2507/2507.01250.md

Title: A Multimessenger Strategy for Downselecting the Orientations of Galactic Close White Dwarf Binaries

URL Source: https://arxiv.org/html/2507.01250

Published Time: Thu, 03 Jul 2025 00:10:10 GMT

Markdown Content: (July 1, 2025)

Abstract

The planned space-based gravitational wave detector, LISA, will provide a fundamentally new means of studying the orbital alignment of close white dwarf binaries. However, due to the inherent symmetry of their gravitational wave signals, a fourfold degeneracy arises in the transverse projections of their angular momentum vectors. In this paper, we demonstrate that by incorporating timing information from electromagnetic observations, such as radial velocity modulations and light curves, this degeneracy can be reduced to twofold.

pacs:

PACS number(s): 95.55.Ym 98.80.Es,95.85.Sz

I introduction

Do the orbital angular momenta of Galactic binaries statistically align with Galactic structures? For over 100 years, astrometric and radial velocity data have served as fundamental observational resources for investigating this question (see, e.g., Chang (1929) for early studies). In a recent detailed study, Agati et al. Agati et al. (2015) analyzed 95 binaries within 18 pc of the Sun and found that their angular momenta are consistent with being randomly oriented.

More recently, in 2023, Tan et al. Tan et al. (2023) examined the orientations of the symmetry axes of 14 planetary nebulae that host (or are inferred to host) short-period binaries around the Galactic bulge. They reported that the axes are not randomly oriented but tend to be parallel to the Galactic plane. Planetary nebulae consist of gas expelled during the formation of white dwarfs, and the observed 14 axes are considered to coincide with the orbital axes of their associated binaries. Tan et al. Tan et al. (2023) proposed that the Galactic magnetic field at the time of binary formation may be responsible for this observed anisotropy. This finding contrasts sharply with the results of Agati et al. Agati et al. (2015), who analyzed local binaries, highlighting the need for further observational studies on this issue.

The Laser Interferometer Space Antenna (LISA) is designed to have optimal sensitivity to gravitational waves (GWs) in the 0.1-10 mHz range and is anticipated to detect approximately 10 4 superscript 10 4 10^{4}10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT Galactic close white dwarf binaries (CWDBs) emitting nearly monochromatic GWs Seoane et al. (2023) (see also Ruan et al. (2020) for Taiji and Luo et al. (2016) for TianQin). At relatively high frequencies above ∼3 similar-to absent 3\sim 3∼ 3 mHz, CWDBs will likely be individually resolved throughout the Galaxy Seoane et al. (2023). In contrast, at lower frequencies, the number of CWDBs per frequency bin is expected to be much greater than unity, with only a small fraction near the Sun being resolvable. The unresolved sources contribute to the confusion foreground noise Seoane et al. (2023).

Due to dissipative effects, most of these CWDBs are expected to have nearly circular orbits. Since both the generation and measurement of GWs are inherently geometrical, LISA will provide a fundamentally new means of studying the orbital orientations of Galactic binaries Seto (2024a, b).

Unfortunately, due to the intrinsic symmetry of their gravitational waveforms, there is a fourfold degeneracy in estimating the transverse projections of orbital orientations, a well-known issue in the GW community (see, e.g., Cornish and Larson (2003) in the context of Galactic binaries). For instance, suppose LISA detects an edge-on CWDB located near the Galactic plane, with its angular momentum aligned parallel to the plane. The fourfold degeneracy would prevent us from determining whether the binary’s orientation is truly parallel or instead perpendicular to the Galactic plane. Consequently, relying solely on LISA data imposes significant limitations on our ability to analyze the orientations of CWDBs Seto (2024a, b).

