conceptually similar to the RV tech-nique. Compared to the traditional EM methods, GWdetections are able to detect such a CBP population inprinciple everywhere in the MW without being affectedby stellar activities, which, in contrast, should be consid-ered rather carefully in EM observations. An even more
1 https://exoplanetarchive.ipac.caltech.edu
exciting prospect is that space-borne GW detectors havethe potential to detect DWDs in nearby galaxies (Korolet al. 2020; Roebber et al. 2020), up to the border ofthe Local Group (Korol et al. 2018). From these we cansee that in the near future, considering the rapid devel-opment of GW astronomy, the first ever extra-galacticplanetary system might be detected by the space-borneGW detectors (Danielski & Tamanini 2020).
In this paper, firstly, we followed the method and pro-cedure presented in Tamanini & Danielski (2019) to dis-cuss the prospects for detecting CBPs around DWDsby using two different space-borne GW detectors, LISAand Taiji. We give a complementary discussion on thepossibility of CBP detections around some known de-tached DWDs with high signal-to-noise ratios (SNRs).Secondly, based on the DWD population from the MockLISA Data Challenge (MLDC) Round 4 (Babak et al.2010), we explore the population of CBP detections inour Galaxy during a 4-year mission time for GW detec-tors. For comparison, recent work has used dedicatedbinary synthesis simulations for the DWD population(Korol et al. 2017; Lamberts et al. 2019), and our re-sults are remarkably consistent with them, yet providinga faster way for assessments. Thirdly, we introduce de-tectable zone for each promising detectable system anddiscuss the distributions of the inner/outer edge of thisarea. Fourthly, we inject two different planet distribu-tions with an occurrence rate of 50% for DWDs to con-strain the total detection rate during a 4-year missiontime. Finally, we briefly discuss the prospects for de-tecting habitable CBPs around DWDs with a simpli-fied model by assuming that the habitable zone bound-ary criteria for main-sequence (MS) stars also apply to
LISA/TAIJI Exoplanets 3
DWDs. These results can provide a crude benchmarkfor upcoming exoplanetary projects and help analyzeplanetary systems after CE phases.
The organization of this paper is as follows. We brieflyintroduce the two space-borne GW detectors that weuse and the construction of their sensitivity curves inSec. 2. In Sec. 3, we overview the method proposed inTamanini & Danielski (2019), and present the character-istics of DWD populations and CBP models used in ourwork for LISA and Taiji. Using the above ingredients,we report detailed analyses and our results on variousaspects of CBP detections in Sec. 4. Finally, we presenta conclusion in Sec. 5.
2. DETECTORS
The era of GW astronomy has begun since the firstdirect detection of GWs, namely GW150914 (Abbottet al. 2016). Generally, the ground-based GW detectorsare sensitive to frequencies between ∼ 10 Hz and a few ofkHz, which has made them succeed in “listening” to nu-merous GWs from merging stellar-mass sources, like bi-nary black holes and binary neutron stars (Abbott et al.2017). For the space-borne detectors, such as LISA andTaiji, the sensitive frequency band ranges from 0.1 mHzto 1 Hz due to their much longer arm lengths and specificoptics. In such a frequency range, the potential GW sig-nals come from different sources, and are considered tohave great astronomical and cosmological significances(Cutler & Thorne 2002; Berti et al. 2005; Klein et al.2016; Shi et al. 2019). In particular, Galactic binariesincluding DWDs are one class of the prominent sourcesemitting continuous GWs in this frequency band.
2.1. LISA and Taiji
LISA mission, proposed by an international collabo-ration of scientists called the LISA Consortium, is anESA-led L3 mission, with NASA as a junior partner, torecord and study gravitational radiation in the millihertzfrequency band (Amaro-Seoane et al. 2017). It consistsof three spacecrafts with 2.5× 106 km arm-lengths trail-ing the Earth and moving in the Earth orbit around theSun. In the sensitive frequency range of LISA, the dom-inant GW sources by numbers will be Galactic DWDs inthe MW (Lamberts et al. 2019). So for any CBPs orbit-ing DWDs, LISA would be a promising tool to indirectlydetect them. As mentioned in Sec. 1, the potential ofLISA to detect the first extra-galactic planetary was dis-cussed by Danielski & Tamanini (2020).
On the other hand, Taiji, whose prototype was startedin 2008, is a Chinese space-borne GW mission similar toLISA. It also consists of three satellites forming a giantequilateral triangle, but with 3 × 106 km arm-lengths,slightly longer than that of LISA (Luo et al. 2020; Ruanet al. 2020; Wang & Han 2021). These satellites areplanned to orbit the Sun in the Earth orbit with approx-imately 20 degrees ahead of the Earth. Taiji also aimsto detect low-frequency GW sources in the frequency
band between 0.1 mHz to 1 Hz. Its sensitivity curve atthe lower frequency range performs slightly better thanLISA, thus, as we will see, it has advantages on the de-tection of DWDs and CBPs.
2.2. Sensitivity curves
When it comes to GW detectors, sensitivity curves areimportant performance guidelines. We can use them asa tool to evaluate what types of sources can be detectedduring the mission. As described in Robson et al. (2019),we know that the sensitivity of GW detectors dependson the GW frequency f , as given by
Sn(f) =10
3L2
[POMS + 2
(1 + cos2 (f/f∗)
) Pacc
(2πf)4
]×[1 +
6
10
( ff∗
)2]+ Sc(f) ,
(1)
where Sn(f) is referred to as the effective noise powerspectral density; L is the arm length of space-borneGW detector; f∗ = c/(2πL) is called the transfer fre-quency. Due to the longer arm length of Taiji (L =3× 106 km), f∗ is a little smaller for Taiji than for LISA(L = 2.5× 106 km). In the expression above, the single-link optical metrology noise is quoted as Pacc, while thesingle test-mass acceleration noise POMS is slightly dif-ferent between the two missions. More details and anal-yses of these parameters can be found in Robson et al.(2019) and Wang & Han (2021).
