This tutorial was generated from a Jupyter notebook that can be downloaded here. If you’d like to reproduce the results in the notebook, or make changes to the code, we recommend downloading this notebook and running it with Jupyter as certain cells (mostly those that change plot styles) are excluded from the tutorials.

Derivations and Equation Reference

This guide explains the origin and derivation of the equations used in LEGWORK functions. Let’s go through each of the modules and build up to an equation for the signal-to-noise ratio for a given LISA source.

At the end of this document (here) is a table that relates each of the functions in LEGWORK to an equation in this document.

Conversions and Definitions (utils)

This section contains a miscellaneous collection of conversions and definitions that are useful in the later derivations. First, the chirp mass of a binary is defined as

\begin{equation} \mathcal{M}_{c} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}, \label{eq:chirpmass} \end{equation}

where \(m_1\) and \(m_2\) are the primary and secondary mass of the binary. This term often shows up in many equations and hence is easier to measure in gravitational wave data analysis than the individual component masses.

Kepler’s third law allows one to convert between orbital frequency, \(f_{\rm orb}\), and the semi-major axis, \(a\), of a binary. For convenience we show it here

\begin{equation} a = \left(\frac{G(m_1 + m_2)}{(2 \pi f_{\rm orb})^{2}}\right)^{1/3},\qquad f_{\rm orb} = \frac{1}{2 \pi} \sqrt{\frac{G(m_1 + m_2)}{a^3}}. \label{eq:kepler3rd} \end{equation}

As we deal with eccentric binaries, the different harmonic frequencies of gravitational wave emission become important. We can write that the relative power radiated into the \(n^{\rm th}\) harmonic for a binary with eccentricity \(e\) is [PetersMathews63] (Eq. 20)

\begin{equation} \begin{aligned} g(n, e) = \frac{n^{4}}{32} & \left\{ \left[ J_{n-2}(n e)-2 e J_{n-1}(n e)+\frac{2}{n} J_{n}(n e)+2 e J_{n+1}(n e)-J_{n+2}(n e)\right]^{2}\right.\\ &\left.+\left(1-e^{2}\right)\left[J_{n-2}(n e)-2 J_{n}(n e)+J_{n+2}(n e)\right]^{2}+\frac{4}{3 n^{2}}\left[J_{n}(n e)\right]^{2}\right\}, \end{aligned} \label{eq:g(n,e)} \end{equation}

where \(J_{n}(v)\) is the Bessel function of the first kind. Thus, the sum of \(g(n, e)\) over all harmonics gives the factor by which the gravitational wave emission is stronger for a binary of eccentricity \(e\) over an otherwise identical circular binary. This enhancement factor is defined by Peters as [PetersMathews63] (Eq. 17)

\begin{equation} F(e) = \sum_{n = 1}^{\infty} g(n, e) = \frac{1 + (73 / 24) e^2 + (37 / 96) e^4}{(1 - e^2)^{7/2}}. \label{eq:eccentricity_enhancement_factor} \end{equation}

Note that \(F(0) = 1\) as one would expect. A useful number to remember is that \(F(0.5) \approx 5.0\), or in words, a binary with eccentricity \(0.5\) loses energy to gravitational waves at a rate about \(5\) times higher than a similar circular binary.

For binary evolution Peters and Mathews introduced two constants that are useful for calculations though without physical meaning. First, from [Peters64] (Eq. 5.9)

\begin{equation} \beta(m_1, m_2) = \frac{64}{5} \frac{G^3}{c^5} m_1 m_2 (m_1 + m_2). \label{eq:beta_peters} \end{equation}

And additionally from [Peters64] (Eq. 5.11)

\begin{equation} c_0(a_0, e_0) = a_0 \frac{(1 - e_0^2)}{e_0^{12/19}} \left(1 + \frac{121}{304} e_0^2\right)^{-870/2299} \label{eq:c0_peters} \end{equation}

where \(a_0\) and \(e_0\) are the initial semi-major axis and eccentricity respectively.

