Binary Stars and Gravitational Waves

Gravitational waves (GWs) offer a unique probe into compact binary systems. Non-merging white dwarf (WD) binaries—systems that emit continuous, nearly monochromatic GWs without merging within a Hubble time—are significant sources in the milli-Hertz frequency range, observable by LISA. Here we summarize results for GW strain amplitude from such binaries and details how LISA determines their distance and sky position, with exhaustive derivations and examples.

1. Derivation of the GW Signal

Consider two white dwarfs with masses $M_1$ and $M_2$, orbiting circularly with period $P$, at a luminosity distance $D$ from Earth. We use the quadrupole approximation in the non-relativistic limit to compute the GW strain.

Step 1: Define Orbital Parameters

The orbital dynamics are foundational to the GW signal. Define the following:

• Orbital Angular Frequency ($\omega$):
  $$\omega =\frac{2\pi }{P},$$
  where $P$ is the orbital period in seconds, and $\omega$ is in radians per second. This relates the frequency of rotation to the time for one complete orbit.
• Total Mass ($M$):
  $$M=M_1+M_2,$$
  the sum of the individual masses in kilograms, governing the system's gravitational interaction.
• Reduced Mass ($\mu$):
  $$\mu =\frac{M_1{M}_2}{M_1+M_2},$$
  a measure of the effective mass for the two-body problem, also in kilograms, which influences the GW amplitude.
• Orbital Separation ($a$): Apply Kepler's third law for a circular orbit:
  $$P=2\pi \sqrt{\frac{a^3}{GM}},$$
  where $G$ is the gravitational constant ($6.67430\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$). Square both sides:
  $$P^2=\frac{4{\pi }^2{a}^3}{GM}.$$
  Rearrange for $a^3$ and take the cube root:
  $$a={\left(\frac{GM{P}^2}{4{\pi }^2}\right)}^{1/3},$$
  yielding the semi-major axis $a$ as function of the period and total mass.
• Orbital Inclination ($\iota$): The angle between the orbital angular momentum vector and the line of sight to the observer. For $\iota =0$, the binary is face-on, while for $\iota =\pi /2$, it is edge-on.

Step 2: Compute the Quadrupole Moment

The GW signal arises from the changing mass quadrupole moment. The orbit lies in a plane that can be tilted with respect to the observer's line of sight. For a general inclination, we first define the positions in the orbital plane and then transform to the observer's frame.

In the orbital plane (with center of mass at the origin), the positions are:

• WD 1 Position (${\mathbf{r}}_1$):
  $${\mathbf{r}}_1=(\frac{M_2}{M}a\cos \omega t,\frac{M_2}{M}a\sin \omega t,0),$$
  where $\frac{M_2}{M}a$ is the distance from the center of mass to WD 1.
• WD 2 Position (${\mathbf{r}}_2$):
  $${\mathbf{r}}_2=(-\frac{M_1}{M}a\cos \omega t,-\frac{M_1}{M}a\sin \omega t,0),$$
  with $\frac{M_1}{M}a$ as the distance to WD 2, and the negative sign ensuring the center of mass condition $M_1{\mathbf{r}}_1+M_2{\mathbf{r}}_2=0$.

The mass quadrupole tensor is:

$$I_{ij}=\sum _k{m}_k(r_k^i{r}_k^j-\frac{1}{3}{\delta }_{ij}{r}_k^2),$$

where ${\delta }_{ij}$ is the Kronecker delta, and $r_k^2=(r_k^x{)}^2+(r_k^y{)}^2+(r_k^z{)}^2$. Compute key components:

• $I_{xx}$:
  $$I_{xx}=M_1{(\frac{M_2}{M}a\cos \omega t)}^2+M_2{(-\frac{M_1}{M}a\cos \omega t)}^2,$$
  $$=M_1\frac{M_2^2{a}^2{\cos }^2\omega t}{M^2}+M_2\frac{M_1^2{a}^2{\cos }^2\omega t}{M^2},$$
  $$=\frac{M_1{M}_2^2+M_2{M}_1^2}{M^2}{a}^2{\cos }^2\omega t=\frac{M_1{M}_2(M_1+M_2)}{M^2}{a}^2{\cos }^2\omega t,$$
  $$=\mu a^2{\cos }^2\omega t.$$
  Adjust for the trace:
  $$r_1^2={\left(\frac{M_2}{M}a\right)}^2({\cos }^2\omega t+{\sin }^2\omega t)={\left(\frac{M_2}{M}a\right)}^2,$$
  $$I_{xx}=\mu a^2{\cos }^2\omega t-\frac{1}{3}\mu a^2{\delta }_{xx}=\mu a^2({\cos }^2\omega t-\frac{1}{3}).$$
• $I_{yy}$:
  $$I_{yy}=\mu a^2{\sin }^2\omega t-\frac{1}{3}\mu a^2=\mu a^2({\sin }^2\omega t-\frac{1}{3}).$$
• $I_{xy}=I_{yx}$:
  $$I_{xy}=M_1(\frac{M_2}{M}a\cos \omega t)(\frac{M_2}{M}a\sin \omega t)+M_2(-\frac{M_1}{M}a\cos \omega t)(-\frac{M_1}{M}a\sin \omega t),$$
  $$=\mu a^2\cos \omega t\sin \omega t.$$