Besides the instrument noise, estimates for the confu-sion noise Sc(f) are also very important. Sc(f) is causedby the unresolved Galactic binaries, and it is associatedwith the design of the space-borne detectors. As de-scribed in Robson et al. (2019), estimates for Sc(f) arewell fit by
Sc(f) =Afixf−7/3e−f
α+βf sin(κf)
×[1 + tanh
(γ(fk − f)
)]Hz−1 ,
(2)
where the fixed amplitude Afix = 9×10−45, the knee fre-quency fk = 0.00113 and other fit parameters are givenfor a 4-year observational time as α = 0.138, β = −221,κ = 521, and γ = 1680. We plot the sensitivity curvesof LISA and Taiji in terms of their characteristic strain,√fSn(f), in Fig. 2. Note that we have assumed a 4-
year nominal mission duration for all the discussionsthroughout this paper. The curve of Taiji taken fromLuo et al. (2020) is combined with Sc(f), which we as-sume to be approximately the same for LISA and Taijidue to their similar designs and configurations. In re-ality, as the noise in the low-frequency band for Taijiis slightly better for LISA, such an assumption puts usbeing conservative for Taiji’s performance, as Taiji willbe able to distinguish more Galactic binaries at thesefrequencies. Future studies could refine this point.
4 Y. Kang, C. Liu, L. Shao
3. METHOD
As mentioned in the Introduction, the GW method fordetections of CBPs was firstly proposed by Tamanini& Danielski (2019). This approach relies on the largeDWD population with orbital periods . 1 hour, whichare expected to be the most numerous GW sources forspace-based mHz GW detectors (Nelemans et al. 2001;Yu & Jeffery 2010; Amaro-Seoane et al. 2017; Lambertset al. 2018; Breivik et al. 2020). Because of the rich-ness of potential sources, GWs could be a powerful toolto detect CBPs around DWDs. In this section, we willpresent our methodology to analyze the problem, whichextends the original one in Tamanini & Danielski (2019).In Sec. 3.1, we describe how we obtain the DWD popula-tion with some reasonable assumptions. In Sec. 3.2, weprovide some details about the CBP injection process.The method for the GW detection of CBPs is discussedin Sec. 3.3.
3.1. DWD population
We consider systems that compose of an exoplanetaround a DWD. For such three-body systems, there isno doubt that the gravity of DWDs dominate the GWsignal when compared with that of exoplanets. There-fore, we provide quantitative estimates and constraintsfor the detection of CBPs in our Galaxy based mainly onthe population of DWDs. To give quick assessments, weobtain the DWD population from the MLDC Round 4,which is designed to demonstrate and encourage theanalysis of different GW sources (Babak et al. 2010).MLDC Round 4 includes a Galactic DWD populationwith ∼ 3.4 × 107 interacting binaries and ∼ 2.6 × 107
detached ones. We abandon the population of the in-teracting systems mainly for two reasons: (i) the chirpmasses of the accreting systems are hard to obtain withGW observations only, and (ii) accreting effects wouldcomplicate our analysis of CBPs. Such a treatment wasadopted in previous work as well (Tamanini & Danielski2019; Danielski et al. 2019), and we leave the accretingsystems for future studies.
Different values of DWD parameters not only leadto different SNRs, but also change the estimations ofdetecting abilities. This was analyzed in Tamanini &Danielski (2019). We will also discuss the detectingabilities with different values of mass and mass ratioin Sec. 4.1. When we perform the calculations to assessthe prospects of the final CBP detections in the MW, weacquire the parameters of each binary from the dataset,2
including the GW frequency f , the frequency derivativef , the ecliptic latitude β, the ecliptic longitude λ, theGW amplitude A, the inclination ι, and the polariza-tion phase ψ. More details of the population and theparameters are presented in Babak et al. (2008, 2010).
2 https://asd.gsfc.nasa.gov/archive/astrogravs/docs/mldc/
0.0001 0.001 0.01 0.1GW Frequency [Hz]
10 22
10 21
10 20
10 19
10 18
10 17
Char
acte
ristic
Stra
in
LISATaiji
Figure 2. Characteristic strains of the detached DWD pop-
ulation, whose SNR > 10 for Taiji, are plotted with grey
circles. The known detached DWDs with high SNRs (listed
in Table 3 and discussed in Sec. 4.2.1) are highlighted with
red stars. Note that the DWD population is plotted based
on a crude assumption that two WDs are equal in mass. We
also plot the sensitivity curves of LISA (blue line) and Taiji
(red line) as given in Sec. 2.2.
As we will see later in Sec. 3.3.1, we can derive the chirpmass M of each system through the use of observedGW frequency f and its time derivative f . By assum-ing that the two WDs are almost equal in mass, we canthen acquire the mass of each WD pair, (m1,m2), andtheir total mass Mb ≡ m1 + m2. We regard this as acrude but reasonable treatment because the mass ratioq = m1/m2 (m1 > m2) under discussion is often con-sidered lower than 3 for detected DWDs (for example,see e.g. Korol et al. 2017). In Sec. 4.1, we will show lit-tle differences in detecting abilities when realistic massratio is considered.
With the above consideration, we re-calculate theSNRs of 31,530 “bright” detached Galactic binaries fromMLDC Round 4 for the two detectors. We find that ap-proximately 2.9× 104 (2.2× 104) detached DWDs haveSNR > 7 for Taiji (LISA) during a 4-year mission. Thenumber becomes 2.5 × 104 (1.6 × 104) for SNR > 10.For the following, we filter out all detached binaries withSNR < 10, for both LISA and Taiji, to get more reli-able estimations and striking contrasts between the twomissions in detecting abilities. Also, a high SNR is gen-erally needed in order to have CBP detections aroundDWDs. In Fig. 2 we show the dimensionless charac-teristic strain of the detached DWD population withSNR > 10 for Taiji. Because of the direct use of theDWD catalog, we could provide faster assessments forcomparisons which are consistent with previous workwithin an order of magnitude (Korol et al. 2017; Lam-berts et al. 2019; Danielski et al. 2019).
