Derived variables

n̂(pnsystem)
n̂(R)
n_hat(pnsystem)
n_hat(R)

The unit vector pointing from object 2 to object 1, when the frame is given by the rotor R. This is equal to

\[n̂(R) = R x̂ R̄\]

λ̂(pnsystem)
λ̂(R)
lambda_hat(pnsystem)
lambda_hat(R)

The unit vector pointing in the direction of the instantaneous velocity of object 1, when the frame is given by the rotor R. This is equal to

\[λ̂(R) = R ŷ R̄\]

This also completes the right-handed triple of $(n̂, λ̂, ℓ̂)$.

ℓ̂(pnsystem)
ℓ̂(R)
ell_hat(pnsystem)
ell_hat(R)

The unit vector pointing along the direction of orbital angular velocity, when the frame is given by the rotor R. This is equal to

\[ℓ̂(R) = R ẑ R̄\]

Ω(pnsystem)
Ω(;v, M=1)
Omega(pnsystem)
Omega(;v, M=1)

Orbital angular frequency.

The parameter v is the PN velocity parameter, and must be passed as a keyword argument — as in Ω(v=0.1). The parameter M is the total mass of the binary. They are related by definition as

\[\Omega \colonequals \frac{v^3}{M}.\]

See also v.

M(pnsystem)
M(M₁, M₂)
total_mass(pnsystem)
total_mass(M1, M2)

Compute the total mass $M₁+M₂$.

μ(pnsystem)
μ(M₁, M₂)
reduced_mass(pnsystem)
reduced_mass(M1, M2)

Compute the reduced mass $(M₁ M₂)/(M₁+M₂)$.

ν(pnsystem)
ν(M₁, M₂)
reduced_mass_ratio(pnsystem)
reduced_mass_ratio(M1, M2)

Compute the reduced mass ratio $(M₁ M₂)/(M₁+M₂)^2$.

Note that the denominator is squared, unlike in the reduced mass μ.

δ(pnsystem)
δ(M₁, M₂)
mass_difference_ratio(pnsystem)
mass_difference_ratio(M1, M2)

Compute mass-difference ratio $(M₁-M₂)/(M₁+M₂)$.

Note that we do not restrict to $M₁ ≥ M₂$ or vice versa; if you prefer that $δ$ always be positive (or always negative), you are responsible for ensuring that.

q(pnsystem)
q(M₁, M₂)
mass_ratio(pnsystem)
mass_ratio(M1, M2)

Compute mass ratio $M₁/M₂$.

Note that we do not restrict to $M₁ ≥ M₂$ or vice versa; if you prefer that $q$ always be greater than or equal to 1 (or vice versa), you are responsible for ensuring that.

ℳ(pnsystem)
ℳ(M₁, M₂)
chirp_mass(pnsystem)
chirp_mass(M1, M2)

Compute the chirp mass ℳ, which determines the leading-order orbital evolution of a binary system due to energy loss by gravitational-wave emission.

The chirp mass is defined as

\[ \mathcal{M} = \frac{(M_1 M_2)^{3/5}} {(M_1 + M_2)^{1/5}}.\]

X₁(pnsystem)
X₁(M₁, M₂)
X1(pnsystem)
X1(M1, M2)

Compute the reduced individual mass $M₁/(M₁+M₂)$.

X₂(pnsystem)
X₂(M₁, M₂)
X2(pnsystem)
X2(M1, M2)

Compute the reduced individual mass $M₂/(M₁+M₂)$.

S⃗₁(pnsystem)

Dimensionful spin vector of object 1.

S⃗₂(pnsystem)

Dimensionful spin vector of object 2.

S⃗(pnsystem)
S⃗(M₁, M₂, χ⃗₁, χ⃗₂)

Total (dimensionful) spin vector $S⃗₁+S⃗₂$.

Σ⃗(pnsystem)
Σ⃗(M₁, M₂, χ⃗₁, χ⃗₂)

Differential spin vector $M(a⃗₂-a⃗₁)$.

χ⃗(pnsystem)
χ⃗(S⃗, M)

Normalized spin vector $S⃗/M²$.

χ⃗ₛ(M₁, M₂, χ⃗₁, χ⃗₂)

Symmetric spin vector $(χ⃗₁+χ⃗₂)/2$.