Binary Evolution (evol)

Circular binaries

For a circular binary, the evolution can be calculated analytically as the rate at which the binary shrinks can be readily integrated. This gives the semi-major axis of a circular binary as a function of time as [Peters64] (Eq. 5.9)

\begin{equation} a(t, m_1, m_2) = [a_0^4 - 4 t \beta(m_1, m_2)]^{1/4}, \label{eq:a_over_time_circ} \end{equation}

where \(a_0\) is the initial semi-major axis and \(\beta\) is defined in Eq. \eqref{eq:beta_peters}. We can use this to also get the frequency evolution by using Kepler’s third law Eq. \eqref{eq:kepler3rd}.

Moreover, we can set the final semi-major axis in Eq. \eqref{eq:a_over_time_circ} equal to zero and solve for the inspiral time ([Peters64] Eq. 5.10)

\begin{equation} t_{\rm merge, circ} = \frac{a_0^4}{4 \beta} \label{eq:t_merge_circular} \end{equation}

Eccentric binaries

Eccentric binaries are more complicated because the semi-major axis and eccentricity both evolve simultaneously and depend on one another. These equations cannot be solved analytically and require numerical integration. Firstly, we can relate \(a\) and \(e\) with [Peters64] (Eq. 5.11)

\begin{equation} a(e) = c_0 \frac{e^{12/19}}{(1 - e^2)} \left(1 + \frac{121}{304} e^2\right)^{870/2299}, \label{eq:a_from_e} \end{equation}

where \(c_0\) is defined in Eq. \eqref{eq:c0_peters} – such that the initial conditions are satisfied. Then we can numerically integrate [Peters64] (Eq. 5.13)

\begin{equation} \frac{\mathrm{d}e}{\mathrm{d}t} = -\frac{19}{12} \frac{\beta}{c_{0}^{4}} \frac{e^{-29 / 19}\left(1-e^{2}\right)^{3 / 2}}{\left[1+(121 / 304) e^{2}\right]^{1181 / 2299}}, \label{eq:dedt} \end{equation}

to find \(e(t)\) and use this in conjunction with Eq. \eqref{eq:a_from_e} to solve for \(a(t)\), which can in turn be converted to \(f_{\rm orb}(t)\).

Furthermore, we can invert this to find the inspiral time by using that \(e \to 0\) when the binary merges which gives [Peters64] (Eq. 5.14)

\begin{equation} t_{\rm merge} = \frac{12}{19} \frac{c_{0}^{4}}{\beta} \int_0^{e_0} \frac{\left[1+(121 / 304) e^{2}\right]^{1181 / 2299}}{e^{-29 / 19}\left(1-e^{2}\right)^{3 / 2}} \mathrm{d}e \label{eq:t_merge_eccentric} \end{equation}

For very small or very large eccentricities we can approximate this integral using the following expressions (given in unlabelled equations after [Peters64] Eq. 5.14)

\begin{equation} t_{\rm merge,\, e^2 \ll 1} = \frac{c_0^4}{4 \beta} \cdot e_0^{48 / 19} \end{equation}
\begin{equation} t_{\rm merge,\, (1 - e^2) \ll 1} = \frac{768}{425} \frac{a_0^4}{4 \beta} (1 - e_0^2)^{7/2} \end{equation}

Gravitational Wave Strains (strain)

Characteristic Strain

The characteristic strain from a binary in the \(n^{\rm th}\) harmonic is defined as follows (e.g. [BarackCutler04] Eq. 56; [FlanaganHughes98] Eq. 5.1)

\begin{equation} h_{c,n}^2 = \frac{1}{(\pi D_L)^2} \left( \frac{2 G}{c^3} \frac{\dot{E_n}}{\dot{f_n}} \right), \label{eq:char_strain_dedf} \end{equation}

where \(D_L\) is the luminosity distance to the binary, \(\dot{E}_n\) is the power radiated in the \(n^{\rm th}\) harmonic and \(\dot{f}_n\) is the rate of change of the \(n^{\rm th}\) harmonic frequency. The power radiated in the \(n^{\rm th}\) harmonic is given by [PetersMathews63] (Eq. 19)

\begin{equation} \dot{E}_n = \frac{32}{5} \frac{G^{4}}{c^5} \frac{m_{1}^{2} m_{2}^{2}\left(m_{1}+m_{2}\right)}{a^{5}} g(n, e), \label{eq:edot_peters} \end{equation}

where \(m_1\) is the primary mass, \(m_2\) is the secondary mass, \(a\) is the semi-major axis of the binary and \(e\) is the eccentricity. Using Eq. \eqref{eq:chirpmass} and Eq. \eqref{eq:kepler3rd}, we can recast Eq. \eqref{eq:edot_peters} in a form more applicable for gravitational wave detections that is a function of only the chirp mass, orbital frequency and eccentricity.