Step 3: Second Time Derivative of the Quadrupole Moment

GWs depend on the second time derivative ${\ddot{I}}_{ij}$:

• ${\ddot{I}}_{xx}$:
  $${\dot{I}}_{xx}=\mu a^2\frac{d}{dt}({\cos }^2\omega t)=\mu a^2\cdot 2\cos \omega t(-\omega \sin \omega t)=-2\mu a^2\omega \cos \omega t\sin \omega t,$$
  $${\ddot{I}}_{xx}=\frac{d}{dt}(-2\mu a^2\omega \cos \omega t\sin \omega t),$$
  $$=-2\mu a^2\omega [(-\omega \sin \omega t)\sin \omega t+\cos \omega t(\omega \cos \omega t)],$$
  $$=-2\mu a^2{\omega }^2(-{\sin }^2\omega t+{\cos }^2\omega t)=-2\mu a^2{\omega }^2\cos 2\omega t,$$
  using the identity ${\cos }^2\omega t-{\sin }^2\omega t=\cos 2\omega t$.
• ${\ddot{I}}_{yy}$:
  $${\ddot{I}}_{yy}=-{\ddot{I}}_{xx}=2\mu a^2{\omega }^2\cos 2\omega t,$$
• ${\ddot{I}}_{xy}$:
  $${\dot{I}}_{xy}=\mu a^2\frac{d}{dt}(\cos \omega t\sin \omega t)=\mu a^2[(-\omega \sin \omega t)\sin \omega t+\cos \omega t(\omega \cos \omega t)],$$
  $$=\mu a^2\omega ({\cos }^2\omega t-{\sin }^2\omega t)=\mu a^2\omega \cos 2\omega t,$$
  $${\ddot{I}}_{xy}=\mu a^2\omega \frac{d}{dt}(\cos 2\omega t)=\mu a^2\omega (-2\omega \sin 2\omega t)=-2\mu a^2{\omega }^2\sin 2\omega t.$$

Step 4: GW Strain Amplitude for Arbitrary Inclination

In the transverse-traceless (TT) gauge, the GW strain for an observer at inclination $\iota$ is:

$$h_{ij}^{\text{TT}}=\frac{2G}{c^{4}D}{\ddot{I}}_{ij}^{\text{TT}},$$

where $c$ is the speed of light ($3\times {10}^8\text{m/s}$), and $D$ is in meters.

When accounting for inclination $\iota$, we need to project the quadrupole tensor onto the observer's reference frame. For an observer along a general direction, the two polarization components become:

$$h_{+}=\frac{G}{c^{4}D}\mu a^2{\omega }^2(1+{\cos }^2\iota )\cos 2\omega t,$$

$$h_{\times }=\frac{2G}{c^{4}D}\mu a^2{\omega }^2\cos \iota \sin 2\omega t.$$

The overall GW strain amplitude scales as:

$$h\sim \frac{G}{c^{4}D}\mu a^2{\omega }^2\sqrt{(1+{\cos }^2\iota {)}^2+4{\cos }^2\iota }.$$

Substitute $a$ and $\omega$:

$$a^2={\left(\frac{GM{P}^2}{4{\pi }^2}\right)}^{2/3},$$

$${\omega }^2=\frac{4{\pi }^2}{P^2},$$

$$h\sim \frac{G}{c^{4}D}\mu {\left(\frac{GM{P}^2}{4{\pi }^2}\right)}^{2/3}\frac{4{\pi }^2}{P^2}\sqrt{(1+{\cos }^2\iota {)}^2+4{\cos }^2\iota }.$$

Simplify:

$$h=\frac{G}{c^{4}D}\mu (GM{)}^{2/3}\frac{4{\pi }^{2/3}}{P^{2/3}}\sqrt{(1+{\cos }^2\iota {)}^2+4{\cos }^2\iota }.$$