χ⃗ₐ(M₁, M₂, χ⃗₁, χ⃗₂)

Antisymmetric spin vector $(χ⃗₁-χ⃗₂)/2$.

χₑ(s)
chi_eff(s)

Effective spin parameter of the system.

Defined as

\[\chi_{\mathrm{eff}} \colonequals \frac{c}{G} \left( \frac{\mathbf{S}_1}{M_1} + \frac{\mathbf{S}_2}{M_2} \right) \cdot \frac{\hat{\mathbf{L}}_{\mathrm{N}}} {M}.\]

χₚ(s)
chi_p(s)

Effective precession spin parameter of the system.

Note that there are two different definitions of this quantity found in the literature. The original definition (converted to the convention where $M_1 \geq M_2$) is

\[\begin{gathered} A_1 = 2 + \frac{3M_2}{2M_1} \\ A_2 = 2 + \frac{3M_1}{2M_2} \\ \chi_{\mathrm{p}} \colonequals \frac{1}{A_1 M_1^2} \mathrm{max}\left(A_1 S_{1\perp}, A_2 S_{2\perp} \right). \end{gathered}\]

In a paper from early in the detection era, the LIGO collaboration used this definition.

However, a more recent paper redefines this essentially as $M_1^2$ times that quantity. Using the convention that $q = M_2/M_1 \leq 1$, the definition may be more compactly written as

\[\chi_{\mathrm{p}} \colonequals \mathrm{max} \left( \chi_{1\perp}, \chi_{2\perp} q \frac{4q+3}{4+3q} \right).\]

Again, a more recent paper by LIGO/Virgo/KAGRA uses this convention.

Because it seems to be the trend, this function uses the latter definition.

S⃗₀⁺(s)
S⃗₀⁺(M₁, M₂, κ₁, κ₂, S⃗₁, S⃗₂)

Defined in Eq. (3.4) of Buonanno et al. (2012):

\[\vec{S}_0^+ = \frac{M}{M_1} \left( \frac{\kappa_1} {\kappa_2} \right)^{1/4} \left( 1 + \sqrt{1 - \kappa_1 \kappa_2} \right)^{1/2} \vec{S}_1 + \frac{M}{M_2} \left( \frac{\kappa_2} {\kappa_1} \right)^{1/4} \left( 1 - \sqrt{1 - \kappa_1 \kappa_2} \right)^{1/2} \vec{S}_2.\]

Note that, currently, $\kappa_1$ and $\kappa_2$ are both assumed to be equal to 1, as is the case for black holes. You can define κ₁ and κ₂ to have other values for your own PNSystem types, and this function will work appropriately.

See also S⃗₀⁻.

S⃗₀⁻(s)
S⃗₀⁻(M₁, M₂, κ₁, κ₂, S⃗₁, S⃗₂)

Defined below Eq. (3.4) of Buonanno et al. (2012):

\[\vec{S}_0^- = \frac{M}{M_1} \left( \frac{\kappa_1} {\kappa_2} \right)^{1/4} \left( 1 - \sqrt{1 - \kappa_1 \kappa_2} \right)^{1/2} \vec{S}_1 + \frac{M}{M_2} \left( \frac{\kappa_2} {\kappa_1} \right)^{1/4} \left( 1 + \sqrt{1 - \kappa_1 \kappa_2} \right)^{1/2} \vec{S}_2.\]

Note that, currently, $\kappa_1$ and $\kappa_2$ are both assumed to be equal to 1, as is the case for black holes. You can define κ₁ and κ₂ to have other values for your own PNSystem types, and this function will work appropriately.

See also S⃗₀⁺.

rₕ₁(s)

Horizon radius of black hole 1.

As defined on page 2, line 4, of Alvi (2001). See the documentation section on "Horizons" for more details.

rₕ₂(s)

Horizon radius of black hole 2.

As defined on page 2, line 4, of Alvi (2001). See the documentation section on "Horizons" for more details.

Ωₕ₁(s)

Horizon angular velocity of black hole 1.

As defined on page 2, line 5, of Alvi (2001). See the documentation section on "Horizons" for more details.

Ωₕ₂(s)

Horizon angular velocity of black hole 2.