\begin{align} \dot{E}_n &= \frac{32}{5} \frac{G^{4}}{c^5} \left(m_{1}^{2} m_{2}^{2}\left(m_{1}+m_{2}\right)\right) g(n, e) \cdot \left(\frac{(2 \pi f_{\rm orb})^{2}}{G(m_1 + m_2)}\right)^{5/3} \\ \dot{E}_n &= \frac{32}{5} \frac{G^{7/3}}{c^5} \frac{m_{1}^{2} m_{2}^{2}}{\left(m_{1}+m_{2}\right)^{2/3}} (2 \pi f_{\rm orb})^{10/3} g(n, e) \\ \dot{E}_n(\mathcal{M}_c, f_{\rm orb}, e) &= \frac{32}{5} \frac{G^{7 / 3}}{c^{5}}\left(2 \pi f_{\mathrm{orb}} \mathcal{M}_{c}\right)^{10 / 3} g(n, e) \label{eq:edot} \end{align}

The last term needed to define the characteristic strain in Eq. \eqref{eq:char_strain_dedf} is the rate of change of the \(n^{\rm th}\) harmonic frequency. We can first apply the chain rule and note that

\begin{equation} \dot{f}_{n} = \frac{\mathrm{d}f_{n}}{\mathrm{d} a} \frac{\mathrm{d} a}{\mathrm{d} t}. \label{eq:fdot_chainrule} \end{equation}

The frequency of the \(n^{\rm th}\) harmonic is simply defined as \(f_n = n \cdot f_{\rm orb}\) and therefore we can find an expression for \(\mathrm{d} {f_{n}} / \mathrm{d} {a}\) by rearranging and differentiating Eq. \eqref{eq:kepler3rd}

\begin{align} f_{n} &= \frac{n}{2 \pi} \sqrt{\frac{G(m_1 + m_2)}{a^3}}, \\ \frac{\mathrm{d}f_{n}}{\mathrm{d} a} &= -\frac{3 n}{4 \pi} \frac{\sqrt{G(m_1 + m_2)}}{a^{5/2}}. \label{eq:dfda} \end{align}

The rate at which the semi-major axis decreases is [Peters64] (Eq. 5.6)

\begin{equation} \frac{\mathrm{d} a}{\mathrm{d} t} = -\frac{64}{5} \frac{G^{3} m_{1} m_{2}\left(m_{1}+m_{2}\right)}{c^{5} a^{3}} F(e). \label{eq:dadt} \end{equation}

Substituting Eq. \eqref{eq:dfda} and Eq. \eqref{eq:dadt} into Eq. \eqref{eq:fdot_chainrule} gives an expression for \(\dot{f}_{n}\)

\begin{align} \dot{f}_n &= -\frac{3 n}{4 \pi} \frac{\sqrt{G(m_1 + m_2)}}{a^{5/2}} \cdot -\frac{64}{5} \frac{G^{3} m_{1} m_{2}\left(m_{1}+m_{2}\right)}{c^{5} a^{3}} F(e), \\ \dot{f}_n &= \frac{48 n}{5 \pi} \frac{G^{7/2}}{c^5} \left(m_1 m_2 (m_1 + m_2)^{3/2}\right) \frac{F(e)}{a^{11/2}}, \end{align}

which, as above with \(\dot{E}_n\), we can recast using Kepler’s third law and the definition of the chirp mass