LISA/TAIJI Exoplanets 5
3.2. Injection of CBP models
When we consider CBPs around a DWD, there is noevidence to claim that every DWD should have suchan exoplanet. Given that no planets have been discov-ered orbiting DWDs so far, we take a bold approach,following Danielski et al. (2019), and set 50% as theoccurrence rate for our synthetic population of CBPsaround DWDs, which is obtained according to the ob-served frequency of WD pollution effect (Koester et al.2014). Note also that even if such CBPs exist, we maymiss these exoplanets using space-borne GW detectorsfor a variety of reasons. Therefore, a combination ofdiverse semi-major axis (a) and CBP mass (Mp) distri-butions have been tested in Danielski et al. (2019), fromwhich we adopt the optimistic and the pessimistic casesas reference points. Notice that there is a difference be-tween the distributions in Danielski et al. (2019) andours in the planet’s orbital inclination i. Instead of set-ting a uniform distribution in cos i, we inject CBPs intothe DWD systems by assuming coplanar circular orbits.There are theoretical indications that CBPs prefer tobe coplanar with their central binaries (Kennedy et al.2012; Foucart & Lai 2013), and coplanar orbits havebeen considered in various other work as well (Dvorak1986; Holman & Wiegert 1999; Eberle et al. 2008; Hong& van Putten 2021). It is certainly advantageous to re-fine the currently quite uncertain CBP population mod-els in future for more accurate predictions, in particularfor a realistic estimate via the GW method. We willshow more details and our detection rates in Sec. 4.2.3.
3.3. Detection of CBPs around DWDs
This subsection briefly introduces the method for theGW detection of a CBP using space-borne GW detec-tors. We follow Tamanini & Danielski (2019) to modelthe perturbation induced by CBPs around DWDs. Wefirst describe some characteristics of the three-body sys-tem in Sec. 3.3.1, and then provide more details aboutthe parameter estimation process in Sec. 3.3.2.
3.3.1. Perturbation due to a CBP
Considering a three-body system composed of a DWDemitting GWs with an exoplanet on the outer orbit (P-type system), we assume that the separation betweenthe planet and the DWD is much greater than the sep-aration between the two WDs. For simplicity, we alsoconsider both these orbits as circular Keplerian orbits.This could root in the binary evolution scenarios. Basedon these assumptions, we obtain the radial velocity ofthe DWD with respect to the common center of mass(CoM),
vrad(t) = −K cosϕ(t) . (3)
We have defined,
K =(2πG
P
)1/3 Mp
(Mb +Mp)2/3
sin i , (4)
ϕ(t) =2πt
P+ ϕ0 , (5)
where P and i are respectively the orbital period andinclination of the CBP, ϕ(t) is the outer orbital phase,and ϕ0 is its initial value at t = 0.
Through the Doppler effect, the observed GW fre-quency changes in the Earth reference frame to,
fobs(t) =
(1 +
vrad(t)
c
)fGW(t) , (6)
where fGW(t) is the GW frequency in the DWD refer-ence frame (twice the DWD orbital frequency). Galac-tic binaries take much longer than the mission time ofGW detectors to merge, and their frequencies are chang-ing very slowly. We can then describe their time evo-lution with a Taylor expansion and neglect the secondand higher-order terms by using,
fGW(t) = f + f t+O(t2), (7)
where f is the initial observed GW frequency, and f isits time derivative, which are related to the chirp massM = (m1m2)3/5/(m1 +m2)1/5 of the DWD system via,
f =96
5π8/3f 11/3
(GMc3
)5/3
. (8)
Finally, by integrating the observed GW frequencyfobs(t), we can obtain the phase at the observer of theGWs,
Ψobs(t) = 2π
∫fobs (t′) dt′ + Ψ0 , (9)
where Ψ0 is the constant initial phase. The final formof the observed phase is given by,
Ψobs(t) =2π(f +
1
2f t)t− Pf
cK sinϕ(t)
− P ft
cK sinϕ(t)− P 2f
2πcK cosϕ(t) .
(10)
With all the equations above, the parameters character-izing the DWD and the perturbation induced by a CBPcan thus be extracted from the GW phase evolution.
3.3.2. Parameter estimation for LISA and Taiji
In low frequency range, LISA and Taiji each can beeffectively seen as a pair of two-arm GW detectors likeLIGO and Virgo, and output two linearly independentsignals, hI(t) and hII(t). We often assume that the noiseis stationary and Gaussian, and then the two signals
6 Y. Kang, C. Liu, L. Shao
in each independent channel can be written as (Cutler1998),
hI,II(t) =
√3
2AI,II(t) cos
[Ψobs(t) + Φp
I,II(t) + ΦD(t)],
(11)where AI,II(t) are amplitudes of GW signals that con-tain the constant intrinsic amplitudes of the waveformand the antenna pattern functions of the detector. Inour case the waveform is approximated by a circularNewtonian binary. The antenna pattern functions arerelated to geometric parameters, including the locationof the source (θS , φS), the orientation of the DWD orbit(θL, φL), and the configuration of the space-borne detec-tor. In Eq. (11), Φp
I,II(t) are the waveform’s polarizationphases induced by the change of the orientation of thedetector. The Doppler phase ΦD(t) is the difference be-tween the phase of the wavefront at the detector and thephase of the wavefront at the Sun. It is further relatedto the Earth-Sun distance and the orbital period of theEarth. The full expressions for all above quantities canbe found in Cutler (1998), Cornish & Larson (2003), andKorol et al. (2017).