As defined on page 2, line 5, of Alvi (2001). See the documentation section on "Horizons" for more details.

sin²θ₁(s)

Sine-squared of angle between spin of black hole 1 and vector to black hole 2.

Compare to Eq. (18) of Alvi (2001). See the documentation section on "Horizons" for more details.

sin²θ₂(s)

Sine-squared of angle between spin of black hole 2 and vector to black hole 1.

Compare to Eq. (18) of Alvi (2001). See the documentation section on "Horizons" for more details.

ϕ̇̂₁(s)

Rate of rotation of black hole 2 about the spin of black hole 1, relative to orbital rotation rate.

This is the rotation rate ϕ̇ as defined in Eq. (19) of Alvi (2001), divided by $v^3 = M\, \Omega$. This division is done to make sure we can track the relative PN order of terms that depend on this.

See the documentation section on "Horizons" for more details.

ϕ̇̂₂(s)

Rate of rotation of black hole 1 about the spin of black hole 2, relative to orbital rotation rate.

This is the rotation rate ϕ̇ as defined in Eq. (19) of Alvi (2001), divided by $v^3 = M\, \Omega$. This division is done to make sure we can track the relative PN order of terms that depend on this.

See the documentation section on "Horizons" for more details.

Î₀₁(s)

Horizon moment of inertia of black hole 1.

This is the moment divided by $ν^2 v^{12}$, as given by Eq. (10) of Alvi (2001). See the documentation section on "Horizons" for more details.

Î₀₂(s)

Horizon moment of inertia of black hole 2.

This is the moment divided by $ν^2 v^{12}$, as given by Eq. (10) of Alvi (2001). See the documentation section on "Horizons" for more details.

κ₁(s)

The "quadrupolar polarisability" of object 1 used by Bohé et al. (2015).

Note that Bohé et al. refer to the closely related (and co-authored) Marsat (2014), who notes above Eq. (4.7) that this is denoted $C_{\mathrm{ES}^2}$ in Levi and Steinhoff (2014), who in turn notes that "we can set ... the Wilson coefficients $C_{\mathrm{ES}^2} = C_{\mathrm{BS}^3} = 1$ for the black hole case."

However, a very similar constant $\kappa_A$ is used in Eq. (2.1) of Buonanno et al. (2012). They say that $\kappa_A=1$ for an isolated black hole, but can deviate from 1 for a black hole in a binary — though those deviations occur at "much higher" order than those they consider (2PN).

See also λ₁.

κ₂(s)

The "quadrupolar polarisability" of object 2 used by Bohé et al. (2015). See κ₁ for more details.

κ₊(s)

Equal to κ₁+κ₂; defined below Eq. (3.28) of Bohé et al. (2015).

κ₋(s)

Equal to κ₁-κ₂; defined below Eq. (3.28) of Bohé et al. (2015).

λ₁(s)

The "quadrupolar polarisability" of object 1 used by Marsat (2014), who notes above Eq. (4.11) that this is denoted $C_{\mathrm{BS}^2}$ in Levi and Steinhoff (2014), who in turn notes that "we can set ... the Wilson coefficients $C_{\mathrm{ES}^2} = C_{\mathrm{BS}^3} = 1$ for the black hole case."

See also κ₁.

λ₂(s)

The "quadrupolar polarisability" of object 2 used by Bohé et al. (2015). See λ₁ for more details.

λ₊(s)

Equal to λ₁+λ₂; defined below Eq. (3.28) of Bohé et al. (2015).

λ₋(s)

Equal to λ₁-λ₂; defined below Eq. (3.28) of Bohé et al. (2015).

Λ̃(pnsystem)
Lambda_tilde(pnsystem)

Effective tidal deformability of the system.

Much as the chirp mass is a particularly measurable combination of the masses of the two components of the binary, this quantity is a particularly measurable combination of the tidal couplings $\Lambda_1$ and $\Lambda_2$ of the two components.

Raithel et al. (2018) define this as

\[\tilde{\Lambda} = \frac{16}{13} \frac{(M_1 + 12 M_2) M_1^4 \Lambda_1 + (M_2 + 12 M_1) M_2^4 \Lambda_2} {M^5}.\]

See also Λ₁ and Λ₂.