\begin{align} \dot{f}_n &= \frac{48 n}{5 \pi} \frac{G^{7/2}}{c^5} \left(m_1 m_2 (m_1 + m_2)^{3/2}\right) F(e) \cdot \left(\frac{(2 \pi f_{\rm orb})^{2}}{G(m_1 + m_2)}\right)^{11/6}, \\ &= \frac{48 n}{5 \pi} \frac{G^{5/3}}{c^5} \frac{m_1 m_2}{(m_1 + m_2)^{1/3}} \cdot (2 \pi f_{\rm orb})^{11/3} \cdot F(e), \\ \dot{f}_n(\mathcal{M}_c, f_{\rm orb}, e) &= \frac{48 n}{5 \pi} \frac{\left(G \mathcal{M}_c \right)^{5/3}}{c^5} (2 \pi f_{\rm orb})^{11/3} F(e) \label{eq:fdot} \end{align}

With definitions of both \(\dot{E}_n\) and \(\dot{f}_n\), we are now in a position to find an expression for the characteristic strain by plugging Eq. \eqref{eq:edot} and Eq. \eqref{eq:fdot} into Eq. \eqref{eq:char_strain}:

\begin{align} h^2_{c,n} &= \frac{1}{(\pi D_L)^2} \left( \frac{2 G}{c^3} \frac{\frac{32}{5} \frac{G^{7 / 3}}{c^{5}}\left(2 \pi f_{\mathrm{orb}} \mathcal{M}_{c}\right)^{10 / 3} g(n, e)}{\frac{48 n}{5 \pi} \frac{\left(G \mathcal{M}_c\right)^{5/3}}{c^5} (2 \pi f_{\rm orb})^{11/3} F(e)} \right) \\ &= \frac{1}{(\pi D_L)^2} \left( \frac{2^{5/3} \pi^{2/3}}{3} \frac{(G \mathcal{M}_c)^{5/3}}{c^3} \frac{1}{f_{\rm orb}^{1/3}} \frac{g(n, e)}{n F(e)} \right) \end{align}

This gives a final simplified expression for the characteristic strain amplitude of a GW source.

\begin{equation} h_{c,n}^2(\mathcal{M}_c, D_L, f_{\rm orb}, e) = \frac{2^{5/3}}{3 \pi^{4/3}} \frac{(G \mathcal{M}_c)^{5/3}}{c^3 D_L^2} \frac{1}{f_{\rm orb}^{1/3}} \frac{g(n, e)}{n F(e)} \label{eq:char_strain} \end{equation}


The strain can be found by dividing the characteristic strain by the square root of twice the number of cycles. This is explained in [FinnThorne00] and in further detail in [MooreColeBerry15]. The physical reasoning behind the idea is that the binary will spend a certain amount of time in the vicinity of some frequency \(f\) and cause a similar gravitational wave strain. This leads to the signal ‘accumulating’ and resulting in a larger signal-to-noise ratio. Therefore, the characteristic strain represents the strain measured by the detector over the duration of the mission, whilst the strain is what is emitted by the binary at each instantaneous moment.

Following this logic, we can convert between characteristic strain and strain with (e.g. [FinnThorne00] see text before Eq. 2.2)

\begin{equation} h_{c, n}^2 = \left(\frac{f_n^2}{\dot{f}_n} \right) h_n^2. \label{eq:strain-charstrain} \end{equation}


Note that this is factor of 2 different from [FinnThorne00]. This is because the factor of 2 is already included in the [RobsonCornishLiu19] sensitivity curve and so we remove it here.

Therefore, using this, in addition to Eq. \eqref{eq:fdot} and Eq. \eqref{eq:char_strain}, we can write an expression for the strain amplitude of gravitational waves in the \(n^{\rm th}\) harmonic

\begin{align} h_n^2 &= \left(\frac{\dot{f}_n}{f_n^2} \right) h_{c, n}^2, \\ h_n^2 &= \left(\frac{48 n}{5 \pi} \frac{\left(G \mathcal{M}_c \right)^{5/3}}{c^5} F(e) \cdot \frac{(2 \pi f_{\rm orb})^{11/3}}{n^2 f_{\rm orb}^2} \right) \left(\frac{2^{5/3}}{3 \pi^{4/3}} \frac{(G \mathcal{M}_c)^{5/3}}{c^3 D_L^2} \frac{1}{n f_{\rm orb}^{1/3}} \frac{g(n, e)}{F(e)} \right), \end{align}