Based on the above analysis, our next step is to sim-ulate the response of LISA and Taiji and perform pa-rameter estimation. We use the Fisher informationapproach, as was employed by Tamanini & Daniel-ski (2019). For each DWD, there are 11 parameters,
λ ={
ln(A),Ψ0, f, f , θS , φS , θL, φL,K, P, ϕ0
}, charac-
terizing the observed GW waveform. The Fisher matrixcan be written as
Γij =2
Sn (f)
∑α=I,II
∫ Tobs
0
(∂hα(t)
∂λi· ∂hα(t)
∂λj
)dt . (12)
We use the one-sided noise power spectral density Sn(f)of the detector from Eq. (1). For each DWD, it ismerely a constant in Eq. (12) because the binary isquasi-monochromatic during the observational time Tobs
as long as fTobs � f .Similarly, the SNR of the signal can be written as,
SNR =
(2
Sn (f)
∑α=I,II
∫ Tobs
0
hα(t)hα(t) dt
)1/2
. (13)
This allows us to scale all results with the SNR by rescal-ing Sn(f) (Tamanini & Danielski 2019). From the in-verse of the Fisher matrix, we can obtain the uncertain-ties and correlations of parameters as the elements ofthe variance-covariance matrix Σij ,
Σij = 〈∆λi∆λj〉 =(Γ−1
)ij. (14)
Cutler (1998) has studied the uncertainties for binaryparameters. We follow his method and determine dif-ferent partial derivatives of Ψobs(t). Since the error ∆f
would be much larger than the signal’s f itself, we adopt
a treatment to simply set the fiducial value f = 0 with-out introducing noticeable changes (Takahashi & Seto2002). Here we only show expressions of the partialderivatives differing from the equations in the Sec. IV ofCutler (1998),
∂KΨobs(t) = −Pfc
sinϕ(t) ,
∂PΨobs(t) = −fcK sinϕ(t) +
2πft
cPK cosϕ(t) ,
∂f Ψobs(t) = 2πt− P
cK sinϕ(t) ,
∂f Ψobs(t) = πt2 − Pt
cK sinϕ(t)− P 2
2πcK cosϕ(t) ,
∂ϕ0Ψobs(t) = −PfcK cosϕ(t) .
(15)
To measure the additional perturbation due to an ex-oplanet, Tamanini & Danielski (2019) paid more atten-tion to the three parameters associated with the CBP,namely K,P, ϕ0. Since the value of ϕ0 is not importantfor our final results, we fix ϕ0 = π/2. We set 30% tobe the detection criterion on both ∆K/K and ∆P/P ,meaning that a detectable CBP is defined with esti-mated parameter precision being better than this value.
4. RESULTS
Now we give detailed analyses and discuss theprospects for detecting exoplanets around DWDs. UsingTaiji as a demonstration, we first give some complemen-tary discussion on the detecting abilities with differentvalues of masses in Sec. 4.1. We compare our final re-sults of the CBP detection between LISA and Taiji inSec. 4.2.
4.1. Effects of masses
From Eq. (4), we can see that when other parame-ters related to the CBP are fixed, such as its angularposition and the distance, the perturbation K causedby the exoplanet is getting smaller with the increase ofthe total mass Mb. Meanwhile, the SNR will be in con-trast higher when we only increase the total mass of theDWD. Therefore, to give a quantitative evaluation, weperform parameter estimation on the same set of sys-tems except that their total masses are ranging from0.2M� to 2.8M�. We assume two WDs are equal inmass and fix the other parameters of the DWD as
Ψ0 = 0 , θS = 1.27 , ϕS = 5 , i = ι =π
3. (16)
The frequency and distance of the system are fixed tof = 5 mHz and dDWD = 10 kpc respectively. Note thatwe derive the orientation of its orbit (θL, ϕL) by invers-ing Eq. (39) and Eq. (40) in Cornish & Larson (2003).
We plot for comparison in Fig. 3 the selection func-tions of Taiji based on these values. The four dotted
LISA/TAIJI Exoplanets 7
0.1 1 10Distance to the binary star [au]
1
10
102
103
Plan
etar
y m
ass [
MJ]
Mb = 0.2 MMb = 1.0 MMb = 1.8 MMb = 2.6 M
Figure 3. Selection functions of Taiji in the mass-separation
parameter space for exoplanets. Four dotted lines in differ-
ent colors denote systems with different total masses. The
dashed horizontal grey line corresponds to the deuterium
burning limit Mp = 13MJ. Circles are the detectable min-
imum planetary masses for each system, and left/right tri-
angles denote the boundary values we set. The peak of each
three-body system is caused by the degeneracy between the
motion of Taiji and the motion of the DWD around the three-
body CoM.
lines in different colors denote DWDs with different to-tal masses, and the dashed horizontal grey line denotesthe deuterium burning limit Mp = 13MJ, which is con-sidered to be the upper limit of the CBP mass (Danielskiet al. 2019). Therefore, it is the area above the dottedlines and below the dashed line that delimits the de-tectable mass-separation parameter space of the CBPfor Taiji. The peak of each line is caused by the degen-eracies between the motion of Taiji, and the motion ofthe DWD in the three-body system around its commonCoM (see the section “Methods” in Tamanini & Daniel-ski 2019). Sharp peaks appear at P = 1 yr (the orbitalperiod of Taiji), while there are also other smaller peaksat higher harmonics of it.
For each system, we have fixed the range of the CBPorbital period to P = 0.025–10 yr, and calculated theCBP’s distance to the CoM by Kepler’s third law,
a3 =GP 2
4π2(Mb +Mp) . (17)
We mark the distance boundaries as the left and righttriangles in Fig. 3. When the period of the CBP is set toa same value, the planetary orbital size gets larger withthe increase of the total mass of the DWD. We find thatdetecting abilities of each system are gradually gettingbetter with the increasing CBP period before becom-ing worse near the period of one year. There must be aminimum value Mmin
p that corresponds to the detectable
101
102
103
SN
R
(a)
0
4
8
12
Mm
inp
[MJ]
(b)
0.5 1.0 1.5 2.0 2.5
Mb [M�]
1
2
3
am
in[a
u]
(c)
0
100
200
SN
R
(d)
2 4 6 8 10 12
q
2
4
6
Mm
inp
[MJ]
(e)
Figure 4. (a) The SNR, (b) the detectable minimum plane-
tary mass, and (c) its corresponding orbital size, as functions
of the total mass for equal-mass DWDs. (d) The SNR and
(e) the detectable minimum planetary mass, as functions of
the mass ratio q for DWDs with a total mass Mb = 1.4M�.