Meanwhile, CWDBs emit electromagnetic (EM) signals in addition to GWs Seoane et al. (2023). In the LISA era, CWDBs will be key observational targets for multimessenger astronomy. Their sky positions can be estimated from the amplitude and Doppler modulations induced by the motion of the GW detectors Cutler (1998). At GW frequencies above f gw∼1 similar-to subscript 𝑓 gw 1 f_{\rm gw}\sim 1 italic_f start_POSTSUBSCRIPT roman_gw end_POSTSUBSCRIPT ∼ 1 mHz, Doppler modulation generally becomes more useful. For an observational time longer than 2 years, the typical size of the error ellipsoid in the sky is estimated to be ∼0.5⁢deg 2⁢(ρ/50)−2⁢(f gw/5⁢mHz)−2 similar-to absent 0.5 superscript deg 2 superscript 𝜌 50 2 superscript subscript 𝑓 gw 5 mHz 2\sim 0.5,{\rm deg}^{2}(\rho/50)^{-2}(f_{\rm gw}/5,{\rm mHz})^{-2}∼ 0.5 roman_deg start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_ρ / 50 ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( italic_f start_POSTSUBSCRIPT roman_gw end_POSTSUBSCRIPT / 5 roman_mHz ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, where ρ 𝜌\rho italic_ρ is the signal-to-noise ratio Takahashi and Seto (2002).

The light curves and the radial velocity data of CWDBs are modulated by orbital motion and should be relatively easy to observe (see, e.g., Burdge et al. (2019)). In particular, nearly edge-on CWDBs will exhibit distinct eclipsing patterns, making them ideal candidates for follow-up EM observations. Indeed, according to Korol et al. Korol et al. (2017), as many as ∼similar-to\sim∼100 eclipsing CWDBs (nearly at edge-on configurations) could be observed simultaneously by LISA and the Vera C. Rubin Observatory. The majority of these ∼100 similar-to absent 100\sim 100∼ 100 CWDBs will be at distances less than ∼4 similar-to absent 4\sim 4∼ 4 kpc and thus belong to the disk component, comprising a small fraction of the ∼10 4 similar-to absent superscript 10 4\sim 10^{4}∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT resolved CWDBs (see also Littenberg et al. (2013)).

In this paper, we explore the possibility of reducing the fourfold degeneracy of the angular momentum vector, by leveraging multimessenger observations of CWDBs. We revisit the time profile of their GW signals, incorporating orbital phase information partially inferred from basic EM data. We then propose a simple method to reduce the fourfold degeneracy to twofold. In our explanation of the fourfold degeneracy, we considered the example of a hypothetical edge-on binary system located near the Galactic plane, with its angular momentum aligned parallel to the plane. Our reduction method enables us to eliminate the two spurious orientation solutions that are perpendicular to the plane. The approach presented in this paper could further motivate deeper follow-up searches for CWDBs detected by LISA, extending beyond those exhibiting eclipsing patterns.

II GW from a circular binary

In this section, we review the GW emission from a circular binary and discuss how to estimate its orbital orientation based on the observed GW signal.

II.1 Orbital motion

Let us consider a circular binary consisting of two stars, α 𝛼\alpha italic_α and β 𝛽\beta italic_β. We denote its orbital separation by a 𝑎 a italic_a and the masses of the stars by m α subscript 𝑚 𝛼 m_{\alpha}italic_m start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT and m β subscript 𝑚 𝛽 m_{\beta}italic_m start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT, respectively. The orbital frequency is given by

f=1 2⁢π⁢(G⁢M T a 3)1/2 𝑓 1 2 𝜋 superscript 𝐺 subscript 𝑀 𝑇 superscript 𝑎 3 1 2 f=\frac{1}{2\pi}\left(\frac{GM_{T}}{a^{3}}\right)^{1/2}italic_f = divide start_ARG 1 end_ARG start_ARG 2 italic_π end_ARG ( divide start_ARG italic_G italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG start_ARG italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT(1)

where the total mass is M T=m α+m β subscript 𝑀 𝑇 subscript 𝑚 𝛼 subscript 𝑚 𝛽 M_{T}=m_{\alpha}+m_{\beta}italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT.