This gives a final simplified expression for the strain amplitude of a GW source.

\begin{equation} h_n^2(\mathcal{M}_c, f_{\rm orb}, D_L, e) = \frac{2^{28/3}}{5} \frac{(G \mathcal{M}_c)^{10/3}}{c^8 D_L^2} \frac{g(n, e)}{n^2} \left(\pi f_{\rm orb} \right)^{4/3} \label{eq:strain} \end{equation}

Amplitude modulation for orbit averaged sources

Because the LISA detectors are not stationary and instead follow an Earth-trailing orbit, the antenna pattern of LISA is not isotropically distributed or stationary. For sources that have a known position, inclination, and polarisation, we can consider the amplitude modulation of the strain due to the average motion of LISA’s orbit. We closely follow the results of [CornishLarson03] to write down the amplitude modulation as

\begin{equation} A_{\rm{mod}}^{2}=\frac{1}{2} \left[\left(1+\cos ^{2} \iota\right)^{2}\left\langle F_{+}^{2}\right\rangle+4 \cos ^{2} \iota\left\langle F_{\times}^{2}\right\rangle\right], \label{eq:amp_mod} \end{equation}

where \(\left\langle F_{+}^{2}\right\rangle\) and \(\left\langle F_{\times}^{2}\right\rangle\), the orbit-averaged detector responses, are defined as

\begin{equation} \left \langle F_{+}^{2} \right\rangle = \frac{1}{4}\big(\cos ^{2} 2 \psi\left\langle D_{+}^{2}\right\rangle -\sin 4 \psi\left\langle D_{+} D_{\times}\right\rangle +\sin ^{2} 2 \psi\left\langle D_{\times}^{2}\right\rangle\big), \label{eq:response_fplus} \end{equation}
\begin{equation} \left\langle F_{\times}^{2} \right\rangle = \frac{1}{4}\big(\cos^{2} 2 \psi \left \langle D_{\times}^{2} \right\rangle +\sin 4 \psi \left \langle D_{+} D_{\times} \right\rangle +\sin ^{2} 2 \psi \left \langle D_{+}^{2} \right\rangle \big), \label{eq:response_fcross} \end{equation}


\begin{equation} \left\langle D_{+} D_{\times} \right\rangle = \frac{243}{512} \cos \theta \sin 2 \phi \left(2 \cos ^{2} \phi-1\right) \left(1+\cos ^{2} \theta\right), \label{eq:d_plus_cross} \end{equation}\begin{equation} \left\langle D_{\times}^{2} \right\rangle = \frac{3}{512}\big(120 \sin ^{2} \theta +\cos ^{2} \theta + 162 \sin ^{2} 2 \phi \cos ^{2} \theta\big), \label{eq:d_cross} \end{equation}\begin{equation} \left\langle D_{+}^{2} \right\rangle = \frac{3}{2048}\big[487+158 \cos ^{2} \theta+7 \cos ^{4} \theta -162 \sin ^{2} 2 \phi\left(1+\cos ^{2} \theta\right)^{2}\big]. \label{eq:d_plus} \end{equation}

In the equations above, the inclination is given by \(\iota\), the right ascension and declination are given by \(\phi\) and \(\theta\) respsectively, and the polarisation is given by \(\psi\).

The orbital motion of LISA smears the source frequency by roughly \(10^{-4}\,\rm{mHz}\) due to the antenna pattern changing as the detector orbits, the Doppler shift from the motion, and the phase modulation from the \(+\) and \(\times\) polarisations in the antenna pattern. Generally, the modulation reduces the strain amplitude because the smearing in frequency reduces the amount of signal build up at the true source frequency.

We note that since the orbit averaging is carried out in Fourier space, this requires the frequency to be monochromatic and thus is only implemented in LEGWORK for quasi-circular binaries. We also note that since the majority of the calculations in LEGWORK are carried out for the full position, polarisation, and inclination averages, we place a pre-factor of \(5/4\) on the amplitude modulation in the software implementation to undo the factor of \(4/5\) which arises from the averaging of Equations \eqref{eq:response_fplus} and \eqref{eq:response_fcross}.

