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Fundamental variables

These constitute the basic quantities needed to fully describe a PN binary system at any instant. All other variables — and ultimately all PN expressions — may be expressed as functions of these. In the docstrings below, pnsystem is an object of any type that subtypes PNSystem.

PostNewtonian.FundamentalVariables.R — Function
R(pnsystem)

Orientation of the binary, as a Rotor.

At any instant, the binary is represented by the right-handed triad $(n̂, λ̂, ℓ̂)$, where $n̂$ is the unit vector pointing from object 2 to object 1, and the instantaneous velocities of the binary's elements are in the $n̂$-$λ̂$ plane. This Rotor will rotate the $x̂$ vector to be along $n̂$, the $ŷ$ vector to be along $λ̂$, and the $ẑ$ vector to be along $ℓ̂$.

Note that the angular velocity associated to R is given by $Ω⃗ = 2 Ṙ R̄ = Ω ℓ̂ + ϖ n̂$. (Any component of $Ω⃗$ along $λ̂$ would violate the condition that the velocities be in the $n̂$-$λ̂$ plane.) Here, the scalar quantity $Ω$ is the orbital angular frequency, and $ϖ$ is the precession angular frequency.

See also n̂, λ̂, ℓ̂, Ω, and 𝛡$=ϖ n̂$.

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PostNewtonian.FundamentalVariables.v — Function
v(pnsystem)
v(;Ω, M=1)

Post-Newtonian velocity parameter. This is related to the orbital angular frequency $\Omega$ as

\[v \colonequals (M\,\Omega)^{1/3},\]

where $M$ is the total mass of the binary.

Note that if you want to pass the value $Ω$ (rather than a PNSystem), you must pass it as a keyword argument — as in v(Ω=0.1).

See also Ω.

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PostNewtonian.FundamentalVariables.Λ₁ — Function
Λ₁(pnsystem)
Lambda1(pnsystem)

Quadrupolar tidal-coupling parameter of object 1 in this system.

We imagine object 1 begin placed in an (adiabatic) external field with Newtonian potential $\phi$, resulting in a tidal field measured by $\partial_i \partial_j \phi$ evaluated at the center of mass of the object. This induces a quadrupole moment $Q_{ij}$ in object 1, which can be related to the tidal field as

\[Q_{ij} = -\frac{G^4}{c^{10}} \Lambda_1 M_1^5 \partial_i \partial_j \phi,\]

where $M_1$ is the mass of object 1. This tidal-coupling parameter $\Lambda_1$ can be related to the Love number $k_2$ (where the subscript 2 refers to the fact that this is for the $\ell=2$ quadrupole, rather than object 2) as

\[\Lambda_1 = \frac{2}{3} \frac{c^{10}}{G^5} \frac{R_1^5}{M_1^5} k_2,\]

where $R_1$ is the radius of object 1. Note that $\Lambda_1$ is dimensionless. For black holes, it is precisely zero; for neutron stars it may range up to 1; more exotic objects may have significantly larger values.

Note that — as of this writing — only NSNS systems can have a nonzero value for this quantity. All other types return 0, which Julia can use to eliminate code that would then be 0. Thus, it is safe and efficient to use this quantity in any PN expression that specializes on the type of pnsystem.

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PostNewtonian.FundamentalVariables.Λ₂ — Function
Λ₂(pnsystem)
Lambda2(pnsystem)

Quadrupolar tidal coupling parameter of object 2 in this system.

See Λ₁ for details about the definition, swapping "object 1" with "object 2".

Note that — as of this writing — only BHNS and NSNS systems can have a nonzero value for this quantity. All other types return 0, which Julia can use to eliminate code that would then be 0. Thus, it is safe and efficient to use this quantity in any PN expression that specializes on the type of pnsystem.

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