We introduce the coordinate system X⁢Y⁢Z 𝑋 𝑌 𝑍 XYZ italic_X italic_Y italic_Z, as shown in Fig. 1. The binary orbits in the X⁢Y 𝑋 𝑌 XY italic_X italic_Y-plane (around the origin), and the orientation vector e→j subscript→𝑒 𝑗{\vec{e}}_{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT of its angular momentum is directed along the Z 𝑍 Z italic_Z-axis. The positions of the two masses are given by

(x α,y α,z α)subscript 𝑥 𝛼 subscript 𝑦 𝛼 subscript 𝑧 𝛼\displaystyle(x_{\alpha},y_{\alpha},z_{\alpha})( italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT )=\displaystyle==a⁢m β M T⁢(cos⁡Φ s⁢(t),sin⁡Φ s⁢(t),0),𝑎 subscript 𝑚 𝛽 subscript 𝑀 𝑇 subscript Φ s 𝑡 subscript Φ s 𝑡 0\displaystyle a\frac{m_{\beta}}{M_{T}}(\cos\Phi_{\rm s}(t),\sin\Phi_{\rm s}(t)% ,0),italic_a divide start_ARG italic_m start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG start_ARG italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG ( roman_cos roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) , roman_sin roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) , 0 ) ,(2) (x β,y β,z β)subscript 𝑥 𝛽 subscript 𝑦 𝛽 subscript 𝑧 𝛽\displaystyle(x_{\beta},y_{\beta},z_{\beta})( italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT )=\displaystyle==−a⁢m α M T⁢(cos⁡Φ s⁢(t),sin⁡Φ s⁢(t),0),𝑎 subscript 𝑚 𝛼 subscript 𝑀 𝑇 subscript Φ s 𝑡 subscript Φ s 𝑡 0\displaystyle-a\frac{m_{\alpha}}{M_{T}}(\cos\Phi_{\rm s}(t),\sin\Phi_{\rm s}(t% ),0),- italic_a divide start_ARG italic_m start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG start_ARG italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG ( roman_cos roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) , roman_sin roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) , 0 ) ,(3)

where the orbital phase of the binary is

Φ s⁢(t)=2⁢π⁢f⁢t+φ s.subscript Φ s 𝑡 2 𝜋 𝑓 𝑡 subscript 𝜑 s\Phi_{\rm s}(t)=2\pi ft+\varphi_{\rm s}.roman_Φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT ( italic_t ) = 2 italic_π italic_f italic_t + italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT .(4)

II.2 Observed GW

Next we discuss GW emission from the binary. Since CWDBs have small post-Newtonian parameters O⁢(10−3)𝑂 superscript 10 3 O(10^{-3})italic_O ( 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ) in the LISA band, we can apply the quadrupole formula (with the exception of possible rare cases Seto (2025)).

Figure 1: Configuration of a binary and an observer. The binary rotates in the X⁢Y 𝑋 𝑌 XY italic_X italic_Y-plane with its orientation vector e→j subscript→𝑒 𝑗{\vec{e}}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT pointing toward the Z 𝑍 Z italic_Z-axis. At the observer, we introduce the orthonormal vectors (e→N,e→θ,e→ϕ)subscript→𝑒 𝑁 subscript→𝑒 𝜃 subscript→𝑒 italic-ϕ({\vec{e}}{N},{\vec{e}}{\theta},{\vec{e}}{\phi})( over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ).