Sensitivity Curves (psd)


For the LISA sensitivity curve, we follow the equations from [RobsonCornishLiu19], which we list here for your convenience.

The effective LISA noise power spectral density is defined as ([RobsonCornishLiu19] Eq. 2)

\begin{equation} S_{\rm n}(f) = \frac{P_n(f)}{\mathcal{R}(f)} + S_c(f), \end{equation}

where \(P_{\rm n}(f)\) is the power spectral density of the detector noise and \(\mathcal{R}(f)\) is the sky and polarisation averaged signal response function of the instrument. Alternatively if we expand out \(P_n(f)\), approximate \(\mathcal{R}(f)\) and simplify we find ([RobsonCornishLiu19] Eq. 1)

\begin{equation} S_{\rm n}(f) = \frac{10}{3 L^2} \left(P_{\rm OMS}(f) + \frac{4 P_{\rm acc}(f)}{(2 \pi f)^4} \right) \left(1 + \frac{6}{10} \left(\frac{f}{f_*} \right)^2 \right) + S_c(f) \label{eq:LISA_Sn} \end{equation}

where \(L = 2.5\,\mathrm{Gm}\) is detector arm length, \(f^* = 19.09 \, \mathrm{mHz}\) is the response frequency,

\begin{equation} P_{\rm OMS}(f) = \left(1.5 \times 10^{-11} \mathrm{m}\right)^{2}\left(1+\left(\frac{2 \mathrm{mHz}}{f}\right)^{4}\right) \mathrm{Hz}^{-1} \end{equation}

is the single-link optical metrology noise ([RobsonCornishLiu19] Eq. 10),

\begin{equation} P_{\rm acc}(f) = \left(3 \times 10^{-15} \mathrm{ms}^{-2}\right)^{2}\left(1+\left(\frac{0.4 \mathrm{mHz}}{f}\right)^{2}\right)\left(1+\left(\frac{f}{8 \mathrm{mHz}}\right)^{4}\right) \mathrm{Hz}^{-1} \end{equation}

is the single test mass acceleration noise ([RobsonCornishLiu19] Eq. 11) and

\begin{equation} S_{c}(f)=A f^{-7 / 3} e^{-f^{\alpha}+\beta f \sin (\kappa f)}\left[1+\tanh \left(\gamma\left(f_{k}-f\right)\right)\right] \mathrm{Hz}^{-1} \end{equation}

is the galactic confusion noise ([RobsonCornishLiu19] Eq. 14), where the amplitude \(A\) is fixed as \(9 \times 10^{-45}\) and the various parameters change over time:


6 months

1 year

2 years

4 years



























We additionally allow other instruments than LISA. We have the TianQin sensitivity curve built in where we use the power spectral density given in [HuangHuKorol+20] Eq. 13.

\begin{equation} \begin{split} S_{N}(f) &= \frac{10}{3 L^{2}}\left[\frac{4 S_{a}}{(2 \pi f)^{4}}\left(1+\frac{10^{-4} H z}{f}\right)+S_{x}\right] \\ & \times\left[1+0.6\left(\frac{f}{f_{*}}\right)^{2}\right] \end{split} \label{eq:tianqin} \end{equation}

Note that this expression includes an extra factor of 10/3 compared Eq. 13 in [HuangHuKorol+20], since [HuangHuKorol+20] absorbs the factor into the waveform but we instead follow the same convention as [RobsonCornishLiu19] for consistency and include it in this ‘effective’ PSD function instead.

Equation to Function Table

The following table gives a list of the functions in the modules and which equation numbers in this document that they come from.










\(f_{\rm orb}\)



\(g(n, e)\)









\(a_{\rm circ}(t), f_{\rm orb, circ}(t)\)



\(t_{\rm merge, circ}\)



\(e(t), a(t), f_{\rm orb}(t)\)



\(t_{\rm merge}\)









\(S_{\rm n}(f)\)






\(\rho_{\rm e, e}\)



\(\rho_{\rm c, e}\)



\(\rho_{\rm e, s}\)



\(\rho_{\rm c, s}\)





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