Binary system parameters

A compact binary system is characterised by the masses and spins of its two components. This page documents the parameterisations implemented in the puddin::binary module and the conventions used throughout the library.

Mass parameters

Given component masses \(m_1 \geq m_2 > 0\), the derived mass parameters are:

Total mass

\[M = m_1 + m_2\]

Mass ratio

\[q = \frac{m_2}{m_1} \in (0, 1]\]

Symmetric mass ratio

\[\eta = \frac{m_1 m_2}{M^2} \in \left(0, \tfrac{1}{4}\right]\]

with \(\eta = 1/4\) for equal-mass systems.

Chirp mass

\[\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{M^{1/5}} = M \eta^{3/5}\]

The chirp mass governs the leading-order phase evolution during inspiral:

\[\dot{f} \propto \mathcal{M}^{5/3} f^{11/3}\]

It is always true that \(\mathcal{M} \leq M\).

Spin parameters

Each black hole carries a dimensionless spin \(\mathbf{a}_i\) with magnitude \(a_i = |\mathbf{a}_i| \in [0,1]\) (normalised to the Kerr maximum \(GM^2/c\)) and tilt angle \(\theta_i \in [0, \pi]\) measured from the orbital angular momentum axis \(\hat{L}\).

Effective inspiral spin

\[\chi_\mathrm{eff} = \frac{m_1 a_1 \cos\theta_1 + m_2 a_2 \cos\theta_2}{M} \in [-1, 1]\]

\(\chi_\mathrm{eff}\) is approximately conserved through inspiral at 1.5 post-Newtonian order and is the dominant spin observable from the inspiral phase of a gravitational-wave signal.

Effective precession spin

\[\chi_p = \max\!\left( a_1 \sin\theta_1,\; \frac{3 + 4q}{4(1 + q)}\, q\, a_2 \sin\theta_2 \right) \in [0, 1]\]

as defined in . \(\chi_p\) characterises the degree of orbital-plane precession driven by in-plane spin components.