We consider an observer at (r,θ,ϕ)𝑟 𝜃 italic-ϕ(r,\theta,\phi)( italic_r , italic_θ , italic_ϕ ) in the spherical coordinate system (see Fig. 1). The polar angle θ 𝜃\theta italic_θ corresponds to the inclination angle. We have θ=π/2 𝜃 𝜋 2\theta=\pi/2 italic_θ = italic_π / 2 for the edge-on and θ={0,π}𝜃 0 𝜋\theta={0,\pi}italic_θ = { 0 , italic_π } for the face-on configurations. The two transverse vectors (e→θ,e→ϕ)subscript→𝑒 𝜃 subscript→𝑒 italic-ϕ({\vec{e}}{\theta},{\vec{e}}{\phi})( over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ) compose the wave’s principle axes (see e.g., Cutler (1998)). Later, we virtually rotate the binary around the binary-observer axis, but continue to fix the reference directions (e→θ,e→ϕ)subscript→𝑒 𝜃 subscript→𝑒 italic-ϕ({\vec{e}}{\theta},{\vec{e}}{\phi})( over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ) as shown in Fig. 1.

As presented in various textbooks (see, e.g., Thorne and Blandford (2017)), the GW signal at the observer is expressed as

h μ⁢ν=h+⁢(t)⁢e μ⁢ν++h×⁢(t)⁢e μ⁢ν×subscript ℎ 𝜇 𝜈 subscript ℎ 𝑡 superscript subscript 𝑒 𝜇 𝜈 subscript ℎ 𝑡 superscript subscript 𝑒 𝜇 𝜈 h_{\mu\nu}=h_{+}(t)e_{\mu\nu}^{+}+h_{\times}(t)e_{\mu\nu}^{\times}italic_h start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT(5)

with the transverse traceless polarization tensors

𝒆+=e→θ⊗e→θ−e→ϕ⊗e→ϕ,𝒆×=e→θ⊗e→ϕ+e→ϕ⊗e→θ formulae-sequence superscript 𝒆 tensor-product subscript→𝑒 𝜃 subscript→𝑒 𝜃 tensor-product subscript→𝑒 italic-ϕ subscript→𝑒 italic-ϕ superscript 𝒆 tensor-product subscript→𝑒 𝜃 subscript→𝑒 italic-ϕ tensor-product subscript→𝑒 italic-ϕ subscript→𝑒 𝜃{\mbox{\boldmath${e}$}}^{+}={\vec{e}}{\theta}\otimes{\vec{e}}{\theta}-{\vec{% e}}{\phi}\otimes{\vec{e}}{\phi},~{}{\mbox{\boldmath${e}$}}^{\times}={\vec{e}% }{\theta}\otimes{\vec{e}}{\phi}+{\vec{e}}{\phi}\otimes{\vec{e}}{\theta}bold_italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⊗ over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT - over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ⊗ over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT , bold_italic_e start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT = over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⊗ over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT + over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ⊗ over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT(6)

or equivalently

h θ⁢θ=−h ϕ⁢ϕ=h+,h θ⁢ϕ=h ϕ⁢θ=h×.formulae-sequence subscript ℎ 𝜃 𝜃 subscript ℎ italic-ϕ italic-ϕ subscript ℎ subscript ℎ 𝜃 italic-ϕ subscript ℎ italic-ϕ 𝜃 subscript ℎ h_{\theta\theta}=-h_{\phi\phi}=h_{+},{}{}h_{\theta\phi}=h_{\phi\theta}=h_{% \times}.italic_h start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT = - italic_h start_POSTSUBSCRIPT italic_ϕ italic_ϕ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_h start_POSTSUBSCRIPT italic_θ italic_ϕ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_ϕ italic_θ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT .(7)

The time-dependent functions h+⁢(t)subscript ℎ 𝑡 h_{+}(t)italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_t ) and h×⁢(t)subscript ℎ 𝑡 h_{\times}(t)italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT ( italic_t ) are given by