More details about the parameters are given in Sec. 4.1.
minimum planetary mass for each system, which we de-note by circles in Fig. 3.
Moreover, we can see that the detectable parameterspace is getting wider with the increase of the totalmass of a binary, and the detectable minimum plane-tary mass is getting smaller. We point out that this canbe explained by the parameter estimation criterion onK, since in most cases the 30% criterion is not applica-ble for ∆P/P , as we can see in the Fig. 2 of Tamanini& Danielski (2019). Eq. (4) tells us that the ampli-
tude K ∝ M−2/3b . After plugging it into Eq. (12) and
Eq. (14), we get ∆K ∝ M−5/3b . Therefore, the de-
tectable minimum planetary masses limited by ∆K/Kare inversely proportional to the total masses of DWDs.We also find similar results in the upper panels of Fig. 4with a wider range of Mb from 0.2M� to 2.8M�. Fromthese we conclude that SNR is the dominating factorin detecting abilities when we change the total mass ofthe DWD. Note that we choose 2.8M� to be the upper
8 Y. Kang, C. Liu, L. Shao
Table 1. Possible detections of CBPs around known DWDs. For each system, amin is the detectable minimum planetary orbital
size based on our parameter estimation. Through the comparative results of LISA and Taiji, we list the detectable minimum
planetary mass Mminp with the SNR for each DWD. For each promising system, we calculate the IDZ and ODZ. Other relevant
properties of these known detached DWDs are in the Appendix.
Taiji LISA
Source amin Mminp IDZ ODZ SNR Mmin
p IDZ ODZ SNR[au] [MJ] [au] [au] [MJ] [au] [au]
ZTF J153932.16+502738.8 1.53 1.39 0.20 2.20 124.51 2.11 0.32 2.12 81.89
SDSS J065133.34+284423.4 1.79 2.34 0.31 2.04 117.18 2.95 0.49 2.00 92.83
SDSS J093506.92+441107.0 2.03 4.76 0.73 2.21 139.66 9.50 1.40 2.12 70.02
SDSS J232230.20+050942.06 1.59 10.47 1.35 1.66 70.72 21.27 – – 34.82
PTF J053332.05+020911.6 1.63 17.94 – – 29.09 37.69 – – 13.84
SDSS J163030.58+423305.7 1.72 104.33 – – 12.38 283.92 – – 4.55
SDSS J092345.59+302805.0 2.06 179.01 – – 11.61 412.89 – – 5.03
limit in this section because we notice that the maxi-mum total mass can reach 2.8M� when we considereda Galactic DWD population from MLDC Round 4 withan equal-mass assumption in the following sections. Al-though 2.8M� may be too high to be the upper limitfor most DWDs, whose total masses usually do not ex-ceed 2M� (Lamberts et al. 2019), it would not alter thequalitative conclusions derived here.
Similarly, by changing the mass ratio q with a fixedtotal mass Mb = 1.4M� of the DWD, we illustrate ourresults in the lower panels of Fig. 4. It supports thatthe detecting abilities are not weakened too much if thedeviation of q from our assumption (equal mass) is rea-sonable, say, q . 3. Based on this, we claim with con-fidence that our main results have captured the majorfeatures of CBP detections.
4.2. Comparison between LISA and Taiji
Now we compare our results about the detections ofCBPs orbiting DWDs between LISA and Taiji. We firstdiscuss the possible detections around known DWDs inSec. 4.2.1. We find the detectable minimum planetarymass Mmin
p of each system to see if it is below the deu-terium burning limit Mp = 13MJ. If so, we will definethis system as the promising system. In Sec. 4.2.2, weshow the distribution of the promising systems in ourGalaxy and discuss the constraint on their detectablezones (DZs), which are described as the circumbinarydistances where the space-borne GW detector has thepossibility to detect CBPs withMp ≤ 13MJ. The distri-butions of the inner/outer edge of the DZ (referred to asIDZ/ODZ) and their dependence on the GW frequencyare plotted for comparison between LISA and Taiji. InSec. 4.2.3, we analyze the detection rates during 4 yearsby injecting different planet distributions. Finally, wediscuss the prospects for the detection of CBPs in hab-itable zones (HZ) around DWDs in Sec. 4.2.4.
4.2.1. Possible detections of CBPs around known DWDs
We first discuss possible detections of CBPs aroundthe known detached DWDs with high SNRs. Daniel-ski et al. (2019) have analyzed one DWD in de-tail (ZTF J153932.16+502738.8). Here we calculatethe expected SNRs for all DWDs in Huang et al.(2020) for LISA and Taiji, and list the results ofthe ones with high SNRs (SNR > 10 for Taiji) inTable 1. We can see that there are four promis-ing systems for Taiji: ZTF J153932.16+502738.8,SDSS J065133.34+284423.4, SDSS J093506.92+441107.0,and SDSS J093506.92+441107.0, of which the first threeare also promising systems for LISA. Comparing the re-sults in Table 1, Taiji obviously has a better performanceon detecting abilities from two aspects: (i) the smallerdetectable minimum planetary masses Mmin
p , and (ii)the wider DZs. This mainly comes from the noticeabledifferences in SNRs between the two missions.