Relate to GW frequency $f=\frac{2}{P}$ (since GWs oscillate at twice the orbital frequency), so $P=\frac{2}{f}$:

$$P^{-2/3}={\left(\frac{2}{f}\right)}^{-2/3}={\left(\frac{f}{2}\right)}^{2/3},$$

$$h=\frac{G}{c^{4}D}\mu (GM{)}^{2/3}4{\pi }^{2/3}{\left(\frac{f}{2}\right)}^{2/3}\sqrt{(1+{\cos }^2\iota {)}^2+4{\cos }^2\iota }.$$

Introducing the chirp mass:

$$M_{\text{chirp}}=\frac{(M_1{M}_2{)}^{3/5}}{(M_1+M_2{)}^{1/5}}={\mu }^{3/5}{M}^{1/5},$$

the strain can be written in the standard form:

$$h=\frac{4}{D}{\left(\frac{G{M}_{\text{chirp}}}{c^3}\right)}^{5/3}(\pi f{)}^{2/3}\mathcal{A}(\iota ),$$

where $\mathcal{A}(\iota )=\sqrt{(1+{\cos }^2\iota {)}^2+4{\cos }^2\iota }/4$ is the inclination-dependent amplitude factor.

Step 4b: Frequency Evolution and Chirp

While many WD binaries can be treated as monochromatic sources, some systems evolve measurably during observation. The orbital decay due to gravitational wave emission causes the frequency to increase over time, producing a "chirping" signal.

The energy loss due to gravitational wave emission is given by:

$$\frac{dE}{dt}=-\frac{32}{5}\frac{G^4}{c^5}(M_1{M}_2{)}^2(M_1+M_2){\left(\frac{2\pi f_{\text{orb}}}{c}\right)}^{10/3}$$

The orbital energy can be related to frequency:

$$E=-\frac{G{M}_1{M}_2}{2a}=-\frac{G{M}_1{M}_2}{2}{\left(\frac{G(M_1+M_2)}{4{\pi }^2}\right)}^{1/3}(2\pi f_{\text{orb}}{)}^{2/3}$$

Taking the time derivative and equating with the energy loss rate:

$$\frac{dE}{dt}=-\frac{G{M}_1{M}_2}{2}{\left(\frac{G(M_1+M_2)}{4{\pi }^2}\right)}^{1/3}\cdot \frac{2}{3}\cdot (2\pi {)}^{2/3}\cdot f_{\text{orb}}^{-1/3}\cdot {\dot{f}}_{\text{orb}}$$

Solving for ${\dot{f}}_{\text{orb}}$:

$${\dot{f}}_{\text{orb}}=\frac{96}{5}{\pi }^{8/3}{\left(\frac{G{M}_{\text{chirp}}}{c^3}\right)}^{5/3}{f}_{\text{orb}}^{11/3}$$

For gravitational waves with $f_{GW}=2f_{orb}$:

$${\dot{f}}_{GW}=2{\dot{f}}_{\text{orb}}=\frac{96{\pi }^{8/3}}{5\cdot 2^{8/3}}\frac{G^{5/3}{M}_{\text{chirp}}^{5/3}}{c^5}{f}_{GW}^{11/3}$$

This frequency derivative is a strong function of both frequency and chirp mass, making it potentially detectable for higher frequency systems with larger masses.

Step 5: GW Polarizations with Inclination

The time-dependent GW polarizations for a binary with inclination $\iota$ are:

$$h_{+}=h_0\frac{1+{\cos }^2\iota }{2}\cos (2\pi ft+\pi \dot{f}{t}^2+{\varphi }_0),$$

$$h_{\times }=h_0\cos \iota \sin (2\pi ft+\pi \dot{f}{t}^2+{\varphi }_0),$$

where $h_0$ is the overall amplitude and ${\varphi }_0$ is an initial phase. Note that for chirping sources, we've included the $\pi\dot{f}t^2$ phase term.

For different inclinations:

• Face-on ($\iota =0$):
  $$h_{+}=h_0\cos (2\pi ft+\pi \dot{f}{t}^2+{\varphi }_0),h_{\times }=h_0\sin (2\pi ft+\pi \dot{f}{t}^2+{\varphi }_0)$$
  The GW has equal plus and cross polarizations, 90° out of phase, creating circular polarization.
• Edge-on ($\iota =\pi /2$):
  $$h_{+}=\frac{h_0}{2}\cos (2\pi ft+\pi \dot{f}{t}^2+{\varphi }_0),h_{\times }=0$$
  Only the plus polarization remains, creating linear polarization with reduced amplitude.