h+⁢(t)subscript ℎ 𝑡\displaystyle h_{+}(t)italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_t )=\displaystyle==−A⁢(cos 2⁡θ+1)⁢cos⁡2⁢Φ o⁢(t)𝐴 superscript 2 𝜃 1 2 subscript Φ o 𝑡\displaystyle-A(\cos^{2}\theta+1)\cos 2\Phi_{\rm o}(t)- italic_A ( roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ + 1 ) roman_cos 2 roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t )(8) h×⁢(t)subscript ℎ 𝑡\displaystyle h_{\times}(t)italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT ( italic_t )=\displaystyle==−2⁢A⁢cos⁡θ⁢sin⁡2⁢Φ o⁢(t)2 𝐴 𝜃 2 subscript Φ o 𝑡\displaystyle-2A\cos\theta\sin 2\Phi_{\rm o}(t)- 2 italic_A roman_cos italic_θ roman_sin 2 roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t )(9)

with the phase Φ o⁢(t)subscript Φ o 𝑡\Phi_{\rm o}(t)roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) and the amplitude A 𝐴 A italic_A at the observer

Φ o⁢(t)subscript Φ o 𝑡\displaystyle\Phi_{\rm o}(t)roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t )=\displaystyle==2⁢π⁢f⁢(t−r/c)+φ s−ϕ 2 𝜋 𝑓 𝑡 𝑟 𝑐 subscript 𝜑 s italic-ϕ\displaystyle 2\pi f(t-r/c)+\varphi_{\rm s}-\phi 2 italic_π italic_f ( italic_t - italic_r / italic_c ) + italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT - italic_ϕ(10) A 𝐴\displaystyle A italic_A=\displaystyle==2⁢G 5/3⁢m α⁢m β⁢M T−1/3⁢(2⁢π⁢f)2/3 r.2 superscript 𝐺 5 3 subscript 𝑚 𝛼 subscript 𝑚 𝛽 superscript subscript 𝑀 𝑇 1 3 superscript 2 𝜋 𝑓 2 3 𝑟\displaystyle\frac{2G^{5/3}m_{\alpha}m_{\beta}M_{T}^{-1/3}(2\pi f)^{2/3}}{r}.divide start_ARG 2 italic_G start_POSTSUPERSCRIPT 5 / 3 end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 3 end_POSTSUPERSCRIPT ( 2 italic_π italic_f ) start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG .(11)

In Fig. 2, the black and gray curves show the typical shapes of the two functions h+=h θ⁢θ subscript ℎ subscript ℎ 𝜃 𝜃 h_{+}=h_{\theta\theta}italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT and h×=h θ⁢ϕ subscript ℎ subscript ℎ 𝜃 italic-ϕ h_{\times}=h_{\theta\phi}italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_θ italic_ϕ end_POSTSUBSCRIPT for θ=31∘𝜃 superscript 31\theta=31^{\circ}italic_θ = 31 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT. In the lower part of the figure, we also show the deformation pattern of the GW at representative epochs. The GW frequency is twice the orbital frequency f 𝑓 f italic_f.

II.3 Fourfold degeneracy

In this section, we discuss the estimation of the orientation vector e→j subscript→𝑒 𝑗{\vec{e}}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT from the observed GW signal. Following conventions, we decompose the vector e→j subscript→𝑒 𝑗{\vec{e}}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT into the inclination angle θ 𝜃\theta italic_θ and the unit projection vector onto the transverse plane (−e→θ subscript→𝑒 𝜃-{\vec{e}}_{\theta}- over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT in the case of Fig. 1). For the latter, the so-called polarization angle ψ 𝜓\psi italic_ψ is often used, which is measured counterclockwise from a reference direction on the plane.

The inclination angle θ 𝜃\theta italic_θ can be observationally determined within the range [0,π]0 𝜋[0,\pi][ 0 , italic_π ] from the two orthogonal polarization coefficients h+subscript ℎ h_{+}italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT and h×subscript ℎ h_{\times}italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT. More specifically, we use their relative amplitudes and phase (which corresponds to ±1/4 plus-or-minus 1 4\pm 1/4± 1 / 4 of the GW cycle). At the face-on configuration (θ=0 𝜃 0\theta=0 italic_θ = 0 or π 𝜋\pi italic_π), the incoming GW becomes circularly polarized, and the orientation vector e→j subscript→𝑒 𝑗{\vec{e}}_{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT can be uniquely determined, although its transverse projection is ill defined. Below, we exclude these two special configurations.