As mentioned in Sec. 3.2, we consider coplanar circu-lar orbit for the CBP in a three-body system. There-fore, despite it seems likely that LISA and Taiji candetect exoplanets down to ∼ 1MJ around these knowndetached DWDs, the planetary orbital inclinations canin fact deviate from the DWD inclination ι, which maylead to a raise in Mmin
p . This could happen due to thedegeneracy between the planetary mass and inclination.On the other hand, if complementary EM observationsin the future could constrain the planetary inclinationwell, we can then derive bounds for the mass of theCBP. Conversely, space-borne GW detectors could alsogive constraints in the mass-separation parameter space(see e.g., Fig. 3), and our results can provide inputs forthe EM exoplanetary projects, which are especially de-sirable for the study on possible synergy between GWand EM observations.
4.2.2. Constraints on the promising systems in our Galaxy
LISA/TAIJI Exoplanets 9
0.1 1.0 10.0Distance to the DWD [kpc]
0.1
1.0
10.0
The
dete
ctab
le m
inim
um C
BP m
ass [
MJ]
13 MJ
1 10 100 1000SNR
0.1 1.0 10.0Distance to the DWD [kpc]
0.1
1.0
10.0
The
dete
ctab
le m
inim
um C
BP m
ass [
MJ]
13 MJ
1 10 100 1000SNR
Figure 5. The relationship between the distance and the detectable minimum CBP mass for Taiji (left) and LISA (right) for
promising systems in MLDC Round 4 during a nominal 4-year mission. The color represents the SNR of each system, and the
dashed horizontal grey line corresponds to the deuterium burning limit.
We mentioned in Sec. 3.1 that SNR > 10 is chosen tobe the threshold for our work, and in MLDC Round 4there are 25,016 (15,903) detached DWDs satisfying thiscriterion for Taiji (LISA). Based on these populations,we find a total of 9,053 (6,718) promising systems atmost during a 4-year observation for Taiji (LISA). Fig-ure 5 shows that most systems are clustered together inthe 1–13MJ mass range and at about 10 kpc from ourSolar system, which is consistent with the distance to theGalactic center. Generally, nearby DWDs could havelower Mmin
p than distant systems due to their higherSNRs. This also explains why the promising systemsare mainly distributed in the top right of Fig. 5.
To go a step further, we plot the distributions ofthe detectable minimum (maximum) period and corre-sponding IDZ (ODZ) in Fig. 6. We see that some valleysappear at multiples of one year, which correspond tothe peaks in Fig. 3, caused by the degeneracies betweensource and detector parameters (see Sec. 4.1). From thebottom panels, we see that Taiji is expected to detectCBPs with the planetary orbital size smaller than 5 au,which is about the distance from the Sun to the Jupiter(5.2 au), while it is a little smaller (4.4 au) for LISA.The constraints on DZs actually reflect the detectingability of the space-borne GW detector we choose, be-cause these results are based on the population of DWDswithout injecting any CBP models. It shows that thebest range of detection is between 0.1 au to 3 au for bothLISA and Taiji.
For each promising system, the dependence of IDZand ODZ on the GW frequency are illustrated in Fig. 7.Our results seem to suggest that systems with higherGW frequencies tend to have wider DZs. An explana-tion for this may come from the higher SNRs in the sen-sitive frequency band, namely 0.1 mHz to 1 Hz for LISAand Taiji (see Sec. 2.1 and Fig. 2). Note that there are
some gaps at distance . 1 au in Fig. 7, which is due toour sample intervals in the parameter estimation processand would not alter the qualitative conclusion derivedhere.
4.2.3. Detection rates for different CBP models
As mentioned in Sec. 3.2, we only consider the copla-nar orbits and the occurrence rate is set to 50% forthe promising detectable systems. Based on the cat-alog of approximately 2.5 × 104 (1.6 × 104) detectabledetached DWDs for Taiji (LISA) in total (see Sec. 4.2.2),our model predicts that the total number of the injectedCBP population is 4,507 (3,322) during the nominal mis-sion span.
Following Danielski et al. (2019), we consider twoscenarios in our sub-stellar object (SSO) injection pro-cesses:
(A) an optimistic case where a follows a log-uniform dis-tribution in the range of 0.1 – 200 au, and Mp is uni-formly distributed in the range of 1M⊕ – 0.08M�,and
(B) a pessimistic scenario where a is uniformly dis-tributed in the range of 0.1 – 200 au, and Mp is uni-formly distributed in the range of 1M⊕ – 0.08M�.
Note that samples with an injected SSO mass 13MJ <M < 0.08M� are discarded because we only focus onthe CBPs with mass Mp ≤ 13MJ in this work. Morediscussions on brown dwarfs with mass 13MJ < M <0.08M� can be found in Danielski et al. (2019)
We find a total of 40 (16) detected CBPs for (A), 2(0) for (B), corresponding to 0.16% (0.10%) and 0.008%(0%) of the total population of detected DWDs over the4-year mission of Taiji (LISA). From these we concludethat the detection rates in our work are essentially inagreement with the results in Danielski et al. (2019), but
10 Y. Kang, C. Liu, L. Shao
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0Planetary Period [year]
1
10
100
1000
Num
ber
Minimum PeriodMaximum Period
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0Planetary Period [year]
1
10
100
1000
Num
ber
Minimum PeriodMaximum Period
1.0 2.0 3.0 4.0 5.0Planetary Distance to the DWD [au]
1
10
100
1000
Num
ber
I DZODZ
1.0 2.0 3.0 4.0 5.0Planetary Distance to the DWD [au]
1
10
100
1000
Num
ber
I DZODZ
Figure 6. The distributions of detectable minimum and maximum period (top panels), and corresponding IDZ and ODZ
(bottom panels). Left panels are for Taiji while right panels are for LISA.
a little bit more pessimistic as a whole due to the differ-ent underlying models and assumptions. Therefore, it isadvantageous to improve CBP models in future for morecomparisons. Although there seems like no detection inscenario (B) for LISA, Taiji can still has a non-zero re-sult on CBP detections. These data again suggest thatTaiji has a better performance on detecting abilities.