2. Extracting Distance and Position with LISA

LISA, with its $a\approx 2.5\times {10}^9\text{m}$ baseline, uses the GW signal to infer $D$, sky position $(\alpha ,\delta )$, and inclination $\iota$.

Distance ($D$) and Inclination ($\iota$)

From:

$$h=\frac{4}{D}{\left(\frac{G{M}_{\text{chirp}}}{c^3}\right)}^{5/3}(\pi f{)}^{2/3}\mathcal{A}(\iota ),$$

LISA measures $h_{+}$ and $h_{\times }$ separately, allowing determination of both $D$ and $\iota$ through:

$$\frac{h_{\times }}{h_{+}}=\frac{2\cos \iota }{1+{\cos }^2\iota },$$

which gives $\iota$, and then:

$$D=\frac{4\mathcal{A}(\iota )}{h}{\left(\frac{G{M}_{\text{chirp}}}{c^3}\right)}^{5/3}(\pi f{)}^{2/3}.$$

Note that for purely monochromatic sources, $M_{\text{chirp}}$ remains degenerate with distance, and may require electromagnetic counterparts or statistical priors to determine independently.

Chirp Measurement and Distance Precision

For binaries where LISA can measure ${\dot{f}}_{GW}$, the chirp mass can be determined independently:

$$M_{\text{chirp}}={(\frac{5c^5}{96{\pi }^{8/3}}\cdot \frac{{\dot{f}}_{GW}}{f_{GW}^{11/3}}\cdot \frac{2^{8/3}}{G^{5/3}})}^{3/5}$$

This breaks the degeneracy between chirp mass and distance, allowing for direct distance determination:

$$D=\frac{4\mathcal{A}(\iota )}{h}{\left(\frac{G{M}_{\text{chirp}}}{c^3}\right)}^{5/3}(\pi f{)}^{2/3}$$

LISA can typically detect frequency evolution when:

• The frequency shift over the observation period is larger than the Fourier bin width: $\dot{f}_{GW} \cdot T_{obs}^2 > 1$
• The signal-to-noise ratio is sufficient (typically SNR $\gtrsim$ 20)
• The binary has higher frequency ($f_{GW} \gtrsim 3$ mHz) and/or larger chirp mass

The precision in distance determination improves dramatically when $\dot{f}_{GW}$ is measurable:

$$\frac{{\sigma }_D}{D}=\sqrt{{\left(\frac{{\sigma }_h}{h}\right)}^2+{\left(\frac{5}{3}\frac{{\sigma }_{M_{\text{chirp}}}}{M_{\text{chirp}}}\right)}^2}$$

For typical WD binaries at mHz frequencies with 4-year observation times, distance precision can improve from factors of many tens of percent to approximately 10-30% when $\dot{f}_{GW}$ is detectable.

Sky Position ($(\alpha ,\delta )$)

LISA leverages several effects for sky localization:

• Doppler Modulation:
  $$f_{\text{obs}}=f(1+\frac{\mathbf{v}\cdot \mathbf{n}}{c}),$$
  where $\mathbf{v}$ is LISA's orbital velocity (~30 km/s around the Sun), and $\mathbf{n}$ is the unit vector to the source, modulating $f$ over its 1-year orbit.
• Amplitude Modulation:
  $$h(t)=F_{+}{h}_{+}+F_{\times }{h}_{\times },$$
  where $F_{+},F_{\times }$ are antenna pattern functions depending on $(\alpha ,\delta )$ and LISA's orientation.
• Sky Localization Area: For short observations, the resolution is limited by the detector arm length $a$:
  $$\mathrm{\Delta }{\mathrm{\Omega }}_{\text{short}}\sim \frac{1}{{\text{SNR}}^2}{\left(\frac{\lambda }{a}\right)}^2,\lambda =\frac{c}{f},$$
  But for long-term observations (≈1 year), LISA's orbital motion provides an effective baseline of $2R_{\text{orbit}}$ (≈ 2 AU):
  $$\mathrm{\Delta }{\mathrm{\Omega }}_{\text{long}}\sim \frac{1}{{\text{SNR}}^2}{\left(\frac{\lambda }{2R_{\text{orbit}}}\right)}^2$$
  where SNR is the signal-to-noise ratio, and $\lambda$ is the GW wavelength.

Example Calculation

For $f=1\text{mHz}$ and short-term observations with LISA's arm length $a\approx 2.5\times {10}^9\text{m}$:

\[ \lambda = \frac{3