Next, in Fig. 1, let us virtually rotate the binary and the associated GW around the binary-observer axis by 90∘superscript 90 90^{\circ}90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT. Under this rotation, the transverse-traceless tensor h μ⁢ν subscript ℎ 𝜇 𝜈 h_{\mu\nu}italic_h start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT changes its sign, due to its spin-2 nature. In terms of the original orthonormal vectors e→θ subscript→𝑒 𝜃\vec{e}{\theta}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT and e→ϕ subscript→𝑒 italic-ϕ\vec{e}{\phi}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT defined in Fig. 1, after the 90∘superscript 90 90^{\circ}90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT rotation, the transverse projection of the orbital orientation becomes −e→ϕ subscript→𝑒 italic-ϕ-\vec{e}_{\phi}- over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT.

Importantly, from Eqs. (8)-(11), this sign change can be absorbed into a phase shift φ s−ϕ→φ s−ϕ+π/2→subscript 𝜑 s italic-ϕ subscript 𝜑 s italic-ϕ 𝜋 2\varphi_{\rm s}-\phi\to\varphi_{\rm s}-\phi+\pi/2 italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT - italic_ϕ → italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT - italic_ϕ + italic_π / 2. Therefore, by adopting the new combination (−e→ϕ,φ s−ϕ+π/2)subscript→𝑒 italic-ϕ subscript 𝜑 s italic-ϕ 𝜋 2(-{\vec{e}}{\phi},\varphi{\rm s}-\phi+\pi/2)( - over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT - italic_ϕ + italic_π / 2 ) for the projected direction and phase, we can generate an observed GW signal identical to that obtained with the original combination (−e→θ,φ s−ϕ)subscript→𝑒 𝜃 subscript 𝜑 s italic-ϕ(-{\vec{e}}{\theta},\varphi{\rm s}-\phi)( - over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT - italic_ϕ ).

Additionally, considering two other solutions corresponding to rotation angles of 180∘superscript 180 180^{\circ}180 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT and 270∘superscript 270 270^{\circ}270 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT, we find a total of four possible candidates, ±e→θ plus-or-minus subscript→𝑒 𝜃\pm{\vec{e}}{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT and ±e→ϕ plus-or-minus subscript→𝑒 italic-ϕ\pm{\vec{e}}{\phi}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT (in the original frame), for the transverse projection of e→j subscript→𝑒 𝑗{\vec{e}}_{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. This fourfold degeneracy is well known in the context of parameter estimation with quadrupolar GWs Cornish and Larson (2003).

Figure 2: Time profiles of the GW signal and the radial velocities. The specific values on the vertical axis are irrelevant to our study. The black curve represents the GW deformation pattern h+=h θ⁢θ=−h ϕ⁢ϕ subscript ℎ subscript ℎ 𝜃 𝜃 subscript ℎ italic-ϕ italic-ϕ h_{+}=h_{\theta\theta}=-h_{\phi\phi}italic_h start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT = - italic_h start_POSTSUBSCRIPT italic_ϕ italic_ϕ end_POSTSUBSCRIPT, while the gray curve corresponds to h×=h θ⁢ϕ=h ϕ⁢θ subscript ℎ subscript ℎ 𝜃 italic-ϕ subscript ℎ italic-ϕ 𝜃 h_{\times}=h_{\theta\phi}=h_{\phi\theta}italic_h start_POSTSUBSCRIPT × end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_θ italic_ϕ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_ϕ italic_θ end_POSTSUBSCRIPT. The dashed lines indicate the radial velocities of the two component stars, α 𝛼\alpha italic_α and β 𝛽\beta italic_β. The ellipses at the bottom illustrate the tidal deformation patterns of the GW at representative epochs.