4.2.4. Prospects for detections of CBPs in habitable zone
around DWDs
The discovery of thousands of exoplanets in pastdecades has been promoting the study of habitabilityand the search for extraterrestrial life (Cockell et al.2016; Kaltenegger 2017; Lingam & Loeb 2018), whichencompass various research methods within the phys-ical, biological, and environmental sciences. Amongmany contemporary habitability metrics, the habitablezone (HZ) forms a fundamental component to assess thepotential habitability of newly discovered exoplanets. Itdescribes the circumstellar distance where water at thesurface of an exoplanet would be in the liquid phase(Kasting et al. 1993), mainly because all life on theEarth requires liquid water directly or indirectly. Giventhat the Earth is the only known planet with life on it, itis reasonable to suppose that such a concept also appliesto exoplanets beyond the Earth.
Most research about HZ has focused on MS stars thatare similar to the Sun (Kasting et al. 1993; Selsis et al.
2007; Lunine et al. 2008; Rushby et al. 2013). But re-cent studies start to discuss the HZ of WDs (Monteiro2010; Agol 2011; Fossati et al. 2012; Barnes & Heller2013). Although, unlike hydrogen-burning stars, theWD cooling makes the HZ moves inwards with time,WDs are still expected to provide a source of energyfor planets in HZ for giga-year (Gyr) durations. As theremnants of MS stars, WDs are as abundant as Sun-likestars in our Galaxy. Most of them are close in size toour Earth with a characteristic luminosity of ∼ 10−4 L�(Agol 2011). So the HZ around WDs is located within∼ 0.01 au where planets must have migrated inwardsafter the CE phases (Debes & Sigurdsson 2002; Livioet al. 2005; Faedi et al. 2011). As noted by Tamanini& Danielski (2019), the detection of such an exoplanetwould help to provide crucial information on migrationtheories, especially around post-CE binaries.
We assume that the HZ boundary estimations for MSstars also apply to DWD systems. Thus we can de-termine the inner/outer edge of the HZ (referred to asIHZ/OHZ) via equations in Selsis et al. (2007),
IHZ = (IHZ� − ainT? − binT 2? )( LL�
)1/2,
OHZ = (OHZ� − aoutT? − boutT2? )( LL�
)1/2,
(18)
LISA/TAIJI Exoplanets 11
0.01 0.1 1.0Planetary Distance to the DWD [au]
1
10GW
Fre
quen
cy [m
Hz]
I DZODZ
1 10 100 1000SNR
0.01 0.1 1.0Planetary Distance to the DWD [au]
1
10
GW F
requ
ency
[mHz
]
I DZODZ
1 10 100 1000SNR
Figure 7. Dependence of IDZ and ODZ on the GW frequency of each promising system. The color represents the SNR of each
system for Taiji (left) and LISA (right). We use the left and right triangles to denote the IDZ and ODZ respectively. Notice
that for each system, its IDZ and ODZ are located in the same horizontal line (i.e. they have the same GW frequency).
where IHZ� and OHZ� are the boundaries in our Solarsystem depending on different fractional cloud cover onthe day side of an exoplanet (see Table 2). As notedin the Sec. 2 of Selsis et al. (2007), clouds can increasethe planetary albedo and reduce the greenhouse warm-ing, which thus moves IHZ� closer to the star. Butfor OHZ� associated with CO2-ice clouds, which dif-fer significantly from H2O-ice particles in the opticalproperties, the cooling effect caused by the increase ofalbedo is weaker than the warming effect caused by thebackscattering of the infrared surface emission (Lunineet al. 2008). As a result, the theoretical OHZ� shouldbe farther for a 100% cloud cover. Other empiricallydetermined constants in Eq. (18) are,
ain = 2.7619× 10−5 au K−1 ,
bin = 3.8095× 10−9 au K−2 ,
aout = 1.3786× 10−4 au K−1 ,
bout = 1.4286× 10−9 au K−2 . (19)
In Eq. (18), L and L� are the primary’s and Solar lu-minosity, respectively, and T? = Teff − 5700 K, wherethe effective temperature of the primary Teff is givenaccording to the Stefan-Boltzmann law via,
Teff =
(L
4πσR2
)1/4
, (20)
with σ being the Stefan-Boltzmann constant and R be-ing the primary’s radius.
As noted by Barnes & Heller (2013), WDs cool rapidlyfor about 3 Gyr, and then maintain a relatively constanttemperature before falling off again at about 7 Gyr. As a
rough estimation, we neglect the distributions of coolingtime and assume that each WD has the same fixed lu-minosity value for which we set it to 1×10−4 L�, with atotal luminosity 2× 10−4 L�. Through the mass-radiusrelation for WDs in the Newtonian case, as provided inFig. 4 of Ambrosino (2020), we calculate Teff of eachpromising system. Based on our assumption, the twoWDs have the same mass in the DWD system, there-fore their Teff values are the same as well. We plugTeff into Eq. (18) to obtain the IHZ and OHZ of eachpromising system. We verify that ODZ is much fartherthan OHZ for each system due to the low luminositiesof WDs, which means that we only need to compare thelimits between OHZ and IDZ. If the IDZ lies closer tothe DWD than the OHZ, we can say that LISA andTaiji are possible to detect a habitable CBP around thissystem.
We list our results in Table 2 for LISA and Taiji dur-ing a 4-year observation. Note that for each cloud coverscheme, we take the same values of IHZ and OHZ asa criterion because the boundary values of HZ are in-sensitive to Teff in our simplified model. Although thedetection numbers do not seem many, it still shows thatsuch a possibility exists. We also have verified thatthe detection number would not increase by more thanone order of magnitude if we set the total luminosity to2 × 10−3 L�, which is larger than what they really arewhen the WD cooling is considered (see e.g. Fig. 1 inBarnes & Heller 2013).