III EM Signals

In this section, we explain how to extract the orbital phase information of a CWDB using basic EM data, such as the time modulation of radial velocities and photometric luminosity. Note that these EM signals are invariant under rotation around the line of sight and, by themselves, do not provide azimuthal information.

III.1 Doppler shifts

We first evaluate the radial velocity components v α subscript 𝑣 𝛼 v_{\alpha}italic_v start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT and v β subscript 𝑣 𝛽 v_{\beta}italic_v start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT for the two stars, α 𝛼\alpha italic_α and β 𝛽\beta italic_β. Taking the time derivatives of Eqs.(2) and (3) and then computing their inner products with the radial unit vector e→N subscript→𝑒 𝑁\vec{e}_{N}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT, we obtain

v α⁢(t)subscript 𝑣 𝛼 𝑡\displaystyle v_{\alpha}(t)italic_v start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_t )=−2⁢π⁢f⁢a⁢m β m T⁢sin⁡θ⁢sin⁡Φ o⁢(t),absent 2 𝜋 𝑓 𝑎 subscript 𝑚 𝛽 subscript 𝑚 𝑇 𝜃 subscript Φ o 𝑡\displaystyle=-2\pi fa\frac{m_{\beta}}{m_{T}}\sin\theta\sin\Phi_{\rm o}(t),= - 2 italic_π italic_f italic_a divide start_ARG italic_m start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG roman_sin italic_θ roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) ,(12) v β⁢(t)subscript 𝑣 𝛽 𝑡\displaystyle v_{\beta}(t)italic_v start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ( italic_t )=2⁢π⁢f⁢a⁢m α m T⁢sin⁡θ⁢sin⁡Φ o⁢(t).absent 2 𝜋 𝑓 𝑎 subscript 𝑚 𝛼 subscript 𝑚 𝑇 𝜃 subscript Φ o 𝑡\displaystyle=2\pi fa\frac{m_{\alpha}}{m_{T}}\sin\theta\sin\Phi_{\rm o}(t).= 2 italic_π italic_f italic_a divide start_ARG italic_m start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG roman_sin italic_θ roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) .(13)

Here, we have ignored the trivial bulk velocity contribution. The velocities vanish, i.e., v α=v β=0 subscript 𝑣 𝛼 subscript 𝑣 𝛽 0 v_{\alpha}=v_{\beta}=0 italic_v start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT = 0, at sin⁡Φ o⁢(t)=0 subscript Φ o 𝑡 0\sin\Phi_{\rm o}(t)=0 roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) = 0. Note that the phase function Φ o⁢(t)subscript Φ o 𝑡\Phi_{\rm o}(t)roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) in Eqs.(12) and (13) is identical to that appearing in Eqs.(2) and (3) for the GW signal. Therefore, at sin⁡Φ o⁢(t)=0 subscript Φ o 𝑡 0\sin\Phi_{\rm o}(t)=0 roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) = 0, the strain deformation satisfies h θ⁢θ=−h ϕ⁢ϕ<0 subscript ℎ 𝜃 𝜃 subscript ℎ italic-ϕ italic-ϕ 0 h_{\theta\theta}=-h_{\phi\phi}<0 italic_h start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT = - italic_h start_POSTSUBSCRIPT italic_ϕ italic_ϕ end_POSTSUBSCRIPT < 0 and h θ⁢ϕ=h ϕ⁢θ=0 subscript ℎ 𝜃 italic-ϕ subscript ℎ italic-ϕ 𝜃 0 h_{\theta\phi}=h_{\phi\theta}=0 italic_h start_POSTSUBSCRIPT italic_θ italic_ϕ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_ϕ italic_θ end_POSTSUBSCRIPT = 0, indicating compression along the ±e→θ plus-or-minus subscript→𝑒 𝜃\pm\vec{e}_{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT directions (see Fig.2).

By comparing the radial velocity profiles with the strain pattern, we can refine the transverse projection of the orientation vector e→j subscript→𝑒 𝑗\vec{e}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT to lie along the ±e→θ plus-or-minus subscript→𝑒 𝜃\pm\vec{e}{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT directions. This reduces the original fourfold degeneracy (±e→θ plus-or-minus subscript→𝑒 𝜃\pm\vec{e}{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT and ±e→ϕ plus-or-minus subscript→𝑒 italic-ϕ\pm\vec{e}{\phi}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT) that arises when using only GW signals.

Since we are dealing with four discrete options (±e→θ plus-or-minus subscript→𝑒 𝜃\pm\vec{e}{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT and ±e→ϕ plus-or-minus subscript→𝑒 italic-ϕ\pm\vec{e}{\phi}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT), we do not need to determine the exact epochs at which sin⁡Φ o⁢(t)=0 subscript Φ o 𝑡 0\sin\Phi_{\rm o}(t)=0 roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) = 0 from the radial velocity data. A phase accuracy of Δ⁢Φ o⁢(t)∼0.1 similar-to Δ subscript Φ o 𝑡 0.1\Delta\Phi_{\rm o}(t)\sim 0.1 roman_Δ roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) ∼ 0.1 would be sufficient for the present scheme.

III.2 Light curve

Next, we discuss a similar scheme based on the periodic variation of the light curve. We may observe signatures induced by strong binary interactions, such as ellipsoidal variations and irradiation effects Maxted (2016). Here, for simplicity and ease of follow-up observations, we focus on the use of eclipse timing.

After straightforward calculations, we find that the transverse distance between the two stars is proportional to

[sin 2⁡Φ o⁢(t)⁢sin 2⁡θ+cos 2⁡θ]1/2.superscript delimited-[]superscript 2 subscript Φ o 𝑡 superscript 2 𝜃 superscript 2 𝜃 1 2[\sin^{2}\Phi_{\rm o}(t)\sin^{2}\theta+\cos^{2}\theta]^{1/2}.[ roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ ] start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT .

For a nearly edge-on configuration with θ∼π/2 similar-to 𝜃 𝜋 2\theta\sim\pi/2 italic_θ ∼ italic_π / 2, sharp drops in the light curve can be observed around the conjunction epochs, where sin⁡Φ o⁢(t)=0 subscript Φ o 𝑡 0\sin\Phi_{\rm o}(t)=0 roman_sin roman_Φ start_POSTSUBSCRIPT roman_o end_POSTSUBSCRIPT ( italic_t ) = 0 Burdge et al. (2019). Therefore, as in the case of velocity curves, by examining the strain pattern during eclipses, we can identify the two candidate orientations ±e→θ plus-or-minus subscript→𝑒 𝜃\pm\vec{e}{\theta}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT, while excluding the spurious solutions ±e→ϕ plus-or-minus subscript→𝑒 italic-ϕ\pm\vec{e}{\phi}± over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT.

IV Summary

LISA is expected to detect approximately 10 4 superscript 10 4 10^{4}10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT CWDBs emitting nearly monochromatic GWs. By geometrically analyzing the waveform from each binary, we can obtain information on its orbital orientation vector, e→j subscript→𝑒 𝑗{\vec{e}}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, in a fundamentally new way. However, due to the intrinsic symmetry of GW emission, there is a fourfold degeneracy in the projected direction of the vector e→j subscript→𝑒 𝑗{\vec{e}}{j}over→ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT onto the transverse plane.

In this paper, by revisiting the time profile of the GW signal in response to orbital motion, we propose a multimessenger strategy to reduce this degeneracy to twofold. The key idea is to identify the strain deformation pattern at specific orbital phases, inferred from EM data such as radial velocity curves and light curves.

Eclipsing patterns in photometric data will serve as the primary observational signature for follow-up identification of CWDBs initially detected by LISA. To expand the sample of binaries available for orientational analysis, an extensive spectroscopic analysis for short-period binaries would be highly beneficial.

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