5. CONCLUSION
In this work, we introduce the two space-borne GWdetectors, LISA and Taiji, and discuss the use of them
12 Y. Kang, C. Liu, L. Shao
Table 2. Detection numbers and percentages of habitable CBPs around the promising systems for LISA and Taiji. We have
mentioned in Sec. 4.2.2 that the total number of promising systems is 9,053 for Taiji, and 6,718 for LISA for a 4-yr mission.
Clouds 0% Clouds 50% Clouds 100%Detector IHZ� OHZ� IHZ OHZ IHZ� OHZ� IHZ OHZ IHZ� OHZ� IHZ OHZ
[au] [au] [au] [au] [au] [au] [au] [au] [au] [au] [au] [au]
0.895 1.67 0.013 0.024 0.72 1.95 0.010 0.028 0.485 2.4 0.0069 0.034
Taiji 3 (0.033%) 5 (0.055%) 13 (0.14%)
LISA 1 (0.015%) 1 (0.015%) 3 (0.45%)
to detect exoplanets. For the GW detection methodoriginally proposed by Tamanini & Danielski (2019), wegive some complementary calculations on the detectingabilities with different values of binary mass and massratio. The conceptual idea using LISA/Taiji to detectexoplanets is similar to the RV technique but has uniqueadvantages, compared to the traditional EM methods.Before quantitatively analyzing the prospects for detect-ing CBPs around DWDs in the whole Galaxy by usingLISA and Taiji, we show that there is a possibility to de-tect CBPs around four known detached DWDs with highSNRs using Taiji, while three systems are also promis-ing for LISA. The minimum detectable masses aroundthese DWDs can be as small as a few of the Jupitermass. Moreover, if EM observations can give more con-straints on these systems in the meantime, e.g. inferringthe orbital inclination of the CBP, GWs may place morerestrictions on the mass of the CBP and the existenceof such a population.
Based on the DWD population from MLDC Round 4,we give quick assessments of CBP detections in thewhole Galaxy during a 4-yr mission time of LISA/Taiji.Our results show that LISA can detect ∼ 6, 000 promis-ing systems, while the number rises to ∼ 9, 000 forTaiji. From the distributions of DZs, we show that thebest range of CBP detections is between 0.1 au to 3 auaround DWDs for both LISA and Taiji. Furthermore,we inject two different planet distributions with an oc-currence rate of 50%, following Danielski et al. (2019),to constrain the total detection rates. Our results are inbold agreement with previous studies, but seem slightlymore pessimistic as a whole due to the different modelsand assumptions we adopted. By assuming that the HZboundary estimations for MS stars also apply to DWDs,we briefly discuss the prospects for detecting habitableCBPs around detached DWDs in a simplified model. It
shows that such a possibility exists though the detectionrates are not large during 4-yr observations.
In addition to the planetary migration theories (Tur-rini et al. 2015), there are also some studies about thesecond-generation formation process which can be usedto explain the existence of nearby exoplanets in thesesystems (Zorotovic & Schreiber 2013; Volschow et al.2014; Schleicher & Dreizler 2014). All our results can ac-tually help analyze planetary systems after CE phasesand provide a useful input for exoplanetary projects.With a rapid development of GW astronomy in the past5 years, we look forward to the synergy with EM obser-vations and the full investigation of such a GW detectionmethod of exoplanets in the near future.
ACKNOWLEDGMENTS
We thank the anonymous referee for suggestions.This work was supported by the National NaturalScience Foundation of China (11975027, 11991053,11721303), the National SKA Program of China(2020SKA0120300), the Young Elite Scientists Sponsor-ship Program by the China Association for Science andTechnology (2018QNRC001), the Max Planck PartnerGroup Program funded by the Max Planck Society, andthe High-Performance Computing Platform of PekingUniversity. YK acknowledges the Hui-Chun Chin andTsung-Dao Lee Chinese Undergraduate Research En-dowment (Chun-Tsung Endowment) at Peking Univer-sity. This research has made use of the NASA ExoplanetArchive, which is operated by the California Instituteof Technology, under contract with the National Aero-nautics and Space Administration under the ExoplanetExploration Program.
Facilities: LISA, Taiji, Exoplanet Archive
APPENDIX
All of the selected known detached DWDs are given in Table 3. We list the heavier and the lighter masses ofthe DWD m1 and m2, the GW frequency f , the luminosity distance dDWD, the ecliptic coordinates (λ, β), and the
LISA/TAIJI Exoplanets 13
inclination angle ι. Most of these parameters above are taken directly from Huang et al. (2020), except the distanceto ZTF J153932.16+502738.8 (with an asterisk). We have corrected it with the result in Burdge et al. (2019).
Table 3. Properties of the selected known detached DWDs (Huang et al. 2020). Some inclination angles are given with a square
bracket due to lack of direct measurements of them. These estimated values are assigned based on the evolutionary stage and
the mass ratio of the system. The asterisk marks a corrected distance from Burdge et al. (2019).
Source m1 m2 f dDWD λ β ι[M�] [M�] [mHz] [kpc] [deg] [deg] [deg]
ZTF J153932.16+502738.8 0.61 0.21 4.82 2.34* 205.03 66.16 84.0
SDSS J065133.34+284423.4 0.49 0.247 2.61 0.933 101.34 5.80 86.9
SDSS J093506.92+441107.0 0.75 0.312 1.68 0.645 130.98 28.09 [60.0]
SDSS J232230.20+050942.06 0.27 0.24 1.66 0.779 353.44 8.46 27.0
PTF J053332.05+020911.6 0.65 0.167 1.62 1.253 82.91 −21.12 72.8
SDSS J163030.58+423305.7 0.76 0.298 0.84 1.019 231.76 63.05 [60.0]
SDSS J092345.59+302805.0 0.76 0.275 0.51 0.299 133.72 14.43 [60.0]
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